Research ArticleInexact Fuzzy Chance-Constrained Fractional Programming forSustainable Management of Electric Power Systems

C Y Zhou 1 G H Huang 2 J P Chen3 and X Y Zhang3

1School of Control and Computer Engineering North China Electric Power University Beijing 102206 China2Institute for Energy Environment and Sustainability Research UR-NCEPU North China Electric Power UniversityBeijing 102206 China3Institute for Energy Environment and Sustainable Communities UR-BNU 3737 Wascana Parkway Regina SK Canada S4S 0A2

Correspondence should be addressed to G H Huang huangiseisorg

Received 14 April 2018 Accepted 8 July 2018 Published 19 November 2018

Academic Editor Ching-Ter Chang

Copyright copy 2018 C Y Zhou et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An inexact fuzzy chance-constrained fractional programming model is developed and applied to the planning of electricpower systems management under uncertainty An electric power system management system involves several processes withsocioeconomic and environmental influencedDue to themultiobjectivemultilayer andmultiperiod features associatedwith thesevarious factors and their interactions extensive uncertaintiesmay exist in the study systemAs an extension of the existing fractionalprogramming approach the inexact fuzzy chance-constrained fractional programming can explicitly address system uncertaintieswith complex presentationsThe approach can not only deal with multiple uncertainties presented as random variables fuzzy setsinterval values and their combinations but also reflect the tradeoff in conflicting objectives between greenhouse gas mitigationand system economic profit Different from using least-cost models a more sustainable management approach is to maximize theratio between clean energy power generation and system cost Results of the case study indicate that useful solutions for planningelectric power systems management practices can be generated

1 Introduction

Sustainable development of electric power systems (EPS)plays a significant role in urban planning At the presentthe major energy sources for electricity generation in manycountries are still nonrenewable fossil fuels which are con-sidered as one of the major contributors to the greenhousegases emissions Renewable energy as an alternative energysource has the characteristics of cleanliness nondepletionand easier operation and maintenance however its cost ofgenerating electricity is higher and the energy source isintermittent and unreliable Thus there are many challengesto identify sustainable management plans for EPS Amongthem most importantly decision-makers need to considerthe tradeoff between economic cost and environmentalimpacts the reflection of dynamic characteristics of facilitycapacity issues as well as uncertainties of input informa-tion such as the forecast values of electricity demands andrenewable resource availabilities Due to these complexities

inexact systems analysis techniques are desired to assistin developing long-term EPS management plans Thereare many techniques to handle system uncertainties suchas interval parameter programming (IPP) [1ndash3] stochasticmathematical programming (SMP) [4ndash7] and fuzzy possi-bilistic programming (FPP) [8ndash12]

In past decades many inexact optimization methodswere developed for energy system planning and manage-ment [13ndash20] Classically some models were formulated assingle-objective Linear Programming (LP) problems aimedat minimization of system cost under specific levels ofenvironmental requirements [21ndash23] For example Sun et al[23] utilized a static deterministic linear model for planningChinarsquos electric power systems development in which realenergy use patterns among interregional energy spillovereffects were examined Since the early 1980s for betterreflecting the multidimensionality of the sustainability goalit was increasingly popular to represent the EPSmanagementproblems within a Multiple Objective Programming (MOP)

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 5794016 13 pageshttpsdoiorg10115520185794016

2 Mathematical Problems in Engineering

framework [24ndash30] For example Han et al [27] presented amultiobjective model for the EPS planning to maximize theexpected system total profit andminimize the financial risk ofhandling uncertain environments Meza et al [26] proposea long-term multiobjective model for the power generationexpansion planning of Mexican electric power system whichcan optimize simultaneously multiple objectives (ie mini-mizes costs environmental impact imported fuel and fuelprice risks) Nevertheless the tradeoff of multiple objectiveswas neglected and the system complexities could not beadequately reflected In order to deal with the conflictobjectives between the economic development and environ-mental protection Fractional Programming (FP) was usedin many management problems [31ndash34] For instance Wanget al [34] developed a multistage joint-probabilistic chance-constrained fractional programming (MJCFP) approach ofSaskatchewan Canada The MJCFP approach aimed tohelp tackle various uncertainties involved in typical electricpower systems and thus facilitate risk-based managementfor climate change mitigation Chen et al [32] advanced anonlinear fractional programming approach for addressingthe environmentaleconomic power dispatch problems inthe thermal power systems Zhang et al [33] put forward afuzzy linear fractional programming approach for optimalirrigation water allocation under uncertainty

FP has the advantages of better reflecting the real prob-lems by optimizing the ratio between the economic and theenvironmental aspects over the conventional single-objectiveormultiobjective optimization programmingmethodsHow-ever few of the earlier studies about EPS are focused onanalyzing interactive relationships among multiple objec-tives the randomness of the parameters and uncertaintiesexisted in multiple levels In addition chance-constrainedprogramming (CCP) with the dual uncertainties (ie aninterval numberwith fuzzy boundaries) parameters is seldomintegrated into the FP optimization framework to deal withthe violation of system constraints exists in the optimizationmodel

Therefore this study aims to develop an inexact fuzzychance-constrained fractional power system planning (IFCF-PSP) IFCF-PSP model will integrate chance-constrainedprogramming fuzzy programming and interval-parameterprogramming within a fractional programming frameworkResults will provide decision support for (i) achieving trade-offs among system violation risk environmental require-ment and system cost (ii) generating flexible capacity expan-sion strategies under different risk levels (iii) providing avariety of power generation and capacity expansion alterna-tives that can help support decision making under changingconditions and (iv) helping decision-makers identify theoptimal EPS management strategies and gain deeper insightsinto system efficiency system cost and system risk underdifferent CO2 emission targets

2 Methodology

The tradeoff of conflicting objectives between CO2 miti-gation and system economic profit is important to powersystems planning (PSP) The linear fractional programming

(LFP) method can be effective in balancing two conflictingobjectives and addressing randomness in the right-handparameters A general LFP problem can be expressed asfollows

max 119891 = sum119899119895=1 119888119895119909119895 + 120572sum119899119895=1 119889119895119909119895 + 120573 (1a)

subject to119899sum119895=1

119886119894119895119909119895 le 119887119894 119894 = 1 2 119898 (1b)

119909119895 ge 0 119895 = 1 2 119899 (1c)

where 119883 is a vector of interval decision variables 119886119894119895 aretechnical coefficients and 119887119894 are right-hand-side parameters120572 and 120573 are constants

In a power system many data available are impreciseParameters in the model can be represented as intervalnumbers andor fuzzy membership functions such thatthe uncertainties can be directly communicated into theoptimization process and resulting solution Interval linearprogramming (ILP) is an effective method to deal withuncertainties existing as interval values without distributioninformation The ILP method is integrated into the LFPmethod to reflect the uncertainty of the model parametersInterval linear fractional programming (ILFP) can be aneffective tool to tackle dual-objective optimization problemsunder uncertainty especially when distribution informationis not known exactly and merely lower and upper boundsare available A general ILFP problem can be expressed asfollows

max 119891plusmn = sum119899119895=1 119888plusmn119895 119909plusmn119895 + 120573plusmnsum119899119895=1 119889plusmn119895 119909plusmn119895 + 120574plusmn (2a)

subject to119899sum119895=1

119886plusmn119894119895119909plusmn119895 le 119887plusmn119894 119894 = 1 2 119898 (2b)

119909plusmn119895 ge 0 119895 = 1 2 119899 (2c)

If 119887plusmn119894 is a random right-hand-side parameter the constraints(2b) can be transformed as follows

Pr119899sum119895=1

119886plusmn119894119895119909plusmn119895 le 119887plusmn119894 ge 120575119894 119894 = 1 2 119898 (3)

This means that the possible region of occurrence forthe left-hand side of each constraint should be containedwithin a satisfactory or tolerable region as defined by thecorresponding right-hand side Thus by incorporating tol-erance measures 120575119894 (0 le 120575119894 le 1) and utilizing the chance-constrained approach the stochastic constraints of (3) can betransformed to their deterministic equivalents as follows

119899sum119895=1

119886plusmn119894119895119909plusmn119895 le 119887plusmn(119901119894)119894 119894 = 1 2 119898 (4)

where 119887(119901119894)119894 = 119865minus1119894 (119901119894) 119894 = 1 2 119898 119901119894 = 1 minus 120575119894 giventhe cumulative distribution function of 119887119894 (ie 119865119894(119887119894)) and

Mathematical Problems in Engineering 3

Mem

bers

hip

Gra

de

1

xaminusij a+

ij

Figure 1 Fuzzy boundaries for interval 119886plusmn

the probability of violating constraint (4) (119901119894) The constraintof the model in the optimization process is changed fromldquorigid satisfactionrdquo to ldquoflexible responserdquo Therefore thescheme can meet the optimization target with flexibility andmaneuverability

However the problem with constraints (4) can onlyreflect the case when A is deterministic In many real-worldproblems the lower and upper bounds of some intervalparameters can rarely be acquired as deterministic valuesInstead they may often be given as subjective informationthat can only be expressed as fuzzy sets This leads todual uncertainties as shown in Figure 1 If both A and Bare uncertain the set of feasible constraints may becomemore complicated To generate a precise analysis of decision-making multiple uncertainties need to be tackled For exam-ple the total carbon dioxide emissions in a certain region canbe described as probability distributions and the statisticsof such a random parameter can be expressed as fuzzy setsThis results in dual uncertainties which can be represented bythe concept of distribution with fuzzy probability (DFP) Inorder to deal with the hybrid uncertainty resulting from fuzzyand stochastic information in constraints and parametersCCP is then integrated into the ILFP method to reflectprobability distribution of fuzzy numbers Models (2a) (2b)and (2c) can be further improved by incorporating fuzzy andchance-constrained techniquesTherefore letRplusmn be the set ofintervals with fuzzy lower and upper bounds and 120595plusmn denotesa set of fuzzy random numbers for fuzzy lower and upperbounds

119899sum119895=1

119886plusmn119894119895119909plusmn119895 le plusmn(119901119894)119894 119894 = 1 2 119898 (5)

Let 119895and be base variables imposed by fuzzy subsets 119860119895

and 119861 then120583119860119895 119895 997888rarr [0 1] (6a)

120583119861 997888rarr [0 1] (6b)

where 120583119860119895 indicates the possibility of consuming a specificamount of resource by activity 119895 and 120583119861 indicates the possible

Mem

bers

hip

Gra

de

1

x

08

05

020

FL(m minus x

)

FR(xminusm )

Figure 2 L-R fuzzy membership function

availability of resource B le means fuzzy inequality Fuzzysubset119873 can be expressed as a LndashR fuzzy number [35]

120583119873 (119909) =

119865119871 (119898 minus 119909120572 ) 119894119891 minusinfin lt 119909 lt 119898 120572 gt 01 119894119891 119909 = 119898119865119877 (119909 minus 119898120575 ) 119894119891 119898 lt 119909 lt +infin 120575 gt 0

(7a)

where 119865119871 and 119865119877 are the shape functions (Figure 2) For alinear case fuzzy subset 119873 can be defined as the followinggeneral format

120583119873 (119909) =

0 119894119891 119909 lt 120572 119900119903 119909 gt 1205721 119894119891 119909 = 1198981 minus 2 |119898 minus 119909|120572 minus 120572 119894119891 120572 lt 119909 lt 120572

(7b)

where [120572 120572] is an interval imposed by fuzzy subset119873 Basedon the method from Nie et al [36] and Leung et al [37] thefuzzy constraints in (5) can be replaced by the following 2119896precise inequalities in which k denotes the number of 120572minus119888119906119905levels119899sum119895=1

119886plusmn119894119895119904119909plusmn119895 le 119887plusmn119894 119904(119901119894) 119894 = 1 2 119898 119904 = 1 2 119896 (8a)

119899sum119895=1

119886plusmn119894119895119904119909plusmn119895 ge 119887plusmn119894 119904(119901119894) 119894 = 1 2 119898 119904 = 1 2 119896 (8b)

where

119886plusmn119894119895119904 = sup (119886plusmn119904119894119895 ) 119886plusmn119904119894119895 isin (119895)120572119904 (8c)

119886plusmn119894119895119904 = inf (119886plusmn119904119894119895 ) 119886plusmn119904119894119895 isin (119895)120572119904 (8d)

119887plusmn119894 119904 = sup (119887plusmn119904119894 ) 119887plusmn119904119894 isin 120572119904

(8e)

119887plusmn119894 119904 = inf (119887plusmn119904119894 ) 119887plusmn119904119894 isin 120572119904

(8f)

120572119904 isin (0 1] (119904 = 1 2 119896) sup(119905) represents the superiorlimit value among set 119905 and inf(119905) denotes the inferior limitvalue among set 119905

4 Mathematical Problems in Engineering

Stochastic information Interval-parameter Fuzzy sets Fractional objective

Chance-constrainedprogramming

Interval-parameterprogramming Fuzzy programming Mixed-integer fractional

programming

Inexact fuzzy chance-constrained fractional programming (IFCFP)

Optimal solutions of IFCFP

Uncertainty Multiple objectives

Figure 3 Framework of the IFCFP approach

Figure 3 presents the framework of the IFCFP methodGive a set of certain values of the 120572 minus 119888119906119905 level to fuzzyparameters 119886119894119895 119894 and 119901119894 and solve models (1a) (1b) and(1c) through the IFCFP approach According to interactivetransform algorithm proposed by Zhu et al [31] the IFCFPmodel can be transformed into two submodels and thencan be solved through the branch-and-bound algorithm andthe method proposed by Charnes et al [38] Comparedwith the existing optimization methods the proposed IFCFPapproach has three characteristics (i) it can tackle ratiooptimization problems (ii) it can handle uncertainties withunknown distribution information by interval parametersand variables and (iii) it can deal with uncertain parameterspresented as fuzzy sets in the objective and the left hand sideaswell as dual uncertainties expressed as the distributionwithfuzzy probability

3 Case Study

To demonstrate its advantages the proposed IFCFP methodis applied to a typical regional electric system managementproblem with representative cost and technical data withina Chinese context In the study system it is assumed thatthere is an independent regional electricity grid whereone nonrenewable resource (coal) and four clean energyresources (natural gas wind solar and hydro) are availablefor electricity generation The decision-makers are responsi-ble for arranging electricity production from those five typesof power-generation to meet the demand of end users Toreflect the dynamic features of the study system three-timeperiods (5 years for each period) are considered in a 15-yearplanning horizon

A sufficient electricity supply at minimum cost is impor-tant to the electric generation expansion planning Thesystem cost 119862plusmn is formulated as a sum of the following

119862plusmn = 1198911plusmn + 1198912plusmn + 1198913plusmn + 1198914plusmn + 1198915plusmn (9a)

(1) the total cost for primary energy supply

1198911plusmn = 119879sum119905=1

119868sum119895=1

119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 (9b)

where 119905 is equal to index for the time periods (119905 = 1 2 119879)119895 is equal to index for the power-generation technology(119895 = 1 2 119869) 119868 = the number of nonrenewable power-generation technology (eg 119895 = 1 coal power 119895 = 2 naturalgas and 119868 lt 119869) 119862119875119864plusmn119905119895 is equal to cost for local primaryenergy supply for power-generation technology 119895 in period119905(103 $TJ)119860119875119864plusmn119905119895 is equal to decision variable and representslocal supply of primary energy resource for power-generationtechnology 119895 in period 119905(TJ)

(2) fixed and variable operating costs for power genera-tion

1198912plusmn = 119879sum119905=1

119869sum119895=1

119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895 (9c)

where 119860119875119866plusmn119905119895 is equal to decision variable and representselectricity generation from power-generation technology 119895 inperiod 119905(GWh)119862119875119866plusmn119905119895 is equal to fixed and variable operationcosts for generating electricity via technology 119895 in period119905(103 $GWh)

(3) cost for capacity expansions

1198913plusmn = 119879sum119905=1

119869sum119895=1

119872sum119898=1

119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898 (9d)

where119898 is equal to index for the capacity expansion options(119898 = 1 2 119872) 119862119864119875plusmn119905119895 is equal to cost for expandingcapacity for generating electricity via technology 119895 in period119905(106 $GW) 119864119862119860plusmn119905119895119898 is equal to capacity expansion optionof power-generation technology 119895 under different expansionprogram 119898 in period 119905(GW) 119884119905119895119898 is equal to binary variableof capacity option 119898 for power-generation technology 119895 inperiod 119905

(4) cost for importing electricity

1198914plusmn = 119879sum119905=1

119862119868119864plusmn119905 times 119860119868119864plusmn119905 (9e)

where 119862119868119864plusmn119905 is equal to cost of importing electricity inperiod 119905(103 $GWh) 119860119868119864plusmn119905 is equal to decision variable and

Mathematical Problems in Engineering 5

represents the shortage amount of electricity needs to beimported in period 119905(GWh)

(5) cost for pollutant mitigation

1198915plusmn = 119879sum119905=1

119868sum119895=1

119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905 (9f)

where 119862119875119872plusmn119905119895 is equal to cost of pollution mitigation ofpower-generation technology 119895 in period 119905(103 $ton) 120578plusmn119905CO2 emission factor in period 119905(103 tonGWh)

Renewable energy resources are intermittent andunreliable which are subject to spatial andor temporal

fluctuations The natural gas generation with low carbondioxide emissions can stabilize the risk of the intermittentand unpredictable nature of renewable energy generationIn this study we assumed natural gas resources as theclean energy power generation Therefore the objective ofthis study is to maximize the ratio between clean energygeneration (including the natural gas) and system costwhile a series of constraints define the interrelationshipsamong the decision variables and system conditionsfactorsThe inexact fuzzy chance-constrained fractional powersystems planning (IFCF-PSP) can be formulated as fol-lows

max119891plusmn = 119862119866plusmn119862plusmn= sum119879119905=1sum119869119895=119868+1119860119875119866plusmn119905119895 + sum119879119905=1 119860119875119866plusmn1199052sum119879119905=1sum119868119895=1 119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 + sum119879119905=1sum119869119895=1 119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895 + sum119879119905=1sum119869119895=1sum119872119898=1 119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898 + sum119879119905=1 119862119868119864plusmn119905 times 119860119868119864plusmn119905 + sum119879119905=1sum119868119895=1 119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905

(10a)

where the clean energy power generation (119862119866plusmn) consists ofthe renewable energies such as wind power solar energy andhydropower (119860119875119866plusmn119905119895 and j = I + 1 J) and the natural gasgeneration in period 119905

The constraints are listed as follows(1) electricity demand constraints119869sum119895=1

119860119875119866plusmn119905119895 + 119860119868119864plusmn119905 ge 119863119872plusmn119905 times (1 + 120579plusmn119905 ) forall119905 (10b)

where119863119872plusmn119905 is equal to local electricity demand (GWh) 120579plusmn119905 isequal to transmission loss in period 119905

(2) capacity limitation constraints for power-generationfacilities

119860119875119866plusmn119905119895 le (119877119862119860plusmn119905119895 + 119872sum119898=1

119864119862119860plusmn119905119895119898 times 119884119905119895119898) times 119878119879119872plusmn119905119895forall119905 119895

(10c)

where 119877119862119860plusmn119905119895 is equal to the current capacity of power-generation technology 119895 in period 119905(GW) 119878119879119872plusmn119905119895 is equal tothe maximum service time of power-generation technology 119895in period 119905(hour)

(3) primary energy availability constraints

119860119875119864plusmn119905119895 le 119880119875119864plusmn119905119895 forall119905 119895 (10d)

119860119875119866plusmn119905119895 times 119903119891plusmn119905119895 le 119860119875119864plusmn119905119895 forall119905 119895 (10e)

where 119880119875119864plusmn119905119895 is equal to available primary energy 119895 inperiod 119905 (119895 = 1 2TJ) 119903119891plusmn119905119895 is equal to energy consumptionconversion rate by power-generation technology 119895 in period119905 (119895 = 1 2TJGWh)

(4) capacity expansion constraints

119877119862119860plusmn119905119895 + 119872sum119898=1

119864119862119860plusmn119905119895119898 times 119884119905119895119898 le 119880119862119860plusmn119905119895 forall119905 119895 (10f)

where 119880119862119860plusmn119905119895 is equal to maximum capacity of generationtechnology 119895 in period 119905(GW)

(5) expansion options constraints

119872sum119898=1

119884119905119895119898 le 1 forall119905 119895 (10g)

119884119905119895119898 = 1 if capacity expansion is undertaken119884119905119895119898 = 0 otherwise(6) import electricity constraints

119860119868119864plusmn119905 le 119880119868119864plusmn119905 forall119905 (10h)

where119880119868119864plusmn119905 is equal to maximum import amount of electric-ity imports in period 119905 (GWh)

(7) renewable energy availability constraints

119869sum119895=119868+1

119860119875119866plusmn119905119895 ge ( 119869sum119895=1

119860119875119866plusmn119905119895)120590plusmn119905 forall119905 (10i)

where 120590plusmn119905 is equal to the minimum proportion of electric-ity generation by renewable energy in the whole power-generation Currently the Renewable Portfolio Standard(RPS) mechanisms have been adopted in several countriesincluding theUnitedKingdom (RenewablesObligation in theUK) Italy Poland Sweden and Belgium and 29 out of 50US states etc According to the governmentrsquos requirement acertain percentage of the electricity generation of the powerenterprises will come from renewable energy sources Theproportion of electricity from renewable sources will usuallyincrease year by year Thus in the model 120590plusmn119905 is used torepresent the lowest ratio of the RPS

(8) pollutants emission constraints

119868sum119895=1

119860119875119866plusmn119905119895 times 120578plusmn119905119895 times (1 minus 120585plusmn119905 ) le 119864119872plusmn119905 (119901119894) forall119905 (10j)

6 Mathematical Problems in Engineering

Table 1 Cost of energy supply and conversion parameters

Periodt=1 t=2 t=3

Energy supply cost (103 $TJ)Coal [256 306] [326 376] [396 446]Natural gas [676 696] [757 777] [838 858]Units of energy carrier per units of electricity generation (TJGWh)Coal [1112 117] [1068 1116] [1015 1062]Natural gas [838 882] [754 792] [679 738]

Table 2 Fuzzy subsets for efficiency coefficient under different 120572 minus 119888119906119905 levelsperiod efficiency coefficient120572 minus 119888119906119905 level 120588+ 120588+ 120588minus 120588minus

0t=1 0125 0235 012 023t=2 014 0175 0136 017t=3 0119 0149 0116 0145

02t=1 0135 0223 013 0218t=2 0142 017 0138 0165t=3 012 0144 0117 014

05t=1 015 0205 0145 02t=2 0144 0162 014 0157t=3 0123 0138 0119 0134

08t=1 0165 0187 016 0182t=2 0147 0154 0143 015t=3 0125 0131 0122 0127

1t=1 0175 0175 017 017t=2 0149 0149 0145 0145t=3 0126 0126 0123 0123

Note efficiency coefficient (120588plusmn119905 = 1 minus 120585plusmn119905 )

where 119864119872119905 is equal to the permitted CO2 emission in period119905(103 ton) 120585plusmn119905 is equal to the efficiency of chemical absorptionor capture and storage of CO2 in period 119905119901119894 = the riskconfidence levels

(9) nonnegativity constraints

119860119875119864plusmn119905119895 119860119875119866plusmn119905119895 119860119868119864plusmn119905 ge 0 forall119905 119895 (10k)

Interval parameters are adopted to address impreciseuncertainties which are generally associated with electricitydemands prices of energy resources costs of capacity expan-sion and many other constraints The real research data areused as input data of the model The detailed descriptionsof cost for energy supply and relative conversion parameterswere illustrated in Table 1 Table 2 gives fuzzy subsets forefficiency coefficient under different120572minus119888119906119905 levels Table 3 liststhe capacity expansion options and capital investment costsfor each facility

4 Results and Discussion

In Figure 4 coal-fired electricity supply would increasesteadily and still play an important role in the power sys-tem due to its high availability and competitive price overthe study planning horizon However increasingly strin-gent emission limits result in idle coal-fired facilities Moreeconomical and environmentally friendly power generationfacilities will be prioritized The risk confidence level (119901119894) isemployed to the constraints of CO2 emission requirementThe decision-maker can adjust the value of the 119901119894 levelaccording to actual needs We assumed that violations ofemission constraints are allowed under three given 119901119894 levels(119901119894 = 001005 and 01 which are normally adopted asthe significance levels) [39] The higher credibility levelwould correspond to a tight environment requirement thusleading to a lower CO2 emission while the lower credibilitylevel would correspond to a relatively relaxed environmentrequirement thus resulting in a higher CO2 emission There-fore changes in 119901119894-level have an impact on coal-fired power

Mathematical Problems in Engineering 7

Table 3 Capacity expansion options and costs for power-generation facilities

Periodt=1 t=2 t=3

Capacity-expansion options (GW)Coal m=1 005 005 005

m=2 010 010 010m=3 015 015 015

Natural gas m=1 010 010 010m=2 015 015 015m=3 020 020 020

Wind power m=1 005 005 005m=2 015 015 015m=3 020 020 020

Solar energy m=1 005 005 005m=2 015 015 015m=3 020 020 020

Hydropower m=1 015 015 015m=2 020 020 020m=3 025 025 025

Capacity expansion cost (106$GW)Coal [577 607] [547 577] [517 547]Natural gas [726 756] [676 726] [626 676]Wind power [1256 1306] [1156 1206] [1056 1106]Solar energy [2668 2768] [2468 2568] [2268 2368]Hydropower [1597 1697] [1497 1597] [1397 1497]

generation The total supply of coal-fired generation woulddecrease from [48069 50720] times 109GWh when 119901119894 = 01to [47175 50685] times 109GWh when 119901119894 = 005 and reach[45938 50313] times 109GWh when 119901119894 = 001 In contrast cleanenergy generation will increase For example solar powergeneration of three periods would rise from [2438 2768] times109GWh when 119901119894 = 01 to [2562 2768] times 109GWh when 119901119894= 005 and reach [2625 2769] times 109GWh when 119901119894 = 001 Asshown in Figure 4 the generation of solar power and naturalgas in clean power facilities are mainly affected by emissionconstraints When 119901119894 = 001 the solar power generation isthe largest and the power generation from natural gas isthe lowest in the three-study 119901119894-levels This is the result ofthe tradeoff between carbon dioxide emissions and systemcosts Developing natural gas electricity requires a relativelylow capital cost but leads to more CO2 emissions and thesolar energy technology needs a low operational cost but anextremely high capital cost Similarly the results under other119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted Duringthe entire planning horizon the ratio objective between cleanenergy generation and the total system cost would be [2153]GWh per $106 which also represents the range of the systemefficiency When the electricity-generation pattern variesunder different 119901119894 levels and within the interval solutionranges the system efficiency would also fluctuate within itssolution range correspondingly Thus the IFCF-PSP resultscan create multiple decision alternatives through adjusting

different combinations of the solutions Table 4 presentsthe typical alternatives where combinations of upperlowerbound values for 119860119875119864plusmn119905119895 119860119875119866plusmn119905119895 119884plusmn119905119895119898 (119895 = 1 forall119905 119898) and119860119875119866plusmn119905119895 119884plusmn119905119895119898 (119895 = 2 3 4 5 forall119905 119898) are examined Underthe same power generation alternative interval solutionof the system cost can be obtained according to intervalprice parameters and corresponds to the interval solution ofsystem efficiency Based on the result of the total electricitygenerated we divided the 12 alternatives into three groups(high A3 A4 A8 and A12 medium A1 A5 A7 and A9 lowA2A6A10 andA11) tomeet the future highmedium or lowelectricity demand

The comparison of the system efficiency under different119901119894 levels shows that A11gtA6 gtA10 gtA2 under low-level totalpower generation group Alternative A11 (corresponding tof+) would lead to the most sustainable option with a systemefficiency as much as [27142999] GWh per $106 whichcorresponds to the highest amount of clean energy electricity(0478 times 106GWh) and a moderate cost ([15936 17612] times109$) Additionally lower-level electricity demands will besatisfied by a total power generation of 1856 106GWhTherefore Alternative A11 is a desirable choice from thepoint of view of resources conservation and environmentalprotection when electricity demand is at the low levelAlternative A6 would also meet low electricity demand andreached the lowest system cost ([15142 16785] times 109$)When Alternative A10 is adopted all the power generation

8 Mathematical Problems in Engineering

020406080

100120140160180200

Submodel f-Submodel f+

Submodel f-Submodel f+

Submodel f-Submodel f+

020406080

100120140160180200

020406080

100120140160180200

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

ＪＣ=01

ＪＣ=005

ＪＣ=001

Pow

er g

ener

atio

n (103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)

Figure 4 Power generation loads for EPS through the IFCF-PSPmodel

activities will reach their lower bound levels leading to thelowest power generation (1764 times 106GWh) Compared withAlternative A6 this alternative may be of less interest todecision-makers due to its lower system efficiency and highersystem cost However it is the solution that is obtainedunder the most stringent emission constraints (ie 119901119894 =001) Alternative A2 is a moderate solution in this group Itssystem cost system efficiency total power generation andclean energy power generation are [15295 16954] times 109$[22062445] GWh per $106 1816 times 106GWh and 0374 times106GWh

In middle-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A7gt A5gt A9 gtA1 In alternative A7middle-level electricity demands will be satisfied by a totalpower generation of 1893 times 106GWh In addition Alternative

A7 has the highest system efficiency ([2661 2941] GWhper 106$) and the largest amount of clean energy powergeneration (0478times 106GWh) in this groupThus AlternativeA7 is considered to be a desirable choice Alternatives A5 andA9 would lead to the same total power generation solution(1895 times 106GWh) They are the two-moderate solutions inthis groupThe system cost obtained from the A1 is the lowest([15718 17413] times 109$) in this group It would lead to thelowest system efficiency ([2148 2379] GWh per 106$) butdue to the largest proportion of fossil-fired power generationA1 has the maximum security for energy supply

In high-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A3gt A12gt A8 gtA4 The same cleanenergy power generation obtained from the 4 alternativesis 0478 106GWh In Alternative A4 all the power gener-ation activities will be equal to their upper-bound valuesat the same time It would lead to the highest system cost([16901 18676] times 109$) but the moderate system efficiency([2559 2828]GWhper $106) In comparisonAlternative A3would lead to the minimum system cost ([16478 18216] times109$) and the highest system efficiency ([2624 2901] GWhper $106) in this group In addition this alternative wouldalso provide the highest clean energy power generationTherefore this alternative is a desirable sustainable optionunder the high electricity demand level Alternatives A8 andA12 can provide modest system efficiencies which would be[2561 283] and [2575 2845] GWh per $106 respectivelyand lower system costs which would be [16892 18666] and[16800 18564] times 109$ Therefore decision-makers who payattention to the stability of the system may be interested inthese two alternatives

The above alternatives represent various options betweeneconomic and environmental tradeoffsWillingness to accepthigh system cost will guarantee meeting the objective ofincreasing the proportion of clean energy A strong desireto acquire low system cost will cause the risk of violatingemission constraints In general the above research resultswere favored by decision makers due to their flexibilityand preference for practical-making decision processes Thefeasible ranges for decision variables under different 119901119894 levelswere useful for decision makers to justify the generatedalternatives directly

Besides the scenario of maximizing the proportion ofclean energy another scenario of minimizing the system costis also analyzed to evaluate the effects of different energy sup-ply policies The optimal-ratio problem presented in Models(10a)ndash(10k) can be converted into a least-cost problem withthe following objective

min119891 = system cost

= 119879sum119905=1

119868sum119895=1

119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 + 119879sum119905=1

119869sum119895=1

119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895+ 119879sum119905=1

119869sum119895=1

119872sum119898=1

119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898

Mathematical Problems in Engineering 9

Table 4 Typical decision alternatives obtained from the IFCF-PSP model solutions

Alternative 119860119875119864plusmn119905119895 119860119875119866

plusmn119905119895119884plusmn119905119895119898

(119895=1 forall119905119898)

119860119875119866plusmn119905119895 119884plusmn119905119895119898

(119895=2345 forall119905119898)

System efficiency(GWh per 106$) System cost (109$) Total power generation

(106GWh)Clean energy powergeneration (106GWh)119901119894 = 01

A1 + - [21482379] (12) [1571817413] (4) 1895 (6) 0374 (11)A2 - - [22062445] (9) [1529516954] (3) 1816 (10) 0374 (11)A3 - + [26242901] (3) [1647818216] (9) 192 (4) 0478 (1)A4 + + [25592828] (6) [16901 18676] (12) 2 (1) 0478 (1)119901119894 = 005A5 + - [22362477] (8) [1578317487] (5) 1912 (5) 0391 (7)A6 - - [23292582] (7) [1514216785] (1) 1807 (11) 0391 (7)A7 - + [26612941] (2) [1625117963] (8) 1893 (8) 0478 (1)A8 + + [2561283] (5) [1689218666] (11) 1999 (2) 0478 (1)119901119894 = 001A9 + - [2174 2406] (11) [16040 17757] (7) 1895 (6) 0386 (9)A10 - - [22972543] (8) [15176 16806] (2) 1764 (12) 0386 (9)A11 - + [27142999] (1) [1593617612] (6) 1856 (9) 0478 (1)A12 + + [25752845] (4) [16800 18564] (10) 1987 (3) 0478 (1)

Table 5 The proportion of clean energy power generation from IFCF-PSP and LS models

Clean power generation ratio () IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3119901119894 = 01 [1802 2432] [2089 2493] [1986 2540] [1623 1870] [1681 1719] [1515 1638]119901119894 = 005 [2067 2465] [2093 2528] [2010 2578] [1624 1870] [1681 1719] [1632 1869]119901119894 = 001 [2155 2530] [2158 2596] [2141 2596] [1648 1893] [1679 1752] [1997 2224]

+ 119879sum119905=1

119862119868119864plusmn119905 times 119860119868119864plusmn119905+ 119879sum119905=1

119868sum119895=1

119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905(11)

Figure 5 shows the proportion of different power gen-eration technologies from the IFCF-PSP and LS modelsunder 119901119894 = 001 over three planning periods The IFCF-PSPmodel leads to a relatively higher percentage of natural gaspower generation in comparison the LS model leads torelatively higher percentages of coal-fired electricity suppliesAccording to the solutions from IFCF-PSPmodel the percentof electricity generated by natural gas facilities would be [5 6] [7 8] and [5 8] in periods 1 2 and 3 respectivelyTheLS model achieves the slightly lower percent (ie 0 0 and[5 6] in periods 1 2 and 3) Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted

According to Table 5 the differences can be foundbetween the results of two models under different 119901119894 levelsThe proportion of clean energy power generation fromIFCF-PSP model is higher than that of the LS model Forexample under 119901119894 = 001 clean energy power generationof the entire region occupied [2155 2530] [2158 2596]and [2141 2596] of the total electricity generation in

the three study periods from IFCF-PSP model which arehigher than the LS model [1648 1993] [1679 1752] and[1997 2224] respectively Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpretedAn increased 119901119894 level represents a higher admissible riskleading to a decreased strictness for the emission constraintsand hence an expanded decision space Therefore undertypical conditions as the risk level becomes lower theproportion of clean energy power generation or renewablepower generation would increase For example in period 2when the 119901119894 level is dropped from 01 to 001 the proportionof clean energy power generation from IFCF-PSP modelwould be increased from [2089 2493] to [2158 2596]Likewise the results under other periods can be similarlyanalyzed The confidence level of constraints satisfaction ismore reliable because the risk level is lowerTherefore in thiscase the decision-makers will be more conservative in theEPS management There is no capacity-expansion would beconducted for the coal-fired facility in three periods sinceits high carbon dioxide emissions and increasingly stringentemission limits On the other hand clean energy electricitysupply would be insufficient for the future energy demandsAccording to Table 6 when 119901119894 takes different values (ie 119901119894 =001119901119894 = 005 and 119901119894 = 01) differences of capacity-expansionoption can be found between the results of two models Inboth models when the risk of violating the constraints ofcarbon emission target is decreased (ie the value of 119901119894-level

10 Mathematical Problems in Engineering

75

79

5 4

74

8

95 4

74

8

9

5 4

81

095 5

83

09

4 4

78

7

7

4 4

coalnatural gashydro

windsolar

83

010

2

5

83

0

94 4

80

5

84 3

82

19 4

4

78

6

84 4

79

5

84 4

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

IFCFP-PSP f- IFCFP-PSP +

LS +LS -

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

Figure 5 Comparison of power-generation patterns from IFCF-PSP and least-system-cost (LS) models under 119901119894-level 001 Note that theinnermost circle in the pie chart represents the first planning period

is larger) the demand for expansion of clean energy willincrease On the other hand the total demand of capacityexpansion in the lower bound model is lower than the upperbound model One of the reasons for this result is that theannual utilization hours of various power generation facilitiesin the lower bound model are higher than the correspondingparameter settings of the upper bound model

The result of LS model has shown that under thecorresponding lower bound parameter settings the capacityof 015 GW would be added to the hydropower facility inperiod 3 when 119901119894 = 001 and 119901119894 = 005 Only a capacity of015 GWwould be added to the wind power facility in period3 While under the corresponding upper bound parametersettings natural gas and solar energy facilities expansionsare not required that is because the former resource is underpenalty of CO2 emission and the capacity cost of the solarpower expansion is much higher

In the IFCF-PSP model clean energy will be given moredevelopment opportunities Take 119901119894 = 001 as an example(corresponding to 119891minus) with a capacity expansion of 02

GW at the beginning of period 1 and period 2 electricitygeneration ofwind power facilitywould rise from 164 103GWin period 1 to 168 103GW in period 2 and with anothercapacity expansion of 02 GW at the beginning of period 3 itstotal electricity generation capacity would reach 172 103GWLikewise to meet the growing energy demand the capacityof 02 GW would be added to the natural gas-fired facility inperiod 1 period 2 and period 3 as a result its total generationcapacity would rise from 954 times 103GW in period 1 to 119 times103GW in period 2 and reach 124 times 103GW in period 3 Inthe planning period 1 due to more lenient emission policiesand more economical cost strategies some of the natural gaspower generation facilities are idleThe capacities of 025 GWwould be added to the hydropower facility in three periodsand the corresponding power generation capacity would be812 times 103GW 875 103GW and 937 times 103GW Generatingelectricity from solar energy facility is more expensive thanother power generation facilities thus when the emissionlimits and electricity demand are relatively low solar energyfacility would not be expanded during first two periods 015

Mathematical Problems in Engineering 11

Table 6 Binary solutions of capacity expansions

119901119894-level Power-generation facility Capacity-expansion option IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3

01 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 [0 1] [0 1] 0 0 0

Wind power m=1 0 0 0 0 0 0m=2 0 [0 1] 0 0 [0 1] 1m=3 [0 1] [0 1] [0 1] 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 [0 1] [0 1] 0 0 0 [0 1]m=2 0 0 0 0 0 0m=3 [0 1] [0 1] 0 [0 1] [0 1] 0

005 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 [0 1] [0 1] [0 1] 0 0 0

Wind power m=1 [0 1] [0 1] 0 0 0 0m=2 0 0 0 0 [0 1] 1m=3 [0 1] [0 1] 0 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] [0 1]m=2 0 0 0 0 0 0m=3 1 1 0 [0 1] [0 1] 0

001 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 1 1 [0 1] [0 1] [0 1]

Wind power m=1 0 0 0 0 0 0m=2 0 0 0 0 0 [0 1]m=3 1 1 [0 1] [0 1] [0 1] [0 1]

Solar energy m=1 0 0 0 0 0 0m=2 0 0 [0 1] 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] 0m=2 0 0 0 0 0 0m=3 1 1 [0 1] [0 1] [0 1] [0 1]

GW of capacity would only be added to the solar energyfacility in period 3

Apparently with the successful application of IFCI-PSP within a typical regional electric system managementproblem solutions obtained could provide useful decisionalternatives under different policies and various energy avail-abilities Compared with the least-cost model the IFCF-PSP model is an effective tool for providing environmentalmanagement schemes with dual objectives In addition theIFCF-PSP has following advantages over the conventionalprogramming methods (a) balancing multiple conflicting

objectives (b) reflecting interrelationships among systemefficiency economic cost and system reliability (c) effec-tively dealing with randomness in both the objective andconstraints and (d) assisting the analysis of diverse decisionschemes associated with various energy demand levels

5 Conclusions

An inexact fuzzy chance-constrained fractional program-ming approach is developed for optimal electric powersystems management under uncertainties In the developed

12 Mathematical Problems in Engineering

model fuzzy chance-constrained programming is incorpo-rated into a fractional programming optimization frame-work The obtained results are useful for supporting EPSmanagement The IFCF-PSP approach is capable of (i)balancing the conflict between two objectives (ii) reflect-ing different electricity generation and capacity expansionstrategies (iii) presenting optimal solutions under differentconstraint violating conditions and (iv) introducing theconcept of fuzzy boundary interval with which the com-plexity of dual uncertainties can be effectively handled Theresults of IFCF-PSP model show that (i) a higher confidencelevel corresponds to a higher proportion of clean energypower generation and a lower economic productivity and(ii) as a result of encouraging environment-friendly energiesthe generation capacities for the natural gas wind-powersolar energy and hydropower facilities would be signifi-cantly increased The solutions obtained from the IFCF-PSPapproach could provide specific energy options for powersystem planning and provide effective management solutionof the electric power system for identifying electricity gener-ation and capacity expansion schemes

This study attempts to develop a modeling framework forratio problems involving fuzzy uncertainties to deal with elec-tric power systems management problemThe results suggestthat it is also applicable to other energy management prob-lems In the future practice IFCF-PSP could be further im-proved through considering more impact factors Forinstance the fuzzymembership functions and the confidencelevel are critical in the decision-making process and thisrequires effective ways to provide appropriate choices fordecision making Such challenges desire further investiga-tions Future research can be aimed at applying the advancedapproach to a more complex real-world electric power sys-tem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the Fundamental ResearchFunds for the Central Universities NCEPU (2018BJ0293)the State Scholarship Fund (201706735025) the NationalKey Research and Development Plan (2016YFC0502800) theNatural Sciences Foundation (51520105013 51679087) andthe 111 Program (B14008) The data used to support thefindings of this study are available from the correspondingauthor upon request

References

[1] G H Huang ldquoA hybrid inexact-stochastic water managementmodelrdquo European Journal of Operational Research vol 107 no1 pp 137ndash158 1998

[2] A Grosfeld-Nir and A Tishler ldquoA stochastic model for themeasurement of electricity outage costsrdquo Energy pp 157ndash1741993

[3] S Wang G Huang and B W Baetz ldquoAn Inexact Probabilistic-Possibilistic Optimization Framework for Flood Managementin a Hybrid Uncertain Environmentrdquo IEEE Transactions onFuzzy Systems vol 23 no 4 pp 897ndash908 2015

[4] G H Huang and D P Loucks ldquoAn inexact two-stage stochasticprogramming model for water resources management underuncertaintyrdquo Civil Engineering and Environmental Systems vol17 no 2 pp 95ndash118 2000

[5] K Darby-Dowman S Barker E Audsley and D Parsons ldquoAtwo-stage stochastic programming with recourse model fordetermining robust planting plans in horticulturerdquo Journal ofthe Operational Research Society vol 51 no 1 pp 83ndash89 2000

[6] J Hu L Sun C H Li X Wang X L Jia and Y P Cai ldquoWaterQuality Risk Assessment for the Laoguanhe River of ChinaUsing a Stochastic SimulationMethodrdquo Journal of Environmen-tal Informatics vol 31 no 2 pp 123ndash136 2018

[7] SWang andG HHuang ldquoAmulti-level Taguchi-factorial two-stage stochastic programming approach for characterization ofparameter uncertainties and their interactions An applicationto water resources managementrdquo European Journal of Opera-tional Research vol 240 no 2 pp 572ndash581 2015

[8] S Chanas and P Zielinski ldquoOn the equivalence of two opti-mization methods for fuzzy linear programming problemsrdquoEuropean Journal of Operational Research vol 121 no 1 pp 56ndash63 2000

[9] J Mula D Peidro and R Poler ldquoThe effectiveness of a fuzzymathematical programming approach for supply chain pro-duction planning with fuzzy demandrdquo International Journal ofProduction Economics vol 128 no 1 pp 136ndash143 2010

[10] B Chen P Li H J Wu T Husain and F Khan ldquoMCFP Amonte carlo simulation-based fuzzy programming approachfor optimization under dual uncertainties of possibility andcontinuous probabilityrdquo Journal of Environmental Informatics(JEI) vol 29 no 2 pp 88ndash97 2017

[11] C Z Huang S Nie L Guo and Y R Fan ldquoInexact fuzzystochastic chance constraint programming for emergency evac-uation in Qinshan nuclear power plant under uncertaintyrdquoJournal of Environmental Informatics (JEI) vol 30 no 1 pp 63ndash78 2017

[12] U S Sakalli ldquoOptimization of Production-Distribution Prob-lem in Supply Chain Management under Stochastic and FuzzyUncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 4389064 29 pages 2017

[13] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval research logistics (NRL) pp 181ndash186 1962

[14] N I Voropai and E Y Ivanova ldquoMulti-criteria decision analysistechniques in electric power system expansion planningrdquo Inter-national Journal of Electrical Power amp Energy Systems vol 24no 1 pp 71ndash78 2002

[15] M Chakraborty and S Gupta ldquoFuzzy mathematical pro-gramming for multi objective linear fractional programmingproblemrdquo Fuzzy Sets and Systems vol 125 no 3 pp 335ndash3422002

[16] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[17] W Li Z BaoGHHuang andY L Xie ldquoAn Inexact CredibilityChance-Constrained Integer Programming for Greenhouse

Mathematical Problems in Engineering 13

GasMitigationManagement in Regional Electric Power Systemunder Uncertaintyrdquo in Journal of Environmental Informatics2017

[18] M Paunovic N M Ralevic V Gajovic et al ldquoTwo-StageFuzzy Logic Model for Cloud Service Supplier Selection andEvaluationrdquoMathematical Problems in Engineering 2018

[19] S Wang G H Huang B W Baetz and B C Ancell ldquoTowardsrobust quantification and reduction of uncertainty in hydro-logic predictions Integration of particle Markov chain MonteCarlo and factorial polynomial chaos expansionrdquo Journal ofHydrology vol 548 pp 484ndash497 2017

[20] SWang B C Ancell GHHuang andBW Baetz ldquoImprovingRobustness ofHydrologic Ensemble PredictionsThroughProb-abilistic Pre- and Post-Processing in Sequential Data Assimila-tionrdquoWater Resources Research 2018

[21] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[22] A Rong and R Lahdelma ldquoFuzzy chance constrained linearprogramming model for optimizing the scrap charge in steelproductionrdquo European Journal of Operational Research vol 186no 3 pp 953ndash964 2008

[23] X Sun J Li H Qiao and B Zhang ldquoEnergy implicationsof Chinarsquos regional development New insights from multi-regional input-output analysisrdquo Applied Energy vol 196 pp118ndash131 2017

[24] M A Quaddus and T N Goh ldquoElectric power generationexpansion Planning with multiple objectivesrdquo Applied Energyvol 19 no 4 pp 301ndash319 1985

[25] C H Antunes A G Martins and I S Brito ldquoA multipleobjective mixed integer linear programming model for powergeneration expansion planningrdquo Energy vol 29 no 4 pp 613ndash627 2004

[26] J L C Meza M B Yildirim and A S M Masud ldquoA modelfor the multiperiod multiobjective power generation expansionproblemrdquo IEEETransactions on Power Systems vol 22 no 2 pp871ndash878 2007

[27] J-HHan Y-C Ahn and I-B Lee ldquoAmulti-objective optimiza-tion model for sustainable electricity generation and CO2 mit-igation (EGCM) infrastructure design considering economicprofit and financial riskrdquo Applied Energy vol 95 pp 186ndash1952012

[28] M Rekik A Abdelkafi and L Krichen ldquoAmicro-grid ensuringmulti-objective control strategy of a power electrical system forquality improvementrdquo Energy vol 88 pp 351ndash363 2015

[29] A Azadeh Z Raoofi and M Zarrin ldquoA multi-objective fuzzylinear programming model for optimization of natural gassupply chain through a greenhouse gas reduction approachrdquoJournal of Natural Gas Science and Engineering vol 26 pp 702ndash710 2015

[30] K Li L Pan W Xue H Jiang and H Mao ldquoMulti-objectiveoptimization for energy performance improvement of residen-tial buildings a comparative studyrdquo Energies vol 10 no 2 p245 2017

[31] H Zhu W W Huang and G H Huang ldquoPlanning of regionalenergy systems An inexact mixed-integer fractional program-ming modelrdquo Applied Energy vol 113 pp 500ndash514 2014

[32] F Chen G H Huang Y R Fan and R F Liao ldquoA nonlinearfractional programming approach for environmentaleconomicpower dispatchrdquo International Journal of Electrical Power ampEnergy Systems vol 78 pp 463ndash469 2016

[33] C Zhang and P Guo ldquoFLFP A fuzzy linear fractional pro-gramming approach with double-sided fuzziness for optimalirrigation water allocationrdquo Agricultural Water Managementvol 199 pp 105ndash119 2018

[34] L Wang G Huang X Wang and H Zhu ldquoRisk-basedelectric power system planning for climate change mitigationthrough multi-stage joint-probabilistic left-hand-side chance-constrained fractional programming A Canadian case studyrdquoRenewable amp Sustainable Energy Reviews vol 82 pp 1056ndash10672018

[35] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[36] X H Nie G H Huang Y P Li and L Liu ldquoIFRP A hybridinterval-parameter fuzzy robust programming approach forwaste management planning under uncertaintyrdquo Journal ofEnvironmental Management vol 84 no 1 pp 1ndash11 2007

[37] Y Leung Spatial analysis and planning under imprecisionElsevier 1st edition 2013

[38] A Charnes W W Cooper and M J L Kirby ldquoChance-constrained programming an extension of statistical methodrdquoinOptimizingmethods in statistics pp 391ndash402 Academic Press1971

[39] Y P Cai G H Huang Z F Yang and Q Tan ldquoIdentificationof optimal strategies for energy management systems planningunder multiple uncertaintiesrdquoApplied Energy vol 86 no 4 pp480ndash495 2009

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Mathematical Problems in Engineering 3

Mem

bers

hip

Gra

de

1

xaminusij a+

ij

Figure 1 Fuzzy boundaries for interval 119886plusmn

the probability of violating constraint (4) (119901119894) The constraintof the model in the optimization process is changed fromldquorigid satisfactionrdquo to ldquoflexible responserdquo Therefore thescheme can meet the optimization target with flexibility andmaneuverability

However the problem with constraints (4) can onlyreflect the case when A is deterministic In many real-worldproblems the lower and upper bounds of some intervalparameters can rarely be acquired as deterministic valuesInstead they may often be given as subjective informationthat can only be expressed as fuzzy sets This leads todual uncertainties as shown in Figure 1 If both A and Bare uncertain the set of feasible constraints may becomemore complicated To generate a precise analysis of decision-making multiple uncertainties need to be tackled For exam-ple the total carbon dioxide emissions in a certain region canbe described as probability distributions and the statisticsof such a random parameter can be expressed as fuzzy setsThis results in dual uncertainties which can be represented bythe concept of distribution with fuzzy probability (DFP) Inorder to deal with the hybrid uncertainty resulting from fuzzyand stochastic information in constraints and parametersCCP is then integrated into the ILFP method to reflectprobability distribution of fuzzy numbers Models (2a) (2b)and (2c) can be further improved by incorporating fuzzy andchance-constrained techniquesTherefore letRplusmn be the set ofintervals with fuzzy lower and upper bounds and 120595plusmn denotesa set of fuzzy random numbers for fuzzy lower and upperbounds

119899sum119895=1

119886plusmn119894119895119909plusmn119895 le plusmn(119901119894)119894 119894 = 1 2 119898 (5)

Let 119895and be base variables imposed by fuzzy subsets 119860119895

and 119861 then120583119860119895 119895 997888rarr [0 1] (6a)

120583119861 997888rarr [0 1] (6b)

where 120583119860119895 indicates the possibility of consuming a specificamount of resource by activity 119895 and 120583119861 indicates the possible

Mem

bers

hip

Gra

de

1

x

08

05

020

FL(m minus x

)

FR(xminusm )

Figure 2 L-R fuzzy membership function

availability of resource B le means fuzzy inequality Fuzzysubset119873 can be expressed as a LndashR fuzzy number [35]

120583119873 (119909) =

119865119871 (119898 minus 119909120572 ) 119894119891 minusinfin lt 119909 lt 119898 120572 gt 01 119894119891 119909 = 119898119865119877 (119909 minus 119898120575 ) 119894119891 119898 lt 119909 lt +infin 120575 gt 0

(7a)

where 119865119871 and 119865119877 are the shape functions (Figure 2) For alinear case fuzzy subset 119873 can be defined as the followinggeneral format

120583119873 (119909) =

0 119894119891 119909 lt 120572 119900119903 119909 gt 1205721 119894119891 119909 = 1198981 minus 2 |119898 minus 119909|120572 minus 120572 119894119891 120572 lt 119909 lt 120572

(7b)

where [120572 120572] is an interval imposed by fuzzy subset119873 Basedon the method from Nie et al [36] and Leung et al [37] thefuzzy constraints in (5) can be replaced by the following 2119896precise inequalities in which k denotes the number of 120572minus119888119906119905levels119899sum119895=1

119886plusmn119894119895119904119909plusmn119895 le 119887plusmn119894 119904(119901119894) 119894 = 1 2 119898 119904 = 1 2 119896 (8a)

119899sum119895=1

119886plusmn119894119895119904119909plusmn119895 ge 119887plusmn119894 119904(119901119894) 119894 = 1 2 119898 119904 = 1 2 119896 (8b)

where

119886plusmn119894119895119904 = sup (119886plusmn119904119894119895 ) 119886plusmn119904119894119895 isin (119895)120572119904 (8c)

119886plusmn119894119895119904 = inf (119886plusmn119904119894119895 ) 119886plusmn119904119894119895 isin (119895)120572119904 (8d)

119887plusmn119894 119904 = sup (119887plusmn119904119894 ) 119887plusmn119904119894 isin 120572119904

(8e)

119887plusmn119894 119904 = inf (119887plusmn119904119894 ) 119887plusmn119904119894 isin 120572119904

(8f)

120572119904 isin (0 1] (119904 = 1 2 119896) sup(119905) represents the superiorlimit value among set 119905 and inf(119905) denotes the inferior limitvalue among set 119905

4 Mathematical Problems in Engineering

Stochastic information Interval-parameter Fuzzy sets Fractional objective

Chance-constrainedprogramming

Interval-parameterprogramming Fuzzy programming Mixed-integer fractional

programming

Inexact fuzzy chance-constrained fractional programming (IFCFP)

Optimal solutions of IFCFP

Uncertainty Multiple objectives

Figure 3 Framework of the IFCFP approach

Figure 3 presents the framework of the IFCFP methodGive a set of certain values of the 120572 minus 119888119906119905 level to fuzzyparameters 119886119894119895 119894 and 119901119894 and solve models (1a) (1b) and(1c) through the IFCFP approach According to interactivetransform algorithm proposed by Zhu et al [31] the IFCFPmodel can be transformed into two submodels and thencan be solved through the branch-and-bound algorithm andthe method proposed by Charnes et al [38] Comparedwith the existing optimization methods the proposed IFCFPapproach has three characteristics (i) it can tackle ratiooptimization problems (ii) it can handle uncertainties withunknown distribution information by interval parametersand variables and (iii) it can deal with uncertain parameterspresented as fuzzy sets in the objective and the left hand sideaswell as dual uncertainties expressed as the distributionwithfuzzy probability

3 Case Study

To demonstrate its advantages the proposed IFCFP methodis applied to a typical regional electric system managementproblem with representative cost and technical data withina Chinese context In the study system it is assumed thatthere is an independent regional electricity grid whereone nonrenewable resource (coal) and four clean energyresources (natural gas wind solar and hydro) are availablefor electricity generation The decision-makers are responsi-ble for arranging electricity production from those five typesof power-generation to meet the demand of end users Toreflect the dynamic features of the study system three-timeperiods (5 years for each period) are considered in a 15-yearplanning horizon

A sufficient electricity supply at minimum cost is impor-tant to the electric generation expansion planning Thesystem cost 119862plusmn is formulated as a sum of the following

119862plusmn = 1198911plusmn + 1198912plusmn + 1198913plusmn + 1198914plusmn + 1198915plusmn (9a)

(1) the total cost for primary energy supply

1198911plusmn = 119879sum119905=1

119868sum119895=1

119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 (9b)

where 119905 is equal to index for the time periods (119905 = 1 2 119879)119895 is equal to index for the power-generation technology(119895 = 1 2 119869) 119868 = the number of nonrenewable power-generation technology (eg 119895 = 1 coal power 119895 = 2 naturalgas and 119868 lt 119869) 119862119875119864plusmn119905119895 is equal to cost for local primaryenergy supply for power-generation technology 119895 in period119905(103 $TJ)119860119875119864plusmn119905119895 is equal to decision variable and representslocal supply of primary energy resource for power-generationtechnology 119895 in period 119905(TJ)

(2) fixed and variable operating costs for power genera-tion

1198912plusmn = 119879sum119905=1

119869sum119895=1

119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895 (9c)

where 119860119875119866plusmn119905119895 is equal to decision variable and representselectricity generation from power-generation technology 119895 inperiod 119905(GWh)119862119875119866plusmn119905119895 is equal to fixed and variable operationcosts for generating electricity via technology 119895 in period119905(103 $GWh)

(3) cost for capacity expansions

1198913plusmn = 119879sum119905=1

119869sum119895=1

119872sum119898=1

119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898 (9d)

where119898 is equal to index for the capacity expansion options(119898 = 1 2 119872) 119862119864119875plusmn119905119895 is equal to cost for expandingcapacity for generating electricity via technology 119895 in period119905(106 $GW) 119864119862119860plusmn119905119895119898 is equal to capacity expansion optionof power-generation technology 119895 under different expansionprogram 119898 in period 119905(GW) 119884119905119895119898 is equal to binary variableof capacity option 119898 for power-generation technology 119895 inperiod 119905

(4) cost for importing electricity

1198914plusmn = 119879sum119905=1

119862119868119864plusmn119905 times 119860119868119864plusmn119905 (9e)

where 119862119868119864plusmn119905 is equal to cost of importing electricity inperiod 119905(103 $GWh) 119860119868119864plusmn119905 is equal to decision variable and

Mathematical Problems in Engineering 5

represents the shortage amount of electricity needs to beimported in period 119905(GWh)

(5) cost for pollutant mitigation

1198915plusmn = 119879sum119905=1

119868sum119895=1

119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905 (9f)

where 119862119875119872plusmn119905119895 is equal to cost of pollution mitigation ofpower-generation technology 119895 in period 119905(103 $ton) 120578plusmn119905CO2 emission factor in period 119905(103 tonGWh)

Renewable energy resources are intermittent andunreliable which are subject to spatial andor temporal

fluctuations The natural gas generation with low carbondioxide emissions can stabilize the risk of the intermittentand unpredictable nature of renewable energy generationIn this study we assumed natural gas resources as theclean energy power generation Therefore the objective ofthis study is to maximize the ratio between clean energygeneration (including the natural gas) and system costwhile a series of constraints define the interrelationshipsamong the decision variables and system conditionsfactorsThe inexact fuzzy chance-constrained fractional powersystems planning (IFCF-PSP) can be formulated as fol-lows

max119891plusmn = 119862119866plusmn119862plusmn= sum119879119905=1sum119869119895=119868+1119860119875119866plusmn119905119895 + sum119879119905=1 119860119875119866plusmn1199052sum119879119905=1sum119868119895=1 119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 + sum119879119905=1sum119869119895=1 119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895 + sum119879119905=1sum119869119895=1sum119872119898=1 119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898 + sum119879119905=1 119862119868119864plusmn119905 times 119860119868119864plusmn119905 + sum119879119905=1sum119868119895=1 119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905

(10a)

where the clean energy power generation (119862119866plusmn) consists ofthe renewable energies such as wind power solar energy andhydropower (119860119875119866plusmn119905119895 and j = I + 1 J) and the natural gasgeneration in period 119905

The constraints are listed as follows(1) electricity demand constraints119869sum119895=1

119860119875119866plusmn119905119895 + 119860119868119864plusmn119905 ge 119863119872plusmn119905 times (1 + 120579plusmn119905 ) forall119905 (10b)

where119863119872plusmn119905 is equal to local electricity demand (GWh) 120579plusmn119905 isequal to transmission loss in period 119905

(2) capacity limitation constraints for power-generationfacilities

119860119875119866plusmn119905119895 le (119877119862119860plusmn119905119895 + 119872sum119898=1

119864119862119860plusmn119905119895119898 times 119884119905119895119898) times 119878119879119872plusmn119905119895forall119905 119895

(10c)

where 119877119862119860plusmn119905119895 is equal to the current capacity of power-generation technology 119895 in period 119905(GW) 119878119879119872plusmn119905119895 is equal tothe maximum service time of power-generation technology 119895in period 119905(hour)

(3) primary energy availability constraints

119860119875119864plusmn119905119895 le 119880119875119864plusmn119905119895 forall119905 119895 (10d)

119860119875119866plusmn119905119895 times 119903119891plusmn119905119895 le 119860119875119864plusmn119905119895 forall119905 119895 (10e)

where 119880119875119864plusmn119905119895 is equal to available primary energy 119895 inperiod 119905 (119895 = 1 2TJ) 119903119891plusmn119905119895 is equal to energy consumptionconversion rate by power-generation technology 119895 in period119905 (119895 = 1 2TJGWh)

(4) capacity expansion constraints

119877119862119860plusmn119905119895 + 119872sum119898=1

119864119862119860plusmn119905119895119898 times 119884119905119895119898 le 119880119862119860plusmn119905119895 forall119905 119895 (10f)

where 119880119862119860plusmn119905119895 is equal to maximum capacity of generationtechnology 119895 in period 119905(GW)

(5) expansion options constraints

119872sum119898=1

119884119905119895119898 le 1 forall119905 119895 (10g)

119884119905119895119898 = 1 if capacity expansion is undertaken119884119905119895119898 = 0 otherwise(6) import electricity constraints

119860119868119864plusmn119905 le 119880119868119864plusmn119905 forall119905 (10h)

where119880119868119864plusmn119905 is equal to maximum import amount of electric-ity imports in period 119905 (GWh)

(7) renewable energy availability constraints

119869sum119895=119868+1

119860119875119866plusmn119905119895 ge ( 119869sum119895=1

119860119875119866plusmn119905119895)120590plusmn119905 forall119905 (10i)

where 120590plusmn119905 is equal to the minimum proportion of electric-ity generation by renewable energy in the whole power-generation Currently the Renewable Portfolio Standard(RPS) mechanisms have been adopted in several countriesincluding theUnitedKingdom (RenewablesObligation in theUK) Italy Poland Sweden and Belgium and 29 out of 50US states etc According to the governmentrsquos requirement acertain percentage of the electricity generation of the powerenterprises will come from renewable energy sources Theproportion of electricity from renewable sources will usuallyincrease year by year Thus in the model 120590plusmn119905 is used torepresent the lowest ratio of the RPS

(8) pollutants emission constraints

119868sum119895=1

119860119875119866plusmn119905119895 times 120578plusmn119905119895 times (1 minus 120585plusmn119905 ) le 119864119872plusmn119905 (119901119894) forall119905 (10j)

6 Mathematical Problems in Engineering

Table 1 Cost of energy supply and conversion parameters

Periodt=1 t=2 t=3

Energy supply cost (103 $TJ)Coal [256 306] [326 376] [396 446]Natural gas [676 696] [757 777] [838 858]Units of energy carrier per units of electricity generation (TJGWh)Coal [1112 117] [1068 1116] [1015 1062]Natural gas [838 882] [754 792] [679 738]

Table 2 Fuzzy subsets for efficiency coefficient under different 120572 minus 119888119906119905 levelsperiod efficiency coefficient120572 minus 119888119906119905 level 120588+ 120588+ 120588minus 120588minus

0t=1 0125 0235 012 023t=2 014 0175 0136 017t=3 0119 0149 0116 0145

02t=1 0135 0223 013 0218t=2 0142 017 0138 0165t=3 012 0144 0117 014

05t=1 015 0205 0145 02t=2 0144 0162 014 0157t=3 0123 0138 0119 0134

08t=1 0165 0187 016 0182t=2 0147 0154 0143 015t=3 0125 0131 0122 0127

1t=1 0175 0175 017 017t=2 0149 0149 0145 0145t=3 0126 0126 0123 0123

Note efficiency coefficient (120588plusmn119905 = 1 minus 120585plusmn119905 )

where 119864119872119905 is equal to the permitted CO2 emission in period119905(103 ton) 120585plusmn119905 is equal to the efficiency of chemical absorptionor capture and storage of CO2 in period 119905119901119894 = the riskconfidence levels

(9) nonnegativity constraints

119860119875119864plusmn119905119895 119860119875119866plusmn119905119895 119860119868119864plusmn119905 ge 0 forall119905 119895 (10k)

Interval parameters are adopted to address impreciseuncertainties which are generally associated with electricitydemands prices of energy resources costs of capacity expan-sion and many other constraints The real research data areused as input data of the model The detailed descriptionsof cost for energy supply and relative conversion parameterswere illustrated in Table 1 Table 2 gives fuzzy subsets forefficiency coefficient under different120572minus119888119906119905 levels Table 3 liststhe capacity expansion options and capital investment costsfor each facility

4 Results and Discussion

In Figure 4 coal-fired electricity supply would increasesteadily and still play an important role in the power sys-tem due to its high availability and competitive price overthe study planning horizon However increasingly strin-gent emission limits result in idle coal-fired facilities Moreeconomical and environmentally friendly power generationfacilities will be prioritized The risk confidence level (119901119894) isemployed to the constraints of CO2 emission requirementThe decision-maker can adjust the value of the 119901119894 levelaccording to actual needs We assumed that violations ofemission constraints are allowed under three given 119901119894 levels(119901119894 = 001005 and 01 which are normally adopted asthe significance levels) [39] The higher credibility levelwould correspond to a tight environment requirement thusleading to a lower CO2 emission while the lower credibilitylevel would correspond to a relatively relaxed environmentrequirement thus resulting in a higher CO2 emission There-fore changes in 119901119894-level have an impact on coal-fired power

Mathematical Problems in Engineering 7

Table 3 Capacity expansion options and costs for power-generation facilities

Periodt=1 t=2 t=3

Capacity-expansion options (GW)Coal m=1 005 005 005

m=2 010 010 010m=3 015 015 015

Natural gas m=1 010 010 010m=2 015 015 015m=3 020 020 020

Wind power m=1 005 005 005m=2 015 015 015m=3 020 020 020

Solar energy m=1 005 005 005m=2 015 015 015m=3 020 020 020

Hydropower m=1 015 015 015m=2 020 020 020m=3 025 025 025

Capacity expansion cost (106$GW)Coal [577 607] [547 577] [517 547]Natural gas [726 756] [676 726] [626 676]Wind power [1256 1306] [1156 1206] [1056 1106]Solar energy [2668 2768] [2468 2568] [2268 2368]Hydropower [1597 1697] [1497 1597] [1397 1497]

generation The total supply of coal-fired generation woulddecrease from [48069 50720] times 109GWh when 119901119894 = 01to [47175 50685] times 109GWh when 119901119894 = 005 and reach[45938 50313] times 109GWh when 119901119894 = 001 In contrast cleanenergy generation will increase For example solar powergeneration of three periods would rise from [2438 2768] times109GWh when 119901119894 = 01 to [2562 2768] times 109GWh when 119901119894= 005 and reach [2625 2769] times 109GWh when 119901119894 = 001 Asshown in Figure 4 the generation of solar power and naturalgas in clean power facilities are mainly affected by emissionconstraints When 119901119894 = 001 the solar power generation isthe largest and the power generation from natural gas isthe lowest in the three-study 119901119894-levels This is the result ofthe tradeoff between carbon dioxide emissions and systemcosts Developing natural gas electricity requires a relativelylow capital cost but leads to more CO2 emissions and thesolar energy technology needs a low operational cost but anextremely high capital cost Similarly the results under other119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted Duringthe entire planning horizon the ratio objective between cleanenergy generation and the total system cost would be [2153]GWh per $106 which also represents the range of the systemefficiency When the electricity-generation pattern variesunder different 119901119894 levels and within the interval solutionranges the system efficiency would also fluctuate within itssolution range correspondingly Thus the IFCF-PSP resultscan create multiple decision alternatives through adjusting

different combinations of the solutions Table 4 presentsthe typical alternatives where combinations of upperlowerbound values for 119860119875119864plusmn119905119895 119860119875119866plusmn119905119895 119884plusmn119905119895119898 (119895 = 1 forall119905 119898) and119860119875119866plusmn119905119895 119884plusmn119905119895119898 (119895 = 2 3 4 5 forall119905 119898) are examined Underthe same power generation alternative interval solutionof the system cost can be obtained according to intervalprice parameters and corresponds to the interval solution ofsystem efficiency Based on the result of the total electricitygenerated we divided the 12 alternatives into three groups(high A3 A4 A8 and A12 medium A1 A5 A7 and A9 lowA2A6A10 andA11) tomeet the future highmedium or lowelectricity demand

The comparison of the system efficiency under different119901119894 levels shows that A11gtA6 gtA10 gtA2 under low-level totalpower generation group Alternative A11 (corresponding tof+) would lead to the most sustainable option with a systemefficiency as much as [27142999] GWh per $106 whichcorresponds to the highest amount of clean energy electricity(0478 times 106GWh) and a moderate cost ([15936 17612] times109$) Additionally lower-level electricity demands will besatisfied by a total power generation of 1856 106GWhTherefore Alternative A11 is a desirable choice from thepoint of view of resources conservation and environmentalprotection when electricity demand is at the low levelAlternative A6 would also meet low electricity demand andreached the lowest system cost ([15142 16785] times 109$)When Alternative A10 is adopted all the power generation

8 Mathematical Problems in Engineering

020406080

100120140160180200

Submodel f-Submodel f+

Submodel f-Submodel f+

Submodel f-Submodel f+

020406080

100120140160180200

020406080

100120140160180200

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

ＪＣ=01

ＪＣ=005

ＪＣ=001

Pow

er g

ener

atio

n (103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)

Figure 4 Power generation loads for EPS through the IFCF-PSPmodel

activities will reach their lower bound levels leading to thelowest power generation (1764 times 106GWh) Compared withAlternative A6 this alternative may be of less interest todecision-makers due to its lower system efficiency and highersystem cost However it is the solution that is obtainedunder the most stringent emission constraints (ie 119901119894 =001) Alternative A2 is a moderate solution in this group Itssystem cost system efficiency total power generation andclean energy power generation are [15295 16954] times 109$[22062445] GWh per $106 1816 times 106GWh and 0374 times106GWh

In middle-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A7gt A5gt A9 gtA1 In alternative A7middle-level electricity demands will be satisfied by a totalpower generation of 1893 times 106GWh In addition Alternative

A7 has the highest system efficiency ([2661 2941] GWhper 106$) and the largest amount of clean energy powergeneration (0478times 106GWh) in this groupThus AlternativeA7 is considered to be a desirable choice Alternatives A5 andA9 would lead to the same total power generation solution(1895 times 106GWh) They are the two-moderate solutions inthis groupThe system cost obtained from the A1 is the lowest([15718 17413] times 109$) in this group It would lead to thelowest system efficiency ([2148 2379] GWh per 106$) butdue to the largest proportion of fossil-fired power generationA1 has the maximum security for energy supply

In high-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A3gt A12gt A8 gtA4 The same cleanenergy power generation obtained from the 4 alternativesis 0478 106GWh In Alternative A4 all the power gener-ation activities will be equal to their upper-bound valuesat the same time It would lead to the highest system cost([16901 18676] times 109$) but the moderate system efficiency([2559 2828]GWhper $106) In comparisonAlternative A3would lead to the minimum system cost ([16478 18216] times109$) and the highest system efficiency ([2624 2901] GWhper $106) in this group In addition this alternative wouldalso provide the highest clean energy power generationTherefore this alternative is a desirable sustainable optionunder the high electricity demand level Alternatives A8 andA12 can provide modest system efficiencies which would be[2561 283] and [2575 2845] GWh per $106 respectivelyand lower system costs which would be [16892 18666] and[16800 18564] times 109$ Therefore decision-makers who payattention to the stability of the system may be interested inthese two alternatives

The above alternatives represent various options betweeneconomic and environmental tradeoffsWillingness to accepthigh system cost will guarantee meeting the objective ofincreasing the proportion of clean energy A strong desireto acquire low system cost will cause the risk of violatingemission constraints In general the above research resultswere favored by decision makers due to their flexibilityand preference for practical-making decision processes Thefeasible ranges for decision variables under different 119901119894 levelswere useful for decision makers to justify the generatedalternatives directly

Besides the scenario of maximizing the proportion ofclean energy another scenario of minimizing the system costis also analyzed to evaluate the effects of different energy sup-ply policies The optimal-ratio problem presented in Models(10a)ndash(10k) can be converted into a least-cost problem withthe following objective

min119891 = system cost

= 119879sum119905=1

119868sum119895=1

119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 + 119879sum119905=1

119869sum119895=1

119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895+ 119879sum119905=1

119869sum119895=1

119872sum119898=1

119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898

Mathematical Problems in Engineering 9

Table 4 Typical decision alternatives obtained from the IFCF-PSP model solutions

Alternative 119860119875119864plusmn119905119895 119860119875119866

plusmn119905119895119884plusmn119905119895119898

(119895=1 forall119905119898)

119860119875119866plusmn119905119895 119884plusmn119905119895119898

(119895=2345 forall119905119898)

System efficiency(GWh per 106$) System cost (109$) Total power generation

(106GWh)Clean energy powergeneration (106GWh)119901119894 = 01

A1 + - [21482379] (12) [1571817413] (4) 1895 (6) 0374 (11)A2 - - [22062445] (9) [1529516954] (3) 1816 (10) 0374 (11)A3 - + [26242901] (3) [1647818216] (9) 192 (4) 0478 (1)A4 + + [25592828] (6) [16901 18676] (12) 2 (1) 0478 (1)119901119894 = 005A5 + - [22362477] (8) [1578317487] (5) 1912 (5) 0391 (7)A6 - - [23292582] (7) [1514216785] (1) 1807 (11) 0391 (7)A7 - + [26612941] (2) [1625117963] (8) 1893 (8) 0478 (1)A8 + + [2561283] (5) [1689218666] (11) 1999 (2) 0478 (1)119901119894 = 001A9 + - [2174 2406] (11) [16040 17757] (7) 1895 (6) 0386 (9)A10 - - [22972543] (8) [15176 16806] (2) 1764 (12) 0386 (9)A11 - + [27142999] (1) [1593617612] (6) 1856 (9) 0478 (1)A12 + + [25752845] (4) [16800 18564] (10) 1987 (3) 0478 (1)

Table 5 The proportion of clean energy power generation from IFCF-PSP and LS models

Clean power generation ratio () IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3119901119894 = 01 [1802 2432] [2089 2493] [1986 2540] [1623 1870] [1681 1719] [1515 1638]119901119894 = 005 [2067 2465] [2093 2528] [2010 2578] [1624 1870] [1681 1719] [1632 1869]119901119894 = 001 [2155 2530] [2158 2596] [2141 2596] [1648 1893] [1679 1752] [1997 2224]

+ 119879sum119905=1

119862119868119864plusmn119905 times 119860119868119864plusmn119905+ 119879sum119905=1

119868sum119895=1

119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905(11)

Figure 5 shows the proportion of different power gen-eration technologies from the IFCF-PSP and LS modelsunder 119901119894 = 001 over three planning periods The IFCF-PSPmodel leads to a relatively higher percentage of natural gaspower generation in comparison the LS model leads torelatively higher percentages of coal-fired electricity suppliesAccording to the solutions from IFCF-PSPmodel the percentof electricity generated by natural gas facilities would be [5 6] [7 8] and [5 8] in periods 1 2 and 3 respectivelyTheLS model achieves the slightly lower percent (ie 0 0 and[5 6] in periods 1 2 and 3) Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted

According to Table 5 the differences can be foundbetween the results of two models under different 119901119894 levelsThe proportion of clean energy power generation fromIFCF-PSP model is higher than that of the LS model Forexample under 119901119894 = 001 clean energy power generationof the entire region occupied [2155 2530] [2158 2596]and [2141 2596] of the total electricity generation in

the three study periods from IFCF-PSP model which arehigher than the LS model [1648 1993] [1679 1752] and[1997 2224] respectively Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpretedAn increased 119901119894 level represents a higher admissible riskleading to a decreased strictness for the emission constraintsand hence an expanded decision space Therefore undertypical conditions as the risk level becomes lower theproportion of clean energy power generation or renewablepower generation would increase For example in period 2when the 119901119894 level is dropped from 01 to 001 the proportionof clean energy power generation from IFCF-PSP modelwould be increased from [2089 2493] to [2158 2596]Likewise the results under other periods can be similarlyanalyzed The confidence level of constraints satisfaction ismore reliable because the risk level is lowerTherefore in thiscase the decision-makers will be more conservative in theEPS management There is no capacity-expansion would beconducted for the coal-fired facility in three periods sinceits high carbon dioxide emissions and increasingly stringentemission limits On the other hand clean energy electricitysupply would be insufficient for the future energy demandsAccording to Table 6 when 119901119894 takes different values (ie 119901119894 =001119901119894 = 005 and 119901119894 = 01) differences of capacity-expansionoption can be found between the results of two models Inboth models when the risk of violating the constraints ofcarbon emission target is decreased (ie the value of 119901119894-level

10 Mathematical Problems in Engineering

75

79

5 4

74

8

95 4

74

8

9

5 4

81

095 5

83

09

4 4

78

7

7

4 4

coalnatural gashydro

windsolar

83

010

2

5

83

0

94 4

80

5

84 3

82

19 4

4

78

6

84 4

79

5

84 4

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

IFCFP-PSP f- IFCFP-PSP +

LS +LS -

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

Figure 5 Comparison of power-generation patterns from IFCF-PSP and least-system-cost (LS) models under 119901119894-level 001 Note that theinnermost circle in the pie chart represents the first planning period

is larger) the demand for expansion of clean energy willincrease On the other hand the total demand of capacityexpansion in the lower bound model is lower than the upperbound model One of the reasons for this result is that theannual utilization hours of various power generation facilitiesin the lower bound model are higher than the correspondingparameter settings of the upper bound model

The result of LS model has shown that under thecorresponding lower bound parameter settings the capacityof 015 GW would be added to the hydropower facility inperiod 3 when 119901119894 = 001 and 119901119894 = 005 Only a capacity of015 GWwould be added to the wind power facility in period3 While under the corresponding upper bound parametersettings natural gas and solar energy facilities expansionsare not required that is because the former resource is underpenalty of CO2 emission and the capacity cost of the solarpower expansion is much higher

In the IFCF-PSP model clean energy will be given moredevelopment opportunities Take 119901119894 = 001 as an example(corresponding to 119891minus) with a capacity expansion of 02

GW at the beginning of period 1 and period 2 electricitygeneration ofwind power facilitywould rise from 164 103GWin period 1 to 168 103GW in period 2 and with anothercapacity expansion of 02 GW at the beginning of period 3 itstotal electricity generation capacity would reach 172 103GWLikewise to meet the growing energy demand the capacityof 02 GW would be added to the natural gas-fired facility inperiod 1 period 2 and period 3 as a result its total generationcapacity would rise from 954 times 103GW in period 1 to 119 times103GW in period 2 and reach 124 times 103GW in period 3 Inthe planning period 1 due to more lenient emission policiesand more economical cost strategies some of the natural gaspower generation facilities are idleThe capacities of 025 GWwould be added to the hydropower facility in three periodsand the corresponding power generation capacity would be812 times 103GW 875 103GW and 937 times 103GW Generatingelectricity from solar energy facility is more expensive thanother power generation facilities thus when the emissionlimits and electricity demand are relatively low solar energyfacility would not be expanded during first two periods 015

Mathematical Problems in Engineering 11

Table 6 Binary solutions of capacity expansions

119901119894-level Power-generation facility Capacity-expansion option IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3

01 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 [0 1] [0 1] 0 0 0

Wind power m=1 0 0 0 0 0 0m=2 0 [0 1] 0 0 [0 1] 1m=3 [0 1] [0 1] [0 1] 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 [0 1] [0 1] 0 0 0 [0 1]m=2 0 0 0 0 0 0m=3 [0 1] [0 1] 0 [0 1] [0 1] 0

005 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 [0 1] [0 1] [0 1] 0 0 0

Wind power m=1 [0 1] [0 1] 0 0 0 0m=2 0 0 0 0 [0 1] 1m=3 [0 1] [0 1] 0 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] [0 1]m=2 0 0 0 0 0 0m=3 1 1 0 [0 1] [0 1] 0

001 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 1 1 [0 1] [0 1] [0 1]

Wind power m=1 0 0 0 0 0 0m=2 0 0 0 0 0 [0 1]m=3 1 1 [0 1] [0 1] [0 1] [0 1]

Solar energy m=1 0 0 0 0 0 0m=2 0 0 [0 1] 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] 0m=2 0 0 0 0 0 0m=3 1 1 [0 1] [0 1] [0 1] [0 1]

GW of capacity would only be added to the solar energyfacility in period 3

Apparently with the successful application of IFCI-PSP within a typical regional electric system managementproblem solutions obtained could provide useful decisionalternatives under different policies and various energy avail-abilities Compared with the least-cost model the IFCF-PSP model is an effective tool for providing environmentalmanagement schemes with dual objectives In addition theIFCF-PSP has following advantages over the conventionalprogramming methods (a) balancing multiple conflicting

objectives (b) reflecting interrelationships among systemefficiency economic cost and system reliability (c) effec-tively dealing with randomness in both the objective andconstraints and (d) assisting the analysis of diverse decisionschemes associated with various energy demand levels

5 Conclusions

An inexact fuzzy chance-constrained fractional program-ming approach is developed for optimal electric powersystems management under uncertainties In the developed

12 Mathematical Problems in Engineering

model fuzzy chance-constrained programming is incorpo-rated into a fractional programming optimization frame-work The obtained results are useful for supporting EPSmanagement The IFCF-PSP approach is capable of (i)balancing the conflict between two objectives (ii) reflect-ing different electricity generation and capacity expansionstrategies (iii) presenting optimal solutions under differentconstraint violating conditions and (iv) introducing theconcept of fuzzy boundary interval with which the com-plexity of dual uncertainties can be effectively handled Theresults of IFCF-PSP model show that (i) a higher confidencelevel corresponds to a higher proportion of clean energypower generation and a lower economic productivity and(ii) as a result of encouraging environment-friendly energiesthe generation capacities for the natural gas wind-powersolar energy and hydropower facilities would be signifi-cantly increased The solutions obtained from the IFCF-PSPapproach could provide specific energy options for powersystem planning and provide effective management solutionof the electric power system for identifying electricity gener-ation and capacity expansion schemes

This study attempts to develop a modeling framework forratio problems involving fuzzy uncertainties to deal with elec-tric power systems management problemThe results suggestthat it is also applicable to other energy management prob-lems In the future practice IFCF-PSP could be further im-proved through considering more impact factors Forinstance the fuzzymembership functions and the confidencelevel are critical in the decision-making process and thisrequires effective ways to provide appropriate choices fordecision making Such challenges desire further investiga-tions Future research can be aimed at applying the advancedapproach to a more complex real-world electric power sys-tem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the Fundamental ResearchFunds for the Central Universities NCEPU (2018BJ0293)the State Scholarship Fund (201706735025) the NationalKey Research and Development Plan (2016YFC0502800) theNatural Sciences Foundation (51520105013 51679087) andthe 111 Program (B14008) The data used to support thefindings of this study are available from the correspondingauthor upon request

References

[1] G H Huang ldquoA hybrid inexact-stochastic water managementmodelrdquo European Journal of Operational Research vol 107 no1 pp 137ndash158 1998

[2] A Grosfeld-Nir and A Tishler ldquoA stochastic model for themeasurement of electricity outage costsrdquo Energy pp 157ndash1741993

[3] S Wang G Huang and B W Baetz ldquoAn Inexact Probabilistic-Possibilistic Optimization Framework for Flood Managementin a Hybrid Uncertain Environmentrdquo IEEE Transactions onFuzzy Systems vol 23 no 4 pp 897ndash908 2015

[4] G H Huang and D P Loucks ldquoAn inexact two-stage stochasticprogramming model for water resources management underuncertaintyrdquo Civil Engineering and Environmental Systems vol17 no 2 pp 95ndash118 2000

[5] K Darby-Dowman S Barker E Audsley and D Parsons ldquoAtwo-stage stochastic programming with recourse model fordetermining robust planting plans in horticulturerdquo Journal ofthe Operational Research Society vol 51 no 1 pp 83ndash89 2000

[6] J Hu L Sun C H Li X Wang X L Jia and Y P Cai ldquoWaterQuality Risk Assessment for the Laoguanhe River of ChinaUsing a Stochastic SimulationMethodrdquo Journal of Environmen-tal Informatics vol 31 no 2 pp 123ndash136 2018

[7] SWang andG HHuang ldquoAmulti-level Taguchi-factorial two-stage stochastic programming approach for characterization ofparameter uncertainties and their interactions An applicationto water resources managementrdquo European Journal of Opera-tional Research vol 240 no 2 pp 572ndash581 2015

[8] S Chanas and P Zielinski ldquoOn the equivalence of two opti-mization methods for fuzzy linear programming problemsrdquoEuropean Journal of Operational Research vol 121 no 1 pp 56ndash63 2000

[9] J Mula D Peidro and R Poler ldquoThe effectiveness of a fuzzymathematical programming approach for supply chain pro-duction planning with fuzzy demandrdquo International Journal ofProduction Economics vol 128 no 1 pp 136ndash143 2010

[10] B Chen P Li H J Wu T Husain and F Khan ldquoMCFP Amonte carlo simulation-based fuzzy programming approachfor optimization under dual uncertainties of possibility andcontinuous probabilityrdquo Journal of Environmental Informatics(JEI) vol 29 no 2 pp 88ndash97 2017

[11] C Z Huang S Nie L Guo and Y R Fan ldquoInexact fuzzystochastic chance constraint programming for emergency evac-uation in Qinshan nuclear power plant under uncertaintyrdquoJournal of Environmental Informatics (JEI) vol 30 no 1 pp 63ndash78 2017

[12] U S Sakalli ldquoOptimization of Production-Distribution Prob-lem in Supply Chain Management under Stochastic and FuzzyUncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 4389064 29 pages 2017

[13] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval research logistics (NRL) pp 181ndash186 1962

[14] N I Voropai and E Y Ivanova ldquoMulti-criteria decision analysistechniques in electric power system expansion planningrdquo Inter-national Journal of Electrical Power amp Energy Systems vol 24no 1 pp 71ndash78 2002

[15] M Chakraborty and S Gupta ldquoFuzzy mathematical pro-gramming for multi objective linear fractional programmingproblemrdquo Fuzzy Sets and Systems vol 125 no 3 pp 335ndash3422002

[16] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[17] W Li Z BaoGHHuang andY L Xie ldquoAn Inexact CredibilityChance-Constrained Integer Programming for Greenhouse

Mathematical Problems in Engineering 13

GasMitigationManagement in Regional Electric Power Systemunder Uncertaintyrdquo in Journal of Environmental Informatics2017

[18] M Paunovic N M Ralevic V Gajovic et al ldquoTwo-StageFuzzy Logic Model for Cloud Service Supplier Selection andEvaluationrdquoMathematical Problems in Engineering 2018

[19] S Wang G H Huang B W Baetz and B C Ancell ldquoTowardsrobust quantification and reduction of uncertainty in hydro-logic predictions Integration of particle Markov chain MonteCarlo and factorial polynomial chaos expansionrdquo Journal ofHydrology vol 548 pp 484ndash497 2017

[20] SWang B C Ancell GHHuang andBW Baetz ldquoImprovingRobustness ofHydrologic Ensemble PredictionsThroughProb-abilistic Pre- and Post-Processing in Sequential Data Assimila-tionrdquoWater Resources Research 2018

[21] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[22] A Rong and R Lahdelma ldquoFuzzy chance constrained linearprogramming model for optimizing the scrap charge in steelproductionrdquo European Journal of Operational Research vol 186no 3 pp 953ndash964 2008

[23] X Sun J Li H Qiao and B Zhang ldquoEnergy implicationsof Chinarsquos regional development New insights from multi-regional input-output analysisrdquo Applied Energy vol 196 pp118ndash131 2017

[24] M A Quaddus and T N Goh ldquoElectric power generationexpansion Planning with multiple objectivesrdquo Applied Energyvol 19 no 4 pp 301ndash319 1985

[25] C H Antunes A G Martins and I S Brito ldquoA multipleobjective mixed integer linear programming model for powergeneration expansion planningrdquo Energy vol 29 no 4 pp 613ndash627 2004

[26] J L C Meza M B Yildirim and A S M Masud ldquoA modelfor the multiperiod multiobjective power generation expansionproblemrdquo IEEETransactions on Power Systems vol 22 no 2 pp871ndash878 2007

[27] J-HHan Y-C Ahn and I-B Lee ldquoAmulti-objective optimiza-tion model for sustainable electricity generation and CO2 mit-igation (EGCM) infrastructure design considering economicprofit and financial riskrdquo Applied Energy vol 95 pp 186ndash1952012

[28] M Rekik A Abdelkafi and L Krichen ldquoAmicro-grid ensuringmulti-objective control strategy of a power electrical system forquality improvementrdquo Energy vol 88 pp 351ndash363 2015

[29] A Azadeh Z Raoofi and M Zarrin ldquoA multi-objective fuzzylinear programming model for optimization of natural gassupply chain through a greenhouse gas reduction approachrdquoJournal of Natural Gas Science and Engineering vol 26 pp 702ndash710 2015

[30] K Li L Pan W Xue H Jiang and H Mao ldquoMulti-objectiveoptimization for energy performance improvement of residen-tial buildings a comparative studyrdquo Energies vol 10 no 2 p245 2017

[31] H Zhu W W Huang and G H Huang ldquoPlanning of regionalenergy systems An inexact mixed-integer fractional program-ming modelrdquo Applied Energy vol 113 pp 500ndash514 2014

[32] F Chen G H Huang Y R Fan and R F Liao ldquoA nonlinearfractional programming approach for environmentaleconomicpower dispatchrdquo International Journal of Electrical Power ampEnergy Systems vol 78 pp 463ndash469 2016

[33] C Zhang and P Guo ldquoFLFP A fuzzy linear fractional pro-gramming approach with double-sided fuzziness for optimalirrigation water allocationrdquo Agricultural Water Managementvol 199 pp 105ndash119 2018

[34] L Wang G Huang X Wang and H Zhu ldquoRisk-basedelectric power system planning for climate change mitigationthrough multi-stage joint-probabilistic left-hand-side chance-constrained fractional programming A Canadian case studyrdquoRenewable amp Sustainable Energy Reviews vol 82 pp 1056ndash10672018

[35] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[36] X H Nie G H Huang Y P Li and L Liu ldquoIFRP A hybridinterval-parameter fuzzy robust programming approach forwaste management planning under uncertaintyrdquo Journal ofEnvironmental Management vol 84 no 1 pp 1ndash11 2007

[37] Y Leung Spatial analysis and planning under imprecisionElsevier 1st edition 2013

[38] A Charnes W W Cooper and M J L Kirby ldquoChance-constrained programming an extension of statistical methodrdquoinOptimizingmethods in statistics pp 391ndash402 Academic Press1971

[39] Y P Cai G H Huang Z F Yang and Q Tan ldquoIdentificationof optimal strategies for energy management systems planningunder multiple uncertaintiesrdquoApplied Energy vol 86 no 4 pp480ndash495 2009

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4 Mathematical Problems in Engineering

Stochastic information Interval-parameter Fuzzy sets Fractional objective

Chance-constrainedprogramming

Interval-parameterprogramming Fuzzy programming Mixed-integer fractional

programming

Inexact fuzzy chance-constrained fractional programming (IFCFP)

Optimal solutions of IFCFP

Uncertainty Multiple objectives

Figure 3 Framework of the IFCFP approach

Figure 3 presents the framework of the IFCFP methodGive a set of certain values of the 120572 minus 119888119906119905 level to fuzzyparameters 119886119894119895 119894 and 119901119894 and solve models (1a) (1b) and(1c) through the IFCFP approach According to interactivetransform algorithm proposed by Zhu et al [31] the IFCFPmodel can be transformed into two submodels and thencan be solved through the branch-and-bound algorithm andthe method proposed by Charnes et al [38] Comparedwith the existing optimization methods the proposed IFCFPapproach has three characteristics (i) it can tackle ratiooptimization problems (ii) it can handle uncertainties withunknown distribution information by interval parametersand variables and (iii) it can deal with uncertain parameterspresented as fuzzy sets in the objective and the left hand sideaswell as dual uncertainties expressed as the distributionwithfuzzy probability

3 Case Study

To demonstrate its advantages the proposed IFCFP methodis applied to a typical regional electric system managementproblem with representative cost and technical data withina Chinese context In the study system it is assumed thatthere is an independent regional electricity grid whereone nonrenewable resource (coal) and four clean energyresources (natural gas wind solar and hydro) are availablefor electricity generation The decision-makers are responsi-ble for arranging electricity production from those five typesof power-generation to meet the demand of end users Toreflect the dynamic features of the study system three-timeperiods (5 years for each period) are considered in a 15-yearplanning horizon

A sufficient electricity supply at minimum cost is impor-tant to the electric generation expansion planning Thesystem cost 119862plusmn is formulated as a sum of the following

119862plusmn = 1198911plusmn + 1198912plusmn + 1198913plusmn + 1198914plusmn + 1198915plusmn (9a)

(1) the total cost for primary energy supply

1198911plusmn = 119879sum119905=1

119868sum119895=1

119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 (9b)

where 119905 is equal to index for the time periods (119905 = 1 2 119879)119895 is equal to index for the power-generation technology(119895 = 1 2 119869) 119868 = the number of nonrenewable power-generation technology (eg 119895 = 1 coal power 119895 = 2 naturalgas and 119868 lt 119869) 119862119875119864plusmn119905119895 is equal to cost for local primaryenergy supply for power-generation technology 119895 in period119905(103 $TJ)119860119875119864plusmn119905119895 is equal to decision variable and representslocal supply of primary energy resource for power-generationtechnology 119895 in period 119905(TJ)

(2) fixed and variable operating costs for power genera-tion

1198912plusmn = 119879sum119905=1

119869sum119895=1

119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895 (9c)

where 119860119875119866plusmn119905119895 is equal to decision variable and representselectricity generation from power-generation technology 119895 inperiod 119905(GWh)119862119875119866plusmn119905119895 is equal to fixed and variable operationcosts for generating electricity via technology 119895 in period119905(103 $GWh)

(3) cost for capacity expansions

1198913plusmn = 119879sum119905=1

119869sum119895=1

119872sum119898=1

119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898 (9d)

where119898 is equal to index for the capacity expansion options(119898 = 1 2 119872) 119862119864119875plusmn119905119895 is equal to cost for expandingcapacity for generating electricity via technology 119895 in period119905(106 $GW) 119864119862119860plusmn119905119895119898 is equal to capacity expansion optionof power-generation technology 119895 under different expansionprogram 119898 in period 119905(GW) 119884119905119895119898 is equal to binary variableof capacity option 119898 for power-generation technology 119895 inperiod 119905

(4) cost for importing electricity

1198914plusmn = 119879sum119905=1

119862119868119864plusmn119905 times 119860119868119864plusmn119905 (9e)

where 119862119868119864plusmn119905 is equal to cost of importing electricity inperiod 119905(103 $GWh) 119860119868119864plusmn119905 is equal to decision variable and

Mathematical Problems in Engineering 5

represents the shortage amount of electricity needs to beimported in period 119905(GWh)

(5) cost for pollutant mitigation

1198915plusmn = 119879sum119905=1

119868sum119895=1

119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905 (9f)

where 119862119875119872plusmn119905119895 is equal to cost of pollution mitigation ofpower-generation technology 119895 in period 119905(103 $ton) 120578plusmn119905CO2 emission factor in period 119905(103 tonGWh)

Renewable energy resources are intermittent andunreliable which are subject to spatial andor temporal

fluctuations The natural gas generation with low carbondioxide emissions can stabilize the risk of the intermittentand unpredictable nature of renewable energy generationIn this study we assumed natural gas resources as theclean energy power generation Therefore the objective ofthis study is to maximize the ratio between clean energygeneration (including the natural gas) and system costwhile a series of constraints define the interrelationshipsamong the decision variables and system conditionsfactorsThe inexact fuzzy chance-constrained fractional powersystems planning (IFCF-PSP) can be formulated as fol-lows

max119891plusmn = 119862119866plusmn119862plusmn= sum119879119905=1sum119869119895=119868+1119860119875119866plusmn119905119895 + sum119879119905=1 119860119875119866plusmn1199052sum119879119905=1sum119868119895=1 119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 + sum119879119905=1sum119869119895=1 119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895 + sum119879119905=1sum119869119895=1sum119872119898=1 119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898 + sum119879119905=1 119862119868119864plusmn119905 times 119860119868119864plusmn119905 + sum119879119905=1sum119868119895=1 119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905

(10a)

where the clean energy power generation (119862119866plusmn) consists ofthe renewable energies such as wind power solar energy andhydropower (119860119875119866plusmn119905119895 and j = I + 1 J) and the natural gasgeneration in period 119905

The constraints are listed as follows(1) electricity demand constraints119869sum119895=1

119860119875119866plusmn119905119895 + 119860119868119864plusmn119905 ge 119863119872plusmn119905 times (1 + 120579plusmn119905 ) forall119905 (10b)

where119863119872plusmn119905 is equal to local electricity demand (GWh) 120579plusmn119905 isequal to transmission loss in period 119905

(2) capacity limitation constraints for power-generationfacilities

119860119875119866plusmn119905119895 le (119877119862119860plusmn119905119895 + 119872sum119898=1

119864119862119860plusmn119905119895119898 times 119884119905119895119898) times 119878119879119872plusmn119905119895forall119905 119895

(10c)

where 119877119862119860plusmn119905119895 is equal to the current capacity of power-generation technology 119895 in period 119905(GW) 119878119879119872plusmn119905119895 is equal tothe maximum service time of power-generation technology 119895in period 119905(hour)

(3) primary energy availability constraints

119860119875119864plusmn119905119895 le 119880119875119864plusmn119905119895 forall119905 119895 (10d)

119860119875119866plusmn119905119895 times 119903119891plusmn119905119895 le 119860119875119864plusmn119905119895 forall119905 119895 (10e)

where 119880119875119864plusmn119905119895 is equal to available primary energy 119895 inperiod 119905 (119895 = 1 2TJ) 119903119891plusmn119905119895 is equal to energy consumptionconversion rate by power-generation technology 119895 in period119905 (119895 = 1 2TJGWh)

(4) capacity expansion constraints

119877119862119860plusmn119905119895 + 119872sum119898=1

119864119862119860plusmn119905119895119898 times 119884119905119895119898 le 119880119862119860plusmn119905119895 forall119905 119895 (10f)

where 119880119862119860plusmn119905119895 is equal to maximum capacity of generationtechnology 119895 in period 119905(GW)

(5) expansion options constraints

119872sum119898=1

119884119905119895119898 le 1 forall119905 119895 (10g)

119884119905119895119898 = 1 if capacity expansion is undertaken119884119905119895119898 = 0 otherwise(6) import electricity constraints

119860119868119864plusmn119905 le 119880119868119864plusmn119905 forall119905 (10h)

where119880119868119864plusmn119905 is equal to maximum import amount of electric-ity imports in period 119905 (GWh)

(7) renewable energy availability constraints

119869sum119895=119868+1

119860119875119866plusmn119905119895 ge ( 119869sum119895=1

119860119875119866plusmn119905119895)120590plusmn119905 forall119905 (10i)

where 120590plusmn119905 is equal to the minimum proportion of electric-ity generation by renewable energy in the whole power-generation Currently the Renewable Portfolio Standard(RPS) mechanisms have been adopted in several countriesincluding theUnitedKingdom (RenewablesObligation in theUK) Italy Poland Sweden and Belgium and 29 out of 50US states etc According to the governmentrsquos requirement acertain percentage of the electricity generation of the powerenterprises will come from renewable energy sources Theproportion of electricity from renewable sources will usuallyincrease year by year Thus in the model 120590plusmn119905 is used torepresent the lowest ratio of the RPS

(8) pollutants emission constraints

119868sum119895=1

119860119875119866plusmn119905119895 times 120578plusmn119905119895 times (1 minus 120585plusmn119905 ) le 119864119872plusmn119905 (119901119894) forall119905 (10j)

6 Mathematical Problems in Engineering

Table 1 Cost of energy supply and conversion parameters

Periodt=1 t=2 t=3

Energy supply cost (103 $TJ)Coal [256 306] [326 376] [396 446]Natural gas [676 696] [757 777] [838 858]Units of energy carrier per units of electricity generation (TJGWh)Coal [1112 117] [1068 1116] [1015 1062]Natural gas [838 882] [754 792] [679 738]

Table 2 Fuzzy subsets for efficiency coefficient under different 120572 minus 119888119906119905 levelsperiod efficiency coefficient120572 minus 119888119906119905 level 120588+ 120588+ 120588minus 120588minus

0t=1 0125 0235 012 023t=2 014 0175 0136 017t=3 0119 0149 0116 0145

02t=1 0135 0223 013 0218t=2 0142 017 0138 0165t=3 012 0144 0117 014

05t=1 015 0205 0145 02t=2 0144 0162 014 0157t=3 0123 0138 0119 0134

08t=1 0165 0187 016 0182t=2 0147 0154 0143 015t=3 0125 0131 0122 0127

1t=1 0175 0175 017 017t=2 0149 0149 0145 0145t=3 0126 0126 0123 0123

Note efficiency coefficient (120588plusmn119905 = 1 minus 120585plusmn119905 )

where 119864119872119905 is equal to the permitted CO2 emission in period119905(103 ton) 120585plusmn119905 is equal to the efficiency of chemical absorptionor capture and storage of CO2 in period 119905119901119894 = the riskconfidence levels

(9) nonnegativity constraints

119860119875119864plusmn119905119895 119860119875119866plusmn119905119895 119860119868119864plusmn119905 ge 0 forall119905 119895 (10k)

Interval parameters are adopted to address impreciseuncertainties which are generally associated with electricitydemands prices of energy resources costs of capacity expan-sion and many other constraints The real research data areused as input data of the model The detailed descriptionsof cost for energy supply and relative conversion parameterswere illustrated in Table 1 Table 2 gives fuzzy subsets forefficiency coefficient under different120572minus119888119906119905 levels Table 3 liststhe capacity expansion options and capital investment costsfor each facility

4 Results and Discussion

In Figure 4 coal-fired electricity supply would increasesteadily and still play an important role in the power sys-tem due to its high availability and competitive price overthe study planning horizon However increasingly strin-gent emission limits result in idle coal-fired facilities Moreeconomical and environmentally friendly power generationfacilities will be prioritized The risk confidence level (119901119894) isemployed to the constraints of CO2 emission requirementThe decision-maker can adjust the value of the 119901119894 levelaccording to actual needs We assumed that violations ofemission constraints are allowed under three given 119901119894 levels(119901119894 = 001005 and 01 which are normally adopted asthe significance levels) [39] The higher credibility levelwould correspond to a tight environment requirement thusleading to a lower CO2 emission while the lower credibilitylevel would correspond to a relatively relaxed environmentrequirement thus resulting in a higher CO2 emission There-fore changes in 119901119894-level have an impact on coal-fired power

Mathematical Problems in Engineering 7

Table 3 Capacity expansion options and costs for power-generation facilities

Periodt=1 t=2 t=3

Capacity-expansion options (GW)Coal m=1 005 005 005

m=2 010 010 010m=3 015 015 015

Natural gas m=1 010 010 010m=2 015 015 015m=3 020 020 020

Wind power m=1 005 005 005m=2 015 015 015m=3 020 020 020

Solar energy m=1 005 005 005m=2 015 015 015m=3 020 020 020

Hydropower m=1 015 015 015m=2 020 020 020m=3 025 025 025

Capacity expansion cost (106$GW)Coal [577 607] [547 577] [517 547]Natural gas [726 756] [676 726] [626 676]Wind power [1256 1306] [1156 1206] [1056 1106]Solar energy [2668 2768] [2468 2568] [2268 2368]Hydropower [1597 1697] [1497 1597] [1397 1497]

generation The total supply of coal-fired generation woulddecrease from [48069 50720] times 109GWh when 119901119894 = 01to [47175 50685] times 109GWh when 119901119894 = 005 and reach[45938 50313] times 109GWh when 119901119894 = 001 In contrast cleanenergy generation will increase For example solar powergeneration of three periods would rise from [2438 2768] times109GWh when 119901119894 = 01 to [2562 2768] times 109GWh when 119901119894= 005 and reach [2625 2769] times 109GWh when 119901119894 = 001 Asshown in Figure 4 the generation of solar power and naturalgas in clean power facilities are mainly affected by emissionconstraints When 119901119894 = 001 the solar power generation isthe largest and the power generation from natural gas isthe lowest in the three-study 119901119894-levels This is the result ofthe tradeoff between carbon dioxide emissions and systemcosts Developing natural gas electricity requires a relativelylow capital cost but leads to more CO2 emissions and thesolar energy technology needs a low operational cost but anextremely high capital cost Similarly the results under other119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted Duringthe entire planning horizon the ratio objective between cleanenergy generation and the total system cost would be [2153]GWh per $106 which also represents the range of the systemefficiency When the electricity-generation pattern variesunder different 119901119894 levels and within the interval solutionranges the system efficiency would also fluctuate within itssolution range correspondingly Thus the IFCF-PSP resultscan create multiple decision alternatives through adjusting

different combinations of the solutions Table 4 presentsthe typical alternatives where combinations of upperlowerbound values for 119860119875119864plusmn119905119895 119860119875119866plusmn119905119895 119884plusmn119905119895119898 (119895 = 1 forall119905 119898) and119860119875119866plusmn119905119895 119884plusmn119905119895119898 (119895 = 2 3 4 5 forall119905 119898) are examined Underthe same power generation alternative interval solutionof the system cost can be obtained according to intervalprice parameters and corresponds to the interval solution ofsystem efficiency Based on the result of the total electricitygenerated we divided the 12 alternatives into three groups(high A3 A4 A8 and A12 medium A1 A5 A7 and A9 lowA2A6A10 andA11) tomeet the future highmedium or lowelectricity demand

The comparison of the system efficiency under different119901119894 levels shows that A11gtA6 gtA10 gtA2 under low-level totalpower generation group Alternative A11 (corresponding tof+) would lead to the most sustainable option with a systemefficiency as much as [27142999] GWh per $106 whichcorresponds to the highest amount of clean energy electricity(0478 times 106GWh) and a moderate cost ([15936 17612] times109$) Additionally lower-level electricity demands will besatisfied by a total power generation of 1856 106GWhTherefore Alternative A11 is a desirable choice from thepoint of view of resources conservation and environmentalprotection when electricity demand is at the low levelAlternative A6 would also meet low electricity demand andreached the lowest system cost ([15142 16785] times 109$)When Alternative A10 is adopted all the power generation

8 Mathematical Problems in Engineering

020406080

100120140160180200

Submodel f-Submodel f+

Submodel f-Submodel f+

Submodel f-Submodel f+

020406080

100120140160180200

020406080

100120140160180200

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

ＪＣ=01

ＪＣ=005

ＪＣ=001

Pow

er g

ener

atio

n (103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)

Figure 4 Power generation loads for EPS through the IFCF-PSPmodel

activities will reach their lower bound levels leading to thelowest power generation (1764 times 106GWh) Compared withAlternative A6 this alternative may be of less interest todecision-makers due to its lower system efficiency and highersystem cost However it is the solution that is obtainedunder the most stringent emission constraints (ie 119901119894 =001) Alternative A2 is a moderate solution in this group Itssystem cost system efficiency total power generation andclean energy power generation are [15295 16954] times 109$[22062445] GWh per $106 1816 times 106GWh and 0374 times106GWh

In middle-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A7gt A5gt A9 gtA1 In alternative A7middle-level electricity demands will be satisfied by a totalpower generation of 1893 times 106GWh In addition Alternative

A7 has the highest system efficiency ([2661 2941] GWhper 106$) and the largest amount of clean energy powergeneration (0478times 106GWh) in this groupThus AlternativeA7 is considered to be a desirable choice Alternatives A5 andA9 would lead to the same total power generation solution(1895 times 106GWh) They are the two-moderate solutions inthis groupThe system cost obtained from the A1 is the lowest([15718 17413] times 109$) in this group It would lead to thelowest system efficiency ([2148 2379] GWh per 106$) butdue to the largest proportion of fossil-fired power generationA1 has the maximum security for energy supply

In high-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A3gt A12gt A8 gtA4 The same cleanenergy power generation obtained from the 4 alternativesis 0478 106GWh In Alternative A4 all the power gener-ation activities will be equal to their upper-bound valuesat the same time It would lead to the highest system cost([16901 18676] times 109$) but the moderate system efficiency([2559 2828]GWhper $106) In comparisonAlternative A3would lead to the minimum system cost ([16478 18216] times109$) and the highest system efficiency ([2624 2901] GWhper $106) in this group In addition this alternative wouldalso provide the highest clean energy power generationTherefore this alternative is a desirable sustainable optionunder the high electricity demand level Alternatives A8 andA12 can provide modest system efficiencies which would be[2561 283] and [2575 2845] GWh per $106 respectivelyand lower system costs which would be [16892 18666] and[16800 18564] times 109$ Therefore decision-makers who payattention to the stability of the system may be interested inthese two alternatives

The above alternatives represent various options betweeneconomic and environmental tradeoffsWillingness to accepthigh system cost will guarantee meeting the objective ofincreasing the proportion of clean energy A strong desireto acquire low system cost will cause the risk of violatingemission constraints In general the above research resultswere favored by decision makers due to their flexibilityand preference for practical-making decision processes Thefeasible ranges for decision variables under different 119901119894 levelswere useful for decision makers to justify the generatedalternatives directly

Besides the scenario of maximizing the proportion ofclean energy another scenario of minimizing the system costis also analyzed to evaluate the effects of different energy sup-ply policies The optimal-ratio problem presented in Models(10a)ndash(10k) can be converted into a least-cost problem withthe following objective

min119891 = system cost

= 119879sum119905=1

119868sum119895=1

119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 + 119879sum119905=1

119869sum119895=1

119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895+ 119879sum119905=1

119869sum119895=1

119872sum119898=1

119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898

Mathematical Problems in Engineering 9

Table 4 Typical decision alternatives obtained from the IFCF-PSP model solutions

Alternative 119860119875119864plusmn119905119895 119860119875119866

plusmn119905119895119884plusmn119905119895119898

(119895=1 forall119905119898)

119860119875119866plusmn119905119895 119884plusmn119905119895119898

(119895=2345 forall119905119898)

System efficiency(GWh per 106$) System cost (109$) Total power generation

(106GWh)Clean energy powergeneration (106GWh)119901119894 = 01

A1 + - [21482379] (12) [1571817413] (4) 1895 (6) 0374 (11)A2 - - [22062445] (9) [1529516954] (3) 1816 (10) 0374 (11)A3 - + [26242901] (3) [1647818216] (9) 192 (4) 0478 (1)A4 + + [25592828] (6) [16901 18676] (12) 2 (1) 0478 (1)119901119894 = 005A5 + - [22362477] (8) [1578317487] (5) 1912 (5) 0391 (7)A6 - - [23292582] (7) [1514216785] (1) 1807 (11) 0391 (7)A7 - + [26612941] (2) [1625117963] (8) 1893 (8) 0478 (1)A8 + + [2561283] (5) [1689218666] (11) 1999 (2) 0478 (1)119901119894 = 001A9 + - [2174 2406] (11) [16040 17757] (7) 1895 (6) 0386 (9)A10 - - [22972543] (8) [15176 16806] (2) 1764 (12) 0386 (9)A11 - + [27142999] (1) [1593617612] (6) 1856 (9) 0478 (1)A12 + + [25752845] (4) [16800 18564] (10) 1987 (3) 0478 (1)

Table 5 The proportion of clean energy power generation from IFCF-PSP and LS models

Clean power generation ratio () IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3119901119894 = 01 [1802 2432] [2089 2493] [1986 2540] [1623 1870] [1681 1719] [1515 1638]119901119894 = 005 [2067 2465] [2093 2528] [2010 2578] [1624 1870] [1681 1719] [1632 1869]119901119894 = 001 [2155 2530] [2158 2596] [2141 2596] [1648 1893] [1679 1752] [1997 2224]

+ 119879sum119905=1

119862119868119864plusmn119905 times 119860119868119864plusmn119905+ 119879sum119905=1

119868sum119895=1

119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905(11)

Figure 5 shows the proportion of different power gen-eration technologies from the IFCF-PSP and LS modelsunder 119901119894 = 001 over three planning periods The IFCF-PSPmodel leads to a relatively higher percentage of natural gaspower generation in comparison the LS model leads torelatively higher percentages of coal-fired electricity suppliesAccording to the solutions from IFCF-PSPmodel the percentof electricity generated by natural gas facilities would be [5 6] [7 8] and [5 8] in periods 1 2 and 3 respectivelyTheLS model achieves the slightly lower percent (ie 0 0 and[5 6] in periods 1 2 and 3) Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted

According to Table 5 the differences can be foundbetween the results of two models under different 119901119894 levelsThe proportion of clean energy power generation fromIFCF-PSP model is higher than that of the LS model Forexample under 119901119894 = 001 clean energy power generationof the entire region occupied [2155 2530] [2158 2596]and [2141 2596] of the total electricity generation in

the three study periods from IFCF-PSP model which arehigher than the LS model [1648 1993] [1679 1752] and[1997 2224] respectively Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpretedAn increased 119901119894 level represents a higher admissible riskleading to a decreased strictness for the emission constraintsand hence an expanded decision space Therefore undertypical conditions as the risk level becomes lower theproportion of clean energy power generation or renewablepower generation would increase For example in period 2when the 119901119894 level is dropped from 01 to 001 the proportionof clean energy power generation from IFCF-PSP modelwould be increased from [2089 2493] to [2158 2596]Likewise the results under other periods can be similarlyanalyzed The confidence level of constraints satisfaction ismore reliable because the risk level is lowerTherefore in thiscase the decision-makers will be more conservative in theEPS management There is no capacity-expansion would beconducted for the coal-fired facility in three periods sinceits high carbon dioxide emissions and increasingly stringentemission limits On the other hand clean energy electricitysupply would be insufficient for the future energy demandsAccording to Table 6 when 119901119894 takes different values (ie 119901119894 =001119901119894 = 005 and 119901119894 = 01) differences of capacity-expansionoption can be found between the results of two models Inboth models when the risk of violating the constraints ofcarbon emission target is decreased (ie the value of 119901119894-level

10 Mathematical Problems in Engineering

75

79

5 4

74

8

95 4

74

8

9

5 4

81

095 5

83

09

4 4

78

7

7

4 4

coalnatural gashydro

windsolar

83

010

2

5

83

0

94 4

80

5

84 3

82

19 4

4

78

6

84 4

79

5

84 4

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

IFCFP-PSP f- IFCFP-PSP +

LS +LS -

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

Figure 5 Comparison of power-generation patterns from IFCF-PSP and least-system-cost (LS) models under 119901119894-level 001 Note that theinnermost circle in the pie chart represents the first planning period

is larger) the demand for expansion of clean energy willincrease On the other hand the total demand of capacityexpansion in the lower bound model is lower than the upperbound model One of the reasons for this result is that theannual utilization hours of various power generation facilitiesin the lower bound model are higher than the correspondingparameter settings of the upper bound model

The result of LS model has shown that under thecorresponding lower bound parameter settings the capacityof 015 GW would be added to the hydropower facility inperiod 3 when 119901119894 = 001 and 119901119894 = 005 Only a capacity of015 GWwould be added to the wind power facility in period3 While under the corresponding upper bound parametersettings natural gas and solar energy facilities expansionsare not required that is because the former resource is underpenalty of CO2 emission and the capacity cost of the solarpower expansion is much higher

In the IFCF-PSP model clean energy will be given moredevelopment opportunities Take 119901119894 = 001 as an example(corresponding to 119891minus) with a capacity expansion of 02

GW at the beginning of period 1 and period 2 electricitygeneration ofwind power facilitywould rise from 164 103GWin period 1 to 168 103GW in period 2 and with anothercapacity expansion of 02 GW at the beginning of period 3 itstotal electricity generation capacity would reach 172 103GWLikewise to meet the growing energy demand the capacityof 02 GW would be added to the natural gas-fired facility inperiod 1 period 2 and period 3 as a result its total generationcapacity would rise from 954 times 103GW in period 1 to 119 times103GW in period 2 and reach 124 times 103GW in period 3 Inthe planning period 1 due to more lenient emission policiesand more economical cost strategies some of the natural gaspower generation facilities are idleThe capacities of 025 GWwould be added to the hydropower facility in three periodsand the corresponding power generation capacity would be812 times 103GW 875 103GW and 937 times 103GW Generatingelectricity from solar energy facility is more expensive thanother power generation facilities thus when the emissionlimits and electricity demand are relatively low solar energyfacility would not be expanded during first two periods 015

Mathematical Problems in Engineering 11

Table 6 Binary solutions of capacity expansions

119901119894-level Power-generation facility Capacity-expansion option IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3

01 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 [0 1] [0 1] 0 0 0

Wind power m=1 0 0 0 0 0 0m=2 0 [0 1] 0 0 [0 1] 1m=3 [0 1] [0 1] [0 1] 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 [0 1] [0 1] 0 0 0 [0 1]m=2 0 0 0 0 0 0m=3 [0 1] [0 1] 0 [0 1] [0 1] 0

005 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 [0 1] [0 1] [0 1] 0 0 0

Wind power m=1 [0 1] [0 1] 0 0 0 0m=2 0 0 0 0 [0 1] 1m=3 [0 1] [0 1] 0 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] [0 1]m=2 0 0 0 0 0 0m=3 1 1 0 [0 1] [0 1] 0

001 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 1 1 [0 1] [0 1] [0 1]

Wind power m=1 0 0 0 0 0 0m=2 0 0 0 0 0 [0 1]m=3 1 1 [0 1] [0 1] [0 1] [0 1]

Solar energy m=1 0 0 0 0 0 0m=2 0 0 [0 1] 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] 0m=2 0 0 0 0 0 0m=3 1 1 [0 1] [0 1] [0 1] [0 1]

GW of capacity would only be added to the solar energyfacility in period 3

Apparently with the successful application of IFCI-PSP within a typical regional electric system managementproblem solutions obtained could provide useful decisionalternatives under different policies and various energy avail-abilities Compared with the least-cost model the IFCF-PSP model is an effective tool for providing environmentalmanagement schemes with dual objectives In addition theIFCF-PSP has following advantages over the conventionalprogramming methods (a) balancing multiple conflicting

objectives (b) reflecting interrelationships among systemefficiency economic cost and system reliability (c) effec-tively dealing with randomness in both the objective andconstraints and (d) assisting the analysis of diverse decisionschemes associated with various energy demand levels

5 Conclusions

An inexact fuzzy chance-constrained fractional program-ming approach is developed for optimal electric powersystems management under uncertainties In the developed

12 Mathematical Problems in Engineering

model fuzzy chance-constrained programming is incorpo-rated into a fractional programming optimization frame-work The obtained results are useful for supporting EPSmanagement The IFCF-PSP approach is capable of (i)balancing the conflict between two objectives (ii) reflect-ing different electricity generation and capacity expansionstrategies (iii) presenting optimal solutions under differentconstraint violating conditions and (iv) introducing theconcept of fuzzy boundary interval with which the com-plexity of dual uncertainties can be effectively handled Theresults of IFCF-PSP model show that (i) a higher confidencelevel corresponds to a higher proportion of clean energypower generation and a lower economic productivity and(ii) as a result of encouraging environment-friendly energiesthe generation capacities for the natural gas wind-powersolar energy and hydropower facilities would be signifi-cantly increased The solutions obtained from the IFCF-PSPapproach could provide specific energy options for powersystem planning and provide effective management solutionof the electric power system for identifying electricity gener-ation and capacity expansion schemes

This study attempts to develop a modeling framework forratio problems involving fuzzy uncertainties to deal with elec-tric power systems management problemThe results suggestthat it is also applicable to other energy management prob-lems In the future practice IFCF-PSP could be further im-proved through considering more impact factors Forinstance the fuzzymembership functions and the confidencelevel are critical in the decision-making process and thisrequires effective ways to provide appropriate choices fordecision making Such challenges desire further investiga-tions Future research can be aimed at applying the advancedapproach to a more complex real-world electric power sys-tem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the Fundamental ResearchFunds for the Central Universities NCEPU (2018BJ0293)the State Scholarship Fund (201706735025) the NationalKey Research and Development Plan (2016YFC0502800) theNatural Sciences Foundation (51520105013 51679087) andthe 111 Program (B14008) The data used to support thefindings of this study are available from the correspondingauthor upon request

References

[1] G H Huang ldquoA hybrid inexact-stochastic water managementmodelrdquo European Journal of Operational Research vol 107 no1 pp 137ndash158 1998

[2] A Grosfeld-Nir and A Tishler ldquoA stochastic model for themeasurement of electricity outage costsrdquo Energy pp 157ndash1741993

[3] S Wang G Huang and B W Baetz ldquoAn Inexact Probabilistic-Possibilistic Optimization Framework for Flood Managementin a Hybrid Uncertain Environmentrdquo IEEE Transactions onFuzzy Systems vol 23 no 4 pp 897ndash908 2015

[4] G H Huang and D P Loucks ldquoAn inexact two-stage stochasticprogramming model for water resources management underuncertaintyrdquo Civil Engineering and Environmental Systems vol17 no 2 pp 95ndash118 2000

[5] K Darby-Dowman S Barker E Audsley and D Parsons ldquoAtwo-stage stochastic programming with recourse model fordetermining robust planting plans in horticulturerdquo Journal ofthe Operational Research Society vol 51 no 1 pp 83ndash89 2000

[6] J Hu L Sun C H Li X Wang X L Jia and Y P Cai ldquoWaterQuality Risk Assessment for the Laoguanhe River of ChinaUsing a Stochastic SimulationMethodrdquo Journal of Environmen-tal Informatics vol 31 no 2 pp 123ndash136 2018

[7] SWang andG HHuang ldquoAmulti-level Taguchi-factorial two-stage stochastic programming approach for characterization ofparameter uncertainties and their interactions An applicationto water resources managementrdquo European Journal of Opera-tional Research vol 240 no 2 pp 572ndash581 2015

[8] S Chanas and P Zielinski ldquoOn the equivalence of two opti-mization methods for fuzzy linear programming problemsrdquoEuropean Journal of Operational Research vol 121 no 1 pp 56ndash63 2000

[9] J Mula D Peidro and R Poler ldquoThe effectiveness of a fuzzymathematical programming approach for supply chain pro-duction planning with fuzzy demandrdquo International Journal ofProduction Economics vol 128 no 1 pp 136ndash143 2010

[10] B Chen P Li H J Wu T Husain and F Khan ldquoMCFP Amonte carlo simulation-based fuzzy programming approachfor optimization under dual uncertainties of possibility andcontinuous probabilityrdquo Journal of Environmental Informatics(JEI) vol 29 no 2 pp 88ndash97 2017

[11] C Z Huang S Nie L Guo and Y R Fan ldquoInexact fuzzystochastic chance constraint programming for emergency evac-uation in Qinshan nuclear power plant under uncertaintyrdquoJournal of Environmental Informatics (JEI) vol 30 no 1 pp 63ndash78 2017

[12] U S Sakalli ldquoOptimization of Production-Distribution Prob-lem in Supply Chain Management under Stochastic and FuzzyUncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 4389064 29 pages 2017

[13] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval research logistics (NRL) pp 181ndash186 1962

[14] N I Voropai and E Y Ivanova ldquoMulti-criteria decision analysistechniques in electric power system expansion planningrdquo Inter-national Journal of Electrical Power amp Energy Systems vol 24no 1 pp 71ndash78 2002

[15] M Chakraborty and S Gupta ldquoFuzzy mathematical pro-gramming for multi objective linear fractional programmingproblemrdquo Fuzzy Sets and Systems vol 125 no 3 pp 335ndash3422002

[16] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[17] W Li Z BaoGHHuang andY L Xie ldquoAn Inexact CredibilityChance-Constrained Integer Programming for Greenhouse

Mathematical Problems in Engineering 13

GasMitigationManagement in Regional Electric Power Systemunder Uncertaintyrdquo in Journal of Environmental Informatics2017

[18] M Paunovic N M Ralevic V Gajovic et al ldquoTwo-StageFuzzy Logic Model for Cloud Service Supplier Selection andEvaluationrdquoMathematical Problems in Engineering 2018

[19] S Wang G H Huang B W Baetz and B C Ancell ldquoTowardsrobust quantification and reduction of uncertainty in hydro-logic predictions Integration of particle Markov chain MonteCarlo and factorial polynomial chaos expansionrdquo Journal ofHydrology vol 548 pp 484ndash497 2017

[20] SWang B C Ancell GHHuang andBW Baetz ldquoImprovingRobustness ofHydrologic Ensemble PredictionsThroughProb-abilistic Pre- and Post-Processing in Sequential Data Assimila-tionrdquoWater Resources Research 2018

[21] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[22] A Rong and R Lahdelma ldquoFuzzy chance constrained linearprogramming model for optimizing the scrap charge in steelproductionrdquo European Journal of Operational Research vol 186no 3 pp 953ndash964 2008

[23] X Sun J Li H Qiao and B Zhang ldquoEnergy implicationsof Chinarsquos regional development New insights from multi-regional input-output analysisrdquo Applied Energy vol 196 pp118ndash131 2017

[24] M A Quaddus and T N Goh ldquoElectric power generationexpansion Planning with multiple objectivesrdquo Applied Energyvol 19 no 4 pp 301ndash319 1985

[25] C H Antunes A G Martins and I S Brito ldquoA multipleobjective mixed integer linear programming model for powergeneration expansion planningrdquo Energy vol 29 no 4 pp 613ndash627 2004

[26] J L C Meza M B Yildirim and A S M Masud ldquoA modelfor the multiperiod multiobjective power generation expansionproblemrdquo IEEETransactions on Power Systems vol 22 no 2 pp871ndash878 2007

[27] J-HHan Y-C Ahn and I-B Lee ldquoAmulti-objective optimiza-tion model for sustainable electricity generation and CO2 mit-igation (EGCM) infrastructure design considering economicprofit and financial riskrdquo Applied Energy vol 95 pp 186ndash1952012

[28] M Rekik A Abdelkafi and L Krichen ldquoAmicro-grid ensuringmulti-objective control strategy of a power electrical system forquality improvementrdquo Energy vol 88 pp 351ndash363 2015

[29] A Azadeh Z Raoofi and M Zarrin ldquoA multi-objective fuzzylinear programming model for optimization of natural gassupply chain through a greenhouse gas reduction approachrdquoJournal of Natural Gas Science and Engineering vol 26 pp 702ndash710 2015

[30] K Li L Pan W Xue H Jiang and H Mao ldquoMulti-objectiveoptimization for energy performance improvement of residen-tial buildings a comparative studyrdquo Energies vol 10 no 2 p245 2017

[31] H Zhu W W Huang and G H Huang ldquoPlanning of regionalenergy systems An inexact mixed-integer fractional program-ming modelrdquo Applied Energy vol 113 pp 500ndash514 2014

[32] F Chen G H Huang Y R Fan and R F Liao ldquoA nonlinearfractional programming approach for environmentaleconomicpower dispatchrdquo International Journal of Electrical Power ampEnergy Systems vol 78 pp 463ndash469 2016

[33] C Zhang and P Guo ldquoFLFP A fuzzy linear fractional pro-gramming approach with double-sided fuzziness for optimalirrigation water allocationrdquo Agricultural Water Managementvol 199 pp 105ndash119 2018

[34] L Wang G Huang X Wang and H Zhu ldquoRisk-basedelectric power system planning for climate change mitigationthrough multi-stage joint-probabilistic left-hand-side chance-constrained fractional programming A Canadian case studyrdquoRenewable amp Sustainable Energy Reviews vol 82 pp 1056ndash10672018

[35] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[36] X H Nie G H Huang Y P Li and L Liu ldquoIFRP A hybridinterval-parameter fuzzy robust programming approach forwaste management planning under uncertaintyrdquo Journal ofEnvironmental Management vol 84 no 1 pp 1ndash11 2007

[37] Y Leung Spatial analysis and planning under imprecisionElsevier 1st edition 2013

[38] A Charnes W W Cooper and M J L Kirby ldquoChance-constrained programming an extension of statistical methodrdquoinOptimizingmethods in statistics pp 391ndash402 Academic Press1971

[39] Y P Cai G H Huang Z F Yang and Q Tan ldquoIdentificationof optimal strategies for energy management systems planningunder multiple uncertaintiesrdquoApplied Energy vol 86 no 4 pp480ndash495 2009

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Mathematical Problems in Engineering 5

represents the shortage amount of electricity needs to beimported in period 119905(GWh)

(5) cost for pollutant mitigation

1198915plusmn = 119879sum119905=1

119868sum119895=1

119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905 (9f)

where 119862119875119872plusmn119905119895 is equal to cost of pollution mitigation ofpower-generation technology 119895 in period 119905(103 $ton) 120578plusmn119905CO2 emission factor in period 119905(103 tonGWh)

Renewable energy resources are intermittent andunreliable which are subject to spatial andor temporal

fluctuations The natural gas generation with low carbondioxide emissions can stabilize the risk of the intermittentand unpredictable nature of renewable energy generationIn this study we assumed natural gas resources as theclean energy power generation Therefore the objective ofthis study is to maximize the ratio between clean energygeneration (including the natural gas) and system costwhile a series of constraints define the interrelationshipsamong the decision variables and system conditionsfactorsThe inexact fuzzy chance-constrained fractional powersystems planning (IFCF-PSP) can be formulated as fol-lows

max119891plusmn = 119862119866plusmn119862plusmn= sum119879119905=1sum119869119895=119868+1119860119875119866plusmn119905119895 + sum119879119905=1 119860119875119866plusmn1199052sum119879119905=1sum119868119895=1 119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 + sum119879119905=1sum119869119895=1 119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895 + sum119879119905=1sum119869119895=1sum119872119898=1 119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898 + sum119879119905=1 119862119868119864plusmn119905 times 119860119868119864plusmn119905 + sum119879119905=1sum119868119895=1 119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905

(10a)

where the clean energy power generation (119862119866plusmn) consists ofthe renewable energies such as wind power solar energy andhydropower (119860119875119866plusmn119905119895 and j = I + 1 J) and the natural gasgeneration in period 119905

The constraints are listed as follows(1) electricity demand constraints119869sum119895=1

119860119875119866plusmn119905119895 + 119860119868119864plusmn119905 ge 119863119872plusmn119905 times (1 + 120579plusmn119905 ) forall119905 (10b)

where119863119872plusmn119905 is equal to local electricity demand (GWh) 120579plusmn119905 isequal to transmission loss in period 119905

(2) capacity limitation constraints for power-generationfacilities

119860119875119866plusmn119905119895 le (119877119862119860plusmn119905119895 + 119872sum119898=1

119864119862119860plusmn119905119895119898 times 119884119905119895119898) times 119878119879119872plusmn119905119895forall119905 119895

(10c)

where 119877119862119860plusmn119905119895 is equal to the current capacity of power-generation technology 119895 in period 119905(GW) 119878119879119872plusmn119905119895 is equal tothe maximum service time of power-generation technology 119895in period 119905(hour)

(3) primary energy availability constraints

119860119875119864plusmn119905119895 le 119880119875119864plusmn119905119895 forall119905 119895 (10d)

119860119875119866plusmn119905119895 times 119903119891plusmn119905119895 le 119860119875119864plusmn119905119895 forall119905 119895 (10e)

where 119880119875119864plusmn119905119895 is equal to available primary energy 119895 inperiod 119905 (119895 = 1 2TJ) 119903119891plusmn119905119895 is equal to energy consumptionconversion rate by power-generation technology 119895 in period119905 (119895 = 1 2TJGWh)

(4) capacity expansion constraints

119877119862119860plusmn119905119895 + 119872sum119898=1

119864119862119860plusmn119905119895119898 times 119884119905119895119898 le 119880119862119860plusmn119905119895 forall119905 119895 (10f)

where 119880119862119860plusmn119905119895 is equal to maximum capacity of generationtechnology 119895 in period 119905(GW)

(5) expansion options constraints

119872sum119898=1

119884119905119895119898 le 1 forall119905 119895 (10g)

119884119905119895119898 = 1 if capacity expansion is undertaken119884119905119895119898 = 0 otherwise(6) import electricity constraints

119860119868119864plusmn119905 le 119880119868119864plusmn119905 forall119905 (10h)

where119880119868119864plusmn119905 is equal to maximum import amount of electric-ity imports in period 119905 (GWh)

(7) renewable energy availability constraints

119869sum119895=119868+1

119860119875119866plusmn119905119895 ge ( 119869sum119895=1

119860119875119866plusmn119905119895)120590plusmn119905 forall119905 (10i)

where 120590plusmn119905 is equal to the minimum proportion of electric-ity generation by renewable energy in the whole power-generation Currently the Renewable Portfolio Standard(RPS) mechanisms have been adopted in several countriesincluding theUnitedKingdom (RenewablesObligation in theUK) Italy Poland Sweden and Belgium and 29 out of 50US states etc According to the governmentrsquos requirement acertain percentage of the electricity generation of the powerenterprises will come from renewable energy sources Theproportion of electricity from renewable sources will usuallyincrease year by year Thus in the model 120590plusmn119905 is used torepresent the lowest ratio of the RPS

(8) pollutants emission constraints

119868sum119895=1

119860119875119866plusmn119905119895 times 120578plusmn119905119895 times (1 minus 120585plusmn119905 ) le 119864119872plusmn119905 (119901119894) forall119905 (10j)

6 Mathematical Problems in Engineering

Table 1 Cost of energy supply and conversion parameters

Periodt=1 t=2 t=3

Energy supply cost (103 $TJ)Coal [256 306] [326 376] [396 446]Natural gas [676 696] [757 777] [838 858]Units of energy carrier per units of electricity generation (TJGWh)Coal [1112 117] [1068 1116] [1015 1062]Natural gas [838 882] [754 792] [679 738]

Table 2 Fuzzy subsets for efficiency coefficient under different 120572 minus 119888119906119905 levelsperiod efficiency coefficient120572 minus 119888119906119905 level 120588+ 120588+ 120588minus 120588minus

0t=1 0125 0235 012 023t=2 014 0175 0136 017t=3 0119 0149 0116 0145

02t=1 0135 0223 013 0218t=2 0142 017 0138 0165t=3 012 0144 0117 014

05t=1 015 0205 0145 02t=2 0144 0162 014 0157t=3 0123 0138 0119 0134

08t=1 0165 0187 016 0182t=2 0147 0154 0143 015t=3 0125 0131 0122 0127

1t=1 0175 0175 017 017t=2 0149 0149 0145 0145t=3 0126 0126 0123 0123

Note efficiency coefficient (120588plusmn119905 = 1 minus 120585plusmn119905 )

where 119864119872119905 is equal to the permitted CO2 emission in period119905(103 ton) 120585plusmn119905 is equal to the efficiency of chemical absorptionor capture and storage of CO2 in period 119905119901119894 = the riskconfidence levels

(9) nonnegativity constraints

119860119875119864plusmn119905119895 119860119875119866plusmn119905119895 119860119868119864plusmn119905 ge 0 forall119905 119895 (10k)

Interval parameters are adopted to address impreciseuncertainties which are generally associated with electricitydemands prices of energy resources costs of capacity expan-sion and many other constraints The real research data areused as input data of the model The detailed descriptionsof cost for energy supply and relative conversion parameterswere illustrated in Table 1 Table 2 gives fuzzy subsets forefficiency coefficient under different120572minus119888119906119905 levels Table 3 liststhe capacity expansion options and capital investment costsfor each facility

4 Results and Discussion

In Figure 4 coal-fired electricity supply would increasesteadily and still play an important role in the power sys-tem due to its high availability and competitive price overthe study planning horizon However increasingly strin-gent emission limits result in idle coal-fired facilities Moreeconomical and environmentally friendly power generationfacilities will be prioritized The risk confidence level (119901119894) isemployed to the constraints of CO2 emission requirementThe decision-maker can adjust the value of the 119901119894 levelaccording to actual needs We assumed that violations ofemission constraints are allowed under three given 119901119894 levels(119901119894 = 001005 and 01 which are normally adopted asthe significance levels) [39] The higher credibility levelwould correspond to a tight environment requirement thusleading to a lower CO2 emission while the lower credibilitylevel would correspond to a relatively relaxed environmentrequirement thus resulting in a higher CO2 emission There-fore changes in 119901119894-level have an impact on coal-fired power

Mathematical Problems in Engineering 7

Table 3 Capacity expansion options and costs for power-generation facilities

Periodt=1 t=2 t=3

Capacity-expansion options (GW)Coal m=1 005 005 005

m=2 010 010 010m=3 015 015 015

Natural gas m=1 010 010 010m=2 015 015 015m=3 020 020 020

Wind power m=1 005 005 005m=2 015 015 015m=3 020 020 020

Solar energy m=1 005 005 005m=2 015 015 015m=3 020 020 020

Hydropower m=1 015 015 015m=2 020 020 020m=3 025 025 025

Capacity expansion cost (106$GW)Coal [577 607] [547 577] [517 547]Natural gas [726 756] [676 726] [626 676]Wind power [1256 1306] [1156 1206] [1056 1106]Solar energy [2668 2768] [2468 2568] [2268 2368]Hydropower [1597 1697] [1497 1597] [1397 1497]

generation The total supply of coal-fired generation woulddecrease from [48069 50720] times 109GWh when 119901119894 = 01to [47175 50685] times 109GWh when 119901119894 = 005 and reach[45938 50313] times 109GWh when 119901119894 = 001 In contrast cleanenergy generation will increase For example solar powergeneration of three periods would rise from [2438 2768] times109GWh when 119901119894 = 01 to [2562 2768] times 109GWh when 119901119894= 005 and reach [2625 2769] times 109GWh when 119901119894 = 001 Asshown in Figure 4 the generation of solar power and naturalgas in clean power facilities are mainly affected by emissionconstraints When 119901119894 = 001 the solar power generation isthe largest and the power generation from natural gas isthe lowest in the three-study 119901119894-levels This is the result ofthe tradeoff between carbon dioxide emissions and systemcosts Developing natural gas electricity requires a relativelylow capital cost but leads to more CO2 emissions and thesolar energy technology needs a low operational cost but anextremely high capital cost Similarly the results under other119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted Duringthe entire planning horizon the ratio objective between cleanenergy generation and the total system cost would be [2153]GWh per $106 which also represents the range of the systemefficiency When the electricity-generation pattern variesunder different 119901119894 levels and within the interval solutionranges the system efficiency would also fluctuate within itssolution range correspondingly Thus the IFCF-PSP resultscan create multiple decision alternatives through adjusting

different combinations of the solutions Table 4 presentsthe typical alternatives where combinations of upperlowerbound values for 119860119875119864plusmn119905119895 119860119875119866plusmn119905119895 119884plusmn119905119895119898 (119895 = 1 forall119905 119898) and119860119875119866plusmn119905119895 119884plusmn119905119895119898 (119895 = 2 3 4 5 forall119905 119898) are examined Underthe same power generation alternative interval solutionof the system cost can be obtained according to intervalprice parameters and corresponds to the interval solution ofsystem efficiency Based on the result of the total electricitygenerated we divided the 12 alternatives into three groups(high A3 A4 A8 and A12 medium A1 A5 A7 and A9 lowA2A6A10 andA11) tomeet the future highmedium or lowelectricity demand

The comparison of the system efficiency under different119901119894 levels shows that A11gtA6 gtA10 gtA2 under low-level totalpower generation group Alternative A11 (corresponding tof+) would lead to the most sustainable option with a systemefficiency as much as [27142999] GWh per $106 whichcorresponds to the highest amount of clean energy electricity(0478 times 106GWh) and a moderate cost ([15936 17612] times109$) Additionally lower-level electricity demands will besatisfied by a total power generation of 1856 106GWhTherefore Alternative A11 is a desirable choice from thepoint of view of resources conservation and environmentalprotection when electricity demand is at the low levelAlternative A6 would also meet low electricity demand andreached the lowest system cost ([15142 16785] times 109$)When Alternative A10 is adopted all the power generation

8 Mathematical Problems in Engineering

020406080

100120140160180200

Submodel f-Submodel f+

Submodel f-Submodel f+

Submodel f-Submodel f+

020406080

100120140160180200

020406080

100120140160180200

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

ＪＣ=01

ＪＣ=005

ＪＣ=001

Pow

er g

ener

atio

n (103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)

Figure 4 Power generation loads for EPS through the IFCF-PSPmodel

activities will reach their lower bound levels leading to thelowest power generation (1764 times 106GWh) Compared withAlternative A6 this alternative may be of less interest todecision-makers due to its lower system efficiency and highersystem cost However it is the solution that is obtainedunder the most stringent emission constraints (ie 119901119894 =001) Alternative A2 is a moderate solution in this group Itssystem cost system efficiency total power generation andclean energy power generation are [15295 16954] times 109$[22062445] GWh per $106 1816 times 106GWh and 0374 times106GWh

In middle-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A7gt A5gt A9 gtA1 In alternative A7middle-level electricity demands will be satisfied by a totalpower generation of 1893 times 106GWh In addition Alternative

A7 has the highest system efficiency ([2661 2941] GWhper 106$) and the largest amount of clean energy powergeneration (0478times 106GWh) in this groupThus AlternativeA7 is considered to be a desirable choice Alternatives A5 andA9 would lead to the same total power generation solution(1895 times 106GWh) They are the two-moderate solutions inthis groupThe system cost obtained from the A1 is the lowest([15718 17413] times 109$) in this group It would lead to thelowest system efficiency ([2148 2379] GWh per 106$) butdue to the largest proportion of fossil-fired power generationA1 has the maximum security for energy supply

In high-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A3gt A12gt A8 gtA4 The same cleanenergy power generation obtained from the 4 alternativesis 0478 106GWh In Alternative A4 all the power gener-ation activities will be equal to their upper-bound valuesat the same time It would lead to the highest system cost([16901 18676] times 109$) but the moderate system efficiency([2559 2828]GWhper $106) In comparisonAlternative A3would lead to the minimum system cost ([16478 18216] times109$) and the highest system efficiency ([2624 2901] GWhper $106) in this group In addition this alternative wouldalso provide the highest clean energy power generationTherefore this alternative is a desirable sustainable optionunder the high electricity demand level Alternatives A8 andA12 can provide modest system efficiencies which would be[2561 283] and [2575 2845] GWh per $106 respectivelyand lower system costs which would be [16892 18666] and[16800 18564] times 109$ Therefore decision-makers who payattention to the stability of the system may be interested inthese two alternatives

The above alternatives represent various options betweeneconomic and environmental tradeoffsWillingness to accepthigh system cost will guarantee meeting the objective ofincreasing the proportion of clean energy A strong desireto acquire low system cost will cause the risk of violatingemission constraints In general the above research resultswere favored by decision makers due to their flexibilityand preference for practical-making decision processes Thefeasible ranges for decision variables under different 119901119894 levelswere useful for decision makers to justify the generatedalternatives directly

Besides the scenario of maximizing the proportion ofclean energy another scenario of minimizing the system costis also analyzed to evaluate the effects of different energy sup-ply policies The optimal-ratio problem presented in Models(10a)ndash(10k) can be converted into a least-cost problem withthe following objective

min119891 = system cost

= 119879sum119905=1

119868sum119895=1

119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 + 119879sum119905=1

119869sum119895=1

119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895+ 119879sum119905=1

119869sum119895=1

119872sum119898=1

119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898

Mathematical Problems in Engineering 9

Table 4 Typical decision alternatives obtained from the IFCF-PSP model solutions

Alternative 119860119875119864plusmn119905119895 119860119875119866

plusmn119905119895119884plusmn119905119895119898

(119895=1 forall119905119898)

119860119875119866plusmn119905119895 119884plusmn119905119895119898

(119895=2345 forall119905119898)

System efficiency(GWh per 106$) System cost (109$) Total power generation

(106GWh)Clean energy powergeneration (106GWh)119901119894 = 01

A1 + - [21482379] (12) [1571817413] (4) 1895 (6) 0374 (11)A2 - - [22062445] (9) [1529516954] (3) 1816 (10) 0374 (11)A3 - + [26242901] (3) [1647818216] (9) 192 (4) 0478 (1)A4 + + [25592828] (6) [16901 18676] (12) 2 (1) 0478 (1)119901119894 = 005A5 + - [22362477] (8) [1578317487] (5) 1912 (5) 0391 (7)A6 - - [23292582] (7) [1514216785] (1) 1807 (11) 0391 (7)A7 - + [26612941] (2) [1625117963] (8) 1893 (8) 0478 (1)A8 + + [2561283] (5) [1689218666] (11) 1999 (2) 0478 (1)119901119894 = 001A9 + - [2174 2406] (11) [16040 17757] (7) 1895 (6) 0386 (9)A10 - - [22972543] (8) [15176 16806] (2) 1764 (12) 0386 (9)A11 - + [27142999] (1) [1593617612] (6) 1856 (9) 0478 (1)A12 + + [25752845] (4) [16800 18564] (10) 1987 (3) 0478 (1)

Table 5 The proportion of clean energy power generation from IFCF-PSP and LS models

Clean power generation ratio () IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3119901119894 = 01 [1802 2432] [2089 2493] [1986 2540] [1623 1870] [1681 1719] [1515 1638]119901119894 = 005 [2067 2465] [2093 2528] [2010 2578] [1624 1870] [1681 1719] [1632 1869]119901119894 = 001 [2155 2530] [2158 2596] [2141 2596] [1648 1893] [1679 1752] [1997 2224]

+ 119879sum119905=1

119862119868119864plusmn119905 times 119860119868119864plusmn119905+ 119879sum119905=1

119868sum119895=1

119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905(11)

Figure 5 shows the proportion of different power gen-eration technologies from the IFCF-PSP and LS modelsunder 119901119894 = 001 over three planning periods The IFCF-PSPmodel leads to a relatively higher percentage of natural gaspower generation in comparison the LS model leads torelatively higher percentages of coal-fired electricity suppliesAccording to the solutions from IFCF-PSPmodel the percentof electricity generated by natural gas facilities would be [5 6] [7 8] and [5 8] in periods 1 2 and 3 respectivelyTheLS model achieves the slightly lower percent (ie 0 0 and[5 6] in periods 1 2 and 3) Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted

According to Table 5 the differences can be foundbetween the results of two models under different 119901119894 levelsThe proportion of clean energy power generation fromIFCF-PSP model is higher than that of the LS model Forexample under 119901119894 = 001 clean energy power generationof the entire region occupied [2155 2530] [2158 2596]and [2141 2596] of the total electricity generation in

the three study periods from IFCF-PSP model which arehigher than the LS model [1648 1993] [1679 1752] and[1997 2224] respectively Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpretedAn increased 119901119894 level represents a higher admissible riskleading to a decreased strictness for the emission constraintsand hence an expanded decision space Therefore undertypical conditions as the risk level becomes lower theproportion of clean energy power generation or renewablepower generation would increase For example in period 2when the 119901119894 level is dropped from 01 to 001 the proportionof clean energy power generation from IFCF-PSP modelwould be increased from [2089 2493] to [2158 2596]Likewise the results under other periods can be similarlyanalyzed The confidence level of constraints satisfaction ismore reliable because the risk level is lowerTherefore in thiscase the decision-makers will be more conservative in theEPS management There is no capacity-expansion would beconducted for the coal-fired facility in three periods sinceits high carbon dioxide emissions and increasingly stringentemission limits On the other hand clean energy electricitysupply would be insufficient for the future energy demandsAccording to Table 6 when 119901119894 takes different values (ie 119901119894 =001119901119894 = 005 and 119901119894 = 01) differences of capacity-expansionoption can be found between the results of two models Inboth models when the risk of violating the constraints ofcarbon emission target is decreased (ie the value of 119901119894-level

10 Mathematical Problems in Engineering

75

79

5 4

74

8

95 4

74

8

9

5 4

81

095 5

83

09

4 4

78

7

7

4 4

coalnatural gashydro

windsolar

83

010

2

5

83

0

94 4

80

5

84 3

82

19 4

4

78

6

84 4

79

5

84 4

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

IFCFP-PSP f- IFCFP-PSP +

LS +LS -

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

Figure 5 Comparison of power-generation patterns from IFCF-PSP and least-system-cost (LS) models under 119901119894-level 001 Note that theinnermost circle in the pie chart represents the first planning period

is larger) the demand for expansion of clean energy willincrease On the other hand the total demand of capacityexpansion in the lower bound model is lower than the upperbound model One of the reasons for this result is that theannual utilization hours of various power generation facilitiesin the lower bound model are higher than the correspondingparameter settings of the upper bound model

The result of LS model has shown that under thecorresponding lower bound parameter settings the capacityof 015 GW would be added to the hydropower facility inperiod 3 when 119901119894 = 001 and 119901119894 = 005 Only a capacity of015 GWwould be added to the wind power facility in period3 While under the corresponding upper bound parametersettings natural gas and solar energy facilities expansionsare not required that is because the former resource is underpenalty of CO2 emission and the capacity cost of the solarpower expansion is much higher

In the IFCF-PSP model clean energy will be given moredevelopment opportunities Take 119901119894 = 001 as an example(corresponding to 119891minus) with a capacity expansion of 02

GW at the beginning of period 1 and period 2 electricitygeneration ofwind power facilitywould rise from 164 103GWin period 1 to 168 103GW in period 2 and with anothercapacity expansion of 02 GW at the beginning of period 3 itstotal electricity generation capacity would reach 172 103GWLikewise to meet the growing energy demand the capacityof 02 GW would be added to the natural gas-fired facility inperiod 1 period 2 and period 3 as a result its total generationcapacity would rise from 954 times 103GW in period 1 to 119 times103GW in period 2 and reach 124 times 103GW in period 3 Inthe planning period 1 due to more lenient emission policiesand more economical cost strategies some of the natural gaspower generation facilities are idleThe capacities of 025 GWwould be added to the hydropower facility in three periodsand the corresponding power generation capacity would be812 times 103GW 875 103GW and 937 times 103GW Generatingelectricity from solar energy facility is more expensive thanother power generation facilities thus when the emissionlimits and electricity demand are relatively low solar energyfacility would not be expanded during first two periods 015

Mathematical Problems in Engineering 11

Table 6 Binary solutions of capacity expansions

119901119894-level Power-generation facility Capacity-expansion option IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3

01 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 [0 1] [0 1] 0 0 0

Wind power m=1 0 0 0 0 0 0m=2 0 [0 1] 0 0 [0 1] 1m=3 [0 1] [0 1] [0 1] 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 [0 1] [0 1] 0 0 0 [0 1]m=2 0 0 0 0 0 0m=3 [0 1] [0 1] 0 [0 1] [0 1] 0

005 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 [0 1] [0 1] [0 1] 0 0 0

Wind power m=1 [0 1] [0 1] 0 0 0 0m=2 0 0 0 0 [0 1] 1m=3 [0 1] [0 1] 0 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] [0 1]m=2 0 0 0 0 0 0m=3 1 1 0 [0 1] [0 1] 0

001 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 1 1 [0 1] [0 1] [0 1]

Wind power m=1 0 0 0 0 0 0m=2 0 0 0 0 0 [0 1]m=3 1 1 [0 1] [0 1] [0 1] [0 1]

Solar energy m=1 0 0 0 0 0 0m=2 0 0 [0 1] 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] 0m=2 0 0 0 0 0 0m=3 1 1 [0 1] [0 1] [0 1] [0 1]

GW of capacity would only be added to the solar energyfacility in period 3

Apparently with the successful application of IFCI-PSP within a typical regional electric system managementproblem solutions obtained could provide useful decisionalternatives under different policies and various energy avail-abilities Compared with the least-cost model the IFCF-PSP model is an effective tool for providing environmentalmanagement schemes with dual objectives In addition theIFCF-PSP has following advantages over the conventionalprogramming methods (a) balancing multiple conflicting

objectives (b) reflecting interrelationships among systemefficiency economic cost and system reliability (c) effec-tively dealing with randomness in both the objective andconstraints and (d) assisting the analysis of diverse decisionschemes associated with various energy demand levels

5 Conclusions

An inexact fuzzy chance-constrained fractional program-ming approach is developed for optimal electric powersystems management under uncertainties In the developed

12 Mathematical Problems in Engineering

model fuzzy chance-constrained programming is incorpo-rated into a fractional programming optimization frame-work The obtained results are useful for supporting EPSmanagement The IFCF-PSP approach is capable of (i)balancing the conflict between two objectives (ii) reflect-ing different electricity generation and capacity expansionstrategies (iii) presenting optimal solutions under differentconstraint violating conditions and (iv) introducing theconcept of fuzzy boundary interval with which the com-plexity of dual uncertainties can be effectively handled Theresults of IFCF-PSP model show that (i) a higher confidencelevel corresponds to a higher proportion of clean energypower generation and a lower economic productivity and(ii) as a result of encouraging environment-friendly energiesthe generation capacities for the natural gas wind-powersolar energy and hydropower facilities would be signifi-cantly increased The solutions obtained from the IFCF-PSPapproach could provide specific energy options for powersystem planning and provide effective management solutionof the electric power system for identifying electricity gener-ation and capacity expansion schemes

This study attempts to develop a modeling framework forratio problems involving fuzzy uncertainties to deal with elec-tric power systems management problemThe results suggestthat it is also applicable to other energy management prob-lems In the future practice IFCF-PSP could be further im-proved through considering more impact factors Forinstance the fuzzymembership functions and the confidencelevel are critical in the decision-making process and thisrequires effective ways to provide appropriate choices fordecision making Such challenges desire further investiga-tions Future research can be aimed at applying the advancedapproach to a more complex real-world electric power sys-tem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the Fundamental ResearchFunds for the Central Universities NCEPU (2018BJ0293)the State Scholarship Fund (201706735025) the NationalKey Research and Development Plan (2016YFC0502800) theNatural Sciences Foundation (51520105013 51679087) andthe 111 Program (B14008) The data used to support thefindings of this study are available from the correspondingauthor upon request

References

[1] G H Huang ldquoA hybrid inexact-stochastic water managementmodelrdquo European Journal of Operational Research vol 107 no1 pp 137ndash158 1998

[2] A Grosfeld-Nir and A Tishler ldquoA stochastic model for themeasurement of electricity outage costsrdquo Energy pp 157ndash1741993

[3] S Wang G Huang and B W Baetz ldquoAn Inexact Probabilistic-Possibilistic Optimization Framework for Flood Managementin a Hybrid Uncertain Environmentrdquo IEEE Transactions onFuzzy Systems vol 23 no 4 pp 897ndash908 2015

[4] G H Huang and D P Loucks ldquoAn inexact two-stage stochasticprogramming model for water resources management underuncertaintyrdquo Civil Engineering and Environmental Systems vol17 no 2 pp 95ndash118 2000

[5] K Darby-Dowman S Barker E Audsley and D Parsons ldquoAtwo-stage stochastic programming with recourse model fordetermining robust planting plans in horticulturerdquo Journal ofthe Operational Research Society vol 51 no 1 pp 83ndash89 2000

[6] J Hu L Sun C H Li X Wang X L Jia and Y P Cai ldquoWaterQuality Risk Assessment for the Laoguanhe River of ChinaUsing a Stochastic SimulationMethodrdquo Journal of Environmen-tal Informatics vol 31 no 2 pp 123ndash136 2018

[7] SWang andG HHuang ldquoAmulti-level Taguchi-factorial two-stage stochastic programming approach for characterization ofparameter uncertainties and their interactions An applicationto water resources managementrdquo European Journal of Opera-tional Research vol 240 no 2 pp 572ndash581 2015

[8] S Chanas and P Zielinski ldquoOn the equivalence of two opti-mization methods for fuzzy linear programming problemsrdquoEuropean Journal of Operational Research vol 121 no 1 pp 56ndash63 2000

[9] J Mula D Peidro and R Poler ldquoThe effectiveness of a fuzzymathematical programming approach for supply chain pro-duction planning with fuzzy demandrdquo International Journal ofProduction Economics vol 128 no 1 pp 136ndash143 2010

[10] B Chen P Li H J Wu T Husain and F Khan ldquoMCFP Amonte carlo simulation-based fuzzy programming approachfor optimization under dual uncertainties of possibility andcontinuous probabilityrdquo Journal of Environmental Informatics(JEI) vol 29 no 2 pp 88ndash97 2017

[11] C Z Huang S Nie L Guo and Y R Fan ldquoInexact fuzzystochastic chance constraint programming for emergency evac-uation in Qinshan nuclear power plant under uncertaintyrdquoJournal of Environmental Informatics (JEI) vol 30 no 1 pp 63ndash78 2017

[12] U S Sakalli ldquoOptimization of Production-Distribution Prob-lem in Supply Chain Management under Stochastic and FuzzyUncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 4389064 29 pages 2017

[13] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval research logistics (NRL) pp 181ndash186 1962

[14] N I Voropai and E Y Ivanova ldquoMulti-criteria decision analysistechniques in electric power system expansion planningrdquo Inter-national Journal of Electrical Power amp Energy Systems vol 24no 1 pp 71ndash78 2002

[15] M Chakraborty and S Gupta ldquoFuzzy mathematical pro-gramming for multi objective linear fractional programmingproblemrdquo Fuzzy Sets and Systems vol 125 no 3 pp 335ndash3422002

[16] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[17] W Li Z BaoGHHuang andY L Xie ldquoAn Inexact CredibilityChance-Constrained Integer Programming for Greenhouse

Mathematical Problems in Engineering 13

GasMitigationManagement in Regional Electric Power Systemunder Uncertaintyrdquo in Journal of Environmental Informatics2017

[18] M Paunovic N M Ralevic V Gajovic et al ldquoTwo-StageFuzzy Logic Model for Cloud Service Supplier Selection andEvaluationrdquoMathematical Problems in Engineering 2018

[19] S Wang G H Huang B W Baetz and B C Ancell ldquoTowardsrobust quantification and reduction of uncertainty in hydro-logic predictions Integration of particle Markov chain MonteCarlo and factorial polynomial chaos expansionrdquo Journal ofHydrology vol 548 pp 484ndash497 2017

[20] SWang B C Ancell GHHuang andBW Baetz ldquoImprovingRobustness ofHydrologic Ensemble PredictionsThroughProb-abilistic Pre- and Post-Processing in Sequential Data Assimila-tionrdquoWater Resources Research 2018

[21] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[22] A Rong and R Lahdelma ldquoFuzzy chance constrained linearprogramming model for optimizing the scrap charge in steelproductionrdquo European Journal of Operational Research vol 186no 3 pp 953ndash964 2008

[23] X Sun J Li H Qiao and B Zhang ldquoEnergy implicationsof Chinarsquos regional development New insights from multi-regional input-output analysisrdquo Applied Energy vol 196 pp118ndash131 2017

[24] M A Quaddus and T N Goh ldquoElectric power generationexpansion Planning with multiple objectivesrdquo Applied Energyvol 19 no 4 pp 301ndash319 1985

[25] C H Antunes A G Martins and I S Brito ldquoA multipleobjective mixed integer linear programming model for powergeneration expansion planningrdquo Energy vol 29 no 4 pp 613ndash627 2004

[26] J L C Meza M B Yildirim and A S M Masud ldquoA modelfor the multiperiod multiobjective power generation expansionproblemrdquo IEEETransactions on Power Systems vol 22 no 2 pp871ndash878 2007

[27] J-HHan Y-C Ahn and I-B Lee ldquoAmulti-objective optimiza-tion model for sustainable electricity generation and CO2 mit-igation (EGCM) infrastructure design considering economicprofit and financial riskrdquo Applied Energy vol 95 pp 186ndash1952012

[28] M Rekik A Abdelkafi and L Krichen ldquoAmicro-grid ensuringmulti-objective control strategy of a power electrical system forquality improvementrdquo Energy vol 88 pp 351ndash363 2015

[29] A Azadeh Z Raoofi and M Zarrin ldquoA multi-objective fuzzylinear programming model for optimization of natural gassupply chain through a greenhouse gas reduction approachrdquoJournal of Natural Gas Science and Engineering vol 26 pp 702ndash710 2015

[30] K Li L Pan W Xue H Jiang and H Mao ldquoMulti-objectiveoptimization for energy performance improvement of residen-tial buildings a comparative studyrdquo Energies vol 10 no 2 p245 2017

[31] H Zhu W W Huang and G H Huang ldquoPlanning of regionalenergy systems An inexact mixed-integer fractional program-ming modelrdquo Applied Energy vol 113 pp 500ndash514 2014

[32] F Chen G H Huang Y R Fan and R F Liao ldquoA nonlinearfractional programming approach for environmentaleconomicpower dispatchrdquo International Journal of Electrical Power ampEnergy Systems vol 78 pp 463ndash469 2016

[33] C Zhang and P Guo ldquoFLFP A fuzzy linear fractional pro-gramming approach with double-sided fuzziness for optimalirrigation water allocationrdquo Agricultural Water Managementvol 199 pp 105ndash119 2018

[34] L Wang G Huang X Wang and H Zhu ldquoRisk-basedelectric power system planning for climate change mitigationthrough multi-stage joint-probabilistic left-hand-side chance-constrained fractional programming A Canadian case studyrdquoRenewable amp Sustainable Energy Reviews vol 82 pp 1056ndash10672018

[35] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[36] X H Nie G H Huang Y P Li and L Liu ldquoIFRP A hybridinterval-parameter fuzzy robust programming approach forwaste management planning under uncertaintyrdquo Journal ofEnvironmental Management vol 84 no 1 pp 1ndash11 2007

[37] Y Leung Spatial analysis and planning under imprecisionElsevier 1st edition 2013

[38] A Charnes W W Cooper and M J L Kirby ldquoChance-constrained programming an extension of statistical methodrdquoinOptimizingmethods in statistics pp 391ndash402 Academic Press1971

[39] Y P Cai G H Huang Z F Yang and Q Tan ldquoIdentificationof optimal strategies for energy management systems planningunder multiple uncertaintiesrdquoApplied Energy vol 86 no 4 pp480ndash495 2009

Hindawiwwwhindawicom Volume 2018

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Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

Table 1 Cost of energy supply and conversion parameters

Periodt=1 t=2 t=3

Energy supply cost (103 $TJ)Coal [256 306] [326 376] [396 446]Natural gas [676 696] [757 777] [838 858]Units of energy carrier per units of electricity generation (TJGWh)Coal [1112 117] [1068 1116] [1015 1062]Natural gas [838 882] [754 792] [679 738]

Table 2 Fuzzy subsets for efficiency coefficient under different 120572 minus 119888119906119905 levelsperiod efficiency coefficient120572 minus 119888119906119905 level 120588+ 120588+ 120588minus 120588minus

0t=1 0125 0235 012 023t=2 014 0175 0136 017t=3 0119 0149 0116 0145

02t=1 0135 0223 013 0218t=2 0142 017 0138 0165t=3 012 0144 0117 014

05t=1 015 0205 0145 02t=2 0144 0162 014 0157t=3 0123 0138 0119 0134

08t=1 0165 0187 016 0182t=2 0147 0154 0143 015t=3 0125 0131 0122 0127

1t=1 0175 0175 017 017t=2 0149 0149 0145 0145t=3 0126 0126 0123 0123

Note efficiency coefficient (120588plusmn119905 = 1 minus 120585plusmn119905 )

where 119864119872119905 is equal to the permitted CO2 emission in period119905(103 ton) 120585plusmn119905 is equal to the efficiency of chemical absorptionor capture and storage of CO2 in period 119905119901119894 = the riskconfidence levels

(9) nonnegativity constraints

119860119875119864plusmn119905119895 119860119875119866plusmn119905119895 119860119868119864plusmn119905 ge 0 forall119905 119895 (10k)

Interval parameters are adopted to address impreciseuncertainties which are generally associated with electricitydemands prices of energy resources costs of capacity expan-sion and many other constraints The real research data areused as input data of the model The detailed descriptionsof cost for energy supply and relative conversion parameterswere illustrated in Table 1 Table 2 gives fuzzy subsets forefficiency coefficient under different120572minus119888119906119905 levels Table 3 liststhe capacity expansion options and capital investment costsfor each facility

4 Results and Discussion

In Figure 4 coal-fired electricity supply would increasesteadily and still play an important role in the power sys-tem due to its high availability and competitive price overthe study planning horizon However increasingly strin-gent emission limits result in idle coal-fired facilities Moreeconomical and environmentally friendly power generationfacilities will be prioritized The risk confidence level (119901119894) isemployed to the constraints of CO2 emission requirementThe decision-maker can adjust the value of the 119901119894 levelaccording to actual needs We assumed that violations ofemission constraints are allowed under three given 119901119894 levels(119901119894 = 001005 and 01 which are normally adopted asthe significance levels) [39] The higher credibility levelwould correspond to a tight environment requirement thusleading to a lower CO2 emission while the lower credibilitylevel would correspond to a relatively relaxed environmentrequirement thus resulting in a higher CO2 emission There-fore changes in 119901119894-level have an impact on coal-fired power

Mathematical Problems in Engineering 7

Table 3 Capacity expansion options and costs for power-generation facilities

Periodt=1 t=2 t=3

Capacity-expansion options (GW)Coal m=1 005 005 005

m=2 010 010 010m=3 015 015 015

Natural gas m=1 010 010 010m=2 015 015 015m=3 020 020 020

Wind power m=1 005 005 005m=2 015 015 015m=3 020 020 020

Solar energy m=1 005 005 005m=2 015 015 015m=3 020 020 020

Hydropower m=1 015 015 015m=2 020 020 020m=3 025 025 025

Capacity expansion cost (106$GW)Coal [577 607] [547 577] [517 547]Natural gas [726 756] [676 726] [626 676]Wind power [1256 1306] [1156 1206] [1056 1106]Solar energy [2668 2768] [2468 2568] [2268 2368]Hydropower [1597 1697] [1497 1597] [1397 1497]

generation The total supply of coal-fired generation woulddecrease from [48069 50720] times 109GWh when 119901119894 = 01to [47175 50685] times 109GWh when 119901119894 = 005 and reach[45938 50313] times 109GWh when 119901119894 = 001 In contrast cleanenergy generation will increase For example solar powergeneration of three periods would rise from [2438 2768] times109GWh when 119901119894 = 01 to [2562 2768] times 109GWh when 119901119894= 005 and reach [2625 2769] times 109GWh when 119901119894 = 001 Asshown in Figure 4 the generation of solar power and naturalgas in clean power facilities are mainly affected by emissionconstraints When 119901119894 = 001 the solar power generation isthe largest and the power generation from natural gas isthe lowest in the three-study 119901119894-levels This is the result ofthe tradeoff between carbon dioxide emissions and systemcosts Developing natural gas electricity requires a relativelylow capital cost but leads to more CO2 emissions and thesolar energy technology needs a low operational cost but anextremely high capital cost Similarly the results under other119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted Duringthe entire planning horizon the ratio objective between cleanenergy generation and the total system cost would be [2153]GWh per $106 which also represents the range of the systemefficiency When the electricity-generation pattern variesunder different 119901119894 levels and within the interval solutionranges the system efficiency would also fluctuate within itssolution range correspondingly Thus the IFCF-PSP resultscan create multiple decision alternatives through adjusting

different combinations of the solutions Table 4 presentsthe typical alternatives where combinations of upperlowerbound values for 119860119875119864plusmn119905119895 119860119875119866plusmn119905119895 119884plusmn119905119895119898 (119895 = 1 forall119905 119898) and119860119875119866plusmn119905119895 119884plusmn119905119895119898 (119895 = 2 3 4 5 forall119905 119898) are examined Underthe same power generation alternative interval solutionof the system cost can be obtained according to intervalprice parameters and corresponds to the interval solution ofsystem efficiency Based on the result of the total electricitygenerated we divided the 12 alternatives into three groups(high A3 A4 A8 and A12 medium A1 A5 A7 and A9 lowA2A6A10 andA11) tomeet the future highmedium or lowelectricity demand

The comparison of the system efficiency under different119901119894 levels shows that A11gtA6 gtA10 gtA2 under low-level totalpower generation group Alternative A11 (corresponding tof+) would lead to the most sustainable option with a systemefficiency as much as [27142999] GWh per $106 whichcorresponds to the highest amount of clean energy electricity(0478 times 106GWh) and a moderate cost ([15936 17612] times109$) Additionally lower-level electricity demands will besatisfied by a total power generation of 1856 106GWhTherefore Alternative A11 is a desirable choice from thepoint of view of resources conservation and environmentalprotection when electricity demand is at the low levelAlternative A6 would also meet low electricity demand andreached the lowest system cost ([15142 16785] times 109$)When Alternative A10 is adopted all the power generation

8 Mathematical Problems in Engineering

020406080

100120140160180200

Submodel f-Submodel f+

Submodel f-Submodel f+

Submodel f-Submodel f+

020406080

100120140160180200

020406080

100120140160180200

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

ＪＣ=01

ＪＣ=005

ＪＣ=001

Pow

er g

ener

atio

n (103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)

Figure 4 Power generation loads for EPS through the IFCF-PSPmodel

activities will reach their lower bound levels leading to thelowest power generation (1764 times 106GWh) Compared withAlternative A6 this alternative may be of less interest todecision-makers due to its lower system efficiency and highersystem cost However it is the solution that is obtainedunder the most stringent emission constraints (ie 119901119894 =001) Alternative A2 is a moderate solution in this group Itssystem cost system efficiency total power generation andclean energy power generation are [15295 16954] times 109$[22062445] GWh per $106 1816 times 106GWh and 0374 times106GWh

In middle-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A7gt A5gt A9 gtA1 In alternative A7middle-level electricity demands will be satisfied by a totalpower generation of 1893 times 106GWh In addition Alternative

A7 has the highest system efficiency ([2661 2941] GWhper 106$) and the largest amount of clean energy powergeneration (0478times 106GWh) in this groupThus AlternativeA7 is considered to be a desirable choice Alternatives A5 andA9 would lead to the same total power generation solution(1895 times 106GWh) They are the two-moderate solutions inthis groupThe system cost obtained from the A1 is the lowest([15718 17413] times 109$) in this group It would lead to thelowest system efficiency ([2148 2379] GWh per 106$) butdue to the largest proportion of fossil-fired power generationA1 has the maximum security for energy supply

In high-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A3gt A12gt A8 gtA4 The same cleanenergy power generation obtained from the 4 alternativesis 0478 106GWh In Alternative A4 all the power gener-ation activities will be equal to their upper-bound valuesat the same time It would lead to the highest system cost([16901 18676] times 109$) but the moderate system efficiency([2559 2828]GWhper $106) In comparisonAlternative A3would lead to the minimum system cost ([16478 18216] times109$) and the highest system efficiency ([2624 2901] GWhper $106) in this group In addition this alternative wouldalso provide the highest clean energy power generationTherefore this alternative is a desirable sustainable optionunder the high electricity demand level Alternatives A8 andA12 can provide modest system efficiencies which would be[2561 283] and [2575 2845] GWh per $106 respectivelyand lower system costs which would be [16892 18666] and[16800 18564] times 109$ Therefore decision-makers who payattention to the stability of the system may be interested inthese two alternatives

The above alternatives represent various options betweeneconomic and environmental tradeoffsWillingness to accepthigh system cost will guarantee meeting the objective ofincreasing the proportion of clean energy A strong desireto acquire low system cost will cause the risk of violatingemission constraints In general the above research resultswere favored by decision makers due to their flexibilityand preference for practical-making decision processes Thefeasible ranges for decision variables under different 119901119894 levelswere useful for decision makers to justify the generatedalternatives directly

Besides the scenario of maximizing the proportion ofclean energy another scenario of minimizing the system costis also analyzed to evaluate the effects of different energy sup-ply policies The optimal-ratio problem presented in Models(10a)ndash(10k) can be converted into a least-cost problem withthe following objective

min119891 = system cost

= 119879sum119905=1

119868sum119895=1

119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 + 119879sum119905=1

119869sum119895=1

119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895+ 119879sum119905=1

119869sum119895=1

119872sum119898=1

119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898

Mathematical Problems in Engineering 9

Table 4 Typical decision alternatives obtained from the IFCF-PSP model solutions

Alternative 119860119875119864plusmn119905119895 119860119875119866

plusmn119905119895119884plusmn119905119895119898

(119895=1 forall119905119898)

119860119875119866plusmn119905119895 119884plusmn119905119895119898

(119895=2345 forall119905119898)

System efficiency(GWh per 106$) System cost (109$) Total power generation

(106GWh)Clean energy powergeneration (106GWh)119901119894 = 01

A1 + - [21482379] (12) [1571817413] (4) 1895 (6) 0374 (11)A2 - - [22062445] (9) [1529516954] (3) 1816 (10) 0374 (11)A3 - + [26242901] (3) [1647818216] (9) 192 (4) 0478 (1)A4 + + [25592828] (6) [16901 18676] (12) 2 (1) 0478 (1)119901119894 = 005A5 + - [22362477] (8) [1578317487] (5) 1912 (5) 0391 (7)A6 - - [23292582] (7) [1514216785] (1) 1807 (11) 0391 (7)A7 - + [26612941] (2) [1625117963] (8) 1893 (8) 0478 (1)A8 + + [2561283] (5) [1689218666] (11) 1999 (2) 0478 (1)119901119894 = 001A9 + - [2174 2406] (11) [16040 17757] (7) 1895 (6) 0386 (9)A10 - - [22972543] (8) [15176 16806] (2) 1764 (12) 0386 (9)A11 - + [27142999] (1) [1593617612] (6) 1856 (9) 0478 (1)A12 + + [25752845] (4) [16800 18564] (10) 1987 (3) 0478 (1)

Table 5 The proportion of clean energy power generation from IFCF-PSP and LS models

Clean power generation ratio () IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3119901119894 = 01 [1802 2432] [2089 2493] [1986 2540] [1623 1870] [1681 1719] [1515 1638]119901119894 = 005 [2067 2465] [2093 2528] [2010 2578] [1624 1870] [1681 1719] [1632 1869]119901119894 = 001 [2155 2530] [2158 2596] [2141 2596] [1648 1893] [1679 1752] [1997 2224]

+ 119879sum119905=1

119862119868119864plusmn119905 times 119860119868119864plusmn119905+ 119879sum119905=1

119868sum119895=1

119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905(11)

Figure 5 shows the proportion of different power gen-eration technologies from the IFCF-PSP and LS modelsunder 119901119894 = 001 over three planning periods The IFCF-PSPmodel leads to a relatively higher percentage of natural gaspower generation in comparison the LS model leads torelatively higher percentages of coal-fired electricity suppliesAccording to the solutions from IFCF-PSPmodel the percentof electricity generated by natural gas facilities would be [5 6] [7 8] and [5 8] in periods 1 2 and 3 respectivelyTheLS model achieves the slightly lower percent (ie 0 0 and[5 6] in periods 1 2 and 3) Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted

According to Table 5 the differences can be foundbetween the results of two models under different 119901119894 levelsThe proportion of clean energy power generation fromIFCF-PSP model is higher than that of the LS model Forexample under 119901119894 = 001 clean energy power generationof the entire region occupied [2155 2530] [2158 2596]and [2141 2596] of the total electricity generation in

the three study periods from IFCF-PSP model which arehigher than the LS model [1648 1993] [1679 1752] and[1997 2224] respectively Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpretedAn increased 119901119894 level represents a higher admissible riskleading to a decreased strictness for the emission constraintsand hence an expanded decision space Therefore undertypical conditions as the risk level becomes lower theproportion of clean energy power generation or renewablepower generation would increase For example in period 2when the 119901119894 level is dropped from 01 to 001 the proportionof clean energy power generation from IFCF-PSP modelwould be increased from [2089 2493] to [2158 2596]Likewise the results under other periods can be similarlyanalyzed The confidence level of constraints satisfaction ismore reliable because the risk level is lowerTherefore in thiscase the decision-makers will be more conservative in theEPS management There is no capacity-expansion would beconducted for the coal-fired facility in three periods sinceits high carbon dioxide emissions and increasingly stringentemission limits On the other hand clean energy electricitysupply would be insufficient for the future energy demandsAccording to Table 6 when 119901119894 takes different values (ie 119901119894 =001119901119894 = 005 and 119901119894 = 01) differences of capacity-expansionoption can be found between the results of two models Inboth models when the risk of violating the constraints ofcarbon emission target is decreased (ie the value of 119901119894-level

10 Mathematical Problems in Engineering

75

79

5 4

74

8

95 4

74

8

9

5 4

81

095 5

83

09

4 4

78

7

7

4 4

coalnatural gashydro

windsolar

83

010

2

5

83

0

94 4

80

5

84 3

82

19 4

4

78

6

84 4

79

5

84 4

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

IFCFP-PSP f- IFCFP-PSP +

LS +LS -

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

Figure 5 Comparison of power-generation patterns from IFCF-PSP and least-system-cost (LS) models under 119901119894-level 001 Note that theinnermost circle in the pie chart represents the first planning period

is larger) the demand for expansion of clean energy willincrease On the other hand the total demand of capacityexpansion in the lower bound model is lower than the upperbound model One of the reasons for this result is that theannual utilization hours of various power generation facilitiesin the lower bound model are higher than the correspondingparameter settings of the upper bound model

The result of LS model has shown that under thecorresponding lower bound parameter settings the capacityof 015 GW would be added to the hydropower facility inperiod 3 when 119901119894 = 001 and 119901119894 = 005 Only a capacity of015 GWwould be added to the wind power facility in period3 While under the corresponding upper bound parametersettings natural gas and solar energy facilities expansionsare not required that is because the former resource is underpenalty of CO2 emission and the capacity cost of the solarpower expansion is much higher

In the IFCF-PSP model clean energy will be given moredevelopment opportunities Take 119901119894 = 001 as an example(corresponding to 119891minus) with a capacity expansion of 02

GW at the beginning of period 1 and period 2 electricitygeneration ofwind power facilitywould rise from 164 103GWin period 1 to 168 103GW in period 2 and with anothercapacity expansion of 02 GW at the beginning of period 3 itstotal electricity generation capacity would reach 172 103GWLikewise to meet the growing energy demand the capacityof 02 GW would be added to the natural gas-fired facility inperiod 1 period 2 and period 3 as a result its total generationcapacity would rise from 954 times 103GW in period 1 to 119 times103GW in period 2 and reach 124 times 103GW in period 3 Inthe planning period 1 due to more lenient emission policiesand more economical cost strategies some of the natural gaspower generation facilities are idleThe capacities of 025 GWwould be added to the hydropower facility in three periodsand the corresponding power generation capacity would be812 times 103GW 875 103GW and 937 times 103GW Generatingelectricity from solar energy facility is more expensive thanother power generation facilities thus when the emissionlimits and electricity demand are relatively low solar energyfacility would not be expanded during first two periods 015

Mathematical Problems in Engineering 11

Table 6 Binary solutions of capacity expansions

119901119894-level Power-generation facility Capacity-expansion option IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3

01 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 [0 1] [0 1] 0 0 0

Wind power m=1 0 0 0 0 0 0m=2 0 [0 1] 0 0 [0 1] 1m=3 [0 1] [0 1] [0 1] 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 [0 1] [0 1] 0 0 0 [0 1]m=2 0 0 0 0 0 0m=3 [0 1] [0 1] 0 [0 1] [0 1] 0

005 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 [0 1] [0 1] [0 1] 0 0 0

Wind power m=1 [0 1] [0 1] 0 0 0 0m=2 0 0 0 0 [0 1] 1m=3 [0 1] [0 1] 0 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] [0 1]m=2 0 0 0 0 0 0m=3 1 1 0 [0 1] [0 1] 0

001 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 1 1 [0 1] [0 1] [0 1]

Wind power m=1 0 0 0 0 0 0m=2 0 0 0 0 0 [0 1]m=3 1 1 [0 1] [0 1] [0 1] [0 1]

Solar energy m=1 0 0 0 0 0 0m=2 0 0 [0 1] 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] 0m=2 0 0 0 0 0 0m=3 1 1 [0 1] [0 1] [0 1] [0 1]

GW of capacity would only be added to the solar energyfacility in period 3

Apparently with the successful application of IFCI-PSP within a typical regional electric system managementproblem solutions obtained could provide useful decisionalternatives under different policies and various energy avail-abilities Compared with the least-cost model the IFCF-PSP model is an effective tool for providing environmentalmanagement schemes with dual objectives In addition theIFCF-PSP has following advantages over the conventionalprogramming methods (a) balancing multiple conflicting

objectives (b) reflecting interrelationships among systemefficiency economic cost and system reliability (c) effec-tively dealing with randomness in both the objective andconstraints and (d) assisting the analysis of diverse decisionschemes associated with various energy demand levels

5 Conclusions

An inexact fuzzy chance-constrained fractional program-ming approach is developed for optimal electric powersystems management under uncertainties In the developed

12 Mathematical Problems in Engineering

model fuzzy chance-constrained programming is incorpo-rated into a fractional programming optimization frame-work The obtained results are useful for supporting EPSmanagement The IFCF-PSP approach is capable of (i)balancing the conflict between two objectives (ii) reflect-ing different electricity generation and capacity expansionstrategies (iii) presenting optimal solutions under differentconstraint violating conditions and (iv) introducing theconcept of fuzzy boundary interval with which the com-plexity of dual uncertainties can be effectively handled Theresults of IFCF-PSP model show that (i) a higher confidencelevel corresponds to a higher proportion of clean energypower generation and a lower economic productivity and(ii) as a result of encouraging environment-friendly energiesthe generation capacities for the natural gas wind-powersolar energy and hydropower facilities would be signifi-cantly increased The solutions obtained from the IFCF-PSPapproach could provide specific energy options for powersystem planning and provide effective management solutionof the electric power system for identifying electricity gener-ation and capacity expansion schemes

This study attempts to develop a modeling framework forratio problems involving fuzzy uncertainties to deal with elec-tric power systems management problemThe results suggestthat it is also applicable to other energy management prob-lems In the future practice IFCF-PSP could be further im-proved through considering more impact factors Forinstance the fuzzymembership functions and the confidencelevel are critical in the decision-making process and thisrequires effective ways to provide appropriate choices fordecision making Such challenges desire further investiga-tions Future research can be aimed at applying the advancedapproach to a more complex real-world electric power sys-tem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the Fundamental ResearchFunds for the Central Universities NCEPU (2018BJ0293)the State Scholarship Fund (201706735025) the NationalKey Research and Development Plan (2016YFC0502800) theNatural Sciences Foundation (51520105013 51679087) andthe 111 Program (B14008) The data used to support thefindings of this study are available from the correspondingauthor upon request

References

[1] G H Huang ldquoA hybrid inexact-stochastic water managementmodelrdquo European Journal of Operational Research vol 107 no1 pp 137ndash158 1998

[2] A Grosfeld-Nir and A Tishler ldquoA stochastic model for themeasurement of electricity outage costsrdquo Energy pp 157ndash1741993

[3] S Wang G Huang and B W Baetz ldquoAn Inexact Probabilistic-Possibilistic Optimization Framework for Flood Managementin a Hybrid Uncertain Environmentrdquo IEEE Transactions onFuzzy Systems vol 23 no 4 pp 897ndash908 2015

[4] G H Huang and D P Loucks ldquoAn inexact two-stage stochasticprogramming model for water resources management underuncertaintyrdquo Civil Engineering and Environmental Systems vol17 no 2 pp 95ndash118 2000

[5] K Darby-Dowman S Barker E Audsley and D Parsons ldquoAtwo-stage stochastic programming with recourse model fordetermining robust planting plans in horticulturerdquo Journal ofthe Operational Research Society vol 51 no 1 pp 83ndash89 2000

[6] J Hu L Sun C H Li X Wang X L Jia and Y P Cai ldquoWaterQuality Risk Assessment for the Laoguanhe River of ChinaUsing a Stochastic SimulationMethodrdquo Journal of Environmen-tal Informatics vol 31 no 2 pp 123ndash136 2018

[7] SWang andG HHuang ldquoAmulti-level Taguchi-factorial two-stage stochastic programming approach for characterization ofparameter uncertainties and their interactions An applicationto water resources managementrdquo European Journal of Opera-tional Research vol 240 no 2 pp 572ndash581 2015

[8] S Chanas and P Zielinski ldquoOn the equivalence of two opti-mization methods for fuzzy linear programming problemsrdquoEuropean Journal of Operational Research vol 121 no 1 pp 56ndash63 2000

[9] J Mula D Peidro and R Poler ldquoThe effectiveness of a fuzzymathematical programming approach for supply chain pro-duction planning with fuzzy demandrdquo International Journal ofProduction Economics vol 128 no 1 pp 136ndash143 2010

[10] B Chen P Li H J Wu T Husain and F Khan ldquoMCFP Amonte carlo simulation-based fuzzy programming approachfor optimization under dual uncertainties of possibility andcontinuous probabilityrdquo Journal of Environmental Informatics(JEI) vol 29 no 2 pp 88ndash97 2017

[11] C Z Huang S Nie L Guo and Y R Fan ldquoInexact fuzzystochastic chance constraint programming for emergency evac-uation in Qinshan nuclear power plant under uncertaintyrdquoJournal of Environmental Informatics (JEI) vol 30 no 1 pp 63ndash78 2017

[12] U S Sakalli ldquoOptimization of Production-Distribution Prob-lem in Supply Chain Management under Stochastic and FuzzyUncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 4389064 29 pages 2017

[13] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval research logistics (NRL) pp 181ndash186 1962

[14] N I Voropai and E Y Ivanova ldquoMulti-criteria decision analysistechniques in electric power system expansion planningrdquo Inter-national Journal of Electrical Power amp Energy Systems vol 24no 1 pp 71ndash78 2002

[15] M Chakraborty and S Gupta ldquoFuzzy mathematical pro-gramming for multi objective linear fractional programmingproblemrdquo Fuzzy Sets and Systems vol 125 no 3 pp 335ndash3422002

[16] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[17] W Li Z BaoGHHuang andY L Xie ldquoAn Inexact CredibilityChance-Constrained Integer Programming for Greenhouse

Mathematical Problems in Engineering 13

GasMitigationManagement in Regional Electric Power Systemunder Uncertaintyrdquo in Journal of Environmental Informatics2017

[18] M Paunovic N M Ralevic V Gajovic et al ldquoTwo-StageFuzzy Logic Model for Cloud Service Supplier Selection andEvaluationrdquoMathematical Problems in Engineering 2018

[19] S Wang G H Huang B W Baetz and B C Ancell ldquoTowardsrobust quantification and reduction of uncertainty in hydro-logic predictions Integration of particle Markov chain MonteCarlo and factorial polynomial chaos expansionrdquo Journal ofHydrology vol 548 pp 484ndash497 2017

[20] SWang B C Ancell GHHuang andBW Baetz ldquoImprovingRobustness ofHydrologic Ensemble PredictionsThroughProb-abilistic Pre- and Post-Processing in Sequential Data Assimila-tionrdquoWater Resources Research 2018

[21] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[22] A Rong and R Lahdelma ldquoFuzzy chance constrained linearprogramming model for optimizing the scrap charge in steelproductionrdquo European Journal of Operational Research vol 186no 3 pp 953ndash964 2008

[23] X Sun J Li H Qiao and B Zhang ldquoEnergy implicationsof Chinarsquos regional development New insights from multi-regional input-output analysisrdquo Applied Energy vol 196 pp118ndash131 2017

[24] M A Quaddus and T N Goh ldquoElectric power generationexpansion Planning with multiple objectivesrdquo Applied Energyvol 19 no 4 pp 301ndash319 1985

[25] C H Antunes A G Martins and I S Brito ldquoA multipleobjective mixed integer linear programming model for powergeneration expansion planningrdquo Energy vol 29 no 4 pp 613ndash627 2004

[26] J L C Meza M B Yildirim and A S M Masud ldquoA modelfor the multiperiod multiobjective power generation expansionproblemrdquo IEEETransactions on Power Systems vol 22 no 2 pp871ndash878 2007

[27] J-HHan Y-C Ahn and I-B Lee ldquoAmulti-objective optimiza-tion model for sustainable electricity generation and CO2 mit-igation (EGCM) infrastructure design considering economicprofit and financial riskrdquo Applied Energy vol 95 pp 186ndash1952012

[28] M Rekik A Abdelkafi and L Krichen ldquoAmicro-grid ensuringmulti-objective control strategy of a power electrical system forquality improvementrdquo Energy vol 88 pp 351ndash363 2015

[29] A Azadeh Z Raoofi and M Zarrin ldquoA multi-objective fuzzylinear programming model for optimization of natural gassupply chain through a greenhouse gas reduction approachrdquoJournal of Natural Gas Science and Engineering vol 26 pp 702ndash710 2015

[30] K Li L Pan W Xue H Jiang and H Mao ldquoMulti-objectiveoptimization for energy performance improvement of residen-tial buildings a comparative studyrdquo Energies vol 10 no 2 p245 2017

[31] H Zhu W W Huang and G H Huang ldquoPlanning of regionalenergy systems An inexact mixed-integer fractional program-ming modelrdquo Applied Energy vol 113 pp 500ndash514 2014

[32] F Chen G H Huang Y R Fan and R F Liao ldquoA nonlinearfractional programming approach for environmentaleconomicpower dispatchrdquo International Journal of Electrical Power ampEnergy Systems vol 78 pp 463ndash469 2016

[33] C Zhang and P Guo ldquoFLFP A fuzzy linear fractional pro-gramming approach with double-sided fuzziness for optimalirrigation water allocationrdquo Agricultural Water Managementvol 199 pp 105ndash119 2018

[34] L Wang G Huang X Wang and H Zhu ldquoRisk-basedelectric power system planning for climate change mitigationthrough multi-stage joint-probabilistic left-hand-side chance-constrained fractional programming A Canadian case studyrdquoRenewable amp Sustainable Energy Reviews vol 82 pp 1056ndash10672018

[35] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[36] X H Nie G H Huang Y P Li and L Liu ldquoIFRP A hybridinterval-parameter fuzzy robust programming approach forwaste management planning under uncertaintyrdquo Journal ofEnvironmental Management vol 84 no 1 pp 1ndash11 2007

[37] Y Leung Spatial analysis and planning under imprecisionElsevier 1st edition 2013

[38] A Charnes W W Cooper and M J L Kirby ldquoChance-constrained programming an extension of statistical methodrdquoinOptimizingmethods in statistics pp 391ndash402 Academic Press1971

[39] Y P Cai G H Huang Z F Yang and Q Tan ldquoIdentificationof optimal strategies for energy management systems planningunder multiple uncertaintiesrdquoApplied Energy vol 86 no 4 pp480ndash495 2009

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Mathematical Problems in Engineering 7

Table 3 Capacity expansion options and costs for power-generation facilities

Periodt=1 t=2 t=3

Capacity-expansion options (GW)Coal m=1 005 005 005

m=2 010 010 010m=3 015 015 015

Natural gas m=1 010 010 010m=2 015 015 015m=3 020 020 020

Wind power m=1 005 005 005m=2 015 015 015m=3 020 020 020

Solar energy m=1 005 005 005m=2 015 015 015m=3 020 020 020

Hydropower m=1 015 015 015m=2 020 020 020m=3 025 025 025

Capacity expansion cost (106$GW)Coal [577 607] [547 577] [517 547]Natural gas [726 756] [676 726] [626 676]Wind power [1256 1306] [1156 1206] [1056 1106]Solar energy [2668 2768] [2468 2568] [2268 2368]Hydropower [1597 1697] [1497 1597] [1397 1497]

generation The total supply of coal-fired generation woulddecrease from [48069 50720] times 109GWh when 119901119894 = 01to [47175 50685] times 109GWh when 119901119894 = 005 and reach[45938 50313] times 109GWh when 119901119894 = 001 In contrast cleanenergy generation will increase For example solar powergeneration of three periods would rise from [2438 2768] times109GWh when 119901119894 = 01 to [2562 2768] times 109GWh when 119901119894= 005 and reach [2625 2769] times 109GWh when 119901119894 = 001 Asshown in Figure 4 the generation of solar power and naturalgas in clean power facilities are mainly affected by emissionconstraints When 119901119894 = 001 the solar power generation isthe largest and the power generation from natural gas isthe lowest in the three-study 119901119894-levels This is the result ofthe tradeoff between carbon dioxide emissions and systemcosts Developing natural gas electricity requires a relativelylow capital cost but leads to more CO2 emissions and thesolar energy technology needs a low operational cost but anextremely high capital cost Similarly the results under other119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted Duringthe entire planning horizon the ratio objective between cleanenergy generation and the total system cost would be [2153]GWh per $106 which also represents the range of the systemefficiency When the electricity-generation pattern variesunder different 119901119894 levels and within the interval solutionranges the system efficiency would also fluctuate within itssolution range correspondingly Thus the IFCF-PSP resultscan create multiple decision alternatives through adjusting

different combinations of the solutions Table 4 presentsthe typical alternatives where combinations of upperlowerbound values for 119860119875119864plusmn119905119895 119860119875119866plusmn119905119895 119884plusmn119905119895119898 (119895 = 1 forall119905 119898) and119860119875119866plusmn119905119895 119884plusmn119905119895119898 (119895 = 2 3 4 5 forall119905 119898) are examined Underthe same power generation alternative interval solutionof the system cost can be obtained according to intervalprice parameters and corresponds to the interval solution ofsystem efficiency Based on the result of the total electricitygenerated we divided the 12 alternatives into three groups(high A3 A4 A8 and A12 medium A1 A5 A7 and A9 lowA2A6A10 andA11) tomeet the future highmedium or lowelectricity demand

The comparison of the system efficiency under different119901119894 levels shows that A11gtA6 gtA10 gtA2 under low-level totalpower generation group Alternative A11 (corresponding tof+) would lead to the most sustainable option with a systemefficiency as much as [27142999] GWh per $106 whichcorresponds to the highest amount of clean energy electricity(0478 times 106GWh) and a moderate cost ([15936 17612] times109$) Additionally lower-level electricity demands will besatisfied by a total power generation of 1856 106GWhTherefore Alternative A11 is a desirable choice from thepoint of view of resources conservation and environmentalprotection when electricity demand is at the low levelAlternative A6 would also meet low electricity demand andreached the lowest system cost ([15142 16785] times 109$)When Alternative A10 is adopted all the power generation

8 Mathematical Problems in Engineering

020406080

100120140160180200

Submodel f-Submodel f+

Submodel f-Submodel f+

Submodel f-Submodel f+

020406080

100120140160180200

020406080

100120140160180200

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

ＪＣ=01

ＪＣ=005

ＪＣ=001

Pow

er g

ener

atio

n (103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)

Figure 4 Power generation loads for EPS through the IFCF-PSPmodel

activities will reach their lower bound levels leading to thelowest power generation (1764 times 106GWh) Compared withAlternative A6 this alternative may be of less interest todecision-makers due to its lower system efficiency and highersystem cost However it is the solution that is obtainedunder the most stringent emission constraints (ie 119901119894 =001) Alternative A2 is a moderate solution in this group Itssystem cost system efficiency total power generation andclean energy power generation are [15295 16954] times 109$[22062445] GWh per $106 1816 times 106GWh and 0374 times106GWh

In middle-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A7gt A5gt A9 gtA1 In alternative A7middle-level electricity demands will be satisfied by a totalpower generation of 1893 times 106GWh In addition Alternative

A7 has the highest system efficiency ([2661 2941] GWhper 106$) and the largest amount of clean energy powergeneration (0478times 106GWh) in this groupThus AlternativeA7 is considered to be a desirable choice Alternatives A5 andA9 would lead to the same total power generation solution(1895 times 106GWh) They are the two-moderate solutions inthis groupThe system cost obtained from the A1 is the lowest([15718 17413] times 109$) in this group It would lead to thelowest system efficiency ([2148 2379] GWh per 106$) butdue to the largest proportion of fossil-fired power generationA1 has the maximum security for energy supply

In high-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A3gt A12gt A8 gtA4 The same cleanenergy power generation obtained from the 4 alternativesis 0478 106GWh In Alternative A4 all the power gener-ation activities will be equal to their upper-bound valuesat the same time It would lead to the highest system cost([16901 18676] times 109$) but the moderate system efficiency([2559 2828]GWhper $106) In comparisonAlternative A3would lead to the minimum system cost ([16478 18216] times109$) and the highest system efficiency ([2624 2901] GWhper $106) in this group In addition this alternative wouldalso provide the highest clean energy power generationTherefore this alternative is a desirable sustainable optionunder the high electricity demand level Alternatives A8 andA12 can provide modest system efficiencies which would be[2561 283] and [2575 2845] GWh per $106 respectivelyand lower system costs which would be [16892 18666] and[16800 18564] times 109$ Therefore decision-makers who payattention to the stability of the system may be interested inthese two alternatives

The above alternatives represent various options betweeneconomic and environmental tradeoffsWillingness to accepthigh system cost will guarantee meeting the objective ofincreasing the proportion of clean energy A strong desireto acquire low system cost will cause the risk of violatingemission constraints In general the above research resultswere favored by decision makers due to their flexibilityand preference for practical-making decision processes Thefeasible ranges for decision variables under different 119901119894 levelswere useful for decision makers to justify the generatedalternatives directly

Besides the scenario of maximizing the proportion ofclean energy another scenario of minimizing the system costis also analyzed to evaluate the effects of different energy sup-ply policies The optimal-ratio problem presented in Models(10a)ndash(10k) can be converted into a least-cost problem withthe following objective

min119891 = system cost

= 119879sum119905=1

119868sum119895=1

119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 + 119879sum119905=1

119869sum119895=1

119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895+ 119879sum119905=1

119869sum119895=1

119872sum119898=1

119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898

Mathematical Problems in Engineering 9

Table 4 Typical decision alternatives obtained from the IFCF-PSP model solutions

Alternative 119860119875119864plusmn119905119895 119860119875119866

plusmn119905119895119884plusmn119905119895119898

(119895=1 forall119905119898)

119860119875119866plusmn119905119895 119884plusmn119905119895119898

(119895=2345 forall119905119898)

System efficiency(GWh per 106$) System cost (109$) Total power generation

(106GWh)Clean energy powergeneration (106GWh)119901119894 = 01

A1 + - [21482379] (12) [1571817413] (4) 1895 (6) 0374 (11)A2 - - [22062445] (9) [1529516954] (3) 1816 (10) 0374 (11)A3 - + [26242901] (3) [1647818216] (9) 192 (4) 0478 (1)A4 + + [25592828] (6) [16901 18676] (12) 2 (1) 0478 (1)119901119894 = 005A5 + - [22362477] (8) [1578317487] (5) 1912 (5) 0391 (7)A6 - - [23292582] (7) [1514216785] (1) 1807 (11) 0391 (7)A7 - + [26612941] (2) [1625117963] (8) 1893 (8) 0478 (1)A8 + + [2561283] (5) [1689218666] (11) 1999 (2) 0478 (1)119901119894 = 001A9 + - [2174 2406] (11) [16040 17757] (7) 1895 (6) 0386 (9)A10 - - [22972543] (8) [15176 16806] (2) 1764 (12) 0386 (9)A11 - + [27142999] (1) [1593617612] (6) 1856 (9) 0478 (1)A12 + + [25752845] (4) [16800 18564] (10) 1987 (3) 0478 (1)

Table 5 The proportion of clean energy power generation from IFCF-PSP and LS models

Clean power generation ratio () IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3119901119894 = 01 [1802 2432] [2089 2493] [1986 2540] [1623 1870] [1681 1719] [1515 1638]119901119894 = 005 [2067 2465] [2093 2528] [2010 2578] [1624 1870] [1681 1719] [1632 1869]119901119894 = 001 [2155 2530] [2158 2596] [2141 2596] [1648 1893] [1679 1752] [1997 2224]

+ 119879sum119905=1

119862119868119864plusmn119905 times 119860119868119864plusmn119905+ 119879sum119905=1

119868sum119895=1

119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905(11)

Figure 5 shows the proportion of different power gen-eration technologies from the IFCF-PSP and LS modelsunder 119901119894 = 001 over three planning periods The IFCF-PSPmodel leads to a relatively higher percentage of natural gaspower generation in comparison the LS model leads torelatively higher percentages of coal-fired electricity suppliesAccording to the solutions from IFCF-PSPmodel the percentof electricity generated by natural gas facilities would be [5 6] [7 8] and [5 8] in periods 1 2 and 3 respectivelyTheLS model achieves the slightly lower percent (ie 0 0 and[5 6] in periods 1 2 and 3) Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted

According to Table 5 the differences can be foundbetween the results of two models under different 119901119894 levelsThe proportion of clean energy power generation fromIFCF-PSP model is higher than that of the LS model Forexample under 119901119894 = 001 clean energy power generationof the entire region occupied [2155 2530] [2158 2596]and [2141 2596] of the total electricity generation in

the three study periods from IFCF-PSP model which arehigher than the LS model [1648 1993] [1679 1752] and[1997 2224] respectively Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpretedAn increased 119901119894 level represents a higher admissible riskleading to a decreased strictness for the emission constraintsand hence an expanded decision space Therefore undertypical conditions as the risk level becomes lower theproportion of clean energy power generation or renewablepower generation would increase For example in period 2when the 119901119894 level is dropped from 01 to 001 the proportionof clean energy power generation from IFCF-PSP modelwould be increased from [2089 2493] to [2158 2596]Likewise the results under other periods can be similarlyanalyzed The confidence level of constraints satisfaction ismore reliable because the risk level is lowerTherefore in thiscase the decision-makers will be more conservative in theEPS management There is no capacity-expansion would beconducted for the coal-fired facility in three periods sinceits high carbon dioxide emissions and increasingly stringentemission limits On the other hand clean energy electricitysupply would be insufficient for the future energy demandsAccording to Table 6 when 119901119894 takes different values (ie 119901119894 =001119901119894 = 005 and 119901119894 = 01) differences of capacity-expansionoption can be found between the results of two models Inboth models when the risk of violating the constraints ofcarbon emission target is decreased (ie the value of 119901119894-level

10 Mathematical Problems in Engineering

75

79

5 4

74

8

95 4

74

8

9

5 4

81

095 5

83

09

4 4

78

7

7

4 4

coalnatural gashydro

windsolar

83

010

2

5

83

0

94 4

80

5

84 3

82

19 4

4

78

6

84 4

79

5

84 4

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

IFCFP-PSP f- IFCFP-PSP +

LS +LS -

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

Figure 5 Comparison of power-generation patterns from IFCF-PSP and least-system-cost (LS) models under 119901119894-level 001 Note that theinnermost circle in the pie chart represents the first planning period

is larger) the demand for expansion of clean energy willincrease On the other hand the total demand of capacityexpansion in the lower bound model is lower than the upperbound model One of the reasons for this result is that theannual utilization hours of various power generation facilitiesin the lower bound model are higher than the correspondingparameter settings of the upper bound model

The result of LS model has shown that under thecorresponding lower bound parameter settings the capacityof 015 GW would be added to the hydropower facility inperiod 3 when 119901119894 = 001 and 119901119894 = 005 Only a capacity of015 GWwould be added to the wind power facility in period3 While under the corresponding upper bound parametersettings natural gas and solar energy facilities expansionsare not required that is because the former resource is underpenalty of CO2 emission and the capacity cost of the solarpower expansion is much higher

In the IFCF-PSP model clean energy will be given moredevelopment opportunities Take 119901119894 = 001 as an example(corresponding to 119891minus) with a capacity expansion of 02

GW at the beginning of period 1 and period 2 electricitygeneration ofwind power facilitywould rise from 164 103GWin period 1 to 168 103GW in period 2 and with anothercapacity expansion of 02 GW at the beginning of period 3 itstotal electricity generation capacity would reach 172 103GWLikewise to meet the growing energy demand the capacityof 02 GW would be added to the natural gas-fired facility inperiod 1 period 2 and period 3 as a result its total generationcapacity would rise from 954 times 103GW in period 1 to 119 times103GW in period 2 and reach 124 times 103GW in period 3 Inthe planning period 1 due to more lenient emission policiesand more economical cost strategies some of the natural gaspower generation facilities are idleThe capacities of 025 GWwould be added to the hydropower facility in three periodsand the corresponding power generation capacity would be812 times 103GW 875 103GW and 937 times 103GW Generatingelectricity from solar energy facility is more expensive thanother power generation facilities thus when the emissionlimits and electricity demand are relatively low solar energyfacility would not be expanded during first two periods 015

Mathematical Problems in Engineering 11

Table 6 Binary solutions of capacity expansions

119901119894-level Power-generation facility Capacity-expansion option IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3

01 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 [0 1] [0 1] 0 0 0

Wind power m=1 0 0 0 0 0 0m=2 0 [0 1] 0 0 [0 1] 1m=3 [0 1] [0 1] [0 1] 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 [0 1] [0 1] 0 0 0 [0 1]m=2 0 0 0 0 0 0m=3 [0 1] [0 1] 0 [0 1] [0 1] 0

005 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 [0 1] [0 1] [0 1] 0 0 0

Wind power m=1 [0 1] [0 1] 0 0 0 0m=2 0 0 0 0 [0 1] 1m=3 [0 1] [0 1] 0 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] [0 1]m=2 0 0 0 0 0 0m=3 1 1 0 [0 1] [0 1] 0

001 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 1 1 [0 1] [0 1] [0 1]

Wind power m=1 0 0 0 0 0 0m=2 0 0 0 0 0 [0 1]m=3 1 1 [0 1] [0 1] [0 1] [0 1]

Solar energy m=1 0 0 0 0 0 0m=2 0 0 [0 1] 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] 0m=2 0 0 0 0 0 0m=3 1 1 [0 1] [0 1] [0 1] [0 1]

GW of capacity would only be added to the solar energyfacility in period 3

Apparently with the successful application of IFCI-PSP within a typical regional electric system managementproblem solutions obtained could provide useful decisionalternatives under different policies and various energy avail-abilities Compared with the least-cost model the IFCF-PSP model is an effective tool for providing environmentalmanagement schemes with dual objectives In addition theIFCF-PSP has following advantages over the conventionalprogramming methods (a) balancing multiple conflicting

objectives (b) reflecting interrelationships among systemefficiency economic cost and system reliability (c) effec-tively dealing with randomness in both the objective andconstraints and (d) assisting the analysis of diverse decisionschemes associated with various energy demand levels

5 Conclusions

An inexact fuzzy chance-constrained fractional program-ming approach is developed for optimal electric powersystems management under uncertainties In the developed

12 Mathematical Problems in Engineering

model fuzzy chance-constrained programming is incorpo-rated into a fractional programming optimization frame-work The obtained results are useful for supporting EPSmanagement The IFCF-PSP approach is capable of (i)balancing the conflict between two objectives (ii) reflect-ing different electricity generation and capacity expansionstrategies (iii) presenting optimal solutions under differentconstraint violating conditions and (iv) introducing theconcept of fuzzy boundary interval with which the com-plexity of dual uncertainties can be effectively handled Theresults of IFCF-PSP model show that (i) a higher confidencelevel corresponds to a higher proportion of clean energypower generation and a lower economic productivity and(ii) as a result of encouraging environment-friendly energiesthe generation capacities for the natural gas wind-powersolar energy and hydropower facilities would be signifi-cantly increased The solutions obtained from the IFCF-PSPapproach could provide specific energy options for powersystem planning and provide effective management solutionof the electric power system for identifying electricity gener-ation and capacity expansion schemes

This study attempts to develop a modeling framework forratio problems involving fuzzy uncertainties to deal with elec-tric power systems management problemThe results suggestthat it is also applicable to other energy management prob-lems In the future practice IFCF-PSP could be further im-proved through considering more impact factors Forinstance the fuzzymembership functions and the confidencelevel are critical in the decision-making process and thisrequires effective ways to provide appropriate choices fordecision making Such challenges desire further investiga-tions Future research can be aimed at applying the advancedapproach to a more complex real-world electric power sys-tem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the Fundamental ResearchFunds for the Central Universities NCEPU (2018BJ0293)the State Scholarship Fund (201706735025) the NationalKey Research and Development Plan (2016YFC0502800) theNatural Sciences Foundation (51520105013 51679087) andthe 111 Program (B14008) The data used to support thefindings of this study are available from the correspondingauthor upon request

References

[1] G H Huang ldquoA hybrid inexact-stochastic water managementmodelrdquo European Journal of Operational Research vol 107 no1 pp 137ndash158 1998

[2] A Grosfeld-Nir and A Tishler ldquoA stochastic model for themeasurement of electricity outage costsrdquo Energy pp 157ndash1741993

[3] S Wang G Huang and B W Baetz ldquoAn Inexact Probabilistic-Possibilistic Optimization Framework for Flood Managementin a Hybrid Uncertain Environmentrdquo IEEE Transactions onFuzzy Systems vol 23 no 4 pp 897ndash908 2015

[4] G H Huang and D P Loucks ldquoAn inexact two-stage stochasticprogramming model for water resources management underuncertaintyrdquo Civil Engineering and Environmental Systems vol17 no 2 pp 95ndash118 2000

[5] K Darby-Dowman S Barker E Audsley and D Parsons ldquoAtwo-stage stochastic programming with recourse model fordetermining robust planting plans in horticulturerdquo Journal ofthe Operational Research Society vol 51 no 1 pp 83ndash89 2000

[6] J Hu L Sun C H Li X Wang X L Jia and Y P Cai ldquoWaterQuality Risk Assessment for the Laoguanhe River of ChinaUsing a Stochastic SimulationMethodrdquo Journal of Environmen-tal Informatics vol 31 no 2 pp 123ndash136 2018

[7] SWang andG HHuang ldquoAmulti-level Taguchi-factorial two-stage stochastic programming approach for characterization ofparameter uncertainties and their interactions An applicationto water resources managementrdquo European Journal of Opera-tional Research vol 240 no 2 pp 572ndash581 2015

[8] S Chanas and P Zielinski ldquoOn the equivalence of two opti-mization methods for fuzzy linear programming problemsrdquoEuropean Journal of Operational Research vol 121 no 1 pp 56ndash63 2000

[9] J Mula D Peidro and R Poler ldquoThe effectiveness of a fuzzymathematical programming approach for supply chain pro-duction planning with fuzzy demandrdquo International Journal ofProduction Economics vol 128 no 1 pp 136ndash143 2010

[10] B Chen P Li H J Wu T Husain and F Khan ldquoMCFP Amonte carlo simulation-based fuzzy programming approachfor optimization under dual uncertainties of possibility andcontinuous probabilityrdquo Journal of Environmental Informatics(JEI) vol 29 no 2 pp 88ndash97 2017

[11] C Z Huang S Nie L Guo and Y R Fan ldquoInexact fuzzystochastic chance constraint programming for emergency evac-uation in Qinshan nuclear power plant under uncertaintyrdquoJournal of Environmental Informatics (JEI) vol 30 no 1 pp 63ndash78 2017

[12] U S Sakalli ldquoOptimization of Production-Distribution Prob-lem in Supply Chain Management under Stochastic and FuzzyUncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 4389064 29 pages 2017

[13] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval research logistics (NRL) pp 181ndash186 1962

[14] N I Voropai and E Y Ivanova ldquoMulti-criteria decision analysistechniques in electric power system expansion planningrdquo Inter-national Journal of Electrical Power amp Energy Systems vol 24no 1 pp 71ndash78 2002

[15] M Chakraborty and S Gupta ldquoFuzzy mathematical pro-gramming for multi objective linear fractional programmingproblemrdquo Fuzzy Sets and Systems vol 125 no 3 pp 335ndash3422002

[16] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[17] W Li Z BaoGHHuang andY L Xie ldquoAn Inexact CredibilityChance-Constrained Integer Programming for Greenhouse

Mathematical Problems in Engineering 13

GasMitigationManagement in Regional Electric Power Systemunder Uncertaintyrdquo in Journal of Environmental Informatics2017

[18] M Paunovic N M Ralevic V Gajovic et al ldquoTwo-StageFuzzy Logic Model for Cloud Service Supplier Selection andEvaluationrdquoMathematical Problems in Engineering 2018

[19] S Wang G H Huang B W Baetz and B C Ancell ldquoTowardsrobust quantification and reduction of uncertainty in hydro-logic predictions Integration of particle Markov chain MonteCarlo and factorial polynomial chaos expansionrdquo Journal ofHydrology vol 548 pp 484ndash497 2017

[20] SWang B C Ancell GHHuang andBW Baetz ldquoImprovingRobustness ofHydrologic Ensemble PredictionsThroughProb-abilistic Pre- and Post-Processing in Sequential Data Assimila-tionrdquoWater Resources Research 2018

[21] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[22] A Rong and R Lahdelma ldquoFuzzy chance constrained linearprogramming model for optimizing the scrap charge in steelproductionrdquo European Journal of Operational Research vol 186no 3 pp 953ndash964 2008

[23] X Sun J Li H Qiao and B Zhang ldquoEnergy implicationsof Chinarsquos regional development New insights from multi-regional input-output analysisrdquo Applied Energy vol 196 pp118ndash131 2017

[24] M A Quaddus and T N Goh ldquoElectric power generationexpansion Planning with multiple objectivesrdquo Applied Energyvol 19 no 4 pp 301ndash319 1985

[25] C H Antunes A G Martins and I S Brito ldquoA multipleobjective mixed integer linear programming model for powergeneration expansion planningrdquo Energy vol 29 no 4 pp 613ndash627 2004

[26] J L C Meza M B Yildirim and A S M Masud ldquoA modelfor the multiperiod multiobjective power generation expansionproblemrdquo IEEETransactions on Power Systems vol 22 no 2 pp871ndash878 2007

[27] J-HHan Y-C Ahn and I-B Lee ldquoAmulti-objective optimiza-tion model for sustainable electricity generation and CO2 mit-igation (EGCM) infrastructure design considering economicprofit and financial riskrdquo Applied Energy vol 95 pp 186ndash1952012

[28] M Rekik A Abdelkafi and L Krichen ldquoAmicro-grid ensuringmulti-objective control strategy of a power electrical system forquality improvementrdquo Energy vol 88 pp 351ndash363 2015

[29] A Azadeh Z Raoofi and M Zarrin ldquoA multi-objective fuzzylinear programming model for optimization of natural gassupply chain through a greenhouse gas reduction approachrdquoJournal of Natural Gas Science and Engineering vol 26 pp 702ndash710 2015

[30] K Li L Pan W Xue H Jiang and H Mao ldquoMulti-objectiveoptimization for energy performance improvement of residen-tial buildings a comparative studyrdquo Energies vol 10 no 2 p245 2017

[31] H Zhu W W Huang and G H Huang ldquoPlanning of regionalenergy systems An inexact mixed-integer fractional program-ming modelrdquo Applied Energy vol 113 pp 500ndash514 2014

[32] F Chen G H Huang Y R Fan and R F Liao ldquoA nonlinearfractional programming approach for environmentaleconomicpower dispatchrdquo International Journal of Electrical Power ampEnergy Systems vol 78 pp 463ndash469 2016

[33] C Zhang and P Guo ldquoFLFP A fuzzy linear fractional pro-gramming approach with double-sided fuzziness for optimalirrigation water allocationrdquo Agricultural Water Managementvol 199 pp 105ndash119 2018

[34] L Wang G Huang X Wang and H Zhu ldquoRisk-basedelectric power system planning for climate change mitigationthrough multi-stage joint-probabilistic left-hand-side chance-constrained fractional programming A Canadian case studyrdquoRenewable amp Sustainable Energy Reviews vol 82 pp 1056ndash10672018

[35] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[36] X H Nie G H Huang Y P Li and L Liu ldquoIFRP A hybridinterval-parameter fuzzy robust programming approach forwaste management planning under uncertaintyrdquo Journal ofEnvironmental Management vol 84 no 1 pp 1ndash11 2007

[37] Y Leung Spatial analysis and planning under imprecisionElsevier 1st edition 2013

[38] A Charnes W W Cooper and M J L Kirby ldquoChance-constrained programming an extension of statistical methodrdquoinOptimizingmethods in statistics pp 391ndash402 Academic Press1971

[39] Y P Cai G H Huang Z F Yang and Q Tan ldquoIdentificationof optimal strategies for energy management systems planningunder multiple uncertaintiesrdquoApplied Energy vol 86 no 4 pp480ndash495 2009

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Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

020406080

100120140160180200

Submodel f-Submodel f+

Submodel f-Submodel f+

Submodel f-Submodel f+

020406080

100120140160180200

020406080

100120140160180200

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3Coal Natural gas Wind power Solar energy Hydropower

ＪＣ=01

ＪＣ=005

ＪＣ=001

Pow

er g

ener

atio

n (103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)Po

wer

gen

erat

ion

(103

GW

h)

Figure 4 Power generation loads for EPS through the IFCF-PSPmodel

activities will reach their lower bound levels leading to thelowest power generation (1764 times 106GWh) Compared withAlternative A6 this alternative may be of less interest todecision-makers due to its lower system efficiency and highersystem cost However it is the solution that is obtainedunder the most stringent emission constraints (ie 119901119894 =001) Alternative A2 is a moderate solution in this group Itssystem cost system efficiency total power generation andclean energy power generation are [15295 16954] times 109$[22062445] GWh per $106 1816 times 106GWh and 0374 times106GWh

In middle-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A7gt A5gt A9 gtA1 In alternative A7middle-level electricity demands will be satisfied by a totalpower generation of 1893 times 106GWh In addition Alternative

A7 has the highest system efficiency ([2661 2941] GWhper 106$) and the largest amount of clean energy powergeneration (0478times 106GWh) in this groupThus AlternativeA7 is considered to be a desirable choice Alternatives A5 andA9 would lead to the same total power generation solution(1895 times 106GWh) They are the two-moderate solutions inthis groupThe system cost obtained from the A1 is the lowest([15718 17413] times 109$) in this group It would lead to thelowest system efficiency ([2148 2379] GWh per 106$) butdue to the largest proportion of fossil-fired power generationA1 has the maximum security for energy supply

In high-level total power generation group the com-parison of the system efficiency under different 119901119894 levelsshows that alternative A3gt A12gt A8 gtA4 The same cleanenergy power generation obtained from the 4 alternativesis 0478 106GWh In Alternative A4 all the power gener-ation activities will be equal to their upper-bound valuesat the same time It would lead to the highest system cost([16901 18676] times 109$) but the moderate system efficiency([2559 2828]GWhper $106) In comparisonAlternative A3would lead to the minimum system cost ([16478 18216] times109$) and the highest system efficiency ([2624 2901] GWhper $106) in this group In addition this alternative wouldalso provide the highest clean energy power generationTherefore this alternative is a desirable sustainable optionunder the high electricity demand level Alternatives A8 andA12 can provide modest system efficiencies which would be[2561 283] and [2575 2845] GWh per $106 respectivelyand lower system costs which would be [16892 18666] and[16800 18564] times 109$ Therefore decision-makers who payattention to the stability of the system may be interested inthese two alternatives

The above alternatives represent various options betweeneconomic and environmental tradeoffsWillingness to accepthigh system cost will guarantee meeting the objective ofincreasing the proportion of clean energy A strong desireto acquire low system cost will cause the risk of violatingemission constraints In general the above research resultswere favored by decision makers due to their flexibilityand preference for practical-making decision processes Thefeasible ranges for decision variables under different 119901119894 levelswere useful for decision makers to justify the generatedalternatives directly

Besides the scenario of maximizing the proportion ofclean energy another scenario of minimizing the system costis also analyzed to evaluate the effects of different energy sup-ply policies The optimal-ratio problem presented in Models(10a)ndash(10k) can be converted into a least-cost problem withthe following objective

min119891 = system cost

= 119879sum119905=1

119868sum119895=1

119862119875119864plusmn119905119895 times 119860119875119864plusmn119905119895 + 119879sum119905=1

119869sum119895=1

119862119875119866plusmn119905119895 times 119860119875119866plusmn119905119895+ 119879sum119905=1

119869sum119895=1

119872sum119898=1

119862119864119875plusmn119905119895 times 119864119862119860plusmn119905119895119898 times 119884119905119895119898

Mathematical Problems in Engineering 9

Table 4 Typical decision alternatives obtained from the IFCF-PSP model solutions

Alternative 119860119875119864plusmn119905119895 119860119875119866

plusmn119905119895119884plusmn119905119895119898

(119895=1 forall119905119898)

119860119875119866plusmn119905119895 119884plusmn119905119895119898

(119895=2345 forall119905119898)

System efficiency(GWh per 106$) System cost (109$) Total power generation

(106GWh)Clean energy powergeneration (106GWh)119901119894 = 01

A1 + - [21482379] (12) [1571817413] (4) 1895 (6) 0374 (11)A2 - - [22062445] (9) [1529516954] (3) 1816 (10) 0374 (11)A3 - + [26242901] (3) [1647818216] (9) 192 (4) 0478 (1)A4 + + [25592828] (6) [16901 18676] (12) 2 (1) 0478 (1)119901119894 = 005A5 + - [22362477] (8) [1578317487] (5) 1912 (5) 0391 (7)A6 - - [23292582] (7) [1514216785] (1) 1807 (11) 0391 (7)A7 - + [26612941] (2) [1625117963] (8) 1893 (8) 0478 (1)A8 + + [2561283] (5) [1689218666] (11) 1999 (2) 0478 (1)119901119894 = 001A9 + - [2174 2406] (11) [16040 17757] (7) 1895 (6) 0386 (9)A10 - - [22972543] (8) [15176 16806] (2) 1764 (12) 0386 (9)A11 - + [27142999] (1) [1593617612] (6) 1856 (9) 0478 (1)A12 + + [25752845] (4) [16800 18564] (10) 1987 (3) 0478 (1)

Table 5 The proportion of clean energy power generation from IFCF-PSP and LS models

Clean power generation ratio () IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3119901119894 = 01 [1802 2432] [2089 2493] [1986 2540] [1623 1870] [1681 1719] [1515 1638]119901119894 = 005 [2067 2465] [2093 2528] [2010 2578] [1624 1870] [1681 1719] [1632 1869]119901119894 = 001 [2155 2530] [2158 2596] [2141 2596] [1648 1893] [1679 1752] [1997 2224]

+ 119879sum119905=1

119862119868119864plusmn119905 times 119860119868119864plusmn119905+ 119879sum119905=1

119868sum119895=1

119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905(11)

Figure 5 shows the proportion of different power gen-eration technologies from the IFCF-PSP and LS modelsunder 119901119894 = 001 over three planning periods The IFCF-PSPmodel leads to a relatively higher percentage of natural gaspower generation in comparison the LS model leads torelatively higher percentages of coal-fired electricity suppliesAccording to the solutions from IFCF-PSPmodel the percentof electricity generated by natural gas facilities would be [5 6] [7 8] and [5 8] in periods 1 2 and 3 respectivelyTheLS model achieves the slightly lower percent (ie 0 0 and[5 6] in periods 1 2 and 3) Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted

According to Table 5 the differences can be foundbetween the results of two models under different 119901119894 levelsThe proportion of clean energy power generation fromIFCF-PSP model is higher than that of the LS model Forexample under 119901119894 = 001 clean energy power generationof the entire region occupied [2155 2530] [2158 2596]and [2141 2596] of the total electricity generation in

the three study periods from IFCF-PSP model which arehigher than the LS model [1648 1993] [1679 1752] and[1997 2224] respectively Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpretedAn increased 119901119894 level represents a higher admissible riskleading to a decreased strictness for the emission constraintsand hence an expanded decision space Therefore undertypical conditions as the risk level becomes lower theproportion of clean energy power generation or renewablepower generation would increase For example in period 2when the 119901119894 level is dropped from 01 to 001 the proportionof clean energy power generation from IFCF-PSP modelwould be increased from [2089 2493] to [2158 2596]Likewise the results under other periods can be similarlyanalyzed The confidence level of constraints satisfaction ismore reliable because the risk level is lowerTherefore in thiscase the decision-makers will be more conservative in theEPS management There is no capacity-expansion would beconducted for the coal-fired facility in three periods sinceits high carbon dioxide emissions and increasingly stringentemission limits On the other hand clean energy electricitysupply would be insufficient for the future energy demandsAccording to Table 6 when 119901119894 takes different values (ie 119901119894 =001119901119894 = 005 and 119901119894 = 01) differences of capacity-expansionoption can be found between the results of two models Inboth models when the risk of violating the constraints ofcarbon emission target is decreased (ie the value of 119901119894-level

10 Mathematical Problems in Engineering

75

79

5 4

74

8

95 4

74

8

9

5 4

81

095 5

83

09

4 4

78

7

7

4 4

coalnatural gashydro

windsolar

83

010

2

5

83

0

94 4

80

5

84 3

82

19 4

4

78

6

84 4

79

5

84 4

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

IFCFP-PSP f- IFCFP-PSP +

LS +LS -

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

Figure 5 Comparison of power-generation patterns from IFCF-PSP and least-system-cost (LS) models under 119901119894-level 001 Note that theinnermost circle in the pie chart represents the first planning period

is larger) the demand for expansion of clean energy willincrease On the other hand the total demand of capacityexpansion in the lower bound model is lower than the upperbound model One of the reasons for this result is that theannual utilization hours of various power generation facilitiesin the lower bound model are higher than the correspondingparameter settings of the upper bound model

The result of LS model has shown that under thecorresponding lower bound parameter settings the capacityof 015 GW would be added to the hydropower facility inperiod 3 when 119901119894 = 001 and 119901119894 = 005 Only a capacity of015 GWwould be added to the wind power facility in period3 While under the corresponding upper bound parametersettings natural gas and solar energy facilities expansionsare not required that is because the former resource is underpenalty of CO2 emission and the capacity cost of the solarpower expansion is much higher

In the IFCF-PSP model clean energy will be given moredevelopment opportunities Take 119901119894 = 001 as an example(corresponding to 119891minus) with a capacity expansion of 02

GW at the beginning of period 1 and period 2 electricitygeneration ofwind power facilitywould rise from 164 103GWin period 1 to 168 103GW in period 2 and with anothercapacity expansion of 02 GW at the beginning of period 3 itstotal electricity generation capacity would reach 172 103GWLikewise to meet the growing energy demand the capacityof 02 GW would be added to the natural gas-fired facility inperiod 1 period 2 and period 3 as a result its total generationcapacity would rise from 954 times 103GW in period 1 to 119 times103GW in period 2 and reach 124 times 103GW in period 3 Inthe planning period 1 due to more lenient emission policiesand more economical cost strategies some of the natural gaspower generation facilities are idleThe capacities of 025 GWwould be added to the hydropower facility in three periodsand the corresponding power generation capacity would be812 times 103GW 875 103GW and 937 times 103GW Generatingelectricity from solar energy facility is more expensive thanother power generation facilities thus when the emissionlimits and electricity demand are relatively low solar energyfacility would not be expanded during first two periods 015

Mathematical Problems in Engineering 11

Table 6 Binary solutions of capacity expansions

119901119894-level Power-generation facility Capacity-expansion option IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3

01 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 [0 1] [0 1] 0 0 0

Wind power m=1 0 0 0 0 0 0m=2 0 [0 1] 0 0 [0 1] 1m=3 [0 1] [0 1] [0 1] 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 [0 1] [0 1] 0 0 0 [0 1]m=2 0 0 0 0 0 0m=3 [0 1] [0 1] 0 [0 1] [0 1] 0

005 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 [0 1] [0 1] [0 1] 0 0 0

Wind power m=1 [0 1] [0 1] 0 0 0 0m=2 0 0 0 0 [0 1] 1m=3 [0 1] [0 1] 0 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] [0 1]m=2 0 0 0 0 0 0m=3 1 1 0 [0 1] [0 1] 0

001 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 1 1 [0 1] [0 1] [0 1]

Wind power m=1 0 0 0 0 0 0m=2 0 0 0 0 0 [0 1]m=3 1 1 [0 1] [0 1] [0 1] [0 1]

Solar energy m=1 0 0 0 0 0 0m=2 0 0 [0 1] 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] 0m=2 0 0 0 0 0 0m=3 1 1 [0 1] [0 1] [0 1] [0 1]

GW of capacity would only be added to the solar energyfacility in period 3

Apparently with the successful application of IFCI-PSP within a typical regional electric system managementproblem solutions obtained could provide useful decisionalternatives under different policies and various energy avail-abilities Compared with the least-cost model the IFCF-PSP model is an effective tool for providing environmentalmanagement schemes with dual objectives In addition theIFCF-PSP has following advantages over the conventionalprogramming methods (a) balancing multiple conflicting

objectives (b) reflecting interrelationships among systemefficiency economic cost and system reliability (c) effec-tively dealing with randomness in both the objective andconstraints and (d) assisting the analysis of diverse decisionschemes associated with various energy demand levels

5 Conclusions

An inexact fuzzy chance-constrained fractional program-ming approach is developed for optimal electric powersystems management under uncertainties In the developed

12 Mathematical Problems in Engineering

model fuzzy chance-constrained programming is incorpo-rated into a fractional programming optimization frame-work The obtained results are useful for supporting EPSmanagement The IFCF-PSP approach is capable of (i)balancing the conflict between two objectives (ii) reflect-ing different electricity generation and capacity expansionstrategies (iii) presenting optimal solutions under differentconstraint violating conditions and (iv) introducing theconcept of fuzzy boundary interval with which the com-plexity of dual uncertainties can be effectively handled Theresults of IFCF-PSP model show that (i) a higher confidencelevel corresponds to a higher proportion of clean energypower generation and a lower economic productivity and(ii) as a result of encouraging environment-friendly energiesthe generation capacities for the natural gas wind-powersolar energy and hydropower facilities would be signifi-cantly increased The solutions obtained from the IFCF-PSPapproach could provide specific energy options for powersystem planning and provide effective management solutionof the electric power system for identifying electricity gener-ation and capacity expansion schemes

This study attempts to develop a modeling framework forratio problems involving fuzzy uncertainties to deal with elec-tric power systems management problemThe results suggestthat it is also applicable to other energy management prob-lems In the future practice IFCF-PSP could be further im-proved through considering more impact factors Forinstance the fuzzymembership functions and the confidencelevel are critical in the decision-making process and thisrequires effective ways to provide appropriate choices fordecision making Such challenges desire further investiga-tions Future research can be aimed at applying the advancedapproach to a more complex real-world electric power sys-tem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the Fundamental ResearchFunds for the Central Universities NCEPU (2018BJ0293)the State Scholarship Fund (201706735025) the NationalKey Research and Development Plan (2016YFC0502800) theNatural Sciences Foundation (51520105013 51679087) andthe 111 Program (B14008) The data used to support thefindings of this study are available from the correspondingauthor upon request

References

[1] G H Huang ldquoA hybrid inexact-stochastic water managementmodelrdquo European Journal of Operational Research vol 107 no1 pp 137ndash158 1998

[2] A Grosfeld-Nir and A Tishler ldquoA stochastic model for themeasurement of electricity outage costsrdquo Energy pp 157ndash1741993

[3] S Wang G Huang and B W Baetz ldquoAn Inexact Probabilistic-Possibilistic Optimization Framework for Flood Managementin a Hybrid Uncertain Environmentrdquo IEEE Transactions onFuzzy Systems vol 23 no 4 pp 897ndash908 2015

[4] G H Huang and D P Loucks ldquoAn inexact two-stage stochasticprogramming model for water resources management underuncertaintyrdquo Civil Engineering and Environmental Systems vol17 no 2 pp 95ndash118 2000

[5] K Darby-Dowman S Barker E Audsley and D Parsons ldquoAtwo-stage stochastic programming with recourse model fordetermining robust planting plans in horticulturerdquo Journal ofthe Operational Research Society vol 51 no 1 pp 83ndash89 2000

[6] J Hu L Sun C H Li X Wang X L Jia and Y P Cai ldquoWaterQuality Risk Assessment for the Laoguanhe River of ChinaUsing a Stochastic SimulationMethodrdquo Journal of Environmen-tal Informatics vol 31 no 2 pp 123ndash136 2018

[7] SWang andG HHuang ldquoAmulti-level Taguchi-factorial two-stage stochastic programming approach for characterization ofparameter uncertainties and their interactions An applicationto water resources managementrdquo European Journal of Opera-tional Research vol 240 no 2 pp 572ndash581 2015

[8] S Chanas and P Zielinski ldquoOn the equivalence of two opti-mization methods for fuzzy linear programming problemsrdquoEuropean Journal of Operational Research vol 121 no 1 pp 56ndash63 2000

[9] J Mula D Peidro and R Poler ldquoThe effectiveness of a fuzzymathematical programming approach for supply chain pro-duction planning with fuzzy demandrdquo International Journal ofProduction Economics vol 128 no 1 pp 136ndash143 2010

[10] B Chen P Li H J Wu T Husain and F Khan ldquoMCFP Amonte carlo simulation-based fuzzy programming approachfor optimization under dual uncertainties of possibility andcontinuous probabilityrdquo Journal of Environmental Informatics(JEI) vol 29 no 2 pp 88ndash97 2017

[11] C Z Huang S Nie L Guo and Y R Fan ldquoInexact fuzzystochastic chance constraint programming for emergency evac-uation in Qinshan nuclear power plant under uncertaintyrdquoJournal of Environmental Informatics (JEI) vol 30 no 1 pp 63ndash78 2017

[12] U S Sakalli ldquoOptimization of Production-Distribution Prob-lem in Supply Chain Management under Stochastic and FuzzyUncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 4389064 29 pages 2017

[13] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval research logistics (NRL) pp 181ndash186 1962

[14] N I Voropai and E Y Ivanova ldquoMulti-criteria decision analysistechniques in electric power system expansion planningrdquo Inter-national Journal of Electrical Power amp Energy Systems vol 24no 1 pp 71ndash78 2002

[15] M Chakraborty and S Gupta ldquoFuzzy mathematical pro-gramming for multi objective linear fractional programmingproblemrdquo Fuzzy Sets and Systems vol 125 no 3 pp 335ndash3422002

[16] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[17] W Li Z BaoGHHuang andY L Xie ldquoAn Inexact CredibilityChance-Constrained Integer Programming for Greenhouse

Mathematical Problems in Engineering 13

GasMitigationManagement in Regional Electric Power Systemunder Uncertaintyrdquo in Journal of Environmental Informatics2017

[18] M Paunovic N M Ralevic V Gajovic et al ldquoTwo-StageFuzzy Logic Model for Cloud Service Supplier Selection andEvaluationrdquoMathematical Problems in Engineering 2018

[19] S Wang G H Huang B W Baetz and B C Ancell ldquoTowardsrobust quantification and reduction of uncertainty in hydro-logic predictions Integration of particle Markov chain MonteCarlo and factorial polynomial chaos expansionrdquo Journal ofHydrology vol 548 pp 484ndash497 2017

[20] SWang B C Ancell GHHuang andBW Baetz ldquoImprovingRobustness ofHydrologic Ensemble PredictionsThroughProb-abilistic Pre- and Post-Processing in Sequential Data Assimila-tionrdquoWater Resources Research 2018

[21] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[22] A Rong and R Lahdelma ldquoFuzzy chance constrained linearprogramming model for optimizing the scrap charge in steelproductionrdquo European Journal of Operational Research vol 186no 3 pp 953ndash964 2008

[23] X Sun J Li H Qiao and B Zhang ldquoEnergy implicationsof Chinarsquos regional development New insights from multi-regional input-output analysisrdquo Applied Energy vol 196 pp118ndash131 2017

[24] M A Quaddus and T N Goh ldquoElectric power generationexpansion Planning with multiple objectivesrdquo Applied Energyvol 19 no 4 pp 301ndash319 1985

[25] C H Antunes A G Martins and I S Brito ldquoA multipleobjective mixed integer linear programming model for powergeneration expansion planningrdquo Energy vol 29 no 4 pp 613ndash627 2004

[26] J L C Meza M B Yildirim and A S M Masud ldquoA modelfor the multiperiod multiobjective power generation expansionproblemrdquo IEEETransactions on Power Systems vol 22 no 2 pp871ndash878 2007

[27] J-HHan Y-C Ahn and I-B Lee ldquoAmulti-objective optimiza-tion model for sustainable electricity generation and CO2 mit-igation (EGCM) infrastructure design considering economicprofit and financial riskrdquo Applied Energy vol 95 pp 186ndash1952012

[28] M Rekik A Abdelkafi and L Krichen ldquoAmicro-grid ensuringmulti-objective control strategy of a power electrical system forquality improvementrdquo Energy vol 88 pp 351ndash363 2015

[29] A Azadeh Z Raoofi and M Zarrin ldquoA multi-objective fuzzylinear programming model for optimization of natural gassupply chain through a greenhouse gas reduction approachrdquoJournal of Natural Gas Science and Engineering vol 26 pp 702ndash710 2015

[30] K Li L Pan W Xue H Jiang and H Mao ldquoMulti-objectiveoptimization for energy performance improvement of residen-tial buildings a comparative studyrdquo Energies vol 10 no 2 p245 2017

[31] H Zhu W W Huang and G H Huang ldquoPlanning of regionalenergy systems An inexact mixed-integer fractional program-ming modelrdquo Applied Energy vol 113 pp 500ndash514 2014

[32] F Chen G H Huang Y R Fan and R F Liao ldquoA nonlinearfractional programming approach for environmentaleconomicpower dispatchrdquo International Journal of Electrical Power ampEnergy Systems vol 78 pp 463ndash469 2016

[33] C Zhang and P Guo ldquoFLFP A fuzzy linear fractional pro-gramming approach with double-sided fuzziness for optimalirrigation water allocationrdquo Agricultural Water Managementvol 199 pp 105ndash119 2018

[34] L Wang G Huang X Wang and H Zhu ldquoRisk-basedelectric power system planning for climate change mitigationthrough multi-stage joint-probabilistic left-hand-side chance-constrained fractional programming A Canadian case studyrdquoRenewable amp Sustainable Energy Reviews vol 82 pp 1056ndash10672018

[35] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[36] X H Nie G H Huang Y P Li and L Liu ldquoIFRP A hybridinterval-parameter fuzzy robust programming approach forwaste management planning under uncertaintyrdquo Journal ofEnvironmental Management vol 84 no 1 pp 1ndash11 2007

[37] Y Leung Spatial analysis and planning under imprecisionElsevier 1st edition 2013

[38] A Charnes W W Cooper and M J L Kirby ldquoChance-constrained programming an extension of statistical methodrdquoinOptimizingmethods in statistics pp 391ndash402 Academic Press1971

[39] Y P Cai G H Huang Z F Yang and Q Tan ldquoIdentificationof optimal strategies for energy management systems planningunder multiple uncertaintiesrdquoApplied Energy vol 86 no 4 pp480ndash495 2009

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Applied MathematicsJournal of

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Mathematical Problems in Engineering 9

Table 4 Typical decision alternatives obtained from the IFCF-PSP model solutions

Alternative 119860119875119864plusmn119905119895 119860119875119866

plusmn119905119895119884plusmn119905119895119898

(119895=1 forall119905119898)

119860119875119866plusmn119905119895 119884plusmn119905119895119898

(119895=2345 forall119905119898)

System efficiency(GWh per 106$) System cost (109$) Total power generation

(106GWh)Clean energy powergeneration (106GWh)119901119894 = 01

A1 + - [21482379] (12) [1571817413] (4) 1895 (6) 0374 (11)A2 - - [22062445] (9) [1529516954] (3) 1816 (10) 0374 (11)A3 - + [26242901] (3) [1647818216] (9) 192 (4) 0478 (1)A4 + + [25592828] (6) [16901 18676] (12) 2 (1) 0478 (1)119901119894 = 005A5 + - [22362477] (8) [1578317487] (5) 1912 (5) 0391 (7)A6 - - [23292582] (7) [1514216785] (1) 1807 (11) 0391 (7)A7 - + [26612941] (2) [1625117963] (8) 1893 (8) 0478 (1)A8 + + [2561283] (5) [1689218666] (11) 1999 (2) 0478 (1)119901119894 = 001A9 + - [2174 2406] (11) [16040 17757] (7) 1895 (6) 0386 (9)A10 - - [22972543] (8) [15176 16806] (2) 1764 (12) 0386 (9)A11 - + [27142999] (1) [1593617612] (6) 1856 (9) 0478 (1)A12 + + [25752845] (4) [16800 18564] (10) 1987 (3) 0478 (1)

Table 5 The proportion of clean energy power generation from IFCF-PSP and LS models

Clean power generation ratio () IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3119901119894 = 01 [1802 2432] [2089 2493] [1986 2540] [1623 1870] [1681 1719] [1515 1638]119901119894 = 005 [2067 2465] [2093 2528] [2010 2578] [1624 1870] [1681 1719] [1632 1869]119901119894 = 001 [2155 2530] [2158 2596] [2141 2596] [1648 1893] [1679 1752] [1997 2224]

+ 119879sum119905=1

119862119868119864plusmn119905 times 119860119868119864plusmn119905+ 119879sum119905=1

119868sum119895=1

119862119875119872plusmn119905119895 times 119860119875119866plusmn119905119895 times 120578plusmn119905(11)

Figure 5 shows the proportion of different power gen-eration technologies from the IFCF-PSP and LS modelsunder 119901119894 = 001 over three planning periods The IFCF-PSPmodel leads to a relatively higher percentage of natural gaspower generation in comparison the LS model leads torelatively higher percentages of coal-fired electricity suppliesAccording to the solutions from IFCF-PSPmodel the percentof electricity generated by natural gas facilities would be [5 6] [7 8] and [5 8] in periods 1 2 and 3 respectivelyTheLS model achieves the slightly lower percent (ie 0 0 and[5 6] in periods 1 2 and 3) Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpreted

According to Table 5 the differences can be foundbetween the results of two models under different 119901119894 levelsThe proportion of clean energy power generation fromIFCF-PSP model is higher than that of the LS model Forexample under 119901119894 = 001 clean energy power generationof the entire region occupied [2155 2530] [2158 2596]and [2141 2596] of the total electricity generation in

the three study periods from IFCF-PSP model which arehigher than the LS model [1648 1993] [1679 1752] and[1997 2224] respectively Similarly the results underother 119901119894 levels (119901119894 = 005 and 119901119894 = 01) can be interpretedAn increased 119901119894 level represents a higher admissible riskleading to a decreased strictness for the emission constraintsand hence an expanded decision space Therefore undertypical conditions as the risk level becomes lower theproportion of clean energy power generation or renewablepower generation would increase For example in period 2when the 119901119894 level is dropped from 01 to 001 the proportionof clean energy power generation from IFCF-PSP modelwould be increased from [2089 2493] to [2158 2596]Likewise the results under other periods can be similarlyanalyzed The confidence level of constraints satisfaction ismore reliable because the risk level is lowerTherefore in thiscase the decision-makers will be more conservative in theEPS management There is no capacity-expansion would beconducted for the coal-fired facility in three periods sinceits high carbon dioxide emissions and increasingly stringentemission limits On the other hand clean energy electricitysupply would be insufficient for the future energy demandsAccording to Table 6 when 119901119894 takes different values (ie 119901119894 =001119901119894 = 005 and 119901119894 = 01) differences of capacity-expansionoption can be found between the results of two models Inboth models when the risk of violating the constraints ofcarbon emission target is decreased (ie the value of 119901119894-level

10 Mathematical Problems in Engineering

75

79

5 4

74

8

95 4

74

8

9

5 4

81

095 5

83

09

4 4

78

7

7

4 4

coalnatural gashydro

windsolar

83

010

2

5

83

0

94 4

80

5

84 3

82

19 4

4

78

6

84 4

79

5

84 4

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

IFCFP-PSP f- IFCFP-PSP +

LS +LS -

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

Figure 5 Comparison of power-generation patterns from IFCF-PSP and least-system-cost (LS) models under 119901119894-level 001 Note that theinnermost circle in the pie chart represents the first planning period

is larger) the demand for expansion of clean energy willincrease On the other hand the total demand of capacityexpansion in the lower bound model is lower than the upperbound model One of the reasons for this result is that theannual utilization hours of various power generation facilitiesin the lower bound model are higher than the correspondingparameter settings of the upper bound model

The result of LS model has shown that under thecorresponding lower bound parameter settings the capacityof 015 GW would be added to the hydropower facility inperiod 3 when 119901119894 = 001 and 119901119894 = 005 Only a capacity of015 GWwould be added to the wind power facility in period3 While under the corresponding upper bound parametersettings natural gas and solar energy facilities expansionsare not required that is because the former resource is underpenalty of CO2 emission and the capacity cost of the solarpower expansion is much higher

In the IFCF-PSP model clean energy will be given moredevelopment opportunities Take 119901119894 = 001 as an example(corresponding to 119891minus) with a capacity expansion of 02

GW at the beginning of period 1 and period 2 electricitygeneration ofwind power facilitywould rise from 164 103GWin period 1 to 168 103GW in period 2 and with anothercapacity expansion of 02 GW at the beginning of period 3 itstotal electricity generation capacity would reach 172 103GWLikewise to meet the growing energy demand the capacityof 02 GW would be added to the natural gas-fired facility inperiod 1 period 2 and period 3 as a result its total generationcapacity would rise from 954 times 103GW in period 1 to 119 times103GW in period 2 and reach 124 times 103GW in period 3 Inthe planning period 1 due to more lenient emission policiesand more economical cost strategies some of the natural gaspower generation facilities are idleThe capacities of 025 GWwould be added to the hydropower facility in three periodsand the corresponding power generation capacity would be812 times 103GW 875 103GW and 937 times 103GW Generatingelectricity from solar energy facility is more expensive thanother power generation facilities thus when the emissionlimits and electricity demand are relatively low solar energyfacility would not be expanded during first two periods 015

Mathematical Problems in Engineering 11

Table 6 Binary solutions of capacity expansions

119901119894-level Power-generation facility Capacity-expansion option IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3

01 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 [0 1] [0 1] 0 0 0

Wind power m=1 0 0 0 0 0 0m=2 0 [0 1] 0 0 [0 1] 1m=3 [0 1] [0 1] [0 1] 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 [0 1] [0 1] 0 0 0 [0 1]m=2 0 0 0 0 0 0m=3 [0 1] [0 1] 0 [0 1] [0 1] 0

005 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 [0 1] [0 1] [0 1] 0 0 0

Wind power m=1 [0 1] [0 1] 0 0 0 0m=2 0 0 0 0 [0 1] 1m=3 [0 1] [0 1] 0 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] [0 1]m=2 0 0 0 0 0 0m=3 1 1 0 [0 1] [0 1] 0

001 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 1 1 [0 1] [0 1] [0 1]

Wind power m=1 0 0 0 0 0 0m=2 0 0 0 0 0 [0 1]m=3 1 1 [0 1] [0 1] [0 1] [0 1]

Solar energy m=1 0 0 0 0 0 0m=2 0 0 [0 1] 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] 0m=2 0 0 0 0 0 0m=3 1 1 [0 1] [0 1] [0 1] [0 1]

GW of capacity would only be added to the solar energyfacility in period 3

Apparently with the successful application of IFCI-PSP within a typical regional electric system managementproblem solutions obtained could provide useful decisionalternatives under different policies and various energy avail-abilities Compared with the least-cost model the IFCF-PSP model is an effective tool for providing environmentalmanagement schemes with dual objectives In addition theIFCF-PSP has following advantages over the conventionalprogramming methods (a) balancing multiple conflicting

objectives (b) reflecting interrelationships among systemefficiency economic cost and system reliability (c) effec-tively dealing with randomness in both the objective andconstraints and (d) assisting the analysis of diverse decisionschemes associated with various energy demand levels

5 Conclusions

An inexact fuzzy chance-constrained fractional program-ming approach is developed for optimal electric powersystems management under uncertainties In the developed

12 Mathematical Problems in Engineering

model fuzzy chance-constrained programming is incorpo-rated into a fractional programming optimization frame-work The obtained results are useful for supporting EPSmanagement The IFCF-PSP approach is capable of (i)balancing the conflict between two objectives (ii) reflect-ing different electricity generation and capacity expansionstrategies (iii) presenting optimal solutions under differentconstraint violating conditions and (iv) introducing theconcept of fuzzy boundary interval with which the com-plexity of dual uncertainties can be effectively handled Theresults of IFCF-PSP model show that (i) a higher confidencelevel corresponds to a higher proportion of clean energypower generation and a lower economic productivity and(ii) as a result of encouraging environment-friendly energiesthe generation capacities for the natural gas wind-powersolar energy and hydropower facilities would be signifi-cantly increased The solutions obtained from the IFCF-PSPapproach could provide specific energy options for powersystem planning and provide effective management solutionof the electric power system for identifying electricity gener-ation and capacity expansion schemes

This study attempts to develop a modeling framework forratio problems involving fuzzy uncertainties to deal with elec-tric power systems management problemThe results suggestthat it is also applicable to other energy management prob-lems In the future practice IFCF-PSP could be further im-proved through considering more impact factors Forinstance the fuzzymembership functions and the confidencelevel are critical in the decision-making process and thisrequires effective ways to provide appropriate choices fordecision making Such challenges desire further investiga-tions Future research can be aimed at applying the advancedapproach to a more complex real-world electric power sys-tem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the Fundamental ResearchFunds for the Central Universities NCEPU (2018BJ0293)the State Scholarship Fund (201706735025) the NationalKey Research and Development Plan (2016YFC0502800) theNatural Sciences Foundation (51520105013 51679087) andthe 111 Program (B14008) The data used to support thefindings of this study are available from the correspondingauthor upon request

References

[1] G H Huang ldquoA hybrid inexact-stochastic water managementmodelrdquo European Journal of Operational Research vol 107 no1 pp 137ndash158 1998

[2] A Grosfeld-Nir and A Tishler ldquoA stochastic model for themeasurement of electricity outage costsrdquo Energy pp 157ndash1741993

[3] S Wang G Huang and B W Baetz ldquoAn Inexact Probabilistic-Possibilistic Optimization Framework for Flood Managementin a Hybrid Uncertain Environmentrdquo IEEE Transactions onFuzzy Systems vol 23 no 4 pp 897ndash908 2015

[4] G H Huang and D P Loucks ldquoAn inexact two-stage stochasticprogramming model for water resources management underuncertaintyrdquo Civil Engineering and Environmental Systems vol17 no 2 pp 95ndash118 2000

[5] K Darby-Dowman S Barker E Audsley and D Parsons ldquoAtwo-stage stochastic programming with recourse model fordetermining robust planting plans in horticulturerdquo Journal ofthe Operational Research Society vol 51 no 1 pp 83ndash89 2000

[6] J Hu L Sun C H Li X Wang X L Jia and Y P Cai ldquoWaterQuality Risk Assessment for the Laoguanhe River of ChinaUsing a Stochastic SimulationMethodrdquo Journal of Environmen-tal Informatics vol 31 no 2 pp 123ndash136 2018

[7] SWang andG HHuang ldquoAmulti-level Taguchi-factorial two-stage stochastic programming approach for characterization ofparameter uncertainties and their interactions An applicationto water resources managementrdquo European Journal of Opera-tional Research vol 240 no 2 pp 572ndash581 2015

[8] S Chanas and P Zielinski ldquoOn the equivalence of two opti-mization methods for fuzzy linear programming problemsrdquoEuropean Journal of Operational Research vol 121 no 1 pp 56ndash63 2000

[9] J Mula D Peidro and R Poler ldquoThe effectiveness of a fuzzymathematical programming approach for supply chain pro-duction planning with fuzzy demandrdquo International Journal ofProduction Economics vol 128 no 1 pp 136ndash143 2010

[10] B Chen P Li H J Wu T Husain and F Khan ldquoMCFP Amonte carlo simulation-based fuzzy programming approachfor optimization under dual uncertainties of possibility andcontinuous probabilityrdquo Journal of Environmental Informatics(JEI) vol 29 no 2 pp 88ndash97 2017

[11] C Z Huang S Nie L Guo and Y R Fan ldquoInexact fuzzystochastic chance constraint programming for emergency evac-uation in Qinshan nuclear power plant under uncertaintyrdquoJournal of Environmental Informatics (JEI) vol 30 no 1 pp 63ndash78 2017

[12] U S Sakalli ldquoOptimization of Production-Distribution Prob-lem in Supply Chain Management under Stochastic and FuzzyUncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 4389064 29 pages 2017

[13] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval research logistics (NRL) pp 181ndash186 1962

[14] N I Voropai and E Y Ivanova ldquoMulti-criteria decision analysistechniques in electric power system expansion planningrdquo Inter-national Journal of Electrical Power amp Energy Systems vol 24no 1 pp 71ndash78 2002

[15] M Chakraborty and S Gupta ldquoFuzzy mathematical pro-gramming for multi objective linear fractional programmingproblemrdquo Fuzzy Sets and Systems vol 125 no 3 pp 335ndash3422002

[16] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[17] W Li Z BaoGHHuang andY L Xie ldquoAn Inexact CredibilityChance-Constrained Integer Programming for Greenhouse

Mathematical Problems in Engineering 13

GasMitigationManagement in Regional Electric Power Systemunder Uncertaintyrdquo in Journal of Environmental Informatics2017

[18] M Paunovic N M Ralevic V Gajovic et al ldquoTwo-StageFuzzy Logic Model for Cloud Service Supplier Selection andEvaluationrdquoMathematical Problems in Engineering 2018

[19] S Wang G H Huang B W Baetz and B C Ancell ldquoTowardsrobust quantification and reduction of uncertainty in hydro-logic predictions Integration of particle Markov chain MonteCarlo and factorial polynomial chaos expansionrdquo Journal ofHydrology vol 548 pp 484ndash497 2017

[20] SWang B C Ancell GHHuang andBW Baetz ldquoImprovingRobustness ofHydrologic Ensemble PredictionsThroughProb-abilistic Pre- and Post-Processing in Sequential Data Assimila-tionrdquoWater Resources Research 2018

[21] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[22] A Rong and R Lahdelma ldquoFuzzy chance constrained linearprogramming model for optimizing the scrap charge in steelproductionrdquo European Journal of Operational Research vol 186no 3 pp 953ndash964 2008

[23] X Sun J Li H Qiao and B Zhang ldquoEnergy implicationsof Chinarsquos regional development New insights from multi-regional input-output analysisrdquo Applied Energy vol 196 pp118ndash131 2017

[24] M A Quaddus and T N Goh ldquoElectric power generationexpansion Planning with multiple objectivesrdquo Applied Energyvol 19 no 4 pp 301ndash319 1985

[25] C H Antunes A G Martins and I S Brito ldquoA multipleobjective mixed integer linear programming model for powergeneration expansion planningrdquo Energy vol 29 no 4 pp 613ndash627 2004

[26] J L C Meza M B Yildirim and A S M Masud ldquoA modelfor the multiperiod multiobjective power generation expansionproblemrdquo IEEETransactions on Power Systems vol 22 no 2 pp871ndash878 2007

[27] J-HHan Y-C Ahn and I-B Lee ldquoAmulti-objective optimiza-tion model for sustainable electricity generation and CO2 mit-igation (EGCM) infrastructure design considering economicprofit and financial riskrdquo Applied Energy vol 95 pp 186ndash1952012

[28] M Rekik A Abdelkafi and L Krichen ldquoAmicro-grid ensuringmulti-objective control strategy of a power electrical system forquality improvementrdquo Energy vol 88 pp 351ndash363 2015

[29] A Azadeh Z Raoofi and M Zarrin ldquoA multi-objective fuzzylinear programming model for optimization of natural gassupply chain through a greenhouse gas reduction approachrdquoJournal of Natural Gas Science and Engineering vol 26 pp 702ndash710 2015

[30] K Li L Pan W Xue H Jiang and H Mao ldquoMulti-objectiveoptimization for energy performance improvement of residen-tial buildings a comparative studyrdquo Energies vol 10 no 2 p245 2017

[31] H Zhu W W Huang and G H Huang ldquoPlanning of regionalenergy systems An inexact mixed-integer fractional program-ming modelrdquo Applied Energy vol 113 pp 500ndash514 2014

[32] F Chen G H Huang Y R Fan and R F Liao ldquoA nonlinearfractional programming approach for environmentaleconomicpower dispatchrdquo International Journal of Electrical Power ampEnergy Systems vol 78 pp 463ndash469 2016

[33] C Zhang and P Guo ldquoFLFP A fuzzy linear fractional pro-gramming approach with double-sided fuzziness for optimalirrigation water allocationrdquo Agricultural Water Managementvol 199 pp 105ndash119 2018

[34] L Wang G Huang X Wang and H Zhu ldquoRisk-basedelectric power system planning for climate change mitigationthrough multi-stage joint-probabilistic left-hand-side chance-constrained fractional programming A Canadian case studyrdquoRenewable amp Sustainable Energy Reviews vol 82 pp 1056ndash10672018

[35] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[36] X H Nie G H Huang Y P Li and L Liu ldquoIFRP A hybridinterval-parameter fuzzy robust programming approach forwaste management planning under uncertaintyrdquo Journal ofEnvironmental Management vol 84 no 1 pp 1ndash11 2007

[37] Y Leung Spatial analysis and planning under imprecisionElsevier 1st edition 2013

[38] A Charnes W W Cooper and M J L Kirby ldquoChance-constrained programming an extension of statistical methodrdquoinOptimizingmethods in statistics pp 391ndash402 Academic Press1971

[39] Y P Cai G H Huang Z F Yang and Q Tan ldquoIdentificationof optimal strategies for energy management systems planningunder multiple uncertaintiesrdquoApplied Energy vol 86 no 4 pp480ndash495 2009

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

75

79

5 4

74

8

95 4

74

8

9

5 4

81

095 5

83

09

4 4

78

7

7

4 4

coalnatural gashydro

windsolar

83

010

2

5

83

0

94 4

80

5

84 3

82

19 4

4

78

6

84 4

79

5

84 4

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

coalnatural gashydro

windsolar

IFCFP-PSP f- IFCFP-PSP +

LS +LS -

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

coalnatural gashydro

Figure 5 Comparison of power-generation patterns from IFCF-PSP and least-system-cost (LS) models under 119901119894-level 001 Note that theinnermost circle in the pie chart represents the first planning period

is larger) the demand for expansion of clean energy willincrease On the other hand the total demand of capacityexpansion in the lower bound model is lower than the upperbound model One of the reasons for this result is that theannual utilization hours of various power generation facilitiesin the lower bound model are higher than the correspondingparameter settings of the upper bound model

The result of LS model has shown that under thecorresponding lower bound parameter settings the capacityof 015 GW would be added to the hydropower facility inperiod 3 when 119901119894 = 001 and 119901119894 = 005 Only a capacity of015 GWwould be added to the wind power facility in period3 While under the corresponding upper bound parametersettings natural gas and solar energy facilities expansionsare not required that is because the former resource is underpenalty of CO2 emission and the capacity cost of the solarpower expansion is much higher

In the IFCF-PSP model clean energy will be given moredevelopment opportunities Take 119901119894 = 001 as an example(corresponding to 119891minus) with a capacity expansion of 02

GW at the beginning of period 1 and period 2 electricitygeneration ofwind power facilitywould rise from 164 103GWin period 1 to 168 103GW in period 2 and with anothercapacity expansion of 02 GW at the beginning of period 3 itstotal electricity generation capacity would reach 172 103GWLikewise to meet the growing energy demand the capacityof 02 GW would be added to the natural gas-fired facility inperiod 1 period 2 and period 3 as a result its total generationcapacity would rise from 954 times 103GW in period 1 to 119 times103GW in period 2 and reach 124 times 103GW in period 3 Inthe planning period 1 due to more lenient emission policiesand more economical cost strategies some of the natural gaspower generation facilities are idleThe capacities of 025 GWwould be added to the hydropower facility in three periodsand the corresponding power generation capacity would be812 times 103GW 875 103GW and 937 times 103GW Generatingelectricity from solar energy facility is more expensive thanother power generation facilities thus when the emissionlimits and electricity demand are relatively low solar energyfacility would not be expanded during first two periods 015

Mathematical Problems in Engineering 11

Table 6 Binary solutions of capacity expansions

119901119894-level Power-generation facility Capacity-expansion option IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3

01 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 [0 1] [0 1] 0 0 0

Wind power m=1 0 0 0 0 0 0m=2 0 [0 1] 0 0 [0 1] 1m=3 [0 1] [0 1] [0 1] 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 [0 1] [0 1] 0 0 0 [0 1]m=2 0 0 0 0 0 0m=3 [0 1] [0 1] 0 [0 1] [0 1] 0

005 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 [0 1] [0 1] [0 1] 0 0 0

Wind power m=1 [0 1] [0 1] 0 0 0 0m=2 0 0 0 0 [0 1] 1m=3 [0 1] [0 1] 0 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] [0 1]m=2 0 0 0 0 0 0m=3 1 1 0 [0 1] [0 1] 0

001 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 1 1 [0 1] [0 1] [0 1]

Wind power m=1 0 0 0 0 0 0m=2 0 0 0 0 0 [0 1]m=3 1 1 [0 1] [0 1] [0 1] [0 1]

Solar energy m=1 0 0 0 0 0 0m=2 0 0 [0 1] 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] 0m=2 0 0 0 0 0 0m=3 1 1 [0 1] [0 1] [0 1] [0 1]

GW of capacity would only be added to the solar energyfacility in period 3

Apparently with the successful application of IFCI-PSP within a typical regional electric system managementproblem solutions obtained could provide useful decisionalternatives under different policies and various energy avail-abilities Compared with the least-cost model the IFCF-PSP model is an effective tool for providing environmentalmanagement schemes with dual objectives In addition theIFCF-PSP has following advantages over the conventionalprogramming methods (a) balancing multiple conflicting

objectives (b) reflecting interrelationships among systemefficiency economic cost and system reliability (c) effec-tively dealing with randomness in both the objective andconstraints and (d) assisting the analysis of diverse decisionschemes associated with various energy demand levels

5 Conclusions

An inexact fuzzy chance-constrained fractional program-ming approach is developed for optimal electric powersystems management under uncertainties In the developed

12 Mathematical Problems in Engineering

model fuzzy chance-constrained programming is incorpo-rated into a fractional programming optimization frame-work The obtained results are useful for supporting EPSmanagement The IFCF-PSP approach is capable of (i)balancing the conflict between two objectives (ii) reflect-ing different electricity generation and capacity expansionstrategies (iii) presenting optimal solutions under differentconstraint violating conditions and (iv) introducing theconcept of fuzzy boundary interval with which the com-plexity of dual uncertainties can be effectively handled Theresults of IFCF-PSP model show that (i) a higher confidencelevel corresponds to a higher proportion of clean energypower generation and a lower economic productivity and(ii) as a result of encouraging environment-friendly energiesthe generation capacities for the natural gas wind-powersolar energy and hydropower facilities would be signifi-cantly increased The solutions obtained from the IFCF-PSPapproach could provide specific energy options for powersystem planning and provide effective management solutionof the electric power system for identifying electricity gener-ation and capacity expansion schemes

This study attempts to develop a modeling framework forratio problems involving fuzzy uncertainties to deal with elec-tric power systems management problemThe results suggestthat it is also applicable to other energy management prob-lems In the future practice IFCF-PSP could be further im-proved through considering more impact factors Forinstance the fuzzymembership functions and the confidencelevel are critical in the decision-making process and thisrequires effective ways to provide appropriate choices fordecision making Such challenges desire further investiga-tions Future research can be aimed at applying the advancedapproach to a more complex real-world electric power sys-tem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the Fundamental ResearchFunds for the Central Universities NCEPU (2018BJ0293)the State Scholarship Fund (201706735025) the NationalKey Research and Development Plan (2016YFC0502800) theNatural Sciences Foundation (51520105013 51679087) andthe 111 Program (B14008) The data used to support thefindings of this study are available from the correspondingauthor upon request

References

[1] G H Huang ldquoA hybrid inexact-stochastic water managementmodelrdquo European Journal of Operational Research vol 107 no1 pp 137ndash158 1998

[2] A Grosfeld-Nir and A Tishler ldquoA stochastic model for themeasurement of electricity outage costsrdquo Energy pp 157ndash1741993

[3] S Wang G Huang and B W Baetz ldquoAn Inexact Probabilistic-Possibilistic Optimization Framework for Flood Managementin a Hybrid Uncertain Environmentrdquo IEEE Transactions onFuzzy Systems vol 23 no 4 pp 897ndash908 2015

[4] G H Huang and D P Loucks ldquoAn inexact two-stage stochasticprogramming model for water resources management underuncertaintyrdquo Civil Engineering and Environmental Systems vol17 no 2 pp 95ndash118 2000

[5] K Darby-Dowman S Barker E Audsley and D Parsons ldquoAtwo-stage stochastic programming with recourse model fordetermining robust planting plans in horticulturerdquo Journal ofthe Operational Research Society vol 51 no 1 pp 83ndash89 2000

[6] J Hu L Sun C H Li X Wang X L Jia and Y P Cai ldquoWaterQuality Risk Assessment for the Laoguanhe River of ChinaUsing a Stochastic SimulationMethodrdquo Journal of Environmen-tal Informatics vol 31 no 2 pp 123ndash136 2018

[7] SWang andG HHuang ldquoAmulti-level Taguchi-factorial two-stage stochastic programming approach for characterization ofparameter uncertainties and their interactions An applicationto water resources managementrdquo European Journal of Opera-tional Research vol 240 no 2 pp 572ndash581 2015

[8] S Chanas and P Zielinski ldquoOn the equivalence of two opti-mization methods for fuzzy linear programming problemsrdquoEuropean Journal of Operational Research vol 121 no 1 pp 56ndash63 2000

[9] J Mula D Peidro and R Poler ldquoThe effectiveness of a fuzzymathematical programming approach for supply chain pro-duction planning with fuzzy demandrdquo International Journal ofProduction Economics vol 128 no 1 pp 136ndash143 2010

[10] B Chen P Li H J Wu T Husain and F Khan ldquoMCFP Amonte carlo simulation-based fuzzy programming approachfor optimization under dual uncertainties of possibility andcontinuous probabilityrdquo Journal of Environmental Informatics(JEI) vol 29 no 2 pp 88ndash97 2017

[11] C Z Huang S Nie L Guo and Y R Fan ldquoInexact fuzzystochastic chance constraint programming for emergency evac-uation in Qinshan nuclear power plant under uncertaintyrdquoJournal of Environmental Informatics (JEI) vol 30 no 1 pp 63ndash78 2017

[12] U S Sakalli ldquoOptimization of Production-Distribution Prob-lem in Supply Chain Management under Stochastic and FuzzyUncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 4389064 29 pages 2017

[13] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval research logistics (NRL) pp 181ndash186 1962

[14] N I Voropai and E Y Ivanova ldquoMulti-criteria decision analysistechniques in electric power system expansion planningrdquo Inter-national Journal of Electrical Power amp Energy Systems vol 24no 1 pp 71ndash78 2002

[15] M Chakraborty and S Gupta ldquoFuzzy mathematical pro-gramming for multi objective linear fractional programmingproblemrdquo Fuzzy Sets and Systems vol 125 no 3 pp 335ndash3422002

[16] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[17] W Li Z BaoGHHuang andY L Xie ldquoAn Inexact CredibilityChance-Constrained Integer Programming for Greenhouse

Mathematical Problems in Engineering 13

GasMitigationManagement in Regional Electric Power Systemunder Uncertaintyrdquo in Journal of Environmental Informatics2017

[18] M Paunovic N M Ralevic V Gajovic et al ldquoTwo-StageFuzzy Logic Model for Cloud Service Supplier Selection andEvaluationrdquoMathematical Problems in Engineering 2018

[19] S Wang G H Huang B W Baetz and B C Ancell ldquoTowardsrobust quantification and reduction of uncertainty in hydro-logic predictions Integration of particle Markov chain MonteCarlo and factorial polynomial chaos expansionrdquo Journal ofHydrology vol 548 pp 484ndash497 2017

[20] SWang B C Ancell GHHuang andBW Baetz ldquoImprovingRobustness ofHydrologic Ensemble PredictionsThroughProb-abilistic Pre- and Post-Processing in Sequential Data Assimila-tionrdquoWater Resources Research 2018

[21] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[22] A Rong and R Lahdelma ldquoFuzzy chance constrained linearprogramming model for optimizing the scrap charge in steelproductionrdquo European Journal of Operational Research vol 186no 3 pp 953ndash964 2008

[23] X Sun J Li H Qiao and B Zhang ldquoEnergy implicationsof Chinarsquos regional development New insights from multi-regional input-output analysisrdquo Applied Energy vol 196 pp118ndash131 2017

[24] M A Quaddus and T N Goh ldquoElectric power generationexpansion Planning with multiple objectivesrdquo Applied Energyvol 19 no 4 pp 301ndash319 1985

[25] C H Antunes A G Martins and I S Brito ldquoA multipleobjective mixed integer linear programming model for powergeneration expansion planningrdquo Energy vol 29 no 4 pp 613ndash627 2004

[26] J L C Meza M B Yildirim and A S M Masud ldquoA modelfor the multiperiod multiobjective power generation expansionproblemrdquo IEEETransactions on Power Systems vol 22 no 2 pp871ndash878 2007

[27] J-HHan Y-C Ahn and I-B Lee ldquoAmulti-objective optimiza-tion model for sustainable electricity generation and CO2 mit-igation (EGCM) infrastructure design considering economicprofit and financial riskrdquo Applied Energy vol 95 pp 186ndash1952012

[28] M Rekik A Abdelkafi and L Krichen ldquoAmicro-grid ensuringmulti-objective control strategy of a power electrical system forquality improvementrdquo Energy vol 88 pp 351ndash363 2015

[29] A Azadeh Z Raoofi and M Zarrin ldquoA multi-objective fuzzylinear programming model for optimization of natural gassupply chain through a greenhouse gas reduction approachrdquoJournal of Natural Gas Science and Engineering vol 26 pp 702ndash710 2015

[30] K Li L Pan W Xue H Jiang and H Mao ldquoMulti-objectiveoptimization for energy performance improvement of residen-tial buildings a comparative studyrdquo Energies vol 10 no 2 p245 2017

[31] H Zhu W W Huang and G H Huang ldquoPlanning of regionalenergy systems An inexact mixed-integer fractional program-ming modelrdquo Applied Energy vol 113 pp 500ndash514 2014

[32] F Chen G H Huang Y R Fan and R F Liao ldquoA nonlinearfractional programming approach for environmentaleconomicpower dispatchrdquo International Journal of Electrical Power ampEnergy Systems vol 78 pp 463ndash469 2016

[33] C Zhang and P Guo ldquoFLFP A fuzzy linear fractional pro-gramming approach with double-sided fuzziness for optimalirrigation water allocationrdquo Agricultural Water Managementvol 199 pp 105ndash119 2018

[34] L Wang G Huang X Wang and H Zhu ldquoRisk-basedelectric power system planning for climate change mitigationthrough multi-stage joint-probabilistic left-hand-side chance-constrained fractional programming A Canadian case studyrdquoRenewable amp Sustainable Energy Reviews vol 82 pp 1056ndash10672018

[35] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[36] X H Nie G H Huang Y P Li and L Liu ldquoIFRP A hybridinterval-parameter fuzzy robust programming approach forwaste management planning under uncertaintyrdquo Journal ofEnvironmental Management vol 84 no 1 pp 1ndash11 2007

[37] Y Leung Spatial analysis and planning under imprecisionElsevier 1st edition 2013

[38] A Charnes W W Cooper and M J L Kirby ldquoChance-constrained programming an extension of statistical methodrdquoinOptimizingmethods in statistics pp 391ndash402 Academic Press1971

[39] Y P Cai G H Huang Z F Yang and Q Tan ldquoIdentificationof optimal strategies for energy management systems planningunder multiple uncertaintiesrdquoApplied Energy vol 86 no 4 pp480ndash495 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

Table 6 Binary solutions of capacity expansions

119901119894-level Power-generation facility Capacity-expansion option IFCF-PSP LSt=1 t=2 t=3 t=1 t=2 t=3

01 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 [0 1] [0 1] 0 0 0

Wind power m=1 0 0 0 0 0 0m=2 0 [0 1] 0 0 [0 1] 1m=3 [0 1] [0 1] [0 1] 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 [0 1] [0 1] 0 0 0 [0 1]m=2 0 0 0 0 0 0m=3 [0 1] [0 1] 0 [0 1] [0 1] 0

005 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 [0 1] [0 1] [0 1] 0 0 0

Wind power m=1 [0 1] [0 1] 0 0 0 0m=2 0 0 0 0 [0 1] 1m=3 [0 1] [0 1] 0 0 0 0

Solar energy m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] [0 1]m=2 0 0 0 0 0 0m=3 1 1 0 [0 1] [0 1] 0

001 Natural gas m=1 0 0 0 0 0 0m=2 0 0 0 0 0 0m=3 1 1 1 [0 1] [0 1] [0 1]

Wind power m=1 0 0 0 0 0 0m=2 0 0 0 0 0 [0 1]m=3 1 1 [0 1] [0 1] [0 1] [0 1]

Solar energy m=1 0 0 0 0 0 0m=2 0 0 [0 1] 0 0 0m=3 0 0 0 0 0 0

Hydropower m=1 0 0 0 0 [0 1] 0m=2 0 0 0 0 0 0m=3 1 1 [0 1] [0 1] [0 1] [0 1]

GW of capacity would only be added to the solar energyfacility in period 3

Apparently with the successful application of IFCI-PSP within a typical regional electric system managementproblem solutions obtained could provide useful decisionalternatives under different policies and various energy avail-abilities Compared with the least-cost model the IFCF-PSP model is an effective tool for providing environmentalmanagement schemes with dual objectives In addition theIFCF-PSP has following advantages over the conventionalprogramming methods (a) balancing multiple conflicting

objectives (b) reflecting interrelationships among systemefficiency economic cost and system reliability (c) effec-tively dealing with randomness in both the objective andconstraints and (d) assisting the analysis of diverse decisionschemes associated with various energy demand levels

5 Conclusions

An inexact fuzzy chance-constrained fractional program-ming approach is developed for optimal electric powersystems management under uncertainties In the developed

12 Mathematical Problems in Engineering

model fuzzy chance-constrained programming is incorpo-rated into a fractional programming optimization frame-work The obtained results are useful for supporting EPSmanagement The IFCF-PSP approach is capable of (i)balancing the conflict between two objectives (ii) reflect-ing different electricity generation and capacity expansionstrategies (iii) presenting optimal solutions under differentconstraint violating conditions and (iv) introducing theconcept of fuzzy boundary interval with which the com-plexity of dual uncertainties can be effectively handled Theresults of IFCF-PSP model show that (i) a higher confidencelevel corresponds to a higher proportion of clean energypower generation and a lower economic productivity and(ii) as a result of encouraging environment-friendly energiesthe generation capacities for the natural gas wind-powersolar energy and hydropower facilities would be signifi-cantly increased The solutions obtained from the IFCF-PSPapproach could provide specific energy options for powersystem planning and provide effective management solutionof the electric power system for identifying electricity gener-ation and capacity expansion schemes

This study attempts to develop a modeling framework forratio problems involving fuzzy uncertainties to deal with elec-tric power systems management problemThe results suggestthat it is also applicable to other energy management prob-lems In the future practice IFCF-PSP could be further im-proved through considering more impact factors Forinstance the fuzzymembership functions and the confidencelevel are critical in the decision-making process and thisrequires effective ways to provide appropriate choices fordecision making Such challenges desire further investiga-tions Future research can be aimed at applying the advancedapproach to a more complex real-world electric power sys-tem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the Fundamental ResearchFunds for the Central Universities NCEPU (2018BJ0293)the State Scholarship Fund (201706735025) the NationalKey Research and Development Plan (2016YFC0502800) theNatural Sciences Foundation (51520105013 51679087) andthe 111 Program (B14008) The data used to support thefindings of this study are available from the correspondingauthor upon request

References

[1] G H Huang ldquoA hybrid inexact-stochastic water managementmodelrdquo European Journal of Operational Research vol 107 no1 pp 137ndash158 1998

[2] A Grosfeld-Nir and A Tishler ldquoA stochastic model for themeasurement of electricity outage costsrdquo Energy pp 157ndash1741993

[3] S Wang G Huang and B W Baetz ldquoAn Inexact Probabilistic-Possibilistic Optimization Framework for Flood Managementin a Hybrid Uncertain Environmentrdquo IEEE Transactions onFuzzy Systems vol 23 no 4 pp 897ndash908 2015

[4] G H Huang and D P Loucks ldquoAn inexact two-stage stochasticprogramming model for water resources management underuncertaintyrdquo Civil Engineering and Environmental Systems vol17 no 2 pp 95ndash118 2000

[5] K Darby-Dowman S Barker E Audsley and D Parsons ldquoAtwo-stage stochastic programming with recourse model fordetermining robust planting plans in horticulturerdquo Journal ofthe Operational Research Society vol 51 no 1 pp 83ndash89 2000

[6] J Hu L Sun C H Li X Wang X L Jia and Y P Cai ldquoWaterQuality Risk Assessment for the Laoguanhe River of ChinaUsing a Stochastic SimulationMethodrdquo Journal of Environmen-tal Informatics vol 31 no 2 pp 123ndash136 2018

[7] SWang andG HHuang ldquoAmulti-level Taguchi-factorial two-stage stochastic programming approach for characterization ofparameter uncertainties and their interactions An applicationto water resources managementrdquo European Journal of Opera-tional Research vol 240 no 2 pp 572ndash581 2015

[8] S Chanas and P Zielinski ldquoOn the equivalence of two opti-mization methods for fuzzy linear programming problemsrdquoEuropean Journal of Operational Research vol 121 no 1 pp 56ndash63 2000

[9] J Mula D Peidro and R Poler ldquoThe effectiveness of a fuzzymathematical programming approach for supply chain pro-duction planning with fuzzy demandrdquo International Journal ofProduction Economics vol 128 no 1 pp 136ndash143 2010

[10] B Chen P Li H J Wu T Husain and F Khan ldquoMCFP Amonte carlo simulation-based fuzzy programming approachfor optimization under dual uncertainties of possibility andcontinuous probabilityrdquo Journal of Environmental Informatics(JEI) vol 29 no 2 pp 88ndash97 2017

[11] C Z Huang S Nie L Guo and Y R Fan ldquoInexact fuzzystochastic chance constraint programming for emergency evac-uation in Qinshan nuclear power plant under uncertaintyrdquoJournal of Environmental Informatics (JEI) vol 30 no 1 pp 63ndash78 2017

[12] U S Sakalli ldquoOptimization of Production-Distribution Prob-lem in Supply Chain Management under Stochastic and FuzzyUncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 4389064 29 pages 2017

[13] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval research logistics (NRL) pp 181ndash186 1962

[14] N I Voropai and E Y Ivanova ldquoMulti-criteria decision analysistechniques in electric power system expansion planningrdquo Inter-national Journal of Electrical Power amp Energy Systems vol 24no 1 pp 71ndash78 2002

[15] M Chakraborty and S Gupta ldquoFuzzy mathematical pro-gramming for multi objective linear fractional programmingproblemrdquo Fuzzy Sets and Systems vol 125 no 3 pp 335ndash3422002

[16] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[17] W Li Z BaoGHHuang andY L Xie ldquoAn Inexact CredibilityChance-Constrained Integer Programming for Greenhouse

Mathematical Problems in Engineering 13

GasMitigationManagement in Regional Electric Power Systemunder Uncertaintyrdquo in Journal of Environmental Informatics2017

[18] M Paunovic N M Ralevic V Gajovic et al ldquoTwo-StageFuzzy Logic Model for Cloud Service Supplier Selection andEvaluationrdquoMathematical Problems in Engineering 2018

[19] S Wang G H Huang B W Baetz and B C Ancell ldquoTowardsrobust quantification and reduction of uncertainty in hydro-logic predictions Integration of particle Markov chain MonteCarlo and factorial polynomial chaos expansionrdquo Journal ofHydrology vol 548 pp 484ndash497 2017

[20] SWang B C Ancell GHHuang andBW Baetz ldquoImprovingRobustness ofHydrologic Ensemble PredictionsThroughProb-abilistic Pre- and Post-Processing in Sequential Data Assimila-tionrdquoWater Resources Research 2018

[21] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[22] A Rong and R Lahdelma ldquoFuzzy chance constrained linearprogramming model for optimizing the scrap charge in steelproductionrdquo European Journal of Operational Research vol 186no 3 pp 953ndash964 2008

[23] X Sun J Li H Qiao and B Zhang ldquoEnergy implicationsof Chinarsquos regional development New insights from multi-regional input-output analysisrdquo Applied Energy vol 196 pp118ndash131 2017

[24] M A Quaddus and T N Goh ldquoElectric power generationexpansion Planning with multiple objectivesrdquo Applied Energyvol 19 no 4 pp 301ndash319 1985

[25] C H Antunes A G Martins and I S Brito ldquoA multipleobjective mixed integer linear programming model for powergeneration expansion planningrdquo Energy vol 29 no 4 pp 613ndash627 2004

[26] J L C Meza M B Yildirim and A S M Masud ldquoA modelfor the multiperiod multiobjective power generation expansionproblemrdquo IEEETransactions on Power Systems vol 22 no 2 pp871ndash878 2007

[27] J-HHan Y-C Ahn and I-B Lee ldquoAmulti-objective optimiza-tion model for sustainable electricity generation and CO2 mit-igation (EGCM) infrastructure design considering economicprofit and financial riskrdquo Applied Energy vol 95 pp 186ndash1952012

[28] M Rekik A Abdelkafi and L Krichen ldquoAmicro-grid ensuringmulti-objective control strategy of a power electrical system forquality improvementrdquo Energy vol 88 pp 351ndash363 2015

[29] A Azadeh Z Raoofi and M Zarrin ldquoA multi-objective fuzzylinear programming model for optimization of natural gassupply chain through a greenhouse gas reduction approachrdquoJournal of Natural Gas Science and Engineering vol 26 pp 702ndash710 2015

[30] K Li L Pan W Xue H Jiang and H Mao ldquoMulti-objectiveoptimization for energy performance improvement of residen-tial buildings a comparative studyrdquo Energies vol 10 no 2 p245 2017

[31] H Zhu W W Huang and G H Huang ldquoPlanning of regionalenergy systems An inexact mixed-integer fractional program-ming modelrdquo Applied Energy vol 113 pp 500ndash514 2014

[32] F Chen G H Huang Y R Fan and R F Liao ldquoA nonlinearfractional programming approach for environmentaleconomicpower dispatchrdquo International Journal of Electrical Power ampEnergy Systems vol 78 pp 463ndash469 2016

[33] C Zhang and P Guo ldquoFLFP A fuzzy linear fractional pro-gramming approach with double-sided fuzziness for optimalirrigation water allocationrdquo Agricultural Water Managementvol 199 pp 105ndash119 2018

[34] L Wang G Huang X Wang and H Zhu ldquoRisk-basedelectric power system planning for climate change mitigationthrough multi-stage joint-probabilistic left-hand-side chance-constrained fractional programming A Canadian case studyrdquoRenewable amp Sustainable Energy Reviews vol 82 pp 1056ndash10672018

[35] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[36] X H Nie G H Huang Y P Li and L Liu ldquoIFRP A hybridinterval-parameter fuzzy robust programming approach forwaste management planning under uncertaintyrdquo Journal ofEnvironmental Management vol 84 no 1 pp 1ndash11 2007

[37] Y Leung Spatial analysis and planning under imprecisionElsevier 1st edition 2013

[38] A Charnes W W Cooper and M J L Kirby ldquoChance-constrained programming an extension of statistical methodrdquoinOptimizingmethods in statistics pp 391ndash402 Academic Press1971

[39] Y P Cai G H Huang Z F Yang and Q Tan ldquoIdentificationof optimal strategies for energy management systems planningunder multiple uncertaintiesrdquoApplied Energy vol 86 no 4 pp480ndash495 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Mathematical Problems in Engineering

model fuzzy chance-constrained programming is incorpo-rated into a fractional programming optimization frame-work The obtained results are useful for supporting EPSmanagement The IFCF-PSP approach is capable of (i)balancing the conflict between two objectives (ii) reflect-ing different electricity generation and capacity expansionstrategies (iii) presenting optimal solutions under differentconstraint violating conditions and (iv) introducing theconcept of fuzzy boundary interval with which the com-plexity of dual uncertainties can be effectively handled Theresults of IFCF-PSP model show that (i) a higher confidencelevel corresponds to a higher proportion of clean energypower generation and a lower economic productivity and(ii) as a result of encouraging environment-friendly energiesthe generation capacities for the natural gas wind-powersolar energy and hydropower facilities would be signifi-cantly increased The solutions obtained from the IFCF-PSPapproach could provide specific energy options for powersystem planning and provide effective management solutionof the electric power system for identifying electricity gener-ation and capacity expansion schemes

This study attempts to develop a modeling framework forratio problems involving fuzzy uncertainties to deal with elec-tric power systems management problemThe results suggestthat it is also applicable to other energy management prob-lems In the future practice IFCF-PSP could be further im-proved through considering more impact factors Forinstance the fuzzymembership functions and the confidencelevel are critical in the decision-making process and thisrequires effective ways to provide appropriate choices fordecision making Such challenges desire further investiga-tions Future research can be aimed at applying the advancedapproach to a more complex real-world electric power sys-tem

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the Fundamental ResearchFunds for the Central Universities NCEPU (2018BJ0293)the State Scholarship Fund (201706735025) the NationalKey Research and Development Plan (2016YFC0502800) theNatural Sciences Foundation (51520105013 51679087) andthe 111 Program (B14008) The data used to support thefindings of this study are available from the correspondingauthor upon request

References

[1] G H Huang ldquoA hybrid inexact-stochastic water managementmodelrdquo European Journal of Operational Research vol 107 no1 pp 137ndash158 1998

[2] A Grosfeld-Nir and A Tishler ldquoA stochastic model for themeasurement of electricity outage costsrdquo Energy pp 157ndash1741993

[3] S Wang G Huang and B W Baetz ldquoAn Inexact Probabilistic-Possibilistic Optimization Framework for Flood Managementin a Hybrid Uncertain Environmentrdquo IEEE Transactions onFuzzy Systems vol 23 no 4 pp 897ndash908 2015

[4] G H Huang and D P Loucks ldquoAn inexact two-stage stochasticprogramming model for water resources management underuncertaintyrdquo Civil Engineering and Environmental Systems vol17 no 2 pp 95ndash118 2000

[5] K Darby-Dowman S Barker E Audsley and D Parsons ldquoAtwo-stage stochastic programming with recourse model fordetermining robust planting plans in horticulturerdquo Journal ofthe Operational Research Society vol 51 no 1 pp 83ndash89 2000

[6] J Hu L Sun C H Li X Wang X L Jia and Y P Cai ldquoWaterQuality Risk Assessment for the Laoguanhe River of ChinaUsing a Stochastic SimulationMethodrdquo Journal of Environmen-tal Informatics vol 31 no 2 pp 123ndash136 2018

[7] SWang andG HHuang ldquoAmulti-level Taguchi-factorial two-stage stochastic programming approach for characterization ofparameter uncertainties and their interactions An applicationto water resources managementrdquo European Journal of Opera-tional Research vol 240 no 2 pp 572ndash581 2015

[8] S Chanas and P Zielinski ldquoOn the equivalence of two opti-mization methods for fuzzy linear programming problemsrdquoEuropean Journal of Operational Research vol 121 no 1 pp 56ndash63 2000

[9] J Mula D Peidro and R Poler ldquoThe effectiveness of a fuzzymathematical programming approach for supply chain pro-duction planning with fuzzy demandrdquo International Journal ofProduction Economics vol 128 no 1 pp 136ndash143 2010

[10] B Chen P Li H J Wu T Husain and F Khan ldquoMCFP Amonte carlo simulation-based fuzzy programming approachfor optimization under dual uncertainties of possibility andcontinuous probabilityrdquo Journal of Environmental Informatics(JEI) vol 29 no 2 pp 88ndash97 2017

[11] C Z Huang S Nie L Guo and Y R Fan ldquoInexact fuzzystochastic chance constraint programming for emergency evac-uation in Qinshan nuclear power plant under uncertaintyrdquoJournal of Environmental Informatics (JEI) vol 30 no 1 pp 63ndash78 2017

[12] U S Sakalli ldquoOptimization of Production-Distribution Prob-lem in Supply Chain Management under Stochastic and FuzzyUncertaintiesrdquoMathematical Problems in Engineering vol 2017Article ID 4389064 29 pages 2017

[13] A Charnes and W W Cooper ldquoProgramming with linearfractional functionalsrdquo Naval research logistics (NRL) pp 181ndash186 1962

[14] N I Voropai and E Y Ivanova ldquoMulti-criteria decision analysistechniques in electric power system expansion planningrdquo Inter-national Journal of Electrical Power amp Energy Systems vol 24no 1 pp 71ndash78 2002

[15] M Chakraborty and S Gupta ldquoFuzzy mathematical pro-gramming for multi objective linear fractional programmingproblemrdquo Fuzzy Sets and Systems vol 125 no 3 pp 335ndash3422002

[16] Y Deng ldquoA threat assessment model under uncertain environ-mentrdquoMathematical Problems in Engineering vol 2015 ArticleID 878024 12 pages 2015

[17] W Li Z BaoGHHuang andY L Xie ldquoAn Inexact CredibilityChance-Constrained Integer Programming for Greenhouse

Mathematical Problems in Engineering 13

GasMitigationManagement in Regional Electric Power Systemunder Uncertaintyrdquo in Journal of Environmental Informatics2017

[18] M Paunovic N M Ralevic V Gajovic et al ldquoTwo-StageFuzzy Logic Model for Cloud Service Supplier Selection andEvaluationrdquoMathematical Problems in Engineering 2018

[19] S Wang G H Huang B W Baetz and B C Ancell ldquoTowardsrobust quantification and reduction of uncertainty in hydro-logic predictions Integration of particle Markov chain MonteCarlo and factorial polynomial chaos expansionrdquo Journal ofHydrology vol 548 pp 484ndash497 2017

[20] SWang B C Ancell GHHuang andBW Baetz ldquoImprovingRobustness ofHydrologic Ensemble PredictionsThroughProb-abilistic Pre- and Post-Processing in Sequential Data Assimila-tionrdquoWater Resources Research 2018

[21] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[22] A Rong and R Lahdelma ldquoFuzzy chance constrained linearprogramming model for optimizing the scrap charge in steelproductionrdquo European Journal of Operational Research vol 186no 3 pp 953ndash964 2008

[23] X Sun J Li H Qiao and B Zhang ldquoEnergy implicationsof Chinarsquos regional development New insights from multi-regional input-output analysisrdquo Applied Energy vol 196 pp118ndash131 2017

[24] M A Quaddus and T N Goh ldquoElectric power generationexpansion Planning with multiple objectivesrdquo Applied Energyvol 19 no 4 pp 301ndash319 1985

[25] C H Antunes A G Martins and I S Brito ldquoA multipleobjective mixed integer linear programming model for powergeneration expansion planningrdquo Energy vol 29 no 4 pp 613ndash627 2004

[26] J L C Meza M B Yildirim and A S M Masud ldquoA modelfor the multiperiod multiobjective power generation expansionproblemrdquo IEEETransactions on Power Systems vol 22 no 2 pp871ndash878 2007

[27] J-HHan Y-C Ahn and I-B Lee ldquoAmulti-objective optimiza-tion model for sustainable electricity generation and CO2 mit-igation (EGCM) infrastructure design considering economicprofit and financial riskrdquo Applied Energy vol 95 pp 186ndash1952012

[28] M Rekik A Abdelkafi and L Krichen ldquoAmicro-grid ensuringmulti-objective control strategy of a power electrical system forquality improvementrdquo Energy vol 88 pp 351ndash363 2015

[29] A Azadeh Z Raoofi and M Zarrin ldquoA multi-objective fuzzylinear programming model for optimization of natural gassupply chain through a greenhouse gas reduction approachrdquoJournal of Natural Gas Science and Engineering vol 26 pp 702ndash710 2015

[30] K Li L Pan W Xue H Jiang and H Mao ldquoMulti-objectiveoptimization for energy performance improvement of residen-tial buildings a comparative studyrdquo Energies vol 10 no 2 p245 2017

[31] H Zhu W W Huang and G H Huang ldquoPlanning of regionalenergy systems An inexact mixed-integer fractional program-ming modelrdquo Applied Energy vol 113 pp 500ndash514 2014

[32] F Chen G H Huang Y R Fan and R F Liao ldquoA nonlinearfractional programming approach for environmentaleconomicpower dispatchrdquo International Journal of Electrical Power ampEnergy Systems vol 78 pp 463ndash469 2016

[33] C Zhang and P Guo ldquoFLFP A fuzzy linear fractional pro-gramming approach with double-sided fuzziness for optimalirrigation water allocationrdquo Agricultural Water Managementvol 199 pp 105ndash119 2018

[34] L Wang G Huang X Wang and H Zhu ldquoRisk-basedelectric power system planning for climate change mitigationthrough multi-stage joint-probabilistic left-hand-side chance-constrained fractional programming A Canadian case studyrdquoRenewable amp Sustainable Energy Reviews vol 82 pp 1056ndash10672018

[35] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[36] X H Nie G H Huang Y P Li and L Liu ldquoIFRP A hybridinterval-parameter fuzzy robust programming approach forwaste management planning under uncertaintyrdquo Journal ofEnvironmental Management vol 84 no 1 pp 1ndash11 2007

[37] Y Leung Spatial analysis and planning under imprecisionElsevier 1st edition 2013

[38] A Charnes W W Cooper and M J L Kirby ldquoChance-constrained programming an extension of statistical methodrdquoinOptimizingmethods in statistics pp 391ndash402 Academic Press1971

[39] Y P Cai G H Huang Z F Yang and Q Tan ldquoIdentificationof optimal strategies for energy management systems planningunder multiple uncertaintiesrdquoApplied Energy vol 86 no 4 pp480ndash495 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 13

GasMitigationManagement in Regional Electric Power Systemunder Uncertaintyrdquo in Journal of Environmental Informatics2017

[18] M Paunovic N M Ralevic V Gajovic et al ldquoTwo-StageFuzzy Logic Model for Cloud Service Supplier Selection andEvaluationrdquoMathematical Problems in Engineering 2018

[19] S Wang G H Huang B W Baetz and B C Ancell ldquoTowardsrobust quantification and reduction of uncertainty in hydro-logic predictions Integration of particle Markov chain MonteCarlo and factorial polynomial chaos expansionrdquo Journal ofHydrology vol 548 pp 484ndash497 2017

[20] SWang B C Ancell GHHuang andBW Baetz ldquoImprovingRobustness ofHydrologic Ensemble PredictionsThroughProb-abilistic Pre- and Post-Processing in Sequential Data Assimila-tionrdquoWater Resources Research 2018

[21] M G Iskander ldquoA suggested approach for possibility andnecessity dominance indices in stochastic fuzzy linear program-mingrdquo Applied Mathematics Letters vol 18 no 4 pp 395ndash3992005

[22] A Rong and R Lahdelma ldquoFuzzy chance constrained linearprogramming model for optimizing the scrap charge in steelproductionrdquo European Journal of Operational Research vol 186no 3 pp 953ndash964 2008

[23] X Sun J Li H Qiao and B Zhang ldquoEnergy implicationsof Chinarsquos regional development New insights from multi-regional input-output analysisrdquo Applied Energy vol 196 pp118ndash131 2017

[24] M A Quaddus and T N Goh ldquoElectric power generationexpansion Planning with multiple objectivesrdquo Applied Energyvol 19 no 4 pp 301ndash319 1985

[25] C H Antunes A G Martins and I S Brito ldquoA multipleobjective mixed integer linear programming model for powergeneration expansion planningrdquo Energy vol 29 no 4 pp 613ndash627 2004

[26] J L C Meza M B Yildirim and A S M Masud ldquoA modelfor the multiperiod multiobjective power generation expansionproblemrdquo IEEETransactions on Power Systems vol 22 no 2 pp871ndash878 2007

[27] J-HHan Y-C Ahn and I-B Lee ldquoAmulti-objective optimiza-tion model for sustainable electricity generation and CO2 mit-igation (EGCM) infrastructure design considering economicprofit and financial riskrdquo Applied Energy vol 95 pp 186ndash1952012

[28] M Rekik A Abdelkafi and L Krichen ldquoAmicro-grid ensuringmulti-objective control strategy of a power electrical system forquality improvementrdquo Energy vol 88 pp 351ndash363 2015

[29] A Azadeh Z Raoofi and M Zarrin ldquoA multi-objective fuzzylinear programming model for optimization of natural gassupply chain through a greenhouse gas reduction approachrdquoJournal of Natural Gas Science and Engineering vol 26 pp 702ndash710 2015

[30] K Li L Pan W Xue H Jiang and H Mao ldquoMulti-objectiveoptimization for energy performance improvement of residen-tial buildings a comparative studyrdquo Energies vol 10 no 2 p245 2017

[31] H Zhu W W Huang and G H Huang ldquoPlanning of regionalenergy systems An inexact mixed-integer fractional program-ming modelrdquo Applied Energy vol 113 pp 500ndash514 2014

[32] F Chen G H Huang Y R Fan and R F Liao ldquoA nonlinearfractional programming approach for environmentaleconomicpower dispatchrdquo International Journal of Electrical Power ampEnergy Systems vol 78 pp 463ndash469 2016

[33] C Zhang and P Guo ldquoFLFP A fuzzy linear fractional pro-gramming approach with double-sided fuzziness for optimalirrigation water allocationrdquo Agricultural Water Managementvol 199 pp 105ndash119 2018

[34] L Wang G Huang X Wang and H Zhu ldquoRisk-basedelectric power system planning for climate change mitigationthrough multi-stage joint-probabilistic left-hand-side chance-constrained fractional programming A Canadian case studyrdquoRenewable amp Sustainable Energy Reviews vol 82 pp 1056ndash10672018

[35] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[36] X H Nie G H Huang Y P Li and L Liu ldquoIFRP A hybridinterval-parameter fuzzy robust programming approach forwaste management planning under uncertaintyrdquo Journal ofEnvironmental Management vol 84 no 1 pp 1ndash11 2007

[37] Y Leung Spatial analysis and planning under imprecisionElsevier 1st edition 2013

[38] A Charnes W W Cooper and M J L Kirby ldquoChance-constrained programming an extension of statistical methodrdquoinOptimizingmethods in statistics pp 391ndash402 Academic Press1971

[39] Y P Cai G H Huang Z F Yang and Q Tan ldquoIdentificationof optimal strategies for energy management systems planningunder multiple uncertaintiesrdquoApplied Energy vol 86 no 4 pp480ndash495 2009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom