Research ArticleA Competitive Swarm Optimizer-Based TechnoeconomicOptimization with Appliance Scheduling in Domestic PV-BatteryHybrid Systems

Bo Wang 1 Yanjing Li1 Fei Yang2 and Xiaohua Xia3

1Key Laboratory of Ministry of Education for Image Processing and Intelligent ControlArticial Intelligence and Automation School Huazhong University of Science and Technology Wuhan 430074 China2Department of Articial Intelligence and Automation School of Electrical Engineering and Automation Wuhan UniversityWuhan 430072 China3Department of Electrical Electronic and Computer Engineering University of Pretoria Pretoria 0028 South Africa

Correspondence should be addressed to Bo Wang wb8517husteducn

Received 19 July 2019 Accepted 28 August 2019 Published 13 October 2019

Guest Editor Chun Wei

Copyright copy 2019 Bo Wang et al is 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

A technoeconomic optimization problem for a domestic grid-connected PV-battery hybrid energy system is investigated Itincorporates the appliance time scheduling with appliance-specic power dispatch e optimization is aimed at minimizingenergy cost maximizing renewable energy penetration and increasing user satisfaction over a nite horizon Nonlinear objectivefunctions and constraints as well as discrete and continuous decision variables are involved To solve the proposed mixed-integernonlinear programming problem at a large scale a competitive swarm optimizer-based numerical solver is designed andemployed e eectiveness of the proposed approach is veried by simulation results

1 Introduction

Making best use of renewable energies has been a topic thatreceives continuous attention [1] e photovoltaic (PV)energy is one of the most concerned renewable energybecause of the ubiquity of solar irradiation and very lowcarbon emission [2] e PV energy generation is thereforeintegrated into power grids in many countries [3] Howeverbecause of PV energyrsquos intermittent nature it is dicult touse PV energy alone to support sustained power demands inthe complicated context such as domestic electrical loads Apopular paradigm to utilize the PV power source is to in-tegrate the PV energy into hybrid energy systems wheremultiple power sources are adopted and dispatched co-operatively [4] ere were an enormous number of studieson hybrid system optimization over the past twenty yearsAdvanced technologies have been applied to the economicpower dispatch problem in hybrid energy systems [5ndash7]Most of such studies focused on power iexclow control

strategies where the demand side was rather consideredconstraints in the system ere were also studies that in-troduced demand-side management into hybrid energysystem management [8 9] whereas these studies hardlyexplored the potentials of controlling both power iexclows andload behaviors In general the potentials of incorporatingthe power dispatch with demand-side management remainto be explored at the current stage Indeed such a problem isdicult because of the complex correlations between powersources and loads and its large-scale nature

In this paper a technoeconomic optimization problem isextended and improved to investigate further energy e-ciency and economic potentials in such a domestic grid-connected PV-battery hybrid energy system based on aseries of previous studies by Tazvinga et al [10ndash15]ere aretwo parts of the interventions the power dispatch thatdecides which power is to be supplied to which load andappliance time scheduling that decides when to activate aspecic domestic appliance to fulll the user requirements

HindawiComplexityVolume 2019 Article ID 4824837 15 pageshttpsdoiorg10115520194824837

(e optimization is implemented over a finite time horizon(ere are three objectives involved in the optimizationFirstly the overall energy cost over the horizon has to beminimized (e energy cost mainly comes from the con-sumed grid power based on the time-of-use (TOU) tariff[16] (e battery wear cost is also integrated into the overallcost Secondly the usage of renewable energy has to bemaximized Given that the overall power demand from theloads is considered constant this objective is transformedinto the grid power consumption minimization (irdly theuser satisfaction has to be maximized An inconvenienceindicator has been proposed by Setlhaolo and Xia [12] so thatthe overall difference between the scheduled applianceoperation and the baseline appliance operation is calculated(e objective is thus introduced by minimizing such aninconvenience indicator A weighted sum approach isemployed to simultaneously optimize the three objectivessubject to a series of system constraints

(emain contributions of this study are listed as followsFirstly the hybrid system design is improved (e majorimprovement from the design perspective is that separatepower dispatch is introduced to each connected applianceinstead of considering the electrical load (consisting of avariety of appliances) as a whole (is is realized by in-troducing additional power lines between each pair of theappliances and power sources (e power dispatch is thusmanaged in a more flexible way In the previous model allappliances are compelled to choose the same power source atone time In the improved system design additional powerlines and switches are deployed such that appliances canchoose the power sources themselves (e supply thatcombines various power sources provides the system moreflexible power dispatch choices For example at peak hoursthe battery bank cannot support heavy loads independentlybecause of its capacity limitation According to the previousdesign the battery bank can only work for a short time in thelate evening otherwise the power demands cannot bematched In the new design the battery bank has a muchlonger possible working time by only supporting a part of theloads which leaves more space for the load schedulingImproved flexibility brings more energy efficiency poten-tials and thus more economic benefits can be achieved

Secondly the mathematical formulation of the tech-noeconomic optimization is improved as the dimension ofdecision variables is minimized (e aforementioned flexi-bility improves at the cost of additional decision variablesie the ONOFF states of the additional switches (enumber of decision variables grows largely as the problemscale grows spatially with the number of switches andlongitudinally with the number of control intervals incomparison with the previous system design To reduce theresulting computational burden the interplay among theswitch behaviors is identified Some switch behavior con-straints are involved for example the battery can onlycharge or discharge at one time and the appliance can onlyhave one active power line connection (e correlatedswitches manifest finite states as a result a set of discretestate variables are introduced (e values of the discretevariables indicate the combination of ONOFF states of the

correlated switches Given the constraints the state variableschoose values from a limited range In this way the numberof variables to describe the complex and interacting switchbehaviors is minimized As the ONOFF states of theswitches are a major part of the decision variables the di-mension of the decision variables is largely reducedtherefore the computational burden is smaller such that theproblem is more promising to be solved within limited time

(irdly an advanced numerical solver is designed for theproposed problem A major difficulty to implement thetechnoeconomic optimization is the solver (e investigatedproblem involves nonlinear objective functions and con-straints as well as continuous and discrete decision vari-ables It is thus a mixed-integer nonlinear programming(MINLP) problem Furthermore as mentioned above thedecision variable can be a large number eg over 700 Aproper solver to such a complicated and large-scale opti-mization problem is thus required An intelligent optimi-zation algorithm namely the competitive swarm optimizer(CSO) is employed to design the numerical solver Chengand Jin firstly proposed the CSO algorithm to solve large-scale optimization problems [17] (e CSO algorithm isdesigned on the basis of the particle swarm optimization(PSO) algorithm with a very different searching mechanismIn the PSO algorithm the term ldquoparticlerdquo is employed torefer to the individual solutions (e particles are charac-terized with two vectors namely the position and velocityvectors (e position vector describes the value of a solutionand the velocity vector the incremental of the value(e PSOupdates the position vector with the velocity vector viainteracting with the global best position in the swarm (thepopulation of individuals) and the personal best position inhistory [18] (e CSO algorithm adopts the position andvelocity vector modelling from PSO but employs a randompairwise competition mechanism such that the loser particlecan learn from the winner particle to update its position andvelocity In this way the CSO algorithm can reduce theopportunity of convergence to local optimum therebymanifesting better and satisfying overall performances thanlarge-scale PSO algorithms(e numerical solver is designedon the pairwise competition concept basis and modified tobetter match the investigated scenarios

A case study is employed to test and verify the effec-tiveness of the proposed approach where the power dispatchand appliance time scheduling on a daily basis are applied toa typical South African household hybrid system Tothoroughly investigate the effectiveness results from threecases where different objective functions are applied withthe new flexible power dispatch and the previous dispatchmethod are illustrated and analyzed For all cases the CSO-based numerical solver is employed and thus the powerdispatch methods are focused and compared

(e remainder of this paper is structured as followsSection 2 introduces the hybrid system component mod-elling Section 3 takes advantage of the component mod-elling to formulate the technoeconomic optimizationproblem Section 4 describes the CSO-based numericalsolver Section 5 shows the case study with simulation resultsand analysis Section 6 draws the conclusion

2 Complexity

2 Domestic PV-Battery Hybrid System

A domestic grid-connected PV-battery hybrid system ishereby employed as the investigated hybrid system (egeneral layout of the PV-battery hybrid system is illustratedin Figure 1 (e main purpose of such a system is to supplythe daily activities of a number of domestic appliances egelectrical water heater (EWH) stove television set andwashing machine (e involved appliances are connected toboth the power grid and the PV system A battery bank isalso introduced to facilitate the power dispatch (e batterybank is able to charge from the power sources which in thiscase are the power grid and PV system and discharge tosupply the appliances In order to make use of all possiblepower sources in the system a power management unit(PMU) is thereby introduced to implement the energyconversion and the power dispatch In this way the PMUmanages the operation of the system including (1) the se-lection of energy flow to support the active appliances (2)the time scheduling of the appliances and (3) the energyconversion and voltagecurrent matching It is clear that thePMU is the central piece of the hybrid system that managesfrom power quality to energy balance An assumption isemployed that the voltagecurrent matching is well main-tained by the PMU Our investigation focuses on the powerscheduling

(e power management diagram is depicted in Figure 2(ere are several components in the PMU From the PVside there are a solar charge controller and inverter (echarge controller integrates a DCDC converter to tune thePV output to match the DC loads including the battery (einverter receives input from the charge controller andbattery and converts the DC power inputs into the AC loadswhich in this case refer to the appliances From the grid sidethere is an AC charger integrating an ACDC converter thatallows the battery be charged by the grid (ere are also anumber of controllable switches to implement switchingcontrol strategies As Figure 2 depicts there is one switch foreach power line (e ONOFF states of the switches controlthe power flows in the system ie the pair of the applianceand its supplier For each connected appliance there are a setof switches that control whether the appliance is supplied bythe PV the battery or the power grid (e arrows on eachpower line indicate the direction of the power flow Asmentioned above the additional switches are deployed suchthat any connected appliance can choose among multiplepower sources by adjusting the ONOFF status of switchesTaking advantage of the developing smart grid and smartbuilding technologies appliances are equipped with opencommunication interfaces which allow the PMU toschedule the activities of the appliances and switches si-multaneously in both wired and wireless manners As aresult the system design in Figure 2 becomes feasible inpractice

Remark 1 (ere are more and more domestic loads thatcan be made DC in the modern daily life for example anAC light bulb can be replaced with a DC light-emittingdiode (LED) bulb In the future it is possible to connect

the DC loads directly to the DC power sources eg thePV and battery bank Such a system may reduce theoperational cost of renewable energy resources How-ever currently most domestic appliances are made ACfor the sake of standardization Consequently the systemdesign mainly focuses on the AC loads at the currentstage therefore all involved power sources are tuned tobe AC suppliers Direct connections between the DCsuppliers and the DC loads can be involved in the futuredesign

Assume that there are n appliances connected to thesystem A set of binary variables are employed to denote theONOFF state of the switches Let k denote the time instantduring operation g1(k) denotes the ONOFF state of theswitch between the PV charge controller and the batteryg2(k) the charge controller and the inverter g3(k) the ACcharger and the battery and g4(k) the battery and the in-verter g21(k) g22(k) g2n(k) denote the switches be-tween the PV and the appliances g41(k) g42(k) g4n(k)

the battery and the appliances and g51(k) g52(k) g5n(k)

the grid and the appliances t denotes the time instant over theoperation P1(k) and P4(k) denote the charging power fromthe PV and the grid respectively P2(k) P4(k) and P5(k)

denote the power outputs of the respective power sourcesIn this system the operation of appliances is managed

together with the power flows In this study the operationmanagement is actually time scheduling as the powers ofappliances are considered known a priori and invariable (etime scheduling is implemented simultaneously with thepower flow management such that the supply can match thedemand and achieve higher energy efficiency potentials

(e mathematical formulations of the behavior for eachcomponent in the system are introduced as follows

21 PV Systems (e PV consists of arrays of solar cells suchthat the solar energy is converted into electrical power (econverted power is proportional to the solar irradiation andthe size of PV panels As an alternative power source to thepower grid the power output is a major concern of the PVsystem It is formulated as follows

Power grid

Powermanagement

unit

Energy storage

Photovoltaicsystem

Appliance

+ndash

Figure 1 General layout of the hybrid system

Complexity 3

Ppv ηpvIpvAc (1)

where Ppv denotes the hourly PV power output (kW) ηpv

denotes the efficiency of the solar cells Ipv denotes thehourly solar irradiation per unit area (kWm2) and Ac

indicates the area of the PV panels that receive solar irra-diation (m2) (ere is an intermittent nature of the PVsystem that is Ipv can be absent at some sampling instantsAn output profile of the PV system is therefore requiredUsually the PV outputs over succeeding 24 hours are pre-dictable [19] Qpv denotes a time period where Ppv gt 0 Fork notin Qpv Ppv is considered to be zero given a very small Ipv APV output profile is usually employed to indicate the hourlyPV power outputs in a day Qpv can be identified via the PVoutput profile

22 Battery Bank (e battery bank charges from otherpower sources and discharges to support electrical loadactivities (e battery bank behaviors are dynamic owing tothe complicated scheduling of both power sources andappliances (e state of charge (SOC) is employed tocharacterize the battery bank status (e dynamics of theSOC can be formulated as follows

Soc(k + 1) Soc(k) + ηBg1(k)P1(k) + ηBηcg3(k)P3(k)

minus g4(k)P4(k)

(2)

where Soc(k) denotes the SOC at sampling instant k g1(k)g3(k) and g4(k) are binary variables that denote the ONOFF state of the respective switches at instant k as depictedin Figure 2 P1(k) P3(k) and P4(k) are the aforementionedpower outputs ηc denotes the energy conversion efficienciesof the AC charger and ηB denotes the battery chargingefficiency during operation Given the current-stage batterysystem limitations the simultaneous charging from two

different sources or simultaneous charging and dischargingare considered unpermitted A constraint of the switchesmust be taken into account

g1(k) + g3(k) + g4(k)le 1 (3)

such that only one of the switches g1 g3 and g4 can beturned on at the same time Following (2) the SOC at a giventime τ can be formulated as follows

Soc(τ) Soc(0) + ηB 1113944

τ

k0g1(k)P1(k) + ηBηc 1113944

τ

k0g3(k)P3(k)

minus 1113944τ

k0g4(k)P4(k)

(4)

where Soc(0) denotes the initial state of the battery bankSoc(k) is subject to the following constraint

Cmin le Soc(k)leC

max (5)

where Cmin and Cmax denote the minimum and maximumavailable capacity (kWh) respectively

(e battery wear level is also evaluated (e wear cost isformulated as follows

JB(τ) φbCD(τ)BC

TH (6)

where CD(τ) 1113936τk0P4(τ) denotes the overall throughput of

the battery bank until instant τ and BCTH denotes thebattery wear cost per 1 kWh from throughput energy inwhich BC denotes the battery cost and TH denotes theoverall throughput energy (e calculation of BCTH can befound from previous studies [15 20 21]

23 Power Grid From the hybrid system viewpoint thegrid supplies infinite and stable electricity at an alternative

AC charge

BT

pv Charge controller

Inverter

Inverter

Appliance_1

Appliance_2

Appliance_n

P1 P2

P3

P4

P5 g51

g52

g5n

g41

g42

g4n

g21

g22

g2n

g3

g1

Load

Figure 2 Schematic of the investigated power management system

4 Complexity

voltage level of 220 V (e grid power comes with a pricewhich results in the major operational costs As men-tioned above a TOU tariff is introduced such that thedemand response can be implemented Let ρ(k) denote theTOU electricity price at time k of a day ρ(k) changesaccording to which period the time k lies within (eoverall operational cost at a given time τ can be formu-lated as follows

Cost(τ) 1113944τ

k0ρ(k) P3(k) + P5(k)1113858 1113859 (7)

24 Appliances (e electrical loads consist of the loadprofiles of all appliances connected to the hybrid system(e boundary identification must be conducted beforesystem design Given the domestic scenario of the in-vestigated system several reasonable simplifications aremade to characterize the demand-side activities Firstlyall appliances within the system require standard ACpower Secondly each appliance is subject to a respectiveconstant operation duration Control strategies can beinvolved to determine the operation of such appliances[22 23]

According to previous studies [12ndash14] from the time-scheduling perspective domestic appliances can be cate-gorized into three types the flexible loads the shiftable loadsand the fixed loads (e flexible loadsrsquo working times can bescheduled freely at any favorable working time (e shift-able loadsrsquo working times can be scheduled within a pref-erable but limited time period (e fixed loadsrsquo workingtimes are fixed unchangeable in any case For an arbitraryappliance from any type the operation duration is constantsuch that the time scheduling can be characterized as theselection of the starting instant

In the investigated system an appliance is connected tothe PV battery bank and grid An appliance i from the nappliances is given As depicted in Figure 2 there is oneswitch for each power source connection (e ONOFFstates of the three switches at instant k are denoted by g2i(k)g4i(k) and g5i(k) (e simultaneous supply from multiplepower sources is unpermitted therefore a switch constraintis introduced as follows

g2i(k) + g4i(k) + g5i(k)le 1 i 1 2 n (8)

Let Sti denote the starting instant of the appliance i andDi the operation duration (e appliance continuouslyoperates until the end instant denoted by Eni Sti + Di minus 1A continuous operation constraint is introduced as Xia andZhang proposed [24]

1113944

Nminus Di+1

k0ui(k)ui(k + 1)ui(k + 2) ui k + Di minus 1( 1113857 1

i 1 2 n

(9)

where ui(k) is a binary variable that indicates whether theappliance i is active at time k ie the time schedule and N

denotes the length of a finite scheduling period In thiscase ui(k) 1 if Sti le kleEni and ui(k) 0 if otherwise Itis notable that the continuous operation constraint isassociated with the actual switch behavior in the followingway

g2i(k) + g4i(k) + g5i(k) 1 if ui(k) 1

g2i(k) + g4i(k) + g5i(k) 0 if ui(k) 01113896 (10)

Let pi(t) with t 0 1 2 Di minus 1 denote the loadprofile over the operation period (0 Di minus 1) of the appliancei Assuming that pi(t) are known a priori the power demandpi(k) can thus be determined if k Si + t pi(k) pi(t)otherwise pi(k) 0

25 System Constraints A series of constraints must beintroduced to the system such that the operation re-quirements are all satisfied and none of the physical laws isviolated

(a) (e energy balance must be fulfilled anytime duringoperation as indicated by the following equation

ηI2P2(k)

ηI4P4(k)

P5(k)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113944n

i1

g2i(k)

g4i(k)

g5i(k)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦pi(k) (11)

where P2(k) P4(k) and P5(k) are the power flowsfrom the PV output battery bank discharge andpower grid supply and ηI2 and ηI4 denote the in-verter efficiency of the PV and battery bank re-spectively For the convenience of calculation alogical state variable swi(k) is employed to describethe combination of g2i(k) g4i(k) and g5i(k) swi(k)

chooses a value from 0 1 2 3 swi(k) 0 indicatesthe state that all switches are turned off and theswitch states equal to 0 swi(k) 1 indicates thatonly g2i(k) 1 swi(k) 2 that only g4i(k) 1 andswi(k) 3 that only g5i(k) 1

(b) (e capacity constraints must be followed such thatthe power flow is kept within the range of thecomponent capacity (e power flows in the systemare thus limited (e power flows P1(k) and P2(k)

are subject to the following constraint

0leP1(k) + P2(k)le ηsPpv(k) (12)

where Ppv(k) as mentioned above is the PV output attime k and ηs denotes the charge controller efficiency(e power flows P2(k) and P4(k) as the supplier aresubject to the following constraints

0le ηI2P2(k)lePI2(k)

0le ηI4P2(k)lePI4(k)1113896 (13)

Complexity 5

where PI2(k) and PI4(k) denote the inverter capacity ofthe PV and battery bank respectively (e power flowsP3(k) and P5(k) are subject to the following constraint

0leg3(k)P3(k) + P5(k)lePmaxG (14)

where PmaxG denotes the allocated grid maximum

power for this grid-connected system (e powerflows P2(k) P4(k) and P5(k) are obtained from theenergy balance equation (11) P1(k) and P3(k) aredecided by the scheduling algorithm and they arepart of the decision variables

(c) (e switch control strategies are also subject toconstraints that prevent infeasible switch behaviors(ese constraints are formulated in preceding sec-tions along with the system component modellingie constraints (3) and (8)ndash(10) Some further as-sociations of the switch behaviors are identified asfollows

g1(k) 1 if P1(k)gt 0

g1(k) 0 if P1(k) 0

g3(k) 1 if P3(k)gt 0

g3(k) 0 if P3(k)gt 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)

g2(k) 1 if 1113937n

i1swi(k) minus 1( 1113857 0

g2(k) 0 if otherwise

g4(k) 1 if 1113937n

i1swi(k) minus 2( 1113857 0

g4(k) 0 if otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

In this way the states of g1(k)minus g4(k) at time k can beidentified from the values of P1(k) P3(k) and swi(k) withi 1 2 n

3 Problem Statement

(e primary management objective of the investigatedoptimization problem is cost minimization Secondly therenewable energy penetration is involved that is the usageof grid power should be minimized as well Owing to theTOU tariff the two objectives manifest certain differencesand must be equally considered in scheduling Furthermorethe user satisfaction in time scheduling is taken into accountAll these objectives are evaluated on a finite horizon basisLet T denote the number of sampling instants (e opti-mization problem is formulated as follows

31 Decision Variables (e involved decision variablesconsist of three parts (1) the switch control strategy decisionvariables (2) the charging power control variables and (3)the appliance time-scheduling decision variables It is giventhat i 1 2 n in the following discussion

(e switch control variables are swi(k) which cancharacterize the status of most switches in the system asconstraints (15) and (16) imply Given constraints (8)ndash(10) itis unnecessary to cover the whole finite horizon For theappliance i swi(k)gt 0 when Sti le kleEni and swi(k) 0 if klies outside the working period (erefore the minimumrequired control variables are swi(k) with k isin [Sti Eni] (edimension of this part is 1113936

ni1Di

(e charging power control variables are P1(k) andP3(k) Given constraints (3) and (15) P1(k)lowastP3(k) 0 atany given instant k therefore the dimension of this part isT

As mentioned above the appliance time scheduling issimplified with the known a priori and constant load profilepi(t) and operation duration Di Taking advantage of theknowledge the scheduling is implemented with decisionvariables Sti (e dimension of this part is n

(e dimension of the optimization is therefore1113936

ni1Di + T + n Comparing with the previous study [15] the

problem is extended to a higher dimension but simplified bytaking advantage of swi(k) and the constraints such that thedimension of the optimization grows slower than theproblem scale

32 Objectives (e cost minimization objective is formu-lated as follows

Jc 1113944T

k0ρ(k) P3(k) + P5(k)1113858 1113859 + φbJB(T) (17)

where φb denotes a weight that indicates the preferableimportance of the battery wearout to the decision-maker

(e renewable energy penetration objective is formu-lated as follows

Je 1113944T

k0P3(k) + P5(k)1113858 1113859 (18)

(e user satisfaction is evaluated via an inconvenienceindicator which is adopted from the study of Setlhaolo andXia [12]

β 1113944n

i1ci 1113944

T

k0u

bli (k) minus ui(k)1113960 1113961

2 (19)

where ci denotes an importance factor of the appliance i andubl

i (k) denotes the baseline time schedule of the appliance i(19) quantifies the difference between the baseline timeschedule and the adopted time schedule Given the systemmodelling and constraints in this study the user satisfactionevaluation is simplified as follows

β

1113944n

i1ci St

bli minus Sti1113960 1113961

2

11139741113972

(20)

where Stbli denotes the baseline starting instant Such a

difference has to be minimized such that the user remainshappy with the optimized appliance time schedule

6 Complexity

33 Technoeconomic Optimization Taking advantage of thepreceding objective functions and constraint formulationsthe technoeconomic optimization problem is obtained byminimizing the following objective function

J swi(k) P1(k) P3(k) Sti( 1113857 λcJc + λeJe + λbβ (21)

subject to the battery dynamics (2) and constraints (3)(8)ndash(10) and (11)ndash(16)

According to the above formulations there arenonlinear objective functions and constraints as well ascontinuous and discrete decision variables in the prob-lem (ey result in a mixed-integer nonlinear pro-gramming (MINLP) problem at a relatively large scale(e general theoretic approach of solving an MINLPproblem remains an open question as a result numericalsolvers are widely employed In the previous study [15]an OTPI toolbox httpswwwinverseproblemconzOPTIindexphpDLDownloadOPTI in MATLAB wasadopted as the numerical solver (e former solver tookquite a large amount of time for calculation In this studythe implementation of intelligent optimization algo-rithms on such a problem is investigated A newly pro-posed algorithm named the competitive swarm optimizer(CSO) is adopted as the numerical solver (e in-troduction to the CSO-based solver comes in the fol-lowing section

4 Numerical Solver Design

41Competitive SwarmOptimizer In the CSO algorithm letx denote a particle and w and l the indices of the winner andloser particles in a pair Assume that it is the G-th iterationand there have been k minus 1 competitions After the k-thcompetition the next-generation winner particle xwk(G +

1) remains the same as xwk(G) (e loser particlexlk(G + 1) namely the position vector is thereby updatedas follows

xlk(G + 1) xlk(G) + Vlk(G + 1) (22)

where Vlk(G + 1) is the next-generation velocity vectorupdated as follows

Vlk(G + 1) rn1(k G)Vlk(G) + rn2(k G) xwk(G) minus xlk(G)1113960 1113961

+ φrn3 xk(G) minus xlk(G)1113960 1113961

(23)

where rn1 rn2 and rn3 are random vectors xk(G) is thecenter of neighborhood filed particles ie a set of par-ticles that are close enough to xlk(G) and φ is theweighting coefficient of xk(G) Such a neighborhood fieldis predefined(ere is a special case that the neighborhoodcovers the whole swarm where xk(G) indicates the globalmean position of the particles at iteration G (e velocityand position vectors are employed for the continuouscases In a discrete case eg the decision variables swi(k)

in this study other update mechanisms must beemployed A crossover mechanism is hereby employed asfollows

xlk(rn G + 1) xwk(rn G)

xlk(rn G + 1) xlk(rn G)1113896 (24)

where rn denotes randomly generated indices of the com-ponents in a particle x and rn the unselected indices (24)suggests that the winner particle xwk(G) selects and copies apart of its components into the next-generation loser particlexlk(G + 1) (e unselected components of xlk(G + 1) re-main the same as xlk(G) In this way the loser particle canlearn from the winner particle

For an MINLP problem there are simultaneouslycontinuous and discrete components in a particle In thiscase the continuous part and discrete part are separated(23) and (22) are implemented on the continuous part and(24) is implemented on the discrete part After the learningprocess the two updated parts are combined again toobtain the particle of next generation (e theoretical proofof the convergence of the CSO algorithm can be referred to[17]

(e pseudocode of the CSO according to the precedingintroduction is illustrated in Algorithm 1 [25]

Remark 2 (e introduced CSO algorithm is mainlydesigned for continuous problems For discrete decisionvariables eg binary variables or integers the discrete PSOalgorithm [26 27] can facilitate the algorithm design (epairwise competition can be further introduced to otherevolutionary algorithms such as differential evolution(DE)

Remark 3 Given the investigated MINLP problem (21) theoriginal CSO algorithm cannot be applied in a straight-forward way Modifications that match the decision vari-ables are to be introduced such that satisfying performancescan be achieved

42 Modified CSO-Based Solver In order to implement theCSO algorithm on a constrained problem a penalty functionis introduced to the original objective function (17) Giventhat there are NC constraints to a problem the penaltyfunction is formulated as follows

Pen 1113944T

k01113944

NC

i0ωPeniPeni(k) (25)

where

Peni(k) 0 if constraint i is obeyed

M if constraint i is violated1113896 (26)

where M is a large positive number and ωPeni is theweighting factor for constraint i A fitness function is therebyformulated with (17) (25) and (26)

f(x) Jc + Pen (27)

In this way for a minimization problem a particlebecomes much less competitive when any of the constraints

Complexity 7

2 Domestic PV-Battery Hybrid System

A domestic grid-connected PV-battery hybrid system ishereby employed as the investigated hybrid system (egeneral layout of the PV-battery hybrid system is illustratedin Figure 1 (e main purpose of such a system is to supplythe daily activities of a number of domestic appliances egelectrical water heater (EWH) stove television set andwashing machine (e involved appliances are connected toboth the power grid and the PV system A battery bank isalso introduced to facilitate the power dispatch (e batterybank is able to charge from the power sources which in thiscase are the power grid and PV system and discharge tosupply the appliances In order to make use of all possiblepower sources in the system a power management unit(PMU) is thereby introduced to implement the energyconversion and the power dispatch In this way the PMUmanages the operation of the system including (1) the se-lection of energy flow to support the active appliances (2)the time scheduling of the appliances and (3) the energyconversion and voltagecurrent matching It is clear that thePMU is the central piece of the hybrid system that managesfrom power quality to energy balance An assumption isemployed that the voltagecurrent matching is well main-tained by the PMU Our investigation focuses on the powerscheduling

(e power management diagram is depicted in Figure 2(ere are several components in the PMU From the PVside there are a solar charge controller and inverter (echarge controller integrates a DCDC converter to tune thePV output to match the DC loads including the battery (einverter receives input from the charge controller andbattery and converts the DC power inputs into the AC loadswhich in this case refer to the appliances From the grid sidethere is an AC charger integrating an ACDC converter thatallows the battery be charged by the grid (ere are also anumber of controllable switches to implement switchingcontrol strategies As Figure 2 depicts there is one switch foreach power line (e ONOFF states of the switches controlthe power flows in the system ie the pair of the applianceand its supplier For each connected appliance there are a setof switches that control whether the appliance is supplied bythe PV the battery or the power grid (e arrows on eachpower line indicate the direction of the power flow Asmentioned above the additional switches are deployed suchthat any connected appliance can choose among multiplepower sources by adjusting the ONOFF status of switchesTaking advantage of the developing smart grid and smartbuilding technologies appliances are equipped with opencommunication interfaces which allow the PMU toschedule the activities of the appliances and switches si-multaneously in both wired and wireless manners As aresult the system design in Figure 2 becomes feasible inpractice

Remark 1 (ere are more and more domestic loads thatcan be made DC in the modern daily life for example anAC light bulb can be replaced with a DC light-emittingdiode (LED) bulb In the future it is possible to connect

the DC loads directly to the DC power sources eg thePV and battery bank Such a system may reduce theoperational cost of renewable energy resources How-ever currently most domestic appliances are made ACfor the sake of standardization Consequently the systemdesign mainly focuses on the AC loads at the currentstage therefore all involved power sources are tuned tobe AC suppliers Direct connections between the DCsuppliers and the DC loads can be involved in the futuredesign

Assume that there are n appliances connected to thesystem A set of binary variables are employed to denote theONOFF state of the switches Let k denote the time instantduring operation g1(k) denotes the ONOFF state of theswitch between the PV charge controller and the batteryg2(k) the charge controller and the inverter g3(k) the ACcharger and the battery and g4(k) the battery and the in-verter g21(k) g22(k) g2n(k) denote the switches be-tween the PV and the appliances g41(k) g42(k) g4n(k)

the battery and the appliances and g51(k) g52(k) g5n(k)

the grid and the appliances t denotes the time instant over theoperation P1(k) and P4(k) denote the charging power fromthe PV and the grid respectively P2(k) P4(k) and P5(k)

denote the power outputs of the respective power sourcesIn this system the operation of appliances is managed

together with the power flows In this study the operationmanagement is actually time scheduling as the powers ofappliances are considered known a priori and invariable (etime scheduling is implemented simultaneously with thepower flow management such that the supply can match thedemand and achieve higher energy efficiency potentials

(e mathematical formulations of the behavior for eachcomponent in the system are introduced as follows

21 PV Systems (e PV consists of arrays of solar cells suchthat the solar energy is converted into electrical power (econverted power is proportional to the solar irradiation andthe size of PV panels As an alternative power source to thepower grid the power output is a major concern of the PVsystem It is formulated as follows

Power grid

Powermanagement

unit

Energy storage

Photovoltaicsystem

Appliance

+ndash

Figure 1 General layout of the hybrid system

Complexity 3

Ppv ηpvIpvAc (1)

where Ppv denotes the hourly PV power output (kW) ηpv

denotes the efficiency of the solar cells Ipv denotes thehourly solar irradiation per unit area (kWm2) and Ac

indicates the area of the PV panels that receive solar irra-diation (m2) (ere is an intermittent nature of the PVsystem that is Ipv can be absent at some sampling instantsAn output profile of the PV system is therefore requiredUsually the PV outputs over succeeding 24 hours are pre-dictable [19] Qpv denotes a time period where Ppv gt 0 Fork notin Qpv Ppv is considered to be zero given a very small Ipv APV output profile is usually employed to indicate the hourlyPV power outputs in a day Qpv can be identified via the PVoutput profile

22 Battery Bank (e battery bank charges from otherpower sources and discharges to support electrical loadactivities (e battery bank behaviors are dynamic owing tothe complicated scheduling of both power sources andappliances (e state of charge (SOC) is employed tocharacterize the battery bank status (e dynamics of theSOC can be formulated as follows

Soc(k + 1) Soc(k) + ηBg1(k)P1(k) + ηBηcg3(k)P3(k)

minus g4(k)P4(k)

(2)

where Soc(k) denotes the SOC at sampling instant k g1(k)g3(k) and g4(k) are binary variables that denote the ONOFF state of the respective switches at instant k as depictedin Figure 2 P1(k) P3(k) and P4(k) are the aforementionedpower outputs ηc denotes the energy conversion efficienciesof the AC charger and ηB denotes the battery chargingefficiency during operation Given the current-stage batterysystem limitations the simultaneous charging from two

different sources or simultaneous charging and dischargingare considered unpermitted A constraint of the switchesmust be taken into account

g1(k) + g3(k) + g4(k)le 1 (3)

such that only one of the switches g1 g3 and g4 can beturned on at the same time Following (2) the SOC at a giventime τ can be formulated as follows

Soc(τ) Soc(0) + ηB 1113944

τ

k0g1(k)P1(k) + ηBηc 1113944

τ

k0g3(k)P3(k)

minus 1113944τ

k0g4(k)P4(k)

(4)

where Soc(0) denotes the initial state of the battery bankSoc(k) is subject to the following constraint

Cmin le Soc(k)leC

max (5)

where Cmin and Cmax denote the minimum and maximumavailable capacity (kWh) respectively

(e battery wear level is also evaluated (e wear cost isformulated as follows

JB(τ) φbCD(τ)BC

TH (6)

where CD(τ) 1113936τk0P4(τ) denotes the overall throughput of

the battery bank until instant τ and BCTH denotes thebattery wear cost per 1 kWh from throughput energy inwhich BC denotes the battery cost and TH denotes theoverall throughput energy (e calculation of BCTH can befound from previous studies [15 20 21]

23 Power Grid From the hybrid system viewpoint thegrid supplies infinite and stable electricity at an alternative

AC charge

BT

pv Charge controller

Inverter

Inverter

Appliance_1

Appliance_2

Appliance_n

P1 P2

P3

P4

P5 g51

g52

g5n

g41

g42

g4n

g21

g22

g2n

g3

g1

Load

Figure 2 Schematic of the investigated power management system

4 Complexity

voltage level of 220 V (e grid power comes with a pricewhich results in the major operational costs As men-tioned above a TOU tariff is introduced such that thedemand response can be implemented Let ρ(k) denote theTOU electricity price at time k of a day ρ(k) changesaccording to which period the time k lies within (eoverall operational cost at a given time τ can be formu-lated as follows

Cost(τ) 1113944τ

k0ρ(k) P3(k) + P5(k)1113858 1113859 (7)

24 Appliances (e electrical loads consist of the loadprofiles of all appliances connected to the hybrid system(e boundary identification must be conducted beforesystem design Given the domestic scenario of the in-vestigated system several reasonable simplifications aremade to characterize the demand-side activities Firstlyall appliances within the system require standard ACpower Secondly each appliance is subject to a respectiveconstant operation duration Control strategies can beinvolved to determine the operation of such appliances[22 23]

According to previous studies [12ndash14] from the time-scheduling perspective domestic appliances can be cate-gorized into three types the flexible loads the shiftable loadsand the fixed loads (e flexible loadsrsquo working times can bescheduled freely at any favorable working time (e shift-able loadsrsquo working times can be scheduled within a pref-erable but limited time period (e fixed loadsrsquo workingtimes are fixed unchangeable in any case For an arbitraryappliance from any type the operation duration is constantsuch that the time scheduling can be characterized as theselection of the starting instant

In the investigated system an appliance is connected tothe PV battery bank and grid An appliance i from the nappliances is given As depicted in Figure 2 there is oneswitch for each power source connection (e ONOFFstates of the three switches at instant k are denoted by g2i(k)g4i(k) and g5i(k) (e simultaneous supply from multiplepower sources is unpermitted therefore a switch constraintis introduced as follows

g2i(k) + g4i(k) + g5i(k)le 1 i 1 2 n (8)

Let Sti denote the starting instant of the appliance i andDi the operation duration (e appliance continuouslyoperates until the end instant denoted by Eni Sti + Di minus 1A continuous operation constraint is introduced as Xia andZhang proposed [24]

1113944

Nminus Di+1

k0ui(k)ui(k + 1)ui(k + 2) ui k + Di minus 1( 1113857 1

i 1 2 n

(9)

where ui(k) is a binary variable that indicates whether theappliance i is active at time k ie the time schedule and N

denotes the length of a finite scheduling period In thiscase ui(k) 1 if Sti le kleEni and ui(k) 0 if otherwise Itis notable that the continuous operation constraint isassociated with the actual switch behavior in the followingway

g2i(k) + g4i(k) + g5i(k) 1 if ui(k) 1

g2i(k) + g4i(k) + g5i(k) 0 if ui(k) 01113896 (10)

Let pi(t) with t 0 1 2 Di minus 1 denote the loadprofile over the operation period (0 Di minus 1) of the appliancei Assuming that pi(t) are known a priori the power demandpi(k) can thus be determined if k Si + t pi(k) pi(t)otherwise pi(k) 0

25 System Constraints A series of constraints must beintroduced to the system such that the operation re-quirements are all satisfied and none of the physical laws isviolated

(a) (e energy balance must be fulfilled anytime duringoperation as indicated by the following equation

ηI2P2(k)

ηI4P4(k)

P5(k)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113944n

i1

g2i(k)

g4i(k)

g5i(k)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦pi(k) (11)

where P2(k) P4(k) and P5(k) are the power flowsfrom the PV output battery bank discharge andpower grid supply and ηI2 and ηI4 denote the in-verter efficiency of the PV and battery bank re-spectively For the convenience of calculation alogical state variable swi(k) is employed to describethe combination of g2i(k) g4i(k) and g5i(k) swi(k)

chooses a value from 0 1 2 3 swi(k) 0 indicatesthe state that all switches are turned off and theswitch states equal to 0 swi(k) 1 indicates thatonly g2i(k) 1 swi(k) 2 that only g4i(k) 1 andswi(k) 3 that only g5i(k) 1

(b) (e capacity constraints must be followed such thatthe power flow is kept within the range of thecomponent capacity (e power flows in the systemare thus limited (e power flows P1(k) and P2(k)

are subject to the following constraint

0leP1(k) + P2(k)le ηsPpv(k) (12)

where Ppv(k) as mentioned above is the PV output attime k and ηs denotes the charge controller efficiency(e power flows P2(k) and P4(k) as the supplier aresubject to the following constraints

0le ηI2P2(k)lePI2(k)

0le ηI4P2(k)lePI4(k)1113896 (13)

Complexity 5

where PI2(k) and PI4(k) denote the inverter capacity ofthe PV and battery bank respectively (e power flowsP3(k) and P5(k) are subject to the following constraint

0leg3(k)P3(k) + P5(k)lePmaxG (14)

where PmaxG denotes the allocated grid maximum

power for this grid-connected system (e powerflows P2(k) P4(k) and P5(k) are obtained from theenergy balance equation (11) P1(k) and P3(k) aredecided by the scheduling algorithm and they arepart of the decision variables

(c) (e switch control strategies are also subject toconstraints that prevent infeasible switch behaviors(ese constraints are formulated in preceding sec-tions along with the system component modellingie constraints (3) and (8)ndash(10) Some further as-sociations of the switch behaviors are identified asfollows

g1(k) 1 if P1(k)gt 0

g1(k) 0 if P1(k) 0

g3(k) 1 if P3(k)gt 0

g3(k) 0 if P3(k)gt 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)

g2(k) 1 if 1113937n

i1swi(k) minus 1( 1113857 0

g2(k) 0 if otherwise

g4(k) 1 if 1113937n

i1swi(k) minus 2( 1113857 0

g4(k) 0 if otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

In this way the states of g1(k)minus g4(k) at time k can beidentified from the values of P1(k) P3(k) and swi(k) withi 1 2 n

3 Problem Statement

(e primary management objective of the investigatedoptimization problem is cost minimization Secondly therenewable energy penetration is involved that is the usageof grid power should be minimized as well Owing to theTOU tariff the two objectives manifest certain differencesand must be equally considered in scheduling Furthermorethe user satisfaction in time scheduling is taken into accountAll these objectives are evaluated on a finite horizon basisLet T denote the number of sampling instants (e opti-mization problem is formulated as follows

31 Decision Variables (e involved decision variablesconsist of three parts (1) the switch control strategy decisionvariables (2) the charging power control variables and (3)the appliance time-scheduling decision variables It is giventhat i 1 2 n in the following discussion

(e switch control variables are swi(k) which cancharacterize the status of most switches in the system asconstraints (15) and (16) imply Given constraints (8)ndash(10) itis unnecessary to cover the whole finite horizon For theappliance i swi(k)gt 0 when Sti le kleEni and swi(k) 0 if klies outside the working period (erefore the minimumrequired control variables are swi(k) with k isin [Sti Eni] (edimension of this part is 1113936

ni1Di

(e charging power control variables are P1(k) andP3(k) Given constraints (3) and (15) P1(k)lowastP3(k) 0 atany given instant k therefore the dimension of this part isT

As mentioned above the appliance time scheduling issimplified with the known a priori and constant load profilepi(t) and operation duration Di Taking advantage of theknowledge the scheduling is implemented with decisionvariables Sti (e dimension of this part is n

(e dimension of the optimization is therefore1113936

ni1Di + T + n Comparing with the previous study [15] the

problem is extended to a higher dimension but simplified bytaking advantage of swi(k) and the constraints such that thedimension of the optimization grows slower than theproblem scale

32 Objectives (e cost minimization objective is formu-lated as follows

Jc 1113944T

k0ρ(k) P3(k) + P5(k)1113858 1113859 + φbJB(T) (17)

where φb denotes a weight that indicates the preferableimportance of the battery wearout to the decision-maker

(e renewable energy penetration objective is formu-lated as follows

Je 1113944T

k0P3(k) + P5(k)1113858 1113859 (18)

(e user satisfaction is evaluated via an inconvenienceindicator which is adopted from the study of Setlhaolo andXia [12]

β 1113944n

i1ci 1113944

T

k0u

bli (k) minus ui(k)1113960 1113961

2 (19)

where ci denotes an importance factor of the appliance i andubl

i (k) denotes the baseline time schedule of the appliance i(19) quantifies the difference between the baseline timeschedule and the adopted time schedule Given the systemmodelling and constraints in this study the user satisfactionevaluation is simplified as follows

β

1113944n

i1ci St

bli minus Sti1113960 1113961

2

11139741113972

(20)

where Stbli denotes the baseline starting instant Such a

difference has to be minimized such that the user remainshappy with the optimized appliance time schedule

6 Complexity

33 Technoeconomic Optimization Taking advantage of thepreceding objective functions and constraint formulationsthe technoeconomic optimization problem is obtained byminimizing the following objective function

J swi(k) P1(k) P3(k) Sti( 1113857 λcJc + λeJe + λbβ (21)

subject to the battery dynamics (2) and constraints (3)(8)ndash(10) and (11)ndash(16)

According to the above formulations there arenonlinear objective functions and constraints as well ascontinuous and discrete decision variables in the prob-lem (ey result in a mixed-integer nonlinear pro-gramming (MINLP) problem at a relatively large scale(e general theoretic approach of solving an MINLPproblem remains an open question as a result numericalsolvers are widely employed In the previous study [15]an OTPI toolbox httpswwwinverseproblemconzOPTIindexphpDLDownloadOPTI in MATLAB wasadopted as the numerical solver (e former solver tookquite a large amount of time for calculation In this studythe implementation of intelligent optimization algo-rithms on such a problem is investigated A newly pro-posed algorithm named the competitive swarm optimizer(CSO) is adopted as the numerical solver (e in-troduction to the CSO-based solver comes in the fol-lowing section

4 Numerical Solver Design

41Competitive SwarmOptimizer In the CSO algorithm letx denote a particle and w and l the indices of the winner andloser particles in a pair Assume that it is the G-th iterationand there have been k minus 1 competitions After the k-thcompetition the next-generation winner particle xwk(G +

1) remains the same as xwk(G) (e loser particlexlk(G + 1) namely the position vector is thereby updatedas follows

xlk(G + 1) xlk(G) + Vlk(G + 1) (22)

where Vlk(G + 1) is the next-generation velocity vectorupdated as follows

Vlk(G + 1) rn1(k G)Vlk(G) + rn2(k G) xwk(G) minus xlk(G)1113960 1113961

+ φrn3 xk(G) minus xlk(G)1113960 1113961

(23)

where rn1 rn2 and rn3 are random vectors xk(G) is thecenter of neighborhood filed particles ie a set of par-ticles that are close enough to xlk(G) and φ is theweighting coefficient of xk(G) Such a neighborhood fieldis predefined(ere is a special case that the neighborhoodcovers the whole swarm where xk(G) indicates the globalmean position of the particles at iteration G (e velocityand position vectors are employed for the continuouscases In a discrete case eg the decision variables swi(k)

in this study other update mechanisms must beemployed A crossover mechanism is hereby employed asfollows

xlk(rn G + 1) xwk(rn G)

xlk(rn G + 1) xlk(rn G)1113896 (24)

where rn denotes randomly generated indices of the com-ponents in a particle x and rn the unselected indices (24)suggests that the winner particle xwk(G) selects and copies apart of its components into the next-generation loser particlexlk(G + 1) (e unselected components of xlk(G + 1) re-main the same as xlk(G) In this way the loser particle canlearn from the winner particle

For an MINLP problem there are simultaneouslycontinuous and discrete components in a particle In thiscase the continuous part and discrete part are separated(23) and (22) are implemented on the continuous part and(24) is implemented on the discrete part After the learningprocess the two updated parts are combined again toobtain the particle of next generation (e theoretical proofof the convergence of the CSO algorithm can be referred to[17]

(e pseudocode of the CSO according to the precedingintroduction is illustrated in Algorithm 1 [25]

Remark 2 (e introduced CSO algorithm is mainlydesigned for continuous problems For discrete decisionvariables eg binary variables or integers the discrete PSOalgorithm [26 27] can facilitate the algorithm design (epairwise competition can be further introduced to otherevolutionary algorithms such as differential evolution(DE)

Remark 3 Given the investigated MINLP problem (21) theoriginal CSO algorithm cannot be applied in a straight-forward way Modifications that match the decision vari-ables are to be introduced such that satisfying performancescan be achieved

42 Modified CSO-Based Solver In order to implement theCSO algorithm on a constrained problem a penalty functionis introduced to the original objective function (17) Giventhat there are NC constraints to a problem the penaltyfunction is formulated as follows

Pen 1113944T

k01113944

NC

i0ωPeniPeni(k) (25)

where

Peni(k) 0 if constraint i is obeyed

M if constraint i is violated1113896 (26)

where M is a large positive number and ωPeni is theweighting factor for constraint i A fitness function is therebyformulated with (17) (25) and (26)

f(x) Jc + Pen (27)

In this way for a minimization problem a particlebecomes much less competitive when any of the constraints

Complexity 7

Ppv ηpvIpvAc (1)

where Ppv denotes the hourly PV power output (kW) ηpv

denotes the efficiency of the solar cells Ipv denotes thehourly solar irradiation per unit area (kWm2) and Ac

indicates the area of the PV panels that receive solar irra-diation (m2) (ere is an intermittent nature of the PVsystem that is Ipv can be absent at some sampling instantsAn output profile of the PV system is therefore requiredUsually the PV outputs over succeeding 24 hours are pre-dictable [19] Qpv denotes a time period where Ppv gt 0 Fork notin Qpv Ppv is considered to be zero given a very small Ipv APV output profile is usually employed to indicate the hourlyPV power outputs in a day Qpv can be identified via the PVoutput profile

22 Battery Bank (e battery bank charges from otherpower sources and discharges to support electrical loadactivities (e battery bank behaviors are dynamic owing tothe complicated scheduling of both power sources andappliances (e state of charge (SOC) is employed tocharacterize the battery bank status (e dynamics of theSOC can be formulated as follows

Soc(k + 1) Soc(k) + ηBg1(k)P1(k) + ηBηcg3(k)P3(k)

minus g4(k)P4(k)

(2)

where Soc(k) denotes the SOC at sampling instant k g1(k)g3(k) and g4(k) are binary variables that denote the ONOFF state of the respective switches at instant k as depictedin Figure 2 P1(k) P3(k) and P4(k) are the aforementionedpower outputs ηc denotes the energy conversion efficienciesof the AC charger and ηB denotes the battery chargingefficiency during operation Given the current-stage batterysystem limitations the simultaneous charging from two

different sources or simultaneous charging and dischargingare considered unpermitted A constraint of the switchesmust be taken into account

g1(k) + g3(k) + g4(k)le 1 (3)

such that only one of the switches g1 g3 and g4 can beturned on at the same time Following (2) the SOC at a giventime τ can be formulated as follows

Soc(τ) Soc(0) + ηB 1113944

τ

k0g1(k)P1(k) + ηBηc 1113944

τ

k0g3(k)P3(k)

minus 1113944τ

k0g4(k)P4(k)

(4)

where Soc(0) denotes the initial state of the battery bankSoc(k) is subject to the following constraint

Cmin le Soc(k)leC

max (5)

where Cmin and Cmax denote the minimum and maximumavailable capacity (kWh) respectively

(e battery wear level is also evaluated (e wear cost isformulated as follows

JB(τ) φbCD(τ)BC

TH (6)

where CD(τ) 1113936τk0P4(τ) denotes the overall throughput of

the battery bank until instant τ and BCTH denotes thebattery wear cost per 1 kWh from throughput energy inwhich BC denotes the battery cost and TH denotes theoverall throughput energy (e calculation of BCTH can befound from previous studies [15 20 21]

23 Power Grid From the hybrid system viewpoint thegrid supplies infinite and stable electricity at an alternative

AC charge

BT

pv Charge controller

Inverter

Inverter

Appliance_1

Appliance_2

Appliance_n

P1 P2

P3

P4

P5 g51

g52

g5n

g41

g42

g4n

g21

g22

g2n

g3

g1

Load

Figure 2 Schematic of the investigated power management system

4 Complexity

voltage level of 220 V (e grid power comes with a pricewhich results in the major operational costs As men-tioned above a TOU tariff is introduced such that thedemand response can be implemented Let ρ(k) denote theTOU electricity price at time k of a day ρ(k) changesaccording to which period the time k lies within (eoverall operational cost at a given time τ can be formu-lated as follows

Cost(τ) 1113944τ

k0ρ(k) P3(k) + P5(k)1113858 1113859 (7)

24 Appliances (e electrical loads consist of the loadprofiles of all appliances connected to the hybrid system(e boundary identification must be conducted beforesystem design Given the domestic scenario of the in-vestigated system several reasonable simplifications aremade to characterize the demand-side activities Firstlyall appliances within the system require standard ACpower Secondly each appliance is subject to a respectiveconstant operation duration Control strategies can beinvolved to determine the operation of such appliances[22 23]

According to previous studies [12ndash14] from the time-scheduling perspective domestic appliances can be cate-gorized into three types the flexible loads the shiftable loadsand the fixed loads (e flexible loadsrsquo working times can bescheduled freely at any favorable working time (e shift-able loadsrsquo working times can be scheduled within a pref-erable but limited time period (e fixed loadsrsquo workingtimes are fixed unchangeable in any case For an arbitraryappliance from any type the operation duration is constantsuch that the time scheduling can be characterized as theselection of the starting instant

In the investigated system an appliance is connected tothe PV battery bank and grid An appliance i from the nappliances is given As depicted in Figure 2 there is oneswitch for each power source connection (e ONOFFstates of the three switches at instant k are denoted by g2i(k)g4i(k) and g5i(k) (e simultaneous supply from multiplepower sources is unpermitted therefore a switch constraintis introduced as follows

g2i(k) + g4i(k) + g5i(k)le 1 i 1 2 n (8)

Let Sti denote the starting instant of the appliance i andDi the operation duration (e appliance continuouslyoperates until the end instant denoted by Eni Sti + Di minus 1A continuous operation constraint is introduced as Xia andZhang proposed [24]

1113944

Nminus Di+1

k0ui(k)ui(k + 1)ui(k + 2) ui k + Di minus 1( 1113857 1

i 1 2 n

(9)

where ui(k) is a binary variable that indicates whether theappliance i is active at time k ie the time schedule and N

denotes the length of a finite scheduling period In thiscase ui(k) 1 if Sti le kleEni and ui(k) 0 if otherwise Itis notable that the continuous operation constraint isassociated with the actual switch behavior in the followingway

g2i(k) + g4i(k) + g5i(k) 1 if ui(k) 1

g2i(k) + g4i(k) + g5i(k) 0 if ui(k) 01113896 (10)

Let pi(t) with t 0 1 2 Di minus 1 denote the loadprofile over the operation period (0 Di minus 1) of the appliancei Assuming that pi(t) are known a priori the power demandpi(k) can thus be determined if k Si + t pi(k) pi(t)otherwise pi(k) 0

25 System Constraints A series of constraints must beintroduced to the system such that the operation re-quirements are all satisfied and none of the physical laws isviolated

(a) (e energy balance must be fulfilled anytime duringoperation as indicated by the following equation

ηI2P2(k)

ηI4P4(k)

P5(k)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113944n

i1

g2i(k)

g4i(k)

g5i(k)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦pi(k) (11)

where P2(k) P4(k) and P5(k) are the power flowsfrom the PV output battery bank discharge andpower grid supply and ηI2 and ηI4 denote the in-verter efficiency of the PV and battery bank re-spectively For the convenience of calculation alogical state variable swi(k) is employed to describethe combination of g2i(k) g4i(k) and g5i(k) swi(k)

chooses a value from 0 1 2 3 swi(k) 0 indicatesthe state that all switches are turned off and theswitch states equal to 0 swi(k) 1 indicates thatonly g2i(k) 1 swi(k) 2 that only g4i(k) 1 andswi(k) 3 that only g5i(k) 1

(b) (e capacity constraints must be followed such thatthe power flow is kept within the range of thecomponent capacity (e power flows in the systemare thus limited (e power flows P1(k) and P2(k)

are subject to the following constraint

0leP1(k) + P2(k)le ηsPpv(k) (12)

where Ppv(k) as mentioned above is the PV output attime k and ηs denotes the charge controller efficiency(e power flows P2(k) and P4(k) as the supplier aresubject to the following constraints

0le ηI2P2(k)lePI2(k)

0le ηI4P2(k)lePI4(k)1113896 (13)

Complexity 5

where PI2(k) and PI4(k) denote the inverter capacity ofthe PV and battery bank respectively (e power flowsP3(k) and P5(k) are subject to the following constraint

0leg3(k)P3(k) + P5(k)lePmaxG (14)

where PmaxG denotes the allocated grid maximum

power for this grid-connected system (e powerflows P2(k) P4(k) and P5(k) are obtained from theenergy balance equation (11) P1(k) and P3(k) aredecided by the scheduling algorithm and they arepart of the decision variables

(c) (e switch control strategies are also subject toconstraints that prevent infeasible switch behaviors(ese constraints are formulated in preceding sec-tions along with the system component modellingie constraints (3) and (8)ndash(10) Some further as-sociations of the switch behaviors are identified asfollows

g1(k) 1 if P1(k)gt 0

g1(k) 0 if P1(k) 0

g3(k) 1 if P3(k)gt 0

g3(k) 0 if P3(k)gt 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)

g2(k) 1 if 1113937n

i1swi(k) minus 1( 1113857 0

g2(k) 0 if otherwise

g4(k) 1 if 1113937n

i1swi(k) minus 2( 1113857 0

g4(k) 0 if otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

In this way the states of g1(k)minus g4(k) at time k can beidentified from the values of P1(k) P3(k) and swi(k) withi 1 2 n

3 Problem Statement

(e primary management objective of the investigatedoptimization problem is cost minimization Secondly therenewable energy penetration is involved that is the usageof grid power should be minimized as well Owing to theTOU tariff the two objectives manifest certain differencesand must be equally considered in scheduling Furthermorethe user satisfaction in time scheduling is taken into accountAll these objectives are evaluated on a finite horizon basisLet T denote the number of sampling instants (e opti-mization problem is formulated as follows

31 Decision Variables (e involved decision variablesconsist of three parts (1) the switch control strategy decisionvariables (2) the charging power control variables and (3)the appliance time-scheduling decision variables It is giventhat i 1 2 n in the following discussion

(e switch control variables are swi(k) which cancharacterize the status of most switches in the system asconstraints (15) and (16) imply Given constraints (8)ndash(10) itis unnecessary to cover the whole finite horizon For theappliance i swi(k)gt 0 when Sti le kleEni and swi(k) 0 if klies outside the working period (erefore the minimumrequired control variables are swi(k) with k isin [Sti Eni] (edimension of this part is 1113936

ni1Di

(e charging power control variables are P1(k) andP3(k) Given constraints (3) and (15) P1(k)lowastP3(k) 0 atany given instant k therefore the dimension of this part isT

As mentioned above the appliance time scheduling issimplified with the known a priori and constant load profilepi(t) and operation duration Di Taking advantage of theknowledge the scheduling is implemented with decisionvariables Sti (e dimension of this part is n

(e dimension of the optimization is therefore1113936

ni1Di + T + n Comparing with the previous study [15] the

problem is extended to a higher dimension but simplified bytaking advantage of swi(k) and the constraints such that thedimension of the optimization grows slower than theproblem scale

32 Objectives (e cost minimization objective is formu-lated as follows

Jc 1113944T

k0ρ(k) P3(k) + P5(k)1113858 1113859 + φbJB(T) (17)

where φb denotes a weight that indicates the preferableimportance of the battery wearout to the decision-maker

(e renewable energy penetration objective is formu-lated as follows

Je 1113944T

k0P3(k) + P5(k)1113858 1113859 (18)

(e user satisfaction is evaluated via an inconvenienceindicator which is adopted from the study of Setlhaolo andXia [12]

β 1113944n

i1ci 1113944

T

k0u

bli (k) minus ui(k)1113960 1113961

2 (19)

where ci denotes an importance factor of the appliance i andubl

i (k) denotes the baseline time schedule of the appliance i(19) quantifies the difference between the baseline timeschedule and the adopted time schedule Given the systemmodelling and constraints in this study the user satisfactionevaluation is simplified as follows

β

1113944n

i1ci St

bli minus Sti1113960 1113961

2

11139741113972

(20)

where Stbli denotes the baseline starting instant Such a

difference has to be minimized such that the user remainshappy with the optimized appliance time schedule

6 Complexity

33 Technoeconomic Optimization Taking advantage of thepreceding objective functions and constraint formulationsthe technoeconomic optimization problem is obtained byminimizing the following objective function

J swi(k) P1(k) P3(k) Sti( 1113857 λcJc + λeJe + λbβ (21)

subject to the battery dynamics (2) and constraints (3)(8)ndash(10) and (11)ndash(16)

According to the above formulations there arenonlinear objective functions and constraints as well ascontinuous and discrete decision variables in the prob-lem (ey result in a mixed-integer nonlinear pro-gramming (MINLP) problem at a relatively large scale(e general theoretic approach of solving an MINLPproblem remains an open question as a result numericalsolvers are widely employed In the previous study [15]an OTPI toolbox httpswwwinverseproblemconzOPTIindexphpDLDownloadOPTI in MATLAB wasadopted as the numerical solver (e former solver tookquite a large amount of time for calculation In this studythe implementation of intelligent optimization algo-rithms on such a problem is investigated A newly pro-posed algorithm named the competitive swarm optimizer(CSO) is adopted as the numerical solver (e in-troduction to the CSO-based solver comes in the fol-lowing section

4 Numerical Solver Design

41Competitive SwarmOptimizer In the CSO algorithm letx denote a particle and w and l the indices of the winner andloser particles in a pair Assume that it is the G-th iterationand there have been k minus 1 competitions After the k-thcompetition the next-generation winner particle xwk(G +

1) remains the same as xwk(G) (e loser particlexlk(G + 1) namely the position vector is thereby updatedas follows

xlk(G + 1) xlk(G) + Vlk(G + 1) (22)

where Vlk(G + 1) is the next-generation velocity vectorupdated as follows

Vlk(G + 1) rn1(k G)Vlk(G) + rn2(k G) xwk(G) minus xlk(G)1113960 1113961

+ φrn3 xk(G) minus xlk(G)1113960 1113961

(23)

where rn1 rn2 and rn3 are random vectors xk(G) is thecenter of neighborhood filed particles ie a set of par-ticles that are close enough to xlk(G) and φ is theweighting coefficient of xk(G) Such a neighborhood fieldis predefined(ere is a special case that the neighborhoodcovers the whole swarm where xk(G) indicates the globalmean position of the particles at iteration G (e velocityand position vectors are employed for the continuouscases In a discrete case eg the decision variables swi(k)

in this study other update mechanisms must beemployed A crossover mechanism is hereby employed asfollows

xlk(rn G + 1) xwk(rn G)

xlk(rn G + 1) xlk(rn G)1113896 (24)

where rn denotes randomly generated indices of the com-ponents in a particle x and rn the unselected indices (24)suggests that the winner particle xwk(G) selects and copies apart of its components into the next-generation loser particlexlk(G + 1) (e unselected components of xlk(G + 1) re-main the same as xlk(G) In this way the loser particle canlearn from the winner particle

For an MINLP problem there are simultaneouslycontinuous and discrete components in a particle In thiscase the continuous part and discrete part are separated(23) and (22) are implemented on the continuous part and(24) is implemented on the discrete part After the learningprocess the two updated parts are combined again toobtain the particle of next generation (e theoretical proofof the convergence of the CSO algorithm can be referred to[17]

(e pseudocode of the CSO according to the precedingintroduction is illustrated in Algorithm 1 [25]

Remark 2 (e introduced CSO algorithm is mainlydesigned for continuous problems For discrete decisionvariables eg binary variables or integers the discrete PSOalgorithm [26 27] can facilitate the algorithm design (epairwise competition can be further introduced to otherevolutionary algorithms such as differential evolution(DE)

Remark 3 Given the investigated MINLP problem (21) theoriginal CSO algorithm cannot be applied in a straight-forward way Modifications that match the decision vari-ables are to be introduced such that satisfying performancescan be achieved

42 Modified CSO-Based Solver In order to implement theCSO algorithm on a constrained problem a penalty functionis introduced to the original objective function (17) Giventhat there are NC constraints to a problem the penaltyfunction is formulated as follows

Pen 1113944T

k01113944

NC

i0ωPeniPeni(k) (25)

where

Peni(k) 0 if constraint i is obeyed

M if constraint i is violated1113896 (26)

where M is a large positive number and ωPeni is theweighting factor for constraint i A fitness function is therebyformulated with (17) (25) and (26)

f(x) Jc + Pen (27)

In this way for a minimization problem a particlebecomes much less competitive when any of the constraints

Complexity 7

voltage level of 220 V (e grid power comes with a pricewhich results in the major operational costs As men-tioned above a TOU tariff is introduced such that thedemand response can be implemented Let ρ(k) denote theTOU electricity price at time k of a day ρ(k) changesaccording to which period the time k lies within (eoverall operational cost at a given time τ can be formu-lated as follows

Cost(τ) 1113944τ

k0ρ(k) P3(k) + P5(k)1113858 1113859 (7)

24 Appliances (e electrical loads consist of the loadprofiles of all appliances connected to the hybrid system(e boundary identification must be conducted beforesystem design Given the domestic scenario of the in-vestigated system several reasonable simplifications aremade to characterize the demand-side activities Firstlyall appliances within the system require standard ACpower Secondly each appliance is subject to a respectiveconstant operation duration Control strategies can beinvolved to determine the operation of such appliances[22 23]

According to previous studies [12ndash14] from the time-scheduling perspective domestic appliances can be cate-gorized into three types the flexible loads the shiftable loadsand the fixed loads (e flexible loadsrsquo working times can bescheduled freely at any favorable working time (e shift-able loadsrsquo working times can be scheduled within a pref-erable but limited time period (e fixed loadsrsquo workingtimes are fixed unchangeable in any case For an arbitraryappliance from any type the operation duration is constantsuch that the time scheduling can be characterized as theselection of the starting instant

In the investigated system an appliance is connected tothe PV battery bank and grid An appliance i from the nappliances is given As depicted in Figure 2 there is oneswitch for each power source connection (e ONOFFstates of the three switches at instant k are denoted by g2i(k)g4i(k) and g5i(k) (e simultaneous supply from multiplepower sources is unpermitted therefore a switch constraintis introduced as follows

g2i(k) + g4i(k) + g5i(k)le 1 i 1 2 n (8)

Let Sti denote the starting instant of the appliance i andDi the operation duration (e appliance continuouslyoperates until the end instant denoted by Eni Sti + Di minus 1A continuous operation constraint is introduced as Xia andZhang proposed [24]

1113944

Nminus Di+1

k0ui(k)ui(k + 1)ui(k + 2) ui k + Di minus 1( 1113857 1

i 1 2 n

(9)

where ui(k) is a binary variable that indicates whether theappliance i is active at time k ie the time schedule and N

denotes the length of a finite scheduling period In thiscase ui(k) 1 if Sti le kleEni and ui(k) 0 if otherwise Itis notable that the continuous operation constraint isassociated with the actual switch behavior in the followingway

g2i(k) + g4i(k) + g5i(k) 1 if ui(k) 1

g2i(k) + g4i(k) + g5i(k) 0 if ui(k) 01113896 (10)

Let pi(t) with t 0 1 2 Di minus 1 denote the loadprofile over the operation period (0 Di minus 1) of the appliancei Assuming that pi(t) are known a priori the power demandpi(k) can thus be determined if k Si + t pi(k) pi(t)otherwise pi(k) 0

25 System Constraints A series of constraints must beintroduced to the system such that the operation re-quirements are all satisfied and none of the physical laws isviolated

(a) (e energy balance must be fulfilled anytime duringoperation as indicated by the following equation

ηI2P2(k)

ηI4P4(k)

P5(k)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113944n

i1

g2i(k)

g4i(k)

g5i(k)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦pi(k) (11)

where P2(k) P4(k) and P5(k) are the power flowsfrom the PV output battery bank discharge andpower grid supply and ηI2 and ηI4 denote the in-verter efficiency of the PV and battery bank re-spectively For the convenience of calculation alogical state variable swi(k) is employed to describethe combination of g2i(k) g4i(k) and g5i(k) swi(k)

chooses a value from 0 1 2 3 swi(k) 0 indicatesthe state that all switches are turned off and theswitch states equal to 0 swi(k) 1 indicates thatonly g2i(k) 1 swi(k) 2 that only g4i(k) 1 andswi(k) 3 that only g5i(k) 1

(b) (e capacity constraints must be followed such thatthe power flow is kept within the range of thecomponent capacity (e power flows in the systemare thus limited (e power flows P1(k) and P2(k)

are subject to the following constraint

0leP1(k) + P2(k)le ηsPpv(k) (12)

where Ppv(k) as mentioned above is the PV output attime k and ηs denotes the charge controller efficiency(e power flows P2(k) and P4(k) as the supplier aresubject to the following constraints

0le ηI2P2(k)lePI2(k)

0le ηI4P2(k)lePI4(k)1113896 (13)

Complexity 5

where PI2(k) and PI4(k) denote the inverter capacity ofthe PV and battery bank respectively (e power flowsP3(k) and P5(k) are subject to the following constraint

0leg3(k)P3(k) + P5(k)lePmaxG (14)

where PmaxG denotes the allocated grid maximum

power for this grid-connected system (e powerflows P2(k) P4(k) and P5(k) are obtained from theenergy balance equation (11) P1(k) and P3(k) aredecided by the scheduling algorithm and they arepart of the decision variables

(c) (e switch control strategies are also subject toconstraints that prevent infeasible switch behaviors(ese constraints are formulated in preceding sec-tions along with the system component modellingie constraints (3) and (8)ndash(10) Some further as-sociations of the switch behaviors are identified asfollows

g1(k) 1 if P1(k)gt 0

g1(k) 0 if P1(k) 0

g3(k) 1 if P3(k)gt 0

g3(k) 0 if P3(k)gt 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)

g2(k) 1 if 1113937n

i1swi(k) minus 1( 1113857 0

g2(k) 0 if otherwise

g4(k) 1 if 1113937n

i1swi(k) minus 2( 1113857 0

g4(k) 0 if otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

In this way the states of g1(k)minus g4(k) at time k can beidentified from the values of P1(k) P3(k) and swi(k) withi 1 2 n

3 Problem Statement

(e primary management objective of the investigatedoptimization problem is cost minimization Secondly therenewable energy penetration is involved that is the usageof grid power should be minimized as well Owing to theTOU tariff the two objectives manifest certain differencesand must be equally considered in scheduling Furthermorethe user satisfaction in time scheduling is taken into accountAll these objectives are evaluated on a finite horizon basisLet T denote the number of sampling instants (e opti-mization problem is formulated as follows

31 Decision Variables (e involved decision variablesconsist of three parts (1) the switch control strategy decisionvariables (2) the charging power control variables and (3)the appliance time-scheduling decision variables It is giventhat i 1 2 n in the following discussion

(e switch control variables are swi(k) which cancharacterize the status of most switches in the system asconstraints (15) and (16) imply Given constraints (8)ndash(10) itis unnecessary to cover the whole finite horizon For theappliance i swi(k)gt 0 when Sti le kleEni and swi(k) 0 if klies outside the working period (erefore the minimumrequired control variables are swi(k) with k isin [Sti Eni] (edimension of this part is 1113936

ni1Di

(e charging power control variables are P1(k) andP3(k) Given constraints (3) and (15) P1(k)lowastP3(k) 0 atany given instant k therefore the dimension of this part isT

As mentioned above the appliance time scheduling issimplified with the known a priori and constant load profilepi(t) and operation duration Di Taking advantage of theknowledge the scheduling is implemented with decisionvariables Sti (e dimension of this part is n

(e dimension of the optimization is therefore1113936

ni1Di + T + n Comparing with the previous study [15] the

problem is extended to a higher dimension but simplified bytaking advantage of swi(k) and the constraints such that thedimension of the optimization grows slower than theproblem scale

32 Objectives (e cost minimization objective is formu-lated as follows

Jc 1113944T

k0ρ(k) P3(k) + P5(k)1113858 1113859 + φbJB(T) (17)

where φb denotes a weight that indicates the preferableimportance of the battery wearout to the decision-maker

(e renewable energy penetration objective is formu-lated as follows

Je 1113944T

k0P3(k) + P5(k)1113858 1113859 (18)

(e user satisfaction is evaluated via an inconvenienceindicator which is adopted from the study of Setlhaolo andXia [12]

β 1113944n

i1ci 1113944

T

k0u

bli (k) minus ui(k)1113960 1113961

2 (19)

where ci denotes an importance factor of the appliance i andubl

i (k) denotes the baseline time schedule of the appliance i(19) quantifies the difference between the baseline timeschedule and the adopted time schedule Given the systemmodelling and constraints in this study the user satisfactionevaluation is simplified as follows

β

1113944n

i1ci St

bli minus Sti1113960 1113961

2

11139741113972

(20)

where Stbli denotes the baseline starting instant Such a

difference has to be minimized such that the user remainshappy with the optimized appliance time schedule

6 Complexity

33 Technoeconomic Optimization Taking advantage of thepreceding objective functions and constraint formulationsthe technoeconomic optimization problem is obtained byminimizing the following objective function

J swi(k) P1(k) P3(k) Sti( 1113857 λcJc + λeJe + λbβ (21)

subject to the battery dynamics (2) and constraints (3)(8)ndash(10) and (11)ndash(16)

According to the above formulations there arenonlinear objective functions and constraints as well ascontinuous and discrete decision variables in the prob-lem (ey result in a mixed-integer nonlinear pro-gramming (MINLP) problem at a relatively large scale(e general theoretic approach of solving an MINLPproblem remains an open question as a result numericalsolvers are widely employed In the previous study [15]an OTPI toolbox httpswwwinverseproblemconzOPTIindexphpDLDownloadOPTI in MATLAB wasadopted as the numerical solver (e former solver tookquite a large amount of time for calculation In this studythe implementation of intelligent optimization algo-rithms on such a problem is investigated A newly pro-posed algorithm named the competitive swarm optimizer(CSO) is adopted as the numerical solver (e in-troduction to the CSO-based solver comes in the fol-lowing section

4 Numerical Solver Design

41Competitive SwarmOptimizer In the CSO algorithm letx denote a particle and w and l the indices of the winner andloser particles in a pair Assume that it is the G-th iterationand there have been k minus 1 competitions After the k-thcompetition the next-generation winner particle xwk(G +

1) remains the same as xwk(G) (e loser particlexlk(G + 1) namely the position vector is thereby updatedas follows

xlk(G + 1) xlk(G) + Vlk(G + 1) (22)

where Vlk(G + 1) is the next-generation velocity vectorupdated as follows

Vlk(G + 1) rn1(k G)Vlk(G) + rn2(k G) xwk(G) minus xlk(G)1113960 1113961

+ φrn3 xk(G) minus xlk(G)1113960 1113961

(23)

where rn1 rn2 and rn3 are random vectors xk(G) is thecenter of neighborhood filed particles ie a set of par-ticles that are close enough to xlk(G) and φ is theweighting coefficient of xk(G) Such a neighborhood fieldis predefined(ere is a special case that the neighborhoodcovers the whole swarm where xk(G) indicates the globalmean position of the particles at iteration G (e velocityand position vectors are employed for the continuouscases In a discrete case eg the decision variables swi(k)

in this study other update mechanisms must beemployed A crossover mechanism is hereby employed asfollows

xlk(rn G + 1) xwk(rn G)

xlk(rn G + 1) xlk(rn G)1113896 (24)

where rn denotes randomly generated indices of the com-ponents in a particle x and rn the unselected indices (24)suggests that the winner particle xwk(G) selects and copies apart of its components into the next-generation loser particlexlk(G + 1) (e unselected components of xlk(G + 1) re-main the same as xlk(G) In this way the loser particle canlearn from the winner particle

For an MINLP problem there are simultaneouslycontinuous and discrete components in a particle In thiscase the continuous part and discrete part are separated(23) and (22) are implemented on the continuous part and(24) is implemented on the discrete part After the learningprocess the two updated parts are combined again toobtain the particle of next generation (e theoretical proofof the convergence of the CSO algorithm can be referred to[17]

(e pseudocode of the CSO according to the precedingintroduction is illustrated in Algorithm 1 [25]

Remark 2 (e introduced CSO algorithm is mainlydesigned for continuous problems For discrete decisionvariables eg binary variables or integers the discrete PSOalgorithm [26 27] can facilitate the algorithm design (epairwise competition can be further introduced to otherevolutionary algorithms such as differential evolution(DE)

Remark 3 Given the investigated MINLP problem (21) theoriginal CSO algorithm cannot be applied in a straight-forward way Modifications that match the decision vari-ables are to be introduced such that satisfying performancescan be achieved

42 Modified CSO-Based Solver In order to implement theCSO algorithm on a constrained problem a penalty functionis introduced to the original objective function (17) Giventhat there are NC constraints to a problem the penaltyfunction is formulated as follows

Pen 1113944T

k01113944

NC

i0ωPeniPeni(k) (25)

where

Peni(k) 0 if constraint i is obeyed

M if constraint i is violated1113896 (26)

where M is a large positive number and ωPeni is theweighting factor for constraint i A fitness function is therebyformulated with (17) (25) and (26)

f(x) Jc + Pen (27)

In this way for a minimization problem a particlebecomes much less competitive when any of the constraints

Complexity 7

where PI2(k) and PI4(k) denote the inverter capacity ofthe PV and battery bank respectively (e power flowsP3(k) and P5(k) are subject to the following constraint

0leg3(k)P3(k) + P5(k)lePmaxG (14)

where PmaxG denotes the allocated grid maximum

power for this grid-connected system (e powerflows P2(k) P4(k) and P5(k) are obtained from theenergy balance equation (11) P1(k) and P3(k) aredecided by the scheduling algorithm and they arepart of the decision variables

(c) (e switch control strategies are also subject toconstraints that prevent infeasible switch behaviors(ese constraints are formulated in preceding sec-tions along with the system component modellingie constraints (3) and (8)ndash(10) Some further as-sociations of the switch behaviors are identified asfollows

g1(k) 1 if P1(k)gt 0

g1(k) 0 if P1(k) 0

g3(k) 1 if P3(k)gt 0

g3(k) 0 if P3(k)gt 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(15)

g2(k) 1 if 1113937n

i1swi(k) minus 1( 1113857 0

g2(k) 0 if otherwise

g4(k) 1 if 1113937n

i1swi(k) minus 2( 1113857 0

g4(k) 0 if otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

In this way the states of g1(k)minus g4(k) at time k can beidentified from the values of P1(k) P3(k) and swi(k) withi 1 2 n

3 Problem Statement

(e primary management objective of the investigatedoptimization problem is cost minimization Secondly therenewable energy penetration is involved that is the usageof grid power should be minimized as well Owing to theTOU tariff the two objectives manifest certain differencesand must be equally considered in scheduling Furthermorethe user satisfaction in time scheduling is taken into accountAll these objectives are evaluated on a finite horizon basisLet T denote the number of sampling instants (e opti-mization problem is formulated as follows

31 Decision Variables (e involved decision variablesconsist of three parts (1) the switch control strategy decisionvariables (2) the charging power control variables and (3)the appliance time-scheduling decision variables It is giventhat i 1 2 n in the following discussion

(e switch control variables are swi(k) which cancharacterize the status of most switches in the system asconstraints (15) and (16) imply Given constraints (8)ndash(10) itis unnecessary to cover the whole finite horizon For theappliance i swi(k)gt 0 when Sti le kleEni and swi(k) 0 if klies outside the working period (erefore the minimumrequired control variables are swi(k) with k isin [Sti Eni] (edimension of this part is 1113936

ni1Di

(e charging power control variables are P1(k) andP3(k) Given constraints (3) and (15) P1(k)lowastP3(k) 0 atany given instant k therefore the dimension of this part isT

As mentioned above the appliance time scheduling issimplified with the known a priori and constant load profilepi(t) and operation duration Di Taking advantage of theknowledge the scheduling is implemented with decisionvariables Sti (e dimension of this part is n

(e dimension of the optimization is therefore1113936

ni1Di + T + n Comparing with the previous study [15] the

problem is extended to a higher dimension but simplified bytaking advantage of swi(k) and the constraints such that thedimension of the optimization grows slower than theproblem scale

32 Objectives (e cost minimization objective is formu-lated as follows

Jc 1113944T

k0ρ(k) P3(k) + P5(k)1113858 1113859 + φbJB(T) (17)

where φb denotes a weight that indicates the preferableimportance of the battery wearout to the decision-maker

(e renewable energy penetration objective is formu-lated as follows

Je 1113944T

k0P3(k) + P5(k)1113858 1113859 (18)

(e user satisfaction is evaluated via an inconvenienceindicator which is adopted from the study of Setlhaolo andXia [12]

β 1113944n

i1ci 1113944

T

k0u

bli (k) minus ui(k)1113960 1113961

2 (19)

where ci denotes an importance factor of the appliance i andubl

i (k) denotes the baseline time schedule of the appliance i(19) quantifies the difference between the baseline timeschedule and the adopted time schedule Given the systemmodelling and constraints in this study the user satisfactionevaluation is simplified as follows

β

1113944n

i1ci St

bli minus Sti1113960 1113961

2

11139741113972

(20)

where Stbli denotes the baseline starting instant Such a

difference has to be minimized such that the user remainshappy with the optimized appliance time schedule

6 Complexity

33 Technoeconomic Optimization Taking advantage of thepreceding objective functions and constraint formulationsthe technoeconomic optimization problem is obtained byminimizing the following objective function

J swi(k) P1(k) P3(k) Sti( 1113857 λcJc + λeJe + λbβ (21)

subject to the battery dynamics (2) and constraints (3)(8)ndash(10) and (11)ndash(16)

According to the above formulations there arenonlinear objective functions and constraints as well ascontinuous and discrete decision variables in the prob-lem (ey result in a mixed-integer nonlinear pro-gramming (MINLP) problem at a relatively large scale(e general theoretic approach of solving an MINLPproblem remains an open question as a result numericalsolvers are widely employed In the previous study [15]an OTPI toolbox httpswwwinverseproblemconzOPTIindexphpDLDownloadOPTI in MATLAB wasadopted as the numerical solver (e former solver tookquite a large amount of time for calculation In this studythe implementation of intelligent optimization algo-rithms on such a problem is investigated A newly pro-posed algorithm named the competitive swarm optimizer(CSO) is adopted as the numerical solver (e in-troduction to the CSO-based solver comes in the fol-lowing section

4 Numerical Solver Design

41Competitive SwarmOptimizer In the CSO algorithm letx denote a particle and w and l the indices of the winner andloser particles in a pair Assume that it is the G-th iterationand there have been k minus 1 competitions After the k-thcompetition the next-generation winner particle xwk(G +

1) remains the same as xwk(G) (e loser particlexlk(G + 1) namely the position vector is thereby updatedas follows

xlk(G + 1) xlk(G) + Vlk(G + 1) (22)

where Vlk(G + 1) is the next-generation velocity vectorupdated as follows

Vlk(G + 1) rn1(k G)Vlk(G) + rn2(k G) xwk(G) minus xlk(G)1113960 1113961

+ φrn3 xk(G) minus xlk(G)1113960 1113961

(23)

where rn1 rn2 and rn3 are random vectors xk(G) is thecenter of neighborhood filed particles ie a set of par-ticles that are close enough to xlk(G) and φ is theweighting coefficient of xk(G) Such a neighborhood fieldis predefined(ere is a special case that the neighborhoodcovers the whole swarm where xk(G) indicates the globalmean position of the particles at iteration G (e velocityand position vectors are employed for the continuouscases In a discrete case eg the decision variables swi(k)

in this study other update mechanisms must beemployed A crossover mechanism is hereby employed asfollows

xlk(rn G + 1) xwk(rn G)

xlk(rn G + 1) xlk(rn G)1113896 (24)

where rn denotes randomly generated indices of the com-ponents in a particle x and rn the unselected indices (24)suggests that the winner particle xwk(G) selects and copies apart of its components into the next-generation loser particlexlk(G + 1) (e unselected components of xlk(G + 1) re-main the same as xlk(G) In this way the loser particle canlearn from the winner particle

For an MINLP problem there are simultaneouslycontinuous and discrete components in a particle In thiscase the continuous part and discrete part are separated(23) and (22) are implemented on the continuous part and(24) is implemented on the discrete part After the learningprocess the two updated parts are combined again toobtain the particle of next generation (e theoretical proofof the convergence of the CSO algorithm can be referred to[17]

(e pseudocode of the CSO according to the precedingintroduction is illustrated in Algorithm 1 [25]

Remark 2 (e introduced CSO algorithm is mainlydesigned for continuous problems For discrete decisionvariables eg binary variables or integers the discrete PSOalgorithm [26 27] can facilitate the algorithm design (epairwise competition can be further introduced to otherevolutionary algorithms such as differential evolution(DE)

Remark 3 Given the investigated MINLP problem (21) theoriginal CSO algorithm cannot be applied in a straight-forward way Modifications that match the decision vari-ables are to be introduced such that satisfying performancescan be achieved

42 Modified CSO-Based Solver In order to implement theCSO algorithm on a constrained problem a penalty functionis introduced to the original objective function (17) Giventhat there are NC constraints to a problem the penaltyfunction is formulated as follows

Pen 1113944T

k01113944

NC

i0ωPeniPeni(k) (25)

where

Peni(k) 0 if constraint i is obeyed

M if constraint i is violated1113896 (26)

where M is a large positive number and ωPeni is theweighting factor for constraint i A fitness function is therebyformulated with (17) (25) and (26)

f(x) Jc + Pen (27)

In this way for a minimization problem a particlebecomes much less competitive when any of the constraints

Complexity 7

33 Technoeconomic Optimization Taking advantage of thepreceding objective functions and constraint formulationsthe technoeconomic optimization problem is obtained byminimizing the following objective function

J swi(k) P1(k) P3(k) Sti( 1113857 λcJc + λeJe + λbβ (21)

subject to the battery dynamics (2) and constraints (3)(8)ndash(10) and (11)ndash(16)

According to the above formulations there arenonlinear objective functions and constraints as well ascontinuous and discrete decision variables in the prob-lem (ey result in a mixed-integer nonlinear pro-gramming (MINLP) problem at a relatively large scale(e general theoretic approach of solving an MINLPproblem remains an open question as a result numericalsolvers are widely employed In the previous study [15]an OTPI toolbox httpswwwinverseproblemconzOPTIindexphpDLDownloadOPTI in MATLAB wasadopted as the numerical solver (e former solver tookquite a large amount of time for calculation In this studythe implementation of intelligent optimization algo-rithms on such a problem is investigated A newly pro-posed algorithm named the competitive swarm optimizer(CSO) is adopted as the numerical solver (e in-troduction to the CSO-based solver comes in the fol-lowing section

4 Numerical Solver Design

41Competitive SwarmOptimizer In the CSO algorithm letx denote a particle and w and l the indices of the winner andloser particles in a pair Assume that it is the G-th iterationand there have been k minus 1 competitions After the k-thcompetition the next-generation winner particle xwk(G +

1) remains the same as xwk(G) (e loser particlexlk(G + 1) namely the position vector is thereby updatedas follows

xlk(G + 1) xlk(G) + Vlk(G + 1) (22)

where Vlk(G + 1) is the next-generation velocity vectorupdated as follows

Vlk(G + 1) rn1(k G)Vlk(G) + rn2(k G) xwk(G) minus xlk(G)1113960 1113961

+ φrn3 xk(G) minus xlk(G)1113960 1113961

(23)

where rn1 rn2 and rn3 are random vectors xk(G) is thecenter of neighborhood filed particles ie a set of par-ticles that are close enough to xlk(G) and φ is theweighting coefficient of xk(G) Such a neighborhood fieldis predefined(ere is a special case that the neighborhoodcovers the whole swarm where xk(G) indicates the globalmean position of the particles at iteration G (e velocityand position vectors are employed for the continuouscases In a discrete case eg the decision variables swi(k)

in this study other update mechanisms must beemployed A crossover mechanism is hereby employed asfollows

xlk(rn G + 1) xwk(rn G)

xlk(rn G + 1) xlk(rn G)1113896 (24)

where rn denotes randomly generated indices of the com-ponents in a particle x and rn the unselected indices (24)suggests that the winner particle xwk(G) selects and copies apart of its components into the next-generation loser particlexlk(G + 1) (e unselected components of xlk(G + 1) re-main the same as xlk(G) In this way the loser particle canlearn from the winner particle

For an MINLP problem there are simultaneouslycontinuous and discrete components in a particle In thiscase the continuous part and discrete part are separated(23) and (22) are implemented on the continuous part and(24) is implemented on the discrete part After the learningprocess the two updated parts are combined again toobtain the particle of next generation (e theoretical proofof the convergence of the CSO algorithm can be referred to[17]

(e pseudocode of the CSO according to the precedingintroduction is illustrated in Algorithm 1 [25]

Remark 2 (e introduced CSO algorithm is mainlydesigned for continuous problems For discrete decisionvariables eg binary variables or integers the discrete PSOalgorithm [26 27] can facilitate the algorithm design (epairwise competition can be further introduced to otherevolutionary algorithms such as differential evolution(DE)

Remark 3 Given the investigated MINLP problem (21) theoriginal CSO algorithm cannot be applied in a straight-forward way Modifications that match the decision vari-ables are to be introduced such that satisfying performancescan be achieved

42 Modified CSO-Based Solver In order to implement theCSO algorithm on a constrained problem a penalty functionis introduced to the original objective function (17) Giventhat there are NC constraints to a problem the penaltyfunction is formulated as follows

Pen 1113944T

k01113944

NC

i0ωPeniPeni(k) (25)

where

Peni(k) 0 if constraint i is obeyed

M if constraint i is violated1113896 (26)

where M is a large positive number and ωPeni is theweighting factor for constraint i A fitness function is therebyformulated with (17) (25) and (26)

f(x) Jc + Pen (27)

In this way for a minimization problem a particlebecomes much less competitive when any of the constraints

Complexity 7

is violated given that a large positive will be added to theobjective function

After competition the loser particle must learn from thewinner particle However the MINLP search space andconstraints in the investigated problem are quite compli-cated (e dynamics of the battery bank charging anddischarging invoke further difficulty to search for the op-timum Consequently the performance of Algorithm 1 isnot satisfying (e following modifications are therebyemployed to improve the performances on this specificproblem

Firstly the learning strategies are modified to improvethe searching efficiency Let f(G) denote the mean fitness ofthe current swarm (at the G-th iteration) After competitionthe fitness of both winner and loser particles is evaluated

(i) If f(xwk(G))gef(G) both winner and loser par-ticles are considered inferior therefore the winnerparticle must learn from the global best particlexgbest(G) while the loser particle must learn fromthe winner

(ii) If f(xlk(G))lef(G) both winner and loser parti-cles are considered superior therefore the winnerparticle is moved into P(G + 1) and the loserparticle implements mutation subject to the geneticalgorithm style

(iii) If f(xwk(G))lef(G)ltf(xlk(G)) the originalCSO learning strategies are applied

Secondly constraints (3) and (15) are employed togenerate battery bank charging and discharging statesnamely the ldquoknowledge-guided solution filterrdquo forallk g4(k) isfirstly identified subject to (16) If g4(k) 1 thenP1(k) P3(k) 0 If g4(k) 0 then P1(k) is randomlygenerated within [0 Ppv(k) minus P2(k)) If g4(k) 0 andP1(k) 0 then P3(k) is randomly generated If the Soc(k)

reaches its upper bound at time k then P1(k) P3(k) 0In this way the charging and discharging decision variablesare guaranteed to be feasible (e knowledge-guided solu-tion filter reduces cost of trial and error during optimizationsuch that the algorithm can persist searching within afeasible space

Remark 4 According to simulations the modified solverconstantly outperforms the original CSO algorithm on theinvestigated problem (ere lacks a theoretical analysis onthe performances whereas a hypothesis is made that thesuperior performances are resulted from the knowledge-guided solution filter Wang and Zheng [28] reported thatexploitation of the algorithm is enhanced by knowledge-based local search Further details and investigations areexpected in future works

Definitionx the particleP the swarmnp the swarm size ie the number of particlesG number of iterationsw and l the indices of winner and loser particlesf(middot) the fitness function assuming that this is a minimization problemTerminal condition the maximum number of iteration Mg is reached

(1) Begin(2) Initialize population P(1) with np particles(3) while G 1 toMg do(4) P(G+ 1)empty(5) while P(G)neempty do(6) Generate two random indices r1 and r2 from np(7) if f(xr1

)lef(xr2) then

(8) w r1 l r2(9) else(10) w r2 l r1(11) end if(12) put xw(G) into P(G + 1)(13) If x is coded as continuous variables update xl(G) with (23) and (22)(14) If x is coded as discrete variables update xl(G) with (24)(15) If x contains both continuous and discrete parts update the two parts separately(16) put the updated loser particle xl(G) into P(G + 1)(17) remove particles xr1

and xr2from P(G)

(18) end while(19) GG+ 1(20) end while(21) choose xbest the particle with the best fitness 1113954f(middot) from PMg(22) Return xbest(23) End

ALGORITHM 1 Pseudo code of the CSO algorithm

8 Complexity

5 Simulation Results and Analysis

51 Case Study (e case study investigates the operation ofa household grid-connected PV-battery hybrid energysystem (e data are retrieved from the South African do-mestic appliance operation studies [13ndash15] (ere are eightappliances connected in the system (e usage profile of theappliances is shown in Table 1 on a daily basis where thereare 144 time slots aka sampling instants Each time slotlasts 10minutes (e scheduling horizon is 24 hours ie awhole day (e adopted usage profile and baseline timeschedule are for the working day scheduling for a typicalSouth African home To emphasize the power of the ap-pliances is measured average power (e baseline timeschedule reflects the preferable time according to the in-habitantrsquos habits For example the inhabitant turns on theelectrical water heater (EWH) twice a day for the hot waterdemand In the morning the EWH is turned on at 5 00 am(the 31-st time slot) operates for two hours and is turned offat 7 00 am (the 42-nd time slot) such that the user can useheated water after breakfast In the afternoon the EWH isturned on again at 5 10 pm (the 104-th time slot) and turnedoff at 7 10 pm (the 115-th time slot) such that the hot waterfor the evening can be ready (is is the most convenientEWH operation plan for the user Similarly the stove mustbe turn on twice for the cooking demands (e other ap-pliances have to be turned on and off only once daily

(ere are several further constraints with the givenscenario Firstly for the shiftable appliances a preferablerange of starting time slots are given in Table 1 (e flexibleappliance can start anytime in a day and the only re-quirement is that the operation must be finished before theend of the horizon(e fixed appliance cannot be scheduledtherefore the preferable range is not applicable (NA)Secondly the washingmachine and electrical dryer work in asequence that is the dryer must start after the washingmachine job is finished In this case it results in an addi-tional constraint to the preceding ones

St4 ge St3 + D3 (28)

(e adopted PV system and battery bank have theirown limitations as shown in Table 2 (e PV system in-tegrates 14 solar panels with the rated power of 025 kW(erefore the overall capacity ie the rated output of thePV system is 35 kW Actually the PV system output at anygiven time slot depends on the solar irradiation profileSuch a profile is possible to forecast 24 hours ahead of thescheduling [19] In the case study the timely PV output isidentified based on an hourly profile in [29] as shownin Table 3 where Qpv is k isin [39 144) (e battery bankconsists of 4 lead-acid batteries each with 12 V ratedvoltage and 105Ah rated capacity that is the overall ca-pacity is 504 kWh (e battery cost is calculated in SouthAfrican rand (ZAR) which is R5826 (e lifespan ofthe battery bank is 1000 cycles at 50 depth of dis-charge (httpwwwtrojanbatterycommarketsrenewable-energy-re) therefore the wear cost per 1 kWh throughputenergy is 5826(1000 lowast 05lowast 504) 2312RkWh φb 01

Table 1 Typical usage profiles and baseline time schedules

Appliances Power (kW) Duration (min)Baseline

Preferable range of StiIndex i Sti Eni

Shiftable

(1) EWH 30 120 31 42 [19 31]

120 104 115 [91 121]

(2) Stove 25 30 32 34 [25 55]

50 113 117 [97 127]

(3) Washing machine 05 60 109 114 [43 133]

(4) Electric dryer 20 30 116 118 [49 139]

Fixed(5) Refrigerator 01 1440 1 144 NA(6) Television set 02 180 104 121 NAFlexible(7) Dishwasher 18 150 116 130 [1 130]

(8) Bread maker 15 150 118 132 [1 130]

Table 2 PV system and battery bank parameters

PV capacity Ppv 35 kWp

Battery bank maximum capacity Cmax 504 kWhBattery bank minimum capacity Cmin 252 kWhBattery bank cost (ZAR) R5826Initial state of the battery bank 60Cmax

AC charger efficiency ηc 85PV charge controller efficiency ηs 90PV inverter efficiency ηI2 95Battery bank inverter efficiency ηI4 95Battery bank charging efficiency ηB 80

Table 3 Hourly PV output

Time slot k [0 39) [39 45) [45 51) [51 57)

Ppv(k) (kWh) 0 015 085 165[57 63) [63 69) [69 75) [75 81) [81 87)

235 29 3 295 255[87 93) [93 99) [99 105) [105 111) [111 144)

2 145 075 01 0

Complexity 9

5 Simulation Results and Analysis

51 Case Study (e case study investigates the operation ofa household grid-connected PV-battery hybrid energysystem (e data are retrieved from the South African do-mestic appliance operation studies [13ndash15] (ere are eightappliances connected in the system (e usage profile of theappliances is shown in Table 1 on a daily basis where thereare 144 time slots aka sampling instants Each time slotlasts 10minutes (e scheduling horizon is 24 hours ie awhole day (e adopted usage profile and baseline timeschedule are for the working day scheduling for a typicalSouth African home To emphasize the power of the ap-pliances is measured average power (e baseline timeschedule reflects the preferable time according to the in-habitantrsquos habits For example the inhabitant turns on theelectrical water heater (EWH) twice a day for the hot waterdemand In the morning the EWH is turned on at 5 00 am(the 31-st time slot) operates for two hours and is turned offat 7 00 am (the 42-nd time slot) such that the user can useheated water after breakfast In the afternoon the EWH isturned on again at 5 10 pm (the 104-th time slot) and turnedoff at 7 10 pm (the 115-th time slot) such that the hot waterfor the evening can be ready (is is the most convenientEWH operation plan for the user Similarly the stove mustbe turn on twice for the cooking demands (e other ap-pliances have to be turned on and off only once daily

(ere are several further constraints with the givenscenario Firstly for the shiftable appliances a preferablerange of starting time slots are given in Table 1 (e flexibleappliance can start anytime in a day and the only re-quirement is that the operation must be finished before theend of the horizon(e fixed appliance cannot be scheduledtherefore the preferable range is not applicable (NA)Secondly the washingmachine and electrical dryer work in asequence that is the dryer must start after the washingmachine job is finished In this case it results in an addi-tional constraint to the preceding ones

St4 ge St3 + D3 (28)

(e adopted PV system and battery bank have theirown limitations as shown in Table 2 (e PV system in-tegrates 14 solar panels with the rated power of 025 kW(erefore the overall capacity ie the rated output of thePV system is 35 kW Actually the PV system output at anygiven time slot depends on the solar irradiation profileSuch a profile is possible to forecast 24 hours ahead of thescheduling [19] In the case study the timely PV output isidentified based on an hourly profile in [29] as shownin Table 3 where Qpv is k isin [39 144) (e battery bankconsists of 4 lead-acid batteries each with 12 V ratedvoltage and 105Ah rated capacity that is the overall ca-pacity is 504 kWh (e battery cost is calculated in SouthAfrican rand (ZAR) which is R5826 (e lifespan ofthe battery bank is 1000 cycles at 50 depth of dis-charge (httpwwwtrojanbatterycommarketsrenewable-energy-re) therefore the wear cost per 1 kWh throughputenergy is 5826(1000 lowast 05lowast 504) 2312RkWh φb 01

Table 1 Typical usage profiles and baseline time schedules

Appliances Power (kW) Duration (min)Baseline

Preferable range of StiIndex i Sti Eni

Shiftable

(1) EWH 30 120 31 42 [19 31]

120 104 115 [91 121]

(2) Stove 25 30 32 34 [25 55]

50 113 117 [97 127]

(3) Washing machine 05 60 109 114 [43 133]

(4) Electric dryer 20 30 116 118 [49 139]

Fixed(5) Refrigerator 01 1440 1 144 NA(6) Television set 02 180 104 121 NAFlexible(7) Dishwasher 18 150 116 130 [1 130]

(8) Bread maker 15 150 118 132 [1 130]

Table 2 PV system and battery bank parameters

PV capacity Ppv 35 kWp

Battery bank maximum capacity Cmax 504 kWhBattery bank minimum capacity Cmin 252 kWhBattery bank cost (ZAR) R5826Initial state of the battery bank 60Cmax

AC charger efficiency ηc 85PV charge controller efficiency ηs 90PV inverter efficiency ηI2 95Battery bank inverter efficiency ηI4 95Battery bank charging efficiency ηB 80

Table 3 Hourly PV output

Time slot k [0 39) [39 45) [45 51) [51 57)

Ppv(k) (kWh) 0 015 085 165[57 63) [63 69) [69 75) [75 81) [81 87)

235 29 3 295 255[87 93) [93 99) [99 105) [105 111) [111 144)

2 145 075 01 0

Complexity 9

such that the usage of the battery is encouraged (e im-portance factors ci 1 for each i ie involved appliancesare considered equally important in this case (e effi-ciencies of the charge controller AC charger and invertersare given as well

(e power grid supply is described as follows (emaximum household current is 60A which is limited by theutility company (e charging power from the grid isconsidered constant in this case which is 5 kW that isP3(k)

can only be 0 or 5 kWh Furthermore the TOU tariff isadopted from the study in [13] as shown in Table 4

52 Simulations Simulations are programmed in C++ withthe following running environment the CPU is Inter Corei3-8100 CPU360GHz the RAM is 16GB and the systemisWindows 10times 64(ree cases are adopted by adjusting theweighting factors in (21)

(i) λe 1 λc 0 and λb 0 such that the optimizationemploys a single objective ie the renewable energypenetration

(ii) λe 0 λc 1 and λb 0 such that only the costminimization objective is optimized

(iii) λe 1 λc 1 and λb 1 such that the multi-objective optimization of (21) is implemented whereJc Je and β are equally considered

In each case there are two demonstrated results Oneresult is from the proposed approach and the other one isfrom the previous power dispatch model [15] as the com-parative results Both results are reported from the average of20 runs taking advantage of the proposed CSO-based opti-mizer In the CSO algorithm the swarm size is 1500 and theiteration number is 10000 (e neighborhood field is definedto be the nearest superior particle and inferior particle (edetails of such a neighborhood field can be referred to [30 31]

53 Results andAnalysis (e results are reported as follows

(i) In the first case the grid power supply is mini-mized to be 14 kWh at the energy cost of R1657and inconvenience indicator of 1425 (e powerdispatch is depicted in Figure 3 In the com-parative result the minimal grid power is1507 kWh at the energy cost of R1429 and in-convenience indicator of 1459 (e power dis-patch is depicted in Figure 4 (e improvement ofthe objective is 71 (e average running time is1843 minutes

(ii) In the second case the cost is minimized to beR706 while the overall power supply from the gridis 1437 kWh (e inconvenience indicator is 1389(e power dispatch is depicted in Figure 5 In thecomparative result the minimal cost is R772 with

Table 4 TOU tariff

Time periods Electricity price HoursPeak hours R22225kWh [08 00 11 00)cup [19 00 21 00)Standard hours R06773kWh [07 00 08 00)cup [11 00 19 00)cup [21 00 23 00)Off-peak hours R03656kWh [00 00 07 00)cup [23 00 24 00)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (h)

0

05

1

15

2

25

3

35

Pow

er (k

W)

f = Je

P1P2P3g3

P4P5

Figure 3 Power dispatch of the new design in case (i)

10 Complexity

the grid power supply of 1557 kWh and in-convenience indicator of 1503 (e power dispatchis depicted in Figure 6 (e improvement of theobjective is 86 (e average running time is1836minutes

(iii) In the third case the weighted sum of all objec-tives is minimized (e optimized objectivefunction value is 3464 when the energy cost isR829 and the grid power supply is 1553 kWh

and the inconvenience indicator is 1082 (epower dispatch is depicted in Figure 7 In thecomparative result the objective function value is3997 where the energy cost is 953 and gridpower is 1878 kWh with the inconvenience in-dicator of 1166 as well (e power dispatch isdepicted in Figure 8 (e improvement of theobjective is 133 (e average running time is1837minutes

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (h)

0

1

2

3

4

5

6

Pow

er (k

W)

f = Jc

P1P2P3g3

P4P5

Figure 5 Power dispatch of the new design in case (ii)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (h)

0

05

1

15

2

25

3

35 f = Je

P1P2P3g3

P4P5

Pow

er (k

W)

Figure 4 Power dispatch of the previous design in case (i)

Complexity 11

From Figures 3 5 and 7 overlaps between differentpower supplies can be observed especially during peakhours while from Figures 4 6 and 8 none of the time slotsallows multiple power supplies In these cases the gridpower outputs are smoothened under the dispatch with theproposed approach From the comparative results suppliesbecome more intermittent because of the contradictionbetween the renewable penetration objective and the supply

constraint Allowing the combination of multiple powersupplies reduces such an intermittent performance whileaccording to the energy and economy performances in casestudies making better use of the renewable energy sources

As a conclusion the proposed approach outperforms theprevious model in all cases When comparing the results ofthe new and previous designs it appears that the flexiblepower dispatch allows more appliances to be scheduled to

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (h)

0

1

2

3

4

5

6

Pow

er (k

W)

f = Je + Jc + beta

P1P2P3g3

P4P5

Figure 7 Power dispatch of the new design in case (iii)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (h)

0

1

2

3

4

5

6

Pow

er (k

W)

f = Jc

P1P2P3g3

P4P5

Figure 6 Power dispatch of the previous design in case (ii)

12 Complexity

standard and off-peak hours resulting in a lower energycost (e objectives Je and Jc manifest a certain level oftrade-off (is is resulted from the employment of the TOUtariff where some off-peak hours may be infeasible for thePV system because of its intermittent nature When com-paring the results among the three cases it can be found thatthe first two cases can achieve lower energy cost and gridpower consumption because of ignoring the inconvenienceindicator β It invokes an interesting topic for future studiesthat how to strike a balance between the conflicted interestsof user satisfaction and energy efficiency in such a hybridenergy system management

6 Conclusions

(is paper investigates a technoeconomic optimizationproblem for a domestic grid-connected PV-battery hybridenergy system via extending and improving a previouslyproposed system design According to the previous designthe power dispatch is decided for the totality of the elec-trical loads In the new model appliances that comprise theelectrical loads are supplied and managed respectively viaadditional power lines and switches from each powersource Furthermore the appliance time scheduling isincorporated into such a flexible power dispatch In thisway the system achieves better energy efficiency andeconomic performances via the technoeconomic optimi-zation (e performances are evaluated by three optimi-zation objectives minimizing energy cost maximizingrenewable energy penetration and increasing user satis-faction over a finite horizon (ere are nonlinear objectivefunctions and constraints as well as discrete and contin-uous decision variables in such an optimization problem

As a result the problem becomes an MINLP one at a largescale which is difficult to solve A competitive swarmoptimizer-based numerical solver is thereby designed andemployed

In order to verify that the new design does improve theperformances a case study is investigated where the powerdispatch and appliance time scheduling on a daily basis areapplied to a typical South African household hybrid system(ere are three optimization cases each with differentobjective functions including only energy cost minimiza-tion only renewable energy penetration maximization anda weighted sum of the three objectives Simulations areapplied in these cases where comparative results are alsoobtained via optimization of the previous system design(esame solver and system configurations are employed In allcases the results from the new design outperform resultsfrom the previous design (e improvement ranges from71 to 133 and manifests that further energy efficiencyand economic benefits can be achieved by the proposedapproach Furthermore the solver generally takes around184minutes to obtain the solution It verifies that theproposed approach has the potential for application in areal-time context

(ere are several future works to investigate based on thecurrent-stage results Firstly the performance evaluationssuch as the battery wear cost and the renewable energypenetration are simplified More practical indicators can beintroduced in the future Secondly uncertainties from theenvironment and user demands are inevitable in practice(e real-time feedback mechanism can be introduced toovercome such uncertainties (irdly the game theory-based power dispatch and load scheduling consideringconflicted interests call further study Lastly the CSO-based

f = Je + Jc + beta

P1P2P3g3

P4P5

0

1

2

3

4

5

6

Pow

er (k

W)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240Time (h)

Figure 8 Power dispatch of the previous design in case (iii)

Complexity 13

numerical solver can be further investigated to improve thealgorithm performance

Data Availability

All relevant data used to support the findings of this studyare included within the article

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Nature ScienceFoundation of China under Grant 61803162 and the Fun-damental Research Funds for the Central Universities underGrant HUST 2017KFYXJJ178

References

[1] J Soares T Pinto F Lezama and H Morais ldquoSurvey oncomplex optimization and simulation for the new powersystems paradigmrdquo Complexity vol 2018 Article ID 234062832 pages 2018

[2] B Zhu H Tazvinga and X Xia ldquoSwitched model predictivecontrol for energy dispatching of a photovoltaic-diesel-batteryhybrid power systemrdquo IEEE Transactions on Control SystemsTechnology vol 23 pp 1229ndash1236 2014

[3] S Shaahid and M Elhadidy ldquoEconomic analysis of hybridphotovoltaicndashdieselndashbattery power systems for residentialloads in hot regionsmdasha step to clean futurerdquo Renewable andSustainable Energy Reviews vol 12 pp 488ndash503 2008

[4] P Nema R Nema and S Rangnekar ldquoA current and futurestate of art development of hybrid energy system using windand pv-solar a reviewrdquo Renewable and Sustainable EnergyReviews vol 13 pp 2096ndash2103 2009

[5] X Li J Lai and R Tang ldquoA hybrid constraints handlingstrategy for multiconstrained multiobjective optimizationproblem of microgrid economicalenvironmental dispatchrdquoComplexity vol 2017 Article ID 6249432 12 pages 2017

[6] T Cheng M Chen Y Wang et al ldquoAdaptive robust methodfor dynamic economic emission dispatch incorporating re-newable energy and energy storagerdquo Complexity vol 2018Article ID 2517987 13 pages 2018

[7] X Chen B Xu and W Du ldquoAn improved particle swarmoptimization with biogeography-based learning strategy foreconomic dispatch problemsrdquo Complexity vol 2018 ArticleID 7289674 15 pages 2018

[8] R Kallel G Boukettaya and L Krichen ldquoDemand sidemanagement of household appliances in stand-alone hybridphotovoltaic systemrdquo Renewable Energy vol 81 pp 123ndash1352015

[9] S Phiri and K Kusakana ldquoDemand side management of agrid connected pv-wt-battery hybrid systemrdquo in Proceedingsof the 2016 International Conference on the Industrial andCommercial Use of Energy (ICUE) pp 45ndash51 IEEE CapeTown South Africa August 2016

[10] H Tazvinga X Xia and J Zhang ldquoMinimum cost solution ofphotovoltaicndashdieselndashbattery hybrid power systems for remoteconsumersrdquo Solar Energy vol 96 pp 292ndash299 2013

[11] H Tazvinga B Zhu and X Xia ldquoEnergy dispatch strategy fora photovoltaicndashwindndashdieselndashbattery hybrid power systemrdquoSolar Energy vol 108 pp 412ndash420 2014

[12] D Setlhaolo and X Xia ldquoOptimal scheduling of householdappliances with a battery storage system and coordinationrdquoEnergy and Buildings vol 94 pp 61ndash70 2015

[13] S M Sichilalu and X Xia ldquoOptimal energy control of grid tiedpvndashdieselndashbattery hybrid system powering heat pump waterheaterrdquo Solar Energy vol 115 pp 243ndash254 2015

[14] D Setlhaolo and X Xia ldquoCombined residential demand sidemanagement strategies with coordination and economicanalysisrdquo International Journal of Electrical Power amp EnergySystems vol 79 pp 150ndash160 2016

[15] F Yang and X Xia ldquoTechno-economic and environmentaloptimization of a household photovoltaic-battery hybridpower system within demand side managementrdquo RenewableEnergy vol 108 pp 132ndash143 2017

[16] C Kagiri E M Wanjiru L Zhang and X Xia ldquoOptimizedresponse to electricity time-of-use tariff of a compressednatural gas fuelling stationrdquo Applied Energy vol 222pp 244ndash256 2018

[17] R Cheng and Y Jin ldquoA competitive swarm optimizer for largescale optimizationrdquo IEEE Transactions on Cybernetics vol 45pp 191ndash204 2014

[18] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofmachine learning C Sammut and G I Webb Eds SpringerBoston MA USA pp 760ndash766 2010

[19] S Leva A Dolara F Grimaccia M Mussetta and E OgliarildquoAnalysis and validation of 24 hours ahead neural networkforecasting of photovoltaic output powerrdquo Mathematics andComputers in Simulation vol 131 pp 88ndash100 2017

[20] H Bindner T Cronin P Lundsager J F ManwellU Abdulwahid and I Baring-Gould Lifetime Modelling ofLead Acid Batteries Forskningscenter Risoe Roskilde Den-mark 2005

[21] C Wei M Benosman and T Kim ldquoOnline parameteridentification for state of power prediction of lithium-ionbatteries in electric vehicles using extremum seekingrdquo In-ternational Journal of Control Automation and Systems 2019In press

[22] C Yin X Huang S Dadras et al ldquoDesign of optimal lightingcontrol strategy based on multi-variable fractional-orderextremum seeking methodrdquo Information Sciences vol 465pp 38ndash60 2018

[23] F Cheng L Qu W Qiao C Wei and L Hao ldquoFault di-agnosis of wind turbine gearboxes based on DFIG statorcurrent envelope analysisrdquo IEEE Transactions on SustainableEnergy vol 10 no 3 pp 1044ndash1053 2018

[24] X Xia and J Zhang ldquoOperation efficiency optimisationmodelling and application of model predictive controlrdquo IEEECAA Journal of Automatica Sinica vol 2 pp 166ndash172 2015

[25] R Cheng ldquoNature inspired optimization of large problemsrdquoPhD thesis University of Surrey Guildford UK 2016

[26] Q Zhao and L-y Zhang ldquoResearch on the effect of Dpso inteam selection optimization under the background of big datardquoComplexity vol 2018 Article ID 1386407 14 pages 2018

[27] J Liu H Ma X Ren T Shi P Li and X Ma ldquo(e con-tinuous-discrete pso algorithm for shape formation problemof multiple agents in two and three dimensional spacerdquoApplied Soft Computing vol 67 pp 409ndash433 2018

[28] L Wang and X-l Zheng ldquoA knowledge-guided multi-ob-jective fruit fly optimization algorithm for the multi-skillresource constrained project scheduling problemrdquo Swarmand Evolutionary Computation vol 38 pp 54ndash63 2018

14 Complexity

[29] H Tazvinga and T Hove PhotovoltaicDieselBattery HybridPower Supply System VDMPublishers Pretoria South Africa2010

[30] Z Wu and T W Chow ldquoNeighborhood field for cooperativeoptimizationrdquo Soft Computing vol 17 pp 819ndash834 2013

[31] N Ao M Zhao Q Li S Qu and Z Wu ldquoNetwork char-acteristics for neighborhood field algorithmsrdquo Neural Com-puting and Applications 2019

Complexity 15

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

the grid power supply of 1557 kWh and in-convenience indicator of 1503 (e power dispatchis depicted in Figure 6 (e improvement of theobjective is 86 (e average running time is1836minutes

(iii) In the third case the weighted sum of all objec-tives is minimized (e optimized objectivefunction value is 3464 when the energy cost isR829 and the grid power supply is 1553 kWh

and the inconvenience indicator is 1082 (epower dispatch is depicted in Figure 7 In thecomparative result the objective function value is3997 where the energy cost is 953 and gridpower is 1878 kWh with the inconvenience in-dicator of 1166 as well (e power dispatch isdepicted in Figure 8 (e improvement of theobjective is 133 (e average running time is1837minutes

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (h)

0

1

2

3

4

5

6

Pow

er (k

W)

f = Jc

P1P2P3g3

P4P5

Figure 5 Power dispatch of the new design in case (ii)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (h)

0

05

1

15

2

25

3

35 f = Je

P1P2P3g3

P4P5

Pow

er (k

W)

Figure 4 Power dispatch of the previous design in case (i)

Complexity 11

From Figures 3 5 and 7 overlaps between differentpower supplies can be observed especially during peakhours while from Figures 4 6 and 8 none of the time slotsallows multiple power supplies In these cases the gridpower outputs are smoothened under the dispatch with theproposed approach From the comparative results suppliesbecome more intermittent because of the contradictionbetween the renewable penetration objective and the supply

constraint Allowing the combination of multiple powersupplies reduces such an intermittent performance whileaccording to the energy and economy performances in casestudies making better use of the renewable energy sources

As a conclusion the proposed approach outperforms theprevious model in all cases When comparing the results ofthe new and previous designs it appears that the flexiblepower dispatch allows more appliances to be scheduled to

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (h)

0

1

2

3

4

5

6

Pow

er (k

W)

f = Je + Jc + beta

P1P2P3g3

P4P5

Figure 7 Power dispatch of the new design in case (iii)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (h)

0

1

2

3

4

5

6

Pow

er (k

W)

f = Jc

P1P2P3g3

P4P5

Figure 6 Power dispatch of the previous design in case (ii)

12 Complexity

standard and off-peak hours resulting in a lower energycost (e objectives Je and Jc manifest a certain level oftrade-off (is is resulted from the employment of the TOUtariff where some off-peak hours may be infeasible for thePV system because of its intermittent nature When com-paring the results among the three cases it can be found thatthe first two cases can achieve lower energy cost and gridpower consumption because of ignoring the inconvenienceindicator β It invokes an interesting topic for future studiesthat how to strike a balance between the conflicted interestsof user satisfaction and energy efficiency in such a hybridenergy system management

6 Conclusions

(is paper investigates a technoeconomic optimizationproblem for a domestic grid-connected PV-battery hybridenergy system via extending and improving a previouslyproposed system design According to the previous designthe power dispatch is decided for the totality of the elec-trical loads In the new model appliances that comprise theelectrical loads are supplied and managed respectively viaadditional power lines and switches from each powersource Furthermore the appliance time scheduling isincorporated into such a flexible power dispatch In thisway the system achieves better energy efficiency andeconomic performances via the technoeconomic optimi-zation (e performances are evaluated by three optimi-zation objectives minimizing energy cost maximizingrenewable energy penetration and increasing user satis-faction over a finite horizon (ere are nonlinear objectivefunctions and constraints as well as discrete and contin-uous decision variables in such an optimization problem

As a result the problem becomes an MINLP one at a largescale which is difficult to solve A competitive swarmoptimizer-based numerical solver is thereby designed andemployed

In order to verify that the new design does improve theperformances a case study is investigated where the powerdispatch and appliance time scheduling on a daily basis areapplied to a typical South African household hybrid system(ere are three optimization cases each with differentobjective functions including only energy cost minimiza-tion only renewable energy penetration maximization anda weighted sum of the three objectives Simulations areapplied in these cases where comparative results are alsoobtained via optimization of the previous system design(esame solver and system configurations are employed In allcases the results from the new design outperform resultsfrom the previous design (e improvement ranges from71 to 133 and manifests that further energy efficiencyand economic benefits can be achieved by the proposedapproach Furthermore the solver generally takes around184minutes to obtain the solution It verifies that theproposed approach has the potential for application in areal-time context

(ere are several future works to investigate based on thecurrent-stage results Firstly the performance evaluationssuch as the battery wear cost and the renewable energypenetration are simplified More practical indicators can beintroduced in the future Secondly uncertainties from theenvironment and user demands are inevitable in practice(e real-time feedback mechanism can be introduced toovercome such uncertainties (irdly the game theory-based power dispatch and load scheduling consideringconflicted interests call further study Lastly the CSO-based

f = Je + Jc + beta

P1P2P3g3

P4P5

0

1

2

3

4

5

6

Pow

er (k

W)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240Time (h)

Figure 8 Power dispatch of the previous design in case (iii)

Complexity 13

numerical solver can be further investigated to improve thealgorithm performance

Data Availability

All relevant data used to support the findings of this studyare included within the article

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Nature ScienceFoundation of China under Grant 61803162 and the Fun-damental Research Funds for the Central Universities underGrant HUST 2017KFYXJJ178

References

[1] J Soares T Pinto F Lezama and H Morais ldquoSurvey oncomplex optimization and simulation for the new powersystems paradigmrdquo Complexity vol 2018 Article ID 234062832 pages 2018

[2] B Zhu H Tazvinga and X Xia ldquoSwitched model predictivecontrol for energy dispatching of a photovoltaic-diesel-batteryhybrid power systemrdquo IEEE Transactions on Control SystemsTechnology vol 23 pp 1229ndash1236 2014

[3] S Shaahid and M Elhadidy ldquoEconomic analysis of hybridphotovoltaicndashdieselndashbattery power systems for residentialloads in hot regionsmdasha step to clean futurerdquo Renewable andSustainable Energy Reviews vol 12 pp 488ndash503 2008

[4] P Nema R Nema and S Rangnekar ldquoA current and futurestate of art development of hybrid energy system using windand pv-solar a reviewrdquo Renewable and Sustainable EnergyReviews vol 13 pp 2096ndash2103 2009

[5] X Li J Lai and R Tang ldquoA hybrid constraints handlingstrategy for multiconstrained multiobjective optimizationproblem of microgrid economicalenvironmental dispatchrdquoComplexity vol 2017 Article ID 6249432 12 pages 2017

[6] T Cheng M Chen Y Wang et al ldquoAdaptive robust methodfor dynamic economic emission dispatch incorporating re-newable energy and energy storagerdquo Complexity vol 2018Article ID 2517987 13 pages 2018

[7] X Chen B Xu and W Du ldquoAn improved particle swarmoptimization with biogeography-based learning strategy foreconomic dispatch problemsrdquo Complexity vol 2018 ArticleID 7289674 15 pages 2018

[8] R Kallel G Boukettaya and L Krichen ldquoDemand sidemanagement of household appliances in stand-alone hybridphotovoltaic systemrdquo Renewable Energy vol 81 pp 123ndash1352015

[9] S Phiri and K Kusakana ldquoDemand side management of agrid connected pv-wt-battery hybrid systemrdquo in Proceedingsof the 2016 International Conference on the Industrial andCommercial Use of Energy (ICUE) pp 45ndash51 IEEE CapeTown South Africa August 2016

[10] H Tazvinga X Xia and J Zhang ldquoMinimum cost solution ofphotovoltaicndashdieselndashbattery hybrid power systems for remoteconsumersrdquo Solar Energy vol 96 pp 292ndash299 2013

[11] H Tazvinga B Zhu and X Xia ldquoEnergy dispatch strategy fora photovoltaicndashwindndashdieselndashbattery hybrid power systemrdquoSolar Energy vol 108 pp 412ndash420 2014

[12] D Setlhaolo and X Xia ldquoOptimal scheduling of householdappliances with a battery storage system and coordinationrdquoEnergy and Buildings vol 94 pp 61ndash70 2015

[13] S M Sichilalu and X Xia ldquoOptimal energy control of grid tiedpvndashdieselndashbattery hybrid system powering heat pump waterheaterrdquo Solar Energy vol 115 pp 243ndash254 2015

[14] D Setlhaolo and X Xia ldquoCombined residential demand sidemanagement strategies with coordination and economicanalysisrdquo International Journal of Electrical Power amp EnergySystems vol 79 pp 150ndash160 2016

[15] F Yang and X Xia ldquoTechno-economic and environmentaloptimization of a household photovoltaic-battery hybridpower system within demand side managementrdquo RenewableEnergy vol 108 pp 132ndash143 2017

[16] C Kagiri E M Wanjiru L Zhang and X Xia ldquoOptimizedresponse to electricity time-of-use tariff of a compressednatural gas fuelling stationrdquo Applied Energy vol 222pp 244ndash256 2018

[17] R Cheng and Y Jin ldquoA competitive swarm optimizer for largescale optimizationrdquo IEEE Transactions on Cybernetics vol 45pp 191ndash204 2014

[18] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofmachine learning C Sammut and G I Webb Eds SpringerBoston MA USA pp 760ndash766 2010

[19] S Leva A Dolara F Grimaccia M Mussetta and E OgliarildquoAnalysis and validation of 24 hours ahead neural networkforecasting of photovoltaic output powerrdquo Mathematics andComputers in Simulation vol 131 pp 88ndash100 2017

[20] H Bindner T Cronin P Lundsager J F ManwellU Abdulwahid and I Baring-Gould Lifetime Modelling ofLead Acid Batteries Forskningscenter Risoe Roskilde Den-mark 2005

[21] C Wei M Benosman and T Kim ldquoOnline parameteridentification for state of power prediction of lithium-ionbatteries in electric vehicles using extremum seekingrdquo In-ternational Journal of Control Automation and Systems 2019In press

[22] C Yin X Huang S Dadras et al ldquoDesign of optimal lightingcontrol strategy based on multi-variable fractional-orderextremum seeking methodrdquo Information Sciences vol 465pp 38ndash60 2018

[23] F Cheng L Qu W Qiao C Wei and L Hao ldquoFault di-agnosis of wind turbine gearboxes based on DFIG statorcurrent envelope analysisrdquo IEEE Transactions on SustainableEnergy vol 10 no 3 pp 1044ndash1053 2018

[24] X Xia and J Zhang ldquoOperation efficiency optimisationmodelling and application of model predictive controlrdquo IEEECAA Journal of Automatica Sinica vol 2 pp 166ndash172 2015

[25] R Cheng ldquoNature inspired optimization of large problemsrdquoPhD thesis University of Surrey Guildford UK 2016

[26] Q Zhao and L-y Zhang ldquoResearch on the effect of Dpso inteam selection optimization under the background of big datardquoComplexity vol 2018 Article ID 1386407 14 pages 2018

[27] J Liu H Ma X Ren T Shi P Li and X Ma ldquo(e con-tinuous-discrete pso algorithm for shape formation problemof multiple agents in two and three dimensional spacerdquoApplied Soft Computing vol 67 pp 409ndash433 2018

[28] L Wang and X-l Zheng ldquoA knowledge-guided multi-ob-jective fruit fly optimization algorithm for the multi-skillresource constrained project scheduling problemrdquo Swarmand Evolutionary Computation vol 38 pp 54ndash63 2018

14 Complexity

[29] H Tazvinga and T Hove PhotovoltaicDieselBattery HybridPower Supply System VDMPublishers Pretoria South Africa2010

[30] Z Wu and T W Chow ldquoNeighborhood field for cooperativeoptimizationrdquo Soft Computing vol 17 pp 819ndash834 2013

[31] N Ao M Zhao Q Li S Qu and Z Wu ldquoNetwork char-acteristics for neighborhood field algorithmsrdquo Neural Com-puting and Applications 2019

Complexity 15

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

From Figures 3 5 and 7 overlaps between differentpower supplies can be observed especially during peakhours while from Figures 4 6 and 8 none of the time slotsallows multiple power supplies In these cases the gridpower outputs are smoothened under the dispatch with theproposed approach From the comparative results suppliesbecome more intermittent because of the contradictionbetween the renewable penetration objective and the supply

constraint Allowing the combination of multiple powersupplies reduces such an intermittent performance whileaccording to the energy and economy performances in casestudies making better use of the renewable energy sources

As a conclusion the proposed approach outperforms theprevious model in all cases When comparing the results ofthe new and previous designs it appears that the flexiblepower dispatch allows more appliances to be scheduled to

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (h)

0

1

2

3

4

5

6

Pow

er (k

W)

f = Je + Jc + beta

P1P2P3g3

P4P5

Figure 7 Power dispatch of the new design in case (iii)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (h)

0

1

2

3

4

5

6

Pow

er (k

W)

f = Jc

P1P2P3g3

P4P5

Figure 6 Power dispatch of the previous design in case (ii)

12 Complexity

standard and off-peak hours resulting in a lower energycost (e objectives Je and Jc manifest a certain level oftrade-off (is is resulted from the employment of the TOUtariff where some off-peak hours may be infeasible for thePV system because of its intermittent nature When com-paring the results among the three cases it can be found thatthe first two cases can achieve lower energy cost and gridpower consumption because of ignoring the inconvenienceindicator β It invokes an interesting topic for future studiesthat how to strike a balance between the conflicted interestsof user satisfaction and energy efficiency in such a hybridenergy system management

6 Conclusions

(is paper investigates a technoeconomic optimizationproblem for a domestic grid-connected PV-battery hybridenergy system via extending and improving a previouslyproposed system design According to the previous designthe power dispatch is decided for the totality of the elec-trical loads In the new model appliances that comprise theelectrical loads are supplied and managed respectively viaadditional power lines and switches from each powersource Furthermore the appliance time scheduling isincorporated into such a flexible power dispatch In thisway the system achieves better energy efficiency andeconomic performances via the technoeconomic optimi-zation (e performances are evaluated by three optimi-zation objectives minimizing energy cost maximizingrenewable energy penetration and increasing user satis-faction over a finite horizon (ere are nonlinear objectivefunctions and constraints as well as discrete and contin-uous decision variables in such an optimization problem

As a result the problem becomes an MINLP one at a largescale which is difficult to solve A competitive swarmoptimizer-based numerical solver is thereby designed andemployed

In order to verify that the new design does improve theperformances a case study is investigated where the powerdispatch and appliance time scheduling on a daily basis areapplied to a typical South African household hybrid system(ere are three optimization cases each with differentobjective functions including only energy cost minimiza-tion only renewable energy penetration maximization anda weighted sum of the three objectives Simulations areapplied in these cases where comparative results are alsoobtained via optimization of the previous system design(esame solver and system configurations are employed In allcases the results from the new design outperform resultsfrom the previous design (e improvement ranges from71 to 133 and manifests that further energy efficiencyand economic benefits can be achieved by the proposedapproach Furthermore the solver generally takes around184minutes to obtain the solution It verifies that theproposed approach has the potential for application in areal-time context

(ere are several future works to investigate based on thecurrent-stage results Firstly the performance evaluationssuch as the battery wear cost and the renewable energypenetration are simplified More practical indicators can beintroduced in the future Secondly uncertainties from theenvironment and user demands are inevitable in practice(e real-time feedback mechanism can be introduced toovercome such uncertainties (irdly the game theory-based power dispatch and load scheduling consideringconflicted interests call further study Lastly the CSO-based

f = Je + Jc + beta

P1P2P3g3

P4P5

0

1

2

3

4

5

6

Pow

er (k

W)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240Time (h)

Figure 8 Power dispatch of the previous design in case (iii)

Complexity 13

numerical solver can be further investigated to improve thealgorithm performance

Data Availability

All relevant data used to support the findings of this studyare included within the article

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Nature ScienceFoundation of China under Grant 61803162 and the Fun-damental Research Funds for the Central Universities underGrant HUST 2017KFYXJJ178

References

[1] J Soares T Pinto F Lezama and H Morais ldquoSurvey oncomplex optimization and simulation for the new powersystems paradigmrdquo Complexity vol 2018 Article ID 234062832 pages 2018

[2] B Zhu H Tazvinga and X Xia ldquoSwitched model predictivecontrol for energy dispatching of a photovoltaic-diesel-batteryhybrid power systemrdquo IEEE Transactions on Control SystemsTechnology vol 23 pp 1229ndash1236 2014

[3] S Shaahid and M Elhadidy ldquoEconomic analysis of hybridphotovoltaicndashdieselndashbattery power systems for residentialloads in hot regionsmdasha step to clean futurerdquo Renewable andSustainable Energy Reviews vol 12 pp 488ndash503 2008

[4] P Nema R Nema and S Rangnekar ldquoA current and futurestate of art development of hybrid energy system using windand pv-solar a reviewrdquo Renewable and Sustainable EnergyReviews vol 13 pp 2096ndash2103 2009

[5] X Li J Lai and R Tang ldquoA hybrid constraints handlingstrategy for multiconstrained multiobjective optimizationproblem of microgrid economicalenvironmental dispatchrdquoComplexity vol 2017 Article ID 6249432 12 pages 2017

[6] T Cheng M Chen Y Wang et al ldquoAdaptive robust methodfor dynamic economic emission dispatch incorporating re-newable energy and energy storagerdquo Complexity vol 2018Article ID 2517987 13 pages 2018

[7] X Chen B Xu and W Du ldquoAn improved particle swarmoptimization with biogeography-based learning strategy foreconomic dispatch problemsrdquo Complexity vol 2018 ArticleID 7289674 15 pages 2018

[8] R Kallel G Boukettaya and L Krichen ldquoDemand sidemanagement of household appliances in stand-alone hybridphotovoltaic systemrdquo Renewable Energy vol 81 pp 123ndash1352015

[9] S Phiri and K Kusakana ldquoDemand side management of agrid connected pv-wt-battery hybrid systemrdquo in Proceedingsof the 2016 International Conference on the Industrial andCommercial Use of Energy (ICUE) pp 45ndash51 IEEE CapeTown South Africa August 2016

[10] H Tazvinga X Xia and J Zhang ldquoMinimum cost solution ofphotovoltaicndashdieselndashbattery hybrid power systems for remoteconsumersrdquo Solar Energy vol 96 pp 292ndash299 2013

[11] H Tazvinga B Zhu and X Xia ldquoEnergy dispatch strategy fora photovoltaicndashwindndashdieselndashbattery hybrid power systemrdquoSolar Energy vol 108 pp 412ndash420 2014

[12] D Setlhaolo and X Xia ldquoOptimal scheduling of householdappliances with a battery storage system and coordinationrdquoEnergy and Buildings vol 94 pp 61ndash70 2015

[13] S M Sichilalu and X Xia ldquoOptimal energy control of grid tiedpvndashdieselndashbattery hybrid system powering heat pump waterheaterrdquo Solar Energy vol 115 pp 243ndash254 2015

[14] D Setlhaolo and X Xia ldquoCombined residential demand sidemanagement strategies with coordination and economicanalysisrdquo International Journal of Electrical Power amp EnergySystems vol 79 pp 150ndash160 2016

[15] F Yang and X Xia ldquoTechno-economic and environmentaloptimization of a household photovoltaic-battery hybridpower system within demand side managementrdquo RenewableEnergy vol 108 pp 132ndash143 2017

[16] C Kagiri E M Wanjiru L Zhang and X Xia ldquoOptimizedresponse to electricity time-of-use tariff of a compressednatural gas fuelling stationrdquo Applied Energy vol 222pp 244ndash256 2018

[17] R Cheng and Y Jin ldquoA competitive swarm optimizer for largescale optimizationrdquo IEEE Transactions on Cybernetics vol 45pp 191ndash204 2014

[18] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofmachine learning C Sammut and G I Webb Eds SpringerBoston MA USA pp 760ndash766 2010

[19] S Leva A Dolara F Grimaccia M Mussetta and E OgliarildquoAnalysis and validation of 24 hours ahead neural networkforecasting of photovoltaic output powerrdquo Mathematics andComputers in Simulation vol 131 pp 88ndash100 2017

[20] H Bindner T Cronin P Lundsager J F ManwellU Abdulwahid and I Baring-Gould Lifetime Modelling ofLead Acid Batteries Forskningscenter Risoe Roskilde Den-mark 2005

[21] C Wei M Benosman and T Kim ldquoOnline parameteridentification for state of power prediction of lithium-ionbatteries in electric vehicles using extremum seekingrdquo In-ternational Journal of Control Automation and Systems 2019In press

[22] C Yin X Huang S Dadras et al ldquoDesign of optimal lightingcontrol strategy based on multi-variable fractional-orderextremum seeking methodrdquo Information Sciences vol 465pp 38ndash60 2018

[23] F Cheng L Qu W Qiao C Wei and L Hao ldquoFault di-agnosis of wind turbine gearboxes based on DFIG statorcurrent envelope analysisrdquo IEEE Transactions on SustainableEnergy vol 10 no 3 pp 1044ndash1053 2018

[24] X Xia and J Zhang ldquoOperation efficiency optimisationmodelling and application of model predictive controlrdquo IEEECAA Journal of Automatica Sinica vol 2 pp 166ndash172 2015

[25] R Cheng ldquoNature inspired optimization of large problemsrdquoPhD thesis University of Surrey Guildford UK 2016

[26] Q Zhao and L-y Zhang ldquoResearch on the effect of Dpso inteam selection optimization under the background of big datardquoComplexity vol 2018 Article ID 1386407 14 pages 2018

[27] J Liu H Ma X Ren T Shi P Li and X Ma ldquo(e con-tinuous-discrete pso algorithm for shape formation problemof multiple agents in two and three dimensional spacerdquoApplied Soft Computing vol 67 pp 409ndash433 2018

[28] L Wang and X-l Zheng ldquoA knowledge-guided multi-ob-jective fruit fly optimization algorithm for the multi-skillresource constrained project scheduling problemrdquo Swarmand Evolutionary Computation vol 38 pp 54ndash63 2018

14 Complexity

[29] H Tazvinga and T Hove PhotovoltaicDieselBattery HybridPower Supply System VDMPublishers Pretoria South Africa2010

[30] Z Wu and T W Chow ldquoNeighborhood field for cooperativeoptimizationrdquo Soft Computing vol 17 pp 819ndash834 2013

[31] N Ao M Zhao Q Li S Qu and Z Wu ldquoNetwork char-acteristics for neighborhood field algorithmsrdquo Neural Com-puting and Applications 2019

Complexity 15

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

standard and off-peak hours resulting in a lower energycost (e objectives Je and Jc manifest a certain level oftrade-off (is is resulted from the employment of the TOUtariff where some off-peak hours may be infeasible for thePV system because of its intermittent nature When com-paring the results among the three cases it can be found thatthe first two cases can achieve lower energy cost and gridpower consumption because of ignoring the inconvenienceindicator β It invokes an interesting topic for future studiesthat how to strike a balance between the conflicted interestsof user satisfaction and energy efficiency in such a hybridenergy system management

6 Conclusions

(is paper investigates a technoeconomic optimizationproblem for a domestic grid-connected PV-battery hybridenergy system via extending and improving a previouslyproposed system design According to the previous designthe power dispatch is decided for the totality of the elec-trical loads In the new model appliances that comprise theelectrical loads are supplied and managed respectively viaadditional power lines and switches from each powersource Furthermore the appliance time scheduling isincorporated into such a flexible power dispatch In thisway the system achieves better energy efficiency andeconomic performances via the technoeconomic optimi-zation (e performances are evaluated by three optimi-zation objectives minimizing energy cost maximizingrenewable energy penetration and increasing user satis-faction over a finite horizon (ere are nonlinear objectivefunctions and constraints as well as discrete and contin-uous decision variables in such an optimization problem

As a result the problem becomes an MINLP one at a largescale which is difficult to solve A competitive swarmoptimizer-based numerical solver is thereby designed andemployed

In order to verify that the new design does improve theperformances a case study is investigated where the powerdispatch and appliance time scheduling on a daily basis areapplied to a typical South African household hybrid system(ere are three optimization cases each with differentobjective functions including only energy cost minimiza-tion only renewable energy penetration maximization anda weighted sum of the three objectives Simulations areapplied in these cases where comparative results are alsoobtained via optimization of the previous system design(esame solver and system configurations are employed In allcases the results from the new design outperform resultsfrom the previous design (e improvement ranges from71 to 133 and manifests that further energy efficiencyand economic benefits can be achieved by the proposedapproach Furthermore the solver generally takes around184minutes to obtain the solution It verifies that theproposed approach has the potential for application in areal-time context

(ere are several future works to investigate based on thecurrent-stage results Firstly the performance evaluationssuch as the battery wear cost and the renewable energypenetration are simplified More practical indicators can beintroduced in the future Secondly uncertainties from theenvironment and user demands are inevitable in practice(e real-time feedback mechanism can be introduced toovercome such uncertainties (irdly the game theory-based power dispatch and load scheduling consideringconflicted interests call further study Lastly the CSO-based

f = Je + Jc + beta

P1P2P3g3

P4P5

0

1

2

3

4

5

6

Pow

er (k

W)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240Time (h)

Figure 8 Power dispatch of the previous design in case (iii)

Complexity 13

numerical solver can be further investigated to improve thealgorithm performance

Data Availability

All relevant data used to support the findings of this studyare included within the article

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Nature ScienceFoundation of China under Grant 61803162 and the Fun-damental Research Funds for the Central Universities underGrant HUST 2017KFYXJJ178

References

[1] J Soares T Pinto F Lezama and H Morais ldquoSurvey oncomplex optimization and simulation for the new powersystems paradigmrdquo Complexity vol 2018 Article ID 234062832 pages 2018

[2] B Zhu H Tazvinga and X Xia ldquoSwitched model predictivecontrol for energy dispatching of a photovoltaic-diesel-batteryhybrid power systemrdquo IEEE Transactions on Control SystemsTechnology vol 23 pp 1229ndash1236 2014

[3] S Shaahid and M Elhadidy ldquoEconomic analysis of hybridphotovoltaicndashdieselndashbattery power systems for residentialloads in hot regionsmdasha step to clean futurerdquo Renewable andSustainable Energy Reviews vol 12 pp 488ndash503 2008

[4] P Nema R Nema and S Rangnekar ldquoA current and futurestate of art development of hybrid energy system using windand pv-solar a reviewrdquo Renewable and Sustainable EnergyReviews vol 13 pp 2096ndash2103 2009

[5] X Li J Lai and R Tang ldquoA hybrid constraints handlingstrategy for multiconstrained multiobjective optimizationproblem of microgrid economicalenvironmental dispatchrdquoComplexity vol 2017 Article ID 6249432 12 pages 2017

[6] T Cheng M Chen Y Wang et al ldquoAdaptive robust methodfor dynamic economic emission dispatch incorporating re-newable energy and energy storagerdquo Complexity vol 2018Article ID 2517987 13 pages 2018

[7] X Chen B Xu and W Du ldquoAn improved particle swarmoptimization with biogeography-based learning strategy foreconomic dispatch problemsrdquo Complexity vol 2018 ArticleID 7289674 15 pages 2018

[8] R Kallel G Boukettaya and L Krichen ldquoDemand sidemanagement of household appliances in stand-alone hybridphotovoltaic systemrdquo Renewable Energy vol 81 pp 123ndash1352015

[9] S Phiri and K Kusakana ldquoDemand side management of agrid connected pv-wt-battery hybrid systemrdquo in Proceedingsof the 2016 International Conference on the Industrial andCommercial Use of Energy (ICUE) pp 45ndash51 IEEE CapeTown South Africa August 2016

[10] H Tazvinga X Xia and J Zhang ldquoMinimum cost solution ofphotovoltaicndashdieselndashbattery hybrid power systems for remoteconsumersrdquo Solar Energy vol 96 pp 292ndash299 2013

[11] H Tazvinga B Zhu and X Xia ldquoEnergy dispatch strategy fora photovoltaicndashwindndashdieselndashbattery hybrid power systemrdquoSolar Energy vol 108 pp 412ndash420 2014

[12] D Setlhaolo and X Xia ldquoOptimal scheduling of householdappliances with a battery storage system and coordinationrdquoEnergy and Buildings vol 94 pp 61ndash70 2015

[13] S M Sichilalu and X Xia ldquoOptimal energy control of grid tiedpvndashdieselndashbattery hybrid system powering heat pump waterheaterrdquo Solar Energy vol 115 pp 243ndash254 2015

[14] D Setlhaolo and X Xia ldquoCombined residential demand sidemanagement strategies with coordination and economicanalysisrdquo International Journal of Electrical Power amp EnergySystems vol 79 pp 150ndash160 2016

[15] F Yang and X Xia ldquoTechno-economic and environmentaloptimization of a household photovoltaic-battery hybridpower system within demand side managementrdquo RenewableEnergy vol 108 pp 132ndash143 2017

[16] C Kagiri E M Wanjiru L Zhang and X Xia ldquoOptimizedresponse to electricity time-of-use tariff of a compressednatural gas fuelling stationrdquo Applied Energy vol 222pp 244ndash256 2018

[17] R Cheng and Y Jin ldquoA competitive swarm optimizer for largescale optimizationrdquo IEEE Transactions on Cybernetics vol 45pp 191ndash204 2014

[18] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofmachine learning C Sammut and G I Webb Eds SpringerBoston MA USA pp 760ndash766 2010

[19] S Leva A Dolara F Grimaccia M Mussetta and E OgliarildquoAnalysis and validation of 24 hours ahead neural networkforecasting of photovoltaic output powerrdquo Mathematics andComputers in Simulation vol 131 pp 88ndash100 2017

[20] H Bindner T Cronin P Lundsager J F ManwellU Abdulwahid and I Baring-Gould Lifetime Modelling ofLead Acid Batteries Forskningscenter Risoe Roskilde Den-mark 2005

[21] C Wei M Benosman and T Kim ldquoOnline parameteridentification for state of power prediction of lithium-ionbatteries in electric vehicles using extremum seekingrdquo In-ternational Journal of Control Automation and Systems 2019In press

[22] C Yin X Huang S Dadras et al ldquoDesign of optimal lightingcontrol strategy based on multi-variable fractional-orderextremum seeking methodrdquo Information Sciences vol 465pp 38ndash60 2018

[23] F Cheng L Qu W Qiao C Wei and L Hao ldquoFault di-agnosis of wind turbine gearboxes based on DFIG statorcurrent envelope analysisrdquo IEEE Transactions on SustainableEnergy vol 10 no 3 pp 1044ndash1053 2018

[24] X Xia and J Zhang ldquoOperation efficiency optimisationmodelling and application of model predictive controlrdquo IEEECAA Journal of Automatica Sinica vol 2 pp 166ndash172 2015

[25] R Cheng ldquoNature inspired optimization of large problemsrdquoPhD thesis University of Surrey Guildford UK 2016

[26] Q Zhao and L-y Zhang ldquoResearch on the effect of Dpso inteam selection optimization under the background of big datardquoComplexity vol 2018 Article ID 1386407 14 pages 2018

[27] J Liu H Ma X Ren T Shi P Li and X Ma ldquo(e con-tinuous-discrete pso algorithm for shape formation problemof multiple agents in two and three dimensional spacerdquoApplied Soft Computing vol 67 pp 409ndash433 2018

[28] L Wang and X-l Zheng ldquoA knowledge-guided multi-ob-jective fruit fly optimization algorithm for the multi-skillresource constrained project scheduling problemrdquo Swarmand Evolutionary Computation vol 38 pp 54ndash63 2018

14 Complexity

[29] H Tazvinga and T Hove PhotovoltaicDieselBattery HybridPower Supply System VDMPublishers Pretoria South Africa2010

[30] Z Wu and T W Chow ldquoNeighborhood field for cooperativeoptimizationrdquo Soft Computing vol 17 pp 819ndash834 2013

[31] N Ao M Zhao Q Li S Qu and Z Wu ldquoNetwork char-acteristics for neighborhood field algorithmsrdquo Neural Com-puting and Applications 2019

Complexity 15

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

numerical solver can be further investigated to improve thealgorithm performance

Data Availability

All relevant data used to support the findings of this studyare included within the article

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Nature ScienceFoundation of China under Grant 61803162 and the Fun-damental Research Funds for the Central Universities underGrant HUST 2017KFYXJJ178

References

[1] J Soares T Pinto F Lezama and H Morais ldquoSurvey oncomplex optimization and simulation for the new powersystems paradigmrdquo Complexity vol 2018 Article ID 234062832 pages 2018

[2] B Zhu H Tazvinga and X Xia ldquoSwitched model predictivecontrol for energy dispatching of a photovoltaic-diesel-batteryhybrid power systemrdquo IEEE Transactions on Control SystemsTechnology vol 23 pp 1229ndash1236 2014

[3] S Shaahid and M Elhadidy ldquoEconomic analysis of hybridphotovoltaicndashdieselndashbattery power systems for residentialloads in hot regionsmdasha step to clean futurerdquo Renewable andSustainable Energy Reviews vol 12 pp 488ndash503 2008

[4] P Nema R Nema and S Rangnekar ldquoA current and futurestate of art development of hybrid energy system using windand pv-solar a reviewrdquo Renewable and Sustainable EnergyReviews vol 13 pp 2096ndash2103 2009

[5] X Li J Lai and R Tang ldquoA hybrid constraints handlingstrategy for multiconstrained multiobjective optimizationproblem of microgrid economicalenvironmental dispatchrdquoComplexity vol 2017 Article ID 6249432 12 pages 2017

[6] T Cheng M Chen Y Wang et al ldquoAdaptive robust methodfor dynamic economic emission dispatch incorporating re-newable energy and energy storagerdquo Complexity vol 2018Article ID 2517987 13 pages 2018

[7] X Chen B Xu and W Du ldquoAn improved particle swarmoptimization with biogeography-based learning strategy foreconomic dispatch problemsrdquo Complexity vol 2018 ArticleID 7289674 15 pages 2018

[8] R Kallel G Boukettaya and L Krichen ldquoDemand sidemanagement of household appliances in stand-alone hybridphotovoltaic systemrdquo Renewable Energy vol 81 pp 123ndash1352015

[9] S Phiri and K Kusakana ldquoDemand side management of agrid connected pv-wt-battery hybrid systemrdquo in Proceedingsof the 2016 International Conference on the Industrial andCommercial Use of Energy (ICUE) pp 45ndash51 IEEE CapeTown South Africa August 2016

[10] H Tazvinga X Xia and J Zhang ldquoMinimum cost solution ofphotovoltaicndashdieselndashbattery hybrid power systems for remoteconsumersrdquo Solar Energy vol 96 pp 292ndash299 2013

[11] H Tazvinga B Zhu and X Xia ldquoEnergy dispatch strategy fora photovoltaicndashwindndashdieselndashbattery hybrid power systemrdquoSolar Energy vol 108 pp 412ndash420 2014

[12] D Setlhaolo and X Xia ldquoOptimal scheduling of householdappliances with a battery storage system and coordinationrdquoEnergy and Buildings vol 94 pp 61ndash70 2015

[13] S M Sichilalu and X Xia ldquoOptimal energy control of grid tiedpvndashdieselndashbattery hybrid system powering heat pump waterheaterrdquo Solar Energy vol 115 pp 243ndash254 2015

[14] D Setlhaolo and X Xia ldquoCombined residential demand sidemanagement strategies with coordination and economicanalysisrdquo International Journal of Electrical Power amp EnergySystems vol 79 pp 150ndash160 2016

[15] F Yang and X Xia ldquoTechno-economic and environmentaloptimization of a household photovoltaic-battery hybridpower system within demand side managementrdquo RenewableEnergy vol 108 pp 132ndash143 2017

[16] C Kagiri E M Wanjiru L Zhang and X Xia ldquoOptimizedresponse to electricity time-of-use tariff of a compressednatural gas fuelling stationrdquo Applied Energy vol 222pp 244ndash256 2018

[17] R Cheng and Y Jin ldquoA competitive swarm optimizer for largescale optimizationrdquo IEEE Transactions on Cybernetics vol 45pp 191ndash204 2014

[18] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofmachine learning C Sammut and G I Webb Eds SpringerBoston MA USA pp 760ndash766 2010

[19] S Leva A Dolara F Grimaccia M Mussetta and E OgliarildquoAnalysis and validation of 24 hours ahead neural networkforecasting of photovoltaic output powerrdquo Mathematics andComputers in Simulation vol 131 pp 88ndash100 2017

[20] H Bindner T Cronin P Lundsager J F ManwellU Abdulwahid and I Baring-Gould Lifetime Modelling ofLead Acid Batteries Forskningscenter Risoe Roskilde Den-mark 2005

[21] C Wei M Benosman and T Kim ldquoOnline parameteridentification for state of power prediction of lithium-ionbatteries in electric vehicles using extremum seekingrdquo In-ternational Journal of Control Automation and Systems 2019In press

[22] C Yin X Huang S Dadras et al ldquoDesign of optimal lightingcontrol strategy based on multi-variable fractional-orderextremum seeking methodrdquo Information Sciences vol 465pp 38ndash60 2018

[23] F Cheng L Qu W Qiao C Wei and L Hao ldquoFault di-agnosis of wind turbine gearboxes based on DFIG statorcurrent envelope analysisrdquo IEEE Transactions on SustainableEnergy vol 10 no 3 pp 1044ndash1053 2018

[24] X Xia and J Zhang ldquoOperation efficiency optimisationmodelling and application of model predictive controlrdquo IEEECAA Journal of Automatica Sinica vol 2 pp 166ndash172 2015

[25] R Cheng ldquoNature inspired optimization of large problemsrdquoPhD thesis University of Surrey Guildford UK 2016

[26] Q Zhao and L-y Zhang ldquoResearch on the effect of Dpso inteam selection optimization under the background of big datardquoComplexity vol 2018 Article ID 1386407 14 pages 2018

[27] J Liu H Ma X Ren T Shi P Li and X Ma ldquo(e con-tinuous-discrete pso algorithm for shape formation problemof multiple agents in two and three dimensional spacerdquoApplied Soft Computing vol 67 pp 409ndash433 2018

[28] L Wang and X-l Zheng ldquoA knowledge-guided multi-ob-jective fruit fly optimization algorithm for the multi-skillresource constrained project scheduling problemrdquo Swarmand Evolutionary Computation vol 38 pp 54ndash63 2018

14 Complexity

[29] H Tazvinga and T Hove PhotovoltaicDieselBattery HybridPower Supply System VDMPublishers Pretoria South Africa2010

[30] Z Wu and T W Chow ldquoNeighborhood field for cooperativeoptimizationrdquo Soft Computing vol 17 pp 819ndash834 2013

[31] N Ao M Zhao Q Li S Qu and Z Wu ldquoNetwork char-acteristics for neighborhood field algorithmsrdquo Neural Com-puting and Applications 2019

Complexity 15

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

[29] H Tazvinga and T Hove PhotovoltaicDieselBattery HybridPower Supply System VDMPublishers Pretoria South Africa2010

[30] Z Wu and T W Chow ldquoNeighborhood field for cooperativeoptimizationrdquo Soft Computing vol 17 pp 819ndash834 2013

[31] N Ao M Zhao Q Li S Qu and Z Wu ldquoNetwork char-acteristics for neighborhood field algorithmsrdquo Neural Com-puting and Applications 2019

Complexity 15

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