A New Optimal Dispatch Method for the Day-Ahead Electricity Market Using a Multi-

objective Evolutionary Approach

Dapeng Li, Anil Pahwa, Fellow, IEEE, Sanjoy Das, and Deepal Rodrigo

Abstract-- In this paper, a novel approach to optimally operatea day-ahead electricity market with considerations of marketpower is proposed. First, this approach formulates the 1-hourbased optimal dispatch problem as a nonlinear constrainedmulti-objective optimization problem which simultaneouslyminimizes fuel cost, emissions and the modified Herfindahl–Hirschman index. Then the Non-dominated Sorting GeneticAlgorithm II (NSGA-II) is used to solve this optimizationproblem. Simulation results illustrate that this algorithm is capable of finding the Pareto-optimal front in a single run. Thebest operation schedule for the whole 24 hours can be chosen bytaking the ramp rate constraints of generators into account. Thisapproach can provide the market operator with much operatingfeasibility and thus is more realistic and feasible.

Index Terms—Economic dispatch, market power, day-aheadelectricity market, multi-objective evolutionary algorithm.

I. INTRODUCTION

HE electric power industry around the world isexperiencing unprecedented restructuring for breaking

traditional monopoly, introducing competition and creating electricity markets. The Independent System Operator playsan important role of coordinating, controling and monitoringthe operation of the regional power system.

The main goal of the traditional economic dispatch is tooperate all the generators at minimum fuel cost subject tovarious system and operating constraints. However, with theClean Air Act Amendments of 1990 applying to theenvironmental protection, electric power companies had toadopt some strategies to reduce the atmospheric emissions of the thermal power plants. These strategies include installationof pollution cleaning equipment, replacement of aged fuel-burners and generator units, using low emission fuels and consideration of emission dispatch when making thegenerating schedule [1-2]. Among these methodologies, thelast one has been receiving increasing attention. In the electricity market, obtaining the optimal dispatch whileconsidering both fuel cost and emission

D. Li, A. Pahwa and S. Das are with the Department of Electrical andComputer Engineering, Kansas State University, Manhattan, KS 66506 USA(e-mail: dapeng@ksu.edu; pahwa@ksu.edu; sdas@ksu.edu).

D. Rodrigo is with the Midwest ISO, Inc., Carmel, IN 46032 USA (e-mail:drodrigo@midwestiso.org).

(economic/environment dispatch) is very important for a market operator. There are two major research directions to achieve this goal: One is to minimize fuel cost while treatingthe emission as a constraint [3-5], the other one is tosimultaneously take the emission as another objective to beminimized in addition to the fuel cost [6-10]. The secondmethod has received a lot of attention in recent years.

During past few years many studies only consideredeconomic/environment dispatch. However, in the deregulatedelectricity market, market power policy is a priority concernof market operators and market participants. Market power isthe ability of a single firm who owns one or more generatorsor a group of these firms to maintain electricity pricesprofitably above competitive levels for a sustained period oftime. One of the most common means of performing marketpower analysis is the analysis of market concentration. A widely used methodology is the Herfindahl–Hirschman index(HHI) [11]. This index is defined as follows:

N

iisHHI

1

2 (1)

where N is the number of market participants and refers to

the percentage market share of each participant. Obviously the HHI for a monopoly firm who has market share of 100%

would be . On the other hand, the HHI would bea small number if there are many participants and noparticipant has more than say 5% market share. If the HHI isless than 1000, the market is not concentrated; if the HHI isbetween 1000 and 1800, the market is moderatelyconcentrated; and if the HHI is greater than 1800, the marketis highly concentrated. Thus, taking the market power intoaccount while doing economic/environmental dispatch to forma multi-objective problem can give decision makers moreoptions to choose an optimal schedule according to realconditions.

is

10000102

In a day-ahead electricity market, firms submit offer schedules to market operators as means of expressingwillingness to supply electricity before a pre-specified deadline, usually 15 hours ahead of the beginning of themarket. Usually firms submit offer schedules based on everyhour of the day. However, the market is consideredcontinuously operating for the entire 24-hour period

T

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transitioning smoothly from the previous day to the currentmarket day. Moreover, in order to keep thermal gradientsinside the turbine within safe limits to avoid shortening thelife of generators, ramp rate constraints of the generators for every two sequential hours have to be considered in thedispatch problem. Hence, choosing an optimal schedule forthe entire 24-hour period rather than each individual hour in isolation is more realistic and useful for a market operator.Dynamic Economic Dispatch (DED), which takes the entirehorizon into account, has received a great deal of attention. Different methods, such as dynamic programming [12-14],dual optimization technique [15], improved LP-based redispatch technique [16, 17] were applied to DED problems.

Over the past few years, artificial intelligence techniquessuch as, tabu search [18], simulated annealing [19], geneticalgorithms [20], evolutionary programming [21-23], particleswarm optimization (PSO) [24], Hopfield neural networks(HNN) [26] and adaptive Hopfield neural networks (AHNN)[27], have provided efficient and powerful approaches toobtain the global or near global optimum in power systemoptimization problems.

Many multi-objective problems are highly convex andnonlinear. Conventional multi-objective optimization methodsconvert multiple objectives into a single one by using a weighting method. However, the preference-based weightingstrategy is subjective to the decision marker. Moreover, itcannot find multiple solutions in one single run. As an alternative, multi-objective evolutionary algorithm does notneed any pre-defined preference information, and are able tofind many different trade-off solutions which are Pareto-optimal. Various evolutionary algorithms, such as Horn etal.’s niched-pareto genetic algorithm (NPGA) [28], Srinivasand Deb’s non-dominated genetic sorting algorithm (NSGA) [29], Zitzler and Thiele’s strength pareto evolutionaryalgorithm (SPEA) [30], Knowles and Corne’s pareto-archivedevolution strategy (PAES) [31], Deb et al.’s non-dominatedgenetic algorithm II (NSGA-II) [32], have been successfully applied to real multi-objective problems. Among those, NSGA-II is one of most efficient algorithms which can find a set of solutions as close as possible to the Pareto-optimal frontwhile keeping these solutions as diverse as possible.

In this paper, a multi-objective optimal dispatch problemwith considerations of market power for a day-ahead electricity market is proposed, and NSGA-II is used for solving this problem. After getting 24 different Pareto-frontsfor each hour before the operating day, decision makers can choose an optimal schedule according to some constraints such as ramp rate and other preferences.

II. PROBLEM FORMULATION

In the day-ahead electricity market, the optimal dispatch for each hour can be formulated as follows.

A. Minimization of fuel cost

The overall fuel cost of the system can be

represented as the sum of a set of quadratic functions of activepower output of generators:

)( GPF

)/($)( 2

1

hcPbPaPF iGiGi

NG

iG ii

(2)

where NG is the number of generators in the system;

and are the cost coefficients of the i-th generator; is

the active power output of the i-th generator.

ia ib

iciGP

B. Minimization of Emission

The total emission of all the pollutants such

as , and particles can be also expressed as the sum

of a set of quadratic functions of active power output of generators [37]:

)( GPE

xNO xSO

)/()exp(10)( 2

1

2 htonPPPPEiii GiiiGiGi

NG

iG

(3) where

iiii ,,, andiare emission cost coefficients of

the i-th generator.

C. Minimization of Modified HHI

Here a modified HHI which represents the dynamic natureof a day-ahead electricity market are used to evaluate the market power of each firm operating in the market [35]. The difference between this modified HHI and traditional HHI is that in this modified formulation only those generators thathave capacity remaining to be offered at a given price-levelwill be considered in calculating the market-share index.Formulation of this modified index can be represented as:

)(

1

2)]([DN

ii DsDHHI (4)

where N(D) is the number of owners whose generators stillhave capacity remaining to be offered to the market; si(D

+) is the revised market share of generator owner i, with capacitystill left to offer into the market. Typical regulatory and legalinterpretations of DHHI are similar to those of the traditional HHI which are presented in the previous section.

D. Equality and Inequality Constraints

(i) Power Balance Constraints

)5(,...,10cos||1

NBiYVVPPNB

jijijijjidG ii

and are the voltage magnitude and the phase angle of the i-

)6(,...,10sin||1

NBiYVVQQNB

jijijijjidG ii

where and are active and reactive output of i-th

generator respectively and are active and reactive

load demand of the i-th bus is the i-jth element of the Y-

bus matrix of the system; is the phase angle of

iGP iGQ

idPidQ

ijY

ij ijY iV

i

434 2007 39th North American Power Symposium (NAPS 2007)

th bu respectively; NB is the number of buses. iV and i for

each bus are obtained using power flow based of Newton-Raphson method.

s

(ii) Generator Capacity LimitsThe generation power of eac

mh

where and re the minimum and maxim

utpu the i-t

owe put

SORTING GENETIC A

op

d Sorting

counter i=1.

rm tP 1

: Perform the Cr t ( proce

N

he sorted

y u

Distance Assignment Procedure: Crowding-sort

Call the number of solutions in F as

generator should lie between aximum limit and minimum limit. That is,

NGiPPPiii GGG 1maxmin (7),...,

NGiQQQiii GGG ,...,1maxmin (8)

min

iGP max

iGP a um active

power o t of h generator, and min

iGQ and max

iGQ are the

minimum and maximum reactive p r out of i-thgenerator respectively.

III. ELITIST NON-DOMINATED LGORITHM

There are two important goals in a multi-objectivetimization [33]:

a. to find a set of solutions as close as possible to the Pareto-optimal front

b. to find a set of solutions as diverse as possible.Deb et al have proposed an elitist Non-dominate

GA (NSGA-II) which uses elite-preservation strategy and diversity-preserving mechanism. It can find a set of solutionshaving a good diversity in the desired space for most complexand nonlinear multi-objective problems. It has been illustratedthat NSGA-II is more efficient compared with other elitist multi-objective evolutionary algorithms such as PAES andSPEA. Generally, the procedure of NSGA-II can be describedas following [32].Step 1: Combine parent and offspring populations and

create ttt QPR . Perform a non-dominated sorting to tR

and ident t fronts: ,iF i=1, 2, …, etc.

Step 2: Set new population 1t . Set a

ify differen

P

Until NFP it |||| 1 , perfo it FP 1 and i=i+1.

Step 3 owding-sor dure belowciF , )

and include the most widely spread ( |1tP ) solutions by

using the crowding distance values in t iF to 1tP .

Step 4: Create offspring population 1tQ from 1 b sing

|

tPthe crowded tournament selection, crossover and mutationoperators.

rowdingC( ciF , )

Step C1: || iFl . For

functi m=1,2,…,M, sort the set

1

each i in the set, first assign 0id .

Step C2: For each objective on

in worse order of mf or, find the sorted indices vector:

),( mm fsortI .

Step C3: For m=1,2,…,M, assign a large distance to the

boundary solutions, or d , and for all other

solutions j=2 to (l-1), assign:

ml

m IId

minmax

11

mm

Im

Im

II ff

ffdd

mj

mj

mj

mj

IV. THE BEST SCHEDULE FOR A DAY-AHEAD ELECTRICITY

MARKET

The model above is based on 1 hour time intervals.However, it is necessary to choose one optimal schedule for continuous time period of 24 hours before the operatingmarket day. Most studies of Dynamic Economic Dispatch(DED) have treated the whole 24 hours as a integral intervalof time and included the ramp rate constraints whichdistinguished the DED from static economic dispatch.However, since the ramping constraints couple the timeintervals, a direct solution of a DED with N units and T timeintervals would require the solution of an optimizationproblem of size TN , a considerably more complex task than the solution of T economic dispatch problems, each with N units [36]. Moreover, there are more combinations ofdifferent schedules than those by taking the whole 24 hours an integral interval of time. Thus with considerations ofoperational feasibility and optimality, it is more realistic andgives the market operator more freedom to choose an optimalschedule for each hour to form a best schedule for the whole 24 hours, that is, the time horizon is divided in to 24 hours andthe dispatch is optimized hour-by-hour, then solved bystochastic algorithms such as GA considering the ramp rateconstraints to get the best schedule for 24 hours. The schematic diagram of the solution algorithm is shown in Fig. 1.

Fig. 1. Schematic diagram of choosing the best schedule for the whole 24 hours for a day-ahead electricity market.

Got all schedules for 24 hours?

Solve the new model using NSGA-II

Save schedules for this hour

Input the ramp rate constraints and decision makers’ preferences

Input data of one hour

Get the best schedule for the whole 24 hours by any stochastic algorithm

Y

N

2007 39th North American Power Symposium (NAPS 2007) 435

V. RESULTS AND DISCUSSION

The standard IEEE 30-bus 6-generator test system is usedto investigate the effectiveness of the proposed model and theefficiency of NSGA-II. The diagram of the test system is shown in Fig. 2. The values of fuel cost and emissioncoefficients are given in Table I and Table II respectively. Table III gives the generator active and reactive power limits.There are six generators which are G1, G2, G3, G4, G5 and G6 at bus 1, 2, 5, 8, 11 and 13 in the system respectively. Inorder to show the effectiveness of NSGA-II, test results ofthree special cases are shown for example. Case 1 considers that each generator belongs to an individual firm. Case 2 considers that there are 3 firms in the market and each one owns two different generators. Case 3 considers that there isonly one firm which owns six generators in the market.

The optimal schedule problem on 1-hour basis was handledas a 3-objective optimization problem where all objectives were optimized simultaneously with NSGA-II. All non-dominated solutions which are optimal schedules for thespecified hour can be attained in a single run. The parametersbelonging to the real-coded NSGA-II used in the simulationsin each case are:

Population size = 80 Maximum generation number=100Crossover probability=0.95Mutation probability = 0.05Distribution index for crossover = 10 Distribution index for mutation = 40

Fig. 2. IEEE 30-Bus Test System [34].

TABLE IGENERATOR FUEL COST COEFFICIENTS

Generator iia ib ic

G1 10 200 100G2 10 150 120G3 20 180 40G4 10 100 60G5 20 180 40G6 10 150 100

TABLE IIGENERATOR EMISSION COEFFICIENTS

Generator ii i i i i

G1 4.091 -5.554 6.490 2e-4 2.857G2 2.543 -6.047 5.638 5e-4 3.333G3 4.528 -5.094 4.586 1e-6 8.000G4 5.326 -3.550 3.380 2e-3 2.000G5 4.258 -5.094 4.586 1e-6 8.000G6 6.131 -5.555 5.151 1e-5 6.667

TABLE IIIGENERATOR ACTIVE AND REACTIVE POWER LIMITS

Generatori

Active power limits(MW)

Reactive power limits(MVar)

Minimum Maximum Minimum MaximumG1 5 150 -20 150G2 5 150 -20 60G3 5 150 -15 62.45G4 5 150 -15 48.73G5 5 150 -10 40G6 5 150 -15 44.72

A. Test with Case 1

For this case, there are six firms which own the sixgenerators individually. The Pareto-optimal front whichconsists of the 80 non-dominated solutions is shown in Fig. 3.From the figure it can be seen that the Pareto-optimalsolutions generally lie on a nonlinear manifold which are indicated by the grey grids. The minimum, maximum and average values of fuel cost, emission and DHHI are given in Table III. It should be noticed that the DHHI lies in a narrowrange while the fuel cost and emission both have a wide range,which means that the market power for this market does notvary much. As we can see, the minimum and maximum DHHIare 1695.03 and 1695.60, which are between 1000 and 1800. Thus this market can be considered as moderatelyconcentrated. Convergence of average value of fuel cost,emission and DHHI are shown in Fig. 4, Fig. 5 and Fig.6 respectively. Obviously the average values of each objective converge after 20 generations.

436 2007 39th North American Power Symposium (NAPS 2007)

605610

615620

625630

635640

645

0.19

0.2

0.21

0.22

0.23

0.241695

1695.1

1695.2

1695.3

1695.4

1695.5

1695.6

1695.7

1695.8

Fuel cost ($/h)Emission (ton/h)

DH

HI

Non-dominated solution

Fig. 3. Non-dominated solutions of the NSGA-II after 100 generations

0 10 20 30 40 50 60 70 80 90 100620

630

640

650

660

670

680

690

Fue

l co

st($

/h)

Fig. 4. Convergence of average fuel cost

0 10 20 30 40 50 60 70 80 90 1000.2

0.205

0.21

0.215

0.22

0.225

0.23

0.235

0.24

0.245

Em

issi

on (

ton/

h)

Fig. 5. Convergence of average emission

0 10 20 30 40 50 60 70 80 90 1001695.2

1695.25

1695.3

1695.35

1695.4

1695.45

1695.5

1695.55

1695.6

1695.65

Generations

DH

HI

Fig. 6. Convergence of average DHHI

TABLE IIIMinimum, maximum and average of non-dominated solutions

Fuel cost ($/h) Emission (ton/h) DHHIMinimum 607.77 0.1970 1695.03Maximum 643.04 0.2264 1695.60Average 623.58 0.2039 1695.35

B. Test with Case 2

For this case, there are three firms and each of them owns two generators, i.e. (G1, G2), (G3, G4) and (G5, G6) belongto the three firms respectively. The Pareto-optimal front whichconsists of the 80 non-dominated solutions is shown in Fig. 7.Notice that more than half of the Pareto-optimal solutions lie on the edge of a nonlinear manifold which are indicated by thegrey grids. The minimum, maximum and average values of fuel cost, emission and DHHI are given in Table IV. It shouldbe also noticed that the DHHI lies in a narrow range while the fuel cost and emission both have a wide range, which meansthat the market power for this market does not vary much.This market is highly concentrated because the minimum and maximum DHHI are 3355.31 and 3356.50, which are greaterthan 1800.

600620

640660

680700

720740

0.18

0.2

0.22

0.24

0.260.28

0.33355

3355.5

3356

3356.5

3357

3357.5

Fuel cost ($/h)Emission (ton/h)

DH

HI

Non-dominated solution

Fig. 7. Non-dominated solutions of the NSGA-II after 100 generations

2007 39th North American Power Symposium (NAPS 2007) 437

TABLE IVMinimum, maximum and average of non-dominated solutions

Fuel cost ($/h) Emission (ton/h) DHHIMinimum 607.22 0.1976 3355.31Maximum 726.67 0.2842 3356.50Average 624.22 0.2106 3355.91

C. Test with Case 3

For this case, all the six generators belong to the only onefirm in the market, which means it owns the whole market or has the market share of 100%. In this case, the DHHI for thisfirm is 10000 which is the maximum. So the optimizationproblem reduces to a 2-objective optimization problem whichminimizes the total fuel cost and emission simultaneously.The Pareto-optimal front which consists of the 80 non-dominated solutions is illustrated in Fig. 8, from which we canobserve that all the Pareto-optimal solutions lie on a plane onwhich the DHHI equals to 10000. Obviously the firm has complete control over the market or is a complete monopoly.The minimum, maximum and average values of fuel cost, emission and DHHI are given in Table V.

605610

615620

625630

635640

0.195

0.2

0.205

0.21

0.215

0.221

1

1

1

1

1

1

x 104

Fuel cost ($/h)Emission (ton/h)

DH

HI

Non-dominated solution

Fig. 8. Non-dominated solutions of the NSGA-II after 100 generations

TABLE VMinimum, maximum and average of non-dominated solutions

Fuel cost ($/h) Emission (ton/h) DHHIMinimum 607.98 0.1982 10000Maximum 638.83 0.2195 10000Average 620.68 0.2034 10000

VI. CONCLUSION

In this paper, a new model with considerations of fuel cost,emission and market power which is evaluated by a modifiedHHI for the optimal dispatch of the day-ahead electricity market is proposed. An elitist Non-dominate Sorting Genetic Algorithm is applied to the optimal dispatch on 1-hour basis.Simulation results demonstrate that NSGA-II is efficient for

solving this multi-objective dispatch.From test results of the above three cases, we can see that

the diversities of non-dominated solutions decrease with the increase of the DHHI. It can be attributed to the increased market shares of the firms in the electricity market. The other interesting observation is that for the final Pareto-optimalsolutions, the DHHI varies in a narrow range compared with the fuel cost and emission, which indicates that marketoperators may pay much attention to the other two objectiveswhile doing schedules. The non-dominated solutions achievedfor each hour give the decision maker much more operating feasibility to choose the best schedule for the whole 24 hoursof the operating day. Future work will focus on usingstochastic algorithms such as genetic algorithms to get the bestschedule for the day in the day-ahead electricity market based on additional conditions such as ramp rate constraints.

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VIII. BIOGRAPHIES

Dapeng Li was born in Hean, China, in August, 1980. He received the B.E. from Xi’an University of Technology, Xi’an, China in 2001 and the M.E.from Shanghai University, Shanghai, China in 2004. He worked as a researchassistant at the Hong Kong Polytechnic University from April 2004 to March2005. He joined the Hygrand Electronic Equipment Company, Shanghai,China working as a power system engineer in May 2005. Currently, he is pursuing the Ph.D. degree at Kansas State University, Manhattan, KS, USA.His research interests are in the areas of computational intelligence applications in power systems and power markets.

Anil Pahwa received the B.E. (honors) degree in Electrical Engineering fromBirla Institute of Technology & Science, Pilani, India in 1975, the M.S. in Electrical Engineering from University of Maine in 1979, and the Ph.D. inElectrical Engineering from Texas A&M University in 1983. Since 1983 he has been with Kansas State University where presently he is Professor in the Electrical and Computer Engineering department. He worked at ABB-ETI inRaleigh, NC during sabbatical from August 1999 to August 2000. Hisresearch interests include distribution automation, distribution systemplanning and analysis, distribution system reliability, and intelligent computational methods for distribution system applications. He is a memberof Eta Kappa Nu, Tau Beta Pi, and ASEE, and Fellow of IEEE.

Sanjoy Das received his B.S. in Electrical Engineering from SambalpurUniversity, Orissa, India in 1987 and the M.S. and Ph.D. in ElectricalEngineering from Louisiana State University, Baton Rouge, in 1994. Between 1994 and 1996 he received his postdoctoral training from the University of California, Berkeley and the Smith-Kettlewell Institute, San Francisco. He has also worked as a research scientist at ITT systems. Since 2001, he has been an Assistant Professor in the Electrical & Computer Engineering department atKansas State University. His research interests include evolutionaryalgorithms, swarm intelligence, multi-objective optimization, neural networksand computational intelligence.

Deepal Rodrigo is the Director of Data Management & Integrity at the Midwest ISO, one of the largest Regional Transmission Organization' s in the world which is responsible for operating the electricity grid and energymarkets in the midwestern states of the United States. Deepal has worked forover 15 years in the electric utility industry with varying responsibilities in the areas of utility operations, planning, software development and informationtechnology management. Deepal holds a PhD and MS in ElectricalEngineering from the Kansas State University, MBA form the University of Colombo and is a Senior Member of the IEEE.

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