2005 IEEE/PES Transmission and DistributionConference & Exhibition: Asia and PacificDalian, China

An Improved PSO Approach for Profit-basedUnit Commitment in Electricity Market

Yuan Xiaohui , Yuan Yanbin ,Wang Cheng and Zhang Xiaopan

Abstract--In this paper, we propose a formulation of the unitcommitment problem based on the profit under the deregulatedelectricity market(PBUC). We express the unit commitmentproblem as a mixed integer nonlinear optimization problem inwhich the objective is to maximize profits for generation companyand the decisions are required to meet all kinds of operatingconstraints. Under the assumption of competitive market andprice forecasting, we developed an improved discrete binaryparticle swarm optimization(PSO) and standard value PSO tosolve this PBUC problem iteratively. A generation company with10 generating units is used to demonstrate the effectiveness of theproposed approach. Simulation results are compared with thoseobtained from reference method.

keywords: particle swarm optimization, profit-based unitcommitment, electricity market

I. NOMENCLATURE

uit The commitment state of unit i at time t.

pt Generation of unit i at time tptD

SRtPiminPimax

Rit,n

Forecasted demand at hour t

Forecasted reserve at hour tMinimum generation of unit iMaximum generation of unit iReserve power of unit i at time t

TiUr/ Minimum on time of unit i

Toff Minimum off time of unit i

Xon Time duration for which unit i has been on at time ti,t

XOf Time duration for which unit i has been off at time tip t

p t Forecasted spot price at time t

pt Forecasted reserve price at time t

f (Pit) Cost function of unit i at time t

f (Pt) = ai(p)2 + biPit +ci

This work was supported by the National Science Foundation of Chinaunder Grant No. 50409010 and No. 50309013.

Yuan Xiaohui,Wang Cheng and Zhang xiaopan are with the Departmentof Hydropower Engineering, Huazhong University of Science & Technology,Wuhan,430074 China (e-mail: yxh7IA163.com).

Yuan Yanbin is with the Department of Resource & EnvironmentEngineering, Wuhan University of Technology, Wuhan,430071 China (e-mail:yybjmA163.com).0-7803-9114-4/05/$20. 00 ©2005 IEEE.

STiProbNT

Start up cost of unit iProbability that the reserve is called and generated.Number of generator unitsNumber of hours

II. INTRODUCTIONT He task of Unit Commitment (UC) involves scheduling the

on/off status, as well as the real power outputs, of thermalunits for use in meeting all kinds of constraints. In a verticallyintegrated utility environment, the UC determines generatingunit schedules in a utility for minimizing the operating costover the time interval as a goal. The deregulation andrestructuring of the electric power industry have created acompetitive open market environment. Generation companiessolve UC not for minimizing total production cost as beforebut for maximizing their own profit under deregulatedenvironment. This problem is called profit-based UCproblem(PBUC). PBUC is defined as a method to schedulegenerators economically based on the forecasted information,such as prices and demand/reserve with an objective tomaximize profit of generation company. It is much moredifficult problem to solve than traditional UC.

The PBUC problem is a mixed integer and continuousnonlinear optimization problem, which is very complex tosolve because of its enormous dimension, a nonlinear objectivefunction, and a large number of constraints. Many solutiontechniques [1-7], such as integer programming, dynamicprogramming, Lagrangian relaxation and genetic algorithm canbe used to solve the PBUC. Because of the inherit limitation ofthese methods, which have some one or another drawback forsolution the PBUC. More recently, meta-heuristic approachbased on swarm intelligent called particle swarm optimization(PSO) is suit to solve PBUC for its good characteristic. Thispaper puts forth a new approach for solving PBUC, animproved particle swarm optimization (IPSO).A PBUC model in the competitive electricity market

environment is established in this paper, considering bothpower and reserve generations. To overcome thedisadvantages of conventional optimization techniques and thedrawbacks of PSO, an improved discrete binary PSO methodis proposed to solve the PBUC problem in electricity market.

This paper is organized as follows. Part III describes thePBUC problem formulation in the competitive environment.Part IV describes the improved PSO method, which was usedto solve the PBUC problem. Part V presents the results of

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illustrative example. Finally, Part VI provides someconclusions.

III. PROFIT-BASED UNIT COMMITMENT FORMULATION

The objective of PBUC is to maximize the generationcompany profit (revenue minus cost) subject to all kinds ofconstraints.

A. Objectivefunction

max Pr ofit = Rv - Tc or min Tc - Rv (1)

In this paper, reserve is paid when only reserve is actuallyused'5l. Therefore, revenue and costs in (1) can be calculatedfrom the following equations:

N T

RV Z[pt *pit+ prob. ptR,]ut (2)i=1 t=

N T

Tc= t [(I - prob).f (Pt) + prob.f (pt + Rt)] ut + ST(I -ut-})uti=1 t=1

B. Subject to thefollowing constraints1 ) Energy constraint

N

SP/UX< PD t= 1,2,..... . ...T(3)i=1

2) Reserve constraint

N

Z,Rtut <SRti=l

t = 1,2,...T (4)

3) Unit generation and reserve limits constraint

imin I - imax

O<Rt<Pimax 'Pmii- mmixPI< Rit < PImax-in

Pt+ <(5)

i = 1,2,...N

4) Unit minimum ON/OFF duration constraint

(X1jon( i, t- |

I(X offl- oin ) (U t-I _ U t ) >O-Ti°)(uj -t.i (6)

-7j0of)(Ut-ut-u).>0

IV. PROPOSED SOLUTION METHOD

A. Overview ofthe PSOParticle swarm optimization (PSO), first introduced byKennedy and Eberhart, is one of the modem heuristicoptimization algorithmsE81. PSO provides a population basedsearch procedure in which individuals called particles changetheir positions with time. The PSO can generate high qualitysolutions within shorter calculation time and stable

convergence characteristic than other stochastic optimizationmethods.

The PSO model consists of a swarm of particles moving inan d-dimensional real-valued space of possible problemsolutions. Every particle has a position

Xi = (Xil1 Xi2 ..., Xid) and a flight velocity

Vi = (v-1, vi2 ,..., Vid) . Moreover, each particle contains its

own best position seen so far Pi = (Pil Pi2 , Pid) and a

global best position Pg = (Pgl Pg2'' Pgd) obtained

through communication with its fellow neighbor particles.At each time step t, the velocity is updated and the particle

is moved to a new position. This new position is simplycalculated as the sum of the previous position and the newvelocity:

t+ = t t+ i

The update of the velocity from the previous velocity to thenew velocity is determined by:

Vik = C) Vik + c1randl(0, 1)(Pik -Xik ) +

c2rand2(0,1)(pgk - xlt) i = 1,2,...n; k = 1,2,...d(8)

whereco is inertia weight factorc1I c2 are acceleration constantrandl(0,1) and rand2(0,1) are uniform random valuebetween[0, 1 ]

Vtk is the velocity of particle i at iteration t

Xlk is the position of particle i at iteration t

B. An ImprovedPSO to PBUCThe original PSO is basically developed for continuousoptimization problems. However, PBUC is formulated asmixed integer nonlinear programming problems. Kennedy andEberhart proposed a discrete binary PSO(BPSO) to solveinteger optimization problems191. However, with a simplemodification, the PSO can be made to operate on binaryproblems, such as UC. In BPSO, Xi in (7) can take on valuesof 0 or 1 only. The velocity Vi will determine a probabilitythreshold. If Vi is higher, the individual is more likely tochoose 1, while lower values favour the 0 choice. Such athreshold requires staying in the range [0,1]. One of thefunctions accomplishing this feature is the sigmoid function,derived as follows:

s(vi (t)) = 1/(1 + e-Vi(t)) (9)

A random number (uniform distribution between 0 and 1) isfirst generated, whereby Xi is set to 1 if the random number isless than the value from the sigmoid function as illustratedbelow.

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I if rand(O,l) < s(V, (t + 1))u1 ( +1) =

0 else(10)

In PBUC problem, u,(t+1) represents the on or off state ofthe generator i at time t+1.

The PBUC is decomposed into two embedded optimizationsub-problems: one the unit on/off status schedule problem withinteger variables that can be solved by the improved discretebinary PSO and the other economic dispatch problem withcontinuous generated power that can be solved by the standardreal value PSO. In calculating process, the real-valued PSOand BPSO are run in parallel, with each updated according to(8) and (10) separately.

In order to obtain high precision solution for PBUC, twooperators which are repair strategy and swap mutation operatorwere adopted to improved BPSO performance:1 ) Repair strategy operatorThis operator repairs solutions that are infeasible regarding theminimum up/down constraints of unit. The process of fixing a

solution is done by evaluating the state of each unit Xon andi,t

X<of . As it was previously defined Xon andX off denotes the

consecutive time that unit i has been up and down at time tseparately. The state of a unit is evaluated starting from hour 0.If at a given hour t the minimum up or down time constraint isviolated, the state (on/off) of the unit at that hour is reversedand Xon orX Off is updated. The process continues until the

last hour has been reached. This operator can be effectivelydealt with the minimum up/down time constraints of unit2) Swap mutation operatorThe swap operator uses the full-load average costs ( a ) of thegenerating units to perform a swap of unit states. The a of aunit is defined as the cost per unit of power when the generatoris at its full capacity. The generating units are ranked by theira in ascending order. This means that for any two generatingunits i and j where i<j the inequality a i< aj holds. Units withlower a should have higher priority to be dispatched. At agiven hour, the operator probabilistically swaps the states oftwo units i and j only if the unit i is ranked better than unit jand the state of the units are off and on respectively. Thisoperator enhanced the algorithm's global optimal performance.

In PSO, the evaluation function value is mainly used toprovide a measure of how the particle performed in theproblem domain. The best particle in swarm should have thehighest profit and also satisfy constraints of the PBUC.Therefore, in the improved PSO, Energy and reserveconstraints can be dealt with through penalty function, so wedefine the evaluation function F as following equation (11).After some treatment for these constraints, the followingobjective functions can be obtained:

T T

minF = -Profit-+ a(71 t)2 + >((Pt2t 1t=l t=l

N

t =min{O,(P - Piui)}i=l

N

q' = min{O, (SR' - Rtut)}i=l

(12)

(13)

where cG is the constraints penalty coefficient.It is of crucial importance to select the penalty coefficient in

the search process of the PSO. Overlarge penalty coefficientwill possibly lead to converge at the local optimum solution.On the other hand, the convergence of algorithm will be poor.Therefore, this paper adopts self-adaptive adjusted strategy tochoose penalty factor cG, which is computed at the kthgeneration defined as cG = c±o+ log(k+l).where cyo is a givenconstant. The pressure on the infeasible solution can beincreased with the number of generations based on the Kuhn-Tucker optimality theorem and penalty function theoremproviding guidelines to choose the penalty term. Thus CTincreases gradually which ensures the satisfaction ofconstraints with the process of the PSO.

V. NUMERICAL EXPERIMENTS

The PBUC problem solution method is implemented in VisualC++6.0. We use a generation company with 10 generatingunits to illustrate the proposed method. In our implementation,energy and reserve are considered simultaneously in theformulation. 24 h scheduling period is considered. Fuel costfunction of each generating unit is estimated into quadraticform as shows in nomenclature. Unit data, forecasted demand,reserve and market prices were given in Tables 1,2 and figure1 as following, which obtained from Reference [7].

TABLE 1 Generators Data

_ _Jitl Uk2iU2itU 3 Unir4 UnitPiSax 455 455 130 130 162Pmin 150 10S 20 20

0,0004: 0.o001 M00210 21 1 0.00398b 1619 1726 16.60 1630 W.70c 00 ff ff;1000 970 700 680 450

m 8 5 56406f~~~~8 _ _ .5 . 6ST 400 500 550 560 900

ig,8 8 d -S4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~hiiUt 6 Unit7 IJf iUii 9 Unito

P. I + I S 80I_55 55 55

Iii 20 I 2 I , 0 .1 ola 0.00712 0.00079 O.00413 0.00222 0.00173b 22.26 27.74 25.92 27.27 27.79__ 370 480 660 665 670mST 170 2603 0 i 30 30

,,I ,3 E ~~~~~~~~~~~~~3 ... .... ......ST I 170 1 0 30 I 30 1 30biL | | l X

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TABLE 2 Forecasted demand and reserve

FORELATED FO ( flHOUX. A

AStRV

I 700 70________2 750 753 50 854 O950 95

6 l1t1 1107 1100 1159 3X1 INo9 1 300 13010 1400A 1411 1450 14512 15(s) ISO13 1400 140______14 I3O 13015 120 12016 1050 ___

17 1001 '104)189 110010

_____ _1200 ___120

20 1400 14021 130) 1;3022 iIO; 11023 900 9024 800 80

2 4 a .i 4 1 I 0 2 2"

Fig. 1. Forecasted spot price for 24 periodIn order to overcome the stochastic nature of the improved

PSO for obtaining better result, each experiment was run 10times, starting with an different random initial populationparticles. The best result is selected as the final result, underthe control parameter settings of the approach are populationsize 30 and maximize iterative generation 2000.

The profit using the improved PSO method is $113018.7and the corresponding profit using the hybrid approachbetween Lagrangian Relaxation (LR) and EvolutionaryProgramming (EP) in Reference [7] is $112818.9. The profitusing the improved PSO approach is 0.2% higher than that ofhybrid method between LR and EP.

VI. CONCLUSIONSIn this paper, we have established an model of the unitcommitment problem based on profit under the deregulationelectricity market environment, which generation company isresponsible for unit commitment in a restructured marketstructure. We have shown that when a competitive market andprice forecasting are assumed the unit commitment problemcan be solved by improved particle swarm optimization. Theproposed method helps generation company to make adecision, how much power and reserve should be sold inmarkets and how to schedule generators in order to receive themaximum profit. The numerical results on the generationcompany with 10 units demonstrate the quick-speedconvergence and higher accuracy of the proposed approach, soit provides a new effective method for solution of the PBUCproblem in electricity market.

VII. REFERENCES[1] Sheble G.B., Fahd G.N. "Unit commitment literature synopsis," IEEE

Trans. on power systems, vol.9, pp.128-135, Jan., 1994.[2] K. S. Swarup, S. Yamashiro. "Unit commitment solution methodology

using genetic algorithm,"IEEE Trans. on power systems, vol.17, pp.87-91, Jan.2002.

[3] Chuan-Ping Cheng, Chih-Wen Liu, Chun-Chang Liu."Unit commitmentby Lagrangian relaxation and genetic algorithms,"IEEE Trans. on powersystems, vol.15, pp.707-714, April 2000.

[4] Chung-Ching Su, Yuan-Yih Hsu. "Fuzzy dynamic programming: anapplication to unit commitment,"IEEE Trans. on power systems, vol.6,pp. 123 141237,July 1991.

[5] Eric H. Allien, Marija D. Ilic. "Reserve markets for power systemsreliability," IEEE Trans. on power systems, vol.15, pp.228-233, Jan.2000.

[6] H. Y. Yamin,S.M. Shahidehpour. "Unit commitment using a hybridmodel between Lagrangian relaxation and genetic algorithm incompetitive electricity markets,"Electric Power System Research,vol.68, pp.83-92, 2004.

[7] Pathom A., Hiroyuki, K., Eiichi, T. "A hybrid LR-EP for solving newprofit-based UC problem under competitive environment," IEEE Trans.on power systems, vol. 18, pp. 229-237,Jan. 2003.

[8] Kennedy J. and Eberhart R.,"Particle swarm optimization,"inProceedings of IEEE Conference on Neural Networks, 1995,pp.1942-1948

[9] Kennedy J.,Eberhart R., "A discrete binary version of the particle swarmoptimization algorithm," in Proc. of the 1997 conference on Systems,Man, and Cybernetics, 1997, pp.4104-4109

VIII. BIOGRAPHIES

Yuan Xiaohui was born in Hubei province in People's Republic of China, onDec. 18, 1971. He received PH.D degree from Huazhong University ofScience & Technology in China in 2002.He works as a post-fellow researcherof the Department of Hydropower Engineering of Huazhong University ofScience & Technology, Wuhan, China. His researchs are in the field ofartificial intelligent with application to power system operation and electricitymarket.

Yuan Yanbin was born in Hubei, China, on Jan. 1,1970. He received Ph. Ddegrees in earth probing and information technology from China Universityof Geosciences in 2000 respectively. He is currently a Professor with theDepartment of Resource & Environment Engineering in Wuhan University ofTechnology. His current research is mainly the application of geographicinformation system (GIS) techniques to digital valley engineering.

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