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Fast solution technique for large-scale un itcommitment problem using genetic algorithmT. Senjyu, H.Yamashiro, K. Shimabukuro, K. Uezato and T. Funabashi

Abstract An approach for a large-scale unit commitment problem is presented. The unitcommitment (UC) problem plays a major role in power systems, because the improvement ofcommitment schedules results in the reduction of operating costs. However, the unit commitmentproblem is one of the most difficult optimisation problems in power systems. because this problembas many constraints. Moreover, search space is vast. To overcome these problems, a new geneticoperator based on unit characteristic classification and unit integation technique are proposed.The proposed algorithm was tested on a reported UC problem. From simulation results, bettersolutions are obtained in comparison with previously reported results. Numerical results forsystems up to 100 units are compared to previously reported results.

index of unit (i= I , 2, ._., )index of time(t= 1,2, ..,, 7)ith unit status at hour f ( I if the unit is ONand 0 if OFF at hour t)output power of ith unit at hour fmaximum output power of ith unitminimum output power of ith unitdemand power at hour Isystem reserve at hour fminimum up time of ith unitminimum down time of ith unitduration during which ith unit is continu-ously ONduration during which ith unit is continu-ouslv OFFstarf-up cost of ith unitfuel cost of ith unittotal costhot start cost o f ith unitcold start cost of ith unitcold start hour of ith unitnumber of unitscheduling period

1 IntroductionIn r a n t years, power demand has been increasing withrapid industrialisation. As electric power systems play amajor role in a modem society, professional engineers areresponsible for proper operation of the generators. Hence,0 EE. 2W 3IE E Proceedky.9 Online no. 20030939doi: 10.1049/ipyd2@930939Paper first received I I th De am ber 2002 and in revised form 5th August 2003.Onllne publishing date: 21 October 2W3T.%njyu, H. Y s m d r o , K. Shunabukuro and K. U a t o are with the Facultyof E n g " e n g , University of lh e Ry&)us, I Senbaru NishihvraihoNakagami. Okinawa 9030213, JapanT. Funabashi is with the Meidensha Corporation. Riverside Building 36-2.NihonbasU Hokozakich o, Chuo-ku, Tokyo, 103-8515.JapanIEE Proc.-Gpner Trmm.Dlrrib .. Vo l 180, No . 6,Nowmber 2003

the unit commitment (UC) problem is the most importantproblem and has been studied over decades.The unit commitment problem involves determining theeconomical operation schedule subject to a number ofoperation constraints. Moreover, this problem has integerand,continuous variables. It is difficult to determine aneconomical operation schedule for this reason. The exactoptimal solution can be obtained by a complete enumera-tion. However, this method cannot be applied to the unitcommitment problem as the search space of this problem isvast. To solve the unit commitment problem, someoptimisation techniques are applied to it. For example,there are Lagrangian relaxation [I], simulated annealing [2]and dynamic programming [3].This paper presents a genetic algorithm (GA) solution tothe large-scale unit commitment problem. Recently, agenetic algorithm has been successfully applied to unitcommitment problems [49]. However, a GA requiresexcessive computation time because it has many popula-tions. Also as genetic operators are operated at random, aninfeasible solution may be generated. To overcome thesedimculties, a new genetic operator and unit integrationtechnique are introduced.From simulation results, an economical operationschedule can be obtained in reasonable computation timein comparison with previously reported results [6, IO].2 Problem formation2.7 Objective functionThe objective of the unit commitment problem is theminimisation of the total cost.2.7.7 Total coat The objective function of the UCproblem consists of fuel cost and start-up cost.

2.7.2 Fuel cost; Fuel cost of thermal unit is expressedasa second-order function of each unit output.F C , ( t )= a, +b,P,(t)+ ci


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2.7.3 Start-up cost; In this paper, simplified start-upcost is applied as

it1

2.2 ConstraintsThe constraints that must be satisfied during the optimisa-tion process are as follows:2.2.7 System power balance: The generated powerfrom all committed units must satisfy the power demand

0 0 1 1 1 . . e . 1 0 0

2.2.2 Spinning reserve: To maintain system reliabil-ity, adequate spinning reserves are required. In this paper,the spinning reserve is 10% of the load,

unit2

NcIi(t)cm: D ( f )+ R ( t )i= I

0 1 1 1 1 . . . e 0 0 0

2.2.3 Unit output limit; Each unit has output range5 c(t) Pym. (7 )

unilN

2.2.4 Minimum up/down time: Once a unit iscommitted/decommitted, there is a minimum time beforeit can be decommited/committd,

0 0 0 1 1 . e * * 1 1 0

3 Proposed methodWhen a genetic algorithm is applied to the UC problem,solution representation of the UC problem is shown inFig. I . Figure 2 shows the system schedule. In this study, weseparated commitment schedule into each unit.

1 2 3 4 5 ........

Fig. 1 Comm itment scheduleunit Nnit 1 unit 2 ...

Fig.2 Genotype75 4

3.7 Unit integration of large-scale problemThe simulation includes runs for 40, 60, 80 and 100 unitssystems. These large-scale problems have many combina-tions, as the results, search space is vast. Hence to achieve aglobal search, the number of candidates of solutions anditerative calculation must be increased. However, thesemethods cannot be applied to the large-scale problem, as itis time-consuming. To solve a large-scale unit commitmentproblem in reasonable computation time, we proposed aunit integration method. The unit parameters for thisproblem are shown in Table 1. For the 20 units problem,the initial IO units systems were duplicated. The problemdata were scaled appropriately for the problem with moreunits. In case of 40 units, there are four sets for every setoften units. To reduce search space, groups of units of thesame characteristics are integrated to one generator. As aresult, 40, 60, 80 and 100 units problems can be reduced tothe IO units problem. The parameters of integrated units arescaled in proportion to the number of units. For example, inthe case of 40 units, maximum output, of Unit, 1 is4SSMW x 4 = 1820MW, minimum output is 150MWx 4 = 600MW. There are no changes in the fuel cost,function coefficient, minimum up/down times and initialstate.3.2 Generating initial populationsIt is difficult to generate a feasible solution when initialpopulations are generated at random. The generation ofinitial populations is camed out by focusing on the daypeak and bottom load. Generally, more generators arestarted up at around the peak load, whereas a fewgenerators are started up at the light load. Hence as shownin Fig. 3, generators are mainly set continuously on statusI around at the peak load to obtain feasible solutionefficiently.Moreover, to exclude the difficulties in terms ofthe minimum up time, the minimum up time for eachunit is also satisfied when unit status is set continuouslyon 1.3.3 copyThe best solution is copied to another one. As a result, thebest solution is saved for the next generation. and can beevolved.3.4 MutationAs standard mutation selects the mutation point randomly,it is difficult to satisfy the minimum up/down timesconstraint. To handle the m i n i up/down timesconstraint, a new mutation operator is adopted. Thisoperator looks for 01 or IO combination in commitmentschedule. Then the mutation operator randomly changesthe combination to 00 or 11.3.5 CrossoverWhen standard crossover is adopted, m i n i up/downtime constraints can be violated. To overcome thisdifficulty,we used a new crossover as shown in Fig. 4. Thiscrossover replacesa commitment schedule with other one inthe populations.3.6 Shift operatorAs shown in Fig. 5 , up time is shifted one hour up or down.3.7 Intelligent mutationFigure 6 shows the operation unit price of Unit 1, 5 and IO .From Fig. 6, we can see that the operation unit price ofUnit I O is the most expensive one, therefore, in case ofdetermination of commitment schedule, commitment of

IE E Proe- Cnur T r a m Dinrib., Vol. 150. No. 6, Noember 2003

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Table 1: Unit parameterUnit 1 Unit 2 Unit 3 Unit4 Unit 5

P y x 455 455 130 130 162V" 150 150 20 20 25ai 1000 970 700 E80 450b, 16.19 17.26 16.60 16.50 19.70C, 0.00048 0.00031 0.002 0.00211 0.00398Tp" 8 8 5 5 6Tp" 8 8 5 5 6hcosti 4500 5000 55 0 560 900c-cost, 9000 10000 1100 1120 1800c-s-hour, 5 5 4 4 4ini state 8 8 -5 -5 -6

0 0 1 1 1 1 1 1 0 0 0

p;""Pya,b,C,7 7 - x

T"hcost,c-cost,c-s-hour,ini state

0 1 1 1 1 1 1 0 0 0 0

Unit 68020

37022.26

0.0071233170

3402

-3

0 0 0 1 1 1 1 1 1 0 0

Unit 78525

48027.740.0007933260

5202

-3

Unit 85510

66025.92

0.004131130

600

-1

Unit 95510

66527.27

0.002221130

600

-1

Unit 105510

67027.790.001731130

600

-1

L I1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 l

I0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 010 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0

Fig.3 Initial populationUnit 1 through to Unit 7 should he given priority overcommitment of Unit 8 through to Unit 10. This operatorlooks for 01 combination in commitment scheduleof Unit 1through to Unit 7. As shown in Fig. 7, if overcommitted, itis decommitted, otherwise it is committed. To obtain lowercost, intelligent mutation is introduced into two parts: thefist. is used as one of the GA operators and the second isonly applied to the best solution solved by the GA at theend of program. An explanation for the second part isdescribed in Section 3.11.3.8 Determination of small unitsFrom Table 1, minimum upidown time of small units(Unit8, 9, 10) is one hour, therefore small units can becommitted/decommitted at. short intervals. As operationunit price of these units are costly, on-time must heIEE Proc-Gener T r m . Dlrtr6.. Yo/.150, No. 6, Noaember 2W3

unit 1 Unit 2 unit N

unit 1 unit 2 unit N. . . .. . . . .. . . . . . . .. . . . . . . . .........U. .. .. . . . . . .. . . ... . . . . .



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outpui, MWFig.6 Operation unit price (Unit I , 5, IO)

oa

x -U .-* -

. . . . .. . . . .. . . . .. . . . .0:fi.-I-

. . . . .. , . . .. . . . .. . . . .

" n d 3 01111111..."4 0 0 1 1 . .QP=sh;tdown

Fig. 7 Intelligent mutationminimised. In this study, to minimise on-time of small units,small units mainly supply the power at peak load.Commitment order of these units were determined basedon cost characteristics. Figure 8shows cost characteristics ofsmall units where we can see that the fuel cost of Unit 8 sthe lowest of all, therefore Unit 8 is committed fist , therebyfollowing Unit 9, and Unit I O having the most expensivefuel cost is committed the last.3.9 ELD calculationThe economic load dispatch is a computational intensivepart in a unit commitment problem. Hence, to savecomputation time, the economic load dispatch is onlyperformed if the given unit commitment schedule is able tosatisfy spinning reserve and minimum up/down timeconstraints. As shown in Fig. 9, we can see that calculationrange of ELD varies directly with generation.3.10 EvaluationFor all violated constraints, a penalty term that the amountof violation of constraints is multiplied by a constant isadded to the total cost.

F = K + a PENA (9)where PENA is the amount of violation of constraints, a isthe penalty multiplier.15 6

unit 6- _ - .- . unit 92.0Dz< 1.8 -69i2 1.6 -

1.4 -mI

output, MWFig. 8 Fuel cost (Unit 8, , IO)

160 -

6 60 -.--ii- 40-B

20 -0 100 200 300 400 500

generationFig. 9 Calculation range oJELD3.11 Improvement of scheduleTable 2 shows the commitment schedule solved by the GA,which is returned to the original size problem, i.e. in thiscase 40 units system from IO units system. There is a lot ofroom for improvement in this schedule because integratedunits have same commitment schedule. To save theovercommitted unit, the intelligent mutation, which ismentioned in Section 3.7, is applied at the end of theprogram. To determine an economical schedule, the unitwith a high operation unit price should be stoppedpreferentially. Table 3 shows the operation unit price ofeach unit, from which it is observed that the decommitmentorder is Unit 1 0 ~ 9 ~ 8 ~ 7 ~ 6 ~ 5 ~ 3 ~4 Simulation dataThe sinlation includes runs for 40,60, O and 100 unitssystems, and scheduling periods are 24 h. For the 20 unitsproblem, the initial 10 units system were duplicated. Theproblem data were scaled appropriately for the problemwith more units. In Table 1, ini state indicates how long theunit has been committed/decommitted. If positive, itindicates the number of hours the unit has been committed,if negative, it indicates the number of hours the unit hasbeen decommitted. Adaptation of genetic operator prob-abilities are shown in Table 4.5 Simulation resultsThe 20 test trials were made for each problem set due to thestochastic nature of the GA, with each run starting with

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Table 3 Operation unit price of each unitUnit Operation Unit Operationunit price, unit price,5IMW $IMWUnit 1 18,60620 Unit 6 27.45460Unit 2 19.53292 Unit 7 33.45421Unit 3 22.24462 Unit 8 38.14715Uni t 4 22.00507 Uni t 9 39.48301Unit 5 23.12254 Uni t 10 40.06697

Table4 GA parametersSelection Elite selectionPopulation size 20COPY 0.3Crossover 0.9Mutation 0.04Shift operator 0.05Intelligent mutation 1.0

different initial populations. The population size was 20genotypes in all runs.The simulation results are shown in Tables2, 5-7. Asshown in Table5, the proposed method gives bettersolution in comparison to other methods [6, IO]. In all testcases, total cost can decrease ahout 0.2-0.13% in compar-ison to reported methods [6, IO], From Table 6, as we adoptcalculation range of ELD and unit integration method, theexecution time is much faster than others [6, IO]. In all testcases, execution time can be reduced by up to 99-99.5%, incomparison to that reported in other techniques [6, IO].Table 7 gives an example of the final commitment schedulefor the 40 units system (total cost: $2247336). According toTable 7, this schedule indicates that all constraints are

satisfied. In Table 7, overcommitments are decommitted byintelligent mutation, as mentioned in Section 3.11. Forexample, at 6 to 7 and 21 hours, unit 9 is shut down asshown in .Table 7. Figure I O shows the execution time.Introduction of intelligent mutation can achieve costreduction. From Fig. IO, the execution time of the reportedmethod increases in a quadratic manner with the increase ofproblem scale. However, as units are integrated at the IOunits problem in all problem cases, execution time increasesin a quasilinear way with the number of units. Incomparison to the results produced by the reportedmethods, the proposed method gives satisfactory solutionin a reasonable computation time.

.... ...... GA [6]---fl---.. EP[10]

(.........roposed method .....................................

10 I I40 60 80 100numberof unitsFig. 106 ConclusionsThis paper presents a genetic algorithm solution to thethermal unit commitment problem. The unit commitmentproblem is one of the most difficult optimisation problemsin power systems as the search space is vast. To reducesearch space, unit integration techniques are proposed. Theinitial populations are often generated randomly. However,

Execution t m i e o reported merlwdy

Table 5 Total codTotal cost. 5

40units 60units 80 units 100unitsBest 2247336 3367637 4491509 5610855worst 2250507 3373038 4497047 5617827Average 2248812 3370133 4493853 5613989GA I61 2251911 3376625 4504933 5627437EP 1101 2249093 3371611 4498479 5623885

Table 6 Computation timeComputation time, s

40units 60units 80units 100 unitsProposed 29 42 60 75MethodCM I61 2697 5840 10036 15733EP 1101 1176 2261 3584 6120758 IEE Proc.-Genn T m m . Dim&.. Vol 150. No . 6 Nowmbn 2003



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it is difficult to generate feasible solutions. To obtainfeasible initial solutions, initial populations are generatedbased on load data.Hence feasible initial solutions can be obtained. Con-straints, output range and operation costs vary with eachunit. Units are classified into several groups based onminimum up/down times constraints. The operationschedule of small units are determined by numericalcalculation based on cost characteristic. Other unitsschedules are determined by the genetic algorithm. Toobtain more optimal operation schedules, new geneticoperators are introduoed. The intelligent mutation performsa local bill-climbing optimisation technique. From simula-tion results, the proposed method can determine satisfactorycommitment schedule in reasonable computation time.7 References

I Svoboda, A.J., Tseng, C.-L., Ki, C., an d Johnson, R.B.: Short-termresource schedding wthramp constraints', IEEE T r m . Power Sysi. ,1997, 12, (I), pp . 7 7 4 3

760 IEE Proc.-Gmer. Trrmnn Dltnib., Vol. I S O . No. 6, o~wnber 003

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