genetic-based unit commitment algorithm

IEEE Transactions on Power Systems, Vol. 1 1 , No. 3, August 1996

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GENETIC-BASED UNIT COMMITMENT ALGORITHM

Tim T. Maifeld Student Member Senior Member

Gerald B. Sheble'

Department of Electrical Engineering Iowa State University

Ames, Iowa 50010

Abstract This paper presents a new unit

commitment scheduling algorithm. The proposed algorithm consist of using a genetic algorithm with domain specific mutation operators. The proposed algorithm can easily accommodate any constraint that can be true costed. Robustness of the proposed algorithm is demonstrated by comparison to a Lagrangian relaxation unit commitment algorithm on three different utilities. Results show the proposed algorithm finds good unit commitment schedules in a reasonable amount of computation time. Included in the appendix is an explanation of the true costing approach.

I. Introduction Unit Commitment (UC) is the problem of

determining the optimal set of generating units within a power system to be used during the next one to seven days. The general UC problem is to minimize operational costs (mainly fuel cost), transition costs (start-uphhut-down costs), and no-load cost (idle, banking or stand-by).

priority list methods [ 11, dynamic programming [2,3,4,5,6], Lagrangian relaxation [7,8,9], artificial intelligence methods [ 10, 1 1,121 and sequential methods [13,14].

Past UC methods include:

95 SM 548-8 PWRS A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the 1995 IEEE/PES Summer Meeting, July 23-27, 1995, Portland, OR. Manuscript submitted December 28, 1994; made available for printing June 20, 1995.

This paper presents domain specific operators that reduce the computation time of the genetic-based UC algorithm (GBUCA). The benefits of using a genetic algorithm (GA) are: a robust optimization technique, easy implementation into concurrent processing, and production of multiple UC schedules.

This paper proceeds as follows. Section I1 presents the GBUCA. Section I11 gives results of the GBUCA. Section IV demonstrates the multiple UC schedules that can be produced with the genetic-based UC algorithm. Section V gives our conclusion. Appendix A explains the true costing approach. A description of the UC problem can be found in [15]. A description of genetic algorithms can be found in[ 161.

11. Genetic-Based UC Implementation The GBUCA implementation consists of

initialization, cost calculations, elitism, reproduction, crossover, standard mutation, economic dispatch (ED) calculations, and intelligent mutation of the UC schedules. A flowchart of the algorithm is given in Figure 1.

An explanation of each part of the genetic- based UC algorithm implementation follows :

The initialization will be explained for one member of the population (one UC schedule). The number of population members used in this research was fifty. A member of the population consists of a matrix with dimension equal to the number of generators by the number of scheduling periods. This matrix represents the odof f status of the generating units. The first step of initialization consist of finding the 10 cheapest economic dispatches for each hour that meet system demand

0885-8950/96/$05.00 0 1995 IEEE

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and a 10% spinning reserve. A member of the population is then created by randomly choosing one of the 10 cheapest economic dispatches for each hour.

. ._ K v- U’

TWS

I 1 calculpte cost of uc schedule

I Elitism

Reproduction CmSSOVCI

Stpndsrd Mutation (Can ED fM the hour that mutation has ommd. )

Turn Generator Off Mutation

(Can ED) I

I intelligent Mutation I (Can I ED fa born mu!ntim occured)

ln-t MutPtion I1 (Can

Figure 1. GA flowchart

The calculation of cost of the UC schedule consists of the following:

1.

2.

If a unit breaks the minimum-up time constraint, the unit is charged as if it were on stand-by for those hours. A temporary matrix is then created from the original UC schedule, except the stand-by hours are set to 1 instead of 0. If a unit breaks the minimum-down time constraint in the temporary matrix created in (l), the unit is charged as if it were on stand- by for the additional number of hours needed to satisfjr the constraint. The temporary matrix then has those hours set to 1 instead of 0.

3. Using the temporary matrix created from (1) and (Z), the start-up cost, shut-down cost, and banking cost are calculated for each unit.

4. The fbel cost for each UC schedule is calculated by summing the ED cost for each hour.

Elitism ensures that the best individuals are never lost in moving from one generation to the next. The elitism subroutine combines the two populations and determines the best results from both populations in order of decreasing fitness value. It then saves distinct members that have the highest fitness into the next generation. The amount of distinction between members was determined by the difference in the generators ordoff matrices.

Reproduction is the mechanism by which the most highly fit members in a population are selected to pass on information to the next population of members. The fitness of each member was determined by taking the inverse of the cost of each members UC schedule and then ordering the population members by increasing cost. Then each member was assigned a fitness according to its rank. The members that were kept for reproduction were determined by roulette wheel selection. This type of reproduction is called rank- based fitness and can be found in [ 171.

Crossover is the primary genetic operator which promotes the exploration of new regions in the search space. Crossover is a structured, yet randomized mechanism of exchanging information between strings. Crossover begins by selecting at random two members previously placed in the mating pool during reproduction. A crossover point is then selected at random and information from one parent, up to the crossover point, is exchanged with the other member. Thus creating two new members for the next generation. An example of the crossover operator is depicted in Figure 2.

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For every generation thereafter, the probability of mutation is exponentially increased [ 181. An adaptive mutation operator is needed because in early generations the members of the population are very distinct and do not need mutation. In later generations when the GA is locating good solutions, a method is needed to keep finding better solutions in these areas. This method is mutation.

Note how the crossover operator eliminates the need to recalculate economic dispatch (ED) for each new UC schedule member.

Before crossover Generator/Hour 1 2 Generator 1 1 1 Generator 2 1 1 Generator 3 0 0 Generator 4 0 0

UC member 1

Generat or/Hour 1 2 Generator 1 0 0 Generator 2 0 0 Generat or 3 1 1 Generator 4 1 1

UC member 2

Assume Crossover-Point = 2

M e r crossover GeneratorHour 1 2 Generator 1 1 1 Generator 2 1 1 Generator 3 0 0 Generator 4 0 0

New UC member 1

Generat or/Hour 1 2 Generator 1 0 0 Generator 2 0 0 Generator 3 1 1 Generator 4 1 1

New UC member 2

3 1 1 0 0

3 0 0 1 1

3 0 0 1 1

3 1 1 0 0

4 1 1 0 0

4 0 0 1 1

4 0 0 1 1

4 1 1 0 0

Figure 2. Crossover operator

Standard mutation is generally thought of as a secondary operator. This operator ensures that no string position will ever be fixed at a certain value through the course of the search. Standard mutation operates by toggling any given binary weight matrix position using the probability of mutation. The probability of mutation used in this research was a hnction of the generation number. Initially a probability for mutation was entered.

ED is the subroutine that calculates the real power output of each generating unit to meet a given load and to minimize the total operating costs.

Turn-off generator mutation - this mutation operator turns off a generator for the scheduling period. The generator to be turned off was randomly determined. This operator allowed the GA to figure out if a generating unit should be off for a scheduling period.

Intelligent mutation I is a operator that realiies that a generator cannot be turned on and off every other hour. So this operator looks for 0 1 or 1 0 combinations in the UC schedule. The mutation operator then randomly changed the combination to 0 0 or 1 1. Intelligent mutation I was applied to half .of the newly created population members. The generator to have this technique applied to it was determined randomly.

Intelligent mutation I1 is a operator that realizes that a generator cannot be turned on and off every other hour. So this operator looks for 0 1 or 1 0 combinations in the UC schedule. The mutation operator then figures out which schedule would be cheaper (00 or 11 or stay the same). Intelligent mutation I1 was applied to half of the newly created population members. The generator to have this technique applied to it was determined randomly. 111. Experimental Results

The proposed GBUCA is tested on three different utilities. Each utility has 9 thermal generators. Inputloutput curves of the generators are modeled as second order polynomials. Schedules for 24 and 48 hours are compared to a Lagrangian relaxation UC program. All

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generating units are initially off at the beginning of the scheduling period. The Lagrangian relaxation unit commitment algorithm was run on a Sun Sparc Station LX while GBUCA was run on a DEC system. To be able to compare computer system execution time an algorithm was created and run on both systems. The computer execution time given in the tables of the GBUCA has been normalized to the equivalent computer execution time it would take on a Sun Sparc Station LX. Spinning reserve was not taken into consideration for the test runs, but would not require any additional computation time for the GBUCA. Results for Big Edison Electric Company, Municipal Electric Company, and Rural Electric Exchange are given in Figures 3, 4 and 5. The number given in parenthesis by GBUCA is the initial random number seed of the program. L.R. Algorithm

24 HOUR UC SCHEDULE

TIME (SEC) $643.1 12.00 77

1 P R Q G ~

GBUCA(.lS) GBUCA(.35)

I TIME'""'

$624,434.50 77 $6 1 7,453 .0§ 81

I L.R. Algorithm I $674,103.00 54 , 83.09 53

GBUCA (. 75) GBUCA(.95)

$61 9,962.46 78 $624,750.03 82

1 GBUCA(.95) I $672,663.72 I 53

GBUCA (. 5 5 ) $673,19 1.9'7 52 GBUCA (.7§) $673,3 59.67 51 -

48 HOUR UC SCHEDULE J

GBUCA (. 1 5 ) GBUCA (. 3 5 ) GBUCA(.55)

COMPUTER EXECUTION

90 94

$246,5 88.66 48 $250,806.04 56 $246,452.50 57

~~

Figure 3. Big Edison Electric Company results

GBUCA(.75) GBUCA(.95)

24 HOUR UC SCHEDULE

$246,854.09 52 $245,971.22 51

COlMPUTER PROGRAM

GBUCA1.75) GBUCA(.95)

GBUCA (. 15) $3 18,434.45 43 GBUCA (. 3 5 ) $3 18,640.83 49

$487,43 1.66 90 $488.676.30 8'7

GBUCA(.55) $3 18,168.5 1 48 GBUCA (.75) $3 18.467.00 43 GBUCA (.95) I $3 18,542.22 I 46 I

48 HOUR UC SCHEDULE

I COST($) EXECUTION PROGRAM I CoMPUTER I

I GBUCA(.55) I $616,444.10 1 79 I

Figure 4. Municipal Electric Company

24 HOUR UC SCHEDULE COMPUTER

TIME (SEC) PROGRAM 1 COST ($) I EXECUTION I

1 L.R. Algorithm I $247,489.00 I 50 1

48 HOUR UC SCHEDULE

1

Figure 5. Rural Electric Exchange

The results show the GBUCA is a robust method that can produce good UC schedules in a reasonable amount of computation time. Computer execution time of the GBUCA will vary from run to run (different random number seed) because it is a probabilistic technique.

IV. Multiple UC Schedules The GBUCA has the ability to produce

multiple UC schedules in one run. Multiple UC schedules would allow an electric utility to compare different schedules that have different amounts of spinning reserve for each hour and choose the UC schedule that best meets the utilities needs. Figure 6 is the Lagrangian relaxation UC schedule, while Figure 7 and Figure 8 are two of the GBUCA schedules. Figure 9 is a comparison of the total hourly generation available for GBUCA schedule 1 and 2.

hr/ 1234567891 11 11 11 11 122222 generator 0 1234567890 1234 gl: 111111111111111111111111 g2: 111111111111111111111111 g3: 111111111111111111111111 84: 011 11 111 11 111 11 11 11 1 lbbb g5: 0001 11 11 11 11 11 11 11 1 lbbbb g6: 000011111111111111111111 g7: 000000000000000000000000 g8: 000000000000000000000000 g9: 000000000000000000000000 Total cost of schedule = $674,103.00 Figure 6. LaGrangian relaxaton UC Schedule

hr/ 1234567891 11 11 11 11 122222 generator 012345678901234 gl : 111111111111111111111111 g2: 111111111111111111111111 g3: 111111111111111111111111 g4: 011111111111111111111111 g5: 000000000000000000000000

g7: 000111111111111111111111

g9: 000000000000000000000000

g6: 000000000000000000000000

g8: 000001 11 11 11 11 11 11 11 lbbb

Total cost of schedule = $672556.59 Figure 7. GBUCA schedule 1

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hr/ 123456789111111111122222 generator 012345678901234 g l : 1111111 11 11 111111 111 llbb g2: 111111111111111111111111 g3: 111111111111111111111111 g4: 011111111111111111111111 g5: 0001 11 11 11 11 11 11 11 1 lbbbb g6: 000000111111111111111111 g7: 000000000000000000000000 g8: 000000000000000000000000 g9: 000000000000000000000000 Total cost of schedule = $673680.74 Figure 8. GBUCA schedule 2

0 = generator is off 1 = generator is on b = generator is banking

GBUCA schedule 1 $672556.59 total generation

hour available 1 1170 2 1170 3 1520 4 1520 5 1870 6 1870 7 1870 8 1870 9 1870 10 1870 11 1870 12 1870 13 1870 14 1870 15 1870 16 1870 17 1870 18 1870 19 1870 20 1870 21 1870 22 1870 23 1870 24 1870

GBUCA schedule 2 $673680.74 total generation available 1170 1170 1410 1410 1410 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985 1985

Figure 9. Comparison of total hourly generation available for GBUCA schedule 1 and GBUCA schedule 2

cost difference = $1 124.15 (2-1) difference in MWs

0 0

-110 -110 -460 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115

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Note GBUCA schedule 1 dispatches generators 1,2,3,4,7,8 while GBUCA schedule 2 dispatches generators 1,2,3,4,5,6. GBUCA schedule 2 is $1124.35 more than GBUCA schedule 1, but GBUCA schedule 2 has more reserve on line for certain hours. If an electric utiIity wanted to sell electricity on the spot market, knowing the opportunity cost to have extra spinning reserves could reap an electric utility large financial rewards.

IV. CoNcLUsloN The GBUCA has three main attributes that

make it very attractive for unit commitment scheduling. The first attribute is that it can consistently find good unit commitment schedules in a reasonable amount of computer execution time. The second attribute is that it can produce many different unit commitment schedules in one run. Multiple unit commitment schedules provides data for an analysis tool that can be used for spinning reserve comparison, security analysis, and transaction evaluation and selection. The third attribute is that the GBUCA is an information algorithm. The more information given to the GBUCA (spinning reserve, crew constraints, emission constraints, etc.) by the true costing approach, the better the GBUCA will be able to understand the problem and be able to come up with a good unit commitment schedule.

It is impossible to predict the future, but with the GBUCA ability to find good unit commitment schedules, reasonable computer execution time, easy implementation into concurrent processing, multiple unit commitment schedules, and ability to handle increased complexity using the true costing approach, it is hard to imagine future unit commitment scheduling programs not using it.

References [l] Shoults, R.R., Chang, S,K., Helmick, S., and Grady, W.M., "A Practical Approach to Unit Commitment, Economic Dispatch, and Savings Allocation for Multiple- Area Pool Operation with IinporExport Constraints," IEEE Transactions on PAS-99, No. 2, pp. 625-633, MarchJAprill980.

[2] Snyder, W.L., Powell, H.D., Jr., and Rayburn, J.C., "Dynamic Programming Approach to Unit

Commitment," IEEE Transactions on PWRS-2, N0.2, pp. 339-350, May 1987.

131 Pang, C. K., Sheble', G.B., and Albuyeh, F., "Evaluation of Dynarmc Programming Based Methods and Multiple Area Representation for The& Unit Commitment," E E E Transactions on Power Apparatus and Systems. Vol. PAS-100, No. 3, pp. 1212-1219, March 1981.

[4] Hobbs, W. J., Hermon, G., Warner, S., and Sheble', G.B., "An Enhanced Dynamic Programming Approach for Unit Commitment," IEEE Transactions on Power Systems, Vol. 3, No. 3, pp. 1201-1205, August 1988.

[ 5 ] Tong, S.K. and Shahidehpour, S.M., "Hydrothermal Unit Commitment with Probabilistic Constraints Using Segmentation Method," IEEE Transactions on PWRS-5, No. 1, pp. 276-282, February 1990.

[6] Pang, C.K. and Chen, H.C., "Optimal Short-Term Thermal Unit Commitment," IEEE Transactions on PAS- 95, No. 4, pp. 1336-1346, Julymecember 1976.

[7] Virmani, S., Imhof, K., and Mukhenjee, S., "Implementation of a LaGrangian Relaxation Based Unit Commitment Problem," IEEE Transactions on PWRS-4, No. 4, pp.1373-1379, November 1989.

181 Zhuang, F. and Galiana, F.D., "Towards a More Rigorous and Practical Unit Commitment by LaGrangan Relaxation," Transactions on Power Systems, Vol 3, No. 2, pp. 763-773, May, 1988.

[9] Tong, S.K. and Shahidehpour, S.M., "Combination of LaGrangian-Relaxation and Linear Programming Approaches for Fuel Constrained Unit Commitment Problems," IEE Proceedings, Vol. 136, Part C, NO. 3, pp. 162-174, May 1989.

[lo] Maifeld, T.T., Coppinger, S., and Sheble', G.B., "Unit Commitment by Genetic Algorithm and Expert System", accepted and to be published in Electric Power Systems Research Journal.

1111 Mokhtari, S., Singh, J., and Wollenberg, B., "A Unit Commitment Expert System," Proceedings of the PICA, pp. 400-405, May 1987.

[12] Ouyang, Z. and Shahidehpour, S.M., "A Hybrid Artificial Neural Network-Dynamic Programming Approach to Unit Commitment, 'I IEEE Transactions on PWFtS, Paper 91 SM 438-2, San Diego, 1991.

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the genetic-base UC algorithm for UC scheduling. But with the less regulated utility environment, utilities are going to be willing to figure out the cost of constraints to try to find better ways (financially) to operate.

A genetic-based UC algorithm reproduces the next generation of members (UC schedules) by selecting the cheapest members of the current population. Adding true costed constraints allows the genetic-based UC algorithm to differentiate between members (UC schedules) and pick the cheaper members to be reproduced in the next generation. Thus, the genetic-based UC algorithm performance increases with the addition of true costed constraints.

113) Lee, F.N., "Short-term Unit Commitment - New Method," IEEE Transactions on PWRS-3, No. 2, pp. 421- 428, May, 1988.

[14] Lee, F.N., "The Application of Commitment Utilization Factor (CW) to Thermal Unit Commitment," IEEE Transactions on PWRS-6, No. 2, pp. 691-698, May. 1991.

[15] Wood, A., and Wollenberg, B., Power Generation Operation and Control, John Wiley and Sons, New York, New York, 1984.

[16] Goldberg, D.E., Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1989.

[17] Brittig, K. and Sheble', G.B., "Refined Genetic Algorithm - Economic Dispatch Example," IEEE PES Winter Power Meeting 1994.

[18] Whitley, D., "The Genitor Algorithm and Selection Pressure: Why Fbnked-Etased Allocation of Reproductive Trial is Best", Proceedings of the Third International Conference on Genetic Algorithms and their Applications, pp 116-123, Arlington, VA June 4-7, 1989.

Appendix A

True Costing Approach A genetic-based algorithm can easily

handle any constraint that can be true costed. True costing is representing a constraint by the actual cost to have the solution valid with the constraint broken. An example of how to apply true costing will be given for the crew constraint. A crew can only turn on one unit at a time at a plant. The genetic-based UC algorithm would not understand the constraint if we said it is not possible to turn on 2 units at a given plant at the same time. We have to true cost the constraint. One way to do this is to have the UC schedule charged approximately what it would cost to transport or fly a crew in from a different plant and have them start the second unit. This allows the genetic-based UC algorithm to figure out the constraint and maybe even find a cheaper UC schedule that was never allowed to exist using the old constraint.

The drawback of using true costing is that a cost has to be determined for a given constraint. In the past this would have emancipated the use of

Tim T. Maifeld (S'90) received his B.A. degree in Mathematics fkom Wartburg College in 1990 and his M.S.E.E. degree from South Dakota State University in 1992. He is currently pursuing his Ph.D. degree at Iowa State University. His research interests include load forecasting, economic dispatch, unit commitment, power interchange transaction evaluation and selection, neural networks, genetic algorithms, and the less regulated utility environment.

Gerald B. Sheble' (M71, SM'85) is an Associate Professor of Electrical Engineering at Iowa State University. He received his B.S. and M.S. degrees in Electrical Engineering from Purdue University, and his Ph.D. degree in Electrical Engineering from Virginia Tech. His research interests include power system optimization and scheduling.

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hd 1234567891 11 11 11 11 122222 generator 012345678901234 gl : 111111111111111111111111 g2: 111111111111111111111111 g3: 111111111111111111111111 g4: 011111111111111111111111 g5: 000000000000000000000000 86: 000000000000000000000000 g7: 000111111111111111111111 g8: 000001 11 11 11 11 11 11 1 llbbb g9: 000000000000000000000000 Total cost of schedule = $672556.59 Figure 1. GBUCA schedule 1

hr/ 123456789111111111122222 generator 012345678901234 gl : 111111111111111 l l l l l l l b b g2: 111111111111111111111111 g3: 11111111111111111111111~ g4: 01111111111111111111llll g5: 0001 11 11 11 11 11 11 11 1 lbbbb g6: 000000111111111111111111 g7: 000000000000000000000000 g8: 000000~0000000000000000 g9: 000000000000000000000000 Total cost of schedule = $673680.74 Figure 2. GBUCA schedule 2

0 = generator is off 1 = generator is on b = generator is banking

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



cost difference = $1 124.15 (2-1) difference in MWs

0 0

-110 -110 -460 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115

Figure 3. Comparison of total hourly generation available for GBUCA schedule 1 and GBUCA schedule 2

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Discussion

L. A. Jamniczky: The paper presents an interesting new application of the genetic algorithm to a power system problem. My comments reflect primarily an interest in the degree of applicability of the new technique to real life situations, and as such, are formulated primarily as questions.

1. Have the authors tested the sensitivity of the algorithm to the values of the different values of tuning parameters used during the selection process? I refer to items such as the fraction of the population to which intelligent mutation I and I1 operators are applied. The experimental results shown indicate that there is a sensitivity of the results to the random number seed used in the program.

2. Given that there is a sensitivity to some parameters (such as the random number seed), how do the authors know that true optimali& has in fact been reached by a solution?

3. The authors note that the GBUCA can easily accommodate true-costed constraints. How would physical constraints such as must run conditions (either due to voltage support requirements or due to fuel take or pay contracts) be modeled? How are the power output limits of unit represented? Can periodic unit de-rates (due, for example, to coal mill outages or condense cooling water temperature increases) be represented?

4. Looking at possible scenarios of the new utility business world we seem to be heading into, the following thoughts come to mind: Can interruptible loads be modeled? (As negative generation?) Can bilateral agreements be modeled? These may be a class of must run type constraints. Can transmission constraints be modeled or, alternately, have the authors considered coupling the GBUCA with an OPF algorithm?

Manuscript received June 28, 1995.

A. Conejo, N. JimBnez, J.M. Arroyo, J . Medina (Universidad de MBlaga, MBlaga, Spain): We wish to commend the authors for their valuable contribution in proposing a genetic algorithm to address the complicated unit commitment problem.

We would like the authors to comment on the following issues :

1.

2.

Once a unit commitment schedule is fixed for a given generation member, which economic dispatch algorithm is used every hour to compute power output of each generating unit to meet the corresponding load and to minimize total operating costs? Is it based on linear programming, or it is based on quadratic programming, or it is just based on a priority list?

Let us consider the crossover policies described be-

3.

4.

low. Let z be the number of columns and y the number of rows of any of the considered matrices representing generation members.

Policy 1. A random number r between 0 and y is generated. Child matrix A is made up with the first r rows of parent matrix A and the last y - r rows of parent matrix B. Conversely, child matrix B is made up with the first r rows of parent matrix B and the last y - T rows of parent matrix A.

Policy 2. A random number s between 0 and x is generated. Child matrix A is made up with the first s columns of parent matrix A and the last x - s columns of parent matrix B. Con- versely, child matrix B is made up with the first s columns of parent matrix B and the last 2 - s columns of parent matrix A.

PoEicy 3. Rows are processed one at a time, all of them in an identical fashion. The procedure is therefore illustrated for row i. A random number U between 0 and 5 is generated. Row i of child matrix A is made up with the first U

elements of row i of parent matrix A and the last I - U elements of row z of parent matrix B. Conversely, row i of child matrix B is made up with the first U elements of row i of parent matrix B and the last x - U elements of row i of parent matrix A.

Pol icy 4. Columns are processed one at a time, all of them in an identical fashion. The procedure is therefore illustrated for column j. A random number v between 0 and 1~ is generated. Col- umn j of child matrix A is made up with the first w elements of column j of parent matrix A and the last 1~ - w elements of column j of parent matrix B. Conversely, column J of child matrix B is made up with the first w elements of column J of parent matrix B and the last g - 2 1 elements of column j of parent matrix A.

Policies 1 and 4 may produce load (spinning reserve or demand constraint violations) infeasibilities which can be solved using heuristic procedures similar to those proposed by the authors in their mutation procedures. On the other hand, policies 2 and 3 may produce time (minimum up time or minimum down time constraint violations) infeasibilities which can be also solved using heuristic procedures. The authors use a crossover policy of type 2. Is there any particular reason for doing so? Have they tried other policies? Do not they think that policy 3 may be computationally more efficient than policy 2?

How many generations have the authors used in their cases tudies?

The CPU time required by the genetic algorithm is similar to the CPU time required by the Lagrangian

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relaxation algorithm in the presented case-studies. If the number of units to be scheduled is much larger, i.e. 50 to 100 units, do the authors think that the CPU time required by both algorithms will be similar?

Once again, we would like to congratulate the authors for their paper.

Manuscript received August 8, 1995.

S . LIN, P . B. LUH, University of Connecticut:

The authors are to be congratulated for their

contributions on a new unit commitment

algorithm. We would appreciate the authors’

comments on the following issues.

1. According to the discussion at the 1995

Summer Meeting, the duality gaps obtained

by using the Lagrangian relaxation technique

are very small. This leads to a result that

the costs of GBUCA are even lower than the

lower bounds of the Lagrangian relaxation

method. We would like to know more

information about the Lagrangian relaxation

method used in the testing. We would also

appreciate if the authors could explain why

the costs of GBUCA are lower than the

lower bounds.

It is stated in the paper that adding the

spinning reserve constraint will not increase

the computation time. We would appreciate

if the authors could explain why this is the

2.

case.

The power systems presented in the testing

are relatively small. Could this algorithm

be extended to a larger system which

contains thermal, hydro and pumped storage

generators? How would the CPU time

increase as the problem size increases?

3.

Manuscript received August 18, 1995.

Get-and B. shebne: The authors are delighted with the number and depth of discussions of this paper. A. Coneju, et. al., pose four insightful questions, my response follows: 1. The economic dispatch used by the GA is based on lambda iteration. However, the economic dispatch is executed only during the initialization step or when the mutation operator changes a unit status for a given hour. However, any optimization technique could be used. Thus, the excessive overhead of ED for each hour for each combination is avoided. 2. Other policies were tried. However, since ED execution is not desired after initialization, to save computer resources, a crossover operator which does not re-execution of ED saves considerable computational effort (time). 3. Fifty (50) generations were used for comparison to the Lagrangian approach. 4. Depending on the solution landscape (solution space) observed during testing, the CPU time for the GA approach is expected to stay similar to the Lagrangian approach. S. Lin and P. B. Luh pose questions eliciting the following responses: 1. The Lagrangian relaxation technique is a standard, commercially available program (no source code), so the authors can not address how the duality gap is calculated. 2. Addition of the spinning reserve constraint does not increase solution time since the vast majority of the ED executions are performed as part of the initialization step and not during the subsequent generations except when the mutation operator requires ED re-execution. 3. The program is written in the standard C language and could handle larger system sizes without any redesign. There is no reason why combined systems of thermal, hydro and pumped storage could not be added. The CPU time for thermal systems would increaselinearly based on the observations during testing. The CPU time for hydro and for pumped storage systems can not be estimated since such systems were not investigated. This author is of the opinion that such estimates should be based on observations. L. Jamniczky poses questions to which the responses are: 1. The algorithm is not sensitive to the random number seed. However, the probability of mutation

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is changed (increased) to maintain the diversity of the genetic material when C ~ Q S S Q V ~ T is no longer dominant. 2. The problem With GA is that the number of generations needed to fmd the optimal solution is not OW. GA is a probabilistic tecMque, akin to rmdom direction search techniques. Only multiple ~ X ~ C U ~ ~ Q I I S of varying number of generations can yield indications that M e r improvements would not be found. 3. All hard constraints are handled by “true costing” for time dependent constraints. Time independent constraints are handled by ED in classical fashion.

constraint is violated. Specifically, the violation ~f a n e key iS to be able to lmodel the Cost if Such a

contract would be m ~ d e k d by including the penalty clause or a spot market price. This author sees IXI

problem with included the additional constraints listed. 4. %his author is including such models and d g o x i h changes as listed for the new business environment. While the coupling Q ~ G A and OPF is needed for trmmission constraints in the old environment, this auth~r does not believe such an approach is appropiate in the new envhment. Auction markets appear to be an easier approach.

Manuscript received October 23, 1995.

genetic-based unit commitment algorithm

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