Short Term Unit Commitment

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Short Term Unit Commitment Using Genetic Algorithms
Dipankar Dasgupta and Douglas R. McGregor Department of Computer Science University of Strathclyde Glasgow G1 1XH U. K. TECHNICAL REPORT NO. IKBS-16-93 August 10, 1993.
Keywords : unit commitment, genetic algorithms, load demand, spinning reserve, scheduling time horizon, multi-period optimization, AFLC, strategy path.

Unit commitment is a complex decision-making process because of multiple constraints which must not be violated while nding the optimal or a near-optimal commitment schedule. This paper discusses the application of genetic algorithms for determining short term commitment order of thermal units in power generation. The objective of the optimal commitment is to determine the on/o states of the units in the system to meet the load demand and spinning reserve requirement at each time period such that the overall cost of generation is minimum, while satisfying various operational constraints. The paper examines the feasibility of using genetic algorithms, and reports preliminary results in determining a near-optimal commitment order of thermal units in a studied power system.
A version of this report will appear in the Proceedings of 5th IEEE International Conference on Tools with Arti cial Intelligence, November 8-11, 1993, Boston, USA.

Abstract

1

1 Introduction
In power industries, fuel expenses constitute a signi cant part of the overall generation costs. In general, there exist di erent types of thermal power units based on fuel used (e.g coal, natural gas, oil), with di erent production costs, generating capacities and characteristics. Figure 1 shows the block diagram of a simple power system coupling the generating units and di erent end users. The system usually operates under continuous variation of consumer load demand. This demand for electricity exhibits such large variations between weekdays and weekends, and between peak and o -peak hours that it is not economical to keep all the generating units continuously on-line. So the demand and the reserve requirement impose global constraints in coupling all active generating units, while the di erent operating characteristics of each unit constitute local constraints. Thus determining which units should be kept on-line and which ones should not, constitutes a di cult problem for operators seeking to minimize the system operational cost.
Generators
G1 G2 G3 G 18 G19 G 20

Power Transmission Loads
L1 L2 L3

and
L 12

Distribution Grids
L 13 L 29 L 30

Figure 1: A simple block diagram showing a power system. Thus unit commitment (UC) problem belongs to the class of complex combinatorial optimization problem. Several mathematical programming techniques have been proposed 14, 4] to solve these time-dependent unit commitment problems. They typically include Complete Priority Ordering (CPO) and Heuristic methods, Dynamic Programming (DP) 10, 13], Method of Local Variations, Mixed Integer Programming, Lagrangian Relaxation 2, 15, 23], Branch and Bound method 4], Bender Decomposition 1], etc. Among all these methods, dynamic programming methods based on a priority list have been used most extensively throughout the power industry. However, di erent strategies for selecting a set of units from priority list have been adopted with dynamic programming to limit the search space and execution time. They include DP-SC (Dynamic Programming-Sequential Combination) 20], DP-TC (Dynamic Programming-Truncated Combination) 19], DP-STC 20] which is a combination of DP-SC and DP-TC approaches, DP-VW (Variable Window-Truncated Dynamic Programming) 17], and a neural-based method DP-ANN 18]. Recently, some researchers have suggested Arti cial Intelligence based techniques to supplement the limitation of mathematical programming methods. These include simulated annealing 22], expert systems 16], heuristic rule-based systems 21] and neural networks 18]; these hybrid approaches have demonstrated some improvement in solving unit commitment problems. However, heuristic and expert system based 2

mathematical approaches require a lot of operator interaction which is troublesome and time-consuming for even a medium-size utility 9]. The paper presents an adaptive search method, called Genetic Algorithm (GA), for optimal or near optimal commitment order of thermal units in power generation. Genetic algorithms are di erent from the above-mentioned classical methods in three ways 7]: they work with a coding of the parameter set rather than with actual parameters and work equally with discrete and continuous functions; they search from a population of points; and they use probabilistic transition rules. These di erences ensure that the success of a genetic algorithm is usually not related to the semantics of any particular problem. Our motivation for applying a genetic algorithm to the unit commitment problem is that it can completely replace classical mathematical programming methods and is easy to implement as a search procedure. The paper will rst give a brief description of genetic algorithms and then explain its implementation to the problem of optimizing commitment order of thermal units in (single area) power systems.

2 Genetic Algorithms
Genetic algorithms 11] (GAs) represent a class of general purpose stochastic search techniques which simulate natural inheritance by genetics and the Darwinian survival of the ttest principle. Genetic algorithms are nding increasing applications in a variety of search, optimization and scheduling problems across a wide spectrum of disciplines 7]. A genetic algorithm has four components that must be designed to solve the problem as contemplated. These components are the syntax of the chromosome, the interpretation and evaluation of the chromosome, and the set of operators to work on the chromosomes. Genetic algorithm works with a population of candidate solutions (chromosomes). The tness of each member of the population (point in search space) is computed by an evaluation function that measures how well an individual performs with respect to the problem domain. The constraints can be incorporated in the evaluation function in the form of penalty terms. The betterperforming members are rewarded and individuals showing poor performance are punished or discarded. So starting with an initial random population, the genetic algorithm exploits the information contained in the present population and explores new individuals by generating o spring in the next generation using genetic operations. By this selective breeding process it eventually reaches to a near-optimal solution with a high probability. A pseudo code outlining the generic genetic algorithm is as follows:
Initialize P(t=0); /* P(0) = initial population */

3

Evaluate members of P(t); While (not termination condition) { Generate P(t+1) from P(t) as follows: { select individuals from P(t) on basis of fitness; Perform genetic operation on those selected; } t = t+1; evaluate members of P(t); }

Several di erent genetic operators have been developed, but usually two (crossover and mutation) are extensively used to produce new chromosomes. The crossover operator exchanges portions of information between the pair of chromosomes. The mutation operator is a secondary operator which randomly changes the values at one or more genes of a selected chromosome in order to search for unexplored space. These operations are shown in gure 2. The workings of simple GAs have been described in detail elsewhere 7].

Crossover

Mutation

Figure 2: One-point crossover and simple mutation operations. Genetic algorithms are suitable for solving problem where the domain knowledge is limited to evaluation procedures that can only measure the quality of any given point in the search space. If the problem space is encoded properly, the genetic search eventually converges to the globally optimal or a near-optimal solution. However, the quality of the solution and the length of time it takes to nd the solution mainly depends on the nature of the problem and the use of GA parameters.

3 Problem Description
Operating under the present competitive environment, unit commitment has become increasingly important in the power industry, because signi cant savings can be accrued from sound commitment decisions. A scheduler to do this must indicate which of the generating units are to be committed (on or o ) in every time interval during the scheduling time horizon. This decision must take into account load 4

forecast information and the economic implications of the startup or shutdown of various units. The transition between their commitment states must satisfy the operating (minimum up and down time) constraints. So there are two types of constraints associated with unit commitment problems. The rst type is the global one resulting from load requirement that couple all the generating units during each time period. The second type of constraint is local, representing the di erent operating restrictions of the individual unit. In order to maintain a certain degree of reliability, some stand-by capacity is necessary that can immediately take over when a running unit breaks down or unexpected load occurs. Hence the amount of spinning reserve is an important factor in ensuring an uninterrupted power supply. There are several spinning reserve policies that have usually been adopted: a xed percentage of the forecast peak demand at every time period, a variable reserve, a reserve slightly greater than the output of the most heavily loaded unit, or a probabilistic reserve constraint known as unit commitment risk may be used to ensure better system reliability. But there should be no signi cant excess of reserve capacity in an economic commitment. However, spinning reserve constraints only provide the lower bound since the total capacities of the committed units may not exactly match the load and spinning reserve. The objective (or cost) function of the unit commitment problem is to determine the state of each unit uti (0 or 1) at each time period t, where unit number i = 1 : : : Umax, and time periods t = 1 : : : Tmax, so that the overall operation cost is a minimum within the scheduling time horizon.

3.1 Objective Function

min

Tmax Umax t=1 i=1

X X uti (AFLC )i + uti(1
such that
Umax i=1

uti 1) Si(xti) + uti 1(1 uti) Di]:::::(1)

For each committed unit, the cost involved is the start-up cost (Si) and the Average Full Load Cost (AFLC) per MWh, according to the unit's maximum capacity

X Pimax

Rt + Lt; ::::::::::::::::::::::::::(2)

where is the maximum output capacity of unit i, Lt is the demand and Rt is the spinning reserve in time period t. The above objective function should satisfy minimum up-time and down-time constraints of generating units. The start-up cost is expressed as a function of the number of hours (xti) the unit has been down and the shut-down cost is considered as a xed amount (Di ) for each unit per shut-down, and these state transition costs are applied in the period when the unit is committed or taken o -line respectively 10]. However, unit commitment decisions based solely on unit-AFLC usually do not provide su cient information about the impact of system load conditions on how 5

Pimax

e ciently (eg. fully) the committed units being utilized while determining the optimal or a near-optimal commitment 12]. An index is used to measure the utility Reserve Requirements of each commitment decision, called Utility Factor = Load Total committed output , while satisfying the global constraint (equ. (2)). This factor helps to compensate the deciency associated with over committed decisions based soly on the classical AFLC of units. So during the performance evaluation, commitment decisions having a low utility factor are penalized accordingly.

4 Implementation Details
The implementation proposed here uses a simple genetic search technique for determining the optimal (a least total cost solution) or a near-optimal commitment schedule for a given study period. The short term commitment is considered with a 24-hour time horizon, which may be repeated using the load pro le of each day. Since the system load varies substantially over a 24-hour period and the cost of operation over this time span depends on the timing and frequency of unit's start-ups and shut-downs, this commitment problem is generally viewed as a multi-period problem where the commitment horizon is divided into a number of periods of shorter length (usually a 1-hour commitment interval). In order to use a genetic algorithm for this problem, the rst step is to encode the commitment space. If the whole planning horizon (24-hour) is encoded in a chromosome by the concatenation of commitment spaces of all time periods, it appears to be possible to determine a complete commitment order for the whole span at a time by performing a global search using the genetic algorithm. But this approach makes the problem hard for the genetic algorithms to solve for the following reasons: Firstly, with the increased number of generating units, the length of the encoded chromosome increases in a higher ratio; for example, a system with 20 generating units, the chromosome length becomes 480 bits when the scheduling span is 24 hours. But with a longer encoded string, genetic algorithms nd di culty in reaching to a near-optimal solution; since a genetic search exploits schemata representing hyperplanes and an increase the size of encoding increases the amount of space the algorithm has to explore in order to nd good schemata 11]. Secondly, since the optimal population size is a function of the string length for better schema processing 8], so with the increased string length a bigger size of population is needed. But this will result in both memory and computational overhead. Thirdly, in addition to the above drawbacks such encoding makes the problem space highly epistasis1. When the epistasis is extremely high, gene values are so dependent on each other that unless a complete set of unique values is found simultaneously, no substantial tness
1A biological term that states the amount of interdependency among genes encoding the chromosome.

6

U1

U2 Ui

U3
= 0

U4
or 1

U5

U6

U

7

U max
on/off

according to the unit and i is unit index.

Figure 3: Chromosomal representation of unit commitment decisions. improvements can be noticed 6]. In this case, since the commitment decisions of previous hours have strong e ect on the decision in successive hours, so encoding these dependent decisions in a single string makes the representation highly epistasis. This increases the complexity of search space and makes it di cult for a genetic search to nd a near-optimal solution in a reasonable time. In this work, the problem is considered as a multi-period process as in practice and a simple genetic algorithm is used for commitment scheduling. Each chromosome is encoded in the form of a position-dependent genes (bit string) representing the number of thermal units available in the system, and the allele value at loci give the state (on/o ) of the units as a commitment decision at each time period, shown in gure 3. Unit commitment decisions satisfying load-reserve requirement and the operating constraints of units are regarded as feasible solutions, and any violation of the constraints is penalized through a penalty function. So the raw tness function is formulated here using a weighted sum of the objective function and values of the penalty function based on the number of constraints violated and the extent of these violations. By choosing suitable weights for the penalty function, it is possible to nd a near-optimal solution to the problem. In our case,

Fitness function = Objective function + Penalty function (L R; Utility Factor; min up; min down),
where L R is the load-reserve requirements, min up and min down are minimumup and minimum-down time constraints of the units. In our implementation here, the following scaling technique is used to normalize tness and to produce a non-negative gure of merit.

Scaled Fitness (I ) = (M I )=(M N ); where I is the individual raw tness, M and N are the relative maximum and minimum tnesses respectively among all individuals in the current generation. The goal is to maximize the scaled tness in order to minimize the cost (or objective) function. The scaled tness is used to determine the probability of selecting of members in the population for breeding. The ow chart of implementing the genetic algorithm for unit commitment is shown in gure 4. For better understanding of the diagram, some of the ow lines connected to the repeatedly used program module (GA routines) and domain information (look-up tables) are nuberbered accordance with the sequence of execution.
7

START

AT T:=1 INITIALIZE POPULATION

RUN GENETIC ALGORITHM FOR FIXED NUMBER OF GENERATIONS OR ACCORDING TO SOME STOPPING CRITERIA

LOOK UP TABLES
INTERACTION WITH DOMAIN INFORMATION FORECASTED LOAD PROFILE THERMAL UNITS CHARACTERISTICS AND OPERATIONAL CONSTRAINTS STATUS OF UNITS

1 OUTPUT A SET OF FEASIBLE COMMITMENT DECISIONS WITH SMALLER ASSOCIATED COSTS AS OPTIONAL STRATEGIES

store status infomation for all strategies

2

4

3

T := T + 1

FOR EACH STRATEGY REPLACE A PERCENTAGE OF POPULATION

OUTPUT THE BEST SUCCESSIVE PATH OF THE STRATEGY

update status infomation for each strategy

5

DETERMINE THE PATH FOR MINIMUM COST STRATEGY AND DISPLAY COMPLETE UNIT COMMITMENT SCHEDULE WITH TOTAL COST

YES

IS T > T_max

NO

STOP

Figure 4: A ow diagram for unit commitment using genetic algorithms. The genetic-based unit commitment program starts with a random initial population (at T=1) and computes the tness of each individual (commitment decision) using the forecasted load demand at each period, and the operating constraints of the units (using lookup tables). Each time the genetic optimizer is called, it runs for a xed number of generations or until the best individual remains unchanged for a long time (here 100 successive generations). Since the unit commitment problem is time-dependent, these piecewise approaches of working forward in time and retaining the best decision, can not be guaranteed to nd the optimal commitment schedule. The reason for this is that a decision with signi cantly higher costs during the early hours of scheduling could lead to signi cant savings later and may produce a lower overall cost commitment schedule. In order to make the genetic-based unit commitment program robust in nding near-optimal solutions, a number of feasible commitment decisions (less than or equal to a prede ned value S) with smaller associated costs are saved at each time period. These strategies2 determine how many possible alternative paths are
A strategy is a sequence of commitment decisions from the starting period to the current period with its accumulated cost.
2

8

Time

t=1 S 11

t=2 S 21 S 22 S 23

t=3 S 31

t=4
converges to same strategy

t=5

t=T

max

S T R A T E G I E S

S 12 S 13

S 32 S 33

S 42 S 43

S 52

S 2 T max

no path found

S 1n

S 2n

S 3n

S 4n

S 5n

S n T max

Figure 5: A number of alternative path (strategy) for nding a near optimal commitment order. available at each period for nding the overall operation cost. Figure 5 illustrates the di erent paths available, where one path converges to the other in midway and another stopped because it could not nd path within the allocated resources. The selection of S is a ective in economical scheduling (in nding an optimal solution), memory requirement and computation time. In order to save computation time the same strategies are carried forward to the next period if the load remains unaltered or varies slightly in the current period such that load-reserve requirements are satis ed by all strategies. If a strategy cannot meet the demand of present period, the genetic optimization process is performed for the period to nd a feasible successor paths with smaller cost. This approach increases the likelihood of nding the path of minimum cumulative cost. These temporary commitment strategies used to update the status information of the units (up-time/down-time counter) to keep track of the units in service or shutdown for a number of successive hours. In the next time period, half of the population is replaced by randomly generated individuals to introduce diversity in the population so that the search for new commitment strategy can proceed according to the load demand. The purpose of keeping half of the previous population is that in most situations the load varies slightly in some successive time intervals and the previous better individuals (commitment strategies) are likely to perform well in the current period. However, if there is a drastic change in load demand, newly generated individuals can explore the commitment space for nding the best solution. The iterative process continues for each period in the scheduling horizon, and the accumulated cost associated with each commitment strategy gives the overall cost for the commitment path. In a time period, if a unit is to be decommitted due to a decrease in load demand, and because of that -if its minimum-up time constraint is violated, then the unit is considered to remain committed (in banking state) until its minimum-up period is completed. -also to tackle sharp rises in load demand in the next period, a look-ahead mechanism is incorporated which decides that the unit will remain committed (in banking state) even though it represents an uneconomical decision for the current period. 9

During these multi-period optimization processes, if in a particular period no feasible solution (strategy) is found, the process is repeated so that at least one feasible solution is found before shifting to the next period. However, in our test example such repetitions are required only in a few occasions in later periods.

5 Experimental Results
The genetic algorithm-based unit commitment program is applied to an example problem which consists of 10 thermal units. The capacities, costs and operating constraints vary greatly among the various generating units in these test systems. Di erent type of load pro les are tested which represent typical operating circumstances in the studied power system. We have considered a short term scheduling where the time horizon is 24-hour and scheduling for an entire day is done in advance which may be repeated using load pro le of each day for a long-term scheduling. In these experiments, the spinning reserve requirement is assumed to be 10% of the expected hourly peak load. A program implementing the algorithm has been run on a SUN (sparc) workstation under UNIX 4.1.1 operating system. The experiment is conducted with a population size of 250 using di erent crossover and mutation rates. For the result reported here (shown in a tabular form), a crossover probability of 78% and mutation rate of 15% were used along with a stochastic remainder selection scheme 3] for reproduction. We also used an elitist scheme which passes the best individual unaltered to the succeeding generation. Each run was allowed to continue up to 500 generations and the strategy path with minimum cumulative cost gives a near-optimal commitment for the whole scheduling period. For this example, table-1 gives the characteristics of and the initial states of the generating units. Table-2 gives commitment schedule for two cases, which were run independently. In rst case, one best solution is saved at each time period and in second case, multiple least cost strategies are saved for determining the minimum cost path. A comparison shows that substential reduction in overall cost can be achieved when the best commitment schedule is determined from multiple least cost strategies. In this table, the second column gives the hourly load demand, the third column shows total requirement after adding spinning reserve, the rest columns give the total output capacity (in MW) of the committed units and the state of units in each case, where `1' is used to indicate a unit is committed, `0' to indicate that a unit is decommitted. The genetic based unit commitment system have been tested under di erent operating conditions in order to evaluate the algorithm's performance. It is observed that the scheduling which produces optimal power output does not always give the overall minimum cost scheduling, and also the minimum cost scheduling is very sensitive to the system parameters and the operating constraints of the generating units.

10

Unit Maximum Min. Up Min. Down No. Capacity Time Time (MW) (hr) (hr) 1 60 3 1 2 80 3 1 3 100 4 2 4 120 4 2 5 150 5 3 6 280 5 2 7 520 8 4 8 150 4 2 9 320 5 2 10 200 5 2

Initial Status (hr) -1 -1 1 5 -7 3 -5 3 -6 -3

b1
85 101 114 94 113 176 267 282 187 227

St Up cost b2 b3 20.588 0.2 20.594 0.2 22.57 0.2 10.65 0.18 18.639 0.18 27.568 0.15 34.749 0.09 45.749 0.09 38.617 0.130 26.641 0.11
e

Sh Down Cost AFLC 15 25 40 32 29 42 75 49 70 62 15.3 16 20.2 20.2 25.6 30.5 32.5 26.0 25.8 27.0

(-) indicates unit is down for hours and positive otherwise. *We used start-up cost = b1;i(1

b3;i :(xt i ) ) + b2;i.

Table 1: Characteristics and initial state of the thermal units.
-------------------------------------------------------------------------------CASE - 1 CASE - 2 When only best strategy Best of five least is saved at each hour cost strategies saved -------------------------------------------------------------------------------Time Load Load + Committed State of Committed State of in Demand Reserve Output units Output Units (hr) (MW) (MW) (MW) (MW) -------------------------------------------------------------------------------1 1459.00 1677.85 1700.00 1011111110 1710.00 1110011111 2 1372.00 1577.80 1710.00 1110111011 1860.00 1110111111 3 1299.00 1493.85 1710.00 1110111011 1550.00 1111101011 4 1280.00 1472.00 1490.00 0111101011 1490.00 0111101011 5 1271.00 1461.65 1470.00 1011101011 1470.00 1011101011 6 1314.00 1511.10 1550.00 1111101011 1550.00 1111101011 7 1372.00 1577.80 1600.00 1101101111 1580.00 1110101111 8 1314.00 1511.10 1600.00 1101101111 1520.00 0110101111 9 1271.00 1461.65 1500.00 1111101110 1500.00 1111101110 10 1242.00 1428.30 1630.00 1111011110 1500.00 1111101110 11 1197.00 1376.55 1420.00 0111011010 1380.00 1011110111 12 1182.00 1359.30 1360.00 1111011001 1360.00 1101110111 13 1154.00 1327.10 1330.00 1001111001 1360.00 1101110111 14 1138.00 1308.70 1410.00 1101111001 1310.00 1111110011 15 1124.00 1292.60 1310.00 1111110011 1310.00 1111110011 16 1095.00 1259.25 1280.00 0110110111 1260.00 1111110110 17 1066.00 1225.90 1260.00 1010110111 1260.00 1111110110 18 1037.00 1192.55 1260.00 1010110111 1200.00 0111110110 19 993.00 1141.95 1180.00 1111100111 1180.00 1111100111 20 978.00 1124.70 1200.00 1111001010 1180.00 1111100111 21 963.00 1107.45 1320.00 0101011010 1180.00 1111100111 22 1022.00 1175.30 1210.00 1101011100 1180.00 1111100111 23 1081.00 1243.15 1340.00 1110111100 1250.00 0111110011 24 1459.00 1677.85 1900.00 1011111111 1680.00 1111011011 ------------------------------------------------------------------------------Cumulative scheduling Cumulative scheduling cost = 940101.65 cost = 877854.32 ------------------------------------------------------------------------------The difference in cost is approximately 7% in these two cases.

Table 2: Unit commitment schedules determined by the genetic algorithm. 11

6 Conclusions
In this paper, we have discussed the application of a genetic algorithm in solving short term unit commitment problems i.e deciding commitment order of units for an entire day in advance. When a set of smaller cost strategies are saved at each hour, the path which gives the overall minimum cost determines the near-optimal commitment schedule. The number of strategies considered here have been found to produce a well comparable commitment schedule to that of existing approaches. The approach appears to be like DP-TC or DP-VW, but with an advantage of being population-based (no window sizing i.e truncation of search space is needed), and having a high probability of nding a solution near to the global optimum. When only the minimum strategy (S=1) is saved at every hour 5], the commitment policy is then similar to the AFLC-based priority order method, but in our GAbased approach it is not necessary to provide priority order list explicitly in order to determine commitment decisions. The major advantages of using GAs are that they can replace the shortcomings of mathematical programming approaches more e ciently. The genetic-based unit commitment system evaluates the priority of the units dynamically considering the system parameters, operating constraints and load pro le at each time period in the scheduling horizon. Though the global optimality is desirable, but in most practical purposes near-optimal (good feasible) solutions are generally su cient. This paper attempts to nd the best schedule from a set of good feasible commitment decisions. Though the example considered here is small and other classical methods can nd commitment order easily, but the GA-based method appears to be simple and a possible alternative approach with some advantages. However, the work is underway to make a quantitive comparison with previous methods and will be reported in the near future3. Further research will also address the commitment problem of large systems with hundreds of units in multiple areas. One possible approach may be to use indirect encoding or use of some grammar rule (as used in other GA applications) for representing a cluster of units in a chromosome. The method presented in this paper can include more of the constraints that are encountered in real-world applications of this type. One disadvantage of this approach is the computational time needed to evaluate the population in each generation, but since genetic algorithms can e cently be implemented in a highly parallel fashion, this drawback becomes less signi cant with its implementation in a parallel machine environment. The rst author gratefully acknowledges the support given by the Government of Assam (India) for awarding State Overseas Scholarship. We also acknowledge
A comparison with other methods will appear in the Journal of IEE Proceedings-C: Generation, Transmission and Distribution.
3

Acknowledgements

12

the valuable discussions with our colleagues in the department and reviewers of the paper for their comments.

References
1] L. F. B. Baptistella and J. C. Geromel. A decomposition approach to problem of unit commitment schedule for hydrothermal systems. IEE Proceedings- Part-C, 127(6):250{258, November 1980. 2] Jonathan F. Bard. Short-Term Scheduling of Thermal-Electric Generators using Lagrangian Relaxation. Operations Research, 36(5):756{766, Sept/Oct. 1988. 3] L. B. Booker. Intelligent behavior as an adaptation to the task environment. PhD thesis, Computer Science, University of Michigan, Ann Arbor, U. S. A, 1982. 4] Arthur I. Cohen and Miki Yoshimura. A Branch-and-Bound Algorithm for Unit Commitment. IEEE Transactions on Power Apparatus and Systems., PAS-102(2):444{449, February 1983. 5] Dipankar Dasgupta. Unit Commitment in Thermal Power Generation using Genetic Algorithms. In Proceedings of the Sixth International Conference on Industrial & Engineering Applications of Arti cial Intelligence and Expert Systems (IEA/AIE-93), pages 374{383, Edinburgh, UK, June 1-4 1993. 6] Yuval Davidor. Epistasis Variance: Suitability of a Representation to Genetic Algorithms. Complex Systems, 4:369{383, 1990. 7] David E. Goldberg. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley., rst edition, 1989. 8] David E. Goldberg. Sizing populations for serial and parallel genetic algorithms. In International Conference on Genetic Algorithms (ICGA-89), pages 70{79, 1989. 9] A. R. Hamdam and K. Mohamed-Nor. Integrating an expert system into a thermal unit-commitment algorithm. IEE Proceedings-C, 138(6):553{559, November 1991. 10] W. J. Hobbs, G. Hermon, S. Warner, and G. B. Sheble. An Advanced Dynamic Programming approach for Unit Commitment. IEEE Transactions on Power Systems, 3(3):1201{1205, August 1988. 11] John H. Holland. Adaptation in Natural and Arti cial Systems. University of Michigan press, Ann Arbor, 1975. 13

12] Fred N. Lee. The application of commitment utilization factor (CUF) to thermal unit commitment. IEEE Transactions on Power Systems., 6(2):691{698, May 1991. 13] P. G. Lowery. Generating Unit Commitment by Dynamic Programming. IEEE Transactions on Power Apparatus and Systems, PAS-85(5):422{426, May 1966. 14] A. Merlin and P. Sandrin. A New Method for Unit Commitment at Electricite De France. IEEE Transactions on Power Apparatus and Systems, PAS102(5):1218{1225, May 1983. 15] John A. Muckstadt and Sherri A. Koenig. An Application of Lagrangian Relaxation to Scheduling in Power Generation Systems. Operations Research, 25(1):387{403, Jan/Feb 1977. 16] Sasan Mukhtari, Jagjit Singh, and Bruce Wollenberg. A Unit Commitment Expert System. IEEE Transactions on Power Systems, 3(1):272{277, February 1988. 17] Z. Ouyang and S. M. Shahidehpour. An Intelligent Dynamic Programming for Unit Commitment Application. IEEE Transactions on Power Systems, 6(3):1203{1209, August 1991. 18] Z. Ouyang and S. M. Shahidehpour. A Hybrid Arti cial Neural NetworkDynamic Programming approach to Unit Commitment. IEEE Transactions on Power Systems, 7(1):236{242, February 1992. 19] C. K. Pang and H. C. Chen. Optimal short-term Thermal Unit Commitment. IEEE Transaction on Power Apparatus and Systems, PAS-95(4):1336{1346, July/August 1976. 20] C. K. Pang, G. B. Sheble', and F. Albuyeh. Evaluation of dynamic programming based methods and multiple area representation for thermal unit commitments. IEEE Transactions on Power Appratus and Systems, PAS-100(3):1212{1218, March 1981. 21] S. K. Tong, S. M. Shahidehpour, and Z. Ouyang. A Heuristic short-term Unit Commitment. IEEE Transactions on Power Systems, 6(3):1210{1216, August 1991. 22] F. Zhuang and F. D. Galiana. Unit Commitment by Simulated Annealing. IEEE Transactions on Power Systems, 5(1):311{317, February 1990. 23] Fulin Zhuang and F. D. Galiana. Towards a more rigorous and practical Unit Commitment by Lagrangian Relaxation. IEEE Transactions on Power Systems, 3(2):763{773, May 1988. 14

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