GENETIC ALGORITHMS SOLUTION TO OPTIMAL MAINTENANCE SCHEDULING (OMS) OF GENERATING UNITS WITH MODIFIED GENETIC OPERATORS S. BASKAR
P. SUBBARAJ
S. TAMILSELVI
M.V.C RAO
Electrical Engineering Department Thiagarajar College of Engineering, India
Faculty of Engineering Technology Multimedia University, Malaysia ABSTRACT
This paper applies Genetic algorithm (GA) to OMS problem with modified genetic operators like uniform integer crossover , integer mutation along with the new genetic operator called string reversal. The main contribution of this paper is ‘probabilistic production simulation’ with “Equivalent energy function method”, which excels in faster computation as well as in accuracy. The performance of the algorithm has been tested on a 21-unit system. The problem is solved for two different objective functions namely, minimizing expected energy production cost and maximizing reserve. GA has been implemented with Integer encoding and Binary for integer encoding. GA results have been compared with the solution by conventional methods. This paper reports the effect of ‘string reversal’ to OMS problem, with an explicit case study and simulation results. Finally, it is concluded that, Integer GA finds the global optimum solution, irrespective of the objective function and faster convergence is possible only with string reversal.
1. INTRODUCTION OMS involves specifying maintenance decision, i.e., when the maintenance of particular generating unit should be begun in a power plant such that the operating costs are minimized and constraints are met. Maintenance scheduling is required to reduce the risk of capacity outage and improve the availability of units. It is the most important operation planning decision in a system, as the badly planned maintenance decision would directly reflect in a rise in un-served energy and any sub-optimality in the schedule would reduce the supply demand gap, making the OMS problem more complicated. The optimal maintenance plan can only lead to substantial savings in production cost and reduction in un-served energy. The maintenance must be performed on a number of generators inside a fixed planning horizon. Taking this aspect, uncertain load and reserve constraints and a hard crew constraint into account, is recognized as a complex combinatorial problem. In past, many different methods have been applied to solve it using various objective functions. Minimization of expected energy production cost [1,2,3,4], minimization of minimum reserve [5,6], and minimizing system unreliability [7,8,9] as objective functions, several mathematical methods and heuristic techniques have been proposed for solving the OMS problem. From the review, it is found mathematicalprogramming methods [1,2,3,7] are unable to cope up with even moderate scale problems because of ‘curse of dimensionality’. Most of the optimization techniques rely on heuristics [5,8,9], able to alleviate the limitations of mathematical programming methods wherein, each
generating unit is considered separately for the selection of its optimal outage interval. Due to the coupling constraints, sequential unit scheduling generation often leads to infeasible solutions despite the existence of feasible solutions. Out of the existing evolutionary techniques to OMS, GA based solution in which reliability criteria of leveling the reserve is investigated[10,11], GA with the combined use of Tabu Search and Simulated Annealing (SA) employed fuel cost minimization[12], and SA method [13], Hybrid evolutionary technique[14], reported for both fuel cost and maintenance cost minimization. From this review it is clear that, the forced outage uncertainty of generating units is not considered in all the proposed evolutionary techniques. Minimization of Expected Energy Production cost incorporating the forced outage rate of the generators is often a more crucial factor than the maintenance cost or fuel cost minimization, because the consideration of uncertain random outage can alone draw out a preventive and planned optimal maintenance schedule. Though it is consuming more computational time for probabilistic production simulation this would reduce the frequency of service interruptions and many undesirable consequences and hence the lifetime of the equipment can be extended. Hence in this paper, performance of GA for minimization of expected energy production cost and maximization of reserve using two different encoding have been done. Applicability of string reversal or inversion mutation has been studied in addition to uniform integer crossover and integer mutation.
Nomenclature i n t mt ai Eit
= Index of generating units = Total number of generating units = Index of periods = Number of planning horizons in weeks = Average operating cost of unit ‘i’ in S/MWh = Expected energy produced by unit i at period t in MWh prt = Net reserve of the t th time interval , which is equal to installed capacity minus load and maintenance unit capacity. pravg = Average reserve of the system which is equal to the sum of net reserve of each time interval and the maintenance period. Xit = State variable Xit = 1, if unit i is offline for maintenance 0, otherwise T = Set of indices of periods in planning horizon. M i = Maintenance length of unit i. Ei = Earliest period for the unit i to start maintenance Y = Set of units to be maintained Yi = Maintenance period till the ith unit Yi-1 = Maintenance period till the (i-1)th unit AM t = Available manpower at period t. M it = Man power required by unit i at period t. Dt = Anticipated Demand at period t in MW Rt = Required reserve at period t. Sit = Set of start time periods k such that if the maintenance of unit i starts at period k that unit will be in maintenance at period t,
Sit={k∈Ti: t - Mi +1≤ k ≤ t}
Pi
= Generating capacity of unit ‘i’ in MW
Li
= Latest period for the unit i to close the maintenance = Set of periods when the maintenance of unit i may start .Ti={t∈T: Ei ≤ t ≤ Li - M i +1} = Set of units which are allowed to be in maintenance in period t, It=[i: t∈T i]
T It
2. PROBLEM DESCRIPTION Production cost Objective function mt n F = Min Σ Σ ait Eit (1-Xit) ------ ( 1) t=1 i=1 tries to minimize the total expected energy production cost over the operational planning period. Reliability Objective Function mt F=Min Σ (prt - p ravg)2/ pravg ------ ( 2 ) t=1 Here, the mostly used levelized reserve capacity method is applied which tries to maximize the reserve, by making each maintenance time interval have the same level of net reserve.
Subject to: Maintenance Window Constraint Σ Xit = 1 ------- (3) t∈Ti It gives that each unit must be maintained exactly once and specifies a time interval during which maintenance on that unit must take place. Consecutive Periods Of Maintenance Ti+Yi-1 Σ Xit = Yi ------ (4) t=T i This Completion of maintenance guarantees that the maintenance for each unit must occupy the required time duration without interruption. Crew Constraint n mt Σ Σ Xik M ik ≤ AM t ------ (5) i∈It k∈sit It depends on the manpower availability and units can be simultaneously maintained at any stage only when the available crew and the required crew for units to be maintained at that stage are same. Demand /Reserve Constraint n Σ Pi (1-Xit) ≥ Dt + Rt ------------ (6) i=1 In order to ensure that the total available power is greater than the demand, even when the random outage occurs, the reserve constraint is imposed.
3. GA IMPLEMENTATION Chromosome is interpreted as maintenance-starting period of the generating units. Fitness is evaluated for the chromosome, which is sum of the objective function value and penalty of the violated constraint. The randomly generated population has selection process, based on the fitness of the individuals to undergo recombination operation like uniform crossover, mutation and thus new generation is evolved with the offspring. GA uses probabilistic transition rules to select someone to be reproduced and someone to die so as to guide their search toward regions of the search space with likely improvement. After several generations, the algorithm converges to a sort of an excellent chromosome. 3.1 Problem encoding The approach of Integer encoding [10] consists of a string of integers, each one of which indicates the maintenance start period of a unit and the string length is equal to number of units. For every unit the
maintenance window varies and to reduce the complexity of handling OMS problem, the start period can be selected in the preferred maintenance window. The integer formulation of the problem can be encoded by using binary code to represent integer variables in the genetic structure called ‘Binary For Integer’ encoding in which case, if the number of variable values is not a power of 2, some of the binary values will be simply redundant. 3.2 Genetic Operators 3.2.1 Crossover With uniform integer crossover [15], in Integer GA, the possibility of infeasible offspring, cannot be due to the violation of soft maintenance window constraint. However with binary crossover, in Binary for integer coding GA, it is not the case and is difficult to generate feasible solution satisfying even soft constraints. 3.2.2 Mutation In binary for integer coding GA, binary mutation is applied. In integer coding GA, gene-undergoing mutation in a chromosome is replaced by a randomly generated integer that obeys soft constraint. In order to investigate the performance of GA to OMS, 50 independent runs are conducted and average number of generations(ANOG) taken to reach the optimum solution are noted. It is found that the number of changes in the fitness value is less and after some generations, the solution algorithm gets saturated when approaching towards the global value causing more computation time to converge. In the plan of improving the search efficiency of the proposed technique, several other genetic operators[16] have been tried and found that Inversion mutation or string reversal work well for the OMS problem. 3.2.3 String Reversal For this OMS problem, string reversal has been employed as a modified genetic operator with which the convergence rate of the proposed solution method is increased. Since for the test system taken, 13 units begin their outages in the first half of the year; and 8 outages begin in the second half of the year, the string should not be easily reversed as it is, to avoid the window constraint violation. Since all the units are required to complete the maintenance in their respective 26-week period, before employing this operator, the chromosome should be divided into thirteen genes as first group (comprising units being maintained in the first half of the year) and eight genes as second group (units allowed to begin maintenance in the next half of the year). In string reversal, some portion (randomly generated) of the string is reversed in both the groups of the parent, causing some of the unit’s startup period to be changed. After reversal the respective index of the parent is replaced by the changed value to produce an offspring.
4. SIMULATION RESULTS Two different maintenance schedules are obtained for two objective functions, cost function and reliability function. For implementing GA, population size of 50 and Roulette wheel type selection are employed. Crossover, mutation, and inversion mutation probabilities are set to 0.8, 0.015, and 0.2 respectively. Programs are developed using Matlab5.3 software package and results are simulated on Pentium III computer@700 MHz. For cost objective function, constraints eqns(3), and (4) are taken. To prove the validity of the GA method, the result has been compared with the solution of dynamic programming(DP) [2] approach. For the implementation of reliability function, constraints such as eqns(3), (4), (6), and with/without crew constraint(5) are considered. For the comparison of GA results, maintenance scheduling is done using deterministic levelised reserve method(LR). Table 1 Comparison of Results for cost Function Solution Method DP Approach Binary for Integer GA Integer GA
Expected Energy Production Cost 252927428.72 249383830.55 249225195.00
Table 2 Comparison of results for reliability function
Solution Method LR method Binary for Integer GA Integer GA
Min Reserve W/o crew Const 4.8419
Min Reserve With crew Const 5.6172
4.6819
5.0496
4.6819
4.9437
From Table 1, the expected energy production cost obtained using DP method is more than the cost that has been found using GA. From Table 2, it is obvious that, the levelised reserve method is not suitable for finding the optimal solution. From Tables 1&2, it is found that Binary for integer GA is failed to find the optimal solution whereas Integer GA is able to obtain the optimal solution to OMS problem, irrespective of the objective function. In Figure 1, the convergence of GA, for the two coding methods, have been compared in a same plot for cost function. 50 independent runs are conducted for both functions and the performance of coding algorithms is compared, with respect to their solution accuracy and mean computation time, in Table 3. For the reliability function, when hard crew constraint is not included, the convergence speed is faster. But, when it is included, the solution algorithm takes more execution time to get into the optimal region. For cost function, since the probabilistic production simulation takes more time, the solution algorithm consumes more
computation time. Thus to reduce the time consumption and improve the search efficiency of GA, modified genetic operator, string reversal has been applied to OMS problem.
both functions, which have been found from integer GA, is tabulated in Table 5. 6.8
8
x 10 2.52
INTEGER GA BINARY FOR INTEGER GA
2.515
6.4 6.2 6
2.51
Best Solutions
MINIMUM COST OF EACH GENERATION
Before string reversal After string reversal
6.6
2.505
5.8 5.6 5.4
2.5 5.2 5
2.495
4.8 1
10 2.49 0 10
10
1
10
2
10
10
2
3
3
10 Number of Generations
10
4
10
5
NUMBER OF GENERATIONS
Figure 1. Comparison of convergence speed of coding methods Table 3 Performance of GA without Inversion mutation Objective Function Reliability Function (With out crew Constraint) Reliability Function (With crew Constraint) Cost Function
Solution Best Mean Worst Time(sec) ANOG Best Mean Worst Time(sec) ANOG Best Mean Worst Time(hrs) ANOG
Binary for integer GA 4.6819 4.7615 4.8256 275.954 857 5.0496 5.4438 5.5434 2.163 17,914 249383830.55 249523451.02 249601154.21 20.66 302
Integer GA 4.6819 4.7098 4.7432 125.96 546 4.9437 4.9826 5.0242 1.2600 12,854 249225195.00 249236323.67 249240281.21 14.7 252
In Figure 2, the convergence characteristics of integer coding GA, with and without inverse mutation are compared for the reliability objective function with crew constraint case. Before applying the modified genetic operator, GA takes more than 10,000 generations to converge. Once the string reversal operator is employed, solution algorithm is able to converge within 4000 generations. The performance of string reversal in improving the convergence rate is demonstrated in Table 4, from which it is clear that, only integer GA has improvement, when string reversal is employed. The Optimal Maintenance Schedule for
Figure 2. Comparison of convergence of GA with/without String reversal operator Table 4 Performance of GA with Inversion mutation Objective Function Cost Function Reliability Function (With crew Constraint)
Solution Best Mean Worst Time(hr) ANOG Best Mean Worst Time(sec) ANOG
Binary for Integer GA 249383830.55 249523451.02 249601154.21 _ 5.0496 5.4438 5.5434 _
Integer GA 249225195.00 249227129.02 249229934.16 8.45 145 4.9437 4.9870 5.1606 703.92 2544
5. CONCLUSION To establish the supremacy of Genetic algorithm technique applied to OMS problem, a medium sized 21unit system is considered and the simulation results are compared with the solution of conventional methods for two objective functions. Solution algorithm is found to be saturated, while approaching the global solution, due to the larger search space and leads to more execution time to converge to optima. To reduce the computation time and to improve the search efficiency of the GA, a modified genetic operator called string reversal has been employed. Though some additional analyses have to be conducted according to the particular features of the OMS problem, implementat ion of string reversal is an effective way to improve the performance of GA computation. Out of two coding methods employed, from the numerical results, Integercoding GA is found satisfactory to OMS problem, irrespective of the objective function.
Table 5 OMS using Integer GA
Unit No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
START UP WEEK FOR MAINTENANCE Reliability Reliability Cost Function Function Function (W/O Crew (With Crew constraint) constraint) 13 14 10 36 48 48 13 12 22 13 18 7 36 43 42 13 24 19 13 21 24 36 37 47 6 4 17 13 23 11 13 3 6 36 34 32 13 10 2 13 19 13 13 8 4 13 25 17 38 48 31 37 51 35 38 52 37 29 49 38 4 9 8
Acknowledgements
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Authors wish to thank the authorities of Thiagarajar college of Engineering, Madurai-15, Tamilnadu, India, for providing necessary facilities to carry out this research work.
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Address of author Dr S. BASKAR, Assistant Professor, Electrical Engineering Department, Thiagarajar College of Engineering, Madurai - 625 015, Tamilnadu, India.
[email protected]
