Genetic algorithms solution to generator maintenance scheduling with modified genetic operators

S. Baskar, P. Subbarai, M.V.C. Rao and S. Tamilselvi

Abstract: The applicability of genetic algorithms (GA) to the generator maintenance scheduling (GMS) problem with modified genetic operators (MGO), such as string reversal and reciprocal exchdnge mutation (REM) is demonstrated. The main contribution is the use of ‘probabilistic production simulation’ (PPS) with an equivalent energy function method, which outperforms other methods in terms of computation time and accuracy. The performance of the algorithm has been tested on 5- and 21-unit test systems with integer encoding, binary for integer encoding, and real encoding. The GMS problem is solved to minimise the expected enera production cost (EEPC) and maximising the reserve objectives under it series of constraints. Results are compared with solution by conventional methods. This paper places in proper perspective the effect of MGO, with an explicit case study and simulation results. It is placed in evidence that only integer coding CA finds the global optimum solution, irrespective of the nature of the objective function and system size. Faster convergence is enhanced with the implementation of MGO for integer CA only.

index of generating units total number of generating units index of periods number of planning horizons, weeks average operating cost of unit i, S/MWh expected energy produced by unit i at period I, MWh net reserve of the tth time interval. which is equal to installed capacity minus load and maintenance unit capacity average reserve of the system which is equal to the sum of net reserve of leach time interval and the maintenance period State variable

{ 0; othenvise set of indices of periods in planning horizon maintenance length of unit i earliest period for unit i to start main- tenance latest period for unit i to close the maintenance

1 ~ if unit i is omine for maintenance

0 IEE, 2003 IEE Proceedings online no. 20030073 dai:l0. 1049/ipgld:20030073 Paper firs1 received 291h Apnl 2002 and in revised form 16th September 2002 S . Baikar and S. Tamilselvi are with the Electrical and Eleclronici Enpnxnng Drpartment. Thiagaarajar College of Engineering. Mildurai ~ 625015. India P. Subbaraj i s with the PSR Engineering College, Sivsksri. Indie M.V.C. Rao i s with the Faculty of Enpneenng and Teclmolog. lvlrliika Campus, Multimedia University. Malaysia

56

Ti

yi- I A M ,

set of periods when the maintenance of unit i may start, Ti={t tT: Ei<r< L,-M,+ I } set of units which are allowed to he in maintenance in period 1. I,=[? IE TI] set of units to be maintained maintenance period till the ith unit maintenance period till the (i-1)th unit available manpower at period I man power required by unit i at period / anticipated demand at period r , MW required reserve at period I 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 pcriod I, S i , = { k t T j : 1-M; +Iskit} generating capacity of unit ‘ i, MW allowable value of expected energy not served (EENS), MWh forced outage rate of unit i

1 Introduction

Power qilality and reliability have become important parameters making generator maintenance scheduling (GMS) [ I ] indispensable for power system operation. The aim is to allocate proper maintenance sequences and timetables to generators, while minimising costs and satisfying maintenance and system demands. This complex combinatorial problem has been studied widely in the past and different methods have been applied to solve it. Mathematical programming methods such as the branch and bound technique [2] are suitable for solving this problem. while other methods such as linear prograni- ming [3], a decomposition approach (DA) [4]. integer

IEE Pro.-Gcw~. Tror,.~rn Di.~wih., Vu/. I50 No. I , J~tflwzr? 2Ml.7

programming [SI, and dynamic programming (DP) [6] cannot accurately simulate power system operations due to the cnrse of dimensionality. Heuristic methods [7-91 are able to alleviate the limitations of mathematical programming methods wherein, each generating unit is considered separately for selection of its optimal outage interval, leading to a local optimum. Intelligent computation methods, such as simulated annealing (SA) [IO], expert systems [Ill, fuzzy systems [I21 and evolutionary optimisa- tion have been applied to the GMS probleni. Genetic algorithms (GAS) [13, 141 have solved the GMS problem using reliability criteria. In [IS] evidence was produced that. search performance is improved by introducing Tabu search to GA+SA for minimisation ofjfuel and main- tenance cost objectives. On a related topic, [IO: 16, 171 have also coilcentrated on fuel cost minimisation.

The minimization of expected energy production cost (EEPC) incorporating ‘q’, which is often a more crucial factor than maintenance or fuel cost minimisation for preventive maintenance, is not considered in existing evolutionary techniques. Although probabilistic production simulation (PPS) consumes more computer time. it does reducc the frequency of service interruptions. Hence. the performance of a GA with ’q’ is investigated, when applied to a cost function.

2 Problem description

This paper considers two ohjcctive functions:

tiies to minimise the EEPC over the operational planning period. E,,(.) is a function of D,, P,. qp

tries to make each maintenance time interval have the same level of net reserve and minimise the minimum reserve. Subject to:

Maintenance window constraint

Consecutive periods of maintenance

Crew constraint

Demand reserve constraint

(3)

(4)

i= I Reliability constraint

EENS, 5 i: (7)

In the GMS problem, (3) and (4) are soft constraints. since the starting period of any unit can easily be selected in the ‘preferred maintenance’ interval. But. (3, (6) and (7) are hard constraints and it is very dificult to generate a feasible solution.

IEE Proc-Gmer. 7ilinmi. Dim&, Vb! I X J h‘o 1. Jmzrury 2lK3

3 GA implementation for GMS

In this paper, the problem is tackled by finding the starting period, i.e. ‘when the maintenance should be begun for each generating unit? [13]. The integer encoding approach [13] consists of a string of integers, each of which indicates the maintenance start period of a unit and the string length is equal to a number of units. Since. the maintenance period varies for every unit, the start period is selected within the specified maintenance window, satisfying soft constraints. The integer formulation of the problem can be encoded using a binary code called ‘Binary for Integer’ representa- tion in which. if the number of variable values is not a power of 2. some of the binary values will simply be redundant. Real encoding, similar to integer coding, is real- valued instead of integer-valued. Before function evaluation (sum of penalty values for each constraint violation along with the extent of violation and the objective function), it is given as an integer. Except for binary for integer coding CA, with conventional uniform crossover [IS], and mutation, the possibility of arising infeasible offspring cannot be due to the violation of soft constraints.

In order to investigate the performance of the proposed algorithm for GMS, initially a test was conducted on a small five-unit system (operating characteristics taken from [4]). I t was found that the number of changes in the fitness value was less and after some generations the sohItion algoiithm saturated, when approaching the global value and took more time to converge. Therefore, it brings into sharper focus. the question of how it will behave for a large- sized system. To improve the search efficiency of the proposed technique, several other genetic operators [I91 havc been tried and the results in Table 1 show that modified genetic operators (MGOs) work well in the GMS problem. Since all units are required to complete main- tenance in their respective 26-week period, before employing MGOs, the chromosome should be divided into 13 genes as the first groiip (comprising units being maintained in the first half of the year) and eight genes as the second group (units allowed to begin maintenance in the second half of the year). In string reversal [19], some portion (randomly generated) of the string is reversed in both groups of the parent. In reciprocal exchange mutation (REM) [19], two positions are selected randomly and swapped with each other.

Table 1: Performance of GA with MGO for 5-unit system

Method !~<900Mwhi Fitness ANOG Time !si value (51

Integer Without 39,99050.009 76.75 307 GA MGO

With 39,99050.009 54.11 216.44 MGO

DA 40,00723.100 - ~

4

Test data was loosely derived from [6] for the cost function and [SI for the reliability function. To implement GA, a population sue (N,) of 50 and roulette wheel type selection were employed. The number of elite individual was kept as 2. Since CA is a randomised procedure, to prove consistency in obtaining optimal solutions, 50 independent

Simulation results for 21-unit test system

57

runs were conducted. All the programs were developed using the MATLAB 6 software package and each problem was simulated on a Pentium 111 700MHz computer.

4.1 Analysis of results For the cost objective function, constraints (3) and (4) are taken. For implementation of the reliability objective function, constraints (3), (4) and (6) with and without the crew constraint ( 5 ) are considered. Crossover rate ( P J and mutation rate (P,) are set to 0.8 and 0.015, respectively. Table 2 summarises the results obtained with the GA for the cost function, and compares the results given in [6] using DP. The EEPC obtained using DP is more than the cost found using CA. Also, it is evident that only integer and real GA are able to find the optimum cost, because of the great reduction in search space. From Table 3, it is obvious that, the levelised reserve method is not suitable for finding an optimal solution. From Tables 2 and 3 it is found that

Table 2 Comparison of GA results for cost function - Solution method EEPC

DP ~52927428.72 Binary for Integer GA 249383830.55 Integer & Real GA 249225195.00

Table 3: Comparisons of results for reliability function

Solution method Min reserve without Min reserve with crew constraint crew constraint

Levelised reserve 4.8419 5.6172 method

Binary for 4.6819 5.0496 integer GA

Integer & Real GA 4.6819 4.9437

Table 4: Performance of GA without MGO

GA using binary for integer coding is unable to reach a global value because of the reduced probability of obtaining the optimum (I/2'Os) and the tendency to create infeasible offspring, after applying genetic operators. Without MGO, 50 independent runs were conducted for both functions and the performance of the coding algorithms compared with respect to their solution accuracy, avcrage number of generations (ANOG), and mean computation time to reach the optimum (depicted in Table 4). Integer GA was able to converge faster than other coding methods. For the reliability function without the crew constraint, convergence speed was faster without MGO. Consequently MGO was implemented only for the cost function and crew constraint cases to improve search efficiency.

4.2 Effect of MGO The string reversal rate was set to (l-Pc)> i.e. 0.2. The number of parents selected to undergo mutation was tnn= Np*Pn~*n. The sharing between mutation and REM of m n parents was decided by a factor called the SDF (sharing decision factor) (Fig. I). The performance of MGO was studied for different SDFs and it was found that can SDF of 50% performed well, as shown in Fig. 2 for the crew constraint case. Fig. 3 and Table 5 demonstrate the improvement in convergence rate of MGO. Table 5 shows that only integer CA is able to converge faster: the other two encoding methods show no improvement in their perfomiance when employing MGOs (Table 6).

5 Conclusions

A non-classical GA was applied using real, integer and binary for integer coding methods. Maintenance schedules were obtained for two different objective functions: cost minimisation and reserve maximisation. This methodology explicitly considers the EENS related to the loss of load probability. To prove the superiority of the optimisation technique applied, two test systems were considered and simulation results of the proposed approach were compared with a solution by conventional methods. The solution algorithm was found to Saturate when approaching the

Objective function Solution Binary for integer GA Integer GA Real GA

Reliability function (without crew constraint) Best

Mea"

Worst

Time 1s)

ANOG

Reliability function (with crew constraint) Best

Mean Worst

Time (h)

ANOG

4.6819 4.6819 4.6819 4.7505 4.7048 4.7444 4.8023 4.7342 4.8233 261.94 117.33 196.51 814 509 960

5.0496 4.9437 4.9437 5.3841 4.9870 5.1606 5.5295 5.1606 5.3002 2.1499 1.228 1.543 17,834 12.534 14,235

Cost function Best 249m830.55 249225195.00 249225195.00 Mean 249422932.40 249226374.20 249326374.20 worst 249501 136.21 249236987.96 249296987.96 Time (h) 20.13 14.46 15.86 ANOG 302 248 272

58 IEE Proc.-Gmer, Trunrnr Dbfrib. Vol. 150. No. I . Joniunj ZOO3

6.4

- 8ntegerGA

binary for integer GA 6.0

6.2

S 6.0 Y)

.- - - 5: 5.8 I

d 5.6

5.4

5.2

5.8

4 5.6

YI

0

- El 3 5.4 D

5.2

5.0

4.8

4 R

-

-

-

-

~

-

6.6

6.4 ' INT GA with MGO ! INT GA without MGO

5.0 5.2 F

5.0 1 4.8 I I I

101 102 1 o3 number of generations

&ff?r o / S D F on pegbrmo,ice of' CA Fig. 2

Table 5 Performance of GA with MGO

4.8 I ' ' ' ' " , , ' ' ' . - ' ' ' , , , I L L ' ' " " " ' 10' 1 02 1 o3 i 04 1 os

number of generations

Fig. 3 Perfbniiance of MGO

global solution for reasonably sized systems, due to the larger search space, thus leading to an external execution time to converge to optima. To reduce cornpulation time and to improve search efficiency of the GA. MGO was employed. Although some additional analyses havc to be conducted due to the particular features of the GMS problem, implementation of MGO was shown to be an effective way to improve thc GA computation performance. Of the three coding methods employed. the numerical results show that the integer-coding GA was most satisfactory for the GMS problem, irrespective of the objective function and system size.

6 Acknowledgments

The authors sincerely thank the authorities of Thiagdrajar College of Engineering, Madurai-625 015. Tamilnadu. India for providing necessary facilities to carry out this research.

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Objective function Solution Binan/ for integer GA Integer GA Real GA

Cost function Best 249383830.55 Mean 249422932.40 worst 249501 136.21 Time ih) ~

ANOG

249225195.00 249225195.00 249226374.20 249226374.20

249236987.96 249228934.56 8.16 140

-

Reliability funnion (with crew constraint) Best 5.3408 4.9437 4.9437 Mean 5.3841 4.9870 4.9870 worst 5.5295 5.1606 5.1606

Time is) ~ 667.538 ANOG 2414

~

Table 6 Generator maintenance schedule for 21-unit system using integer GA

cost 13 36 13 13 36 13 13 36 6 13 13 36 13 13 13 13 38 37 38 29 Reliability 14 48 12 18 43 24 21 37 4 23 3 34 10 19 8 25 48 51 52 49 Crew 10 48 22 7 42 19 24 47 17 1 1 6 32 2 13 4 17 31 35 37 38

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