- Email: [email protected]
Reliability based generator maintenance scheduling using hybrid evolutionary approach
Electrical Power and Energy Systems 42 (2012) 434–439
Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Reliability based generator maintenance scheduling using hybrid evolutionary approach Ehsan Reihani a,⇑, Ali Sarikhani b, Moez Davodi a,b, Mehdi Davodi a,b a b
Asrar Institute of Higher Education, Mashhad, Iran Florida International University, Miami, FL 33174, USA
a r t i c l e
i n f o
Article history: Received 16 May 2009 Received in revised form 27 March 2012 Accepted 9 April 2012 Available online 4 June 2012 Keywords: Generator maintenance scheduling Hybrid evolutionary algorithm Extremal optimization Levelised risk method
a b s t r a c t With the growth of electrical energy demand, providing reliable energy without interruption has become very important nowadays. Maintenance scheduling of generating units is one of the crucial factors in delivering reliable electrical energy to the vital industrial and urban loads. As number of generating units and constraints over their operation is increasing, there is growing need for developing new methods for planning optimal outage of generating units for maintenance. This paper presents a hybrid evolutionary algorithm to tackle the reliability based generator maintenance scheduling problem. Uncertainties in the generating units and the load variations are included so that a more realistic scheduling is obtained. Maintenance scheduling problem is a large scale constrained optimization problem with a large number of variables which needs novel methods to cope with it. A new local search method which is derived from Extremal Optimization (EO) and Genetic Algorithm (GA) is presented to tackle the problem. The proposed method can be used as a local optimizer to further improve the potential solutions in the GA. The proposed method, Hill Climbing Technique (HCT), GA and their hybrid approaches are applied to the IEEE Reliability Test System (RTS) and the obtained results are discussed. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Providing electric energy to the consumers without interruption is of high importance in the electricity industry. The maintenance of generating units has a fundamental role in the reliable delivery of the electrical energy. Therefore an accurate maintenance schedule is necessary to ensure the safe and sound operation of generating units. Maintenance scheduling establishes a preventive outage schedule within a specific time horizon in order to optimize a given objective under the specified operating constraints. Maintenance goals can be generally categorized into three groups; technological, power system and external (economic and social) [1]. From the technological point of view basic maintenance goals are increasing the equipment reliability, improving the efficiency of operation and extending the working life of the generating units. From the power system operator’s perspective the maintenance goals are minimizing the fuel consumption and achieving the required power system reliability and from an external standpoint the maintenance goals are minimizing the cost of generated electrical energy, achieving the required security of supplying the consumers with electrical energy and postponing the investment into ⇑ Corresponding author. Tel.: +98 9155214600. E-mail addresses: [email protected] (E. Reihani), asari001@fiu.edu (A. Sarikhani), [email protected] (M. Davodi), [email protected] (M. Davodi). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.04.018
new generation facilities. The constraints can be also places into three groups, time constraints, reserve constraints and resource constraints. Precedence and maintenance window for units are examples of time constraints. Satisfying the load and providing enough reserve are related to the reserve constraint. Available crew and spare parts at each period of time are concerned with the resource constraints. Utilities spend billions of dollars per year for maintenance. The reliability of system operation and production cost is affected by the maintenance outage of generating units. Suboptimal maintenance schedules lead to higher production cost and lower system reliability. Generator maintenance scheduling is so important that other short and long term plannings such as unit commitment, generation dispatch and generation planning are affected by outage timetable of the generating units. There are generally two criteria for maintenance scheduling; reliability and operating cost. The operating cost includes production cost and maintenance cost. The operating cost mainly consists of the fuel cost. As the production cost requires many approximations, operating cost is an insensitive criterion for maintenance scheduling [2]. The power system’s electricity supply reliability is closely related to it capacity reserve which is directly influenced by the generating unit outage. The risk of the system’s supply being not sufficient may be increased during the scheduled maintenance outage. There are some definitions for measuring reliability in power system which can be used as the objective function for maintenance
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E. Reihani et al. / Electrical Power and Energy Systems 42 (2012) 434–439
Nomenclature Cn,t Ce,nt epi i lpi It Lt Le,t N m
capacity of nth unit at time stage t effective load carrying capability of nth unit at time stage t earliest period for maintenance of unit i to begin index of generating units latest period for maintenance of unit i to end set of the units which are allowed to be in maintenance in period t anticipated load demand for stage t equivalent load at time stage t total number of generating units risk characteristic coefficient
scheduling. Loss of load probability and expected energy not supplied are among the definitions. Levelising the reserve in the planning horizon is a reliability objective function which assumes a deterministic objective function and levels the reserve in the scheduling time. This method does not consider the probabilistic nature of generating units and load variations and therefore the levelised risk method is introduced which considers uncertainties. In this method an effective load carry capability of a generating unit is involved in the maintenance scheduling problem instead of a unit rating. Also the daily variations of load are considered and hence the drawback of using the maximum daily load as the representative load is overcome. Levelising the risk can be realized by maximizing the minimum reserve during the time periods in the planning horizon.
Mit ODj Pn(X) qn r(t) t T Tp MSit X
manpower needed by unit i at period t outage duration of unit j cumulative probability of the system outage capacity forced outage rate of nth unit reserve in period t index of periods set of indices of periods in planning horizon number of stages in maintenance scheduling interval 1; if unit i starts maintenance in period t 0; otherwise outage capacity
The power generating unit’s effective load carrying capacity is given in the following equation:
C e ¼ C m lnðp þ qeC=m Þ where p ¼ 1 q
ð4Þ
The next step in the levelised risk method is calculating the equivalent load. The equivalent load is a single load that if happens the number of times in a maintenance scheduling interval produces the same risk as if the varying peak load in the stage. If, for example, the maintenance interval is 1 week in duration then seven daily peaks produce the same interval risk as a single load encounters seven times. The equivalent load Le is obtained by the following relation: Tp X Le ¼ m ln eLj =m
!, Tp
ð5Þ
j¼1
2. Problem formulation The levelised risk method uses a random reliability objective function which levels the risk in the planning horizon. The Loss of Load Probability (LOLP) is a suitable reliability index to evaluate the risk in each period and hence levelising the risk yields the following equation:
LOLP i ¼ LOLP j
j 2 t; t ¼ 1; 2; . . . ; T
ð1Þ
The levelised risk method involves calculating the effective load carry capability of a generating unit and equivalent load in each stage. The first step in calculating these components is building the outage capacity probability table (COPT) .This is a table of capacity levels and the associated probabilities of existence. The cumulative probability of outages can be also calculated which is useful for obtaining the reliability indices. The table can be calculated by the following relations:
Pn ðXÞ ¼ Pn1 ðXÞð1 qn Þ þ Pn1 ðX C n Þqn Pn1 ðX C n Þ ¼ 1 X 6 C n
ð2Þ
In the above relation, Pn(X) is the cumulative probability of the system outage capacity when the new nth unit with capacity of Cn and forced outage rate of qn is added. An important parameter called risk characteristic coefficient m, is introduced which is the corresponding change of generating unit’s outage capacity in megawatts when the system’s risk or P(X) changes by a factor of e. The coefficient m can be calculated by the following relation:
m¼
XB XA ln½PðX A Þ=PðX B Þ
ð3Þ
The parameters XA and XB are chosen in the COPT that their corresponding probabilities are close to 0.1 and 0.0001 respectively.
In the above relation, Tp is the number of stages in maintenance scheduling interval. If an interval is supposed to be 1 week, then there are seven stages in an interval and Tp is equal to 7 and Lj is the peak load in each stage. Leveling the risk can be realized by minimizing the sum of the squares of the reserves in the planning period. The reserve in each interval is an indication of risk and can be obtained by subtracting the available effective load carrying capacity of generating units and the equivalent load in each interval. Two sets are defined here. Firstly let Sit be the set of start time periods such that if the maintenance of unit i starts at period k that unit will be in maintenance at period t, so Sit = {k e Ti:t di + 1 6 k 6 t}. Secondly, let It be the set of units which are allowed to be in maintenance in period t, so It = {i:t e Ti}. Thus the objective function can be stated as follows:
8 0 0 1 12 9 =
t
ð6Þ
it
P In the objective function, the component i2I C e;it is the sum of P P produced power of all units. The term, i2It reprek2Sit MSik C e;ik sents the loss of generation due to pre-scheduled outages. Hence P P P gives the reserve the term, i2I C e;it i2It k2Sit MSik C e;ik Le;t in stage t. The test case study is subject to the following constraint: Resource constraint, which puts a ban on hydro units’ outages at times of agricultural irrigation. In other words, hydro units are not maintained in weeks 10–25. Also, thermal units cannot go under maintenance in hot season namely weeks 23–38. Eq. (6) defines a general mathematical model for the GMS problem formulated as an optimization problem with integer
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E. Reihani et al. / Electrical Power and Energy Systems 42 (2012) 434–439
representation. In this model the objective function is minimized if the reserve is distributed uniformly in the planning horizon. The case study considered is IEEE reliability test system which includes 33 units in a one year horizon [3]. Each unit must be maintained once in a year for a specific duration without interruption considering the constraints imposed on each unit.
integer string with the length of 33 where each string stands for the start week for the outage of that unit. An initial population of 80 individuals is created. The fitness of each individual is calculated using an evaluation function. The evaluation function used in this paper is the weighted sum of the objective function and penalty functions for considering the constraints violation. The objective value can be written as:
3. Solution techniques
Objective value ¼ SSR
Due to the importance of generator maintenance scheduling, GMS problem has been widely studied in the past. Mathematical methods such as integer programming and dynamic programming are applied to the GMS problem. The mathematical methods incur from the curse of dimensionality. These methods cannot also tackle with nonlinearity and undifferentiability of objective functions. Heuristic methods are also suitable for the GMS problem if there is specific information about the problem. A number of meta-heuristic and artificial intelligence based approaches are presented in the literature which do not have the limitations of the above mentioned methods. In [4] Simulated Annealing (SA) is applied to the GMS problem which could address the medium and large scale problems. Genetic Algorithm (GA) with binary representation is applied to the GMS problem. Later, the method was improved by using GA with integer representation which reduced the search space and execution time of the GA. It is shown that Tabu Search (TS) outperforms than GA, SA and TS/ SA hybrid. Hybrid Evolutionary technique method is proposed in [5] where the genetic algorithm works as the evolutionary approach and different local search methods such as Hill Climbing Technique (HCT), TS and SA were used as local search. It is shown that TS as the local optimizer yields the best results in their proposed case study. Also HCT is used in conjunction with the Evolutionary Programming (EP) to find a feasible solution in the neighborhood of the new infeasible solutions. In [6] combination of GA and SA is used where acceptance probability of SA is included in the algorithm as a criterion for the survival of individuals during evolution process. It is shown there that GA/SA is more robust to the parameters compared to GA and SA. Among the tested approaches, the inoculated GA/SA gives the best average performance. In this paper, GA and HCT are applied to the GMS problem. Moreover a novel local search approach which is based on GA and EO is developed and used as a local optimizer to further improve the individuals fitness. 4. Implementation of hybrid evolutionary algorithms 4.1. Genetic algorithm Genetic algorithm is an iterative optimization method based on natural evolution [7]. In this method, an initial population of candidate solutions, called individuals, is randomly created. In each iteration, a number of genetic operators, derived from those found in natural evolution, are applied to the individuals and new individuals are generated. Thus, the genetic components change through generations and so explore the search/solution space. A selection scheme is applied to the new individuals and the fittest members pass through to future iterations. This process is continued until the termination criterion is satisfied. This criterion could be either the maximum number of iterations or when it appears that the value of the objective function is stable. 4.1.1. GA implementation In this paper a software package which includes the genetic algorithm routines is used [8]. An individual is represented by an
ð7Þ
In the above relation SSR stands for the sum of squares of the reserve in the planning horizon. The maintenance window constraint is embedded in the representation of an individual. If no constraint is considered, the reserve is distributed uniformly in the planning horizon and gives the lowest band to the SSR. The reserve in this state is called the ideal reserve which is calculated by the following relation for a one year horizon:
P P P 52 j C e;j j C e;j ODj j Le Ideal reserve ¼ 52
ð8Þ
4.2. Hill climbing technique Hill climbing is an optimization algorithm which looks for the better solutions in the neighborhood of the current solution. It is usually used as a local search which improves the individuals. Definition of the neighbor is problem specific. In our test case study, the start week of each unit in an individual is incremented and a new individual is created. The best member in the neighboring individuals is selected. HCT or any other local search can be applied to the individual in each locating of GA (i.e. initial population, mating pool, offspring created by crossover operator and offspring created by mutation). 4.3. Extremal optimization Extremal optimization is a local search heuristic method which is proposed recently [9,10]. In this method an individual is divided into components and a fitness value is allocated to each component. Then the component with the worst fitness is found and a random value is allocated to that component. This process is continued until the fitness of the individual is improved. In our test case study, a component stands for the maintenance start week of a unit. Therefore there are 33 components in each individual. Fitness of component should be defined in a way that improvement in the component fitness leads to the improvement in the individual fitness. If the reserve in each period is close to the ideal reserve, the reserve is distributed uniformly in the planning horizon and hence the individual fitness has the highest value. Therefore an individual has a high fitness value if the outage weeks of the units are arranged such that the reserve in each week is close to the ideal reserve. So the proximity to the ideal reserve can be defined as the component objective function. This proximity is defined as the standard deviation from the ideal reserve in outage duration of each unit. The component objective value can be written as follows:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uPcðiÞþodðiÞ1 u ðrðtÞ ideal reserveÞ2 t t¼cðiÞ odðiÞ
ð9Þ
The first term in the above relation represents the standard deviation of the reserve in outage duration of each unit from the ideal reserve. The term c(i) represents the start week for maintenance of unit i. The second term is added to consider total component constraint violation in outage duration of the component. If the maintenance start week of a unit in an individual is such that the reserve in the outage duration of that unit is so far from the
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ideal reserve, that component has a high participation in the in lowness of the individual fitness. Therefore, correcting the maintenance start week of that unit has a high effect on the individual fitness. 4.4. The proposed approach (EO/GA) In the EO method, a random value is allocated to the component with the lowest fitness value. There is no consideration to the outcome of such a move. If this move is done intelligently, then the fitness value of the individual improves greatly. Moreover if the combination of the maintenance start weeks of several units is improved, this combination has a higher effect on the individual fitness. Due to high number of different permutations of several components, population based algorithms are the right choice for
Select an individual
Divide the individual into components Find all the components fitness values
C1
C2
C3
…
Cn
C1 + 1
C2 + 1
C3 + 1
…
Cn + 1
C1 + 2
C2 + 2
C3 + 2
…
Cn + 2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C1 − 1
C2 − 1
C3 − 1
…
Cn − 1
Fig. 2. Members of neighbor pool for an individual.
finding the best result. As discussed, GA is one the best population based search algorithms with easy implementation. Therefore GA is used to improve a number of components in an individual. The GA operators work only on the selected components. For example consider an individual with n components where m components are selected to be improved by GA. The selected components are changed in the initial population and other components are left unchanged. By using this method, the selected components with the least fitness values are improved, hence improving the individual fitness greatly. The proposed method algorithm is shown in Fig. 1. 5. Discussion and results
Select some of the components with the lowest fitness values
5.1. HCT and EO/GA applications In this section, application of HCT and EO/GA are discussed. For HCT, a random individual is created and a pool of individuals is
Create an initial population in which only the selected components are changed
Select some of the individuals for breeding
Perform crossover and mutation operators only on the selected components
Select some of the best individuals for replacement
NO
Termination criterion satisfied?
Yes Select the individual with the best fitness value Fig. 1. The proposed EO/GA algorithm.
Table 1 Maintenance schedule for the best result. Unit no.
Outage week
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
1 21 17 20 1 43 22 8 17 3 15 39 9 14 43 9 40 4 45 24 6 10 34 31 31 28 27 27 37 26 40 40 11
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E. Reihani et al. / Electrical Power and Energy Systems 42 (2012) 434–439
600
Reserve (MW)
500 400 300 200 100 5
10
15
20
25
30
35
40
45
50
Week Fig. 3. Reserve (MW) in each week for the best maintenance schedule.
6
Average of objective function
7.8
x 10
7.75 7.7 7.65 7.6 7.55 7.5 7.45 7.4
HCTM
EOGAM
HCTIP
EOGAIP
Search Algorithms Fig. 4. Comparison of different search methods in different locations.
produced from the random individual. The components of the neighbor individuals are additions of the main components. For example, suppose an individual in the first row of Fig. 2. Each row is produced by incrementing each component of the above row and this process is continued until each component value takes each week of a year and therefore there are 52 members in a neighbor pool. Then, a member with the highest fitness value is selected and mutated to create a new individual. The same process is repeated for the new individual. There are 50 iterations for improvement of an individual in our simulation. The average value of objective functions over 10 runs is 83,145,754. In the next simulation, EO/GA is applied to the GMS problem. In this method, a random individual is created and half of the components with least fitness values are selected for optimization in a genetic algorithm. The component fitness values of the final result of GA are calculated and optimization of the worst half of components is repeated. This process is continued for 50 iterations. The mean of objective values over 10 runs is 8,763,486 which show a great improvement over single HCT. 5.2. HCT and EO/GA as a local optimizer The GA with different local search methods is applied to the GMS problem. The local search methods are used to improve the individual fitness of GA. The hill climbing technique is used in the initial population to prepare an optimized population of individuals. It is also applied on the best member of offspring population to improve the individual with the best fitness for replacement with the worst individual of previous population, the replacement strategy called elitism. In the GA algorithm, the crossover rate is 0.7 and one individual is selected for mutation and the algorithm is run for 50 iterations. In the HCT algorithm, a total number of
52 members are defined as the neighbor for each individual. Each component of an individual pool in the neighbor is created by incrementing the previous component. The member in the neighbor pool with the highest fitness is used to replace the main individual. The mean of final results of objective function values for 10 runs is 7,783,191 for GA. If the initial population is optimized by HCT, the mean of objective functions over 10 runs improve to 7,577,043. Thus application of HCT in the initial population improves the mean of objective values by 4.2%. Combination of EO and GA or EO/GA is also applied to the individuals in the initial population. The parameters of GA in EO/GA are as the same as the main GA and half of the components are selected for optimization in EO/GA. The mean of objective function values for ten runs is 7,442,926 which is 4.5% better than GA and 1.8% better than HCT. Application of the above local search methods on the individual selected for next iterations gives almost the same result. In order to investigate the capability of the EO/GA algorithm, this algorithm and HCT have been simultaneously applied both to the initial population and offspring created by mutation operator. The simulations are run for 100 iterations. The obtained results are 7,293,550 and 7,538,200 for EO/GA and HCT respectively. It can be easily seen that EO/GA outperforms HCT by 3.2%. The best result out of the simulated methods is obtained by applying EO/GA both in initial population and offspring after the mutation operator. The maintenance schedule and the reserve for this schedule for this result are given in Table 1 and Fig. 3 respectively. 5.3. Comparisons of search algorithms Application of search algorithms is summarized in Fig. 4. In this figure, the performance above discussed search methods is depicted. Labels HCTM and EOGAM, stand for application of HCT
E. Reihani et al. / Electrical Power and Energy Systems 42 (2012) 434–439 Table 2 Execution time for different algorithms. Algorithm
Execution time (min)
HCTIP HCTM HCT HCTIPM GA EOGA EOGAM EOGAIP EOGAIPM
0.37 0.43 5.9 55 0.18 6 9.8 15.6 1512
and EOGA on the best offspring after the mutation operator respectively. HCTIP and EOGAIP refer to application of HCT and EOGA on all individuals in the initial population. It can be easily seen from figure that EOGA outperforms HCT in initial population and EOGA is slightly better than HCT when applied to solely one individual. When applied on all offspring individuals EOGA proves that it superior to HCT by 3.2%. The algorithm time executions are shown in Table 2. Application of EO/GA algorithm on all individuals both in initial population and offspring takes the longest time as this algorithm is iterated in the main GA. If execution time is not an important factor, the proposed algorithm can be used with much better results than other algorithms. 6. Conclusion In this paper a hybrid evolutionary algorithm is applied to the Generator Maintenance Scheduling (GMS) problem. Genetic algorithm is selected as the evolutionary algorithm and several local search methods are used to improve the individuals. A new local search method which is derived from EO and GA is developed and applied to the GMS problem. The new local search method concentrates on the individual unit schedules and improves them. This is done by allocating a fitness value to each unit and improving the fitness by using another genetic algorithm. The proposed
439
method and hill climbing technique and their applications as local optimizers are applied to the GMS problem in different locations of the GA. The locations are in the initial population and the offspring created by the mutation operator. For application of each local search method, 10 runs are executed. The results also show that the proposed method is superior to HCT. Also when used as local search method, EOGA yields better results than HCT in both locations. Acknowledgment The authors would like to thank Dr.Oloomi and Dr.Banejad for their invaluable contributions at Shahrood University of Technology. References [1] Endrenyi J et al. The present status of maintenance strategies and the impact of maintenance on reliability. IEEE Trans Power Syst 2001;16(4):638–46. [2] Wang Y, Handschin E. A new genetic algorithm for preventive unit maintenance scheduling of power systems. Int J Electr Power Energy Syst 2000;22(5):343–8. [3] IEEE Committee report, IEEE reliability test system, IEEE transactions on power apparatus and systems, vol. PAS-98, no. 6; November/December 1979. p. 2047–54. [4] A simulated annealing based approach to solve the generator maintenance scheduling problem. In: Electric power system research, vol. 81(7); July 2011, p. 1283–91. [5] El-Sharkh MY, El-Keib AA. An evolutionary programming-based solution methodology for power system generation and transmission maintenance scheduling. Electr Power Syst Res 2003;65(1):35–40. [6] Dahal KP, Nopasit Chakpitak. Generator maintenance scheduling in power systems using metaheuristic-based hybrid approaches. In: Electric power systems research; 2007. p. 771–9. [7] Hart William E, Krasnogor N, Smith JE, editors. Recent advances in memetic algorithms, vol. 166. Springer; 2005. [8] Andrew Chipperfield, Peter Fleming, Hartmut Pohlheim, Carlos Fonseca. Genetic algorithm toolbox for use with MATLABÒ, version 1.2, Department of Automatic Control and System Engineering, University of Sheffield; 1997. [9] Boettcher S, Percus AG. Nature’s way of optimizing. Artif Intell 2000;119: 275–286. [10] Boettcher S, Percus AG. Extremal optimization: an evolutionary local search algorithm. In: Proceedings of the 8th INFORMS computing society conference; 2003.
Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Reliability based generator maintenance scheduling using hybrid evolutionary approach Ehsan Reihani a,⇑, Ali Sarikhani b, Moez Davodi a,b, Mehdi Davodi a,b a b
Asrar Institute of Higher Education, Mashhad, Iran Florida International University, Miami, FL 33174, USA
a r t i c l e
i n f o
Article history: Received 16 May 2009 Received in revised form 27 March 2012 Accepted 9 April 2012 Available online 4 June 2012 Keywords: Generator maintenance scheduling Hybrid evolutionary algorithm Extremal optimization Levelised risk method
a b s t r a c t With the growth of electrical energy demand, providing reliable energy without interruption has become very important nowadays. Maintenance scheduling of generating units is one of the crucial factors in delivering reliable electrical energy to the vital industrial and urban loads. As number of generating units and constraints over their operation is increasing, there is growing need for developing new methods for planning optimal outage of generating units for maintenance. This paper presents a hybrid evolutionary algorithm to tackle the reliability based generator maintenance scheduling problem. Uncertainties in the generating units and the load variations are included so that a more realistic scheduling is obtained. Maintenance scheduling problem is a large scale constrained optimization problem with a large number of variables which needs novel methods to cope with it. A new local search method which is derived from Extremal Optimization (EO) and Genetic Algorithm (GA) is presented to tackle the problem. The proposed method can be used as a local optimizer to further improve the potential solutions in the GA. The proposed method, Hill Climbing Technique (HCT), GA and their hybrid approaches are applied to the IEEE Reliability Test System (RTS) and the obtained results are discussed. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Providing electric energy to the consumers without interruption is of high importance in the electricity industry. The maintenance of generating units has a fundamental role in the reliable delivery of the electrical energy. Therefore an accurate maintenance schedule is necessary to ensure the safe and sound operation of generating units. Maintenance scheduling establishes a preventive outage schedule within a specific time horizon in order to optimize a given objective under the specified operating constraints. Maintenance goals can be generally categorized into three groups; technological, power system and external (economic and social) [1]. From the technological point of view basic maintenance goals are increasing the equipment reliability, improving the efficiency of operation and extending the working life of the generating units. From the power system operator’s perspective the maintenance goals are minimizing the fuel consumption and achieving the required power system reliability and from an external standpoint the maintenance goals are minimizing the cost of generated electrical energy, achieving the required security of supplying the consumers with electrical energy and postponing the investment into ⇑ Corresponding author. Tel.: +98 9155214600. E-mail addresses: [email protected] (E. Reihani), asari001@fiu.edu (A. Sarikhani), [email protected] (M. Davodi), [email protected] (M. Davodi). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.04.018
new generation facilities. The constraints can be also places into three groups, time constraints, reserve constraints and resource constraints. Precedence and maintenance window for units are examples of time constraints. Satisfying the load and providing enough reserve are related to the reserve constraint. Available crew and spare parts at each period of time are concerned with the resource constraints. Utilities spend billions of dollars per year for maintenance. The reliability of system operation and production cost is affected by the maintenance outage of generating units. Suboptimal maintenance schedules lead to higher production cost and lower system reliability. Generator maintenance scheduling is so important that other short and long term plannings such as unit commitment, generation dispatch and generation planning are affected by outage timetable of the generating units. There are generally two criteria for maintenance scheduling; reliability and operating cost. The operating cost includes production cost and maintenance cost. The operating cost mainly consists of the fuel cost. As the production cost requires many approximations, operating cost is an insensitive criterion for maintenance scheduling [2]. The power system’s electricity supply reliability is closely related to it capacity reserve which is directly influenced by the generating unit outage. The risk of the system’s supply being not sufficient may be increased during the scheduled maintenance outage. There are some definitions for measuring reliability in power system which can be used as the objective function for maintenance
435
E. Reihani et al. / Electrical Power and Energy Systems 42 (2012) 434–439
Nomenclature Cn,t Ce,nt epi i lpi It Lt Le,t N m
capacity of nth unit at time stage t effective load carrying capability of nth unit at time stage t earliest period for maintenance of unit i to begin index of generating units latest period for maintenance of unit i to end set of the units which are allowed to be in maintenance in period t anticipated load demand for stage t equivalent load at time stage t total number of generating units risk characteristic coefficient
scheduling. Loss of load probability and expected energy not supplied are among the definitions. Levelising the reserve in the planning horizon is a reliability objective function which assumes a deterministic objective function and levels the reserve in the scheduling time. This method does not consider the probabilistic nature of generating units and load variations and therefore the levelised risk method is introduced which considers uncertainties. In this method an effective load carry capability of a generating unit is involved in the maintenance scheduling problem instead of a unit rating. Also the daily variations of load are considered and hence the drawback of using the maximum daily load as the representative load is overcome. Levelising the risk can be realized by maximizing the minimum reserve during the time periods in the planning horizon.
Mit ODj Pn(X) qn r(t) t T Tp MSit X
manpower needed by unit i at period t outage duration of unit j cumulative probability of the system outage capacity forced outage rate of nth unit reserve in period t index of periods set of indices of periods in planning horizon number of stages in maintenance scheduling interval 1; if unit i starts maintenance in period t 0; otherwise outage capacity
The power generating unit’s effective load carrying capacity is given in the following equation:
C e ¼ C m lnðp þ qeC=m Þ where p ¼ 1 q
ð4Þ
The next step in the levelised risk method is calculating the equivalent load. The equivalent load is a single load that if happens the number of times in a maintenance scheduling interval produces the same risk as if the varying peak load in the stage. If, for example, the maintenance interval is 1 week in duration then seven daily peaks produce the same interval risk as a single load encounters seven times. The equivalent load Le is obtained by the following relation: Tp X Le ¼ m ln eLj =m
!, Tp
ð5Þ
j¼1
2. Problem formulation The levelised risk method uses a random reliability objective function which levels the risk in the planning horizon. The Loss of Load Probability (LOLP) is a suitable reliability index to evaluate the risk in each period and hence levelising the risk yields the following equation:
LOLP i ¼ LOLP j
j 2 t; t ¼ 1; 2; . . . ; T
ð1Þ
The levelised risk method involves calculating the effective load carry capability of a generating unit and equivalent load in each stage. The first step in calculating these components is building the outage capacity probability table (COPT) .This is a table of capacity levels and the associated probabilities of existence. The cumulative probability of outages can be also calculated which is useful for obtaining the reliability indices. The table can be calculated by the following relations:
Pn ðXÞ ¼ Pn1 ðXÞð1 qn Þ þ Pn1 ðX C n Þqn Pn1 ðX C n Þ ¼ 1 X 6 C n
ð2Þ
In the above relation, Pn(X) is the cumulative probability of the system outage capacity when the new nth unit with capacity of Cn and forced outage rate of qn is added. An important parameter called risk characteristic coefficient m, is introduced which is the corresponding change of generating unit’s outage capacity in megawatts when the system’s risk or P(X) changes by a factor of e. The coefficient m can be calculated by the following relation:
m¼
XB XA ln½PðX A Þ=PðX B Þ
ð3Þ
The parameters XA and XB are chosen in the COPT that their corresponding probabilities are close to 0.1 and 0.0001 respectively.
In the above relation, Tp is the number of stages in maintenance scheduling interval. If an interval is supposed to be 1 week, then there are seven stages in an interval and Tp is equal to 7 and Lj is the peak load in each stage. Leveling the risk can be realized by minimizing the sum of the squares of the reserves in the planning period. The reserve in each interval is an indication of risk and can be obtained by subtracting the available effective load carrying capacity of generating units and the equivalent load in each interval. Two sets are defined here. Firstly let Sit be the set of start time periods such that if the maintenance of unit i starts at period k that unit will be in maintenance at period t, so Sit = {k e Ti:t di + 1 6 k 6 t}. Secondly, let It be the set of units which are allowed to be in maintenance in period t, so It = {i:t e Ti}. Thus the objective function can be stated as follows:
8 0 0 1 12 9 =
t
ð6Þ
it
P In the objective function, the component i2I C e;it is the sum of P P produced power of all units. The term, i2It reprek2Sit MSik C e;ik sents the loss of generation due to pre-scheduled outages. Hence P P P gives the reserve the term, i2I C e;it i2It k2Sit MSik C e;ik Le;t in stage t. The test case study is subject to the following constraint: Resource constraint, which puts a ban on hydro units’ outages at times of agricultural irrigation. In other words, hydro units are not maintained in weeks 10–25. Also, thermal units cannot go under maintenance in hot season namely weeks 23–38. Eq. (6) defines a general mathematical model for the GMS problem formulated as an optimization problem with integer
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representation. In this model the objective function is minimized if the reserve is distributed uniformly in the planning horizon. The case study considered is IEEE reliability test system which includes 33 units in a one year horizon [3]. Each unit must be maintained once in a year for a specific duration without interruption considering the constraints imposed on each unit.
integer string with the length of 33 where each string stands for the start week for the outage of that unit. An initial population of 80 individuals is created. The fitness of each individual is calculated using an evaluation function. The evaluation function used in this paper is the weighted sum of the objective function and penalty functions for considering the constraints violation. The objective value can be written as:
3. Solution techniques
Objective value ¼ SSR
Due to the importance of generator maintenance scheduling, GMS problem has been widely studied in the past. Mathematical methods such as integer programming and dynamic programming are applied to the GMS problem. The mathematical methods incur from the curse of dimensionality. These methods cannot also tackle with nonlinearity and undifferentiability of objective functions. Heuristic methods are also suitable for the GMS problem if there is specific information about the problem. A number of meta-heuristic and artificial intelligence based approaches are presented in the literature which do not have the limitations of the above mentioned methods. In [4] Simulated Annealing (SA) is applied to the GMS problem which could address the medium and large scale problems. Genetic Algorithm (GA) with binary representation is applied to the GMS problem. Later, the method was improved by using GA with integer representation which reduced the search space and execution time of the GA. It is shown that Tabu Search (TS) outperforms than GA, SA and TS/ SA hybrid. Hybrid Evolutionary technique method is proposed in [5] where the genetic algorithm works as the evolutionary approach and different local search methods such as Hill Climbing Technique (HCT), TS and SA were used as local search. It is shown that TS as the local optimizer yields the best results in their proposed case study. Also HCT is used in conjunction with the Evolutionary Programming (EP) to find a feasible solution in the neighborhood of the new infeasible solutions. In [6] combination of GA and SA is used where acceptance probability of SA is included in the algorithm as a criterion for the survival of individuals during evolution process. It is shown there that GA/SA is more robust to the parameters compared to GA and SA. Among the tested approaches, the inoculated GA/SA gives the best average performance. In this paper, GA and HCT are applied to the GMS problem. Moreover a novel local search approach which is based on GA and EO is developed and used as a local optimizer to further improve the individuals fitness. 4. Implementation of hybrid evolutionary algorithms 4.1. Genetic algorithm Genetic algorithm is an iterative optimization method based on natural evolution [7]. In this method, an initial population of candidate solutions, called individuals, is randomly created. In each iteration, a number of genetic operators, derived from those found in natural evolution, are applied to the individuals and new individuals are generated. Thus, the genetic components change through generations and so explore the search/solution space. A selection scheme is applied to the new individuals and the fittest members pass through to future iterations. This process is continued until the termination criterion is satisfied. This criterion could be either the maximum number of iterations or when it appears that the value of the objective function is stable. 4.1.1. GA implementation In this paper a software package which includes the genetic algorithm routines is used [8]. An individual is represented by an
ð7Þ
In the above relation SSR stands for the sum of squares of the reserve in the planning horizon. The maintenance window constraint is embedded in the representation of an individual. If no constraint is considered, the reserve is distributed uniformly in the planning horizon and gives the lowest band to the SSR. The reserve in this state is called the ideal reserve which is calculated by the following relation for a one year horizon:
P P P 52 j C e;j j C e;j ODj j Le Ideal reserve ¼ 52
ð8Þ
4.2. Hill climbing technique Hill climbing is an optimization algorithm which looks for the better solutions in the neighborhood of the current solution. It is usually used as a local search which improves the individuals. Definition of the neighbor is problem specific. In our test case study, the start week of each unit in an individual is incremented and a new individual is created. The best member in the neighboring individuals is selected. HCT or any other local search can be applied to the individual in each locating of GA (i.e. initial population, mating pool, offspring created by crossover operator and offspring created by mutation). 4.3. Extremal optimization Extremal optimization is a local search heuristic method which is proposed recently [9,10]. In this method an individual is divided into components and a fitness value is allocated to each component. Then the component with the worst fitness is found and a random value is allocated to that component. This process is continued until the fitness of the individual is improved. In our test case study, a component stands for the maintenance start week of a unit. Therefore there are 33 components in each individual. Fitness of component should be defined in a way that improvement in the component fitness leads to the improvement in the individual fitness. If the reserve in each period is close to the ideal reserve, the reserve is distributed uniformly in the planning horizon and hence the individual fitness has the highest value. Therefore an individual has a high fitness value if the outage weeks of the units are arranged such that the reserve in each week is close to the ideal reserve. So the proximity to the ideal reserve can be defined as the component objective function. This proximity is defined as the standard deviation from the ideal reserve in outage duration of each unit. The component objective value can be written as follows:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uPcðiÞþodðiÞ1 u ðrðtÞ ideal reserveÞ2 t t¼cðiÞ odðiÞ
ð9Þ
The first term in the above relation represents the standard deviation of the reserve in outage duration of each unit from the ideal reserve. The term c(i) represents the start week for maintenance of unit i. The second term is added to consider total component constraint violation in outage duration of the component. If the maintenance start week of a unit in an individual is such that the reserve in the outage duration of that unit is so far from the
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ideal reserve, that component has a high participation in the in lowness of the individual fitness. Therefore, correcting the maintenance start week of that unit has a high effect on the individual fitness. 4.4. The proposed approach (EO/GA) In the EO method, a random value is allocated to the component with the lowest fitness value. There is no consideration to the outcome of such a move. If this move is done intelligently, then the fitness value of the individual improves greatly. Moreover if the combination of the maintenance start weeks of several units is improved, this combination has a higher effect on the individual fitness. Due to high number of different permutations of several components, population based algorithms are the right choice for
Select an individual
Divide the individual into components Find all the components fitness values
C1
C2
C3
…
Cn
C1 + 1
C2 + 1
C3 + 1
…
Cn + 1
C1 + 2
C2 + 2
C3 + 2
…
Cn + 2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C1 − 1
C2 − 1
C3 − 1
…
Cn − 1
Fig. 2. Members of neighbor pool for an individual.
finding the best result. As discussed, GA is one the best population based search algorithms with easy implementation. Therefore GA is used to improve a number of components in an individual. The GA operators work only on the selected components. For example consider an individual with n components where m components are selected to be improved by GA. The selected components are changed in the initial population and other components are left unchanged. By using this method, the selected components with the least fitness values are improved, hence improving the individual fitness greatly. The proposed method algorithm is shown in Fig. 1. 5. Discussion and results
Select some of the components with the lowest fitness values
5.1. HCT and EO/GA applications In this section, application of HCT and EO/GA are discussed. For HCT, a random individual is created and a pool of individuals is
Create an initial population in which only the selected components are changed
Select some of the individuals for breeding
Perform crossover and mutation operators only on the selected components
Select some of the best individuals for replacement
NO
Termination criterion satisfied?
Yes Select the individual with the best fitness value Fig. 1. The proposed EO/GA algorithm.
Table 1 Maintenance schedule for the best result. Unit no.
Outage week
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
1 21 17 20 1 43 22 8 17 3 15 39 9 14 43 9 40 4 45 24 6 10 34 31 31 28 27 27 37 26 40 40 11
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600
Reserve (MW)
500 400 300 200 100 5
10
15
20
25
30
35
40
45
50
Week Fig. 3. Reserve (MW) in each week for the best maintenance schedule.
6
Average of objective function
7.8
x 10
7.75 7.7 7.65 7.6 7.55 7.5 7.45 7.4
HCTM
EOGAM
HCTIP
EOGAIP
Search Algorithms Fig. 4. Comparison of different search methods in different locations.
produced from the random individual. The components of the neighbor individuals are additions of the main components. For example, suppose an individual in the first row of Fig. 2. Each row is produced by incrementing each component of the above row and this process is continued until each component value takes each week of a year and therefore there are 52 members in a neighbor pool. Then, a member with the highest fitness value is selected and mutated to create a new individual. The same process is repeated for the new individual. There are 50 iterations for improvement of an individual in our simulation. The average value of objective functions over 10 runs is 83,145,754. In the next simulation, EO/GA is applied to the GMS problem. In this method, a random individual is created and half of the components with least fitness values are selected for optimization in a genetic algorithm. The component fitness values of the final result of GA are calculated and optimization of the worst half of components is repeated. This process is continued for 50 iterations. The mean of objective values over 10 runs is 8,763,486 which show a great improvement over single HCT. 5.2. HCT and EO/GA as a local optimizer The GA with different local search methods is applied to the GMS problem. The local search methods are used to improve the individual fitness of GA. The hill climbing technique is used in the initial population to prepare an optimized population of individuals. It is also applied on the best member of offspring population to improve the individual with the best fitness for replacement with the worst individual of previous population, the replacement strategy called elitism. In the GA algorithm, the crossover rate is 0.7 and one individual is selected for mutation and the algorithm is run for 50 iterations. In the HCT algorithm, a total number of
52 members are defined as the neighbor for each individual. Each component of an individual pool in the neighbor is created by incrementing the previous component. The member in the neighbor pool with the highest fitness is used to replace the main individual. The mean of final results of objective function values for 10 runs is 7,783,191 for GA. If the initial population is optimized by HCT, the mean of objective functions over 10 runs improve to 7,577,043. Thus application of HCT in the initial population improves the mean of objective values by 4.2%. Combination of EO and GA or EO/GA is also applied to the individuals in the initial population. The parameters of GA in EO/GA are as the same as the main GA and half of the components are selected for optimization in EO/GA. The mean of objective function values for ten runs is 7,442,926 which is 4.5% better than GA and 1.8% better than HCT. Application of the above local search methods on the individual selected for next iterations gives almost the same result. In order to investigate the capability of the EO/GA algorithm, this algorithm and HCT have been simultaneously applied both to the initial population and offspring created by mutation operator. The simulations are run for 100 iterations. The obtained results are 7,293,550 and 7,538,200 for EO/GA and HCT respectively. It can be easily seen that EO/GA outperforms HCT by 3.2%. The best result out of the simulated methods is obtained by applying EO/GA both in initial population and offspring after the mutation operator. The maintenance schedule and the reserve for this schedule for this result are given in Table 1 and Fig. 3 respectively. 5.3. Comparisons of search algorithms Application of search algorithms is summarized in Fig. 4. In this figure, the performance above discussed search methods is depicted. Labels HCTM and EOGAM, stand for application of HCT
E. Reihani et al. / Electrical Power and Energy Systems 42 (2012) 434–439 Table 2 Execution time for different algorithms. Algorithm
Execution time (min)
HCTIP HCTM HCT HCTIPM GA EOGA EOGAM EOGAIP EOGAIPM
0.37 0.43 5.9 55 0.18 6 9.8 15.6 1512
and EOGA on the best offspring after the mutation operator respectively. HCTIP and EOGAIP refer to application of HCT and EOGA on all individuals in the initial population. It can be easily seen from figure that EOGA outperforms HCT in initial population and EOGA is slightly better than HCT when applied to solely one individual. When applied on all offspring individuals EOGA proves that it superior to HCT by 3.2%. The algorithm time executions are shown in Table 2. Application of EO/GA algorithm on all individuals both in initial population and offspring takes the longest time as this algorithm is iterated in the main GA. If execution time is not an important factor, the proposed algorithm can be used with much better results than other algorithms. 6. Conclusion In this paper a hybrid evolutionary algorithm is applied to the Generator Maintenance Scheduling (GMS) problem. Genetic algorithm is selected as the evolutionary algorithm and several local search methods are used to improve the individuals. A new local search method which is derived from EO and GA is developed and applied to the GMS problem. The new local search method concentrates on the individual unit schedules and improves them. This is done by allocating a fitness value to each unit and improving the fitness by using another genetic algorithm. The proposed
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method and hill climbing technique and their applications as local optimizers are applied to the GMS problem in different locations of the GA. The locations are in the initial population and the offspring created by the mutation operator. For application of each local search method, 10 runs are executed. The results also show that the proposed method is superior to HCT. Also when used as local search method, EOGA yields better results than HCT in both locations. Acknowledgment The authors would like to thank Dr.Oloomi and Dr.Banejad for their invaluable contributions at Shahrood University of Technology. References [1] Endrenyi J et al. The present status of maintenance strategies and the impact of maintenance on reliability. IEEE Trans Power Syst 2001;16(4):638–46. [2] Wang Y, Handschin E. A new genetic algorithm for preventive unit maintenance scheduling of power systems. Int J Electr Power Energy Syst 2000;22(5):343–8. [3] IEEE Committee report, IEEE reliability test system, IEEE transactions on power apparatus and systems, vol. PAS-98, no. 6; November/December 1979. p. 2047–54. [4] A simulated annealing based approach to solve the generator maintenance scheduling problem. In: Electric power system research, vol. 81(7); July 2011, p. 1283–91. [5] El-Sharkh MY, El-Keib AA. An evolutionary programming-based solution methodology for power system generation and transmission maintenance scheduling. Electr Power Syst Res 2003;65(1):35–40. [6] Dahal KP, Nopasit Chakpitak. Generator maintenance scheduling in power systems using metaheuristic-based hybrid approaches. In: Electric power systems research; 2007. p. 771–9. [7] Hart William E, Krasnogor N, Smith JE, editors. Recent advances in memetic algorithms, vol. 166. Springer; 2005. [8] Andrew Chipperfield, Peter Fleming, Hartmut Pohlheim, Carlos Fonseca. Genetic algorithm toolbox for use with MATLABÒ, version 1.2, Department of Automatic Control and System Engineering, University of Sheffield; 1997. [9] Boettcher S, Percus AG. Nature’s way of optimizing. Artif Intell 2000;119: 275–286. [10] Boettcher S, Percus AG. Extremal optimization: an evolutionary local search algorithm. In: Proceedings of the 8th INFORMS computing society conference; 2003.