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Genetic algorithm optimisation of the maintenance scheduling of generating units in a power system
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Reliability Engineering and System Safety 93 (2008) 757–767 www.elsevier.com/locate/ress
Genetic algorithm optimisation of the maintenance scheduling of generating units in a power system Andrija Volkanovskia,, Borut Mavkoa, Tome Bosˇ evskib, Anton Cˇausˇ evskib, Marko Cˇepina a
Reactor Engineering Division, Jozˇef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia Faculty of Electrical Engineering, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia
b
Received 12 December 2006; received in revised form 15 March 2007; accepted 18 March 2007 Available online 24 March 2007
Abstract A new method for optimisation of the maintenance scheduling of generating units in a power system is developed. Maintenance is scheduled to minimise the risk through minimisation of the yearly value of the loss of load expectation (LOLE) taken as a measure of the power system reliability. The proposed method uses genetic algorithm to obtain the best solution resulting in a minimal value of the annual LOLE value for the power system in the analysed period. The operational constraints for generating units are included in the method. The proposed algorithm was tested on a Macedonian power system and the obtained results were compared with the results received from the approximate methodology. The results show the improved reliability of a power system with the maintenance schedule obtained by the new method compared to the results from the approximate methodology. r 2007 Elsevier Ltd. All rights reserved. Keywords: Genetic algorithm; Optimisation; Maintenance scheduling; Safety; Loss of load expectation
1. Introduction In order to ensure the overall power system reliability, it is necessary to maintain generating units after a certain period of service. The sub-optimal maintenance schedules contribute to a higher production costs and decrease the system reliability. The maintenance schedule affects many short-term and long-term planning functions, including unit commitment, fuel scheduling, optimal utilisation of water resources, long-term power system development planning, reliability calculations and production costs. The sub-optimal schedule could affect each of these functions adversely. Such considerations encourage the development of a strategy that is able to formulate an optimised timetable for the maintenance sequence of the generators concerned. An optimal generator maintenance schedule increases the operating system reliability, reduces power generating cost, and extends the generator lifetime. Additionally, the optimised maintenance schedules could potentially defer Corresponding author. Tel.: +386 1 5885 307; fax: +386 1 5885 377.
E-mail address: [email protected] (A. Volkanovski). 0951-8320/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2007.03.027
some capital expenditure for the new plants and allow critical maintenance work to be performed. The maintenance scheduling (MS) problem received considerable attention in the past. A mixed integer programming model, utilising combination of the implicit enumeration and branch-and-bound techniques was formulated [1,2]. This method made a number of assumptions and solved only relatively small problems [3]. The use of the decomposition approach was also demonstrated [4,5]. The branch-and-bound approach and simulated annealing were implemented for solving the model [6,7]. The levelling incremental risks method was proposed [8] and was further extended for multi-area maintenance planning [9]. Such single objective methods may not meet the requirements of utility planners, whose most important task is determination of the best compromise solution of the considered objectives. The multi-objective type of the maintenance planning problem [10–13] seeks the solutions obtained by optimisation of two alternative objective criteria: costs and reliability. In order to overcome the limitations corresponding to the non-linear objectives and constraints of the maintenance planning problem and increased computational time, a number of artificial intelligence approaches
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have been studied [14], utilising different optimisation and hybrid methods developed with combination of several optimisation techniques [15–18]. Knowledge based and expert system models have been proposed for producing advice in inspection and maintenance planning [19,20] and fuzzy logic for calculation of imprecision [21–23]. The MS of power units in a power system share certain specific features with MS activities of safety systems in nuclear power plants. Methods for MS in nuclear power plants integrate probabilistic safety assessment as the tool for assessing and improving plant safety with optimisation techniques such as genetic algorithms (GAs) or simulated annealing [24–27]. Integer programming, branch-and-bound techniques and dynamic programming are generally not suitable for the non-linear objectives and constraints of generator maintenance schedule problem and their computational time grows prohibitively with problem size. Heuristic methods require significant operator input and may even fail to find feasible solution. Main improvements of the proposed method comparing to the previous approaches are: usage of exact, nonsimplified probability models for the generating units and inclusion of multiple constraints in the optimisation with possibility of simple addition of supplementary constraints if necessary. The method is applicable to real size power systems with short calculation times and possibility of providing multiple solutions satisfying given set of constraints. Additional original feature of the proposed method is the initialisation of optimisation technique with solution provided by approximate method. Main aspects for further improvement of the proposed method are: usage of other reliability indices for reliability assessment of a power system, for example EENS (expected energy not supplied) integrally or separately. The proposed method could be developed to solve the multi-objective type of the maintenance planning problem accounting costs together with reliability and inclusion of network constraints and network failures. Additionally data uncertainty could be incorporated in calculations and results. Section 2 gives general description of the problem and definition of used reliability indices. In Section 3 definition and description of the used model for generating unit reliability are given. Description and algorithm for building of capacity outage probability table are given in Section 4. Operating and planning constraints of MS problem are given in Section 5. Description of used optimisation technique (GA), initialisation procedure, problem encoding and decoding are given in Section 6. Obtained results of the analysis and conclusions are given in Sections 7 and 8, respectively.
objectives and constraints. The loss of load expectation (LOLE), also known as loss of load probability (LOLP), represents the system risk index used in the method. The annual value of LOLE of power system is an objective function requiring minimisation in the analysed period min LOLEa ðX NG Þ,
(1)
where LOLEa is a calculated annual loss of load expectation in days/year for X NG maintenance schedules of NG generators, X NG the maintenance schedule of unit X given as day/week when maintenance of unit X starts, NG the number of generating units in power system. The objective function is subjected to operating and planning constraints. The number of the independent variables in a model is determined by the number of units in the power system considered in the planning horizon. The basic approach for evaluating the LOLE for particular configuration of capacities consists of three parts as shown in Fig. 1. The capacity model is formed in the direct analytical methods by creating a capacity outage probability table. This table represents the capacity outage states of the generating system together with the probabilities of each state. The load model can either be the daily peak load variation curve (DPLVC), which only includes the peak loads of each day, or the load duration curve (LDC), which represents the hourly variation of the load. The risk indices, LOLE, are evaluated by convolution of the capacity (also known as generation) and load model, obtaining the expected number of days in the specified period in which the daily peak load will exceed the available capacity (LOLEp Þ LOLEp ¼
n X
Pi ðC i oLi Þdays=period,
(2)
i¼1
where LOLEp is the value of LOLE in period p with n days, C i the available capacity on day i, Li the forecast peak load on day i, Pi ðC i oLi Þ the probability of loss of load on day i, obtained directly from the capacity outage cumulative probability table. Loss of load method provides a consistent and sensitive measure of power generating system reliability. In the developed method, the value of weekly LOLEp is calculated, using Eq. (2) and n ¼ 7, and
2. Problem statement The MS of generating units in a power system is a largescale combinatorial optimisation problem with non-linear
Fig. 1. Conceptual tasks in evaluation of generating power system reliability.
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those values are summed to calculate the annual LOLEa : LOLEa ¼
52 X
LOLEp .
(3)
p¼1
The DPLVC is used to evaluate LOLE indices giving a risk expressed in number of days during the period of study when the load will exceed available capacity. The calculated indices measure the overall adequacy of the power generating system to cover total system load, not taking into account separate load points, transmission constraints or energy available in the system. Load forecast is normally predicted on previous load change, and its uncertainty can be described by a probability distribution whose parameters can be determined from past experience, load forecast model and possible subjective evaluation. The uncertainty of load forecasting can be included in the risk computations by dividing the load forecast probability distribution into class intervals with area corresponding to the probability that the load is the class interval mid-value. The LOLE is computed for each load represented by the class interval and multiplied by the probability of existence of the load. The sum of these products represents the LOLE for the forecast load. The load forecast uncertainty, besides unit reliability, might be the single most important parameter in operating capacity reliability evaluation.
Fig. 2. Two-state model for a base load unit.
Fig. 3. Four-state model for planning studies.
service. Scheduled outages must be considered separately. Eq. (4) is derived from equation for the time-depended probabilities of a single repairable component given by P0 ðtÞ ¼
m leðlþmÞt þ , lþm lþm
(6)
P1 ðtÞ ¼
l leðlþmÞt , lþm lþm
(7)
3. Model of generating station The probability of finding the generating station in forced outage at some distant time in the future represents the basic parameter used in static capacity evaluation for modelling generating unit reliability. This probability is defined as unit unavailability (U) and in power system applications it is known as the unit forced outage rate (FOR), defined as l r r f ¼ ¼ ¼ lþm rþm T m P ½down_time P ¼ P , ½down_time þ ½up_time
U ¼ FOR ¼
U ¼ FOR ¼
FOH , FOH þ SH
where P0 ðtÞ and P1 ðtÞ are the probabilities of being found in the operating state and failed state, respectively, as a function of time given that the system started at time t ¼ 0 in the operating state. The limiting state probabilities can be evaluated from Eqs. (6) and (7) as t ! 1: P1 ð1Þ ¼ U ¼ FOR ¼
ð4Þ
(5)
where U is the unit unavailability, l the expected failure rate, m the expected repair rate, m the mean time to failure: MTTF ¼ 1=l, r the mean time to repair: MTTR ¼ 1=m, m þ r the mean time between failures: MTBF ¼ 1=f , f the cycle frequency: f ¼ 1=T, T the cycle time: T ¼ 1=f ¼ m þ r, FOH the forecasted outage hours, SH the service hours. Terms FOH and SH are used in power system applications for downtime and uptime, respectively. The concept of unavailability as illustrated in Eqs. (4) and (5) is associated with the simple two-state model shown in Fig. 2. This model is directly applicable to a base load generating unit, which is either operating or forced out of
l . lþm
(8)
For generating unit with relatively long operating cycles, the unavailability (FOR), calculated with Eqs. (4) and (5) is an adequate estimator of the probability that the unit under similar conditions will not be available for service in the future. When the demand cycle is short, as in the case of a peaking or intermittent operating unit, four-state model, shown in Fig. 3 and corresponding equations can be used [28]. In Fig. 3, T represents the average reserve shut-down time between periods of need, D is the average in-service time per occasion of demand and P is the probability of starting failure. 4. Capacity outage probability table Capacity outage probability table is an array of capacity levels, or corresponding capacity out of service and the associated probabilities of existence. The associated prob-
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ability of existence is the probability of exactly the indicated amount of capacity being out of service. For capacity model is used the cumulative probability of existence and it is equal to sum of probabilities corresponding to capacity on outage equal to or greater than the indicated amount. Capacity model can be created using a convolution algorithm [29,30], which is a recursive algorithm of adding or removing units in the model and calculating the cumulative probability of a particular capacity outage state. The modified convolution algorithm can be used for a multi-state unit, i.e. a unit that can exist in one or more derated or partial output states. In case of two state units, cumulative probability of a particular capacity outage state of X (MW), after a unit of capacity C (MW) and forced outage rate U is added, is given by equation PðX Þ ¼ ð1 UÞP0 ðX Þ þ ðUÞP0 ðX CÞ,
(9)
0
where P ðX Þ denotes the cumulative probability of the capacity outage state of X (MW) before unit is added, PðX Þ denotes the cumulative probability of the capacity outage state of X (MW) after the unit is added. The above expression is initialised by setting P0 ðX Þ ¼ 1:0 for X pC and P0 ðX Þ ¼ 0 otherwise and P0 ðX CÞ ¼ 1:0 for X pC. Eq. (9) can be modified as follows to include multi-state unit representation: PðX Þ ¼
n X
U i P0 ðX C i Þ,
(10)
completes the maintenance without interruption. The maintenance completion constraints also ensure that a unit is maintained just once during the planning horizon. The electric utilities have limited maintenance manpower and that limitation should be considered in solution. The crew constraints depend on the available manpower and they specify the number of units, which can be simultaneously maintained by the same crew. In many instances, a precedence constraint exists that specifies the maintenance sequence of the units. For example, one generator should be on maintenance after finalisation of another unit in same location due to the accessibility of machinery or staff. The precedence constraint specifies the order in which the maintenance of the units has to be performed. Due to the connection of hydro systems of some hydro generators, linkage of maintenance of those generators should be taken into account (in case of run-of-the-river plants or when dam capacity following plants is small). The resource (capacity) constraint ensures that the maintenance outage capacity does not exceed the gross reserve during any stage of maintenance. The capacity constraint is modelled by the limiting maximum value of calculated period LOLEp . The additional constraints can be considered (like network contingencies) and implemented into the algorithm. The system analyses of the Macedonian power system showed that the maintenance of the generating units cannot result in the network overload or other contingencies connected with the power transfer capacity of transmission network [31,32].
i¼1
where n is the number of unit states, C i the capacity outage of state i for the unit being added, U i the unit unavailability of state i for the unit being added. When n ¼ 2, this equation reduces to Eq. (9). When generating units are removed from system for scheduled maintenance they are neither available for service nor for failure. New capacity model of system is created using recursive algorithm for unit removal: P0 ðX Þ ¼
PðX Þ ðUÞP0 ðX CÞ , ð1 UÞ
(11)
where P0 ðX CÞ ¼ 1:0 for X pC. In case of multi-state unit the equation is as follows: P PðX Þ ni¼2 U i P0 ðX C i Þ 0 P ðX Þ ¼ , (12) U1 where P0 ðX C i Þ ¼ 1:0 for X pC. 5. Maintenance schedule constraints The scheduled start of the maintenance should satisfy operating and planning constraints and meet the objective criteria. Each generating unit has to be maintained in the given maintenance window specifying earliest and latest times for start of the maintenance of a unit. The maintenance completion constraints ensure that once a unit is removed from the system for maintenance, it
6. Genetic algorithms A genetic algorithm (GA) can be seen as a search algorithm that is based on the concepts of natural selection and genetic inheritance. It searches an optimal solution to the problems by manipulating a population of strings (chromosomes) that represent different potential solutions, each corresponding to a sample point from the search space. For each generation, all the populations are evaluated based on their fitness. An individual with a larger fitness has a higher chance of evolving into the next generation. By searching many peaks simultaneously, GA reduces the possibility of trapping into a local minimum. GA works with a coding of parameters instead of parameters themselves. The coding of parameters helps the genetic operator to evolve the current state into the next state with minimum computations. GA evaluates the fitness of each string to guide its search instead of the explicit optimisation function. There is no need for computations of derivatives or other auxiliary knowledge. Finally, GA explores the search space where the probability of finding improved performance is high. Therefore, GA is applicable and effective for solving maintenance schedule problems [33,34]. 6.1. Problem encoding In GA, parameters are represented by a string structure called chromosomes. In presented method, real-coded GA
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was used, where each chromosome corresponds to the specific maintenance schedule of units in the power system. The GA always deals with a set of chromosomes called a population. Transforming chromosomes from a population, new population, corresponding to the next generation is obtained. There are three genetic operators to do this: selection, crossover, and mutation. Values of the xk phenotypes in the chromosome are double precision real numbers in interval between zero and one: xk 2 ½0:0; 1:0. Therefore, it is necessary to encode values of MS for generators from initialisation procedure and decode values of chromosomes to maintenance times (periods) in order to calculate their fitness. Values of X i maintenance schedule, received from the approximate initialisation procedure, are encoded to GA’s chromosomes normalising their value in the interval between 0 and 1. The normalisation is done by the division of their value with the number of periods in analysed period m:
The individual unit capacities are arranged in descending order and then added to weekly peak loads. The biggest unit capacity is added to smallest peak load, and this procedure is repeated until all units are arranged in the load diagram. The four major steps in the procedure are:
(1) Arrange generating units by size, with the largest first and the smallest last. (2) Schedule the largest generating unit for maintenance during periods of lowest load, consistent with the minimum and maximum time constraints. (3) Adjust weekly peak load by the generating unit capacity on maintenance. (4) Repeat steps 2–4 until all generating units are scheduled.
Xi , (13) m where m ¼ 52 in case when week is taken as period. Decoding of real values of phenotypes to maintenance times (periods for start of maintenance) is done using decoding procedure. In case of the independent maintenance the decoding is performed as follows:
xk ¼
X g ¼ ninðfirst þ ððlast maintenance firstÞ þ 1Þ xk Þ, (14) where X g is the maintenance time for units received from GA converted to time periods (weeks), xk the values of the maintenance time for units received from GA, first the earliest time period for start of maintenance of unit, last the latest time (week) for terminating the maintenance of a unit, nin the elemental intrinsic FORTRAN NINT function—returns the nearest integer to the argument x, maintenance the maintenance length of unit. During decoding process, using Eq. (14), the constraints of the earliest and the latest time period for start of maintenance of unit and necessary time for finishing maintenance are taken into account. 6.2. An initialisation procedure To improve the effectiveness of the GA in solving the maintenance schedule a separate initialisation procedure was applied. The approach used for the initialisation procedure is to reduce the total installed capacity by the expected capacity loss (product of the unit capacity and its availability) and then schedule the maintenance on a constant reserve basis. The constant reserve maintenance procedure [35] can be written as follows: Constant ¼ weekly peak load þ weekly capacity on maintenance.
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Fig. 4. The flowchart of the optimisation procedure.
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This approximate MS planning technique is simple and straightforward but it cannot include the constraints, which are normal for the operating power system. The maintenance schedule X i calculated by this procedure represents the initialisation value for the GA. During the initialisation of GA, only part of starting population is substituted with results received from approximate MS. Several substitution rates are tested comparing obtained results.
constraint l. The constant penalisation value for W l ¼ 2 was used in the developed method, but the penalty term corresponding to constraint justification can be different for different limitations and its magnitude can model the importance of satisfying that constraint. The other additional constraints can be added to model, including network constraints or energy limitations. 6.4. Selection
6.3. Chromosome fitness Fitness is a quantity related to the chromosome and it enables comparison between them. The decoding process using Eq. (16) ensures that the constraints corresponding to time limitations are satisfied. Other constraints, corresponding to manpower limitation, precedence constraint and unit linkage have to be checked for violations, using the penalty method, which degrades the fitness in cases with violated constraint. The fitness for chromosome is defined by equation !1 X F i ¼ LOLEa Wl , (15) l2VC i
where LOLEa is the objective function related to chromosome i, VC i is set of violated constraints associated to chromosome i, W l is the penalty term corresponding to
Improvement of the average fitness of the population is achieved through selection of individuals as parents from the completed population. The selection is performed in such a way that chromosomes having higher fitness are more likely to be selected as parents. Bearing in mind that some of the individuals may have significantly higher fitness compared to the others, the next generation may be constituted of the large number of identical individuals. This poses a limitation on the population diversity, meaning that the search space will be reduced. One can omit such a situation by using relative fitness. Let F min and F max be the minimal and the maximal fitness in the population, respectively. Then the relative fitness for chromosome i is defined as G i Gi ¼
F i F min . F max F min
(16)
Table 1 Calculated values of the forecasted outage rate for the thermal power plants in Macedonia power system Power plant
Bitola 1
Bitola 2
Bitola 3
Oslomej
Negotino
Average
FOR mean
0.011157
0.016569
0.009685
0.13292
0.008038
0.035674
Initial pop. substitution
Annual LOLE days/year
Calculation min
9 9 20 20 50 50
0.129 0.132 0.129 0.132 0.124 0.129 0.127 0.128
11.2 2.3 11.16 2.31 11.18 2.45 12.05 2.4
9 9 20 20 50 50
0.256 0.256 0.256 0.258 0.249 0.258 0.250 0.258
11.61 2.41 12.26 2.41 11.56 2.56 11.56 2.38
Table 2 The results for the Macedonian power system for years 2000 and 2001 for different scenarios Size of population GA Scenario for year 2000 1 (500) 2 (100) 3 (500) 4 (100) 5 (500) 6 (100) 7 (500) 8 (100)
No substitution–initiation No substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation
1 2 3 4 5 6 7 8
No substitution–initiation No substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation
(500) (100) (500) (100) (500) (100) (500) (100)
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Therefore, the likelihood of selecting a chromosome as a parent is a function of its fitness relative to the sum fitness of all chromosomes. To further improve the evolving process, the GA can carry over the best individuals from the completed population to the new population set (principle of elitism).
any position (the crossover point) and exchanges the substrings between the two chromosomes. After the crossover is performed, the new chromosomes are added to the new population set. The mutation is specifically applied to increase population diversity. Mutation involves randomly selecting genes within the chromosomes and assigning them random values within the corresponding predefined interval. In order not to destroy good genetic code, non-uniform mutation has to be applied. In such a manner, in later stages of GA optimisation process, the interval for random selection of genes’ values is narrowed.
6.5. Crossover and mutation
6.6. GA parameters
After the selection, the GA picks a pair of selected chromosomes in order to create two new chromosomes. The GA applies a random generator to cut the strings at
GA requires definition of a number of parameters, which can affect the efficiency of the search process in several ways. The population size should be large enough to create
There are several selection techniques. The roulette selection method is used here. In this method, the relative weight of each chromosome’s fitness is calculated as Gi pi ¼ PN POP k¼1
Gk
.
(17)
Fig. 5. The peak loads and the power of generators in maintenance for each week in years 2000 and 2001.
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sufficient diversity covering the possible solution space. Generally, one cannot know the optimal value in advance. Clearly, a more complex problem domain requires a larger population due to the larger possible combination of variables. Another user-defined criterion is the point at which the optimisation process terminates. GA with fixed number of generations is used in the developed method, for
which it is assumed that the search process has covered a sufficient search space. Other parameters, such as crossover probability, mutation rate, selection, and crossover mechanisms, seem to affect the GA process less significantly when evaluated over a larger number of generations. The flowchart of the developed optimisation procedure is shown in Fig. 4.
Fig. 6. The maintenance schedule in years 2002 and 2001.
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7. Analysis and results The proposed method was tested using the Macedonian power system data for years 2000 and 2001. The input considered 29 generating units in the power generating system. The calculated values of forecasted outage rate of thermal power plants in Macedonian power system used in analysis are given in Table 1. The length of the planning horizon was one year with value of LOLEp calculated for each week. The method included the following constraints: the maintenance completion, crew, precedence, reserve and resource constraints. Multiple simulations were done, chaining number of populations in GA and analysing cases with and without initialisation of GA with approximate MS planning technique. Results are given in Table 2. Results show improved MS for generators using initialisation of GA with approximate MS planning technique, particularly in case of small populations. Substitution of the population should be done carefully, because higher rates of substitution can limit search space of GA resulting with higher value of annual LOLE. Calculation time is proportional to
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number of populations and no improvement of objective function was observed with the increase of number of populations above 500. The peak loads and the power of generators on maintenance are shown in Fig. 5 for each week for year 2000 and 2001, respectively. The detailed maintenance schedule for years 2000 and 2001 (the green area in Fig. 5) is given in Fig. 6. Thermal power plants, which have similar size and FOR, are put in the periods of lower loads and smaller generators in hydro power plants are consecutively added to them. The results shown in Figs. 5 and 6 look similar to those obtained by constant reserve maintenance procedure but it should be noted that in the MS of the units their size and corresponding FOR was taken into account. The evolution of annual LOLE in years 2000 and 2001 with approximate MS planning technique is given in Fig. 7. The results show the effect of providing a good start-up population to genetic GA especially in case of small number of generations. With the increase of the number of generations similar results are obtained by standard initiation procedure, with random number generator, and with approximate MS planning technique. A comparison
Fig. 7. Evolution of the annual LOLE in years 2002 and 2001.
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Table 3 Comparison of the results for the Macedonian power system Annual LOLE days/year Input year 2000 1 Approximate method 2 New method
0.229 0.124
Input year 2001 1 Approximate method 2 New method
0.312 0.249
of results obtained from the approximate method and the new method are given in Table 3. The results show that the new method has annual value of LOLE about 1 day/year smaller than the approximate method, taking account all constraints of MS operations.
8. Conclusions A new method for optimisation of the MS of generating units in a power system is developed. Maintenance is scheduled to minimise the risk through minimisation of the yearly value of the LOLE taken as a measure of the power system reliability. The proposed method uses GA to obtain the best solution resulting in a minimal value of the annual loss of load expectation value for the power system in the analysed period. The operational constraints for generating units are included in the method. The proposed algorithm was tested on a Macedonian power system and the obtained results were compared with the results received from the approximate methodology. The results show the improved reliability of a power system with the maintenance schedule obtained by the new method compared to the results from the approximate methodology. The new method is developed in a way that enables easy variations of optimised function and inclusion of additional constraints. Different reliability measures and constraints for the system can be used (e.g. loss of energy expectation LOEE) depending on the system characteristics and the operators requests. Minimising the total generator operating cost as the objective function can be additionally modelled and implemented in the algorithm, if the additional data of the power generating units is available, including price of energy sources for the power plants in the analysed period and the price of electrical energy on market for the selected period. Network contingencies can be also additionally included in the model. The algorithm of the new method and its corresponding computer code offer a promising approach for the MS of generating units. The method enables improvement of the reliability with no substantial expenses for the owner and operators of the power system.
Acknowledgements This research was partly supported by the Slovenian research Agency (contract number 1000-05-310016). Authors acknowledge support from Faculty of Electrical Engineering—Skopje, Republic of Macedonia.
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ARTICLE IN PRESS A. Volkanovski et al. / Reliability Engineering and System Safety 93 (2008) 757–767 [20] Choueiry B, Sekine Y. Knowledge based method for power generators maintenance scheduling. In: Proceedings of the expert systems applications to power systems (ESAP), Stockholm-Helsinki; 1988. p. 9.7–14. [21] Lin C, Huang C, Lee S. An optimal generators maintenance approach using fuzzy dynamic programming. In: Proceedings of the IEEE/PES summer meeting, Seattle; 1992. p. 401–10. [22] Jovanovic A, Maile K. Management of remaining life of power plant components by means of intelligent software tools developed at MPA Stuttgart. Int J Press Vessel Piping 1996;66: 367–79. [23] Bretthaner G, Gamaleja T, Handschin E, Neumann U, Hoffmann W. Integrated maintenance scheduling system for electrical energy systems. IEEE Trans Power Deliv 1998;132:655–60. [24] Cˇepin M. Optimization of safety equipment outages improves safety. Reliab Eng Syst Safety 2002;77:71–80. [25] Cˇepin M, Borut M. A dynamic fault tree. Reliab Eng Syst Safety 2002;75:83–91. [26] Cˇepin M, Martorell S. Evaluation of allowed outage time considering a set of plant configurations. Reliab Eng Syst Safety 2002;78: 256–66.
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[27] Cˇepin M, Borut M. Probabilistic safety assessment improves surveillance requirements in technical specification. Reliab Eng Syst Safety 1997;56:69–77. [28] IEEE Committee, A four state model for estimation of outage risk for units in peaking service. IEEE Trans PAS-91 1972;618–27. [29] Billinton R. Power system reliability evaluation. New York: Gordon and Breach; 1970. [30] Billinton R, Allan RN. Reliability evaluation of power systems. New York: Plenum Press; 1996. [31] Volkanovski A. Optimization of generators in power system using DC-model and simplex methodology. Seminar work, Electrotehnical faculty—Skopje, Skopje; 2003. [32] Volkanovski A. Analysis of Macedonian power system reliability. Seminar work, Electrotehnical faculty—Skopje, Skopje; 2002. [33] Martorell S, Carlos S, Sa´nchez A, Serradell V. Constrained optimisation of test intervals using a steady-state genetic algorithm. Reliab Eng System Safety 2000;67(3):215–32. [34] Martorell S, Sa´nchez A, Carlos S, Serradell V. Alternatives and challenges in optimizing industrial safety using genetic algorithms. Reliab Eng Syst Safety 2004;86(1):25–38. [35] Stoll H. Least-cost electric utility planning. New York: Wiley; 1989.
Reliability Engineering and System Safety 93 (2008) 757–767 www.elsevier.com/locate/ress
Genetic algorithm optimisation of the maintenance scheduling of generating units in a power system Andrija Volkanovskia,, Borut Mavkoa, Tome Bosˇ evskib, Anton Cˇausˇ evskib, Marko Cˇepina a
Reactor Engineering Division, Jozˇef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia Faculty of Electrical Engineering, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia
b
Received 12 December 2006; received in revised form 15 March 2007; accepted 18 March 2007 Available online 24 March 2007
Abstract A new method for optimisation of the maintenance scheduling of generating units in a power system is developed. Maintenance is scheduled to minimise the risk through minimisation of the yearly value of the loss of load expectation (LOLE) taken as a measure of the power system reliability. The proposed method uses genetic algorithm to obtain the best solution resulting in a minimal value of the annual LOLE value for the power system in the analysed period. The operational constraints for generating units are included in the method. The proposed algorithm was tested on a Macedonian power system and the obtained results were compared with the results received from the approximate methodology. The results show the improved reliability of a power system with the maintenance schedule obtained by the new method compared to the results from the approximate methodology. r 2007 Elsevier Ltd. All rights reserved. Keywords: Genetic algorithm; Optimisation; Maintenance scheduling; Safety; Loss of load expectation
1. Introduction In order to ensure the overall power system reliability, it is necessary to maintain generating units after a certain period of service. The sub-optimal maintenance schedules contribute to a higher production costs and decrease the system reliability. The maintenance schedule affects many short-term and long-term planning functions, including unit commitment, fuel scheduling, optimal utilisation of water resources, long-term power system development planning, reliability calculations and production costs. The sub-optimal schedule could affect each of these functions adversely. Such considerations encourage the development of a strategy that is able to formulate an optimised timetable for the maintenance sequence of the generators concerned. An optimal generator maintenance schedule increases the operating system reliability, reduces power generating cost, and extends the generator lifetime. Additionally, the optimised maintenance schedules could potentially defer Corresponding author. Tel.: +386 1 5885 307; fax: +386 1 5885 377.
E-mail address: [email protected] (A. Volkanovski). 0951-8320/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2007.03.027
some capital expenditure for the new plants and allow critical maintenance work to be performed. The maintenance scheduling (MS) problem received considerable attention in the past. A mixed integer programming model, utilising combination of the implicit enumeration and branch-and-bound techniques was formulated [1,2]. This method made a number of assumptions and solved only relatively small problems [3]. The use of the decomposition approach was also demonstrated [4,5]. The branch-and-bound approach and simulated annealing were implemented for solving the model [6,7]. The levelling incremental risks method was proposed [8] and was further extended for multi-area maintenance planning [9]. Such single objective methods may not meet the requirements of utility planners, whose most important task is determination of the best compromise solution of the considered objectives. The multi-objective type of the maintenance planning problem [10–13] seeks the solutions obtained by optimisation of two alternative objective criteria: costs and reliability. In order to overcome the limitations corresponding to the non-linear objectives and constraints of the maintenance planning problem and increased computational time, a number of artificial intelligence approaches
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have been studied [14], utilising different optimisation and hybrid methods developed with combination of several optimisation techniques [15–18]. Knowledge based and expert system models have been proposed for producing advice in inspection and maintenance planning [19,20] and fuzzy logic for calculation of imprecision [21–23]. The MS of power units in a power system share certain specific features with MS activities of safety systems in nuclear power plants. Methods for MS in nuclear power plants integrate probabilistic safety assessment as the tool for assessing and improving plant safety with optimisation techniques such as genetic algorithms (GAs) or simulated annealing [24–27]. Integer programming, branch-and-bound techniques and dynamic programming are generally not suitable for the non-linear objectives and constraints of generator maintenance schedule problem and their computational time grows prohibitively with problem size. Heuristic methods require significant operator input and may even fail to find feasible solution. Main improvements of the proposed method comparing to the previous approaches are: usage of exact, nonsimplified probability models for the generating units and inclusion of multiple constraints in the optimisation with possibility of simple addition of supplementary constraints if necessary. The method is applicable to real size power systems with short calculation times and possibility of providing multiple solutions satisfying given set of constraints. Additional original feature of the proposed method is the initialisation of optimisation technique with solution provided by approximate method. Main aspects for further improvement of the proposed method are: usage of other reliability indices for reliability assessment of a power system, for example EENS (expected energy not supplied) integrally or separately. The proposed method could be developed to solve the multi-objective type of the maintenance planning problem accounting costs together with reliability and inclusion of network constraints and network failures. Additionally data uncertainty could be incorporated in calculations and results. Section 2 gives general description of the problem and definition of used reliability indices. In Section 3 definition and description of the used model for generating unit reliability are given. Description and algorithm for building of capacity outage probability table are given in Section 4. Operating and planning constraints of MS problem are given in Section 5. Description of used optimisation technique (GA), initialisation procedure, problem encoding and decoding are given in Section 6. Obtained results of the analysis and conclusions are given in Sections 7 and 8, respectively.
objectives and constraints. The loss of load expectation (LOLE), also known as loss of load probability (LOLP), represents the system risk index used in the method. The annual value of LOLE of power system is an objective function requiring minimisation in the analysed period min LOLEa ðX NG Þ,
(1)
where LOLEa is a calculated annual loss of load expectation in days/year for X NG maintenance schedules of NG generators, X NG the maintenance schedule of unit X given as day/week when maintenance of unit X starts, NG the number of generating units in power system. The objective function is subjected to operating and planning constraints. The number of the independent variables in a model is determined by the number of units in the power system considered in the planning horizon. The basic approach for evaluating the LOLE for particular configuration of capacities consists of three parts as shown in Fig. 1. The capacity model is formed in the direct analytical methods by creating a capacity outage probability table. This table represents the capacity outage states of the generating system together with the probabilities of each state. The load model can either be the daily peak load variation curve (DPLVC), which only includes the peak loads of each day, or the load duration curve (LDC), which represents the hourly variation of the load. The risk indices, LOLE, are evaluated by convolution of the capacity (also known as generation) and load model, obtaining the expected number of days in the specified period in which the daily peak load will exceed the available capacity (LOLEp Þ LOLEp ¼
n X
Pi ðC i oLi Þdays=period,
(2)
i¼1
where LOLEp is the value of LOLE in period p with n days, C i the available capacity on day i, Li the forecast peak load on day i, Pi ðC i oLi Þ the probability of loss of load on day i, obtained directly from the capacity outage cumulative probability table. Loss of load method provides a consistent and sensitive measure of power generating system reliability. In the developed method, the value of weekly LOLEp is calculated, using Eq. (2) and n ¼ 7, and
2. Problem statement The MS of generating units in a power system is a largescale combinatorial optimisation problem with non-linear
Fig. 1. Conceptual tasks in evaluation of generating power system reliability.
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those values are summed to calculate the annual LOLEa : LOLEa ¼
52 X
LOLEp .
(3)
p¼1
The DPLVC is used to evaluate LOLE indices giving a risk expressed in number of days during the period of study when the load will exceed available capacity. The calculated indices measure the overall adequacy of the power generating system to cover total system load, not taking into account separate load points, transmission constraints or energy available in the system. Load forecast is normally predicted on previous load change, and its uncertainty can be described by a probability distribution whose parameters can be determined from past experience, load forecast model and possible subjective evaluation. The uncertainty of load forecasting can be included in the risk computations by dividing the load forecast probability distribution into class intervals with area corresponding to the probability that the load is the class interval mid-value. The LOLE is computed for each load represented by the class interval and multiplied by the probability of existence of the load. The sum of these products represents the LOLE for the forecast load. The load forecast uncertainty, besides unit reliability, might be the single most important parameter in operating capacity reliability evaluation.
Fig. 2. Two-state model for a base load unit.
Fig. 3. Four-state model for planning studies.
service. Scheduled outages must be considered separately. Eq. (4) is derived from equation for the time-depended probabilities of a single repairable component given by P0 ðtÞ ¼
m leðlþmÞt þ , lþm lþm
(6)
P1 ðtÞ ¼
l leðlþmÞt , lþm lþm
(7)
3. Model of generating station The probability of finding the generating station in forced outage at some distant time in the future represents the basic parameter used in static capacity evaluation for modelling generating unit reliability. This probability is defined as unit unavailability (U) and in power system applications it is known as the unit forced outage rate (FOR), defined as l r r f ¼ ¼ ¼ lþm rþm T m P ½down_time P ¼ P , ½down_time þ ½up_time
U ¼ FOR ¼
U ¼ FOR ¼
FOH , FOH þ SH
where P0 ðtÞ and P1 ðtÞ are the probabilities of being found in the operating state and failed state, respectively, as a function of time given that the system started at time t ¼ 0 in the operating state. The limiting state probabilities can be evaluated from Eqs. (6) and (7) as t ! 1: P1 ð1Þ ¼ U ¼ FOR ¼
ð4Þ
(5)
where U is the unit unavailability, l the expected failure rate, m the expected repair rate, m the mean time to failure: MTTF ¼ 1=l, r the mean time to repair: MTTR ¼ 1=m, m þ r the mean time between failures: MTBF ¼ 1=f , f the cycle frequency: f ¼ 1=T, T the cycle time: T ¼ 1=f ¼ m þ r, FOH the forecasted outage hours, SH the service hours. Terms FOH and SH are used in power system applications for downtime and uptime, respectively. The concept of unavailability as illustrated in Eqs. (4) and (5) is associated with the simple two-state model shown in Fig. 2. This model is directly applicable to a base load generating unit, which is either operating or forced out of
l . lþm
(8)
For generating unit with relatively long operating cycles, the unavailability (FOR), calculated with Eqs. (4) and (5) is an adequate estimator of the probability that the unit under similar conditions will not be available for service in the future. When the demand cycle is short, as in the case of a peaking or intermittent operating unit, four-state model, shown in Fig. 3 and corresponding equations can be used [28]. In Fig. 3, T represents the average reserve shut-down time between periods of need, D is the average in-service time per occasion of demand and P is the probability of starting failure. 4. Capacity outage probability table Capacity outage probability table is an array of capacity levels, or corresponding capacity out of service and the associated probabilities of existence. The associated prob-
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ability of existence is the probability of exactly the indicated amount of capacity being out of service. For capacity model is used the cumulative probability of existence and it is equal to sum of probabilities corresponding to capacity on outage equal to or greater than the indicated amount. Capacity model can be created using a convolution algorithm [29,30], which is a recursive algorithm of adding or removing units in the model and calculating the cumulative probability of a particular capacity outage state. The modified convolution algorithm can be used for a multi-state unit, i.e. a unit that can exist in one or more derated or partial output states. In case of two state units, cumulative probability of a particular capacity outage state of X (MW), after a unit of capacity C (MW) and forced outage rate U is added, is given by equation PðX Þ ¼ ð1 UÞP0 ðX Þ þ ðUÞP0 ðX CÞ,
(9)
0
where P ðX Þ denotes the cumulative probability of the capacity outage state of X (MW) before unit is added, PðX Þ denotes the cumulative probability of the capacity outage state of X (MW) after the unit is added. The above expression is initialised by setting P0 ðX Þ ¼ 1:0 for X pC and P0 ðX Þ ¼ 0 otherwise and P0 ðX CÞ ¼ 1:0 for X pC. Eq. (9) can be modified as follows to include multi-state unit representation: PðX Þ ¼
n X
U i P0 ðX C i Þ,
(10)
completes the maintenance without interruption. The maintenance completion constraints also ensure that a unit is maintained just once during the planning horizon. The electric utilities have limited maintenance manpower and that limitation should be considered in solution. The crew constraints depend on the available manpower and they specify the number of units, which can be simultaneously maintained by the same crew. In many instances, a precedence constraint exists that specifies the maintenance sequence of the units. For example, one generator should be on maintenance after finalisation of another unit in same location due to the accessibility of machinery or staff. The precedence constraint specifies the order in which the maintenance of the units has to be performed. Due to the connection of hydro systems of some hydro generators, linkage of maintenance of those generators should be taken into account (in case of run-of-the-river plants or when dam capacity following plants is small). The resource (capacity) constraint ensures that the maintenance outage capacity does not exceed the gross reserve during any stage of maintenance. The capacity constraint is modelled by the limiting maximum value of calculated period LOLEp . The additional constraints can be considered (like network contingencies) and implemented into the algorithm. The system analyses of the Macedonian power system showed that the maintenance of the generating units cannot result in the network overload or other contingencies connected with the power transfer capacity of transmission network [31,32].
i¼1
where n is the number of unit states, C i the capacity outage of state i for the unit being added, U i the unit unavailability of state i for the unit being added. When n ¼ 2, this equation reduces to Eq. (9). When generating units are removed from system for scheduled maintenance they are neither available for service nor for failure. New capacity model of system is created using recursive algorithm for unit removal: P0 ðX Þ ¼
PðX Þ ðUÞP0 ðX CÞ , ð1 UÞ
(11)
where P0 ðX CÞ ¼ 1:0 for X pC. In case of multi-state unit the equation is as follows: P PðX Þ ni¼2 U i P0 ðX C i Þ 0 P ðX Þ ¼ , (12) U1 where P0 ðX C i Þ ¼ 1:0 for X pC. 5. Maintenance schedule constraints The scheduled start of the maintenance should satisfy operating and planning constraints and meet the objective criteria. Each generating unit has to be maintained in the given maintenance window specifying earliest and latest times for start of the maintenance of a unit. The maintenance completion constraints ensure that once a unit is removed from the system for maintenance, it
6. Genetic algorithms A genetic algorithm (GA) can be seen as a search algorithm that is based on the concepts of natural selection and genetic inheritance. It searches an optimal solution to the problems by manipulating a population of strings (chromosomes) that represent different potential solutions, each corresponding to a sample point from the search space. For each generation, all the populations are evaluated based on their fitness. An individual with a larger fitness has a higher chance of evolving into the next generation. By searching many peaks simultaneously, GA reduces the possibility of trapping into a local minimum. GA works with a coding of parameters instead of parameters themselves. The coding of parameters helps the genetic operator to evolve the current state into the next state with minimum computations. GA evaluates the fitness of each string to guide its search instead of the explicit optimisation function. There is no need for computations of derivatives or other auxiliary knowledge. Finally, GA explores the search space where the probability of finding improved performance is high. Therefore, GA is applicable and effective for solving maintenance schedule problems [33,34]. 6.1. Problem encoding In GA, parameters are represented by a string structure called chromosomes. In presented method, real-coded GA
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was used, where each chromosome corresponds to the specific maintenance schedule of units in the power system. The GA always deals with a set of chromosomes called a population. Transforming chromosomes from a population, new population, corresponding to the next generation is obtained. There are three genetic operators to do this: selection, crossover, and mutation. Values of the xk phenotypes in the chromosome are double precision real numbers in interval between zero and one: xk 2 ½0:0; 1:0. Therefore, it is necessary to encode values of MS for generators from initialisation procedure and decode values of chromosomes to maintenance times (periods) in order to calculate their fitness. Values of X i maintenance schedule, received from the approximate initialisation procedure, are encoded to GA’s chromosomes normalising their value in the interval between 0 and 1. The normalisation is done by the division of their value with the number of periods in analysed period m:
The individual unit capacities are arranged in descending order and then added to weekly peak loads. The biggest unit capacity is added to smallest peak load, and this procedure is repeated until all units are arranged in the load diagram. The four major steps in the procedure are:
(1) Arrange generating units by size, with the largest first and the smallest last. (2) Schedule the largest generating unit for maintenance during periods of lowest load, consistent with the minimum and maximum time constraints. (3) Adjust weekly peak load by the generating unit capacity on maintenance. (4) Repeat steps 2–4 until all generating units are scheduled.
Xi , (13) m where m ¼ 52 in case when week is taken as period. Decoding of real values of phenotypes to maintenance times (periods for start of maintenance) is done using decoding procedure. In case of the independent maintenance the decoding is performed as follows:
xk ¼
X g ¼ ninðfirst þ ððlast maintenance firstÞ þ 1Þ xk Þ, (14) where X g is the maintenance time for units received from GA converted to time periods (weeks), xk the values of the maintenance time for units received from GA, first the earliest time period for start of maintenance of unit, last the latest time (week) for terminating the maintenance of a unit, nin the elemental intrinsic FORTRAN NINT function—returns the nearest integer to the argument x, maintenance the maintenance length of unit. During decoding process, using Eq. (14), the constraints of the earliest and the latest time period for start of maintenance of unit and necessary time for finishing maintenance are taken into account. 6.2. An initialisation procedure To improve the effectiveness of the GA in solving the maintenance schedule a separate initialisation procedure was applied. The approach used for the initialisation procedure is to reduce the total installed capacity by the expected capacity loss (product of the unit capacity and its availability) and then schedule the maintenance on a constant reserve basis. The constant reserve maintenance procedure [35] can be written as follows: Constant ¼ weekly peak load þ weekly capacity on maintenance.
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Fig. 4. The flowchart of the optimisation procedure.
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This approximate MS planning technique is simple and straightforward but it cannot include the constraints, which are normal for the operating power system. The maintenance schedule X i calculated by this procedure represents the initialisation value for the GA. During the initialisation of GA, only part of starting population is substituted with results received from approximate MS. Several substitution rates are tested comparing obtained results.
constraint l. The constant penalisation value for W l ¼ 2 was used in the developed method, but the penalty term corresponding to constraint justification can be different for different limitations and its magnitude can model the importance of satisfying that constraint. The other additional constraints can be added to model, including network constraints or energy limitations. 6.4. Selection
6.3. Chromosome fitness Fitness is a quantity related to the chromosome and it enables comparison between them. The decoding process using Eq. (16) ensures that the constraints corresponding to time limitations are satisfied. Other constraints, corresponding to manpower limitation, precedence constraint and unit linkage have to be checked for violations, using the penalty method, which degrades the fitness in cases with violated constraint. The fitness for chromosome is defined by equation !1 X F i ¼ LOLEa Wl , (15) l2VC i
where LOLEa is the objective function related to chromosome i, VC i is set of violated constraints associated to chromosome i, W l is the penalty term corresponding to
Improvement of the average fitness of the population is achieved through selection of individuals as parents from the completed population. The selection is performed in such a way that chromosomes having higher fitness are more likely to be selected as parents. Bearing in mind that some of the individuals may have significantly higher fitness compared to the others, the next generation may be constituted of the large number of identical individuals. This poses a limitation on the population diversity, meaning that the search space will be reduced. One can omit such a situation by using relative fitness. Let F min and F max be the minimal and the maximal fitness in the population, respectively. Then the relative fitness for chromosome i is defined as G i Gi ¼
F i F min . F max F min
(16)
Table 1 Calculated values of the forecasted outage rate for the thermal power plants in Macedonia power system Power plant
Bitola 1
Bitola 2
Bitola 3
Oslomej
Negotino
Average
FOR mean
0.011157
0.016569
0.009685
0.13292
0.008038
0.035674
Initial pop. substitution
Annual LOLE days/year
Calculation min
9 9 20 20 50 50
0.129 0.132 0.129 0.132 0.124 0.129 0.127 0.128
11.2 2.3 11.16 2.31 11.18 2.45 12.05 2.4
9 9 20 20 50 50
0.256 0.256 0.256 0.258 0.249 0.258 0.250 0.258
11.61 2.41 12.26 2.41 11.56 2.56 11.56 2.38
Table 2 The results for the Macedonian power system for years 2000 and 2001 for different scenarios Size of population GA Scenario for year 2000 1 (500) 2 (100) 3 (500) 4 (100) 5 (500) 6 (100) 7 (500) 8 (100)
No substitution–initiation No substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation
1 2 3 4 5 6 7 8
No substitution–initiation No substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation Substitution–initiation
(500) (100) (500) (100) (500) (100) (500) (100)
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Therefore, the likelihood of selecting a chromosome as a parent is a function of its fitness relative to the sum fitness of all chromosomes. To further improve the evolving process, the GA can carry over the best individuals from the completed population to the new population set (principle of elitism).
any position (the crossover point) and exchanges the substrings between the two chromosomes. After the crossover is performed, the new chromosomes are added to the new population set. The mutation is specifically applied to increase population diversity. Mutation involves randomly selecting genes within the chromosomes and assigning them random values within the corresponding predefined interval. In order not to destroy good genetic code, non-uniform mutation has to be applied. In such a manner, in later stages of GA optimisation process, the interval for random selection of genes’ values is narrowed.
6.5. Crossover and mutation
6.6. GA parameters
After the selection, the GA picks a pair of selected chromosomes in order to create two new chromosomes. The GA applies a random generator to cut the strings at
GA requires definition of a number of parameters, which can affect the efficiency of the search process in several ways. The population size should be large enough to create
There are several selection techniques. The roulette selection method is used here. In this method, the relative weight of each chromosome’s fitness is calculated as Gi pi ¼ PN POP k¼1
Gk
.
(17)
Fig. 5. The peak loads and the power of generators in maintenance for each week in years 2000 and 2001.
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sufficient diversity covering the possible solution space. Generally, one cannot know the optimal value in advance. Clearly, a more complex problem domain requires a larger population due to the larger possible combination of variables. Another user-defined criterion is the point at which the optimisation process terminates. GA with fixed number of generations is used in the developed method, for
which it is assumed that the search process has covered a sufficient search space. Other parameters, such as crossover probability, mutation rate, selection, and crossover mechanisms, seem to affect the GA process less significantly when evaluated over a larger number of generations. The flowchart of the developed optimisation procedure is shown in Fig. 4.
Fig. 6. The maintenance schedule in years 2002 and 2001.
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7. Analysis and results The proposed method was tested using the Macedonian power system data for years 2000 and 2001. The input considered 29 generating units in the power generating system. The calculated values of forecasted outage rate of thermal power plants in Macedonian power system used in analysis are given in Table 1. The length of the planning horizon was one year with value of LOLEp calculated for each week. The method included the following constraints: the maintenance completion, crew, precedence, reserve and resource constraints. Multiple simulations were done, chaining number of populations in GA and analysing cases with and without initialisation of GA with approximate MS planning technique. Results are given in Table 2. Results show improved MS for generators using initialisation of GA with approximate MS planning technique, particularly in case of small populations. Substitution of the population should be done carefully, because higher rates of substitution can limit search space of GA resulting with higher value of annual LOLE. Calculation time is proportional to
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number of populations and no improvement of objective function was observed with the increase of number of populations above 500. The peak loads and the power of generators on maintenance are shown in Fig. 5 for each week for year 2000 and 2001, respectively. The detailed maintenance schedule for years 2000 and 2001 (the green area in Fig. 5) is given in Fig. 6. Thermal power plants, which have similar size and FOR, are put in the periods of lower loads and smaller generators in hydro power plants are consecutively added to them. The results shown in Figs. 5 and 6 look similar to those obtained by constant reserve maintenance procedure but it should be noted that in the MS of the units their size and corresponding FOR was taken into account. The evolution of annual LOLE in years 2000 and 2001 with approximate MS planning technique is given in Fig. 7. The results show the effect of providing a good start-up population to genetic GA especially in case of small number of generations. With the increase of the number of generations similar results are obtained by standard initiation procedure, with random number generator, and with approximate MS planning technique. A comparison
Fig. 7. Evolution of the annual LOLE in years 2002 and 2001.
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Table 3 Comparison of the results for the Macedonian power system Annual LOLE days/year Input year 2000 1 Approximate method 2 New method
0.229 0.124
Input year 2001 1 Approximate method 2 New method
0.312 0.249
of results obtained from the approximate method and the new method are given in Table 3. The results show that the new method has annual value of LOLE about 1 day/year smaller than the approximate method, taking account all constraints of MS operations.
8. Conclusions A new method for optimisation of the MS of generating units in a power system is developed. Maintenance is scheduled to minimise the risk through minimisation of the yearly value of the LOLE taken as a measure of the power system reliability. The proposed method uses GA to obtain the best solution resulting in a minimal value of the annual loss of load expectation value for the power system in the analysed period. The operational constraints for generating units are included in the method. The proposed algorithm was tested on a Macedonian power system and the obtained results were compared with the results received from the approximate methodology. The results show the improved reliability of a power system with the maintenance schedule obtained by the new method compared to the results from the approximate methodology. The new method is developed in a way that enables easy variations of optimised function and inclusion of additional constraints. Different reliability measures and constraints for the system can be used (e.g. loss of energy expectation LOEE) depending on the system characteristics and the operators requests. Minimising the total generator operating cost as the objective function can be additionally modelled and implemented in the algorithm, if the additional data of the power generating units is available, including price of energy sources for the power plants in the analysed period and the price of electrical energy on market for the selected period. Network contingencies can be also additionally included in the model. The algorithm of the new method and its corresponding computer code offer a promising approach for the MS of generating units. The method enables improvement of the reliability with no substantial expenses for the owner and operators of the power system.
Acknowledgements This research was partly supported by the Slovenian research Agency (contract number 1000-05-310016). Authors acknowledge support from Faculty of Electrical Engineering—Skopje, Republic of Macedonia.
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