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A fuzzy evolutionary programming-based solution methodology for security-constrained generation maintenance scheduling
Electric Power Systems Research 67 (2003) 67 /72 www.elsevier.com/locate/epsr
A fuzzy evolutionary programming-based solution methodology for security-constrained generation maintenance scheduling M.Y. El-Sharkh a, A.A. El-Keib b,*, H. Chen c a
b
Department of Electrical and Computer Engineering, University of South Alabama, Mobile, AL 36688, USA Department of Electrical and Computer Engineering, The University of Alabama, Tuscaloosa, AL 35487-0286, USA c Department of Computer Science, The University of Alabama, Tuscaloosa, AL 35487-0286, USA Received 26 November 2002; accepted 21 February 2003
Abstract A traditional mathematical model for maintenance scheduling of power generation systems may give an optimal schedule for a power system with known conditions. A change of the system condition due to uncertainties or sudden changes may render the resulting optimal schedule unsuitable or inapplicable for the power system under study. This paper presents a fuzzy model and an evolutionary programming-based solution technique for the security-constrained maintenance scheduling (MS) problem of generation systems with uncertainties in the load and fuel and maintenance costs. The proposed technique results are fuzzy optimal cost range that reflects the problem uncertainties. The technique solves a decomposed maintenance model of two interrelated subproblems, namely the maintenance and the security-constrained economic dispatch problem. Test results on the IEEE 30-bus system with six generating units reported in this paper are quite encouraging. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Power systems; Optimization; Evolutionary programming; Maintenance scheduling; Fuzzy modeling
1. Introduction Maintenance scheduling as a tool to control the system cost and reliability plays a very important role in power system cost and risk management. The MS problem can be described as determining the optimal starting time for each preventive maintenance outage in a weekly period for 1 year in advance, while satisfying the system constraints and maintaining system reliability. The problem is a minimization one that has two types of decision variables; one is continuous and the other is integer. The continuous variables are the generator output, while the integer variables are the maintenance scheduling variables that define the maintenance starting dates and the maintenance duration for each unit. Solving an optimization problem that contains such variables makes it extremely difficult if not impossible to reach the global or the near-global optimum. Especially when the problem involves a * Corresponding author. E-mail address: [email protected] (A.A. El-Keib).
number of uncertainties. The maintenance scheduling problem has uncertainties associated with it that include: . load forecast, which is subject to increase or decrease by 2 /5% . fuel price and labors fees that are subject to change due to the fuel market prices . amount of reserve and purchased power from neighboring utilities, which could change from time to time . available resources, which also are variant from time to time . maintenance crew availability, which is dependent on the maintenance time and the weather conditions. Solving the model of the MS problem by the existing techniques [1 /8] may give the solution for specified conditions of the power system. If the system condition is changed, for example the system load is increased, the optimal solution may become unsuitable or inapplicable. To remedy this problem, the MS model should incorporate the above mentioned uncertainties. There are two options to handle the uncertainties involved in
0378-7796/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-7796(03)00076-2
68
M.Y. El-Sharkh et al. / Electric Power Systems Research 67 (2003) 67 /72
the maintenance scheduling problem: (1) probabilistic modeling [3]; and (2) fuzzy modeling. While probabilistic modeling can handle different kinds of uncertainties, it does not have the ability to represent the qualitative and subjective uncertainties. For example, the probabilistic modeling can deal with the statistical uncertainty, such as changing of the load by ‘9/3%’, but it is unable to represent the qualitative uncertainty such as the change of the load is ‘approximately 9/3%’ or linguistic variables, such as the change in the load is ‘quite big’. The MS problem has subjective uncertainties, such as dependency of crew members’, availability on the weather conditions and the availability of the material needed for each maintenance outage. In addition to the fact that probabilistic modeling cannot simulate the subjective uncertainties, it also uses the convolution technique to calculate the probability density function of the state variables. The complexity of the convolution technique adds more mathematical burden to the solution of the maintenance scheduling problem. On the other hand, a fuzzy modeling has the ability to capture statistical, qualitative and subjective uncertainties in the maintenance problem model through the use of membership functions. Such function determines the degree of relevance and inclusion of each element included in a fuzzy set. The membership function can take a value between 1 and 0. The closer the membership value of an element to 1, the more it belongs to the predefined fuzzy set. Using membership functions and fuzzy rules, fuzzy logic can implement the human experiences and simulate different kinds of uncertainties. This paper proposes the use of fuzzy logic techniques to handle the MS problem uncertainties and a technique based on evolutionary programming (EP) to find a nearglobal optimal solution for the resulting fuzzy optimization problem. Section 2 of the paper describes the MS problem and the fuzzy mathematical model. Section 3 presents the proposed fuzzy EP-based solution technique. The test results and a discussion of results are included in Sections 4 and 5, respectively. Section 6 presents conclusions.
2. A fuzzy model for the maintenance scheduling The mathematical model of the problem in fuzzy form is stated as follows: XX X Min ait Hi (Pit )+(1yit ) Cit +yit (1) t
i
Subject to: 1 for xi 5 t5 xi di yit 0 otherwise
i
(2)
X
yit nc
Öt
(3)
i
X
yit 1
Öi
(4)
t
5Pit 5Pmax Öi; Öt Pmin i i X Pit (1yit )Dt Öt
(5) (6)
i
jFk j5Fkmax
k fKg
(7)
where ait is the fuzzy number to represent the fuel price; Hi , the fuzzy quadratic function to represent the heat rate of unit i; Pit , the fuzzy number to represent the output power from unit i at time t; Cit is the fuzzy number to represent the maintenance cost of unit i at time t; yit , 1 if unit i on maintenance at time period t (weeks) and 0 otherwise; nc, the total number of available maintenance crews; xi , the starting time of maintenance for unit i; di , the duration of maintenance for unit i ; Dt , the fuzzy number to represent the system load; Fk is the fuzzy number to represent the line flows; Pmax the maximum output power from unit i ; Pmin , the i i minimum output power from unit i ; Fmax is the k maximum allowable power flow on transmission line k ; and K is the set of the system transmission and tie lines. The first part of the objective Eq. (1) represents the fuzzy production cost, which is described using fuzzy fuel price and quadratic heat rate curves. The second part of the objective function is the fuzzy maintenance cost. For the same unit, this cost can vary from one week to another depending on the available resources, weather conditions and maintenance crew availability. The scheduling constraints are as follows: / Constraint (2) represents the maintenance window. It is set to 1 year for each unit. / Constraint (3) is the crew constraint; it only assigns one crew to one unit at a time. / Constraint (4) limits the number of outages for the same unit to one time during the time horizon. The power system constraints are as follows: / Constraint (5) is to ensure that each unit is operating within its output limits. / Constraint (6) is to guarantee that the on-line units can meet the system demand. / Constraint (7) is the line flow constraint. To simulate the demand uncertainty, a triangular membership function is used as shown in Fig. 1(a). The nominal load value has the maximum grade 1 of the membership function. The change in the load is represented by /Dd1 and /Dd2. The load membership function can be expressed as follows:
M.Y. El-Sharkh et al. / Electric Power Systems Research 67 (2003) 67 /72
69
Fig. 1. The demand and cost membership functions.
8 0 > > > > x (D Dd1 ) > > > > < Dd
xBDDd1 DDd1 5x BD
1
mp 1 > > D Dd2 x > > > > > Dd2 > : 0
xD
(8)
D Bx5 DDd2 xDDd2
where x represents the load value. The uncertainties associated with the maintenance cost are due to the change of market prices, availability of the spare parts, the weather conditions and the availability of the maintenance crew. The maintenance cost uncertainty can also be simulated by a triangular membership function, where the most probable cost value for each unit has the maximum membership value. The production cost is a fuzzy number, which depends on the price of fuel at the time of maintenance. The fuel price subjectively depends on the current market prices. Therefore, the maintenance and the production costs are fuzzy in nature and they can be approximated by a triangular fuzzy number, as shown in Fig. 1(b). Mathematically, the cost membership function is represented by Eq. (9) 8 0 xBCi DC1 > > > > x (C DC ) > i 1 > > Ci DC1 5 xBCi > > DC1 < xCi mc 1 (9) > > C DC x > 2 > i > Ci Bx5Ci DC2 > > > DC2 > : 0 xCi DC2 where /DC1 and /DC2 represent the change in the cost and x represent the cost.
3. The proposed evolutionary programming (EP)-based solution methodology Evolutionary programming can be traced back to the early 1950s when Turing discovered a relationship between machine learning and evolution [9 /11]. Later, Bremermann, Box, Friedberg and others put the bases for the evolutionary computation as a tool for machine learning and an optimization technique. Great attention
was given to EP as a powerful tool when Fogal, Burgin, Atmar and others used it to create artificial intelligence to predict the events of a finite state machines (FSM) on the bases of old observations. During the 1980s, evolutionary programming, with advances of the computer performance, was used to solve difficult real-world optimization problems. In the power systems area, EP has been used to solve a number of power systems problems [11]. Evolutionary programming is a search optimization method. It moves from one solution to another using a probabilistic search technique. The EP starts with random individuals. Each individual represents a complete solution for the problem under study. The individuals are moved from one generation or iteration to the other after passing through two main steps, the mutation and the competition. During the mutation step, a Gaussian random variable with uniform probability is added to all the current individuals. The competition steps is a probabilistic selection scheme used to assign a weight to each offspring individual according to a comparison between a current individual and other randomly chosen one. It may happen that the new solution is infeasible. Therefore, using EP alone may require a long time to reach the optimal solution or it may get trapped in a local optimum. This proposed technique uses the hill-climbing technique (HCT) [12,13] in conjunction with EP to move new infeasible solutions into the feasible region. Using the EP search ability and the feasibility watch by the HCT motivates the decomposition of the MS problem into two interrelated subproblems and their sequential solutions. These subproblems are the maintenance subproblem with integer variables and the security-constrained economic dispatch problem with continuous variables. Referring to the MS model given in Section 2, the maintenance subproblem consists of the second part of the objective function subject to constraints Eqs. (2) /(4). The power system subproblem consists of the first part of the objective function subject to constraints Eqs. (5) /(7). Using the HCT to derive the infeasible solution to the feasible region avoids the need to add any additional constraints to the subproblems. The power system subproblem is solved using a fuzzy security-constrained economic dispatch program. The flows on the system
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M.Y. El-Sharkh et al. / Electric Power Systems Research 67 (2003) 67 /72
transmission lines are calculated using the DC power flow. Generation redispatch is used to remove line violations. It utilizes generation shift factors. Following is the proposed solution algorithm: 1)
Generate crisp random solutions for the staring time of maintenance of each generating unit. Si fxg
2)
3)
4)
i 1; . . . ; m
j 1; . . . ; k
5)
6) 7)
8)
9)
11)
12)
(10)
where x is a set of maintenance starting times for each unit; and m , the number of parents in the current generation. For each individual in the current generation, calculate the on/off weeks for each unit yit using the starting time xi and the given maintenance duration for each unit. The random solution is expected to satisfy both the maintenance and system constraints. To check the feasibility of the randomly generated solution for the power system subproblem, a fuzzy economic dispatch program and a DC load flow are used. For each individual in the current generation, calculate the overall fuzzy objective function value using Eq. (1). Mutate each individual and assign it to Sim according to Eq. (11). Sim;j Si;j N(0; bi v(Si )zj )
10)
13)
Wi
N X
Wi;j
(12)
j1
where N is a randomly generated competition number; Wi,j , either 0 or 1 depending on the competition of the individual with another individual selected randomly from the population. The value of Wi,j can be calculated as follows: 1 if v(Si )5v(Sp ) Wi;j (13) 0 otherwise
(11)
where Sij is the j th schedule of the ith individual; k , the number of generating units to be maintained in the current individual; N (m, s2), the Gaussian random variable with mean m and variance s2; bi , a constant to scale v (Si ); and zj , an offset to guarantee a minimum amount of variance. Check the feasibility of each new individual against the constraints of the maintenance subproblem. If there is no violation go to step 6. Otherwise go to step 11. For the new individual, calculate the on and off weeks for each unit as described in step 2. Using the feasible solution for the maintenance subproblem, solve the fuzzy dispatch problem to obtain the membership function of the optimal output and the production cost. If no feasible solution for the dispatch problem was found go to step 11. Using the membership function of the output of each generating unit, check for line flow constraint violation. If none is identified go to step 10, otherwise continue to step 9. Redispatch the units to relieve the overloaded lines. If one or more of the line violations persists go to step 11, otherwise continue to step 10.
Calculate the overall fuzzy objective function for the feasible solution using Eq. (1) and go to step 12. Use the hill-climbing algorithm to drive the infeasible individuals into feasibility. If no feasible solution can be found go back to step 4. Assign a fitness score v (Si ) to each individual Si m (i /1,. . ., 2m ). The score is assigned equal to the membership function of its overall fuzzy cost function. Using Eq. (12), calculate a weight Wi for each individual Si , i/1,. . ., 2m . These weights are to be calculated during a random competition between individuals. Comparison of the different score memberships functions is carried out using the fuzzy comparison technique proposed by the authors [13].
14)
15)
with p/[2mu1/1], p "/i and u1 /U (0, 1) Rank the solution Si (i/1,. . ., 2m ) in descending order according to their values of Wi (if more than one solution has the same Wi , use the actual score of v (Si ) to rank them). Use the first m solutions along with their score values v(Si ) as a new generation for the potential optimal solution. Check for convergence. Criteria used for convergence include the maximum generation number and the average/maximum fitness ratio being less than a predetermined small value. If convergence is achieved, stop; otherwise go to step 4.
4. Test results The proposed method was tested for solving the security-constrained fuzzy MS problem on the IEEE 30-bus system with six generating units and 41 transmission lines. Because of the lack of maintenance cost data for the test system, the authors have selected the costs given in [13]. The ability of the technique to handle varying maintenance costs was tested using maintenance costs that were different for different weeks. Different maintenance periods of 1/3 weeks were used for different generating units. The weekly load curve was created as a percent of the peak load of the system. An
M.Y. El-Sharkh et al. / Electric Power Systems Research 67 (2003) 67 /72
:/5% uniform reserve throughout the maintenance horizon is considered. Reserve is accounted for by increasing the system load by the specified amount of reserve. The parameters used are one maintenance crew, :/5.0% reserve and 52 weeks horizon. To validate its performance, the proposed EP-based technique was compared against the complementary decision variables structure technique (CDV) through testing on three cases on the IEEE 30-bus system. The proposed EPbased technique produced better results than the CDV approach [13]. This case is solved for two different scenarios, the first assumes normal line loadings and the second considers the system behavior under limited lines thermal capacity. The results are given in Tables 1 and 2.
5. Discussion of results Testing revealed that optimal EP parameters for the test system of three individuals and 1000 generation were most suitable for the test case. Higher values of these parameters did not improve the solution. For Case 1, with :/5% reserve and normal line capacity limits, the proposed technique found the optimal solution at generation number 753 with cost components given in Table 1. The effect of line limited capacity was indicated by the increase of the cost ranges, as given in Table 1. The maintenance schedule for each generating unit for both cases is shown in Table 2. Limited line thermal capacities produced a larger number of line flow violations, narrowed the solution space and made it a challenging task to find the optimal solution. The algorithm first tries to remove the violation by rescheduling generation. If the violations persisted, the algorithm finds a new maintenance schedule to remove the line violation(s). As shown in Table 2, the maintenance schedule of unit 1 /3 did not change. However, the maintenance schedule for unit 4/6 was shifted to one that would satisfy the line flow constraints. Shifting the maintenance schedule had to be, of course, at the expense of increasing the maintenance and production costs. With the new maintenance schedule obtained in Case 2, generation had to be redispatched to remove Table 1 The fuzzy EP solution
Production
Case 1
Case 2
{$6062865.06 $6343355.93}
{$6063977.63 $6344443.53} {$85403.00 $90860.00}
Maintenance {$84126.00 $90760.00} Total cost
{$6146991.06 $6434115.93}
{$6149380.63 $6435303.53}
71
Table 2 Maintenance period of system generating unit with :/5% reserve Unit
1
2
3
4
5
6
Case 1 Case 2
34 /36 34 /36
29, 31 29, 31
31, 32 31, 32
28 37
33 27
39 39
some of the line violations that in turn caused, as expected, the production cost to increase (Table 1).
6. Conclusion This paper presented a fuzzy model and a new technique to solve the fuzzy security-constrained MS problem for power generation systems. The approach handles uncertainties associated with the system load over the maintenance horizon and the maintenance and fuel cost variations due to market prices changes. The approach takes advantage of the search ability of the evolutionary programming and the feasibility watch of the hill-climbing technique to achieve a near-optimal solution. The generated solution is a fuzzy range for each of the maintenance and production costs that reflect the problem uncertainties. Test results on the IEEE 30-bus system indicated the viability of the proposed approach and its potential to solve the MS problem with various types of uncertainties.
References [1] M.K.C. Marwali, S.M. Shahidehpour, Integrated generation and transmission maintenance scheduling with network constraints, IEEE Transactions on Power Systems 13 (3) (1998) 1063. [2] M.K.C. Marwali, Transmission and generation maintenance scheduling with different time scales in power systems, Ph.D. Dissertation, Illinois Institute of Technology, 1997. [3] M. Shahidehpour, M. Marwali, Maintenance Scheduling in Restructured Power Systems, Kluwer, New York, 2000. [4] M.Y. El-Sharkh, R. Yasser, A.A. El-Keib, Optimal maintenance scheduling for power generation systems */a literature review. Maintenance and Reliability Conference. Proceedings MARCON ’98, vol. 1, 1998, pp. 20.01 /20.10. [5] J.P. Stremel, Maintenance scheduling for generation system planning, IEEE Transactions on Power Apparatus and Systems PAS-100 (3) (1981) 4010 /4019. [6] Z.A. Yamayee, K. Sidenblad, A computationally efficient optimal maintenance scheduling method, IEEE Transactions on Power Apparatus and Systems PAS-102 (2) (1983) 330 /338. [7] C.E. Lin, C.J. Huang, C.L. Huang, S.Y. Lee, An optimal generator maintenance approach using fuzzy dynamic programming, IEEE PES Summer Meeting, 92 SM 401-0 PWRS, Seattle, July 1992. [8] S.J. Huang, Maintenance scheduling of generating units by genetic-enhanced fuzzy systems, International Conference on Intelligent System Application to Power System, Seoul, 1997, pp. 541 /545. [9] D.B. Fogel, Evolutionary Computation Toward a New Philosophy of Machine Intelligence, IEEE Press.
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[10] T. Back, U. Hammel, H.P. Schwefel, Evolutionary computation: comments on the history and current state, IEEE Transactions on Evolutionary Computation 1 (1) (1997) 3 /17. [11] Miranda, D. Srinivasan, L.M. Proenca, Evolutionary computation in power system, Electrical Power and Energy Systems 20 (2) (1998) 89 /98.
[12] P.H. Winston, Artificial Intelligence, third ed, Addison-Wesley, New York, 1993. [13] M.Y. El-Sharkh, A fuzzy evolutionary programming-based methodology for maintenance scheduling of generation and transmission systems, Ph.D. Dissertation, Electrical and Computer Engineering Department, The University of Alabama, 2002.
A fuzzy evolutionary programming-based solution methodology for security-constrained generation maintenance scheduling M.Y. El-Sharkh a, A.A. El-Keib b,*, H. Chen c a
b
Department of Electrical and Computer Engineering, University of South Alabama, Mobile, AL 36688, USA Department of Electrical and Computer Engineering, The University of Alabama, Tuscaloosa, AL 35487-0286, USA c Department of Computer Science, The University of Alabama, Tuscaloosa, AL 35487-0286, USA Received 26 November 2002; accepted 21 February 2003
Abstract A traditional mathematical model for maintenance scheduling of power generation systems may give an optimal schedule for a power system with known conditions. A change of the system condition due to uncertainties or sudden changes may render the resulting optimal schedule unsuitable or inapplicable for the power system under study. This paper presents a fuzzy model and an evolutionary programming-based solution technique for the security-constrained maintenance scheduling (MS) problem of generation systems with uncertainties in the load and fuel and maintenance costs. The proposed technique results are fuzzy optimal cost range that reflects the problem uncertainties. The technique solves a decomposed maintenance model of two interrelated subproblems, namely the maintenance and the security-constrained economic dispatch problem. Test results on the IEEE 30-bus system with six generating units reported in this paper are quite encouraging. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Power systems; Optimization; Evolutionary programming; Maintenance scheduling; Fuzzy modeling
1. Introduction Maintenance scheduling as a tool to control the system cost and reliability plays a very important role in power system cost and risk management. The MS problem can be described as determining the optimal starting time for each preventive maintenance outage in a weekly period for 1 year in advance, while satisfying the system constraints and maintaining system reliability. The problem is a minimization one that has two types of decision variables; one is continuous and the other is integer. The continuous variables are the generator output, while the integer variables are the maintenance scheduling variables that define the maintenance starting dates and the maintenance duration for each unit. Solving an optimization problem that contains such variables makes it extremely difficult if not impossible to reach the global or the near-global optimum. Especially when the problem involves a * Corresponding author. E-mail address: [email protected] (A.A. El-Keib).
number of uncertainties. The maintenance scheduling problem has uncertainties associated with it that include: . load forecast, which is subject to increase or decrease by 2 /5% . fuel price and labors fees that are subject to change due to the fuel market prices . amount of reserve and purchased power from neighboring utilities, which could change from time to time . available resources, which also are variant from time to time . maintenance crew availability, which is dependent on the maintenance time and the weather conditions. Solving the model of the MS problem by the existing techniques [1 /8] may give the solution for specified conditions of the power system. If the system condition is changed, for example the system load is increased, the optimal solution may become unsuitable or inapplicable. To remedy this problem, the MS model should incorporate the above mentioned uncertainties. There are two options to handle the uncertainties involved in
0378-7796/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-7796(03)00076-2
68
M.Y. El-Sharkh et al. / Electric Power Systems Research 67 (2003) 67 /72
the maintenance scheduling problem: (1) probabilistic modeling [3]; and (2) fuzzy modeling. While probabilistic modeling can handle different kinds of uncertainties, it does not have the ability to represent the qualitative and subjective uncertainties. For example, the probabilistic modeling can deal with the statistical uncertainty, such as changing of the load by ‘9/3%’, but it is unable to represent the qualitative uncertainty such as the change of the load is ‘approximately 9/3%’ or linguistic variables, such as the change in the load is ‘quite big’. The MS problem has subjective uncertainties, such as dependency of crew members’, availability on the weather conditions and the availability of the material needed for each maintenance outage. In addition to the fact that probabilistic modeling cannot simulate the subjective uncertainties, it also uses the convolution technique to calculate the probability density function of the state variables. The complexity of the convolution technique adds more mathematical burden to the solution of the maintenance scheduling problem. On the other hand, a fuzzy modeling has the ability to capture statistical, qualitative and subjective uncertainties in the maintenance problem model through the use of membership functions. Such function determines the degree of relevance and inclusion of each element included in a fuzzy set. The membership function can take a value between 1 and 0. The closer the membership value of an element to 1, the more it belongs to the predefined fuzzy set. Using membership functions and fuzzy rules, fuzzy logic can implement the human experiences and simulate different kinds of uncertainties. This paper proposes the use of fuzzy logic techniques to handle the MS problem uncertainties and a technique based on evolutionary programming (EP) to find a nearglobal optimal solution for the resulting fuzzy optimization problem. Section 2 of the paper describes the MS problem and the fuzzy mathematical model. Section 3 presents the proposed fuzzy EP-based solution technique. The test results and a discussion of results are included in Sections 4 and 5, respectively. Section 6 presents conclusions.
2. A fuzzy model for the maintenance scheduling The mathematical model of the problem in fuzzy form is stated as follows: XX X Min ait Hi (Pit )+(1yit ) Cit +yit (1) t
i
Subject to: 1 for xi 5 t5 xi di yit 0 otherwise
i
(2)
X
yit nc
Öt
(3)
i
X
yit 1
Öi
(4)
t
5Pit 5Pmax Öi; Öt Pmin i i X Pit (1yit )Dt Öt
(5) (6)
i
jFk j5Fkmax
k fKg
(7)
where ait is the fuzzy number to represent the fuel price; Hi , the fuzzy quadratic function to represent the heat rate of unit i; Pit , the fuzzy number to represent the output power from unit i at time t; Cit is the fuzzy number to represent the maintenance cost of unit i at time t; yit , 1 if unit i on maintenance at time period t (weeks) and 0 otherwise; nc, the total number of available maintenance crews; xi , the starting time of maintenance for unit i; di , the duration of maintenance for unit i ; Dt , the fuzzy number to represent the system load; Fk is the fuzzy number to represent the line flows; Pmax the maximum output power from unit i ; Pmin , the i i minimum output power from unit i ; Fmax is the k maximum allowable power flow on transmission line k ; and K is the set of the system transmission and tie lines. The first part of the objective Eq. (1) represents the fuzzy production cost, which is described using fuzzy fuel price and quadratic heat rate curves. The second part of the objective function is the fuzzy maintenance cost. For the same unit, this cost can vary from one week to another depending on the available resources, weather conditions and maintenance crew availability. The scheduling constraints are as follows: / Constraint (2) represents the maintenance window. It is set to 1 year for each unit. / Constraint (3) is the crew constraint; it only assigns one crew to one unit at a time. / Constraint (4) limits the number of outages for the same unit to one time during the time horizon. The power system constraints are as follows: / Constraint (5) is to ensure that each unit is operating within its output limits. / Constraint (6) is to guarantee that the on-line units can meet the system demand. / Constraint (7) is the line flow constraint. To simulate the demand uncertainty, a triangular membership function is used as shown in Fig. 1(a). The nominal load value has the maximum grade 1 of the membership function. The change in the load is represented by /Dd1 and /Dd2. The load membership function can be expressed as follows:
M.Y. El-Sharkh et al. / Electric Power Systems Research 67 (2003) 67 /72
69
Fig. 1. The demand and cost membership functions.
8 0 > > > > x (D Dd1 ) > > > > < Dd
xBDDd1 DDd1 5x BD
1
mp 1 > > D Dd2 x > > > > > Dd2 > : 0
xD
(8)
D Bx5 DDd2 xDDd2
where x represents the load value. The uncertainties associated with the maintenance cost are due to the change of market prices, availability of the spare parts, the weather conditions and the availability of the maintenance crew. The maintenance cost uncertainty can also be simulated by a triangular membership function, where the most probable cost value for each unit has the maximum membership value. The production cost is a fuzzy number, which depends on the price of fuel at the time of maintenance. The fuel price subjectively depends on the current market prices. Therefore, the maintenance and the production costs are fuzzy in nature and they can be approximated by a triangular fuzzy number, as shown in Fig. 1(b). Mathematically, the cost membership function is represented by Eq. (9) 8 0 xBCi DC1 > > > > x (C DC ) > i 1 > > Ci DC1 5 xBCi > > DC1 < xCi mc 1 (9) > > C DC x > 2 > i > Ci Bx5Ci DC2 > > > DC2 > : 0 xCi DC2 where /DC1 and /DC2 represent the change in the cost and x represent the cost.
3. The proposed evolutionary programming (EP)-based solution methodology Evolutionary programming can be traced back to the early 1950s when Turing discovered a relationship between machine learning and evolution [9 /11]. Later, Bremermann, Box, Friedberg and others put the bases for the evolutionary computation as a tool for machine learning and an optimization technique. Great attention
was given to EP as a powerful tool when Fogal, Burgin, Atmar and others used it to create artificial intelligence to predict the events of a finite state machines (FSM) on the bases of old observations. During the 1980s, evolutionary programming, with advances of the computer performance, was used to solve difficult real-world optimization problems. In the power systems area, EP has been used to solve a number of power systems problems [11]. Evolutionary programming is a search optimization method. It moves from one solution to another using a probabilistic search technique. The EP starts with random individuals. Each individual represents a complete solution for the problem under study. The individuals are moved from one generation or iteration to the other after passing through two main steps, the mutation and the competition. During the mutation step, a Gaussian random variable with uniform probability is added to all the current individuals. The competition steps is a probabilistic selection scheme used to assign a weight to each offspring individual according to a comparison between a current individual and other randomly chosen one. It may happen that the new solution is infeasible. Therefore, using EP alone may require a long time to reach the optimal solution or it may get trapped in a local optimum. This proposed technique uses the hill-climbing technique (HCT) [12,13] in conjunction with EP to move new infeasible solutions into the feasible region. Using the EP search ability and the feasibility watch by the HCT motivates the decomposition of the MS problem into two interrelated subproblems and their sequential solutions. These subproblems are the maintenance subproblem with integer variables and the security-constrained economic dispatch problem with continuous variables. Referring to the MS model given in Section 2, the maintenance subproblem consists of the second part of the objective function subject to constraints Eqs. (2) /(4). The power system subproblem consists of the first part of the objective function subject to constraints Eqs. (5) /(7). Using the HCT to derive the infeasible solution to the feasible region avoids the need to add any additional constraints to the subproblems. The power system subproblem is solved using a fuzzy security-constrained economic dispatch program. The flows on the system
70
M.Y. El-Sharkh et al. / Electric Power Systems Research 67 (2003) 67 /72
transmission lines are calculated using the DC power flow. Generation redispatch is used to remove line violations. It utilizes generation shift factors. Following is the proposed solution algorithm: 1)
Generate crisp random solutions for the staring time of maintenance of each generating unit. Si fxg
2)
3)
4)
i 1; . . . ; m
j 1; . . . ; k
5)
6) 7)
8)
9)
11)
12)
(10)
where x is a set of maintenance starting times for each unit; and m , the number of parents in the current generation. For each individual in the current generation, calculate the on/off weeks for each unit yit using the starting time xi and the given maintenance duration for each unit. The random solution is expected to satisfy both the maintenance and system constraints. To check the feasibility of the randomly generated solution for the power system subproblem, a fuzzy economic dispatch program and a DC load flow are used. For each individual in the current generation, calculate the overall fuzzy objective function value using Eq. (1). Mutate each individual and assign it to Sim according to Eq. (11). Sim;j Si;j N(0; bi v(Si )zj )
10)
13)
Wi
N X
Wi;j
(12)
j1
where N is a randomly generated competition number; Wi,j , either 0 or 1 depending on the competition of the individual with another individual selected randomly from the population. The value of Wi,j can be calculated as follows: 1 if v(Si )5v(Sp ) Wi;j (13) 0 otherwise
(11)
where Sij is the j th schedule of the ith individual; k , the number of generating units to be maintained in the current individual; N (m, s2), the Gaussian random variable with mean m and variance s2; bi , a constant to scale v (Si ); and zj , an offset to guarantee a minimum amount of variance. Check the feasibility of each new individual against the constraints of the maintenance subproblem. If there is no violation go to step 6. Otherwise go to step 11. For the new individual, calculate the on and off weeks for each unit as described in step 2. Using the feasible solution for the maintenance subproblem, solve the fuzzy dispatch problem to obtain the membership function of the optimal output and the production cost. If no feasible solution for the dispatch problem was found go to step 11. Using the membership function of the output of each generating unit, check for line flow constraint violation. If none is identified go to step 10, otherwise continue to step 9. Redispatch the units to relieve the overloaded lines. If one or more of the line violations persists go to step 11, otherwise continue to step 10.
Calculate the overall fuzzy objective function for the feasible solution using Eq. (1) and go to step 12. Use the hill-climbing algorithm to drive the infeasible individuals into feasibility. If no feasible solution can be found go back to step 4. Assign a fitness score v (Si ) to each individual Si m (i /1,. . ., 2m ). The score is assigned equal to the membership function of its overall fuzzy cost function. Using Eq. (12), calculate a weight Wi for each individual Si , i/1,. . ., 2m . These weights are to be calculated during a random competition between individuals. Comparison of the different score memberships functions is carried out using the fuzzy comparison technique proposed by the authors [13].
14)
15)
with p/[2mu1/1], p "/i and u1 /U (0, 1) Rank the solution Si (i/1,. . ., 2m ) in descending order according to their values of Wi (if more than one solution has the same Wi , use the actual score of v (Si ) to rank them). Use the first m solutions along with their score values v(Si ) as a new generation for the potential optimal solution. Check for convergence. Criteria used for convergence include the maximum generation number and the average/maximum fitness ratio being less than a predetermined small value. If convergence is achieved, stop; otherwise go to step 4.
4. Test results The proposed method was tested for solving the security-constrained fuzzy MS problem on the IEEE 30-bus system with six generating units and 41 transmission lines. Because of the lack of maintenance cost data for the test system, the authors have selected the costs given in [13]. The ability of the technique to handle varying maintenance costs was tested using maintenance costs that were different for different weeks. Different maintenance periods of 1/3 weeks were used for different generating units. The weekly load curve was created as a percent of the peak load of the system. An
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:/5% uniform reserve throughout the maintenance horizon is considered. Reserve is accounted for by increasing the system load by the specified amount of reserve. The parameters used are one maintenance crew, :/5.0% reserve and 52 weeks horizon. To validate its performance, the proposed EP-based technique was compared against the complementary decision variables structure technique (CDV) through testing on three cases on the IEEE 30-bus system. The proposed EPbased technique produced better results than the CDV approach [13]. This case is solved for two different scenarios, the first assumes normal line loadings and the second considers the system behavior under limited lines thermal capacity. The results are given in Tables 1 and 2.
5. Discussion of results Testing revealed that optimal EP parameters for the test system of three individuals and 1000 generation were most suitable for the test case. Higher values of these parameters did not improve the solution. For Case 1, with :/5% reserve and normal line capacity limits, the proposed technique found the optimal solution at generation number 753 with cost components given in Table 1. The effect of line limited capacity was indicated by the increase of the cost ranges, as given in Table 1. The maintenance schedule for each generating unit for both cases is shown in Table 2. Limited line thermal capacities produced a larger number of line flow violations, narrowed the solution space and made it a challenging task to find the optimal solution. The algorithm first tries to remove the violation by rescheduling generation. If the violations persisted, the algorithm finds a new maintenance schedule to remove the line violation(s). As shown in Table 2, the maintenance schedule of unit 1 /3 did not change. However, the maintenance schedule for unit 4/6 was shifted to one that would satisfy the line flow constraints. Shifting the maintenance schedule had to be, of course, at the expense of increasing the maintenance and production costs. With the new maintenance schedule obtained in Case 2, generation had to be redispatched to remove Table 1 The fuzzy EP solution
Production
Case 1
Case 2
{$6062865.06 $6343355.93}
{$6063977.63 $6344443.53} {$85403.00 $90860.00}
Maintenance {$84126.00 $90760.00} Total cost
{$6146991.06 $6434115.93}
{$6149380.63 $6435303.53}
71
Table 2 Maintenance period of system generating unit with :/5% reserve Unit
1
2
3
4
5
6
Case 1 Case 2
34 /36 34 /36
29, 31 29, 31
31, 32 31, 32
28 37
33 27
39 39
some of the line violations that in turn caused, as expected, the production cost to increase (Table 1).
6. Conclusion This paper presented a fuzzy model and a new technique to solve the fuzzy security-constrained MS problem for power generation systems. The approach handles uncertainties associated with the system load over the maintenance horizon and the maintenance and fuel cost variations due to market prices changes. The approach takes advantage of the search ability of the evolutionary programming and the feasibility watch of the hill-climbing technique to achieve a near-optimal solution. The generated solution is a fuzzy range for each of the maintenance and production costs that reflect the problem uncertainties. Test results on the IEEE 30-bus system indicated the viability of the proposed approach and its potential to solve the MS problem with various types of uncertainties.
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[10] T. Back, U. Hammel, H.P. Schwefel, Evolutionary computation: comments on the history and current state, IEEE Transactions on Evolutionary Computation 1 (1) (1997) 3 /17. [11] Miranda, D. Srinivasan, L.M. Proenca, Evolutionary computation in power system, Electrical Power and Energy Systems 20 (2) (1998) 89 /98.
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