Opf With Enhanced GA
Content
Applied Mathematics and Computation 155 (2004) 391–405 www.elsevier.com/locate/amc
A solution to the optimal power flow using genetic algorithm M.S. Osman a, M.A. Abo-Sinna
b,*
, A.A. Mousa
b
a
b
High technological Institute, 10th Ramadan city, Egypt Department of Basic Engineering Science, Faculty Of Engineering, Moenoufia University, Shebin El-Kom, Egypt
Abstract Optimal power flow (OPF) is one of the main functions of power generation operation and control. It determines the optimal setting of generating units. It is therefore of great importance to solve this problem as quickly and accurately as possible. This paper presents the solution of the OPF using genetic algorithm technique. This paper proposes a new methodology for solving OPF. This methodology is divided into two parts. The first part employs the genetic algorithm (GA) to obtain a feasible solution subject to desired load convergence, while the other part employs GA to obtain the optimal solution. The main goal of this paper is to verify the viability of using genetic algorithm to solve the OPF problem simultaneously composed by the load flow and the economic dispatch problem. Six buses system are used to highlight the goodness of this solution technique. 2003 Elsevier Inc. All rights reserved. Keywords: Nonlinear programming; Genetic algorithms; Load flow; Economic dispatch
1. Introduction The economic load dispatching (ELD) problem is one of key problems in power system operation and planning. The ELD problem may be expressed by minimizing the fuel cost of generators units under constraints depending on
*
Corresponding author. E-mail address: [email protected] (M.A. Abo-Sinna).
0096-3003/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00785-9
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load variations, the output of generators has to be changed to meet the balance between loads and generation therefore, the ELD is one of the fundamental problems in power system operation and planning. The primary objective of the economic dispatch (ED) problem is to find a set of active powers delivered by the committed generators to satisfy at any time the required demand subject to unit technical limits at the production cost. It is therefore of great importance to solve this problem as quickly and accurately as possible. Conventional techniques offer good results but when the search space is nonlinear and has discontinuities these techniques become difficult to solve with a slow convergence ratio and not always seeking to the optimal solution. New numerical methods are then needed to cope with these difficulties specially, those with high-speed search to the optimal and not being trapped in local minima. On the other hand, there has recently been a great deal of interest in promising genetic algorithm (GA) [1,3–6] and its application to various disciplines including power system planning operation and control. Genetic algorithms are also being applied to a wide range of optimization and learning problems in many domains. Genetic algorithms lend themselves well to power system optimization since they are known to exhibit robustness, require to auxiliary information, and can offer significant advantages in a solution methodology and optimization performance. In this paper we formulate optimal power flow (OPF) problem in Section 2. Section 3 describes genetic algorithm techniques. In Section 4, we review some of constraint-handling techniques in genetic algorithm. In Section 5, we define the OPF problem variables. Section 6 describes the proposed algorithm. In Section 7, sample six-bus system are solved and the results are introduced in Section 8. Finally Section 9 describes the main features of the proposed algorithm.
2. Optimal power flow formulation [11] An OPF problem is generally formulated as Min s:t:
f ðxÞ gi ðxÞ ¼ 0;
i ¼ 1; . . . ; q
hj ðxÞ 6 0;
j ¼ q þ 1; . . . ; m:
and
The objective function for the OPF reflects the costs associated with generating power in the system. The quadratic cost model for generation of power will be utilized: 2 CPGi ¼ ai þ bi PGi þ ci PGi
where PGi is the amount of generation in megawatts at generator i. The objective function for the entire power system can then be written as the sum of the quadratic cost model at each generator.
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f ðxÞ ¼ Ct ¼
393
Ng X 2 ðai þ bi PGi þ ci PGi Þ i
This objective function will minimize the total system costs, and does not necessarily minimize the costs for a particular area within the power system, where Ct is the total generation cost; ai , bi , ci , the cost function coefficients of unit; PGi the real power generation of unit i; Ng , the total number of generation units and i ¼ 1; 2; . . . ; Ng . The OPF equality constraints [7,9,11] reflect the physics of the power system as well as the desired voltage set points throughout the system. The physics of the power system are enforced through the power flow equations which require that the net injection of real and reactive power at each bus sum to zero. Therefore gðxÞ is DPp ¼ PGp Pcp
NB X
Vp Vq Ypq cosðdp dq Hpq Þ
q¼1
DQp ¼ QGp Qcp
NB X
Vp Vq Ypq sinðdp dq Hpq Þ
q¼1
where PGp , QGp are the real and reactive power generations at bus P ; Pcp , Qcp the real and reactive power demands at bus P ; VP , the voltage magnitude at bus P ; Vq , the voltage magnitude at bus q; dp , the voltage angle at bus p; dq ; the voltage angle at bus q; YPq , the admittance magnitude; Hpq , the admittance angle; NB , the total number of buses; P ¼ 1; 2; . . . ; NB and q ¼ 1; 2; . . . ; NB . The inequality constraints of the OPF reflect the limits on physical devices in the power system as well as the limits created to ensure system security. Physical devices that require enforcement of limits include generators, tap changing transformers, and phase shifting transformers. PG min 6 PGp 6 PG max ;
QG min 6 QGp 6 QG max ;
Vmin 6 VP 6 Vmax ;
dmin 6 dp 6 dmax
3. Genetic algorithm (GA) GA, invented by Holland [6] in the early 1970s, is a stochastic global search method that mimics the metaphor of natural biological evaluation. GAs operates on a population of candidate solutions encoded to finite bit string called chromosome. In order to obtain optimality, each chromosome exchanges information by using operators borrowed from natural genetic to produce the better solution. Fig. 1 shows outline of GAs for optimization problems. The GAs differ from other optimization and search procedures in four ways:
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Fig. 1. Outline of GAs for optimization problems.
(1) GAs work with a coding of the parameter set, not the parameters themselves. Therefore GAs can easily handle the integer or discrete variables. (2) GAs search from a population of points, not a single point. Therefore GAs can provide a globally optimal solutions. (3) GAs use only objective function information, not derivatives or other auxiliary knowledge. Therefore GAs can deal with the non-smooth, noncontinuous and non-differentiable functions which are actually existed in a practical optimization problem. (4) GAs use probabilistic transition rules, not deterministic rules. 4. Review of constraint-handling techniques [3–6] One of the major components of any evolutionary system is the evaluation function. Evaluation functions are used for assign a quality measure for individuals in a population. Whereas evolutionary computation techniques assume the existence of an (efficient) evaluation function for feasible individuals, there is no uniform methodology for handling (i.e., evaluating) unfeasible ones. The simplest approach, incorporated by evaluation strategies and the version of evolutionary programming (for numerical optimization problems), is to reject unfeasible solutions. But several other methods for handling unfeasible individuals have emerged recently. We review such methods (using a domain of nonlinear programming problems). For an excellent full review of constrainthandling techniques in genetic algorithm, the reader is referred to [4]. 4.1. Methods based on penalty functions The penalty function method is widely used in the mathematical programming literature. It essentially adds to the objective function some terms which punish a solution that is not feasible.
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A general mathematical problem is defined as follows: Max
f ðxÞ
s:t:
x ¼ ðx1 ; x2 ; . . . ; xn Þ 2 X Rn
x
gi ðxÞ 6 0;
i ¼ 1; 2; . . . ; k
hi ðxÞ ¼ 0;
i ¼ k þ 1; . . . ; m
Let F Rn be the feasible set for the above problem. Applying the idea of penalty functions, the above constrained optimization problem can be transformed into an unconstrained optimization problem. The objective function of the unconstrained optimization problem, which will be used as the fitness function in the associated genetic algorithm designed to solve the initial constrained problem, has the following format: f ðxÞ; if x 2 F evalðxÞ ¼ f ðxÞ þ penaltyðxÞ; otherwise where penalty (x) is zero, if no violation occurs, and is positive, otherwise. Usually, the penalty function is based on the distance of the solution form the feasible region F, or on the effort to ‘‘repair’’ the solution, i.e., to force it into F. The former case is the most popular one; in many methods a set of functions fj ð1 6 j 6 mÞ is used to construct the penalty, where the function fj ðxÞ measures the violation of the jth constraint in the following way: maxf0; gj ðxÞg if 1 6 j 6 k fj ðxÞ ¼ if k þ 1 6 j 6 m jhj ðxÞj How the penalty function is designed and applied to infeasible solutions may differ in important details across problems. 4.1.1. Static penalty function The static penalty function assumes that for every constraint we establish a family of intervals which determine an appropriate penalty coefficient Rij [4]. It works as follows: (1) for each constraint, create several (l) levels of violation (these levels measure the degree of violation, e.g., slightly or heavily); (2) for each level of violation and for each constraint, create a penalty coefficient Rij < 0 (i ¼ 1; 2; . . . ; l, j ¼ 1; . . . ; m); higher degree of violation requires heavier punishment (i.e., larger Rij ). The evaluation function has the following structure: evalðxÞ ¼ f ðxÞ þ
m X j¼1
Rij fj2 ðxÞ
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where the fj ðxÞ are as defined above and m is the number of constraints in the problem the central issue in this method is the determination of the relative magnitudes of the coefficients fRij j1 6 i 6 l; 1 6 j 6 mg. The weakness of the method is in the number of parameters. For m constraints the method requires mð2l þ 1Þ parameters in total: m parameters to establish number of intervals for each constraint, l parameters for each constraint, defining the boundaries of the intervals (levels of violation), and l parameters for each constraint representing the penalty coefficient Rij . 4.1.2. Dynamic penalty function Dynamic penalty function method differs from the previous one in that it punishes ‘‘harder’’ as the number of generations increases. The implementation of this method is through the following evaluation function: evalðxÞ ¼ f ðxÞ þ ðC T Þ
a
m X
fjb ðxÞ
J ¼1
where C, a and b are constants. A reasonable choice for these parameters is C ¼ 0:5, a ¼ b ¼ 2 i.e., evalðxÞ ¼ f ðxÞ þ ð0:5T Þ
2
m X
fj2 ðxÞ
J ¼1
The method requires much smaller number of parameters than the first method. Also, instead of defining several levels of violation, the pressure on a infeasible solutions is increased due to the ðC T Þ component of the penalty term: towards the end of the process (for high values of the generation number t), this component assumes large values. 4.1.3. Rejection of unfeasible individuals This ‘‘death penalty functions’’ method is a popular option in many evolutionary techniques like evolutionary strategies or evolutionary programming, this method rejects all infeasible solutions in the population. Thus, under this method, if in some current population infeasible solutions result after the GA operators are applied, these are simply eliminated and replaced by randomly drawn new solutions. 4.2. Repair methods [4] Repair algorithms enjoy a particular popularity in the evolutionary computation community. The main problem facing this method is locating an initial reference point for the purpose for initialization. The proposed algorithm described later can overcome this problem easily and it is relatively easy to repair an infeasible solution individual.
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5. Optimal load flow via GA (OPFGA) To solve optimal load flow via GA, variables defined as follow are needed 2 3 v 6 d 7 vp ; 8 bus 6¼ slack dp ; 8 pv bus 6 7 x¼6 ; ; d¼ 7 where v ¼ 4 pG 5 vf ; otherwise 0; otherwise QG PGp ; pG ¼ 0;
QGp ; pv bus ; QG ¼ otherwise 0;
8 pv bus otherwise
The slack bus generation level is calculated so as to meet the active and reactive power balance of power system using PGI ¼
NL X J
PCJ þ PL
N g 1 X
PGi
i
where PL and QL are the active and reactive losses and NL is the total number of buses. The search space for genetic algorithm is the limits assigned to each variable: • voltage magnitudes: 0.9–1.1 pu for pQ buses and 0.95–1.05 pu for pv buses; • unit active power; the specified maximum and minimum values; • unit reactive power; this is limited to ±75% of the unit maximum active power (equal to 0.8 unit power factor).
6. Modified co-evolutionary genetic algorithm (M-COGA) The main idea behind this algorithm is based on the concept of repair algorithm (repair infeasible solution) by introducing new repairing function. But the present algorithm consists of two sub-algorithm, the first one search for initial reference point that satisfy the constraint with the desired precision. On the other hand, the second one search for the optimal solution for the optimization problem. The present method propose a new repairing function (repairing infeasible solution) which work efficiently with any type of problems the result produced by this approach are compared against other numerical and evolutionary optimization techniques in several engineering design problems with different kinds of constraints. The results obtained show that the new approach can consistently outperform the other techniques using relatively small sub population, and without a significant in terms of performance.
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6.1. The proposed algorithm The nonlinear programming problem can be defined as follows:
NLPP :
optimize f ðxÞ s:t : F ¼ fx 2 Rn jgi ðxÞ 6 0; i ¼ 1; 2; . . . ; q and hj ðxÞ ¼ 0 j ¼ q þ 1; . . . ; mg S ¼ fx 2 Rn jlðxi Þ 6 xi 6 uðxi Þ; i ¼ 1; 2; . . . ; ng
where the set S Rn defines the search space and the set F S defines a feasible part of the search space. At any point x 2 F, the constraint gk ðÞ satisfy gk ðxÞ ¼ 0 are called active constraints at x. By extension, equality constraints hj ðÞ are called active at all points of F. Nonlinear equation hj ðxÞ ¼ 0 requires an additional parameter (w) to define the precision of the system [4]. All nonlinear equations hj ðxÞ ¼ 0 ðfor j ¼ q þ 1; . . . ; mÞ are replaced by pair of inequalities: w 6 hj ðxÞ 6 w so we deal only with nonlinear inequalities. In general, it is impossible to develop a deterministic method for the NLP in the global optimization category, which would be better than the exhaustive search. As stated by Gregory [2] ItÕs unrealistic to expect to find one general NLP code thatÕs going to work for every kind of nonlinear model. Instead, you try to select a code that fits the problem you are solving. If your problem doesnÕt fit in any category expect ‘‘general’’, or if you insist on a globally optimal solution (except when there is no chance of encountering multiple local optima), you should be prepared to have use a method that boils down to exhaustive search, i.e., you have an intractable problem. The proposed algorithm is divided into two parts. The first part employs the genetic algorithm (GA) to obtain a feasible solution subject to desired load convergence, while the other part employs GA to obtain the optimal solution. The proposed algorithm combines concept of co-evolution, repairing procedure (repairing infeasible solution) and elitist strategy. The main idea behind it is repairing infeasible solutions to satisfy the feasible region, using an elitist strategy to produce a faster convergence of the algorithm to the optimal solution of the problem. Repairing procedure repairs the infeasible points to satisfy the feasible space F. The working procedure of the algorithm is described in the following manner.
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399
6.2. Solution representation The algorithm uses a floating point representation for potential solutions. Each generation contain both feasible and infeasible individuals and we distinguish between them using flag pointer assigned to each individual. 6.3. Initialization stage The population vectors in the first generation are initialized randomly satisfying the search space S (the lower and upper bounds), while elitist individual is initialized by zero. The first sub-algorithm initially locating an initial feasible point that satisfying all constraint with the desired precision, since the second sub-algorithm needs at least one feasible point as reference point to enter the evolution process. 6.4. Initial feasible point The first sub-algorithm starts with a determined precision parameter which enable it to get a reference point. For every generation we search for updated reference point, updated reference point represents the individual with the more violation of constraints in each generation. So we locate an initial feasible point nðtÞ that satisfying all constraint with the desired precision to enter the evolution process (i.e., complete the algorithm procedure). 6.5. Repairing infeasible individuals The idea of this technique is to separate any feasible individuals in a population from those that are infeasible by repairing infeasible individuals. This approach co-evolves the population of infeasible individuals until they become feasible individuals in this approach, feasible individuals (z) are generated on a segment defined by two points feasible individuals (i.e., initial reference point nðtÞ 2 F) and infeasible individuals (x). But the segment may be extended equally on both sides determined by a user specified parameter l. Thus, a new feasible individual is expressed as z1 ¼ c:x þ ð1 cÞ:fðtÞ z2 ¼ ð1 cÞ:x þ c:fðtÞ where c ¼ ð1 þ 2lÞd l and d 2 ½0; 1 is a random generated number. Fig. 2 gives schematic view of possible sampling region. For example, l ¼ 0:5 samples point that lie on an interval that extends 0.5d on either side of the interval between the two points nðtÞ 2 F, x 2 I. On the other hand, for l ¼ 0:0, is equivalent samples point between the two points.
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µ.d
d
Reference point
µ.d
Infeasible point
Fig. 2. Possible sampling region.
8 While (z is infeasible) { < z1 ¼ c:x þ ð1 cÞ:fðtÞ Create random point z as: z2 ¼ ð1 cÞ:x þ c:fðtÞ : Check feasibility where c ¼ ð1 þ 2lÞd l } Now all individual are in the feasible space. 6.6. Elitist strategy Using an elitist strategy to produce a faster convergence of the algorithm to the optimal solution of the problem. The elitist individual represents the more fit point of the population. The use of elitist individual guarantees that the best fitness individual never increase (minimization problem) from one generation to the next generation (towards the end of the process). 6.7. Evolution process stage The algorithm uses the objective function to evaluate fitness functions for each individual. The algorithm applies tournament selection procedure/roulette wheel selection to select the new population as previously discussed. 6.8. Stopping rule The algorithm is terminated for either one of the following conditions is satisfied: • The maximum number of generations is achieved. • When the genotypes (the genotypes structures) of the population of individuals converges convergence of the genotype structure occur when all bit
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401
positions in all strings are identical. In this case, crossover will have no further effect.
7. Application of OPFGA The proposed algorithm (Fig. 3) were implemented on the six buses system, which is depicted in Fig. 4, which taken from Ref. [10], where NB ¼ 6, number of equality constraints ¼ 2NB ¼ 12 and number of variables ¼ 20. Appendix A contains information on the six-bus sample power system. The test results obtained via OPFGA was found subject to: load flow convergence (DPp and DQp < :0001) which compared with those from the solution of the classical economic dispatch and standard load flow (ED + LF) and those published by Weber [10] and also the solution of the optimal solution via simulated annealing (OPFSA)[8]. Where in OPFSA the optimal solution was found subject to: load flow convergence (DPp and DQp < 0:01). Table 1 shows GA parameter used for this simulation with the precision parameter used in this algorithm.
Fig. 3. The structure of M_COGA.
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Fig. 4. One-line diagram of six-bus system. Table 1 The GA parameters Population size Mutation rate Crossover rate Crossover operator Selection operator Precision parameter User specified parameter
50 0.01 0.9 Single point Roulette wheel 0.0001 0.1
8. Results Table 2 depicts the number of iterations and the large amount of computing effort required to achieve the overall solution. This was expected due to GA is a stochastic global search method. This code used for these studies was implemented in C++ on Pentium III 450 MHz with 128 MB RAM computer. As can be seen from Tables 3–5 a better ED was obtained without violating line limits all lines have technical limit 100 MVA, except line 5 whose limit of 50 MVA [10]. 9. Conclusions This paper presents a new application of the GA technique in power system problems, it is interesting to note that GA is useful as an optimization technique Table 2 Computing effort
Six-bus system
Iterations
Time (s)
200
31
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403
Table 3 Economic dispatch for the six-bus test system Classical optimization methods Unit 1 (MW) Unit 2 (MW) Unit 3 (MW) Unit 4 (MW) Cost ($/h) Losses (MW) Load flow convergence: (DPp and DQp ) Violating quantities
Non-classical optimization methods
ED + LF
Weber
OPFSA
OPFGA
99.74 216.17 50.00 250.00 7860 15.91
160.39 133.39 143.00 169.00 8062 5.38
131.80 190.98 109.15 178.24 7938 6.33 <0.01
152.3252 151.6563 1.180913 1.870893 7987.176416 9.2088 <0.0001
2
0
0
0
Table 4 Load flow solution of the six-bus system Bus
Voltage
Angle
Generation
Demand
MW
MVA
MW
MVA
1 2 3 4 5 6
1.001621 0.987467 1.010965 0.974785 0.973218 0.967386
0.0 1.717332 )4.203926 )0.01340 )2.619713 )4.438276
152.23252 151.65663 118.0913 187.0893 0.0 0.0
84.7635 )26.5032 112.1499 )25.6332 0.0 0.0
100 100 100 100 100 100
20 20 20 20 50 10
Table 5 Lines power flow of the six-bus system Line
From
To
P (MW)
Losses (MW)
1 2 3 4 5 6 7
1 2 1 3 4 3 4
2 4 5 5 5 6 6
)15.2385 35.5621 67.5701 )7.9323 44.3846 26.0327 77.7561
0.864 0.5198 1.9581 1.089 0.9822 0.9583 2.8374
Overall losses
9.2088
to solve OPF. The method employs GA to get a feasible point that satisfy the equality constraints with the desired precision. Due to space limitation, this paper only reports the results of a six-bus system; however, the approach in general works for a general network for any number of buses. OPFGA has the advantage not to calculate differential equations neither the Jacobean matrix unlike classical methods. This fact permits the definition of
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any type of objective functions regardless of mathematical condition of continuity, concavities, etc. The main disadvantages of this proposal is the large computing time required to obtain the optimal solution this situation was expected because GA is a stochastic search method. The proposed method that combines concept of co-evolution, repairing procedure, elitist strategy has applied to the OPF problem and obtained very reasonable results. This method would be very useful for power planner and/or operator to treat not only the cost but also with environmental objective of power system.
Appendix A It contains information on the six-bus sample power system. The line characteristics of the system are shown in Table A.1. The bus characteristics of the system are shown in Table A.2. The economic information of the system is shown in Table A.3.
Table A.1 Line characteristics for six-bus system From bus
To bus
Resistance [P.U]
Reactance [P.U]
Line charging [P.U]
Line limit [MVA]
1 1 2 3 3 4 4
2 5 4 5 6 5 6
0.04 0.04 0.04 0.04 0.04 0.04 0.04
0.08 0.08 0.08 0.08 0.08 0.08 0.08
0.02 0.02 0.02 0.02 0.02 0.02 0.02
100 100 100 100 100 50 100
Table A.2 Bus characteristics for six-bus system Bus number
Load [MW]
Load [MVAR]
Min. generation [MW]
Max. generation [MW]
1 2 3 4 5 6
100 100 100 100 100 100
20 20 20 20 50 10
50 50 50 50 0 0
250 250 250 250 0 0
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405
Table A.3 Economic information for six-bus system Generator bus
a ½h$
$ b ½MW h
c ½MW$ 2 h
1 2 3 4
105 96 105 94
12.0 9.6 13.0 9.4
0.012 0.0096 0.0130 0.0094
References [1] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley Publishing Company, 1989. [2] J. Gregory, Nonlinear programming FAQ, Usenet sci.answers. Available from ftp:// rtfm.mit.edu/pub/usenet/sci.answers/nonlinear-programming-faq, 1995. [3] M. Gen, R. Cheng, Genetic Algorithms and Engineering Optimization, John Wily & Sons, New York, 2000. [4] Z. Michalewicz, A survey of constraint handling techniques in evolutionary computation methods, in: Proceeding of the Fourth Annual Conference on Evolutionary Programming, MIT Press, Cambridge, MA, 1995, pp. 135–155. [5] Z. Michalewicz, Schoenauer, Evolutionary algorithms for constrained parameter optimization problems, Evolutionary Computation 4 (1) (1996) 1–32. [6] Z. Michalewicz, Genetic Algorithms + Data Structures ¼ Evolution Programs, third ed., Springer-Verlag, 1996. [7] I.J. Nagrath, Modern Power System Analysis, Tata McGraw-Hill, 1999. [8] C.A. Roa-Sepulveda, B.J. Pavez-lazo, A solution to the optimal power flow using simulated annealing, Electrical Power and Energy Systems 25 (2003) 47–57. [9] W.D. Stevenson, Elements of Power System Analysis, fourth ed., McGraw-Hill, New York, 1982. [10] J.D. Weber, Implementation of a Newton-based optimal power flow into a power system simulation environment, Submitted in partial fulfillment of the requirements for the Degree of Master of Science in Electrical Engineering in the Graduate College of the University of Illinois at Urbana-Champaign; Urbana-Illinois. Available from http://energy.ece.uiuc.edu/jamie (1997). [11] A.J. Wood, B.F. Wollenberg, Power Generation Operation and Control, Wiley and Sons, 1996.
A solution to the optimal power flow using genetic algorithm M.S. Osman a, M.A. Abo-Sinna
b,*
, A.A. Mousa
b
a
b
High technological Institute, 10th Ramadan city, Egypt Department of Basic Engineering Science, Faculty Of Engineering, Moenoufia University, Shebin El-Kom, Egypt
Abstract Optimal power flow (OPF) is one of the main functions of power generation operation and control. It determines the optimal setting of generating units. It is therefore of great importance to solve this problem as quickly and accurately as possible. This paper presents the solution of the OPF using genetic algorithm technique. This paper proposes a new methodology for solving OPF. This methodology is divided into two parts. The first part employs the genetic algorithm (GA) to obtain a feasible solution subject to desired load convergence, while the other part employs GA to obtain the optimal solution. The main goal of this paper is to verify the viability of using genetic algorithm to solve the OPF problem simultaneously composed by the load flow and the economic dispatch problem. Six buses system are used to highlight the goodness of this solution technique. 2003 Elsevier Inc. All rights reserved. Keywords: Nonlinear programming; Genetic algorithms; Load flow; Economic dispatch
1. Introduction The economic load dispatching (ELD) problem is one of key problems in power system operation and planning. The ELD problem may be expressed by minimizing the fuel cost of generators units under constraints depending on
*
Corresponding author. E-mail address: [email protected] (M.A. Abo-Sinna).
0096-3003/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00785-9
392
M.S. Osman et al. / Appl. Math. Comput. 155 (2004) 391–405
load variations, the output of generators has to be changed to meet the balance between loads and generation therefore, the ELD is one of the fundamental problems in power system operation and planning. The primary objective of the economic dispatch (ED) problem is to find a set of active powers delivered by the committed generators to satisfy at any time the required demand subject to unit technical limits at the production cost. It is therefore of great importance to solve this problem as quickly and accurately as possible. Conventional techniques offer good results but when the search space is nonlinear and has discontinuities these techniques become difficult to solve with a slow convergence ratio and not always seeking to the optimal solution. New numerical methods are then needed to cope with these difficulties specially, those with high-speed search to the optimal and not being trapped in local minima. On the other hand, there has recently been a great deal of interest in promising genetic algorithm (GA) [1,3–6] and its application to various disciplines including power system planning operation and control. Genetic algorithms are also being applied to a wide range of optimization and learning problems in many domains. Genetic algorithms lend themselves well to power system optimization since they are known to exhibit robustness, require to auxiliary information, and can offer significant advantages in a solution methodology and optimization performance. In this paper we formulate optimal power flow (OPF) problem in Section 2. Section 3 describes genetic algorithm techniques. In Section 4, we review some of constraint-handling techniques in genetic algorithm. In Section 5, we define the OPF problem variables. Section 6 describes the proposed algorithm. In Section 7, sample six-bus system are solved and the results are introduced in Section 8. Finally Section 9 describes the main features of the proposed algorithm.
2. Optimal power flow formulation [11] An OPF problem is generally formulated as Min s:t:
f ðxÞ gi ðxÞ ¼ 0;
i ¼ 1; . . . ; q
hj ðxÞ 6 0;
j ¼ q þ 1; . . . ; m:
and
The objective function for the OPF reflects the costs associated with generating power in the system. The quadratic cost model for generation of power will be utilized: 2 CPGi ¼ ai þ bi PGi þ ci PGi
where PGi is the amount of generation in megawatts at generator i. The objective function for the entire power system can then be written as the sum of the quadratic cost model at each generator.
M.S. Osman et al. / Appl. Math. Comput. 155 (2004) 391–405
f ðxÞ ¼ Ct ¼
393
Ng X 2 ðai þ bi PGi þ ci PGi Þ i
This objective function will minimize the total system costs, and does not necessarily minimize the costs for a particular area within the power system, where Ct is the total generation cost; ai , bi , ci , the cost function coefficients of unit; PGi the real power generation of unit i; Ng , the total number of generation units and i ¼ 1; 2; . . . ; Ng . The OPF equality constraints [7,9,11] reflect the physics of the power system as well as the desired voltage set points throughout the system. The physics of the power system are enforced through the power flow equations which require that the net injection of real and reactive power at each bus sum to zero. Therefore gðxÞ is DPp ¼ PGp Pcp
NB X
Vp Vq Ypq cosðdp dq Hpq Þ
q¼1
DQp ¼ QGp Qcp
NB X
Vp Vq Ypq sinðdp dq Hpq Þ
q¼1
where PGp , QGp are the real and reactive power generations at bus P ; Pcp , Qcp the real and reactive power demands at bus P ; VP , the voltage magnitude at bus P ; Vq , the voltage magnitude at bus q; dp , the voltage angle at bus p; dq ; the voltage angle at bus q; YPq , the admittance magnitude; Hpq , the admittance angle; NB , the total number of buses; P ¼ 1; 2; . . . ; NB and q ¼ 1; 2; . . . ; NB . The inequality constraints of the OPF reflect the limits on physical devices in the power system as well as the limits created to ensure system security. Physical devices that require enforcement of limits include generators, tap changing transformers, and phase shifting transformers. PG min 6 PGp 6 PG max ;
QG min 6 QGp 6 QG max ;
Vmin 6 VP 6 Vmax ;
dmin 6 dp 6 dmax
3. Genetic algorithm (GA) GA, invented by Holland [6] in the early 1970s, is a stochastic global search method that mimics the metaphor of natural biological evaluation. GAs operates on a population of candidate solutions encoded to finite bit string called chromosome. In order to obtain optimality, each chromosome exchanges information by using operators borrowed from natural genetic to produce the better solution. Fig. 1 shows outline of GAs for optimization problems. The GAs differ from other optimization and search procedures in four ways:
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Fig. 1. Outline of GAs for optimization problems.
(1) GAs work with a coding of the parameter set, not the parameters themselves. Therefore GAs can easily handle the integer or discrete variables. (2) GAs search from a population of points, not a single point. Therefore GAs can provide a globally optimal solutions. (3) GAs use only objective function information, not derivatives or other auxiliary knowledge. Therefore GAs can deal with the non-smooth, noncontinuous and non-differentiable functions which are actually existed in a practical optimization problem. (4) GAs use probabilistic transition rules, not deterministic rules. 4. Review of constraint-handling techniques [3–6] One of the major components of any evolutionary system is the evaluation function. Evaluation functions are used for assign a quality measure for individuals in a population. Whereas evolutionary computation techniques assume the existence of an (efficient) evaluation function for feasible individuals, there is no uniform methodology for handling (i.e., evaluating) unfeasible ones. The simplest approach, incorporated by evaluation strategies and the version of evolutionary programming (for numerical optimization problems), is to reject unfeasible solutions. But several other methods for handling unfeasible individuals have emerged recently. We review such methods (using a domain of nonlinear programming problems). For an excellent full review of constrainthandling techniques in genetic algorithm, the reader is referred to [4]. 4.1. Methods based on penalty functions The penalty function method is widely used in the mathematical programming literature. It essentially adds to the objective function some terms which punish a solution that is not feasible.
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A general mathematical problem is defined as follows: Max
f ðxÞ
s:t:
x ¼ ðx1 ; x2 ; . . . ; xn Þ 2 X Rn
x
gi ðxÞ 6 0;
i ¼ 1; 2; . . . ; k
hi ðxÞ ¼ 0;
i ¼ k þ 1; . . . ; m
Let F Rn be the feasible set for the above problem. Applying the idea of penalty functions, the above constrained optimization problem can be transformed into an unconstrained optimization problem. The objective function of the unconstrained optimization problem, which will be used as the fitness function in the associated genetic algorithm designed to solve the initial constrained problem, has the following format: f ðxÞ; if x 2 F evalðxÞ ¼ f ðxÞ þ penaltyðxÞ; otherwise where penalty (x) is zero, if no violation occurs, and is positive, otherwise. Usually, the penalty function is based on the distance of the solution form the feasible region F, or on the effort to ‘‘repair’’ the solution, i.e., to force it into F. The former case is the most popular one; in many methods a set of functions fj ð1 6 j 6 mÞ is used to construct the penalty, where the function fj ðxÞ measures the violation of the jth constraint in the following way: maxf0; gj ðxÞg if 1 6 j 6 k fj ðxÞ ¼ if k þ 1 6 j 6 m jhj ðxÞj How the penalty function is designed and applied to infeasible solutions may differ in important details across problems. 4.1.1. Static penalty function The static penalty function assumes that for every constraint we establish a family of intervals which determine an appropriate penalty coefficient Rij [4]. It works as follows: (1) for each constraint, create several (l) levels of violation (these levels measure the degree of violation, e.g., slightly or heavily); (2) for each level of violation and for each constraint, create a penalty coefficient Rij < 0 (i ¼ 1; 2; . . . ; l, j ¼ 1; . . . ; m); higher degree of violation requires heavier punishment (i.e., larger Rij ). The evaluation function has the following structure: evalðxÞ ¼ f ðxÞ þ
m X j¼1
Rij fj2 ðxÞ
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where the fj ðxÞ are as defined above and m is the number of constraints in the problem the central issue in this method is the determination of the relative magnitudes of the coefficients fRij j1 6 i 6 l; 1 6 j 6 mg. The weakness of the method is in the number of parameters. For m constraints the method requires mð2l þ 1Þ parameters in total: m parameters to establish number of intervals for each constraint, l parameters for each constraint, defining the boundaries of the intervals (levels of violation), and l parameters for each constraint representing the penalty coefficient Rij . 4.1.2. Dynamic penalty function Dynamic penalty function method differs from the previous one in that it punishes ‘‘harder’’ as the number of generations increases. The implementation of this method is through the following evaluation function: evalðxÞ ¼ f ðxÞ þ ðC T Þ
a
m X
fjb ðxÞ
J ¼1
where C, a and b are constants. A reasonable choice for these parameters is C ¼ 0:5, a ¼ b ¼ 2 i.e., evalðxÞ ¼ f ðxÞ þ ð0:5T Þ
2
m X
fj2 ðxÞ
J ¼1
The method requires much smaller number of parameters than the first method. Also, instead of defining several levels of violation, the pressure on a infeasible solutions is increased due to the ðC T Þ component of the penalty term: towards the end of the process (for high values of the generation number t), this component assumes large values. 4.1.3. Rejection of unfeasible individuals This ‘‘death penalty functions’’ method is a popular option in many evolutionary techniques like evolutionary strategies or evolutionary programming, this method rejects all infeasible solutions in the population. Thus, under this method, if in some current population infeasible solutions result after the GA operators are applied, these are simply eliminated and replaced by randomly drawn new solutions. 4.2. Repair methods [4] Repair algorithms enjoy a particular popularity in the evolutionary computation community. The main problem facing this method is locating an initial reference point for the purpose for initialization. The proposed algorithm described later can overcome this problem easily and it is relatively easy to repair an infeasible solution individual.
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5. Optimal load flow via GA (OPFGA) To solve optimal load flow via GA, variables defined as follow are needed 2 3 v 6 d 7 vp ; 8 bus 6¼ slack dp ; 8 pv bus 6 7 x¼6 ; ; d¼ 7 where v ¼ 4 pG 5 vf ; otherwise 0; otherwise QG PGp ; pG ¼ 0;
QGp ; pv bus ; QG ¼ otherwise 0;
8 pv bus otherwise
The slack bus generation level is calculated so as to meet the active and reactive power balance of power system using PGI ¼
NL X J
PCJ þ PL
N g 1 X
PGi
i
where PL and QL are the active and reactive losses and NL is the total number of buses. The search space for genetic algorithm is the limits assigned to each variable: • voltage magnitudes: 0.9–1.1 pu for pQ buses and 0.95–1.05 pu for pv buses; • unit active power; the specified maximum and minimum values; • unit reactive power; this is limited to ±75% of the unit maximum active power (equal to 0.8 unit power factor).
6. Modified co-evolutionary genetic algorithm (M-COGA) The main idea behind this algorithm is based on the concept of repair algorithm (repair infeasible solution) by introducing new repairing function. But the present algorithm consists of two sub-algorithm, the first one search for initial reference point that satisfy the constraint with the desired precision. On the other hand, the second one search for the optimal solution for the optimization problem. The present method propose a new repairing function (repairing infeasible solution) which work efficiently with any type of problems the result produced by this approach are compared against other numerical and evolutionary optimization techniques in several engineering design problems with different kinds of constraints. The results obtained show that the new approach can consistently outperform the other techniques using relatively small sub population, and without a significant in terms of performance.
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6.1. The proposed algorithm The nonlinear programming problem can be defined as follows:
NLPP :
optimize f ðxÞ s:t : F ¼ fx 2 Rn jgi ðxÞ 6 0; i ¼ 1; 2; . . . ; q and hj ðxÞ ¼ 0 j ¼ q þ 1; . . . ; mg S ¼ fx 2 Rn jlðxi Þ 6 xi 6 uðxi Þ; i ¼ 1; 2; . . . ; ng
where the set S Rn defines the search space and the set F S defines a feasible part of the search space. At any point x 2 F, the constraint gk ðÞ satisfy gk ðxÞ ¼ 0 are called active constraints at x. By extension, equality constraints hj ðÞ are called active at all points of F. Nonlinear equation hj ðxÞ ¼ 0 requires an additional parameter (w) to define the precision of the system [4]. All nonlinear equations hj ðxÞ ¼ 0 ðfor j ¼ q þ 1; . . . ; mÞ are replaced by pair of inequalities: w 6 hj ðxÞ 6 w so we deal only with nonlinear inequalities. In general, it is impossible to develop a deterministic method for the NLP in the global optimization category, which would be better than the exhaustive search. As stated by Gregory [2] ItÕs unrealistic to expect to find one general NLP code thatÕs going to work for every kind of nonlinear model. Instead, you try to select a code that fits the problem you are solving. If your problem doesnÕt fit in any category expect ‘‘general’’, or if you insist on a globally optimal solution (except when there is no chance of encountering multiple local optima), you should be prepared to have use a method that boils down to exhaustive search, i.e., you have an intractable problem. The proposed algorithm is divided into two parts. The first part employs the genetic algorithm (GA) to obtain a feasible solution subject to desired load convergence, while the other part employs GA to obtain the optimal solution. The proposed algorithm combines concept of co-evolution, repairing procedure (repairing infeasible solution) and elitist strategy. The main idea behind it is repairing infeasible solutions to satisfy the feasible region, using an elitist strategy to produce a faster convergence of the algorithm to the optimal solution of the problem. Repairing procedure repairs the infeasible points to satisfy the feasible space F. The working procedure of the algorithm is described in the following manner.
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6.2. Solution representation The algorithm uses a floating point representation for potential solutions. Each generation contain both feasible and infeasible individuals and we distinguish between them using flag pointer assigned to each individual. 6.3. Initialization stage The population vectors in the first generation are initialized randomly satisfying the search space S (the lower and upper bounds), while elitist individual is initialized by zero. The first sub-algorithm initially locating an initial feasible point that satisfying all constraint with the desired precision, since the second sub-algorithm needs at least one feasible point as reference point to enter the evolution process. 6.4. Initial feasible point The first sub-algorithm starts with a determined precision parameter which enable it to get a reference point. For every generation we search for updated reference point, updated reference point represents the individual with the more violation of constraints in each generation. So we locate an initial feasible point nðtÞ that satisfying all constraint with the desired precision to enter the evolution process (i.e., complete the algorithm procedure). 6.5. Repairing infeasible individuals The idea of this technique is to separate any feasible individuals in a population from those that are infeasible by repairing infeasible individuals. This approach co-evolves the population of infeasible individuals until they become feasible individuals in this approach, feasible individuals (z) are generated on a segment defined by two points feasible individuals (i.e., initial reference point nðtÞ 2 F) and infeasible individuals (x). But the segment may be extended equally on both sides determined by a user specified parameter l. Thus, a new feasible individual is expressed as z1 ¼ c:x þ ð1 cÞ:fðtÞ z2 ¼ ð1 cÞ:x þ c:fðtÞ where c ¼ ð1 þ 2lÞd l and d 2 ½0; 1 is a random generated number. Fig. 2 gives schematic view of possible sampling region. For example, l ¼ 0:5 samples point that lie on an interval that extends 0.5d on either side of the interval between the two points nðtÞ 2 F, x 2 I. On the other hand, for l ¼ 0:0, is equivalent samples point between the two points.
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µ.d
d
Reference point
µ.d
Infeasible point
Fig. 2. Possible sampling region.
8 While (z is infeasible) { < z1 ¼ c:x þ ð1 cÞ:fðtÞ Create random point z as: z2 ¼ ð1 cÞ:x þ c:fðtÞ : Check feasibility where c ¼ ð1 þ 2lÞd l } Now all individual are in the feasible space. 6.6. Elitist strategy Using an elitist strategy to produce a faster convergence of the algorithm to the optimal solution of the problem. The elitist individual represents the more fit point of the population. The use of elitist individual guarantees that the best fitness individual never increase (minimization problem) from one generation to the next generation (towards the end of the process). 6.7. Evolution process stage The algorithm uses the objective function to evaluate fitness functions for each individual. The algorithm applies tournament selection procedure/roulette wheel selection to select the new population as previously discussed. 6.8. Stopping rule The algorithm is terminated for either one of the following conditions is satisfied: • The maximum number of generations is achieved. • When the genotypes (the genotypes structures) of the population of individuals converges convergence of the genotype structure occur when all bit
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positions in all strings are identical. In this case, crossover will have no further effect.
7. Application of OPFGA The proposed algorithm (Fig. 3) were implemented on the six buses system, which is depicted in Fig. 4, which taken from Ref. [10], where NB ¼ 6, number of equality constraints ¼ 2NB ¼ 12 and number of variables ¼ 20. Appendix A contains information on the six-bus sample power system. The test results obtained via OPFGA was found subject to: load flow convergence (DPp and DQp < :0001) which compared with those from the solution of the classical economic dispatch and standard load flow (ED + LF) and those published by Weber [10] and also the solution of the optimal solution via simulated annealing (OPFSA)[8]. Where in OPFSA the optimal solution was found subject to: load flow convergence (DPp and DQp < 0:01). Table 1 shows GA parameter used for this simulation with the precision parameter used in this algorithm.
Fig. 3. The structure of M_COGA.
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Fig. 4. One-line diagram of six-bus system. Table 1 The GA parameters Population size Mutation rate Crossover rate Crossover operator Selection operator Precision parameter User specified parameter
50 0.01 0.9 Single point Roulette wheel 0.0001 0.1
8. Results Table 2 depicts the number of iterations and the large amount of computing effort required to achieve the overall solution. This was expected due to GA is a stochastic global search method. This code used for these studies was implemented in C++ on Pentium III 450 MHz with 128 MB RAM computer. As can be seen from Tables 3–5 a better ED was obtained without violating line limits all lines have technical limit 100 MVA, except line 5 whose limit of 50 MVA [10]. 9. Conclusions This paper presents a new application of the GA technique in power system problems, it is interesting to note that GA is useful as an optimization technique Table 2 Computing effort
Six-bus system
Iterations
Time (s)
200
31
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Table 3 Economic dispatch for the six-bus test system Classical optimization methods Unit 1 (MW) Unit 2 (MW) Unit 3 (MW) Unit 4 (MW) Cost ($/h) Losses (MW) Load flow convergence: (DPp and DQp ) Violating quantities
Non-classical optimization methods
ED + LF
Weber
OPFSA
OPFGA
99.74 216.17 50.00 250.00 7860 15.91
160.39 133.39 143.00 169.00 8062 5.38
131.80 190.98 109.15 178.24 7938 6.33 <0.01
152.3252 151.6563 1.180913 1.870893 7987.176416 9.2088 <0.0001
2
0
0
0
Table 4 Load flow solution of the six-bus system Bus
Voltage
Angle
Generation
Demand
MW
MVA
MW
MVA
1 2 3 4 5 6
1.001621 0.987467 1.010965 0.974785 0.973218 0.967386
0.0 1.717332 )4.203926 )0.01340 )2.619713 )4.438276
152.23252 151.65663 118.0913 187.0893 0.0 0.0
84.7635 )26.5032 112.1499 )25.6332 0.0 0.0
100 100 100 100 100 100
20 20 20 20 50 10
Table 5 Lines power flow of the six-bus system Line
From
To
P (MW)
Losses (MW)
1 2 3 4 5 6 7
1 2 1 3 4 3 4
2 4 5 5 5 6 6
)15.2385 35.5621 67.5701 )7.9323 44.3846 26.0327 77.7561
0.864 0.5198 1.9581 1.089 0.9822 0.9583 2.8374
Overall losses
9.2088
to solve OPF. The method employs GA to get a feasible point that satisfy the equality constraints with the desired precision. Due to space limitation, this paper only reports the results of a six-bus system; however, the approach in general works for a general network for any number of buses. OPFGA has the advantage not to calculate differential equations neither the Jacobean matrix unlike classical methods. This fact permits the definition of
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any type of objective functions regardless of mathematical condition of continuity, concavities, etc. The main disadvantages of this proposal is the large computing time required to obtain the optimal solution this situation was expected because GA is a stochastic search method. The proposed method that combines concept of co-evolution, repairing procedure, elitist strategy has applied to the OPF problem and obtained very reasonable results. This method would be very useful for power planner and/or operator to treat not only the cost but also with environmental objective of power system.
Appendix A It contains information on the six-bus sample power system. The line characteristics of the system are shown in Table A.1. The bus characteristics of the system are shown in Table A.2. The economic information of the system is shown in Table A.3.
Table A.1 Line characteristics for six-bus system From bus
To bus
Resistance [P.U]
Reactance [P.U]
Line charging [P.U]
Line limit [MVA]
1 1 2 3 3 4 4
2 5 4 5 6 5 6
0.04 0.04 0.04 0.04 0.04 0.04 0.04
0.08 0.08 0.08 0.08 0.08 0.08 0.08
0.02 0.02 0.02 0.02 0.02 0.02 0.02
100 100 100 100 100 50 100
Table A.2 Bus characteristics for six-bus system Bus number
Load [MW]
Load [MVAR]
Min. generation [MW]
Max. generation [MW]
1 2 3 4 5 6
100 100 100 100 100 100
20 20 20 20 50 10
50 50 50 50 0 0
250 250 250 250 0 0
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Table A.3 Economic information for six-bus system Generator bus
a ½h$
$ b ½MW h
c ½MW$ 2 h
1 2 3 4
105 96 105 94
12.0 9.6 13.0 9.4
0.012 0.0096 0.0130 0.0094
References [1] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley Publishing Company, 1989. [2] J. Gregory, Nonlinear programming FAQ, Usenet sci.answers. Available from ftp:// rtfm.mit.edu/pub/usenet/sci.answers/nonlinear-programming-faq, 1995. [3] M. Gen, R. Cheng, Genetic Algorithms and Engineering Optimization, John Wily & Sons, New York, 2000. [4] Z. Michalewicz, A survey of constraint handling techniques in evolutionary computation methods, in: Proceeding of the Fourth Annual Conference on Evolutionary Programming, MIT Press, Cambridge, MA, 1995, pp. 135–155. [5] Z. Michalewicz, Schoenauer, Evolutionary algorithms for constrained parameter optimization problems, Evolutionary Computation 4 (1) (1996) 1–32. [6] Z. Michalewicz, Genetic Algorithms + Data Structures ¼ Evolution Programs, third ed., Springer-Verlag, 1996. [7] I.J. Nagrath, Modern Power System Analysis, Tata McGraw-Hill, 1999. [8] C.A. Roa-Sepulveda, B.J. Pavez-lazo, A solution to the optimal power flow using simulated annealing, Electrical Power and Energy Systems 25 (2003) 47–57. [9] W.D. Stevenson, Elements of Power System Analysis, fourth ed., McGraw-Hill, New York, 1982. [10] J.D. Weber, Implementation of a Newton-based optimal power flow into a power system simulation environment, Submitted in partial fulfillment of the requirements for the Degree of Master of Science in Electrical Engineering in the Graduate College of the University of Illinois at Urbana-Champaign; Urbana-Illinois. Available from http://energy.ece.uiuc.edu/jamie (1997). [11] A.J. Wood, B.F. Wollenberg, Power Generation Operation and Control, Wiley and Sons, 1996.
