ISSN 0973-4562 Volume 1 Number 3 (2006) pp. 409-426 (c) Research India Publications
http://www.ripublication.com/ijaer.htm
Economic Power Dispatch Using
Fuzzy-Genetic Algorithm
A. Laoufi1, A Hazzab1 and M. Rahli2 1
University Center of Bechar B.P 417 BECHAR (08000) ALGERIA
2
University of Sciences and Technology of Oran. ORAN (31000) ALGERIA E-mails: [email protected]. [email protected] and [email protected]
Abstract
This paper presents a new approach for solving the optimal power flow problem. In this approach we propose to use fuzzy logic techniques to dynamically control parameter settings of genetic algorithms (GA’s). We describe the Dynamic Parametric GA: a GA that uses a fuzzy knowledge-based system to control GA parameters such as the mutation and the crossover probability. We then introduce a technique for automatically design and tune the fuzzy knowledgebase system using GA’s. The proposed method was applied to a practical 14-bus system to show its feasibility and capabilities. The numerical and graphical results show that the proposed approach is faster and more robust than the GA’s.
Keywords: Economic dispatch problem. Genetic algorithm. Fuzzy logic.
Optimal power flow problem. Adaptation.
Introduction
The basic objective of economic dispatch problem (ED) of electric power generation is to schedule the committed generating unit outputs so as to meet the load demand at minimum operating cost while satisfying all unit and system equality and inequality constraints [1]. Theoretically and in practice there are several mathematical methods to optimize the distribution of the generated power dispatching. The application of the artificial intelligence has proved its efficacy when applied to the optimization of objective functions [2].
effective strategy over a broad range of problems. GA’s are stochastic optimization techniques based on the principle of natural adaptation in artificial systems. They contain potential features of efficient search strategies like random search. hill climbing statistical sampling and competition (survival of the fittest). Feasibility and application of GA’s to power systems can be effectively addressed when the following features of optimal solution computational advantages problem reduction, simplicity generalization can be identified for the problem under consideration [3].
The selecting parameter settings for GA’s. such as population size and crossover rates is often left to the GA user. Empirical and theoretical results about the effect of GA control parameter settings on GA performance and how to set them to improve GA performance have emerged in the literature [4, 5, 6]. Because the interaction of GA control parameters with GA performance is known to be complex. we assert that GA control is a prime target for a fuzzy system approach. In this paper. we propose a Dynamic Parametric GA; a GA that uses a fuzzy knowledge-based system to dynamically control GA parameters. such as crossover rates and mutation rates in a GA for the ED’s problem. Obtaining the fuzzy knowledge-based system used in the Dynamic Parametric GA can be accomplished in several ways. Design can be done by an expert on GA control by an automatic fuzzy design technique or by both (providing the automatic technique can incorporate the expert’s knowledge). Using an automatic technique can uncover new high performance GA control strategies. The choice of a fuzzy knowledge-based system to represent control strategies was made not only because it is easy to inject expert knowledge into them. but because new knowledge obtained by an automatic technique is equally accessible. This new knowledge may ultimately lead to a better understanding of the complex relationship between the control parameters and the performance. The goal of this work is three-fold: to explore approaches for fusing fuzzy control techniques with GA’s to better understand the relationship between GA control parameters and their effects on GA performance. and to improve GA performance for the ED’s problem.
The accuracy of the proposed algorithm has been established on 14 Bus test system [1, 20]. The results are compared both with those of the simple static GA to examine the strength of our method.
Economic Dispatch Using GA
The problem of the economic dispatch [14, 15, 16] which exist to minimize the cost of production of the real power can generally be stated as follows:
min
( )
∑
= n i i i P F 1 (1) equation (1) is subject to the constraint of equality in real and reactive power balance.∑
= = − − n i L ch i Q Q Q 1 0 (3)The inequality of real and reactive power limits on the generator outputs are:
max min i i i P P P ≤ ≤ (4) mx i i i Q Q Qmin ≤ ≤ (5)
where Fi
( )
Pi is the individual generation production cost in terms of its real power generation Pi.i
P and Qi : real and reactive power generation for unit i respectively.
n : number of generators in the system. ch
P and Qch : total real and reactive current system load demand respectively. L
P and QL : total real and reactive system transmission losses respectively.
The thermal plant can be expressed as input-output models (cost function). where the input is the fuel cost and the output is the power output of each unit. In practice, the cost function could be represented by a quadratic function [1, 14, 15]:
2
)
( i i i i i i i P a bP c P
F = + + (6)
The nonlinear programming problem can be formally stated as [1, 2].
Minimized f(x) (7)
Subject to m linear and/or nonlinear equality constraints:
0 )
(x =
hi where i=1,!,m (8)
and (p−m) linear and/or nonlinear inequality constraints: 0
)
(x ≥
gi where i=m+1,!,p (9)
The sequential unconstrained minimization technique (SUMT) algorithm has been developed to solve the nonlinear programming problem stated by equations (7)-(9), in which the objective function f(x) and inequality constraints gi(x) can be nonlinear functions of the independent variables. The equality constraints hi(x) must be a linear function of the independent variables if convergence to the solution of the nonlinear programming problem is to be guaranteed [1,21].
The GA’s can alleviate this convergence condition indeed by using the GA’s it is not necessary to subject that the equality constraints hi(x) must be a linear function
because the convergence to the solution in this case is always guarantied.
(
) ( )
( )
( )
∑
∑
+ = = + + = p m i k i k m m i k i k k k k x g r x h r x f r x P 1 ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( 1 1 , (10)where the weighting factors r are positive and form a monotonically decreasing sequence of values
{
r/r° >r1>...>0}
[1, 21].For the use of the GA’s in this problem, we propose the following sequence:
(
)
= = Otherwise / 0 mod if 0 0 k r PS k, r rk . (11) where: g N k=1,!, 1 0= r ; gN : maximum number of generation of the GA’s;
PS : the population size of the GA which must have a suitable value.
Note that Fiacco and Me.Cormick [1, 21] originally have chosen the function of the inequality constraints in the form of an added ‘barrier’:
( )
( )
(
)
∑
( )
( ) + = = p m i k i k x g x g G 1 1 (12) Since each time one or more gi(x(k))→0 from the feasible region ( ( ))→∞) (k x g
G ;
hence the concept of a barrier as r(k) is reduced. The effect of the berrier is reduced
and x may move closer to an inequality constraint boundary. As mentioned before, other possible choices exist for G(g(x(k))) such as:
( )
( )
(
)
{
( )
}
2 1 ) ( , 0 min∑
+ = = p m i k i k x g x g G (13) Or ( )( )
(
)
(
( )
)
∑
∑
+ = + = = − = p m i k i p m i k i k x g x g x g G 1 ) ( 1 ) ( ) ( 1 ln ln (14)The final form of the fitness or the penalty function becomes [1, 21]:
Adaptation of the GA’s Parameters by A Fuzzy Rule-Based System
This section proposes the Dynamic Parametric GA. The GA that uses a fuzzy knowledge based system to control GA parameters dynamically. Figure 1 shows the proposed GA system. Inputs to the fuzzy knowledge-based system for GA control can be any combination of GA performance measures or current control settings, outputs can be any of the GA control parameters such as the six mentioned in [7]. Example inputs might be (average fitness)/(best fitness) current population size or current mutation rate. Rules in the fuzzy knowledge-based system can reason about these measures and prescribe some control action. DeJong and Goldberg have been researching the effects of population size on GA performance. We can encode the rules or intuitive notions presented by them as fuzzy rules in a fuzzy knowledge-based system. For example as the current population grows, the sensitivity to mutation rate decreases and the best mutation rate to use also decreases. The interpolative properties of fuzzy systems give the fuzzy knowledge-base system the ability to provide control values in novel situations and in the event of conflicting rules. For example a typical fuzzy control scheme may include the following relations involving population size.
IF (average fitness)/(best fitness) is big then population size should increase. IF (worst fitness)/(average fitness) is small then population size should decrease. IF mutation is small AND population is small then population size should increase. It is easy to apply classical non-linear optimization methods to dynamically change the learning rate or momentum coefficient of neural networks [8], but not to control GA parameters. Other approaches to dynamically control GA parameters have also been presented [6]. The framework of the dynamic parametric GA provides a method to unify some of the notions resulting from this previous work. Because much of the knowledge of GA controls parameters how to control them and their effects on performance is mainly qualitative in nature. The fuzzy knowledge-based approach is well suited to representing it. In addition, there are numerous automatic fuzzy design techniques which can make use of a priori application knowledge and further improve the system [9, 10].
Engine
Fuzzy Rule base
GA
Figure 1: Dynamic Parametric GA
Automatic Design of the Fuzzy Knowledge-Based System for
Dynamic Parametric GA
In the previous section we proposed the framework to control GA parameters dynamically. The next question is how to design the fuzzy knowledge-based system which plays an essential role in the dynamic parametric GA. While it is possible to manually design a fuzzy knowledge-based system in the dynamic parametric GA system for GA control. It can be rather difficult although much literature on the subject of GA control has appeared. Our initial attempts at using this information to manually construct a fuzzy system for genetic control were unfruitful. For this reason we decided to turn to an automatic technique which has been demonstrated to be able to design fuzzy systems for other control applications [10]. In this section we introduce the method for automatically designing fuzzy systems that was previously mentioned. We then show how to use this technique to design an optimal fuzzy knowledge-based system for the dynamic parametric GA. Figure 2 shows the diagram by using an automatic technique. Relevant relations and membership functions can be automatically determined and may offer insight to understanding the complex interaction between GA control parameters and GA performance.
Figure 2: GA approach to auto design the fuzzy knowledge based system in the
dynamic Parametric GA
Figure 3: Fully overlapped membership functions.
Center Center
10011000 11011010 membership function Chromosome (MFC)
output rule ID 00001010 (a) Fuzzy input variable Fuzzy output
variable output rule ID
MFC1 …
.
MFCN1 MFC1 …MFCN2 ORC1 .. ORCN1.N2
(b)
Figure 4: Gene representation of the fuzzy system: (a) two chromosome types, (b)
construct gene map
The maximum number of rules in the system is equal to the number of possible combinations of input sets. Each rule has the possibility of generating a single rule for each output variable. For example a system with three inputs which are all composed of three fuzzy sets and two output variables could have up to 3
3
2× or 54 rules. Each of the possible rules in the system could have as its consequent part one output fuzzy set per output dimension. For example if there are two output variables, each of the two rules produced by a given combination of input variables could be assigned exactly one or none of the output fuzzy sets from each of the output variables. This is specified by a rule identification number (ruleID). For example, an output variable which is composed of five fuzzy sets could lead to six possible rules for a given set of input variables; either set one, two, three, four, five or none of the sets could be assigned to a rule. In the case where no rule is assigned, the rule is discarded from the system to address the coding problem. We first define a composite chromosome as a set of parameters that represent a higher level entity such as a membership function’s shape parameters, i.e. center in Figure 3 and output rule ID (respectively defined as
Center
( )
θ µMFC and ORC in Figure 4(a). These composite chromosomes are concatenated together to form the entire fuzzy system representation (see Figure 4(b)).
As an initial demonstration of our method we propose using a fuzzy system which takes three input variables and produces two output variables; (best fitness)/(average fitness), (best fitness)/(worst fitness), and change in the best fitness since last control action are the inputs and crossover rate change ∆Pc, and mutation rate change ∆Pm are the outputs. All input and output variables are divided into three and five fuzzy sets respectively. The total number of parameters in our fuzzy system coding is (six membership function center parameters for all input axis) + (eight membership function center parameters for all output axis) + 54 (rule ID parameters). The crossover and mutation parameters were also restricted to change at most by half of their current value and were bounded by
[
0.5 1.5]
and[
0.5 1.5]
.The calculated crossover and mutation rate at the th
j generation are given as follows:
( )
c c( )
1 c j P P P =∆ × (16)( )
m m( )
1 m j P P P =∆ × (17)Where Pc
( )
1 and Pm( )
1 are the crossover and mutation rate at the first generation that are chosen initially, ∆Pc and ∆Pm are the outputs of the fuzzy adaptor.System Studies
To validate our findings we compared the results of the proposed GA with the results the simple static GA. on the economic dispatch problem for a 14-bus test system [1, 16]. The single-line diagram of this system is shown in Figure 4 and the detailed data are given in table 1 and 2.
The fuel cost equations (in US $/h) for the two generators are
( )
( )
( )
+ = + = + = 5 2 5 5 5 2 2 2 2 2 1 2 1 1 1 8 . 1 0625 . 0 5 . 1 00175 . 0 0 . 2 00375 . 0 g g g g g g g g g P P P F P P P F P P P F :and the constraints on the reel power (MW) and reactive power (MVAR) are : and the constraints are:
200
50≤Pg1≤ 20≤ Pg2 ≤80
250 20≤ 1≤ − Qg −20≤Qg2 ≤100 80 15 ≤ 5 ≤ − Qg
300
=
chP
(MW)106
=
chQ
(MVAR)By using Newton-Raphson’s method [1, 20], the obtained real (PL) and reactive (QL) transmission line losses are respectively 18.35 MW and 25 MVAR.
The transmission line data in p.u. (per unit) are given in table 1 and table 2 [20].
1 3 6 5 1 1 2 1 0 9 1 4 1 1 7 8 4 2 3
Figure 5: Single-line diagram of 14 bus test system
Table 1: Bus data in p.u
N° Bus type Real power Reactive power
Table 2: Transmission line data in p.u.
p-q Impedance Line charging 1-2 0.01938+j0.05917 j0.0264 1-5 0.05403+j0.22304 j0.0246 2-3 0.04699+j0.19797 j0.0219 2-4 0.05811+j0.17632 j0.0187 2-5 0.05695+j0.17388 j0.0170 3-4 0.06701+j0.17103 j0.0170 4-5 0.01335+j0.04211 j0.0006 4-7 0.00000+j0.02091 j0.0000 4-9 0.00000+j0.55618 j0.0000 5-6 0.00000+j0.25202 j0.0000 6-11 0.09498+j0.19890 j0.0000 6-12 0.12291+j0.25581 j0.0000 6-13 0.06615+j0.13027 j0.0000 7-8 0.00000+j0.17615 j0.0000 7-9 0.00000+j0.11001 j0.000 9-10 0.03181+j0.08450 j0.000 9-14 0.12711+j0.27038 j0.000 10-14 0.08205+j0.19207 j0.000 12-13 0.22092+j0.19980 j0.000 1314 0.17093+j0.34802 j0.000
GA Performance Indices
The next step is to find an appropriate fitness function for the task. The objective of the automatic fuzzy system design algorithm is to improve the performance of the GA controls.In [4], DeJong assembled a set of five functions to study the behaviour and effectiveness of GA’s as function optimizers. The five functions were carefully selected to represent a wide cross section of function families. He designed two measures to quantify the performance of the GA’s online performance to measure ongoing performance and offline performance to measure convergence. Online performance is the running average of all evaluations performed up to a given time (equation (16)). Offline performance is the running average of the best performance value up to a given time (equation (17)).
where:
( )
i kfe , is the objective function values obtained at generation i for k=1,!,PS.
( )
jfe
*
is the best function values obtained until a given generation i for j=1,!,i.
Optimization Results
For designing the fuzzy knowledge based system, the transmission line losses are calculated and maintained constant (PL =13.36MW). The power balance equationwill become:PG1+PG2+PG5 =313.36MW. The initial points are:
MW P MW P MW PG10=120 , G20=60 , G50=40
The simple static GA or meta-level GA that was used to design the fuzzy systems had fixed parameters of population size=50, crossover rate=0.6 mutation rate=0.01 and used an elitist selection strategy. The meta-level GA was allowed to evaluate 00 the dynamic parametric GA’s. After the meta-level GA completed, we select the best fuzzy system.The rule base optimized is presented in table 3.The system used in our experiments used triangular membership functions. Defuzzification of the outputs was performed using the fuzzy centroid method [12].For example, the fuzzy rule #12 of changing population size means:
IF X1 is Zero (Z) and X2 is Negative (N) and X3 is Positive (P). Then ∆Pc is B1m
and ∆Pm is B2m.Where each input variable (X1, X2 and X3) have tree membership
functions.
The optimized offline controller had an initial crossover and mutation rates of 0.7 and 0.99 respectively.The optimized fuzzy membership functions of the inputs and outputs of the fuzzy adaptor are given in figure 6, 7, 8, 9 and 10.
Figure 6: The optimized fuzzy membership functions of the first input of the fuzzy
Figure 7: The optimized fuzzy membership functions of the second input of the fuzzy
adaptor.
Figure 8: The optimized fuzzy membership functions of the third input of the fuzzy
adaptor.
Figure 9: The optimized fuzzy membership functions of the first output of the fuzzy
Figure 10: The optimized fuzzy membership functions of the second output of the
fuzzy adaptor.
Simulation Results
First case:
We will take into account only real constraints.
First Variant
The transmission line losses are calculated and maintained constant (PL =13.36 MW)
The power balance equation will become:
36 . 313 3 2 1+ G + G = G P P P MW
The initial points are:
MW P MW P MW PG10=120 , G20=60 , G50=40
The results of the real generated optimal power and minimum fuel cost are given in table 4.
Results for online and offline comparisons of the simple static GA (static DeJong parameter settings) and the dynamic parametric GA are given in figures 11 and 12.
Figure 11: Comparison of online performance of the DPGA-FLC and the SGA
(case 1, 1st variant).
Figure 12: Comparison of offline performance of the DAGA-FLC and the SGA
(case 1. 1st variant)
Second Variant
The transmission line losses are considered as a linear function of real generated power. The coefficients were calculated by the Gauss-seidel’s method:
2 1 0.03828 08996 . 0 G G L P P P = +
The power balance equation will become therefore:
259 96172 . 0 91004 . 0 PG1+ PG2 = MW
We take the same initial points as the first variant. The results of the real generated optimal power minimum fuel cost transmission line losses and computing time are given in table 4 .
where PGiOpt and QGiOpt are the optimal value of the real and reactive power of ith generator in MW and MVAR respectively. And:
Table 4: Real constraints only
DPGA-FLC SGA
1st variant 2nd variant 1st variant 2nd variant
Opt G P1 156.52 128.55 133.74 133.49 Opt G P 2 40.21 55.18 77.81 51.73 Opt G P 3 17.36 49.99 39.53 33.98 L P 13.36 8.43 13.36 6.76 opt F 809.22 8.27.46 816.29 877.55 Second Case
We will take into account real and reactive constraints with the same considerations in the first case; In the two variants we will take the same initial points::
MW P MW P MW PG10=120 , G20=60 , G50=40 MVAR Q MVAR Q MVAR QG10=30 , G20=20 , G50=20 MVAR Q MVAR Q MVAR QG10=30 , G20=20 , G50=20
The results of real and reactive generated optimal power total real transmission losses minimum fuel cost and computing time are given in table 5.
Table 5: Real and reactive constraints
DPGA-FLC SGA
1st variant 2nd variant 1st variant 2nd variant
Conclusion
This paper has proposed a method for controlling GA’s using fuzzy logic techniques. This framework allows one to qualitatively express GA control strategies based on experience or intuition. To make Dynamic Parametric GA’s accessible to all. we have also presented an automatic fuzzy design technique, which is based on GA’s. This technique was in turn used to design an optimal fuzzy system for GA control. The result was a Dynamic Parametric GA controlled by a fuzzy system that exhibited better performance than a simple static GA. This Dynamic Parametric GA was then evaluated on different cases for the economic dispatch and without additional optimization. out performed a simple static GA. This indicates that the rules found by the automatic design technique may be universally applicable to control GA’s in other optimization tasks. In addition, because the representation of control knowledge is based on a fuzzy rule base, knowledge about GA control can be easily extracted after optimization.
We also would like to point out that using the meta-level GA to find the optimized fuzzy system for the dynamic parameterized GA did consume many hours of computation time. However, because the performance of the final system transferred to another task or system without additional tuning may suggest that if a good knowledge-base were found, then optimizing the fuzzy knowledge-based system for the dynamic parameterized GA need not be performed very often. The performance of our method demonstrated through its evaluation on the 14-bus power system. The test results have shown the proposed genetic algorithm can provide better solutions than the simple static genetic lgorithm.
References
[1] Rahli M. Pirotte P (1999) Optimal load flow using sequential unconstrained minimization technique (SUMT) method under power transmission losses minimization. Electric Power Systems Research 52: 61-64.
[2] Michael LEE. TAKAGI Hiddeyki (1993) Dynamic Control of Genetic Algorithms using Fuzzy Logic Techniques. 5th Int’1 Conf. On Genetic Algorithms (ICGA’93). Urbana-Champaign IL:. 76-83
[3] Grenfenstette JJ (1986) Optimization of control parameters for genetic algorithms IEEE Transaction On Systems Man. and Cybernetics. Vol 16: 122-128.
[4] DeJong KA (1985) Genetic algorithms: A 10 year perspective Proceedings of an International Conference of Genetic Algorithms and their Applications. J Greffenstette. Editor, Pittsburgh. 169-177.
[6] Goldberg DE (1989) Genetic algorithms and Walsh functions: Part II. Deception and its analysis. Complex System. Vol. 3: 153-171
[7] DeJong KA (1975) An analysis of the behaviour of a class of genetic adaptive systems Ph.D. dissertation. University of Michigan.
[8] Goldberg DE. al. (1990) Messy genetic algorithms revisited. Studies in mixed size and scale Complex System Vol. 4: 415-444.
[9] Goldberg DE. Richardson J (1987) Genetic algorithms with sharing for multimodal functions optimization Proc. Second 2nt Conf Genetic Algorithms: 41-49.
[10] Goldberg DE. (1989) Genetic Algorithms in Search. Optimization and Machine Learning Reading MA. Addison Wesley.
[11] Fogarty TC.. (1989) Varying the probability of mutation in genetic algorithms Proc. Third Int. Conf. Genetic Algorithms: 104-109.
[12] Zadeh. LA (1992). The calculus of fuzzy if/then rules. AI Expert:.22-27
[13] Aoki K. Nishikori A. (1984). An algorithm for Constrained Load Flow. IEEE Trans. on PAS: Vol. Pas 103. N° 5.
[14] Aoki K. Nishikori A. (1984). An algorithm for Constrained Load Flow. IEEE Trans. on PAS: Vol. Pas 103. N° 5.
[15] Sasson AM. Merill HM. (1974) Some Applications of Optimization Technics to Power Systems Problems. IEEE Trans on PAS: Vol Pas 62 N° 7
[16] Iwamoto S. Tamura Y. (1981) A Load Flow Calculation Method for Ill-Conditionned Power System IEEE Trans on PAS Vol. Pas 1 N° 4 April.
[17] Freris LL. and Sasson AM. (1968). “Investigation of the load-flow problem.IEE 115: 1459-1470.
[18] Sheirah MA. Eteiba MB. El Sayed MA (1986). On-line adaptive optimal load flow. Electric Power Systems Research pp: 205-214.
[19] Back T (1991) Self-adaptation in genetic algorithms. Proceedings of the first European Conference on Artificial Life. Paris. France. pp 263-271.
[20] DeJong KA (1980) Adaptive system design: a genetic approach. IEEE Trans Syst. Man. and Cybernetics Vol. 10. N° 9. pp. 566-574.
[21] Stagg GW. EL Abiadh AH. (1983) Computer methods in power system analysis. McGraw-Hill Book Company. New York.
