A Hybrid GA-PSO Approach Based on Similarity for Various Types of Economic Dispatch Problems

The Economic Dispatch (ED) problem is one of the non-linear optimization problems in electrical power systems in which the main objective is to reduce the total power generation cost, while satisfying various equality and inequality constraints. Traditionally in ED problems, the cost function for generating units has been approximated as a quadratic function [1]. So far, many optimization methods have been used to overcome the ED problem. These methods can be classified as classical optimization, artificial neural networks and heuristic algorithms. Linear programming and quadratic programming are used as classical optimization methods [2–4]. Other approaches based on artificial neural networks and heuristic algorithms have been proposed to solve various ED problems. They include Hopfield neural network [5,6], improved Hopfield neural network [7], artificial neural network [8], improved ant colony search algorithm [9], particle swarm optimization algorithm [10], artificial immune [11], genetic algorithm [12] and particle swarm optimization with crazy particles [13]. Although these heuristic algorithms do not always guarantee the global best solutions, they can achieve a fast and near global optimal solution [13]. Genetic algorithm and particle swarm optimization (PSO) are the well-known examples of the modern heuristic algorithms, which can be used to solve nonlinear and non-continuous optimization problems. The PSO, was first motivated from the simulation of social behavior by Kennedy and Eberhart [14]. In this paper a new hybrid GA-PSO based on similarity method in order to solve the ED problems are proposed. The arithmetic crossover operator is used as crossover operator in the genetic algorithm. It can be defined to produce a new child as linear combination of two chromosomes. As randomly generated a coefficient is used to produce new child. In this method, a new approach for obtaining a coefficient is proposed benefitting from the similarity of parents chromosomes. The obtained results show that the proposed method has been applied successfully in various ED problems.


Introduction
The Economic Dispatch (ED) problem is one of the non-linear optimization problems in electrical power systems in which the main objective is to reduce the total power generation cost, while satisfying various equality and inequality constraints.Traditionally in ED problems, the cost function for generating units has been approximated as a quadratic function [1].
So far, many optimization methods have been used to overcome the ED problem.These methods can be classified as classical optimization, artificial neural networks and heuristic algorithms.Linear programming and quadratic programming are used as classical optimization methods [2][3][4].Other approaches based on artificial neural networks and heuristic algorithms have been proposed to solve various ED problems.They include Hopfield neural network [5,6], improved Hopfield neural network [7], artificial neural network [8], improved ant colony search algorithm [9], particle swarm optimization algorithm [10], artificial immune [11], genetic algorithm [12] and particle swarm optimization with crazy particles [13].Although these heuristic algorithms do not always guarantee the global best solutions, they can achieve a fast and near global optimal solution [13].Genetic algorithm and particle swarm optimization (PSO) are the well-known examples of the modern heuristic algorithms, which can be used to solve nonlinear and non-continuous optimization problems.The PSO, was first motivated from the simulation of social behavior by Kennedy and Eberhart [14].
In this paper a new hybrid GA-PSO based on similarity method in order to solve the ED problems are proposed.The arithmetic crossover operator is used as crossover operator in the genetic algorithm.It can be defined to produce a new child as linear combination of two chromosomes.As randomly generated a coefficient is used to produce new child.In this method, a new approach for obtaining a coefficient is proposed benefitting from the similarity of parents chromosomes.The obtained results show that the proposed method has been applied successfully in various ED problems.

Classical Economic Dispatch Problem
Minimizing the total fuel cost of the thermal power plants while satisfying the demand power and power system constraints is aimed at to overcome the ED problem.The fuel cost effect is shown in Fig. 1.

Fig. 1. Fuel cost effect
The ED problem can be formulated mathematically in (1) and ( 2): , where i a , i b and i c are the cost coefficients of the i th generator.Generation output of each generator limit values have been given in (3) and (4): min max , 1, 2,..., , where D is power equilibrium and D P represents the total demand power.

Economic Dispatch Problem with Valve point effects
The fuel cost characteristics of each generator are represented with the sinusoidal term added to the classical ED problem which models the valve point effect and the problem given in ( 5) and ( 6) [1]: where i a , i b and i c are the cost coefficients and i e , i f are the constants of the valve point effects of the i th generator.P i min and P i max represent the minimum limit and maximum limit on active power generation of each generator respectively: min max , 1, 2,..., , where D is power equilibrium, D P and L P represent the total demand power, and total transmission lines respectively.In this study, the transmission losses are ignored for the test system.The valve point effects are illustrated in Fig. 2.

Generation loading
Fuel Cost with valve-point without valve-point

Generator ramp-rate limits
The unit output cannot be adjusted instantaneously as soon as the load changes.Ramp-rate limits, i.e up-rate limit URi, down-rate limit DRi and previous hour generation Pi o restrict the operating region of all the online units.When the generator ramp-rate limits are considered, the operating limits of the i th generating unit are modified as follows [1] ( ) ( )

Prohibited operating zones
The ED problem with prohibited zones minimizes the total fuel cost based on quadratic functions.This function presents the regions where the operation is not allowed and this can be represented as the inequality constraints given in (10).Here the power balance equilibrium (8) and limit values of each generator (7)

Proposed method
Evolutionary algorithms aim at reaching the global minimum or maximum value for the problems also wished to be overcome using the proposed method.They have some superiority over each other.The PSO provides a faster convergence rate than the GA early in the run.The PSO and GA are represented based upon the population characterizing a particular solution that is same for the both algorithms.A hybrid technique in order to utilize the effectiveness and uniqueness of these two algorithms can be implemented to achieve a high performance [15].
Grimaldi et al proposed a hybrid technique called as Genetical Swarm Optimization.In this hybrid algorithm, for each iteration, the population is divided into two parts and each part is evolved by its own operators with the two techniques respectively.These two parts are then recombined in the updated population, which is again divided randomly into another two new parts in the next iteration for another run of genetic or particle swarm operators [15].
In real coded genetic algorithm do not require coding and encoding procedures, therefore the computation time and complexity are highly reduced.Also, genetic operators such as crossover and mutation have significant impact on its performance.Michalewicz [16] described a arithmetic crossover operator.It linearly combines two parent chromosome vectors to produce two new children according to the following equations: where a is a random weighting factor selected by the user.
In this method, one child is created by the father and mother.The main drawback of this crossover operator is that a random crossover constant is used.In other words, in applications, there are always difficulties to select a perfect weighting factor.In this study, a constant is obtained from the similarity of parents chromosomes [17].
Similarity can be defined between two or more objects as problems associated.Similarity is an important concept that needs to be calculated in all fields of science.However, this concept can be calculated by the people as shareholding or observational, by machines is an issue that is very difficult to calculate.If we can make similar measurements, we can classify a new object into the group and predict the behavior of the new object.In this respect Nosofsky proposed the Generalized Context Model (GCM) [18] where the probability was calculated as follows for x warning ( ) where b k is a response bias and h k (x) is given by a summed similarity between x and every stored exemplars of category C k, [20] ( ) where N k is number of exemplars in category C k .The γ parameter reflects the amount of determinism in responding.The quantity h k (x) can be interpreted as a measure of category similarity and therefore as a measure of evidence that stimulus x belongs to this category [20].
The variables in parent's chromosomes in a cluster are defined as follows: where k m is the mother and k f is the father chromosomes.Unit load differences of mother and father chromosomes for i th generator could be stated as follows ( ), 1 .
The generator distance between any two unit loads is calculated by Euclidean norm as ( ) In GCM, the similarity of two unit loads could be calculated as

Economic dispatch problem with prohibited zones and ramp-rate limits
ED problem with prohibited operating zones and ramp-rate limits are included in this paper.The second test case has 6-generating units [13] and a total load of 1263 MW.All the units have prohibited zones and ramp-rate limit constraints.Power losses are considered for this system using the B-matrix from [13].A power system with 6 generators data are given in Table 1, 2 and 3.In this system, PSO, PSO crazy, RGA and hybrid GA-PSO methods were used in economic dispatch.According to the results obtained using the hybrid GA-PSO method for economic dispatch is more advantageous than the other methods.In power system with six generators, if economic dispatch is realized by using the hybrid GA-PSO, a gain of fuel costs can be achieved in total.Also the total line loss is decreased.

Economic dispatch with valve point effects
This test case consists of 13-generating units with quadratic cost function combined with the effects of valve point loading.The units data (upper and lower bounds) along with the cost coefficients for the fuel cost (a,b, c, e and f) for the 13 generators with valve-point loading are given in [1].A power system with 13 generators data are given in Table 5.In this power system economic dispatch is realized by using the CEP, FEP, MFEP, IFEP, PS, and hybrid GA-PSO methods.Economic dispatch results are given in Table 7.When economic dispatch is done by using the hybrid GA-PSO, in both total fuel costs and reducing the total line losses significant gains can be achieved.

Economic dispatch with only fuel options
A power system with 18 generators data are given in Table 8 [19].This power system is loaded with different load demands that are given in Table 9.In this power system economic dispatch is made by using different methods and loads.Economic dispatch results are given in Table 10.The solution presented in this paper based on a hybrid GA-PSO to solve various ED problems.If economic dispatch is done by using proposed method, total fuel costs significant gains can be achieved.

Conclusions
This paper introduces a new approach based on similarity a hybrid GA-PSO to solve various economic dispatch problems of the power system.It has been applied to 6, 13 and 18 generators test system.The results have been compared to Ref. [13], [1] and [19].The results in various type economic dispatch problems demonstrate the feasibility and effectiveness of the proposed approach in minimizing cost of the generation.As a result, it has been shown that the proposed method improves the convergence and performs better when compared with the [1,13,19].

S
, value can be used instead of a coefficient in arithmetic crossover and varies between zero and one a

Fig. 3 .
Fig. 3. Flow chart of the GA-PSO based on similarity (11)considered.The total transmission losses are given in(11):− are lower and upper bound of the k th prohibited zone unit i and z i the number of prohibited zones of unit i th

Table 1 .
Generator operating limits and cost coefficients of the test system

Table 2 .
B-loss coefficients of the test system

Table 3 .
Prohibited zones and ramp-rate limits data of the test system

Table 4 .
Generator loading, loss, power output and fuel cost determined by different methods

Table 5 .
Generator operating limits and cost coefficients of the test system

Table 6 .
Generator loading and fuel cost determined by the proposed method

Table 7 .
Comparison of proposed method

Table 8 .
Generator operating limits and cost coefficients of the test system

Table 9 .
Generator loading and fuel cost determined by the proposed method for different load demands

Table 10 .
Comparison of proposed method for different load demands