Solution to economic dispatch problem with valve-point loading effect by using catfish PSO algorithm

Abstract

This paper proposes application of a catfish particle swarm optimization (PSO) algorithm to economic dispatch (ED) problems. The ED problems considered in this paper include valve-point loading effect, power balance constraints, and generator limits. The conventional PSO and catfish PSO algorithms are applied to three different test systems and the solutions obtained are compared with each other and with those reported in literature. The comparison of solutions shows that catfish PSO outperforms the conventional PSO and other methods in terms of solution quality though there is a slight increase in computational time.

Keywords

economic dispatch (ED) / valve point loading / catfish particle swarm optimization (PSO) / optimization

Introduction

The objective of the economic dispatch (ED) problem is to schedule the committed generators in such a way that the overall cost of generation is minimized. The various equality and inequality constraints have to be satisfied to reach this optimal minimum solution. This optimal solution allocates generation levels to the units in a mix, such that system load is supplied entirely and most economically [1]. If valve-point loading is taken into account, the objective function includes a trigonometric term in addition to the quadratic terms that makes it highly nonlinear. This highly nonlinear equation cannot be directly solved by using conventional methods. Thus power system engineers have turned toward non-conventional methods such as evolutionary optimization techniques for promising solutions to problems including valve-point loading. These evolutionary algorithms can find the near global or global optimal solutions irrespective of the shape of the objective function.

Several classical optimization techniques such as the gradient method [2], lambda iteration method [3], linear programming [4], quadratic programming, nonlinear programming [5], Lagrangian relaxation algorithm and dynamic programming [6] are in practice to solve ED problems. Many artificial intelligence methods such as the genetic algorithm [7], Tabu Search [8], Hopfield neural network [9], ant colony optimization [10], different types of evolutionary programming (EP) techniques [11], bacterial foraging (BF) and hybrid bacterial foraging algorithm particle swarm optimization (BFA-PSO) [12], and so on, have also been proposed to solve ED problems.

Several variants of PSO have appeared in literature to solve many optimization problems ever since it was proposed by Kennedy and Eberhart in 1995 [13]. A self-adaptive chaotic PSO for short-term hydroelectric system scheduling in deregulated environment has been proposed in Ref. [14]. This algorithm introduces the chaos mapping and an adaptive scaling term into the PSO algorithm to increase its convergence rate and precision. A new PSO that takes into account not only its own previous best position and its previous best position of the group, but also its own previous worst position and its previous worst position of the group to adjust its current position has been proposed to solve ED problems [15]. There are various other methods that are used to solve ED problems [16–20]. This paper introduces catfish effects on PSO and it is named as catfish particle swarm optimization (PSO) [21].

Formulation of ED problem

The objective function of the ED problem is to minimize the fuel cost of thermal power plants for a given load demand when subjected to various constraints.

Objective function

The objective function of the ED problem is the quadratic fuel cost equation of the thermal units given as

(1)\mathrm{Min}f={\displaystyle \sum _{j=\mathrm{1}}^{n_{}}{F}_j\left(P_j\right)=}{\displaystyle \sum _{j=\mathrm{1}}^{n_{}}\left(a_j+b_j{P}_j+c_j{P}_j^{\mathrm{2}}\right)},

where n is the total number of generating units; P_j , the power generated by the jth generating unit in MW; F_j (P_j), the fuel cost of the jth generating unit in $/h; a_j, b_j and c_j are cost coefficients of jth generator.

The objective function, when taking into account the valve-point loading effect, becomes

(2)$$\begin{gathered} \mathrm{Min}f={\displaystyle \sum _{j=\mathrm{1}}^n{F}_j\left(P_j\right)} \\ ={\displaystyle \sum _{j=\mathrm{1}}^{n_{}}\left(a_j+b_j{P}_j+c_j{P}_j^{\mathrm{2}}+\left|e_j\sin \left(f_j\left(P_j^{\min }-P_j\right)\right)\right|\right)}, \end{gathered}$$

where e_j and f_j are constants of the valve-point effect of generators.

Optimization constraints

The equality and inequality constraints for the ED problem are real power balance criterion, and real power generation limits as given by

(3){\displaystyle \sum _{j=\mathrm{1}}^n{P}_j=P_{\mathrm{D}}},

(4)P_j^{\min }\le P_j\le P_j^{\max },

where P_j is the generation of the jth generating unit in MW, P_D is the total power demand in MW, P_j^{\min } and P_j^{\max } are the minimum and maximum power generation limits of the jth generator.

Implementation of PSO and catfish PSO to ED problems

The PSO is initialized with a group of random particles. Each particle is treated as a point in an n-dimensional search space and each single particle is a potential solution in the search space. Each particle flies around in the search space with a velocity that is constantly updated by the own experience of the particle and the experience of the neighbors of the particle. The particles move until a satisfactory solution is found.

The ith particle is represented as X_i= (x_i1,…x_ij,… x_in ) and their randomly generated initial velocities are V_i=( v_i1 ,…, v_ij, …, v_in). The population is represented as X= (X₁, X₂, …,X_{N_p}). The best previous position of the ith particle is P_i= (p_i1, p_i2…, p_in) and the global best position is P_G= (p_G1, p_G2…, p_Gn).

PSO

The search procedure of the conventional PSO algorithm is given by the following steps.

Step 1 Initialization: The initial particles are chosen randomly and would attempt to cover the entire parameter space uniformly. The uniform probability distribution for all random variables is assumed. The decision variable x_ij is directly coded as a real value within its corresponding lower-upper bounds. The initialization process generates N_P individuals X_i randomly:

(5)X_i=X_{\min }+{\rho }_i\left(X_{\max }-X_{\min }\right),\hspace{1em}i=\mathrm{1},\mathrm{...},N_p,

where X_min and X_max are the minimum and maximum limits of X which are initialized at starting; and ρ_i , a vector of random numbers in the range of [0,1].Similarly, the initial velocities are also chosen randomly and would attempt to cover the entire parameter space uniformly.

(6)V_i=V_{\min }+{\rho }_i\left(V_{\max }-V_{\min }\right),\hspace{1em}i=\mathrm{1},\mathrm{...},N_p.

The maximum and minimum velocities V_max and V_min are calculated as

(7{{\rm\tf="fp1"\char"0061}})V_{\max }={\displaystyle \frac{X_{\max }-X_{\min }}{R}},

(7{{\rm\tf="fp1"\char"0062}})V_{\min }=-V_{\max },

where R is the percentage change for the population and can be taken in the range of [10,20]. The n variables of each particle are the powers generated by each generator satisfying the inequality (generation limits) and equality (power balance) constraints and hence form a feasible solution.

Step 2 Finding bests: The fitness value of each particle is found and the particle with the best fitness is named as particle best, p_best and best fitness among the best of these particles is taken as global best, g_best. The best of the particles and global best are updated in every iteration after obtaining the new particles by adding velocities.

Step 3 Updating velocity and position: To modify the position of each particle, it is necessary to calculate the velocity of each particle in the next stage. In this velocity updating process, the values of parameters such as w, c₁ and c₂ should be determined in advance. The velocity and position of the particle are updated according to Eqs. (8) and (9) respectively.

(8)V_i^t=w^t{V}_i^{t-\mathrm{1}}+c_{\mathrm{1}}{r}_{\mathrm{1}}\left(P_i^{t-\mathrm{1}}-X_i^{t-\mathrm{1}}\right)+c_{\mathrm{2}}{r}_{\mathrm{2}}\left(P_G^{t-\mathrm{1}}-X_i^{t-\mathrm{1}}\right),

(9)X_i^t=X_i^{t-\mathrm{1}}+V_i^t,

where V_i^t is the velocity of particle i at iteration t, c₁ and c₂ are the cognitive and social acceleration coefficients respectively, r₁ and r₂ are the random numbers between 0 and 1, X_i^t is the position of particle i at iteration t, P_i^t is the best position of particle i until iteration t, and P_G^t is the best position of the group until iteration t, and w is the weighting function defined as

(10)w^t=w_{\max }-{\displaystyle \frac{\left(w_{\max }-w_{\min }\right)t}{t_{\max }}},

where w_max and w_min are the initial and final weights, and t_max is the maximum number of iterations.

Step 4 Stop criterion: The search procedure can be stopped when the current iteration number reaches the predetermined maximum iteration (generation) number or when there is no improvement in the minimum fitness value, g_best, between successive iterations. The g_best in the last iteration gives the optimal generations for the given problem.

Catfish PSO

In catfish PSO, a particle swarm is randomly initialized in a first step, and the particles are distributed over the D-dimensional search space. Each particle is updated by following two values. The first one is the best solution (fitness) it has achieved so far, called p_best. The other value tracked by the PSO is the global best value obtained so far by all particles in the population and is called g_best. The position and velocity of each particle are updated by Eqs. (8) to (10) (same as PSO).

If the distance between g_best and the surrounding particles is short, each particle is considered a part of the cluster around g_best and will only move a very short distance in the next generation. To avoid this premature convergence, catfish particles are introduced to replace 10% of the original particles with the worst fitness value of the swarm [21]. Here in this ED problem, catfish particles are taken to be those whose values are maximum and minimum of the generator limits. The concept is as follows. By the PSO reasoning, the entire swarm tends to converge toward the best particle. The best particle itself will not be motivated to improve its position further. Thus there is a tendency for premature convergence. Any improvement will take place only through random perturbations. By introducing catfishes at the boundary of the search space, there is a motivation for the entire swarm to be steered toward a more favorable region. Thus, the catfish particles are essential for the success of a given optimization task.

Numerical results

The proposed algorithm has been implemented in MATLAB 7 computing environment on a Pentium Duo-Core computer with CPU 2.10 GHz and 2 GB RAM memory. Three test systems with 3, 13 and 40 thermal units including valve loading effect are solved to test the proposed algorithm. The data for the three test systems are taken from Ref. [11]. In all performed simulations, the stopping criterion was maximum iterations. The obtained results of the proposed catfish PSO are compared with PSO and those of the recently published ED solution methods. The parameters of the proposed catfish PSO to solve the ED problem in the three test cases are: Population size= 40, c₁ = 2 and c₂ = 2.

Test case 1: 3-unit test system

This system consists of 3 generating units with a load demand of 850 MW. Table 1 shows the comparison of the best and mean fuel costs of 50 trail runs obtained using the catfish PSO and PSO with those reported in the literature. As observed from Table 1, these results are the same as those obtained by other methods.

Table 2 gives the solution and computation time of catfish PSO and PSO. As observed from Table 2, the solution obtained by both these methods is the same for the 3-unit system. The average time of the catfish PSO for this test system is 0.064 s compared to 0.063 s for conventional PSO, which is a completely reasonable computation time to solve this ED problem. The extra time taken by the catfish algorithm resulted from the fact that the entire swarm needs to be moved on the introduction of the catfish particles.

Table 3 lists the frequency of attaining a cost within specific ranges out of 50 runs for each of the catfish PSO and PSO. The catfish PSO converges to near optimal solutions more number of times compared with the PSO.

Figure 1 depicts the convergence characteristics of the catfish PSO and PSO for test case 1. From Fig. 1, it can be observed that both these algorithms converge to the same solution.

Test case 2: 13-unit test system

This system consists of 13 generating units with a load demand of 1500 MW. Table 4 demonstrates the comparison of the best and mean fuel costs of 50 trail runs obtained using catfish PSO and PSO with those reported in the literature. It can be seen from Table 4 that the best and mean solutions of the proposed catfish PSO are better than those of the other ED solution methods. Table 5 gives the solution and computation time between catfish PSO and PSO. This test system has a larger and more complex solution space than the previous one, and, as a result, the difference between different ED solutions can be better revealed in case 2. The minimum cost obtained by catfish PSO is 17969 $/h whereas that obtained by PSO is 17974 $/h. This indicates that catfish PSO is better than PSO. The catfish PSO method takes an average execution time of 3.64 s and the PSO method takes 3.59 s. The extra time taken is caused by the introduction of the catfish particles. Table 6 presents the relative frequency of convergence for 50 trials. Catfish PSO converges to near optimal solutions more number of times compared with PSO.

Figure 2 exhibits the convergence characteristics of catfish PSO and PSO. It is observed from Fig. 2 that the two algorithms converge in a similar manner.

Test case 3: 40-unit test system

This system consists of 40 generating units with a load demand of 10500 MW. Table 7 shows the comparison of the best and mean fuel costs of 50 trail runs obtained using catfish PSO and PSO with those reported in the literature. It can be seen from Table 7 that the best and mean solutions of the proposed catfish PSO are better than those of the other ED solutions. Table 8 gives the solution and computation time between catfish PSO and PSO. This test system has an even larger and more complex solution space than both of the above cases, and, consequently, the difference between different ED solutions can be better revealed in case 3, too. The minimum cost obtained by catfish PSO is 121683.7 $/h whereas that obtained by PSO is 121818.04 $/h. This shows that catfish PSO is better than PSO. The catfish PSO method takes an average execution time of 8.54 s and the PSO method takes 8.23 s. The extra time taken resulted from the introduction of the catfish particles. Table 9 shows the relative frequency of convergence for 50 trials. Catfish PSO converges to near optimal solutions more number of times compared with PSO. Figure 3 displays the convergence characteristics of catfish PSO and PSO. It can be observed from Fig. 3 that catfish PSO converges to a lower value than PSO.

Results analysis

A comparison of conventional PSO with other approaches is made to evaluate the performance of the proposed catfish PSO algorithm. Negligible difference in the performance of these algorithms is seen in the 3-unit test system. However, as the system size increases, the differences between their performances become prominent. In the three cases studied, the proposed catfish PSO outperforms other methods.

Conclusions

In this paper, the catfish PSO method has been implemented for solving the ED problems with valve point loading effects. With its implementation, the cost of generation is reduced though there is a slight increase in computation time due to replacement of the worst particles with catfish particles. The comparisons of the solutions for the test cases considered have proved the efficiency of the proposed method in finding near optimal or optimal solutions. It can be concluded that the catfish particles introduced into the PSO algorithm avoids premature convergence and enhances the searching capability.

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