A New Hybrid Optimization Algorithm for Solving Economic Load Dispatch Problem with Valve-Point Effect Seyyed H. Elyas, Student Member, IEEE, Paras Mandal, Senior Member, IEEE, Ashraf U. Haque, Member, IEEE, Annarita Giani, and Tzu-Liang (Bill) Tseng
Abstract—This paper presents an efficient approach for solving the economic load dispatch (ELD) problem with valvepoint effect using a new hybrid optimization algorithm. The main aim for solving ELD problem is to schedule the output of the committed generating units in order to meet the system load under various operating constraints. Since ELD is a non-linear and non-convex problem, stochastic search algorithms are considered as appropriate solutions. In this paper, the proposed new hybrid optimization algorithm is based on Clonal Selection Algorithm (CSA) that uses the positive features of two other optimization techniques, Gases Brownian Motion Optimization (GBMO) and Particle Swarm Optimization (PSO), for local search and improving the quality of initial population, respectively. To validate the efficiency of the proposed hybrid method, termed as PG-Clonal in this paper, we tested it on two systems considering different constraints, and the obtained results are compared with the results of existing stochastic search algorithms available in the literature. The test results demonstrate the effectiveness of the proposed new hybrid PGClonal method in solving the ELD problem efficiently. Index Terms—Clonal selection algorithm, economic load dispatch, gases Brownian motion optimization, particle swarm optimization, valve-point effect.
I. INTRODUCTION
E
conomic load dispatch (ELD) is considered as one of the most significant optimization problems in the operation and planning of power systems [1]. The primary objective of the ELD is to determine the output power of generating units and make the best generation schedule in order to meet the load demand at minimum operating cost under numerous operating constraints [2]-[4]. ELD is a non-convex and nonlinear constrained optimization problem in which technical This work was supported in part by the U.S. National Science Foundation under grant NSF-DUE-1246050. S. H. Elyas and P. Mandal are with the Department of Electrical and Computer Engineering, University of Texas at El Paso, TX 79968 USA (email:
[email protected];
[email protected]). A. Haque is with the Power Study Group, Teshmont Consultants LP, Calgary, AB, Canada (e-mail:
[email protected]). A. Giani is with Energy and Infrastructure Decision Group, Los Alamos National Laboratory, Los Alamos, NM 87545 USA (e-mail:
[email protected]). B. Tseng is with the Department of Industrial, Manufacturing and Systems Engineering, University of Texas at El Paso, TX 79968 USA (e-mail:
[email protected]).
constraint such as valve-point effect increases its complexity. In the classical ELD problem, the cost function for each generator is approximated by a simple differentiable quadratic function and the dynamic behavior associated with valve effects is not taken into consideration. The fuel cost function of generating units can be modeled in a more practical fashion by including a sinusoidal term in the original objective function as the valve-point effect. Due to the valve-point effect, higher order non-linearity and ripples in the cost function result in a non-convex, non-smooth and complex fuel cost function. ELD is becoming a more challenging optimization problem during the past few decades, and various deterministic optimization methods have been applied to solve such type of problems, such as linear programming, nonlinear programming, dynamic programming, and gradient method. However, because of the nature of this problem, different heuristic methods have been presented in recent years, such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Pattern Search (PS), Evolutionary Programming (EP), and Simulated Annealing (SA) [5]-[9]. This paper contributes to solve ELD problem considering valve-point effect using a new hybrid optimization algorithm, which comprises the combination of Clonal Selection Algorithm (CSA) with PSO and Gases Brownian Optimization (GBMO). A review of literature suggests that GBMO has never been applied in solving ELD problem in combination with CSA and PSO, thus making this paper innovative. Hereinafter, the proposed new hybrid optimization method will be termed as PG-Clonal, in which P stands for PSO, G stands for GBMO and Clonal stands for CSA. CSA is a simple and efficient evolutionary algorithm [10]-[12]. The advantage of using PSO and GBMO into the proposed PG-Clonal method is such that PSO is applied during the initialization phase and GBMO is utilized during the local search phase, thus making the proposed hybrid PG-Clonal algorithm more efficient. Despite the fact that CSA is really fast in converging, it has a lot of flexibilities in combination with other stochastic search algorithms. In addition, this algorithm has advantages, such as finding the true global optimum regardless of the initial parameter values. In order to increase the efficiency of CSA, PSO is used in the beginning of each iteration to improve the quality of initial population. The proposed optimization technique uses both global and local search techniques in order
to explore the search space with maximum m accuracy. Thus, turbulent rotational motion operator, presennted in GBMO, is used as a powerful local search engine in the proposed combined optimization method. GBMO is coonsidered as a new and robust optimization technique, whicch has technical advantages in global and local search coompared to other heuristic optimization algorithms. Howeverr, it has not been applied to power system operation applicatioons. In this paper, GBMO’s local search engine is applied to C CSA to improve its capability to find the global solution. The rest of this paper is organized as foollows: Section II provides the formulation of the ELD problem m in detail. Section III describes the procedure of the proposed hybrid PG-Clonal optimization algorithm. Section IV presennts the numerical results. Finally, Section V concludes the majjor findings of the paper. II. PROBLEM FORMULATION N In power system operations, the main objjective of solving the ELD problem is to determine the most ecoonomic loading of the generating units so that the total load deemand is met. The economic load dispatch problem can be described as the optimization process and can be summarized as follows:
Fig. 1. Cost curve for generating units with and without considering the valve-point effecct.
2- Active power generation limiit
Pnmin ≤ Pn ≤ Pnmax
(5)
where Pmin and Pmax are minimum and maximum generation limits of unit n, reespectively. III. PROPOSED HYBRID OPTIMIZA ATION ALGORITHM FOR ECONOMIC LOAD DISPAT TCH PROBLEM
In this paper, an efficient hybrid method m comprising of PSO, GBMO, and CSA is proposed to sollve the ELD problem with Min[ Ft ( p)] = Fn ( pn ) (1) generation constraints. The follo owing first and second n =1 subsections present brief descriptio ons of CSA and GBMO In the classical ELD, Fn(pn) is considered aas a quadratic and algorithms, and the third subsectio on describes the proposed polynomial function which is formulated by hybrid algorithm PG-Clonal. 2 Fn (pn ) = an + bn pn + cn pn (2) A. Clonal Selection Algorithm (CS SA) N
∑
where Fn(pn) and pn are the fuel cost functtion ($/h) and the active output power (MW) of unit n, respectiively, and Fn(p) is the total generation cost ($/h). In (2), an, bn, and cn are cost coefficients of unit n. The input-outputt curve of each generating unit with valve loading effectss has remarkable differences compared with the smooth cost fuunction. The generating units with multi-valve steaam turbines have a lot of variation in the fuel-cost functions. Sinnce the valve point results in the ripples as shown in Fig. 1, the objective function includes the superposition of sinusoidaal functions and quadratic function. Therefore, (2) should be rreplaced by the (3) to incorporate the valve-point effects. 2
Fn (pn ) = an + bn pn + cn pn
m + en sin( fn (pnmin − pi )
(3)
where en and fn are the coefficients of valve-point effect of the nth unit, respectively. The aforementioned objective function iss subjected to the equality and inequality constraints, which aree listed as follows: 1- Active power balance constraint N
∑P = P n
D
+PL
(4)
n=1
where PD (MW) is the total assumeed system demand and PL (MW) is the total active ppower loss of the system. In this paper, PL is ignoored in the ELD problem for simplicity.
The Artificial Immune System m (AIS) is a powerful computational intelligence method based on the biological immune system and the natural defeense mechanism of human body. As an antigen, such as bacteriium and virus, invades the human body, the biological immun ne system will select the antibodies that can effectively recognize and destroy the antigen. The selection mechanism of the immunity system y of the antibodies with operation is based on the affinity relation to the invading antigen. In AIS, CSA is an efficient nspired by the biological optimization method, which is in immunity system selection mechanism [11]. This method has successfully been applied to optimization and pattern recognition domains [10]-[12]. Witth the above explanation, step-by-step procedure of CSA can be b summarized as follows: 1- Randomly produce the initiaal population within the allowable ranges in the probleem space. Each individual of the population, so called antibody in CSA, is a ptimization problem. The candidate solution for the op number of initial antibodies in n the population is denoted by I. The iteration number it is set to zero (it=0). a In optimization 2- Determine the affinity of the antibodies. problems, an antibody affin nity corresponds to the evaluation of the objective function for the given antibody. 3- Sort antibodies based on their affinity values and select ve the highest affinity. “i” (i
