Elsevier

Energy

Volume 77, 1 December 2014, Pages 372-381
Energy

Solving non-convex economic dispatch problem via backtracking search algorithm

https://doi.org/10.1016/j.energy.2014.09.009Get rights and content

Highlights

  • Non-convex ED (Economic dispatch) problems have been solved.

  • BSA (Backtracking search algorithm) is used to solve economic dispatch problems.

  • The valve-point effects and transmission loss are considered in the problem.

  • The results confirmed BSA's capability in solving ED problems well and robustly.

Abstract

This paper presents BSA (backtracking search algorithm) for solving of ED (economic dispatch) problems (both convex and non-convex) with both the valve-point effects in the generator cost function and the transmission network loss considered. BSA is a new evolutionary algorithm for solving of numerical optimization problems; it uses a single control parameter and two crossover and mutation strategies for powerful exploration of the problem's search space. Four test systems (with 3, 6, 20, and 40 generators) are the case studies verifying the method's robustness and effectiveness. The results confirm that compared with existing well-known methods and especially in large-scale test systems, the proposed algorithm is the better approach to solving ED problems.

Introduction

The problem of ED (economic dispatch) is a basic consideration to optimizing power system operation. ED determines the power shared among the generating units of power system to meet electrical demand while minimizing cost and satisfying system constraints.

In a convex ED problem, the cost function of a generating unit is considered as a quadratic function. Practical and non-convex ED problems, however, contain non-convex cost functions that are due to the valve-point effect of the generating units. Classical methods have been adopted to solve conventional ED problems (i.e., containing convex cost functions) but instead produce non-optimal solutions because of the non-convexity/non-linearity of practical ED problems [1]. Dynamic programming, for example, has been proposed in addressing non-convex ED problems because it does not restrict the form of the cost function; the increased dimension of the problem, however, may demand higher computational efforts [2]. Classical methods include interior point [3], quadratic programming [4], linear programming [5], Lagrangian relaxation algorithm [6], dynamic programming [7], and lambda iteration [8].

Unlike classical methods, metaheuristic methods are better options because they can handle more constraints and are able to explore the search domain effectively in finding the optimum; they include ICA (imperialist competitive algorithm) [9], CS (cuckoo search) [1], DE (differential evolution) [10], ABC (artificial bee colony) [11], PSO (particle swarm optimization) [12], TLBO (Teaching–learning-based optimization) [13], SOA (seeker optimization algorithm) [14], MGSO (modified group search optimizer) [15], GA (genetic algorithm) [16], and HBMO (honey bee mating algorithm) [17]. DE is especially very effective because it does not need derivative information from the cost function; instead it sub-optimally or prematurely converges [17]. Other drawbacks associated with metaheuristics are high sensitivity to the control parameters, long computational time, and slow convergence to approximately optimum solution [18].

Recent hybrid methods overcome those drawbacks, able to handle the high complexities of practical ED problems. One method might be adopted for its high convergence, another for its provision of a suitable initial guess for the problem. The hybrid methods are combinations of either two or more metaheuristic methods or metaheuristic with classical techniques. Combinations of PSO with DE [17], GA with API [19], GA-LI [20], CPSO-SQP [21], and FCASO-SQP [22] perform better as hybrids than individually.

BSA (Backtracking search algorithm) is a new evolutionary algorithm developed by Civicioglu [23]. It has been successfully applied to many high-dimensional multimodal optimization benchmarks. Statistical analysis of its results confirms its superiority and performance over several other widely used evolutionary methods of optimization.

This paper proposes BSA as an approach to solving both convex and non-convex ED problems. The transmission network loss is modeled in consideration of the network topology whereas the valve-point effect is considered for accurate modeling of the generator cost. BSA's performance in solving ED problems is compared with other popular methods in terms of solution quality. The paper is organized next as follows: Section 2 presents the proposed method's algorithm, Section 3 provides the mathematical model of the ED problem considering the transmission loss, Section 4 presents the method's application to four test systems (the case studies), and Section 5 presents the results analysis.

Section snippets

BSA (Backtracking search optimization algorithm)

Evolutionary algorithms use the techniques inspired by natural or biological evolutions such as mutation, crossover, and selection to generate solutions for optimization problems. BSA (Backtracking search algorithm) is also an evolutionary algorithm for solving of constrained optimization problems. It was developed to overcome some of the drawbacks of evolutionary methods; e.g., high sensitivity to the control parameters, time-consuming computation, and premature convergence. Its structure is

Formulating ED and implementing BSA

ED is a mathematical optimization process in which the economic power sharing among the generating units is determined to minimize an objective function. The objective function is usually the cost of the generating units. The ED problem is considered a single-objective optimization problem. Its two main parts are as follows.

Simulation results

The proposed method's robustness and capability were validated through four case studies. The transmission loss is considered to model the electric network whereas the valve-point effect is incorporated for accuracy of the cost model of each generating unit. The code was written in MATLAB (R2010b) and executed on a personal computer with Pentium 2.70 GHz processor and 2 GB RAM.As mentioned in Section 2, BSA has only one control parameter named “mixrate”. This parameter controls the number of

Discussion of the results of the test systems

To evaluate the proposed method's performance in solving ED problems, the results are compared in terms of economic effect, convergence characteristics, and robustness.

Conclusion

The proposed BSA (backtracking search algorithm) method of optimization ably solved the problem of non-convex ED (economic dispatch) that considered transmission network loss and valve-point effects. The four case studies varied in size and complexity were used to apply the proposed method. The results obtained by BSA were compared with other classical and evolutionary methods in terms of solution quality. In the small test systems (3 and 6 units), BSA converged to the same optimal in all the

Acknowledgments

The authors thank the technical and financial assistance of UM Power Energy Dedicated Advanced Centre (UMPEDAC) and the High Impact Research Grant (H-16001-00-D000032).

References (41)

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