Complexity

Volume 2018, Article ID 2591341, 18 pages

https://doi.org/10.1155/2018/2591341

## A Novel Method for Economic Dispatch with Across Neighborhood Search: A Case Study in a Provincial Power Grid, China

^{1}Guizhou Key Laboratory of Intelligent Technology in Power System, College of Electrical Engineering, Guizhou University, Guiyang 550025, China^{2}State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan 430074, China^{3}Guizhou Electric Power Grid Dispatching and Control Center, Guiyang 550002, China^{4}Key Laboratory of Network Control & Intelligent Instrument, Chongqing University of Posts and Telecommunications, Ministry of Education, Chongqing 400065, China

Correspondence should be addressed to Guojiang Xiong; moc.liamxof@eegnoixjg

Received 8 July 2018; Accepted 23 October 2018; Published 4 November 2018

Guest Editor: Zhile Yang

Copyright © 2018 Guojiang Xiong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Economic dispatch (ED) is of cardinal significance for the power system operation. It is mathematically a typical complex nonlinear multivariable strongly coupled optimization problem with equality and inequality constraints, especially considering the valve-point effects. In order to effectively solve the problem, a simple yet very young and efficient population-based algorithm named across neighborhood search (ANS) is implemented in this paper. In ANS, a group of individuals collaboratively navigate through the search space for obtaining the optimal solution by simultaneously searching the neighborhoods of multiple superior solutions. Four benchmark test cases with diverse complexities and characteristics are firstly employed to comprehensively verify the feasibility and effectiveness of ANS. The experimental and comparison results fully demonstrate the superiority of ANS in terms of the final solution quality, convergence speed, robustness, and statistics. In addition, the sensitivities of ANS to variations of population size and across-search degree are studied. Furthermore, ANS is applied to a practical provincial power grid of China. All the comparison results consistently indicate that ANS is highly competitive and can be used as a promising alternative for ED problems.

#### 1. Introduction

Economic dispatch (ED), playing an important role in the power system operation and planning, has received significant attention in recent years. The purpose of ED is to schedule the committed generating unit outputs so as to simultaneously minimize the operating cost and meet the load demand of a power system while satisfying all the equality and inequality constraints [1]. Traditionally, an approximate quadratic function is utilized to make the mathematical formulation of ED problem convex to reduce the computational difficulty. However, in practice, on one hand, the multi-valve steam turbines make the input–output curves of generators inherently present highly nonlinear characteristic. On the other hand, faults in the machines themselves or the associated auxiliaries prohibit generators from operating in some zones [2]. Therefore, the solution space of ED problem with the presence of valve-point effects and prohibited operating zones is highly nonlinear and discontinuous, making the optimization hard to be tractable. In this context, conventional solution methods including linear programming, Lagrange relaxation, nonlinear programming, quadratic programming, dynamic programming, and interior point method are likely to encounter dire difficulties and challenges mainly due to their heavy imposition of various restrictions such as continuity, convexity, and differentiability on the objective functions, and high sensitivity to the initial values of involved optimized variables.

As a promising alternative to the conventional solution methods, metaheuristic methods for ED problems have attracted considerable attention recently. They have no strict requirements on the form of optimization problems and can avoid the influences of the initial condition sensitivity and gradient information. Up to now, the successfully implemented metaheuristic methods include simulated annealing [3], genetic algorithm [4, 5], particle swarm optimization [6–8], differential evolution [9, 10], artificial bee colony [11, 12], harmony search [13–17], biogeography-based optimization [18–22], teaching-learning-based optimization [23–25], firefly algorithm [26], crisscross optimization algorithm [27, 28], bat algorithm [29], grey wolf optimizer [30, 31], cuckoo search [32–34], ant lion optimizer [35], exchange market algorithm [36], symbiotic organisms search [37, 38], backtracking search algorithm [39, 40], interior search algorithm [41], whale optimization algorithm [42], mine blast algorithm [43], and hybrid methods [44–56].

The abovementioned metaheuristic methods have verified their efficacy in solving the ED problems. Regardless of the achieved promising results, the no free lunch theorem [57] indicates that there is no specific method which can be adopted as a gold standard for all kinds of optimization problems. Namely, there is no single universal superior method that, theoretically, always performs best in solving the ED problems. Therefore, there are still some possibilities to attempt new ones to provide more alternatives, which inspires the authors to apply a recently developed metaheuristic method named across neighborhood search (ANS) [58] to obtain high quality solutions for the ED problems.

As a simple yet versatile metaheuristic method, ANS is motivated by two common straightforward assumptions existing in different population-based algorithms: that searching around a superior solution has a higher probability to find another better solution and that high-quality solutions possess good solution components. In this context, ANS, following the law of parsimony, attempts to simultaneously search across the neighborhoods of multiple superior solutions to get as many potential good solution components as possible. The merits of ANS are its simple structure, ease of implementation, and strong robustness. In this paper, ANS is employed for the ED problems. The main contributions of this work are as follows.

(1) Four benchmark test cases with diverse complexities and characteristics are firstly used to verify the feasibility and effectiveness of ANS comprehensively. The superior performance of ANS is experimentally verified by comparing with four popular population-based algorithms and some recently proposed ED solution methods.

(2) The sensitivities of ANS to variations of population size and across-search degree are empirically investigated.

(3) ANS is finally applied to a practical provincial power grid of China. Its performance is further verified. In addition, the experimental results reflect the necessity and importance of the power construction policy of “replacing small power plants with large ones” in China.

The remainder of this paper is organized as follows. Section 2 briefly introduces the mathematical formulation of ED problems. In Section 3, ANS is described. Next, in Section 4, the flowchart of ANS in solving the ED problems is illustrated. In Section 5, four benchmark test cases are employed to verify ANS. ANS is then applied to a practical provincial power grid of China in Section 6. Finally, Section 7 is devoted to conclusions and future work.

#### 2. Problem Formulation

##### 2.1. Objective Function

The mathematical model of ED can be formulated as follows [59]:where is the total generation cost (in $/h), is the number of operating generators, is the active power output of the* i*-th generator (in MW), , is the generation cost function of the* i*-th generator (in $/h), , and are the number of equality constraints and inequality constraints, respectively, is the* j*-th equality constraint, , and is the* j*-th inequality constraint, .

The objective function of traditional ED problem is approximately formulated as follows [1, 2, 4]: where , , and are cost coefficients of the* i*-th generator.

In practice, modelling valve-point effects is necessary and can be formulated as follows [60]:where and are valve-point effects coefficients of the* i*-th generator and is the minimum active power generation limit of the* i*-th generator (in MW).

##### 2.2. Equality and Inequality Constraints

###### 2.2.1. Active Power Balance Constraint

The total active generated power must be equal to the sum of the total system demand () and the total transmission network loss (): where is commonly calculated using the following B-coefficient method [9]:where , , and are loss coefficients.

###### 2.2.2. Generation Capacity Constraints

The active power output of each generator should be within its minimum and maximum limits:where is the maximum active power generation limit of the* i*-th generator (in MW).

###### 2.2.3. Ramp Rate Limits Constraints

The adjustment of active power output of each generator should be in an acceptable range:where is the previous active power output of the* i*-th generator and and are the up-ramp and down-ramp limits of the* i*-th generator, respectively.

###### 2.2.4. Prohibited Operating Zones Constraints

Generators should avoid operating in prohibited zones [2]:where is the number of prohibited operating zones of the* i*-th generator and and are the lower bound and upper bound of the* k*-th prohibited zone of the* i*-th generator, respectively.

#### 3. Across Neighborhood Search

ANS is a very young population-based algorithm proposed by Wu [58] in 2016. ANS, following the law of parsimony and showing good performance compared with other methods [61], attempts to simultaneously search across the neighborhoods of multiple superior solutions to achieve as many potential good solution components as possible. At the same time, it needs to dynamically maintain and update the superior solutions to guarantee the advancement and convergence of the population. The main difference between ANS and other population-based algorithms is that other algorithms mainly utilize some operations such as crossover and mutation to generate new solutions, whereas ANS directly searches across the neighborhoods of multiple superior solutions to produce new solutions.

Like other population-based algorithms, ANS starts with a population of individuals representing the potential solutions. Each individual () consists of variables and it is initialized aswhere , rand(0,1) is a uniformly distributed random real number in (0,1), and and are the lower bound and upper bound of the* d*-th dimension, respectively.

In ANS, a group of individuals collaboratively navigate through the search space for obtaining the optimal solution. Each individual searches across the neighborhoods of multiple superior solutions. These superior solutions, being archived in a collection where is the cardinality and is generally set to be the population size , are directly derived from the individuals’ best positions found so far. The searching strategy is as follows:where is a pool used to record the randomly selected ( is called across-search degree, ) dimensions for individual , () is a randomly superior solution selected from the superior solution collection , and is a Gaussian random value with mean zero and standard deviation which is usually set to be 0.5.

It can be seen from (10) that each individual, on one hand, searches across the neighborhood of the individual’s best position achieved so far. On the other hand, it also simultaneously searches across the neighborhoods of other individuals’ best positions found so far. After updating individual , , its own superior solution will be replaced by if has a better fitness value.

The main procedure of ANS is given in Algorithm 1. It can be seen that the individuals are guided by multiple superior solutions and the structure, following the law of parsimony, is very simple, making the implementation easy.