Complexity

Volume 2019, Article ID 4727168, 11 pages

https://doi.org/10.1155/2019/4727168

## An External Archive-Based Constrained State Transition Algorithm for Optimal Power Dispatch

^{1}The School of Information Science and Engineering, Central South University, Changsha 410083, China^{2}The Institute of Electronics, Communications and Information Technology (ECIT), the School of Electronics, Electrical Engineering and Computer Science (EEECS), Queen’s University Belfast, Belfast BT7 1NN, UK

Correspondence should be addressed to Guanbo Jia; ku.ca.buq@aij.g

Received 13 July 2018; Accepted 18 September 2018; Published 3 January 2019

Guest Editor: Zhile Yang

Copyright © 2019 Xiaojun Zhou 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

This paper proposes an external archive-based constrained state transition algorithm (EA-CSTA) with a preference trade-off strategy for solving the power dispatch optimization problem in the electrochemical process of zinc (EPZ). The optimal power dispatch problem aims to obtain the optimal current density schedule to minimize the cost of power consumption with some rigorous technology and production constraints. The current density of each production equipment in different power stages is restricted by technology and production requirements. In addition, electricity price and current density are considered comprehensively to influence the cost of power consumption. In the process of optimization, technology and production restrictions are difficult to be satisfied, which are modeled as nonconvex equality constraints in the power dispatch optimization problem. Moreover, multiple production equipment and different power supply stages increase the amount of decision variables. In order to solve this problem, an external archive-based constrained state transition algorithm (EA-CSTA) is proposed. The external archive strategy is adopted for maintaining the diversity of solutions to increase the probability of finding the optima of power dispatch optimization problem. Moreover, a preference trade-off strategy is designed to improve the global search performance of EA-CSTA, and the translation transformation in state transition algorithm is modified to improve the local search ability of EA-CSTA. Finally, the experimental results indicate that the proposed method is more efficient compared with other approaches in previous papers for the optimal power dispatch. Furthermore, the proposed method significantly reduces the cost of power consumption, which not only guides the production process of zinc electrolysis but also alleviates the pressure of the power grid load.

#### 1. Introduction

Hydrometallurgical zinc is the main production approach of zinc, accounting for more than 80% of zinc production in the world [1]. The electrochemical process of zinc plays an important role in the hydrometallurgical process of zinc [2]. The power consumption decided by current efficiency and cell voltage is an important economic indicator of electrolytic zinc process. In the process, zinc is deposited in the zinc sulfate solution under the action of direct current. Current efficiency and cell voltage are influenced by current density directly. If the current density is too low, the current efficiency will drop sharply, and zinc deposited on cathodes will be dissolved [3]. If the current density is too high, the temperature of the electrolytic cell will rise and impurities in the solution will increase. At the same time, the increased current density will definitely lead to high cell voltage, which affects the power consumption. The relationship between current density and current efficiency is nonlinear, which may not cause current efficiency to increase as expected. In addition, the current density is also limited by the maximum current strength that the plate can withstand. Due to the complexity of zinc electrolysis process, it is difficult to find a suitable current density for the optimal power dispatch.

Previously, power can be supplied at a constant current density without changing the price of electricity. However, the power sector adopts time-based pricing [4], which means that the price of electricity is high at the peak of electricity consumption, while low at the valley of electricity consumption. If the production process is running at the lowest price period every day, there is no doubt that the daily output of zinc will not be satisfied. It is the same on the basic electricity price period. If the production process is running at the highest price every day, the daily output of zinc will be satisfied. But, this does not achieve the goal of minimizing the cost of power consumption. Our idea is to reasonably allocate the power consumption at different electricity price periods to minimize the cost of power consumption. Therefore, it is necessary to find an optimal power dispatch in different pricing periods [5]. The optimal power dispatch based on time-sharing price counting policy can be formulated as a constrained optimization problem (COP), which is to minimize electricity bills when production and technology constraints are satisfied.

In this paper, the main challenges of optimizing power dispatch are given as follows. (i)Nonconvex equality constraint function: in the hydrometallurgical process of zinc, the equality constraint function is related to the daily output of zinc, which need to satisfy consumer demand. Due to the small feasible search space, it is difficult to satisfy equality constraint in the search process(ii)Multiple decision variables: the number of decision variables is decided by the number of production equipment and power supply stages. The value of decision variable depends on not only the electricity price of different periods but also the production and technology requirements

In the literatures, many methods have been designed for solving optimal power dispatch in the zinc electrolysis process. Yang et al. [5, 6] proposed backpropagation and Hopfield neural network for optimal power dispatch. Li and Gui [4] solved power dispatch optimization problem by an improved particle swarm optimization algorithm. Gui et al. [7] designed a hybrid particle swarm algorithm to solve the power dispatch optimization problem. Han et al. [8] tackled this problem by two-stage constrained state transition algorithm. Although these methods can obtain good solutions, the cost of power consumption can be lower by further optimizing the current density. Based on the study of those literatures, the optimal solution can be improved from the daily output of zinc and the current density of each production equipment in different power supply stages.

The optimal power dispatch in EPZ can be formulated as a constrained optimization problem (COP). Some basic techniques have been applied to solve COPs, such as adaptive penalty function technique [9], adaptive trade-off model [10], and Deb’s rules [11]. Hybrid techniques in which two or more strategies are integrated to solve COPs have been designed, such as Deb-penalty technique [8]. Improved version techniques have been applied for solving COPs, like improved ()-constrained differential evolution [12] and improved adaptive trade-off model [13]. However, these techniques rarely consider the diversity of solutions from the perspective of feasible and infeasible solutions. Moreover, it is difficult to find a good solution since the power dispatch model contains the nonconvex equality constraint and multiple decision variables. So maintaining the diversity of feasible solutions and infeasible solutions can be instructive to find a better solution.

In this paper, the constrained state transition algorithm based on external archive with preference trade-off strategy is proposed for the power dispatch problem in electrolytic zinc process. The external archive-based constrained state transition algorithm (EA-CSTA) is different from the constrained state transition algorithm (CSTA) [8] on selecting solutions. The CSTA selects only a current best solution from a set of candidate solutions, while the EA-CSTA adopts an external archive to store multiple potential solutions. In addition, a novel constraint-handling technique, called preference trade-off strategy, is proposed to select solutions from both feasible and infeasible candidates. The novelty and the main contributions of the paper can be summarized as follows. (1)An external archive strategy is designed to save multiple potential feasible and infeasible solutions. The EA-CSTA achieves the state transition by selecting several potential solutions saved in an external archive, which increases not only the diversity of solutions but also the probability of finding the global solution. In order to expand the search scope of the candidate solutions, translation transformation in STA [14] is modified to share information among potential solutions(2)The preference trade-off strategy in the proposed method contains both preference and trade-off. Firstly, it is able to adjust the number of feasible and infeasible solutions. Secondly, strategies are different in dealing with feasible and infeasible candidates, which avoid the direct comparison of feasible and infeasible candidates. Also, it increases the diversity of these selected solutions in an auxiliary manner. Some preference strategies are adopted to select solutions, such as adding a penalty factor to normalization. The normalization is capable of dealing with the different scale between cost of power consumption and production constraints(3)The proposed method is successfully applied to solve the power dispatch optimization problem in EPZ that can bring significant economic profits to the metallurgy industry. In addition, it is conductive to relieve the pressure not only on the power grid but also on the peak power consuming period of power industry

The remainder of the paper is organized as follows. Section 2 introduces the preliminary knowledge of power dispatch model and constraint-handling techniques. In Section 3, the proposed constrained STA with external archive and preference trade-off strategy is elaborated. Results and discussions are presented in Section 4. Finally, Section 5 draws a conclusion of this paper and gives the possible future work.

#### 2. Preliminaries

In this section, the power dispatch model is expressed in detail. It contains the objective function which is minimizing the cost of power consumption, technology, and production constraints. Then, some classical and effective constraint-handling techniques are described.

##### 2.1. Problem Formulation

The electrochemical process of zinc is a considerable large amount of power consumption process, which accounts for 80% of the total electrical energy consumption in the hydrometallurgy process of zinc. To encourage customers to consume more power in the valley-load period and less power in the peak-load period, the power sector adopts time-based pricing strategy as shown in Table 1. It means that the price of electricity is high at the peak of electricity consumption, while low at the valley of electricity consumption. The cost of power consumption will be decreased in the case that the electrochemical process of zinc consumes a small amount of electricity in the period of high price and vice versa. However, if the current density is too high or too low, it will not only affect the power consumption but also influence the product quality. To prevent low quality of product and excessive power consumption, it is essentially desired to seek for suitable current density in different pricing periods.
where (kW) decided by voltage and current is the power consumption of th price period, which can be formulated as (2), (h) is the duration of th price, is the electricity price (RMB/kW·h) at th period, and is the number of the price periods; Fc0 is the basic tariff charge of electrochemical process of zinc.
where (V) and (A) are the voltage and current of th price period in the th plant, respectively, is the number of cells in the th plant, and is the number of plants.
where and are obtained by recursive least squares method, (A/m^{2}) is the current density th price period in the th plant, denotes the number of plates in a cell in the th plant, and (m^{2}) is the area of negative plate.