A conglomerated ion-motion and crisscross search optimizer for electric power load dispatch
Introduction
In power system parlance, one of the key issues is to meet the real power demand completely without violating any constraint at a minimum possible cost [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]. This key issue is known as economic load dispatch or electric power load dispatch (EPLD) problem [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]. In the literature, the EPLD problem has been studied thoroughly and divided into two groups. In group one, the operating cost function of the EPLD problem is modeled as a convex continuous quadratic function [2]. In group two, the EPLD problem formulations consider the non-convex, dis-continuous and non-differentiable operating cost function with more practical constraints [3]. For optimizing the convex EPLD problem, the commonly employed traditional mathematical optimization techniques are Newton methods, linear programming, Lambda-iteration method, branch and bound method, Lagrangian relaxation method, dynamic programming etc. [1], [2], [3], [4]. In the case of the non-convex EPLD problems, the traditional mathematical optimization algorithms fail due to the NP-hard nature of such EPLD problems. Due to the combinatorial nature of non-convex EPLD problems, soft computing techniques become the most suitable aspirant to solve such type of EPLD problems.
One of the most commonly employed soft computing algorithm for EPLD problem is particle swarm optimization (PSO) and different variants of it [1], [2], [3], [4], [5], [6], [7], [8], [9]. Gaing [1] employed a PSO on different variants of EPLD problem thereby considering various nonlinear characteristics of the generator such as ramp rate limits (RRL), prohibited operating zones (POZ) and non-smooth cost functions. Researchers further developed another variants of PSO which includes: improved PSO (IPSO) [2], hybrid PSO with a real-valued mutation (RVM) (PSO-RVM) [3], hybrid PSO with a gravitational search algorithm (GSA) PSOGSA [4], a novel hybrid algorithm of differential evolution (DE) and PSO (DPD) [5], improved orthogonal design PSO (IODPSO) [6], improved random drift PSO (IRDPSO) [7] and orthogonal learning competitive swarm optimizer (OLCSO) [8]. The recent version of PSO is phasor PSO (PPSO) [9]. In PPSO [9] the parameters of PSO have been tuned by using the sine and cosine functions. The genetic algorithm (GA) also have been developed to solve EPLD problems [10], [11], [12]. The popular variants of GA for EPLD are: improved genetic algorithm with multiplier updating (IGA_MU) [10], a hybrid GA with special class of ant colony optimization (ACO) called API (GA-API) [11] and hybrid GA with differential evaluation (DE) and pattern search (PS) (GA–DE–PS) [12].
Another keen interest of researchers is the differential evaluation (DE) algorithm [13], [14], [15], [16], [17], [18] for EPLD problems. The alternative and improved DE strategies are: modified DE (MDE) [13], hybrid DE with biogeography-based optimization (BBO) (DE/BBO) [14], shuffled DE (SDE) [15], colonial competitive DE (CCDE) [16], improved DE (IDE) [17] and a continuous greedy randomized adaptive search procedure (C-GRASP) with self-adaptive DE (C-GRASP-DE) [18]. The harmony search optimizer also finds its application in EPLD problems as hybrid harmony search (HS) with arithmetic crossover (ACHS) [19], tournament-based HS (THS) [20] and natural updated HS (NTHS) [21].
In [22] a chaotic bat algorithm (CBA) was developed for EPLD problems. A -modified bat algorithm (-MBA) was proposed to solve the EPLD problem considering medium to large-scale power systems by Kavousi-Fard and Khosravi [23]. The -MBA searches in polar coordinate instead of Cartesian search space. Recently, grey wolf optimization (GWO) [24] also grabbed the focus of researchers for solving the EPLD problems. The recent variants of GWO are hybrid GWO (HGWO) [25] and ameliorated GWO (AGWO) [26]. Oppositional based learnings have been introduced in quasi-oppositional TLBO (QOTLBO) [27], oppositional real coded chemical reaction optimization (ORCCRO) [28], oppositional invasive weed optimization (OIWO) [29] and Opposition-based krill herd algorithm (OKHA) [30].
Some other intuitive, modified and hybrid optimizers for EPLD problem are: fire-fly (FA) [31] algorithm, multi-strategy ensemble BBO (MsEBBO) [32], immune algorithm for economic dispatch problem (IA_EDP) [33], one rank cuckoo search algorithm (ORCSA) [34], hybrid with cross-entropy method and sequential quadratic programming (CE-SQP) [35], exchange market algorithm (EMA) [36], rooted tree optimization (RTO) [37], social spider algorithm (SSA) [38], modified Symbiotic Organisms Search (MSOS) [39], backtracking search algorithm (BSA) [40], lightning flash algorithm (LFA) [41], modified crow search algorithm (MCSA) [42] and hybrid artificial algae algorithm (HAAA) [43]. In [44], Çelik and Öztürk proposed a hybrid symbiotic organism search and simulated annealing (HSOS-SA) technique. In HSOS-SA, symbiotic organism search performs exploratory search and simulated annealing searches for the exploitation.
Although, a lot of soft computing optimization techniques have been implemented to solve the EPLD problem none of them can claim to be the best algorithm. Some of them give good exploration but suffers in terms of exploitation and vice-versa. Some soft computing or metaheuristic techniques provide fast convergence but stick to local solutions. Some of them converge towards global optimum but with a very slow rate of convergence. Despite a lot of improvements in soft computing optimization techniques, still researchers are trying to develop an algorithm which will be a better performer in terms of desirable features like; excellent balance between exploration and exploitation, computationally fast and converging towards global optima. Further, it is also observed from the literature review that one of the effective ways to obtain the outstanding algorithm (i.e. having an excellent balance between exploration and exploitation, computationally fast and converging towards global optima) is the hybridization or conglomeration of two algorithms for solving the optimization problems [3], [4], [5], [15], [18], [19], [30], [35], [43].
In the search of a desirable metaheuristic algorithm to solve optimization problems, in this paper, a new conglomerated optimizer based on ion-motion and crisscross search optimizers for non-linear, discontinuous and constrained electric power load dispatch problem (EPLD) has been proposed. In the proposed optimizer, an improved mathematical model of the liquid phase of ion motion optimizer (IMO) has been proposed to achieve better exploration capability. The IMO with the proposed modified liquid phase is termed as modified IMO (MIMO) in this paper. After confirming the superiority of MIMO over IMO while solving the benchmark standard functions and EPLD problems, the proposed MIMO has been conglomerated with crisscross optimizer (CSO) [45]. In the proposed conglomerated optimizer, the proposed MIMO explores the search space and CSO provides a local search. In this way, the proposed conglomerated optimizer comes with an excellent balance between exploration and exploitation capabilities and provides fast convergence towards a global solution. The proposed conglomerated optimizer has been abbreviated as C-MIMO–CSO.
The ion motion optimization (IMO) was proposed by Javidy et al. [46]. The IMO is a population-based nature-inspired algorithm and simply works on the principle of attraction and repulsion of cations and anions in nature [46]. The optimization procedure of IMO includes two phases: liquid and crystal phases. In the liquid phase, the exploratory search is performed by ions (i.e. both cation and anions) considering that ‘all cations move towards best anion and all anions move towards best cation’. In the liquid phase of IMO the effect of repulsion among the same ions and effect of attraction force among different ions is completely ignored. In the crystal phase of IMO the exploitation of search space is performed if best cation and best anion fitness are more than half of worst fitness of cation and anion [46].
The contributions of the paper are listed as follows.
- I.
A modified ion motion optimization (MIMO) is proposed considering the effect of repulsion among the same type of ions and attraction among different type of ions in the liquid phase of IMO. The effectiveness of developed MIMO has been tested on both the unconstraint benchmark and constraint electric power load dispatch (EPLD) problems.
- II.
A new conglomerated optimizer based on modified ion-motion and crisscross optimizers (C-MIMO–CSO) for EPLD has been proposed.
- III.
A new proportionate power-sharing based heuristic approach is used to handle equality constraint along with exterior penalty method and operation in prohibited operating zones is avoided by a heuristic approach.
The rest of the paper is organized in total of five sections. Section 2 gives the formulation of EPLD problems, Section 3 discusses constraint handling schemes for EPLD problems, Section 4 develops and discusses proposed IMO, MIMO and conglomerated optimizer for EPLD problem, Section 5 provides a detailed discussion on results and Section 6 concludes the paper.
Section snippets
The EPLD problem
The detailed mathematical formulation of EPLD problem as a non-linear, discontinuous and constrained optimization problem is given in the following sections:
Constraint handling strategies for EPLD problem
The constraint handling strategies are discussed in ensuing subsections.
The solution methodology for electric power load dispatch problem
In this section, the foundation of ion-motion optimization (IMO) is provided briefly. Afterward, the conglomerated optimizer has been developed and applied to solve electric power load dispatch problem.
Results and discussions
This section discusses the results obtained by proposed algorithms for both CEC’05 benchmark functions and EPLD problems in detail. The IMO, MIMO and C-MIMO–CSO algorithms are run for 50 times for EPLD problems and 25 times (as per the guidelines of IEEE CEC’05 benchmark functions) for benchmark functions due to stochastic nature of these algorithms. The maximum number of function evaluations (MAXNFE) is considered as the termination criteria for each algorithm. The Algorithm 3 iterates for 30
Conclusion and future scope
The ion-motion algorithm has been successfully implemented on six variants of economic power load dispatch problem. It is observed that the ion-motion optimization algorithm is very prone to premature convergence. To overcome this issue, a modified version of ion-motion optimization has been proposed and named as modified ion-motion optimization algorithm. The modified ion-motion optimization algorithm is developed by introducing the concept of random repulsion force and considering the effect
Declaration of Competing Interest
No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.105641.
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