Hybrid Search Method for Optimization in Real Parameter space

Abstract

Many real world problems need to optimize available resources to meet the desired objectives. It is an essential part for the better performance of a system. Optimization helps the planners to have suitable efficient options appropriate for their various applications and needs. Although, the optimization process has been characterized in distinct scenarios, yet the objective is always to obtain a set of decision variables in the feasible search space that maximizes beneficial objectives of the system. A practical optimization problem involves simultaneous optimization of several incommensurable and often competing objectives. In such a case, there is no single optimal solution, but rather a set of possible solutions. These solutions are optimal in the wider sense that no other solutions in the search space are superior to them, when considering all the objectives together. Broadly, multi-objective optimization techniques are grouped under two major categories: non-interactive and interactive. In the non-interactive method, a global performance function of the objectives is identified and optimized with respect to the constraints. On the other hand, in the interactive method, a local preference function or trade-off among objectives is identified by interacting with decision makers (DMs) and the solution process proceeds gradually towards a global satisfactory solution. The solution procedure for these problems involves two parts: Firstly, a setup to generate feasible solution and secondly, a tool to select compromise solution that best suits the conflicting objectives of the problem. Nowadays, the use of optimization algorithms is an essential part of the multiobjective solution approach, as these are responsible to generate potential solution of considered problem in feasible search space. The Evolutionary Algorithm (EA) is an effective tool to explore the problem's search area. Among the available optimization techniques, the EAs are widely accepted. The initial development phase of EA's was dominated by Evolutionary Programming (EP), due to its single operator based mechanism. Later on, Particle Swarm Optimization (PSO) algorithm has received wide attention from the research community due to better exploration and exploitation capabilities. Recently, the Differential Evolution (DE) algorithm has gotten popular among the research community, due to its well balanced diversification and intensification process. Owing to improve the performance of a standard algorithm, hybridization is the possible alternative. It means that although, an appropriate standard algorithm results in better performance and still its performance can be further improved if standard algorithm is combined with specific heuristics or operators that incorporate domain knowledge. Furthermore, hybrid algorithms emphasize on the complementary advantage of population-based search (exploration) and their refinement (exploitative) by local search techniques. The exploration provides a consistent estimate of the global optimum, while the exploitative part concentrates on the search effort around the best solution. The combined method extracts the virtues of the parent approaches and tries to materialize effcient solution approach. Depending on the introduction of local search technique along with population based method in the search process, the hybrid algorithms are classified as before/after/interleaved procedure or mixture of these. On the other hand, the application of non-linear dynamics has recently drawn attention in the field of optimization. One of the applications is the use of chaos map to select algorithm-dependent parameters replacing uniform random variables. The empirical studies have shown that chaotic sequences have a high level of mixing capability and thus it is expected that their use to replace fixed procedures, may result in a better balance of the exploration and exploitation. So, in the context of improving performance of EP, PSO and DE, a useful diversity is ensured by combining deterministic chaotic sequence with them. Among the wide variety of optimization applications, power system scheduling is a realistic active application of the optimization process. The power dispatch problem from the field of power system optimization presents a challenging task to EAs. The dispatch problems are categorized as single objective, such as Economic Load Dispatch (ELD), Minimum Emission Dispatch (MED) and Multi-objective Load Dispatch (MOLD). The ELD problem aims to minimize the power generation cost, whereas the MED is directed to minimize the gaseous pollutant emission. The MOLD intends to minimize the two conflicting objectives power generation cost and pollutant emission simultaneously. Apart, from these objectives the system associated constraints of ramp-rate limits, avoiding prohibited operating zone, valve point loading effect and multiple fuel source options are also considered for practical realization. Therefore, in the above stated power system dispatch problem, the motive is to obtain a generation schedule in the feasible search space that minimizes associated objectives, while satisfying the system operational constraints. After careful study of the recent reported work in literature, it has been observed that: no single optimization technique suits best for various kinds of real life problems, direct search methods are better at local exploitation of solution but are sensitive to initial guess and step size. The random search methods EP, PSO and DE have proven their effectiveness as a better exploration, search process, but still they have limitations. The performance issues of these techniques are improved by hybridizing these techniques with local search techniques or adding heuristics. Another alternative incorporated is the application of chaotic sequence to modify the evolution and selection operators used in EP, PSO and DE algorithms. Apart from this, the multi-objective problem handling procedures and constraint handling techniques is another direction of research. In the light of aforementioned gaps, the main contributions of this research are: The evolutionary programming algorithm is coordinated along with Powell's pattern search and chaotic sequence and a technique chaotic evolutionary programming and Powell's search is proposed. Further, the predator prey optimization (PPO) performance is improved using concepts of collaborative behavior, psychology along with local search techniques. The proposed algorithms are termed as synergic predator prey optimization (SPPO), adaptive predator prey optimization (APPO) and y and walk predator prey optimization (FWPO). Lastly, the unification of differential evolution algorithm with Powell's search and chaotic sequence is proposed as chaotic differential evolution and Powell's search (CDEPS). The formulated hybrid algorithms, namely: CEPPS, SPPO, APPO, FWPO and CDEPS have been validated using the standard test problems. The nature of functions selected for experimentation is uni-modal, multi-modal, separable, non-separable, continuous and discontinuous. In order to validate the formulated algorithms on real world problem, the economic dispatch problems for the multi fuel system, Taiwan power system consisting of 40 generators and Korean power system having 140 generators are implemented. Finally, multi-objective load dispatch has been considered for performance validation. The limitations of weighting method to handle conflicting objective of multi-objective problem is elevated by using it as an interactive approach. In addition, surrogate worth trade off method (SWT) is incorporated in the research work to handle multi-objective problem, in interactive and non-interactive manner. The constraints being an essential constituent of real world power system problems, the constraints are handled either by using the concept of slack generator or by incorporation of power distribution. In this thesis, the research work is organized in seven chapters. In the first chapter, the importance of optimization, eminent milestones in the development of optimization procedures is presented. The review of approaches to solve multi-objective optimization problems has been reviewed. The power system dispatch problem being a realistic application of optimization, the significant contributions of various authors related to dispatch problems and its relevant topics have been briey reviewed in this chapter. The requisite theoretical, mathematical, and computational backgrounds of di_erent local and global optimization techniques are reviewed which are utilized in the present study to determine the solution of optimization problems. Second chapter, introduces a synergic predator-prey optimization (SPPO) algorithm to solve economic load dispatch problem for thermal units with practical aspects. The basic PPO model comprises prey and predator as essential components. In the SPPO, the decision making of prey is bifurcated into corroborative and impeded parts. It comprises four behaviors namely inertial, cognitive, collective swarm intelligence, and prey's individual and neighborhood concern of predator. In order to, improve the quality of prey swarm, which influence the convergence rate, opposition based initialization is used. In order to, verify robustness of proposed algorithm general benchmark problems and small, medium, and large power generation test power systems are simulated. The chapter three, incorporates chaotic evolutionary programming and pattern search (CEPPS) as solution procedure, In CEPPS, the chaotic mutation operator is used in the evolution phase. Similarly, the selection operation is also a chaotic sequence guided. In order to, ensure exploitation, Powell's pattern search operator is included, which operates under stochastic process. The surrogate worth trade off approach is utilized to select a compromised solution from the Pareto front, which suits both the considered objectives in the feasible search space. The generalized test functions and realistic MOLD problem is considered to investigate the performance of the proposed solution approach. The performance comparison of CEPPS and chaotic evolutionary programming, applying on generalized benchmark test functions, shows that CEPPS has better search capabilities. However, while implementing dispatch problems, the CEPPS has resulted in premature convergence. Fourth chapter, presents adaptive predator-prey optimization (APPO) to solve MOLD problem with objectives of operating cost and pollutant emission. In APPO, the fear factor from predator is based on the cognitive and social behavior of prey, to ensure continuous mobility (magnitude and direction) of prey, resulting in better diversification of solutions in the domain throughout the search process. In this chapter, the multi- objective optimization problem is handled by weighting method, whereby the weight pattern assigned to the objectives has been undertaken as decision variables. This results in non-inferior solutions at each swarm move. In order to select a best-compromised solution, fuzzy theory is applied. The performance of the proposed algorithm is investigated on power system test problems. The proposed method provides better results in terms of fuel cost and pollutant emission. In addition, the better satisfaction level of both the conflicting objectives, well distributed Pareto front, acceptable solution in a single trial run and insensitivity to parameter variations is observed in comparison to other existing methods reported in literature. The chapter five, presents a fuzzy surrogate worth trade-off approach to decide the preferred solution among the non-dominated solutions' set. The uniformity of non-dominated solutions' set is maintained exploiting a quality measure approach. This chapter describes a novel foraging activity paradigm to generate non-dominated solutions. The presented approach comprises of fly and walk behavior of preys. The direction of turn heuristic of prey handles system's equality constraint. The search algorithm balances exploration and exploitation and handles system constraints autonomously. The performance of the proposed algorithm is investigated using generalized benchmark functions and multi-fuel, medium and large power system MTPLD problems. The experimental results show that the proposed approach is robust, depends on least parameters and retains Pareto quality in independent trials while generating non-dominated solutions in a trial run. Sixth chapter, utilizes a chaotic differential evolution and Powell's pattern search algorithm (CDEPS) is proposed to solve MOLD problem. The chaotic differential evolution method is responsible for the diversification and Powell's pattern search is dedicated to exploitation. Further, the performance of two CDEPS variants based on Gauss map and Tent map is investigated. Another objective of this chapter is to introduce SWT as a non- interactive approach to solve MOLD problem. The performance analysis is done using generalized benchmark test functions and complex MTPLD problems. The Wilcoxon's test is used to analyze the experimental results. The exhaustive analysis shows that the Tent map based CDEPS has better ability to generate quality generation schedule with faster convergence rate. Finally, the chapter seven, presents a brief summary and conclusions of all the chapters and the recommendation for further research.