Colonial competitive differential evolution: An experimental study for optimal economic load dispatch

Differential evolution (DE) algorithm is a population-based algorithm designed for global optimization of the optimization problems. This paper proposes a different DE algorithm based on mathematical modeling of socio-political evolution which is called Colonial Competitive Differential Evolution (CCDE). The two typical CCDE algorithms are benchmarked on three well-known test functions, and the results are verified by a comparative study with two original DE algorithms which include DE/best/1 and DE/rand/2. Also, the effectiveness of CCDE algorithms is tested on Economic Load Dispatch (ELD) problem including 10, 15, 40, and 140-unit test systems. In this study, the constraints and operational limitations, such as valve-point loading, transmission losses, ramp rate limits, and prohibited operating zones are considered. The comparative results show that the CCDE algorithms have good performance and are reliable tools in solving ELD problem.     

A hybrid differential evolution algorithm based on particle swarm optimization for nonconvex economic dispatch problems

This paper presents the design and application of an efficient hybrid heuristic search method to solve the practical economic dispatch problem considering many nonlinear characteristics of power generators, and their operational constraints, such as transmission losses, valve-point effects, multi-fuel options, prohibited operating zones, ramp rate limits and spinning reserve. These practical operation constraints which can usually be found at the same time in realistic power system operations make the economic load dispatch problem a nonsmooth optimization problem having complex and nonconvex features with heavy equality and inequality constraints.The proposed approach combines in the most effective way the properties of two of the most popular evolutionary optimization techniques now in use for power system optimization, the Differential Evolution (DE) and Particle Swarm Optimization (PSO) algorithms. To improve the global optimization property of DE, the PSO procedure is integrated as additional mutation operator.The effectiveness of the proposed algorithm (termed DEPSO) is demonstrated by solving four kinds of ELD problems with nonsmooth and nonconvex solution spaces. The comparative results with some of the most recently published methods confirm the effectiveness of the proposed strategy to find accurate and feasible optimal solutions for practical ELD problems.     

Solving non-convex/non-smooth economic load dispatch problems via an enhanced particle swarm optimization

Economic load dispatch (ELD) problems have been an important issue in optimal operation and planning of power system. Characterized by non-convex/non-smooth properties and various practical constraints, the ELD problems are difficult to solve using conventional optimization techniques. In this paper, an improved orthogonal design particle swarm optimization (IODPSO) algorithm is presented for solving the single-area and multi-area ELD problems with nonlinear characteristics of the generators, such as valve-point effects, prohibited operating zones, ramp rate limits and multiple fuels. In the IODPSO algorithm, an orthogonal designed method is used to construct a promising exemplar. Multiple auxiliary vector generating strategies are proposed to enhance the efficiency and effectiveness of orthogonal design operations. A tent chaotic map is employed for the adaptation of the acceleration coefficients, thus improving the proposed algorithm's robustness and global search capabilities. In addition, we designed a repair method to handle the practical constraints. Six cases of ELD problems with different characteristics are utilized to benchmark the proposed algorithm. Experimental results demonstrate that IODPSO algorithm is a promising approach for solving the non-convex/non-smooth ELD problems.     

Differential evolution with Gaussian mutation for combined heat and power economic dispatch

This paper presents differential evolution with Gaussian mutation to solve the complex non-smooth non-convex combined heat and power economic dispatch (CHPED) problem. Valve-point loading and prohibited operating zones of conventional thermal generators are taken into account. Differential evolution (DE) is a simple yet powerful global optimization technique. It exploits the differences of randomly sampled pairs of objective vectors for its mutation process. This mutation process is not suitable for complex multimodal optimization. This paper proposes Gaussian mutation in DE which improves search efficiency and guarantees a high probability of obtaining the global optimum without significantly impairing the simplicity of the structure of DE. The effectiveness of the proposed method has been verified on five test problems and three test systems. The results of the proposed approach are compared with those obtained by other evolutionary methods. It is found that the proposed differential evolution with Gaussian mutation-based approach is able to provide better solution.     

The application of one rank cuckoo search algorithm for solving economic load dispatch problems

In this paper, a one rank cuckoo search algorithm (ORCSA) is proposed for solving economic load dispatch (ELD) problems. The main objective of the ELD problem is to minimize total cost of thermal generators while satisfying power balance constraint, prohibited operating zones, ramp rate constraints and operating limits of generators. Moreover, the generating units considered in this paper have different characteristics such as quadratic fuel cost function, nonconvex fuel cost function and multiple fuel options. The proposed ORCSA method has been developed by performing two modifications on the original cuckoo search algorithm (CSA) to improve optimal solution quality and computational time. The first modification is to merge new solution generated from both Lévy flights and replacement a fraction of egg together and to evaluate and rank the solutions at once only. A bound by best solution mechanism has been used in the second modification for properly handling the inequality constraints. The proposed ORCSA method has been tested on different systems with different characteristics of thermal units and constraints. The results obtained by ORCSA have been compared to those from other methods available in the literature and the result comparison has indicated that the ORCSA method can obtain better solution quality than many other methods. Therefore, the proposed ORCSA can be a very effective and efficient method for solving ELD problems.     

Water cycle algorithm-based economic dispatcher for sequential and simultaneous objectives including practical constraints

The article presents an efficient methodology based-on water cycle algorithm (WCA) to solve single and multiple objectives of economic load dispatch (ELD) aiming to generate the optimal value of the active generated power for each unit. Three objectives are adopted for optimisation either sequentially or concurrently; they are: (i) fuel cost considering valve-ripple effect, (ii) emission rate, and (iii) total network loss. The generating unit prohibited zones along with ramp rate limits and generating unit power limits specify the inequality constraints of the problem while maintaining system power balance. Usually, optimisation of simultaneous multiple objectives produces set of non-dominated Pareto-front solutions. To help the decision maker, the best compromise solution is carefully picked among optimal Pareto-front points. The proposed WCA-based methodology is demonstrated on three test cases with various complexities and under number of objective scenarios. Numerical results and further subsequent comparisons to other challenging optimisers indicate the viability and confirm the strength of the proposed WCA-based ELD method.     

Synergic predator-prey optimization for economic thermal power dispatch problem

This paper introduces a synergic predator-prey optimization (SPPO) algorithm to solve economic load dispatch (ELD) problem for thermal units with practical aspects. The basic PPO model comprises prey and predator as essential components. SPPO uses collaborative decision for movement and direction of prey and maintains diversity in the swarm due to fear factor of predator, which acts as the baffled state of preys’ mind. 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. The prey particle memorizes its best and not-best positions as experiences. In this research work, to improve the quality of prey swarm, which influence convergence rate, opposition based initialization is used. To verify robustness of proposed algorithm general benchmark problems and small, medium, and large power generation test power system are simulated. These test systems have non-linear behavior due to multi-fuel options and practical constraints. The constraints of prohibited operating zone and ramp rate limits of power generators’ are handled using heuristics. Newton–Raphson procedure is exploited to attain the transmission losses using load flow analysis. The outcomes of SPPO are compared with the results described in literature and are found satisfactory.     

An adaptive differential evolutionary algorithm incorporating multiple mutation strategies for the economic load dispatch problem

As growing the demand for electrical energy, economic load dispatch (ELD) has become one of the most important and complex issues in the operation of power systems. Owing to the confined optimum convergence and the additional constraints, it does not proficient to crack such problems by the predictable optimization algorithms. In this paper, a self-adaptable differential evolution algorithm integrating with multiple mutation strategies (ADE-MMS) is proposed for the ELD problems. In order to improve the exploration and exploitation capabilities of the original differential evolution algorithm (DE), ADE-MMS has three extensions to DE. Firstly, four types of advanced vectors generated by the different methods are employed in the mutation strategies. Secondly, a self-adaptable selection mechanism for the multiple mutation strategies is implemented in the iterations. Thirdly, the main control parameters are updated according to the fitness value under the tolerance threshold. Additionally, an effective repair method is proposed to handle the equality constraints of the ELD problems. ADE-MMS not only improve the convergence speed of the original DE but also keep equilibrium state between the exploration and the exploration. A tolerance threshold for the main control parameters makes the original DE more adaptive. Moreover, the modified equality constraints handling method is benefit to meet the equality constraints and minimize the impact on the algorithm. The performances of four DE algorithms are tested on the ten ELD problems with diverse complexities. Experimental results and comparisons with other recently reported ELD algorithms confirm that ADE-MMS is capable of obtaining excellent and feasible solutions. It reveal that ADE-MMS has good potential to solvating the ELD problems.     

Symbiotic organisms search algorithm for dynamic economic dispatch with valve-point effects

Abstract

In this study, symbiotic organisms search (SOS) algorithm is proposed to solve the dynamic economic dispatch with valve-point effects problem, which is one of the most important problems of the modern power system. Some practical constraints like valve-point effects, ramp rate limits and prohibited operating zones have been considered as solutions. Proposed algorithm was tested on five different test cases in 5 units, 10 units and 13 units systems. The obtained results have been compared with other well-known metaheuristic methods reported before. Results show that proposed algorithm has a good convergence and produces better results than other methods.     

Self-adaptive check and repair operator-based particle swarm optimization for the multidimensional knapsack problem

The multidimensional knapsack problem (MKP) is a combinatorial optimization problem belonging to the class of NP-hard problems. This study proposes a novel self-adaptive check and repair operator (SACRO) combined with particle swarm optimization (PSO) to solve the MKP. The traditional check and repair operator (CRO) uses a unique pseudo-utility ratio, whereas SACRO dynamically and automatically changes the alternative pseudo-utility ratio as the PSO algorithm runs. Two existing PSO algorithms are used as the foundation to support the novel SACRO methods, the proposed SACRO-based algorithms were tested using 137 benchmark problems from the OR-Library to validate and demonstrate the efficiency of SACRO idea. The results were compared with those of other population-based algorithms. Simulation and evaluation results show that SACRO is more competitive and robust than the traditional CRO. The proposed SACRO-based algorithms rival other state-of-the-art PSO and other algorithms. Therefore, changing different types of pseudo-utility ratios produces solutions with better results in solving MKP. Moreover, SACRO can be combined with other population-based optimization algorithms to solve constrained optimization problems.     

DPD: An intelligent parallel hybrid algorithm for economic load dispatch problems with various practical constraints

The Economic Load Dispatch (ELD) problem has attracted much attention in the field of electric power system. This paper proposes a novel parallel hybrid optimization methodology aimed at solving ELD problem with various generator constraints. The proposed approach combines the Differential Evolution (DE) and Particle Swarm Optimization (PSO). Initially the whole population (in increasing order of fitness) is divided into three groups - Inferior Group, Mid Group and Superior Group. DE is employed in the inferior and superior groups, whereas PSO is used in the mid-group. The proposed method is called DPD as it uses DE-PSO-DE on a population in parallel manner. Two strategies namely Elitism (to retain the best obtained values so far) and Non-redundant search (to improve the solution quality) have been employed in DPD cycle. Moreover, the suitable mutation strategy for both DEs used in DPD is investigated over a set of 8 popular mutation strategies. Combination of 8 mutation strategies generated 64 different variants of DPD. Top 4 DPDs are investigated through IEEE CEC 2006 functions. Based on the performance analysis, best DPD is reported and further used in solving four different typical test systems of ELD problem. Numerical and graphical results indicate the efficiency, convergence characteristic and robustness of proposed DPD.     

Estimation of distribution and differential evolution cooperation for large scale economic load dispatch optimization of power systems

Economic Load Dispatch (ELD) is an important and difficult optimization problem in power system planning. This article aims at addressing two practically important issues related to ELD optimization: (1) analyzing the ELD problem from the perspective of evolutionary optimization; (2) developing effective algorithms for ELD problems of large scale. The first issue is addressed by investigating the fitness landscape of ELD problems with the purpose of estimating the expected performance of different approaches. To address the second issue, a new algorithm named “Estimation of Distribution and Differential Evolution Cooperation” (ED-DE) is proposed, which is a serial hybrid of two effective evolutionary computation (EC) techniques: estimation of distribution and differential evolution. The advantages of ED-DE over the previous ELD optimization algorithms are experimentally testified on ELD problems with the number of generators scaling from 10 to 160. The best solution records of classical 13 and 40-generator ELD problems with valve points, and the best solution records of 10, 20, 40, 80 and 160-generator ELD problems with both valve points and multiple fuels are updated in this work. To further evaluate the efficiency and effectiveness of ED-DE, we also compare it with other state-of-the-art evolutionary algorithms (EAs) on typical function optimization tasks.     

Augmented Lagrange Hopfield network initialized by quadratic programming for economic dispatch with piecewise quadratic cost functions and prohibited zones

This paper proposes a method based on quadratic programming (QP) and augmented Lagrange Hopfield network (ALHN) for solving economic dispatch (ED) problem with piecewise quadratic cost functions and prohibited zones. The ALHN method is a continuous Hopfield neural network with its energy function based on augmented Lagrange function which can properly deal with constrained optimization problems. In the proposed method, the QP method is firstly used to determine the fuel cost curve for each unit and initialize for the ALHN method, then a heuristic search is used for repairing prohibited zone violations, and the ALHN method is finally applied for solving the problem if any violations found. The proposed method has been tested on different systems and the obtained results are compared to those from many other methods in the literature. The result comparison has indicated that the proposed method has obtained better solution quality than many other methods. Therefore, the proposed QP-ALHN method could be a favorable method for solving the ED problem with piecewise quadratic cost functions and prohibited zones.     

基于动态惩罚因子的改进蚱蜢算法求解经济负荷调度和经济排放联合调度问题

电力生产装置运行中各种燃料的成本逐步增加,需要最小化成本函数以求解此类复杂经济负荷调度问题.鉴于此,提出一种基于动态惩罚因子的改进蚱蜢算法求解经济负荷调度(economic load dispatch, ELD)问题和经济排放联合调度(combined economic emission dispatch, CEED)问题.为了提高蚱蜢算法(grasshopper optimization algorithm, GOA)性能,提出一种改进的混合蚱蜢算法(hybrid grasshopper optimization algorithm, HGOA),将重力搜索算子和鸽群搜索算子-地标算子加入GOA中,增强算法的搜索能力,平衡算法的勘探和开发.同时,为了更好地解决ELD和CEED问题中的约束问题,提出6个惩罚函数,包括2个V型函数、反正切函数、反正弦函数、线性函数和二次函数,并使用动态惩罚策略代替传统的固定值惩罚策略.选取3个ELD问题案例和4个CEED问题案例验证所提出方法的有效性,实验结果表明, HGOA相较于其他元启发式算法在求解质量上表现更好,且动态惩罚策略比固定值惩罚策略效果更...     

Mixed integer programming of multiobjective hydro-thermal self scheduling

This paper presents a method for hydro-thermal self scheduling (HTSS) problem in a day-ahead joint energy and reserve market. The HTSS is modeled in the form of multiobjective framework to simultaneously maximize GENCOs profit and minimize emissions of thermal units. In the proposed model the valve loading effects which is a nonlinear problem by itself is linearized. Also a dynamic ramp rate of thermal units is used instead of a fix rate leading to more realistic formulation of HTSS. Furthermore, the multi performance curves of hydro units is developed and prohibited operating zones (POZs) of thermal unit are considered in HTSS problem. Also, in the proposed framework, the mixed integer nonlinear programming (MINLP) of HTSS is converted to mixed integer programming (MIP) problem that can be effectively solved by optimization softwares even for real size power systems. The lexicographic optimization and hybrid augmented-weighted ?-constraint technique is implemented to generate Pareto optimal solutions. The best compromised solution is adopted either by using a fuzzy approach or by considering arbitrage opportunities to achieve more profit. Finally, the effectiveness of the proposed method is studied based on the IEEE 118-bus system.     

Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect

The growing costs of fuel and operation of power generating units warrant improvement of optimization methodologies for economic dispatch (ED) problems. The practical ED problems have non-convex objective functions with equality and inequality constraints that make it much harder to find the global optimum using any mathematical algorithms. Modern optimization algorithms are often meta-heuristic, and they are very promising in solving nonlinear programming problems. This paper presents a novel approach to determining the feasible optimal solution of the ED problems using the recently developed Firefly Algorithm (FA). Many nonlinear characteristics of power generators, and their operational constraints, such as generation limitations, prohibited operating zones, ramp rate limits, transmission loss, and nonlinear cost functions, were all contemplated for practical operation. To demonstrate the efficiency and applicability of the proposed method, we study four ED test systems having non-convex solution spaces and compared with some of the most recently published ED solution methods. The results of this study show that the proposed FA is able to find more economical loads than those determined by other methods. This algorithm is considered to be a promising alternative algorithm for solving the ED problems in practical power systems.     

Uncertainty management in multiobjective hydro-thermal self-scheduling under emission considerations

In this paper, a stochastic multiobjective framework is proposed for a day-ahead short-term Hydro Thermal Self-Scheduling (HTSS) problem for joint energy and reserve markets. An efficient linear formulations are introduced in this paper to deal with the nonlinearity of original problem due to the dynamic ramp rate limits, prohibited operating zones, operating services of thermal plants, multi-head power discharge characteristics of hydro generating units and spillage of reservoirs. Besides, system uncertainties including the generating units’ contingencies and price uncertainty are explicitly considered in the stochastic market clearing scheme. For the stochastic modeling of probable multiobjective optimization scenarios, a lattice Monte Carlo simulation has been adopted to have a better coverage of the system uncertainty spectrum. Consequently, the resulting multiobjective optimization scenarios should concurrently optimize competing objective functions including GENeration COmpany's (GENCO's) profit maximization and thermal units’ emission minimization. Accordingly, the ɛ-constraint method is used to solve the multiobjective optimization problem and generate the Pareto set. Then, a fuzzy satisfying method is employed to choose the most preferred solution among all Pareto optimal solutions. The performance of the presented method is verified in different case studies. The results obtained from ɛ-constraint method is compared with those reported by weighted sum method, evolutionary programming-based interactive Fuzzy satisfying method, differential evolution, quantum-behaved particle swarm optimization and hybrid multi-objective cultural algorithm, verifying the superiority of the proposed approach.     

求解电力系统经济调度问题的改进粒子群优化算法

电力系统经济调度问题是电力系统中的一个重要的研究课题,针对该问题,提出一种改进粒子群优化(ODPSO)算法.改进算法在搜索前期,采用广义的反向学习策略,使算法能够快速地靠近较优的搜索区域,从而提高收敛速度;在搜索后期,借鉴差分进化算法的进化机制设计改进的变异和交叉策略,对当前种群的最优粒子进行更新,从而提高种群的多样性,进而协助算法获得全局最优解.为了验证改进粒子群优化算法的有效性,对CEC2006提出的22个基准约束测试函数进行仿真,结果表明改进算法相比其他算法在寻优精度和稳定性上更具优势.最后,将改进算法应用于考虑机组爬坡速率约束、机组禁行区域约束以及电力平衡约束的两个电力系统经济调度问题,取得了令人满意的结果.     

基于混合蛙跳算法的电力系统经济负荷分配

针对带阀点效应的经济负荷分配(ELD)问题高维、非凸、非线性的特点,应用混合蛙跳算法(SF-LA)解决电力系统ELD问题。该算法结合了模因演算算法(MA)和粒子群优化(PSO)算法二者的优点,在确保全局收敛和满足约束条件下,能够快速有效地搜索到最优解。通过对多个ELD问题实例进行仿真计算,并与参考文献做比较,结果表明:SFLA对于解决电力系统ELD问题是有效、可行的。