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.     

Improved differential evolution approach based on cultural algorithm and diversity measure applied to solve economic load dispatch problems

Evolutionary algorithms (EAs) are general-purpose stochastic search methods that use the metaphor of evolution as the key element in the design and implementation of computer-based problems solving systems. During the past two decades, EAs have attracted much attention and wide applications in a variety of fields, especially for optimization and design. EAs offer a number of advantages: robust and reliable performance, global search capability, little or no information requirement, and others. Among various EAs, differential evolution (DE), which characterized by the different mutation operator and competition strategy from the other EAs, has shown great promise in many numerical benchmark problems and real-world optimization applications. The potentialities of DE are its simple structure, easy use, convergence speed and robustness. To improve the global optimization property of DE, in this paper, a DE approach based on measure of population's diversity and cultural algorithm technique using normative and situational knowledge sources is proposed as alternative method to solving the economic load dispatch problems of thermal generators. The traditional and cultural DE approaches are validated for two test systems consisting of 13 and 40 thermal generators whose nonsmooth fuel cost function takes into account the valve-point loading effects. Simulation results indicate that performance of the cultural DE present best results when compared with previous optimization approaches in solving economic load dispatch problems.     

基于快速自适应差分进化算法的电力系统经济负荷分配

提出一种求解复杂电力系统经济负荷分配问题的快速自适应差分进化算法(FSADE).从矢量运算角度对变异算子进行分析,提出了一种改进的变异算子,大大提高了算法的收敛速率.根据个体的进化过程,引入自学习机制,对个体的变异和交叉概率常数进行自适应地调整,提高了算法的鲁棒性.3个不同规模的算例仿真结果表明,与其他4种典型智能优化算法相比, FSADE具有更好的计算精度和计算速度,是一种求解电力系统经济负荷分配问题的有效方法.     

Modified neo-fuzzy neuron-based approach for economic and environmental optimal power dispatch

At the central energy management center in a power system, the real time controls continuously track the load changes and endeavor to match the total power demand with total generation in such a manner that the operating cost is minimized while all the operating constraints are satisfied. However, due to the strict government regulations on environmental protection, operation at minimum cost is no longer the only criterion for dispatching electrical power. The idea behind the environmentally constrained economic dispatch formulation is to estimate the optimal generation schedule of generating units in such a manner that fuel cost and harmful emission levels are both simultaneously minimized for a given load demand. Conventional optimization techniques become very time consuming and computationally extensive for such complex optimization tasks. These methods are hence not suitable for on-line use. Neural networks and fuzzy systems can be trained to generate accurate relations among variables in complex non-linear dynamical environment, as both are model-free estimators. The existing synergy between these two fields has been exploited in this paper for solving the economic and environmental dispatch problem on-line. A multi-output modified neo-fuzzy neuron (NFN), capable of real time training is proposed for economic and environmental power generation allocation.This model is found to achieve accurate results and the training is observed to be faster than other popular neural networks. The proposed method has been tested on medium-sized sample power systems with three and six generating units and found to be suitable for on-line combined environmental economic dispatch (CEED).     

Solution of economic load dispatch using hybrid chemical reaction optimization approach

In this paper, a new and efficient optimization technique based on hybridization of chemical reaction optimization (CRO) with differential evolution (DE) is developed and demonstrated to solve the ELD problem with thermal cost function having valve point loading effect together with and without multiple fuel options and with and without considering prohibited operating zone and ramp rate constraint. When valve-point effects, multi-fuel operations and the constraints of prohibited operating zone and ramp rate are taken into account, ELD problem become more complex than conventional ELD problem. To show the priority of the proposed algorithm, it is implemented on six different test systems for solving ELD problems. Comparative studies are carried out to examine the effectiveness of the proposed HCRO-DE approach with conventional DE, CRO and the other algorithms reported in the literature. The simulation results show that the proposed HCRO-DE method is capable of obtaining better quality solutions than DE, CRO and the other well popular optimization techniques.     

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.     

Chaotic differential evolution methods for dynamic economic dispatch with valve-point effects

The dynamic economic dispatch (DED), with the consideration of valve-point effects, is a complicated non-linear constrained optimization problem with non-smooth and non-convex characteristics. In this paper, three chaotic differential evolution (CDE) methods are proposed based on the Tent equation to solve DED problem with valve-point effects. In the proposed methods, chaotic sequences are applied to obtain the dynamic parameter settings in DE. Meanwhile, a chaotic local search (CLS) operation for solving DED problem is designed to help DE avoiding premature convergence effectively. Finally, in order to handle the complicated constraints with efficiency, new heuristic constraints handling methods and feasibility based selection strategy are embedded into the proposed CDE methods. The feasibility and effectiveness of the proposed CDE methods are demonstrated for two test systems. The simulation results reveal that, compared with DE and those other methods reported in literatures recently, the proposed CDE methods are capable of obtaining better quality solutions with higher efficiency.     

求解动态优化问题的多种群竞争差分进化算法

针对动态优化问题（DOP）的求解,提出结合多种群方法和竞争策略的差分进化算法（DECS）。首先,将一个种群作为侦测种群,通过监测种群中所有个体的评价值和种群维度来判断环境是否发生变化。其次,将余下多个种群作为搜索种群,独立搜索环境中的最优值。在搜索过程中,引入排除规则,避免多个搜索种群聚集在同一个局部最优的邻域。在迭代若干代后对各搜索种群执行竞争操作,保留评估值最优个体所在的种群并对该种群的下一代个体生成采用量子个体生成机制,而对其他搜索种群重新初始化。最后,利用7个测试函数的49个动态变化问题对DECS进行验证,并将实验结果与人工免疫算法（Dopt-aiNet）、复位粒子群优化（rPSO）算法、改进差分进化（MDE）算法进行比较。实验结果表明,在49个问题上,DECS有34个问题的平均离线误差期望小于Dopt-aiNet算法,所有问题的平均离线误差期望都小于rPSO算法和MDE算法,因此DECS对DOP求解动态优化问题是可行的。     

Differential evolution approach for optimal reactive power dispatch

Differential evolution based optimal reactive power dispatch for real power loss minimization in power system is presented in this paper. The proposed methodology determines control variable settings such as generator terminal voltages, tap positions and the number of shunts to be switched, for real power loss minimization in the transmission system. The problem is formulated as a mixed integer nonlinear optimization problem. A generic penalty function method, which does not require any penalty coefficient, is employed for constraint handling. The formulation also checks for the feasibility of the optimal control variable setting from a voltage security point of view by using a voltage collapse proximity indicator. The algorithm is tested on standard IEEE 14, IEEE 30, and IEEE 118-Bus test systems. To show the effectiveness of proposed method the results are compared with Particle Swarm Optimization and a conventional optimization technique – Sequential Quadratic Programming.     

Environmental economic dispatch in multi-area power system employing improved differential evolution with fuzzy selection

In a deregulated multi-area electrical power system the objective is to determine the most economical generation dispatch strategy that could satisfy the area load demands, the tie-line limits and other operating constraints. Usually, economic dispatch (ED) deals only with the cost minimization, but minimization of emission content has also become an equally important concern due to the mandatory requirement of pollution reduction for environmental protection. Environmental economic dispatch (EED) is a complex multi-objective optimization (MOO) problem with conflicting goals. Normally a fuzzy ranking is employed to rank the large number of Pareto solutions obtained after solving a MOO problem. But in this paper the preference of the decision maker (DM) is used to guide the search and to select the population for the next generation. An improved differential evolution (DE) method is proposed where the selection operation is modified to reduce the complexity of multi-attribute decision making with the help of a fuzzy framework. Solutions are assigned a fuzzy rank on the basis of their level of satisfaction for different objectives before the population selection and then the fuzzy rank is used to select and pass on better solutions to the next generation. A well distributed Pareto-front is obtained which presents a large number of alternate trade-off solutions for the power system operator. A momentum operation is also included to prevent stagnation and to create Pareto diversity. Studies are carried out on three test cases and results obtained are found to be better than some previous literature.     

电力系统经济负荷分配的族群优化方法

经济负荷分配（Economic Load Dispatch,ELD）是电力系统中一种重要的优化问题,它可归为一类高维、离散、非线性的多约束函数优化问题。针对这类问题,提出了一种基于线性截取策略的改进族群进化算法——EGEA/LT,并使用EGEA/LT对IEEE的3机、6机和15机3个仿真系统进行了优化实验,将实验结果与其他典型算法优化结果进行比较,说明了EGEA/LT是一种求解ELD问题的有效方法。     

Multi-elite guide hybrid differential evolution with simulated annealing technique for dynamic economic emission dispatch

This paper proposes an improved multi-objective differential evolutionary algorithm named multi-objective hybrid differential evolution with simulated annealing technique (MOHDE-SAT) to solve dynamic economic emission dispatch (DEED) problem. The proposed MOHDE-SAT integrates the orthogonal initialization method into the differential evolution, which enlarges the population diversity at the beginning of population evolution. In addition, modified mutation operator and archive retention mechanisms are used to control convergence rate, and simulated annealing technique and entropy diversity method are utilized to adaptively monitor the population diversity as the evolution proceeds, which can properly avoid the premature convergence problem. Furthermore, the MOHDE-SAT is applied on the thermal system with a heuristic constraint handling method, and obtains more desirable results in comparison to those alternatives established recently. The obtained results also reveal that the proposed MOHDE-SAT can provide a viable way for solving DEED problems.     

Tournament-based harmony search algorithm for non-convex economic load dispatch problem

This paper proposes a tournament-based harmony search (THS) algorithm for economic load dispatch (ELD) problem. The THS is an efficient modified version of the harmony search (HS) algorithm where the random selection process in the memory consideration operator is replaced by the tournament selection process to activate the natural selection of the survival-of-the-fittest principle and thus improve the convergence properties of HS. The performance THS is evaluated with ELD problem using five different test systems: 3-units generator system; two versions of 13-units generator system; 40-units generator system; and large-scaled 80-units generator system. The effect of tournament size (t) on the performance of THS is studied. A comparative evaluation between THS and other existing methods reported in the literature are carried out. The simulation results show that the THS algorithm is capable of achieving better quality solutions than many of the well-popular optimization methods.     

An improved constrained differential evolution algorithm for unmanned aerial vehicle global route planning

This paper formulates the global route planning problem for the unmanned aerial vehicles (UAVs) as a constrained optimization problem in the three-dimensional environment and proposes an improved constrained differential evolution (DE) algorithm to generate an optimal feasible route. The flight route is designed to have a short length and a low flight altitude. The multiple constraints based on the realistic scenarios are taken into account, including maximum turning angle, maximum climbing/gliding slope, terrain, forbidden flying areas, map and threat area constraints. The proposed DE-based route planning algorithm combines the standard DE with the level comparison method and an improved strategy is proposed to control the satisfactory level. To show the high performance of the proposed method, we compare the proposed algorithm with six existing constrained optimization algorithms and five penalty function based methods. Numerical experiments in two test cases are carried out. Our proposed algorithm demonstrates a good performance in terms of the solution quality, robustness, and the constraint-handling ability.     

朴素差分进化算法

针对变异算子学习方式的单一性,提出一种朴素变异算子,其基本思想是向优秀的个体靠近,同时远离较差个体,其实现方式是设计一种缩放因子调整策略,如果三个随机个体在某维上比较接近,则缩放因子变小,反之变大.在实验过程中通过平均适应度评价次数、成功运行次数和加速比等指标表明,基于朴素变异算子的差分进化算法能有效提高算法的收敛速度和健壮性.     

An improved adaptive differential evolution optimizer for non-convex Economic Dispatch Problems

The present paper proposes an improved adaptive differential evolution algorithm, the IL-SHADE algorithm, to solve Economic Dispatch Problems (EDPs) taking into account practical constraints, such as transmission network losses, ramp rate limit, prohibited operation zone and valve point effect. The IL-SHADE algorithm is introduced as an improved version of the L-SHADE algorithm (Success-History based Adaptive Differential Evolution algorithm with Linear population size reduction). The proposed algorithm is first tested on eight CEC’05 standard benchmark test functions. Then, the efficiency of the proposed optimizer is demonstrated by solving different practical EDPs related to three IEEE power test systems, the IEEE 6-unit, 40-unit and 140-unit test systems. The comparison with various recent state-of-the-art approaches proves that IL-SHADE outperforms the L-SHADE and other cited approaches. Finally, the Wilcoxon sign rank test is used to validate the results.     

基于混合双种群差分进化的电力系统经济负荷分配

针对电力系统经济负荷分配本质上的非线性约束优化问题,提出一种双种群混合差分进化算法.采用两个种群且以较小的计算量实现目标函数的寻优并保持解的可行性,同时引入单纯型法来提高算法的局部搜索能力.基于典型算例对该算法的进化行为进行测试,并通过仿真和比较验证了所提出算法的有效性.     

Design of optimal PID controller using hybrid differential evolution and particle swarm optimization with an aging leader and challengers

This paper presents a new algorithm designed to find the optimal parameters of PID controller. The proposed algorithm is based on hybridizing between differential evolution (DE) and Particle Swarm Optimization with an aging leader and challengers (ALC-PSO) algorithms. The proposed algorithm (ALC-PSODE) is tested on twelve benchmark functions to confirm its performance. It is found that it can get better solution quality, higher success rate in finding the solution and yields in avoiding unstable convergence. Also, ALC-PSODE is used to tune PID controller in three tanks liquid level system which is a typical nonlinear control system. Compared to different PSO variants, genetic algorithm (GA), differential evolution (DE) and Ziegler–Nichols method; the proposed algorithm achieve the best results with least standard deviation for different swarm size. These results show that ALC-PSODE is more robust and efficient while keeping fast convergence.     

基于协同差分进化的多出救点应急物资调度

主要利用差分进化算法来研究时间约束下的多出救点应急物资调度优化问题。针对传统差分进化算法搜索速度慢、易陷入局部最优解的缺点,提出一个并行协同差分进化算法,将该算法应用于时间约束下的多出救点应急物资调度优化,建立相应的数学模型,在此基础上设计相应的算法。实例验证表明,同遗传算法、标准差分进化算法相比,该算法在解决具有时间约束的多出救点应急物资调度优化问题方面具有较快的搜索速度和较好的寻优能力。