Kinetic gas molecule optimization for nonconvex economic dispatch problem

This paper presents kinetic gas molecule optimization (KGMO) algorithm to solve economic dispatch problems with non-smooth/non-convex cost functions. KGMO is based on kinetic energy and the natural motion of gas molecules. The effectiveness of the proposed method has been verified on four different non-convex economic dispatch problems with valve-point effects, prohibited operating zones with transmission losses, multiple fuels with valve point effects and the large-scale Korean power system with valve-point effects and prohibited operating zones. The results of the proposed approach are compared with those obtained by other evolutionary methods. It is found that the proposed KGMO based approach is able to provide better solution.     

Modified Particle Swarm Optimization for Non-smooth Non-convex Combined Heat and Power Economic Dispatch

This article presents modified particle swarm optimization to solve the non-smooth non-convex combined heat and power economic dispatch problem. Valve-point loading and prohibited operating zones of conventional thermal generators are taken into account. Particle swarm optimization performs well for small-dimensional and less complicated problems but fails to locate global minima for complex multi-minima functions. This article proposes Gaussian random variables in the velocity term, which improves search efficiency and guarantees a high probability of obtaining the global optimum without significantly impairing the speed of convergence and the simplicity of the structure of particle swarm optimization. The effectiveness of the proposed method has been verified on two test systems. The results of the proposed approach are compared with those obtained by other evolutionary methods. It is found that the proposed modified particle swarm optimization based approach is able to provide a better solution.     

An Efficient Modified HPSO-TVAC-Based Dynamic Economic Dispatch of Generating Units

Abstract

This paper proposes a novel particle swarm optimization (PSO) algorithm with population reduction, which is called modified new self-organizing hierarchical PSO with jumping time-varying acceleration coefficients (MNHPSO-JTVAC). The proposed method is used for solving well-known benchmark functions, as well as non-convex and non-smooth dynamic economic dispatch (DED) problems for a 24?h time interval in two different test systems. Operational constraints including the prohibited operating zones (POZs), the transmission losses, the ramp-rate limits and the valve-point effects are considered in solving the DED problem. The obtained numerical results show that the MNHPSO-JTVAC algorithm is very suitable and competitive compared to other algorithms and have the capacity to obtain better optimal solutions in solving the non-convex and non-smooth DED problems compared to the other variants of PSO and the state of the art optimization algorithms proposed in recent literature. The source codes of the HPSO-TVAC algorithms and supplementary data for this paper are publicly available at https://github.com/ebrahimakbary/MNHPSO-JTVAC.     

An Improved Particle Swarm Optimization for Nonconvex Economic Dispatch Problems

This paper presents an efficient approach for solving economic dispatch (ED) problems with nonconvex cost functions using an improved particle swarm optimization (IPSO). Although the particle swarm optimization (PSO) approaches have several advantages suitable to heavily constrained nonconvex optimization problems, they still can have the drawbacks such as local optimal trapping due to premature convergence (i.e., exploration problem), insufficient capability to find nearby extreme points (i.e., exploitation problem), and lack of efficient mechanism to treat the constraints (i.e., constraint handling problem). This paper proposes an improved PSO framework employing chaotic sequences combined with the conventional linearly decreasing inertia weights and adopting a crossover operation scheme to increase both exploration and exploitation capability of the PSO. In addition, an effective constraint handling framework is employed for considering equality and inequality constraints. The proposed IPSO is applied to three different nonconvex ED problems with valve-point effects, prohibited operating zones with ramp rate limits as well as transmission network losses, and multi-fuels with valve-point effects. Additionally, it is applied to the large-scale power system of Korea. Also, the results are compared with those of the state-of-the-art methods.      

Non-convex economic dispatch with heuristic load patterns, valve point loading effect, prohibited operating zones, ramp-rate limits and spinning reserve constraints using harmony search algorithm

This paper attempts to investigate the applicability of harmony search algorithm (HSA) to solve extremely challenging non-convex economic load dispatch problem with valve point loading effect, prohibited operating zones, ramp-rate limits, spinning reserve constrains and transmission losses involving variations of consumer load patterns. The performance of the proposed approach HSA has been tested successfully on the standard 6-bus, IEEE-14 bus and IEEE-30 bus system with several heuristic load patterns. The results of this study reveals that the proposed approach is able to find appreciable economical load dispatch solutions than those of improved fast evolutionary program and particle swarm optimization. Besides this, the transmission line losses are also considerably reduced and the computation time is reasonably even and less when compared to other methods.     

An Improved Particle Swarm Optimization for the Combined Heat and Power Dynamic Economic Dispatch Problem

—This study presents a novel improved particle swarm optimization algorithm to solve the combined heat and power dynamic economic dispatch problem. This problem is formulated as a challenging non-convex and non-linear optimization problem considering practical characteristics, such as valve-point effects, transmission losses, ramp-rate limits, mutual dependency of power and heat, spinning reserve requirements, and transmission security constraints. The proposed method combines classical particle swarm optimization with a chaotic mechanism, time-variant acceleration coefficients, and a self-adaptive mutation scheme to prevent premature convergence and improve solution quality. Moreover, multiple efficient constraint handling strategies are employed to deal with complex constraints. The effectiveness of the proposed improved particle swarm optimization for solving the combined heat and power dynamic economic dispatch problem is validated on three different test systems, and the results are compared with those of other variants of particle swarm optimization as well as other methods reported in the literature. The numerical results demonstrate the superiority of improved particle swarm optimization in solving the combined heat and power dynamic economic dispatch problem while strictly satisfying all the constraints.     

Iteration PSO with time varying acceleration coefficients for solving non-convex economic dispatch problems

This paper presents a novel heuristic algorithm for solving economic dispatch (ED) problems, by employing iteration particle swarm optimization with time varying acceleration coefficients (IPSO-TVAC) method. Due to the effect of valve-points and prohibited operation zones (POZs) in the generating units’ cost functions, ED problem is a non-linear and non-convex optimization problem. The problem even may be more complicated if transmission losses are taken into account. The effectiveness of the proposed method is examined and validated by carrying out extensive tests on three different test systems. Valve-point effects, POZs, ramp-rate constraints and transmission losses are modeled. Numerical results show that the IPSO-TVAC method has a good convergence property. Furthermore, the generation costs of the IPSO-TVAC method are lower than other optimization algorithms reported in recent literature.     

A particle swarm optimization for economic dispatch with nonsmooth cost functions

This work presents a new approach to economic dispatch (ED) problems with nonsmooth cost functions using a particle swarm optimization (PSO) technique. The practical ED problems have nonsmooth cost functions with equality and inequality constraints that make the problem of finding the global optimum difficult using any mathematical approaches. A modified PSO (MPSO) mechanism is suggested to deal with the equality and inequality constraints in the ED problems. A constraint treatment mechanism is devised in such a way that the dynamic process inherent in the conventional PSO is preserved. Moreover, a dynamic search-space reduction strategy is devised to accelerate the optimization process. To show its efficiency and effectiveness, the proposed MPSO is applied to test ED problems, one with smooth cost functions and others with nonsmooth cost functions considering valve-point effects and multi-fuel problems. The results of the MPSO are compared with the results of conventional numerical methods, Tabu search method, evolutionary programming approaches, genetic algorithm, and modified Hopfield neural network approaches.     

计及阀点效应负荷经济分配的杂交粒子群算法

提出了一种用于求解复杂的非凸、非线性具有阀点效应的火电有功负荷经济分配问题的杂交粒子群算法（HPSO）。HPSO通过粒子追随自己找到的最优解和整个群的最优解来完成优化,并在此基础上将遗传算法的杂交思想引入到PSO算法当中,使其避免局部最优。算例的仿真结果表明：本文的算法有效、可行,可望应用于更广泛的优化问题。     

Artificial bee colony optimization for multi-area economic dispatch

This paper presents artificial bee colony optimization for solving multi-area economic dispatch (MAED) problem with tie line constraints considering transmission losses, multiple fuels, valve-point loading and prohibited operating zones. Artificial bee colony optimization is a swarm-based algorithm inspired by the food foraging behavior of honey bees. The effectiveness of the proposed algorithm has been verified on three different test systems, both small and large, involving varying degree of complexity. Compared with differential evolution, evolutionary programming and real coded genetic algorithm, considering the quality of the solution obtained, the proposed algorithm seems to be a promising alternative approach for solving the MAED problems in practical power system.     

Optimization of economic load dispatch of higher order general cost polynomials and its sensitivity using modified particle swarm optimization

This paper presents a modified particle swarm optimization (MPSO) for constrained economic load dispatch (ELD) problem. Real cost functions are more complex than conventional second order cost functions when multi-fuel operations, valve-point effects, accurate curve fitting, etc., are considering in deregulated changing market. The proposed modified particle swarm optimization (PSO) consists of problem dependent variable number of promising values (in velocity vector), unit vector and error–iteration dependent step length. It reliably and accurately tracks a continuously changing solution of the complex cost function and no extra concentration/effort is needed for the complex higher order cost polynomials in ELD. Constraint management is incorporated in the modified PSO. The modified PSO has balance between local and global searching abilities, and an appropriate fitness function helps to converge it quickly. To avoid the method to be frozen, stagnated/idle particles are reset. Sensitivity of the higher order cost polynomials is also analyzed visually to realize the importance of the higher order cost polynomials for the optimization of ELD. Finally, benchmark data sets and methods are used to show the effectiveness of the proposed method.     

Solving dynamic economic dispatch problem with security constraints using bacterial foraging PSO-DE algorithm

This paper presents a heuristic optimization methodology, namely, Bacterial foraging PSO-DE (BPSO-DE) algorithm by integrating Bacterial Foraging Optimization Algorithm (BFOA), Particle Swarm Optimization (PSO) and Differential Evolution (DE) for solving non-smooth non-convex Dynamic Economic Dispatch (DED) problem. The DED problem exhibits non-smooth, non-convex nature due to valve-point loading effects, ramp rate limits, spinning reserve capacity, prohibited operating zones and security constraints. The proposed hybrid method eliminates the problem of stagnation of solution with the incorporated PSO and DE operators in original bacterial foraging algorithm. It achieves global cost by selecting the bacterium with good foraging strategies. The bacteria with good foraging strategies are obtained in the updating process of every chemo-tactic step by the PSO operator. The DE operator fine tunes the solution obtained through bacterial foraging and PSO operator. A 3- and 7-unit systems for static economic dispatch, a 26-bus, 6-generator test system and an IEEE 39-bus, 10-unit New England test systems are considered to show the effectiveness of the proposed method over other methods reported in the literature.     

Combined heat and power economic dispatch using opposition-based group search optimization

This paper presents opposition-based group search optimization to solve non-smooth non-convex combined heat and power economic dispatch problem. Valve-point loading and prohibited operating zones of conventional thermal generators are taken into account. Group search optimization inspired by the animal searching behavior is a biologically realistic algorithm. Opposition-based group search optimization has been used here to improve the effectiveness and quality of the solution. The proposed opposition-based group search optimization employs opposition-based learning for population initialization and also for iteration wise update operation. The effectiveness of the proposed method has been verified on four test systems. The results of the proposed approach are compared with those obtained by other evolutionary methods. It is found that the proposed opposition-based group search optimization based approach is able to provide better solution.     

Economic load dispatch using θ-PSO

This paper presents a θ-particle swarm optimization (θ-PSO) based algorithm to solve constrained economic load dispatch (ELD) problems of thermal plants. The proposed methodology easily takes care of practical constraints such as transmission losses, ramp rate limits, and prohibited operating zones, and also deals with non-smoothness of cost function arising due to the use of valve point effects. The main evaluation mechanism of classical particle swarm optimization (CPSO) is modified in the novel θ-PSO by replacing the velocity vector with a phase angle vector. So, the positions are decided by the mapping of phase angles in θ-PSO. The performance of proposed algorithm has been tested on systems possessing 6, 13, 15, and 40 generating units involving varying degree of complexity. Detailed comparisons under equitable conditions are provided with prevalent approaches to measure the efficiency of the proposed algorithm. The findings affirm that the method outperforms the existing techniques in terms of solution quality and computational efficiency and can be a promising alternative approach for solving the ELD problems in practical power system.     

Multiple tabu search algorithm for economic dispatch problem considering valve-point effects

This paper presents a multiple tabu search (MTS) algorithm to solve the economic dispatch (ED) problem by taking valve-point effects into consideration. The practical ED problem with valve-point effects is represented as a non-smooth optimization problem with equality and inequality constraints that make the problem of finding the global or near global optimum difficult. The proposed MTS algorithm is the sequential execution of individual tabu search (TS) algorithm simultaneously by only one personal microcomputer. The MTS algorithm introduces additional techniques for improvement of search process, such as initialization, adaptive searches, multiple searches, replacing and restarting process. To show its effectiveness, the MTS is applied to test two studied systems consisting of 13 and 40 power generating units with valve-point effects. The optimized results by MTS are compared with those of conventional approaches, such as simulated annealing (SA), genetic algorithm (GA), TS algorithm and particle swarm optimization (PSO). Studied results confirm that the proposed MTS approach is capable of obtaining higher quality solution efficiently and lowest computational time.     

Dynamic economic dispatch using harmony search algorithm with modified differential mutation operator

Dynamic economic dispatch (DED), which is a complex non-linear constrained optimization problem, has a pivotal role in power system operation. It is one of the prime functions of power generation and control, where the aim is to operate an electrical power system most economically while the system operation is within its security limits. This problem possess non-convex characteristic when generation unit valve-point effects are considered. This paper proposes to solve DED problem with valve-point effects, using a modified form of recently developed differential harmony search algorithm. A five- and ten-unit system with non-smooth fuel cost function is used to establish the effectiveness of the proposed method over various other methods. It is shown that the proposed method is capable of providing better quality solutions.     

ICPSO算法及其在经济负荷分配中的应用

提出一种改进的混沌粒子群优化ICPSO(improved chaotic particle swarm optimization)算法,用于求解非线性、非凸、不连续等复杂约束条件的电力系统经济负荷分配。通过修正粒子群迭代的行动策略,并引入Tent混沌映射加强部分粒子的全局搜索能力,可以提高优化算法的全局搜索性能。最后将该算法应用于3机6母线的电力系统经济负荷分配中,在计及阀点效应的情况下,分别以考虑网损和忽略网损为例进行仿真。仿真结果表明,该算法有较快的收敛速度和较强的全局搜索能力,验证了算法的有效性和优越性。     

Combined economic and emission dispatch problems using biogeography-based optimization

In this work, biogeography-based optimization (BBO) is presented for solving different constrained economic load dispatch (ELD) problems combined with economic emission aspects in power system. Nonlinear characteristics of generators like valve point discontinuities, ramp rate limits and prohibited operating zones are considered in the problem. The simulation results show that the proposed BBO algorithm based solutions prove to be the best near-global optimal as compared to the solutions based on Newton–Raphson, Tabu search, genetic algorithm (GA), non-dominated sorting genetic algorithm (NSGA), fuzzy logic controlled genetic algorithm (FCGA), particle swarm optimization (PSO) and differential evolution (DE).     

求解经济调度问题的改进粒子群算法

提出了一种求解电力系统经济调度问题的改进粒子群算法。该算法考虑了机组的爬坡速率、工作死区等多种约束条件,并计及了网损。该算法以粒子群算法为基础,提出了新的修补策略对违反各种约束条件的粒子进行积极的修正,并与罚函数技术相结合,使粒子尽可能地在可行解区域或尽量接近可行解的区域内寻优。由于大大减少了粒子在非可行解区域内寻优的概率,因而有效地提高了算法的精度和速度。仿真算例的结果表明,该算法具有速度快、精度高和收敛性好的特点。     

A novel modified hybrid PSOGSA based on fuzzy logic for non-convex economic dispatch problem with valve-point effect

Gravitational Search Algorithm (GSA) is a novel stochastic optimization method inspired by the law of gravity and interaction between masses. This paper proposes a novel modified hybrid Particle Swarm Optimization (PSO) and GSA based on fuzzy logic (FL) to control ability to search for the global optimum and increase the performance of the hybrid PSOGSA. In order to test the performance of the modified hybrid PSOGSA based on FL (FPSOGSA), it has been applied to solve the well-known 23 benchmark test functions. In order to evaluate the efficiency and performance of the proposed approach, standard power systems including IEEE 5-machines 14-bus, IEEE 6-machines 30-bus, 13 and 40 unit test systems are used. These are non-convex economic dispatch problems including the valve-point effect and are computed with and without the losses. The results obtained from the proposed FPSOGSA approach are compared with those of the other heuristic techniques in the literature. The results of the comparison demonstrate that the proposed approach can converge to the near optimal solution and improve the performance of the standard hybrid PSOGSA approach.