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.     

A new rooted tree optimization algorithm for economic dispatch with valve-point effect

This work proposes a new optimization method called root tree optimization algorithm (RTO). The robustness and efficiency of the proposed RTO algorithm is validated on a 23 standard benchmark nonlinear functions and compared with well-known methods by addressing the same problem. Simulation results show effectiveness of the proposed RTO algorithm in term of solution quality and convergence characteristics. In order to evaluate the effectiveness of the proposed method, 3-unit, 30 Bus IEEE, 13-unit and 15-units are used as case studies with incremental fuel cost functions. The constraints include ramp rate limits, prohibited operating zones and the valve point effect. These constraints make the economic dispatch (ED) problem a non-convex minimization problem with constraints. Simulation results obtained by the proposed algorithm are compared with the results obtained using other methods available in the literature. Based on the numerical results, the proposed RTO algorithm is able to provide better solutions than other reported techniques in terms of fuel cost and robustness.     

A hybrid multi-agent based particle swarm optimization algorithm for economic power dispatch

This paper presents a new multi-agent based hybrid particle swarm optimization technique (HMAPSO) applied to the economic power dispatch. The earlier PSO suffers from tuning of variables, randomness and uniqueness of solution. The algorithm integrates the deterministic search, the Multi-agent system (MAS), the particle swarm optimization (PSO) algorithm and the bee decision-making process. Thus making use of deterministic search, multi-agent and bee PSO, the HMAPSO realizes the purpose of optimization. The economic power dispatch problem is a non-linear constrained optimization problem. Classical optimization techniques like direct search and gradient methods fails to give the global optimum solution. Other Evolutionary algorithms provide only a good enough solution. To show the capability, the proposed algorithm is applied to two cases 13 and 40 generators, respectively. The results show that this algorithm is more accurate and robust in finding the global optimum than its counterparts.     

Efficient hybrid optimization approach for emission constrained economic dispatch with nonsmooth cost curves

Power plants usually operate on the strategy of economic dispatch (ED) regardless of emissions produced. Environmental considerations have become one of the major management concerns. Under these circumstances, the alternative strategy of environmental/economic dispatch (EED) is becoming more and more desirable for not only resulting in great economical benefit, but also reducing the pollutants emission.Based on the literature survey, few attempts have been made at considering valve-point effects for the realistic environmental/economic dispatch (EED) problem. This paper proposes a new efficient hybrid differential evolution algorithm with harmony search (DE–HS) to solve the multiobjective environmental/economic dispatch (EED) problems that feature nonsmooth cost curves. The proposed approach combines in the most effective way the properties of differential evolution (DE) and harmony search (HS) algorithms. To enhance the local search capability of the original DE method, the fresh individual generation mechanism of the HS is utilized.Numerical results for three case studies have been presented to illustrate the performance and applicability of the proposed hybrid method. 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 EED problems.     

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.     

Enhanced cross-entropy method for dynamic economic dispatch with valve-point effects

This paper proposes an enhanced cross-entropy (ECE) method to solve dynamic economic dispatch (DED) problem with valve-point effects. The cross-entropy (CE) method, originated from an adaptive variance minimization algorithm for estimating probabilities of rare events, is a generic approach to combinatorial and multi-extremal optimization. Exploration capability of CE algorithm is enhanced in this paper by using chaotic sequence and the resultant ECE is applied to DED with valve-point effects. The performance of the proposed ECE method is rigorously tested for optimality, convergence, robustness and computational efficiency on a 10-unit test system. Additional test cases with different load patterns and increased number of generators are also solved by ECE. Numerical results show that the proposed ECE approach finds high-quality solutions reliably with faster convergence. It outperforms CE and all the previous approaches.     

A novel stochastic search method for the solution of economic dispatch problems with non-convex fuel cost functions

This paper develops a Novel Stochastic Search (NSS) method for the solution of economic dispatch problems with non-convex fuel cost functions. The NSS solution procedure consists of three steps, namely Direct Search (DS), Goal Neighborhood Approximation (GNA) and Marginal Cost Dispatch (MCD). The DS step identifies a set of feasible solutions in accordance with prescribed equality and inequality constraints. The GNA step processes those feasible solutions to identify an appropriate direction for searching the global optimal solution. Finally, in the MCD step, the marginal cost of each generating unit is regulated in order to establish the global optimal solution. The proposed NSS scheme is applied to solve three examples systems of increasing complexity. The results are compared to those obtained using the conventional Simulated Annealing (SA), Genetic Algorithm (GA), and Evolutionary Programming (EP) methods. The results demonstrate that the NSS method provides a fast, robust and highly effective scheme for the solution of economic dispatch.     

Application of pattern search method to power system valve-point economic load dispatch

Direct search (DS) methods are evolutionary algorithms used to solve constrained optimization problems. DS methods do not require any information about the gradient of the objective function at hand, while searching for an optimum solution. One of such methods is pattern search (PS) algorithm. This study presents a new approach based on a constrained pattern search algorithm to solve well-known power system economic load dispatch problem (ELD) with valve-point effect. For illustrative purposes, the proposed PS technique has been applied to various test systems to validate its effectiveness. Furthermore, convergence characteristics and robustness of the proposed method has been assessed and investigated through comparison with results reported in literature. The outcome is very encouraging and proves that pattern search (PS) is very applicable for solving power system economic load dispatch problem.     

Quantum genetic algorithm for dynamic economic dispatch with valve-point effects and including wind power system

An optimization algorithm is proposed in this paper to solve the problem of the economic dispatch that includes wind power generation using quantum genetic algorithm (QGA). In additional to the detail introduction for models of general economic dispatch as well as their associated constraints, the effect of wind power generation is also included in this paper. On the other hand, the use of quantum genetic algorithms to solve the process of economic dispatch is also discussed and real scenarios are used for simulation tests later on. After comparing the algorithm used in this paper with several other algorithms commonly used to solve optimization problems, the results show that the algorithm used in this paper is able to find the optimal solution most quickly and accurately (i.e. to obtain the minimum cost for power generation in the shortest time). At the end, the impact to the total cost saving for the power generation after adding (or not adding) wind power generation is also discussed. The actual operating results prove that the algorithm proposed in this paper is economical and practical as well as superior. They are quite valuable for further research.     

A hybrid genetic algorithm and bacterial foraging approach for dynamic economic dispatch problem

Dynamic economic dispatch (DED) is one of the most significant non-linear complicated problems showing non-convex characteristic in power systems. This is due to the effect of valve-points in the generating units’ cost functions, the ramp-rate limits and transmission losses. Hence, proposing an effective solution method for this optimization problem is of great interest. The original bacterial foraging (BF) optimization algorithm suffers from poor convergence characteristics for larger constrained problems. To overcome this drawback, a hybrid genetic algorithm and bacterial foraging (HGABF) approach is presented in this paper to solve the dynamic economic dispatch problem considering valve-point effects, ramp-rate limits and transmission losses. The HGABF approach can be derived by integrating BF algorithm and genetic algorithm (GA), so that the BF’s drawback can be treated before employing it to solve the complex and high dimensioned search space of the DED problem. To illustrate the effectiveness of the HGABF approach, several test systems with different numbers of generating units are used. The results of HGABF approach are compared with those obtained by other published methods employing same test systems. These results show the effectiveness and the superiority of the introduced method over other published methods.     

Equal embedded algorithm for economic load dispatch problem with transmission losses

This paper presents equal embedded algorithm (EEA) to solve the economic dispatch (ED) problem with quadratic and cubic fuel cost functions and transmission losses. The proposed algorithm involves selection of lambda values, then the expressions of output powers of generators are derived in terms of lambda by interpolation and finally optimal value of lambda is evaluated from the power balance equation by Muller method. The proposed method is implemented and tested by considering 3, 15 and 26 generators to solve the ED problem. Simulation results such as quality of solution, convergence characteristic and computation time of the proposed method are compared with some existing methods like genetic algorithm (GA), particle swarm optimization (PSO) and Lambda iterative method. It is observed from different case studies that the proposed EEA algorithm provides the qualitative solution with less computational time irrespective of the size of the system.     

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。     

Dynamic economic dispatch using artificial immune system for units with valve-point effect

Dynamic economic dispatch (DED) is an important dynamic problem in power system operation and control. The objective of the problem is to schedule power generation for the online units over a time horizon, satisfying the unit and ramp-rate constraints. In this paper, clonal selection based artificial immune system (AIS) algorithm is used to solve the dynamic economic dispatch problem for generating units with valve-point effect. The feasibility of the proposed method is validated with ten and five unit test systems for a period of 24 h. Results obtained with the proposed approach are compared with other techniques in the literature. The results obtained substantiate the robustness and proficiency of the proposed methodology over other existing techniques in terms of solution quality and computational efficiency.     

Hybrid genetic algorithm for economic dispatch with valve-point effect

This paper presents an efficient method for solving the economic dispatch problem (EDP) through combination of genetic algorithm (GA), the sequential quadratic programming (SQP) technique, uniform design technique, the maximum entropy principle, simplex crossover and non-uniform mutation. The proposed hybrid technique uses GA as the main optimizer, the SQP to fine tune in the solution of the GA run. Based on the maximum entropy principle, the cost function of EDP is approximated by using a smooth and differentiable function to improve the performance of the SQP. An initial population obtained by using uniform design exerts optimal performance of the proposed hybrid algorithm. The effectiveness of the proposed method is validated by carrying out extensive tests on two different EDP with incremental fuel-cost function taking into account the valve-point loadings effects. The result shows that the proposed hybrid genetic algorithm improves the solution accuracy and reliability compared to other techniques for EDP considering valve-point effects.     

Application of pattern search method to power system security constrained economic dispatch with non-smooth cost function

Direct search methods are evolutionary algorithms used to solve optimization problems. (DS) methods do not require any information about the gradient of the objective function at hand while searching for an optimum solution. One of such methods is Pattern Search (PS) algorithm. This paper presents a new approach based on a constrained pattern search algorithm to solve a security constrained power system economic dispatch problem (SCED) with non-smooth cost function. Operation of power systems demands a high degree of security to keep the system satisfactorily operating when subjected to disturbances, while and at the same time it is required to pay attention to the economic aspects. Pattern recognition technique is used first to assess dynamic security. Linear classifiers that determine the stability of electric power system are presented and added to other system stability and operational constraints. The problem is formulated as a constrained optimization problem in a way that insures a secure-economic system operation. Pattern search method is then applied to solve the constrained optimization formulation. In particular, the method is tested using three different test systems. Simulation results of the proposed approach are compared with those reported in literature. The outcome is very encouraging and proves that pattern search (PS) is very applicable for solving security constrained power system economic dispatch problem (SCED). In addition, valve-point effect loading and total system losses are considered to further investigate the potential of the PS technique. Based on the results, it can be concluded that the PS has demonstrated ability in handling highly nonlinear discontinuous non-smooth cost function of the SCED.     

Differential evolution algorithm for emission constrained economic power dispatch problem

In this paper, a differential evolution (DE) algorithm is developed to solve emission constrained economic power dispatch (ECEPD) problem. Traditionally electric power systems are operated in such a way that the total fuel cost is minimized regardless of emissions produced. With increased requirements for environmental protection, alternative strategies are required. The proposed algorithm attempts to reduce the production of atmospheric emissions such as sulfur oxides and nitrogen oxides, caused by the operation of fossil-fueled thermal generation. Such reduction is achieved by including emissions as a constraint in the objective of the overall dispatching problem. A simple constraint approach to handle the system constraints is proposed. The performance of the proposed algorithm is tested on standard IEEE 30-bus system and is compared with conventional methods. The results obtained demonstrate the effectiveness of the proposed algorithm for solving the emission constrained economic power dispatch problem.     

A hybrid HNN-QP approach for dynamic economic dispatch problem

This paper introduces a solution of the dynamic economic dispatch (DED) problem using a hybrid approach of Hopfield neural network (HNN) and quadratic programming (QP). The hybrid algorithm is based on using enhanced HNN; to solve the static part of the problem; the QP algorithm for solving the dynamic part of the DED. This technique guarantees the global optimality of the solution due to its look-ahead capability. The new algorithm is applied and tested to an example from the literature and the solution is then compared with that obtained by some other techniques to prove the superiority and effectiveness of the proposed algorithm.     

Solving non-convex economic dispatch problem with valve point effects using modified group search optimizer method

This paper presents a novel solution based on the group search optimizer (GSO) methodology in order to determine the feasible optimal solution of the economic dispatch (ED) problem considering valve loading effects. The basic disadvantage of the original GSO algorithm is the fact that it gives a near-optimal solution rather than an optimal one in a limited runtime period. In this paper, a new modified group search optimizer (MGSO) is presented for improving the scrounger and ranger operators of GSO. The proposed MGSO is applied on different test systems and compared with most of the recent methodologies. The results show the effectiveness of the proposed method and prove that MGSO can be applicable for solving the power system economic load dispatch problem, especially in large scale power systems.     

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.     

A new approach to secure economic power dispatch

This article presents a new nonlinear convex network flow programming model and algorithm for solving the on-line economic power dispatch with N and N−1 security. Based on the load flow equations, a new nonlinear convex network flow model for secure economic power dispatch is set up and then transformed into a quadratic programming model, in which the search direction in the space of the flow variables is to be solved. The concept of maximum basis in a network flow graph was introduced so that the constrained quadratic programming model was changed into an unconstrained quadratic programming model which was then solved by the reduced gradient method. The proposed model and its algorithm were examined numerically with an IEEE 30-bus test system on an ALPHA 400 Model 610 machine. Satisfactory results were obtained.