Interval-based differential evolution approach for combined economic emission load dispatch

A novel and effective method for solving economic load dispatch (ELD) problem is presented in this work by integrating the ? constrained method with differential evolution (DE) algorithm. The ? constrained method sets us free from selecting the appropriate value for the penalty coefficient in the objective function, which is difficult and often chosen empirically. Differential evolution with ? constrained method (?DE) has proven to be effective in solving non-linear constrained optimisation problems. Here, the effectiveness of ?DE, in solving ELD problems is investigated by using three different ELD problems that take the valve-point loading effect into consideration. The proposed approach provides very high quality solutions which are very competitive to other existing techniques to ELD considering the valve-point effect.

Summary In this article, Artificial Cooperative Search (ACS) algorithm is incorporated with the quadratic approximation (QA) operator to solve the multi-objective economic emission load dispatch (EELD) problems with different generation units. ACS is a Swarm Intelligence–based metaheuristic algorithm, based on the interaction between prey and predator organisms in a habitat, which is effective at global search; however, it does not perform so well at exploring promising regions. The QA operator, on the other hand, is a non-derivative–based efficient local search method that finds the minimum of a quadratic hyperspace passing through three points in a D-dimensional space. Solving the EELD problems with the hybridized ACS-QA algorithm, as being proposed in the present article, leads to more accurate results with fewer function evaluations. Also, multi-objectivity of the problem is handled by transforming it into a single-objective problem by using the weighted sum method. The efficiency of the proposed ACS-QA algorithm is tested in comparison to the algorithms existing in literature by implementing it on six different benchmark optimization problems. Afterward, the proposed ACS-QA algorithm and the ACS algorithm are implemented on multi-objective EELD problems with different generation units. The results are compared with the solutions in literature utilizing different metaheuristic optimization algorithms. Both studies firmly showed that the ACS-QA algorithm is able to find more accurate results even though it uses fewer function evaluation calls.

This paper attempts to propose particle swarm optimization algorithm with smart inertia factor (PSO-SIF) to solve the problem of economic emission load dispatch (EELD) in thermal power plants. The aim of EELD solution is synchronous reduction of both fuel costs and emission level. EELD problem is a non-linear and non-convex problem which uses evolutionary algorithms as efficient to solve such problems due to its complicated characteristics. PSO-SIF algorithm is an efficient and robust algorithm to exploit universal optimal point of optimization problems. Tests implemented on 10-unite systems, considering valve-point loading effects, network losses with Multi-objective functions such as system fuel cost and emission level, to show PSO-SIF algorithm capabilities in solving EELD problems. Results obtained from present investigation indicated universal optimal point exploitation of the problems and superiority of the proposed algorithm compared to other new and efficient algorithms.

Solving non-convex economic load dispatch problem via artificial cooperative search algorithm

Using suitable solution techniques to the unit commitment (UC) and economic load dispatch (ELD) problems play major role on finding an optimal power scheduling in electrical power system. The UC and ELD problems are to decide when and which generating units to commit and decommit, in order to minimize the total fuel cost over specified period subject to a large number of difficult constraints. Lagrangian relaxation (LR) has been widely used to solve UC and ELD. Recently bundling method (BM) has appeared to be one of the best multipliers updated techniques for LR. This method has promising and the fastest convergence characteristics for UC and ELD problems. For that reason, this paper is to solve the UC and ELD problems via LR, BM and interior point method. This paper explores also the best possible way of updating Lagrangian multipliers with minimum computation time using fast programming techniques.

Economical dispatch of power is a significant real time optimization problem in the operation of power system, which allocates the loads on the committed generating units to reduce the generation cost, while meeting all the inequality and equality constraints. Traditionally, economic load dispatch problems have been formulated as convex optimization problem and various classical methods such as gradient methods, base-point participation factor method, etc. have been used for solving them. The traditional formulation is a quadratic input-output curve or piece wise linear cost curve, which are continuous and convex, however, real generating units have valve loading effects, which make the cost curve discontinuous and non-convex and hence the economic dispatch problem is no longer simple to handle. Thus, the classical methods cannot be employed for non-convex problems. The non-convex, non-linear and discontinuous economic dispatch (ED) problem can be better solved by metaheuristic approaches. Such approaches often require tuning of their respective evolutionary parameters. Although, solution to such type of problem have already been solved by different metaheuristic optimization techniques in previous studies as well, however, the algorithm employed in this paper to solve Power Economic Load Dispatch (PELD) problem is adaptive real coded quantum-inspired evolutionary algorithm, which does not require tuning of evolutionary parameters and performs better than some of the existing techniques on standard 6-bus 3 machine system and IEEE 14 bus 5 unit system.

To overcome the challenges of conventional power systems, such as increasing power demand, requirements of stability and reliability, and increasing integration of renewable energy sources, the concept of microgrids was introduced and is currently one of the most important solutions for solving the mentioned problems. Generally, microgrids have two operating modes, namely grid-connected and islanded modes. Based on the literature and its unique characteristics, the islanded mode is more challenging than the other one. In this paper, a new self-adaptive comprehensive differential evolution (SACDE) algorithm is proposed for solving economic load dispatch (ELD) and combined economic emission dispatch (CEED) problems, achieving optimal power consumption in isolated microgrids. Initially, SACDE is employed for solving the ELD problem as a single-objective function, meaning that the operational cost is just considered as the objective function, and thereby, the resources are scheduled accordingly. Then, a multi-objective platform based on SACDE is also proposed to solve the CEED problem. It means two objective functions, including operational cost and emission, are simultaneously optimized. For evaluating the performance of the proposed method, three different scenarios under various cases are considered. According to the results, when SACDE is employed to solve the single objective function (cost minimization) problem, it has better performance than other methods. In terms of the bi-objective scheme (cost and emission minimization), SACDE is significantly superior to the price penalty factor technique which is frequently used in previous studies.

An optimal solution to the unit commitment (UC) and economic load dispatch (ELD) problems lead to remarkable saving in the power system operational cost. UC and ELD main problems are to find the maximum profit and fast computation simulation time for the scheduling program. Lagrangian relaxation (LR) has become one of the best solution methods in solving the UC and ELD problems. The approach is claimed to be more efficient and easier than other methods in solving large-scale problems. When applying LR, the dual function needed to be updated at the higher level. Lately, bundling method (BM) is one of the best solution methods to Lagrangian dual problem. This paper presents LR technique, BM and interior point method for solving the UC and ELD problems. It will also explore the best possible way of updating Lagrangian multipliers with minimum computation time using fast programming techniques.

This paper is an extension made in biogeography based optimization (BBO) by modifying its migration models to make it more realistic. This BBO technique is then applied on simple economic load dispatch (ELD) problem and ELD with valve point effect to analyze the effect of different migration models. Our analysis concludes that the quadratic and sinusoidal models provide the best result for the ELD and ELD valve point effect problem respectively.

An ɛ-dominance-based multiobjective genetic algorithm for economic emission load dispatch optimization problem

Economic Load Dispatch (ELD) optimization is an important and difficult task in power system planning. Previously, most of the research mainly focused on proposing various evolutionary algorithms (EAs) to pursue better results of ELD problems. However, few comprehensive analysis of the effects of various constraint handling techniques (CHTs) on the performance of EA-based techniques are available so far. In this paper, we try to fill this gap by experimentally testing the algorithmic variants of combining four effective and widely used EAs with three CHTs. From the experimental results on the ELD problems with valve-point and those problems with both valve-point and multiple-fuel effect, several important conclusions can be achieved, including 1) for the low scale ELD problem with valve-point only, the selection of EAs is more important than CHTs; 2) for the large scale problems, CHTs play crucial roles; 3) the appropriate combination of EA and CHT is helpful to achieve better performance. This study is also expected to provide solid basis for further strengthening the robustness of EAs for ELD optimization. It is also interesting to observe that the experimental results obtained in this paper are much better than those of the previous effective ELD optimization algorithms.

This research paper uses the golden jackal optimization (GJO), a novel meta-heuristic algorithm, to address power system economic load dispatch (ELD) problems. The GJO emulates the hunting behavior of golden jackals. GJO algorithm uses the cooperative attacking behavior of golden jackals to tackle complicated optimization problems efficaciously. The objective of ELD problem is to distribute power system load requirement to the different generators with a minimum total fuel cost of generation. ELD problems are highly complex, non-linear, and non-convex optimization problems while considering constraints namely valve point loading effect (VPL) and prohibited operating zones (POZs). The proposed GJO algorithm is applied to solve complex, non-linear, and non-convex ELD problems. Six different test systems having 6, 10, 13, 40, and 140 generators with various constraints are used to validate the usefulness of the suggested GJO method. Simulation outcomes of the test system are compared with various algorithms reported in the algorithms such as particle swarm optimization (PSO), ant colony optimization (ACO), and backtracking search algorithm (BSA). Results show that the proposed GJO algorithm produces minimal fuel cost and has good convergence in solving ELD problems of power system engineering.

This paper proposes a new hybrid algorithm of combining the conventional Particle Swarm Optimization (PSO) algorithm with Differential Evolution (DE) strategy, named by the authors as Particle Swarm Differential Evolution Optimization (PSDEO) to enhance better balance between local and global search abilities, while solving the economic load dispatch (ELD) problems considering all practical complex constraints and higher order non-smooth cost functions. Considering advantages and disadvantages of both the methods, PSO exploits and DE explores the search space in PSDEO more efficiently to improve the quality of solution and convergence speed. The PSDEO algorithm is tested for benchmark ELD problems, and results show an improved performance of the proposed PSDEO.

Economic Load Dispatch (ELD) problems are nonlinear constrained problems which occupy an important role in the economic operation of power system. Recently, as an alternative to the conventional mathematical approaches, evolutionary algorithms have been given much attention by researchers due to their ability to find good solutions in ELD problems. In this paper, a biogeography based-optimization approach is validated. Biogeography is the study of the geographical distribution of biological organisms. Furthermore, a new approach of the biogeography based-optimization methodology combined with predator-prey approach were proposed and validated for two test systems consisting of 15 and 20 generation units (thermal generators), respectively. Simulation results are compared with those of other studies reported in the literature.

A multi-objective particle swarm optimization (MOPSO) approach for multi-objective economic load dispatch problem in power system is presented in this paper. The economic load dispatch problem is a non-linear constrained multi-objective optimization problem. The proposed MOPSO approach handles the problem as a multi-objective problem with competing and non-commensurable fuel cost, emission and system loss objectives and has a diversity-preserving mechanism using an external memory (call “repository”) and a geographically-based approach to find widely different Pareto-optimal solutions. In addition, fuzzy set theory is employed to extract the best compromise solution. Several optimization runs of the proposed MOPSO approach were carried out on the standard IEEE 30-bus test system. The results revealed the capabilities of the proposed MOPSO approach to generate well-distributed Pareto-optimal non-dominated solutions of multi-objective economic load dispatch. Comparison with Multi-objective Evolutionary Algorithm (MOEA) showed the superiority of the proposed MOPSO approach and confirmed its potential for solving multi-objective economic load dispatch.