Economic Load Dispatch (ELD) is one of the essential components in power system control and operation. Although conventional ELD formulation can be solved using mathematical programming techniques, modern power system introduces new models of the power units which are non-convex, non-differentiable, and sometimes non-continuous. In order to solve such non-convex ELD problems, in this paper we propose a new approach based on the Social Spider Algorithm (SSA). The classical SSA is modified and enhanced to adapt to the unique characteristics of ELD problems, e.g., valve-point effects, multi-fuel operations, prohibited operating zones, and line losses. To demonstrate the superiority of our proposed approach, five widely adopted test systems are employed and the simulation results are compared with the state-of-the-art algorithms. In addition, the parameter sensitivity is illustrated by a series of simulations. The simulation results show that SSA can solve ELD problems effectively and efficiently.
Besides the above non-EA approaches, many EA methods have also been developed to solve various formulations of ELD. Orero and Irving proposed a simple Genetic Algorithm (GA) to solve ELD with POZ @cite_3 . Besides the standard GA, this work also devised a deterministic crowding GA model to solve the problem. Chiang developed an improved GA with the multiplier updating scheme for ELD with VPE and MFO @cite_28 . In this work, the proposed GA is incorporated with an improved evolutionary direction operator. In addition, the tailor-made migration operator efficiently searches the solution space. He proposed a hybrid GA approach to solve ELD with VPE @cite_9 . The algorithm proposed is a hybrid GA with differential evolution (DE) and sequential quadratic programming (SQP). Sinha developed an Evolutionary Programming (EP) method to solve ELP with VPE @cite_18 . Pereira-Neto proposed an Evolutionary Strategy (ES) method to solve ELP with VPE and POZ @cite_24 . DE has also been adapted to solve ELD @cite_43 @cite_0 .
@cite_18 @cite_28 @cite_9 @cite_3 @cite_24 @cite_43 @cite_0
1968481105 2106334424 2053900989 1991640459
Abstract An efficient hybrid genetic algorithm (HGA) approach for solving the economic dispatch problem (EDP) with valve-point effect is presented in this paper. The proposed method combines the GA algorithm with the differential evolution (DE) and sequential quadratic programming (SQP) technique to improve the performance of the algorithm. GA is the main optimizer, while the DE and SQP are used to fine tune in the solution of the GA run. To improve the performance of the SQP, the cost function of EDP is approximated by using a smooth and differentiable function based on the maximum entropy principle. An initial population obtained by using uniform design exerts optimal performance of the proposed hybrid algorithm. The combined algorithm is validated for two test systems consisting of 13 and 40 thermal units whose incremental fuel cost function takes into account the valve-point loading effects. The proposed combined method outperforms other algorithms reported in literatures (EP, EP–SQP, PSO, PSO–SQP) for EDP considering valve-point effects. Evolutionary algorithms (EAs) are often well-suited for optimization problems involving several, often conflicting objectives. Since 1985, various evolutionary approaches to multiobjective optimization have been developed that are capable of searching for multiple solutions concurrently in a single run. However, the few comparative studies of different methods presented up to now remain mostly qualitative and are often restricted to a few approaches. In this paper, four multiobjective EAs are compared quantitatively where an extended 0 1 knapsack problem is taken as a basis. Furthermore, we introduce a new evolutionary approach to multicriteria optimization, the strength Pareto EA (SPEA), that combines several features of previous multiobjective EAs in a unique manner. It is characterized by (a) storing nondominated solutions externally in a second, continuously updated population, (b) evaluating an individual's fitness dependent on the number of external nondominated points that dominate it, (c) preserving population diversity using the Pareto dominance relationship, and (d) incorporating a clustering procedure in order to reduce the nondominated set without destroying its characteristics. The proof-of-principle results obtained on two artificial problems as well as a larger problem, the synthesis of a digital hardware-software multiprocessor system, suggest that SPEA can be very effective in sampling from along the entire Pareto-optimal front and distributing the generated solutions over the tradeoff surface. Moreover, SPEA clearly outperforms the other four multiobjective EAs on the 0 1 knapsack problem. We introduce a simple evolution scheme for multiobjective optimization problems, called the Pareto Archived Evolution Strategy (PAES). We argue that PAES may represent the simplest possible nontrivial algorithm capable of generating diverse solutions in the Pareto optimal set. The algorithm, in its simplest form, is a (1 + 1) evolution strategy employing local search but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors. (1 + 1)-PAES is intended to be a baseline approach against which more involved methods may be compared. It may also serve well in some real-world applications when local search seems superior to or competitive with population-based methods. We introduce (1 + λ) and (μ | λ) variants of PAES as extensions to the basic algorithm. Six variants of PAES are compared to variants of the Niched Pareto Genetic Algorithm and the Nondominated Sorting Genetic Algorithm over a diverse suite of six test functions. Results are analyzed and presented using techniques that reduce the attainment surfaces generated from several optimization runs into a set of univariate distributions. This allows standard statistical analysis to be carried out for comparative purposes. Our results provide strong evidence that PAES performs consistently well on a range of multiobjective optimization tasks. Mutation operators such as Gaussian, Levy and Cauchy have been used with evolutionary programming (EP). According to the no free lunch theorem, it is impossible for EP with a single mutation operator to outperform always. For example, Classical EP (CEP) with Gaussian mutation is better at searching in a local neighborhood while the Fast EP (FEP) with the Cauchy mutation performs better over a larger neighborhood. Motivated by these observations, we propose an ensemble approach where each mutation operator has its associated population and every population benefits from every function call. This approach enables us to benefit from different mutation operators with different parameter values whenever they are effective during different stages of the search process. In addition, the recently proposed Adaptive EP (AEP) using Gaussian (ACEP) and Cauchy (AFEP) mutations is also evaluated. In the AEP, the strategy parameter values are adapted based on the search performance in the previous few generations. The performance of ensemble is compared with a mixed mutation strategy, which integrates several mutation operators into a single algorithm as well as against the AEP with a single mutation operator. Improved performance of the ensemble over the single mutation-based algorithms and mixed mutation algorithm is verified using statistical tests.