Non-convex Economic Dispatch Research Articles

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

According to no free lunch (NFL) theorem, a metaheuristic optimization method is superior to other metaheuristic optimization methods when it has focused on specific class of optimization problems. Thus, this paper focuses on developing artificial cooperative search (ACS) optimization algorithm to solve economic dispatch (ED) problems more precisely with less complexity than other metaheuristic optimization methods. ACS is a recently developed two population search algorithm based on coevolution process with high probability of finding optimal solution in complex and non-convex optimization problems. This merit is provided by balancing exploration of the problem's search space and exploitation of better results through use of two advanced evolutionary operators and only one control parameter. The constraint handling strategy of the proposed method for solving economic power load dispatch problems is to generate and work with feasible solutions along all the optimization iterations without any mismatch between the total amount of electric power generation and electricity demand plus network transmission loss. Unlike the penalty method, this strategy is unaffected by parameter setting of applied optimization method that complicates its applicability for solving economic power load dispatch problems. The feasibility of ACS for solving ED problem is tested on different lossy non-convex test systems of various sizes and complexities. The practical aspects such as satisfaction of power demand constraint, generation limits of generators and value-point loading effect are undertaken to solve ED problem in medium to relatively large-scale electric power systems. Obtained results confirm the ACS's capability in converging to a better solution highly robust within the reasonable computational time in all independent trials; all these as compared with other optimization algorithms reported in the literature for solving lossy non-convex ED problems. The results are analyzed statistically in terms of solution quality and computational efficiency. The statistical analyses reveal that ACS is a potential method to solve economic power load dispatch problems as it provides higher quality solution in comparison with other optimization algorithms.

Combined heat and power economic dispatch problem using advanced modified particle swarm optimization

In this paper, a novel optimization algorithm based on Particle Swarm Optimization (PSO) is proposed for solving the non-convex non-linear economic dispatch (ED) problem in a system integrated with combined heat and power. In the proposed method, which is known as Advanced Modified PSO (AMPSO), one third of the population is updated based on conventional PSO, and the remaining members are updated randomly based on the boundaries of variables. In order to improve the efficiency of the algorithm, the required coefficients were estimated by the Taguchi method. The optimization problem is the nonlinear non-convex ED in which the effects of steam valves and the power loss have been included. The proposed algorithm is applied on different systems, and the results are compared with those of some well-regarded methods developed by other researchers. The results show the superiority of the proposed AMPSO in comparison to other methods.

ED-SHO: A framework for solving nonlinear economic load power dispatch problem using spotted hyena optimizer

This paper presents the application of recently developed metaheuristic optimization algorithm, the spotted hyena optimizer, for solving both convex and non-convex economic dispatch problems. The proposed algorithm has been tested on various test systems (i.e. 6, 10, 20, and 40 generators systems) and compared with other well-known approaches to demonstrate its effectiveness and efficiency. The results show that the proposed algorithm is able to solve economic load power dispatch problem and converge toward the optimum with low computational efforts.

Iterated Local Search Solution for the Non-convex Economic Dispatch Problem

Iterated Local Search Solution for the Non-convex Economic Dispatch Problem

Solving nonconvex economic thermal power dispatch problem with multiple fuel system and valve point loading effect using fuzzy reinforcement learning

Solving nonconvex economic thermal power dispatch problem with multiple fuel system and valve point loading effect using fuzzy reinforcement learning

QUASI-OPPOSITIONAL SYMBIOTIC ORGANISMS SEARCH ALGORITHM FOR DIFFERENT ECONOMIC LOAD DISPATCH PROBLEMS

In this paper, an effective meta-heuristic technique called Quasi-Oppositional Symbiotic Organisms Search is applied for solving non-convex economic dispatch problems. Symbiotic Organisms Search is a soft computing technique, inspired by organisms in the ecosystem. This technique is implemented for improving the solution quality in minimum time. In order to improve convergence rate, quasi-reflected numbers are used here instead of pseudo-random numbers. Different equality and inequality constraints such as transmission loss, load demand, prohibited operating zone, generator operating limits and boundary of ramp rate are considered here. Presence of multiple fuels and valve point are also considered in some cases. This algorithm is applied to four different test systems. Simulation results are compared with many recently developed optimization techniques to show the superiority and consistency of this method. Simulation results also show that the computational efficiency of this algorithm is much better than the other meta-heuristic methods available in the literature.

Improved Artificial Cooperative Search Algorithm for Solving Non-convex Economic Dispatch Problems with Valve-point Effects

This paper presents Improved Artificial Cooperative Search (IACS) algorithm for solving economic dispatch problems considering the valve point effects, ramp rate limits, transmission losses and prohibited operation zones. In order to improve the solution quality and increase the search efficiency, a novel perturbation scheme called “Global best guided chaotic local search” is proposed and incorporated into ACS algorithm. The effectiveness of the proposed IACS algorithm has been benchmarked with twelve widely known optimization test problems. In order to assess the performance of the proposed algorithm on non-convex optimization problems, four case studies related to highly nonlinear economic dispatch problems have been solved . Results retrieved from IACS algorithm have been compared with literature approaches in terms of minimum, maximum and average generation cost values. Comparison results indicate that IACS produces more economical power load than those of other optimizers available in the literature

Semidefinite programming solution of economic dispatch problem with non-smooth, non-convex cost functions

The paper presents a solution to economic dispatch (ED) problems with non-convex, non-smooth fuel cost functions, which characterize practical generating units. A method involving a unified semidefinite programming (SDP) formulation of different ED problems through cost function decomposition was presented. The solution of the resulting rank-relaxed SDP problem was refined to achieve the rank constraint using the method of convex iteration and branch-and-bound technique. The SDP method was investigated on some test problems in the literature. The results showed that the SDP method compared favorably with other methods, and can efficiently solve non-convex and non-smooth ED problems.

Embedded Immune-Evolutionary Programming for Economic Dispatch of Generators with Prohibited Operating Zones

Economic Dispatch (ED) is one of the popular power system problems. It is solved by the grid system operators (GSOs) to find the cheapest production cost for dispatching power from generations to the demands (loads). It is crucial to have cheap production cost and at the same time satisfying all system constraints such as generating units’ operating limits, transmission lines capacity and spinning reserve. ED of generators with prohibited operating zones is considered as a nonconvex ED as it has inequality constraints that need to be considered. This inequality constraint is the operating limits of generating units that considering the allowable and prohibited operating zones. The generating units must be ensured, that they are only operated within the allowable limits to avoid problem with the equipments that related to them. Some generating units will experience high vibration at certain level of operating limits and will burn more fuel to maintain at certain level of load. This paper presents an embedded technique to solve ED of generators problem with prohibited operating zones. The cloning process of Artificial Immune System (AIS) is inserted into Evolutionary Programming (EP) algorithm to form a new technique termed as Embedded Immune-Evolutionary Programming (EIEP). The proposed embedded technique has been tested on the IEEE 26-Bus Reliability Test System (RTS) with three different conditions of load. Besides that the results produced by the embedded technique has been compared with two single techniques which are AIS and EP. It is found that, the results of ED of generators with prohibited operating zones produced by EIEP is better than the two single techniques in terms of low total production cost.

A modified crow search algorithm (MCSA) for solving economic load dispatch problem

This paper presents a novel evolutionary optimization algorithm namely the modified crow search algorithm (MCSA) for solving the non-convex economic load dispatch (ELD) problem which improves the crow search algorithm (CSA) by an innovative selection of the crows and adaptive adjustment of the flight length. MCSA is a population-based technique based on the intelligent behavior of the crows in finding food sources. In MCSA, each crow saves its food in hiding-places for the time it needs. Also, each crow searches environment to find the better foods by stealthily following other crows to discover their hiding-places. The proposed MCSA develops the search capability of crows in the original CSA and introduces a new way by which a destination is selected by a crow to follow. To indicate the applicability of MCSA in the ELD problem, it is applied on five different well-known test systems. The results are compared in terms of the solution quality, robustness, and computing time with other methods implying that the proposed method has a superior performance than the other techniques.

An enhanced moth-flame optimizer for solving non-smooth economic dispatch problems with emissions

This paper proposes an Enhanced Moth-Flame Optimization (EMFO) algorithm for solving the non-convex economic dispatch (ED) problem with valve point effects and emissions. It determines the optimal generation schedule of generating units by minimizing both fuel cost and emission simultaneously while the system constraints are achieved. The moth-flame optimization (MFO) is a recent nature-inspired method, which is based on the navigation mechanism called transverse orientation of Moths in space. The EMFO combines the merits of the traditional MFO and levy flight by concentration the search space. The usage of Lévy-flight has the prominent properties to increase the diversity of population. The effectiveness of the proposed EMFO method is proven on using 10 benchmark functions and 3 standard test systems consisting of 6, 40 and a large scale 80 generating units with non-convex fuel cost functions. The capability of the proposed algorithm is verified also for single and multi-objective studied cases and its results are compared with several well-known previous techniques. The results confirm the high performance of the proposed EMFO method for finding the optimal economic generation scheduling at acceptable low emission levels.

On the non-convex economic power dispatch problem using artificial bee colony

On the non-convex economic power dispatch problem using artificial bee colony

Solving non-convex economic dispatch with prohibited zones using artificial fish swarm optimization

In this paper, a novel approach is proposed to solve the non-convex and discontinuous economic dispatch (ED) problem of power system with thermal power plants. All the practical constraints (loss constraint, generators ramp rate constraints and network constraints) are considered for solving the ED problem. Here, the proposed ED problem is solved by considering the generators with valve point loading (VPL) effects and prohibited operating zones (POZs) effects. In this paper, to solve this practical ED problem, an evolutionary based Artificial Fish Swarm Optimization Algorithm (AFSOA) is utilized. The AFSOA is a global search algorithm based on the characteristics of fish swarm and its autonomous model. The detailed algorithm with its flow chart is presented in this paper. To show the effectiveness of the proposed ED approach, 3 test systems (3, 6 and 20 generating unit systems) are considered. The obtained results are compared with other algorithms reported in the literature.

Heat Transfer Search Algorithm for Non-convex Economic Dispatch Problems

This paper presents Heat Transfer Search (HTS) algorithm for the non-linear economic dispatch problem. HTS algorithm is based on the law of thermodynamics and heat transfer. The proficiency of the suggested technique has been disclosed on three dissimilar complicated economic dispatch problems with valve point effect; prohibited operating zone; and multiple fuels with valve point effect. Test results acquired from the suggested technique for the economic dispatch problem have been fitted to that acquired from other stated evolutionary techniques. It has been observed that the suggested HTS carry out superior solutions.

Improved Artificial Bee Colony Algorithm for Non-Convex Economic Power Dispatch Problem

Improved Artificial Bee Colony Algorithm for Non-Convex Economic Power Dispatch Problem

A Non-convex Economic Dispatch Problem with Valve Loading Effect Using a New Modified $$\beta $$ β -Hill Climbing Local Search Algorithm

In this paper, the problem of economic load dispatch (ELD) is tackled using a recently introduced local search-based method called $$\beta $$ -hill climbing optimizer. In a power system, the ELD problem is tackled by arranging a set of generation units’ outputs in a specific order to minimize the cost of the operating fuel and to match the power system load demand. This goal is achieved by satisfying all the power balance equality constraints and power output inequality constraints. $$\beta $$ -hill climbing algorithm is a new local search algorithm which uses an intelligent stochastic operator called $$\beta $$ -operator to escape the trap of local optima. The proposed method is evaluated using five real-world ELD benchmarks which vary in terms of complexity and size. The sensitivity analysis to study the effect of the proposed method parameters is conducted based on eight different convergence cases. The evaluation results of the proposed method are compared with 45 state-of-the-art methods using the same tested ELD benchmarks. Interestingly, the proposed method is able to produce a very closed-to-optimum result for almost all the tested ELD systems and the best result for one of them.

Orthogonal learning competitive swarm optimizer for economic dispatch problems

Power system economic dispatch (ED), mathematically, is a typical complex nonlinear multivariable strongly coupled optimization problem with equality and inequality constraints, especially considering the valve-point effects. In this paper, a novel variant of competitive swarm optimizer (CSO) referred to as OLCSO is proposed to solve both convex and non-convex ED problems. In the canonical CSO, the loser particle in each pair updates its position by learning from the winner particle. Such a learning strategy is easy to implement, but is not efficient enough because even a bad particle can contain useful information while a good particle may have misleading features. To solve this issue, an orthogonal learning (OL) strategy based on orthogonal experimental design is developed for CSO to quickly discover more useful information that contained in the winner and the loser particles. The OL strategy can easily construct a more promising solution to provide a more systematic search strategy for OLCSO. A set of 24 numerical benchmark functions and three ED cases with diverse complexities and characteristics are used to comprehensively verify the performance of OLCSO. The experimental results and comparison results consistently indicate that OLCSO can be used as a competitive alternative for ED problems.

A Distributed Randomized Gradient-Free Algorithm for the Non-Convex Economic Dispatch Problem

In this paper, a distributed randomized gradient-free algorithm (DRGF) is employed to solve the complex non-convex economic dispatch problem whose non-convex constraints include valve-point loading effects, prohibited operating zones, and multiple fuel options. The DRGF uses the Gauss approximation, smoothing parameters, and a random sequence to construct distributed randomized gradient-free oracles. By employing a consensus procedure, generation units can gather local information through local communication links and then process the economic dispatch data in a distributed iteration format. Based on the principle of projection optimization, a projection operator is adopted in the DRGF to deal with the discontinuous solution space. The effectiveness of the proposed approach in addressing the non-convex economic dispatch problem is demonstrated by simulations implemented on three standard test systems.

A New Sine Cosine Optimization Algorithm for Solving Combined Non-Convex Economic and Emission Power Dispatch Problems

The current study presents a modified sine cosine optimization (MSCO) algorithm for solving the non-smooth environmental/economic power dispatch problem. In the proposed MSCO algorithm, random search agents’ population is initialized in the search domain for simultaneous optimization of both the combined economic and environmental objectives. Added to that, the proposed MSCO proposes an opposition strategy to preserve the diversity of solutions purposefully. Hence, the Pareto optimal solutions are customized according to the Pareto front concepts. These solutions are evolved using a modified version of the sine cosine algorithm (SCA), where the best agent is selected randomly from the stored Pareto solutions. Furthermore the parameter-based tuning mechanism is designed to improve the balance between the exploration and exploitation abilities. The correctness and effectiveness of the proposed MSCO are validated through experiments results and comparisons on EELD problem. Simulations were conducted on two test systems with non-smooth fuel cost and emission issues. The first system constitutes 6-unit benchmarking system, while the second one constitutes 10- units, and their results are compared with the results of other optimization techniques that were reported in the literature. The numerical comparisons reveal the robustness and effectiveness of the proposed MSCO algorithm.

Improved Fireworks Algorithm with Chaotic Sequence Operator for Large-Scale Non-convex Economic Load Dispatch Problem

Fireworks algorithm (FWA) is one of the recently developed meta-heuristic techniques, which has shown potential to solve non-shifted optimization functions. However, several engineering optimization problems have shifted optima where the performance of the algorithm heavily degrades owing to the intrinsic limitations of FWA. This paper proposes an Improved FWA with Chaotic Sequence Operator (IFWA-CSO) to efficiently solve diverse engineering optimization problems having shifted optima. The proposed algorithm is validated on benchmark mathematical functions. The IFWA-CSO algorithm is then proposed to solve complex economic dispatch problem of power systems considering several practical constraints such as valve-point loading effect, ramp rate limits, prohibited operating zones, etc. The application results of the proposed method on 140 generator system are presented and compared with existing methods available in literature.