A Direct Search Method to solve Economic Dispatch Problem with Valve-Point Effect

Economic dispatch (ED) is one of the optimization problems in power system operation and planning. The practical ED problems have non-smooth cost functions with equality and inequality constraints. It becomes more complicated problem when valve point effects of generator are considered and results multiple local minima. This makes the ED problems are difficult to find global optimum solution when using any mathematical approach due to non-convex cost function. The Classical particle swarm optimization (PSO) has ability to reach near global optimum solution by tuning some parameters but always stuck at local minima. In this paper, an iteration particle swarm optimization (IPSO) is proposed to solve ED problems with valve point effects. The proposed algorithm is introduced a new parameter in the original velocity equation of PSO in order to enrich searching behaviour, solution quality and avoid being trapped at local minima. The proposed IPSO algorithm has been implemented on two test power systems (consisting 3 and 13 generating units) to validate its effectiveness. The simulation results confirmed that IPSO algorithm has better convergence characteristic and more robust compared to PSO and some published results.

The objective of the economic dispatch problem of electrical power generation is to schedule the committed generating unit's output so as to meet the required load demand while satisfying the system equality and inequality constraints. This paper solves the combined economic emission dispatch problem by Min-Max and Max-Max price penalty factor approaches using Lagrange's method. The price penalty factor for the combined economic emission dispatch (CEED) is the ratio of fuel cost to emission value. The literature papers reported that most of them use Max-Max Price penalty factor for the CEED problem. This paper analyses Min-Max and Max-Max price penalty factors for CEED problem. This problem is solved for a five generator system by considering the valve point effect for both fuel cost and emission function. The total fuel cost by considering valve point effect of the combined economic emission dispatch problem is less when using Min-Max price penalty factor approach in comparison to Max-Max price penalty factor.

The Economic Dispatch (ED) is one of the most important optimization problems in power systems the ultimate goal of the ED is to minimize the cost of operations in a power generation. In this paper, the Woodpecker Mating Algorithm (WMA) is used to solve the ED problem considering the nonlinear properties of generators such as valve point effects (VPE), prohibited operating zones (POZ), ramp rate limits, multiple fuel options, and transmission loss. The WMA algorithm is a novel metaheuristic algorithm inspired by the mating behavior of woodpeckers and sound intensity (a physical quantity). The WMA is implemented on six test systems of different operational dimensions and characteristics to show its capacity for solving the ED problem. The results are compared with the latest and most efficient methods introduced in the literature. Proving the efficiency of the WMA to solve the ED problem, simulation results are promising and offer the optimal fuel cost of production.

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.

A hybrid genetic algorithm approach based on differential evolution for economic dispatch with valve-point effect

In this paper, an effective and reliable variant of particle swarm optimization with time-varying acceleration coefficients (PSO-TVAC) technique is proposed for the economic load dispatch problem considering valve-point loading effect. Main objective of the economic load dispatch (ELD) problem is to minimize the fuel cost by allocating the generation of the committed units subjected to the equality and inequality constraints. Equation of the economic dispatch objective function gets modified with the addition of a new parameter which represents the effect of valve-point loading. While exploring the use of PSO and its other variants to the economic dispatch problem, a number of research works have not considered the transmission losses properly. This paper represents the usefulness of the proposed PSO-TVAC algorithm to significantly reduce the fuel cost while taking into account the effect of transmission losses along with the non-convex characteristic due to valve loading. The results have been demonstrated for three-generator and ten-generator test systems.

This article presents a new optimization approach called the Wild Horse Optimizer (WHO). It is designed to be user-friendly, effective, and to converge rapidly when addressing economic dispatch (ED). Given that ED is a nonlinear and complex optimization problem, the primary goal of the WHO is to minimize overall generation costs while adhering to the system's equality and inequality constraints. The Combined Economic Emission Dispatch (CEED) problem is also considered, which considers the environmental impacts stemming from the emissions of gaseous pollutants produced by fossil-fueled power plants. To demonstrate the capabilities of the WHO, various networks with 3, 6, 10, 13, 15, and 40 stations have been tested and applied to resolve multiple ED scenarios, including factors such as valve point effects, losses, and prohibited operating zones. The simulation results indicate that the WHO's approach is significantly more effective than the methods found in current academic literature. This analysis highlights the strengths of the WHO's strategies compared to alternatives, suggesting they may offer better solutions to the challenges identified in previous studies.

This paper proposes a new differential evolution optimization (DE) strategy namely, Self-Realized Differential Evolution (SRDE) for solving the economic dispatch (ED) problem with non-smooth cost functions in power systems. The proposed SRDE is in the structure of differential evolution owning new mutation operation and selection mechanism. An effective constraint handling method is presented in the suggested stochastic search technique. The proposed approach has been examined and tested with the numerical results of ED problems with forty-generation units including ramp rate limits, prohibited operating zones and valve-point loading effects also ten-generation units with multiple fuel options. The results of the proposed technique are compared with that of other techniques reported in the literature. For both the cases, the proposed algorithm outperforms the solution reported for the existing algorithms. In addition, the promising results show the robustness, fast convergence and efficiency of the proposed technique.

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

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

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.

The Economic Dispatch (ED) goal is to reduce the cost of fuel for any viable demand of load, though maintain optimum power output at the same time. The practical problem of ED includes valve-point effect and piecewise cost functions due to multiple fuels. Due to presence of valve-point effect and piecewise cost function ED modelled as complex problem. It is challenging to find global finest point by using classic methods. In this paper, Spotted Hyena Optimization (SHO) is used to solve complex ED problem. SHO is a new most eristic algorithm motivated by spotted hyena behavior. The key idea behind SHO algorithm is social connection between spotted hyenas and their collective behaviour. SHO algorithm basic steps are looking for prey, surrounding and attacking prey. This SHO algorithm has been evaluate on 10-unit system with practical constraints for solution of complex ED problem. For given test system four case studies are performed for four different load demands. The test system results are compared with recently published journals results to check the consistency of SHO for solving complex ED problem. The comparative results proven that SHO is better optimization technique to solve complex ED problem.

Power utilities in general use Lagrangian method to solve economic dispatch (ED) problem for ease of implementation. This is possible only if the cost function of generating units is convex. Owing to valve-point effect exhibited by multi-valve steam turbines, the cost function is non-convex. The Maclaurin series-based Lagrangian method is proposed to solve complicated, non-convex and non-linear ED problems. In this method, the rectified sinusoid function is represented by Maclaurin sine series expansion and is solved using the Lagrangian method. The problem is solved iteratively using lambda iteration method. The proposed methodology is validated using IEEE 30-bus test system. In addition, the effectiveness of the new algorithm is demonstrated with 3, 13 and 40 generator test systems available in the literature. The results obtained substantiate the applicability of the proposed method for solving ED problems with non-smooth cost functions at par with stochastic search techniques generally used.

Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect

In the current scenario Economic Dispatch problem has taken a suitable twist as the public has become increasingly concerned with environmental matters. The dispatch of systems to minimise pollutants, as well as to achieve minimum cost are the two main constraints to be dealt with. In this paper, Global optimization techniques such as Honey Bee’s and Simulated Annealing techniques are employed together (HB-SA) to solve the Combined static Economic and Emission dispatch (CEED) problems in power system. This paper clearly is justified with the results, separately obtained for the above two techniques and is also provided with the results on applying both the algorithms separately. With practical consideration, Economic Dispatch will have non-smooth cost functions with equality and inequality constraints that make the problem, a large-scale highly constrained non linear optimization problem. As to demonstrate the effectiveness of the proposed method, it is subjected to the CEED problems considering (or) handling valve point loading effects. Comparison with other optimization techniques showed the superiority of the proposed HB-SA approach and confirmed its potential for solving nonlinear CEED problems.