Solving economic dispatch problem using MiPower by lambda iteration method

In past two decades solution to the Dynamic Economic Load Dispatch (DELD) problem considering emission and transmission losses along with Valve Point Loading (VPL) effect has been investigated by researchers using several meta-heuristic techniques. Numerical techniques such as lambda iteration method, dynamic programming etc. have been unable to solve the ELD problem with VPL effect. This paper presents a systematic performance evaluation of meta-heuristic techniques named as Particle Swarm Optimization and the Genetic Algorithm for combined economic emission dispatch problem considering Valve Point Loading (VPL) to meet 24 hr. load profile on IEEE 9 bus test case with 3 generator units.

This chapter discusses the application of linear programming (LP) techniques to find the optimal solution of the economic dispatch (ED) problem without considering transmission losses. The ED problem is concerned with optimizing the power generated by several generating units. The objective is to find the optimal power produced by each unit to supply the required load at minimum total cost. The generation cost associated with each unit is usually in the form of a quadratic or cubic function of the power produced. To apply LP, these nonlinear cost functions have to be linearized. The optimal solution is then determined by LP based on the approximate linear model. Piece-wise linearization methodology is adopted in this chapter. To evaluate the performance of the linearization method, a comprehensive set of benchmark test problems is used. LP solutions of linearized ED problems are compared with several other techniques from the literature. The LP technique with piece-wise linearization shows an overall competitive advantage in terms of total cost, solution time, and load satisfaction.

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.

Single period economic dispatch cannot handle the intertemporal constraints in multi-period environment. To cope with this issue, the extension of economic dispatch over multiple time intervals (i.e., dynamic economic dispatch) has been introduced that considers the intertemporal constraints between different time intervals. Another issue is determining the most economical generation dispatch that could supply the area demand without violating the tie-line capacity, which cannot be solved by conventional economic dispatch problems. However, this study shows that the most economic schedule of power generation cannot satisfy echo-system expectation; therefore, making a compromise between fuel cost and environmental issues, a hot-button subject in industrialized nations, seems to be crucial. To reach the goals a bi-objective multi-area dynamic economic dispatch approach, which can handle intertemporal and multi-area constraints concurrently, is proposed to assist power system operators more and more. Finally, a hybrid algorithm, namely gray wolf optimizer-particle swarm optimization is introduced to solve the proposed problem and also a set of benchmark problems. By implementing the proposed approach on two small (10-unit, three areas) and large (40-unit, four areas) scale test systems, about 3.1% and 3.3% improvement in generation cost is obtained, respectively compare to the best reported results in the literature.

Ant lion optimizer (ALO) is a newly developed population-based search algorithm inspired by hunting mechanism of antlions and based on five steps of hunting the ants, i.e., the random walk of ants, building traps, entrapment of ants in traps, catching preys and re-building traps. This paper presents the application of ALO algorithm for the solution of non-convex and dynamic economic load dispatch problem of electric power system. The performance of ALO algorithm is tested for economic load dispatch problem of four IEEE benchmarks of small-scale power systems, and the results are verified by a comparative study with lambda iteration method, particle swarm optimization algorithm, genetic algorithm, artificial bee colony, evolutionary programming and Grey Wolf optimizer (GWO). Comparative results show that the performance of ant lion optimizer algorithm is better than recently developed GWO algorithm and other well-known heuristics and meta-heuristics search algorithms.

Moth flame optimization algorithm is novel nature inspired heuristic paradigm inspired by navigation method of moths in nature and based on the concept that the moth eventually converges toward the light. This paper presents the application of MFO algorithm for the solution of non-convex and dynamic economic load dispatch problem of electric power system. The performance of MFO algorithm is tested for non-convex, convex and dynamic economic load dispatch problem of seven IEEE benchmarks and the results are verified by a comparative study with lambda iteration method, particle swarm optimization algorithm, genetic algorithm (GA), artificial bee colony, evolutionary programming (EP) and grey wolf optimizer (GWO). Also, in the proposed research, the impact of renewable energy sources (i.e. wind and solar) has been taken into consideration along with conventional thermal power generating units. Also, critical analysis has been made for percentage cost saving with due consideration of solar and wind power units and it has been experimentally observed that the addition of renewable energy sources to conventional thermal power system results in significant cost saving. Comparative results show that the performance of Moth Flame Optimizer Algorithm is better than recently developed GWO algorithm and other well known heuristics and meta-heuristics search algorithms.

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.

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

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.

The major issue in power system is economic dispatch (ED) problem. The Economic dispatch problem is aim to distribute electrical power generators in some active electrical system to minimize the total of cost of generators. Here, the purpose of this paper is to achieve the combination of power output, maximum total of demand with the transmission losses, minimum the total cost of generating unit and design web based application to simplify the calculation of economic dispatch problem for user at PT. Indonesia Power. This paper introduces a solution to ED problem using Dr. Xin She Yang developed firefly algorithm. The result obtained by the proposed algorithm have been compared with the result from the data source and that results is more optimal using firefly algorithm than the results of the data source. Based on the result, it can be concluded that this web based application is very useful to user in calculating the economic dispatch problem with transmission losses at PT. Indonesia Power.

Economic Power Dispatch is a critical component of the power system network (EPD). It assigns required load to online producing units while accounting for system constraints. The fundamental purpose of EPD is to lower overall power generation costs. For example, the bi-objective Combined Emission- Economic Dispatch (CE-ED) problem addresses the environmental impacts of fossil fuel power plant emissions. The bi-objective challenge reduces both fuel costs and emissions. Generators are simulated using quadratic cost functions to match their output restrictions. The non-linear, non-convex objective functions and equality and inequality requirements make the EPD issue more feasible. Recent evolutionary algorithms have improved results, especially for complex optimization problems. These methods identify global optimum solutions in non-convex and large solution domains. Recent algorithms search the solution space at random utilising numerous candidate solutions. Evolving algorithms must be able to get answers by exploring the feasible zone rather than researching the complex region. So the optimization process is speedier and consumes less calculation time while still finding the global optimum. The traditional technique fails due to premature convergence.

Nowadays, using the units with multiple fuel options is one of the ways to reduce the generation costs and increase the reliability in modern power plants. In such power plants, the minimum generation cost is achieved via selecting the optimal fuel types. Here, the fuelling conditions and limitations of different fuel types indeed have determinative roles, because they not only can significantly influence the system operation point and optimal generation dispatch, but also they certainly affect the fuel consumption cost and the electricity price, as well. This study aims to develop and solve a novel type of multi‐area economic dispatch with multiple fuel options, which also considers the mathematical model of the fuelling constraints. Thus, the system operating conditions are modelled in a more realistic way in which all the models of the power plants as well as their fuelling systems are well regarded. This consequently yields a more accurate generation dispatch that can indeed reduce the generation costs, increase the fuelling reliability and subsequently increase the entire system reliability in the actual power systems in which the fuelling constraints are essential and determinative.

Optimizing multiple objectives is a common challenge in complex decision-making scenarios, where improving one objective often comes at the expense of another. In power system generation, two critical objectives must be simultaneously minimized: the total generation cost and the total emissions released. This study introduces a multi-objective optimization method developed to address these dual objectives. The proposed approach aims to identify the optimal solution to the economic dispatch problem, balancing the trade-off between minimizing generation costs and reducing environmental impact by lowering total emissions. Heuristic optimization is used to solve this multi-objective combined economic and emission dispatch (MOCEED) problems through the implementation of Multi-Objective Evolutionary Programming (MOEP). In this study, the Weighted Sum Method (WSM) is combined with the Evolutionary Programming (EP) technique to minimize both objectives. The developed approach is validated on the IEEE 30-Bus RTS, which consists of six generators. The effectiveness of the developed technique is evaluated and compared the results against scenarios without optimization. Simulations are performed using MATLAB software, and the results demonstrate that the proposed Multi-Objective Evolutionary Programming (MOEP) successfully identifies the optimal generator outputs, achieving significant reductions in both total generation cost and emissions. These results highlight the method's capability to provide superior solutions for multi-objective economic dispatch problems.

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.