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본 논문은 운전금지영역을 가진 이차 볼록 발전비용 함수를 적용하는 경제급전의 최적화 문제에 대한 결정론적 최적화 알고리즘을 제안하였다. 제안된 알고리즘은 운전금지구역을 가진 발전기는 운전금지구역을 벗어나도록 분할하고, 초기치 P _{i} larrow P _{i} ^{max}에 대해 발전단가가 큰 순서대로 발전량을 감소시키고, {} _{max} LEFT { F(P _{i} )-F(P _{i} - beta ) RIGHT } > _{min} LEFT { F(P _{j} + beta ) RIGHT LEFT. -F(P _{j} ) RIGHT } ,`i != j,`beta =1.0,`0.1,`0.01,`0.001에 대해 P _{i} larrow `P _{i} - beta ,`P _{j} ` larrow P _{j} + beta 의 교환 최적화 과정을 수행하였다. 제안된 방법을 15-발전기의 3가지 사례에 적용한 결과 간단하면서도 항상 동일한 결과로 휴리스틱 알고리즘들에 비해 최적의 결과를 나타내었다.

This paper proposes a deterministic optimization algorithm to solve economic load dispatch problem with quadratic convex fuel cost function. The proposed algorithm primarily partitions a generator with prohibited zones into multiple generators so as to place them afield the prohibited zone. It then sets initial values to P _{i} larrow P _{i} ^{max} and reduces power generation costs of those incurring the maximum unit power cost. It finally employs a swap optimization process of P _{i} larrow `P _{i} - beta ,`P _{j} ` larrow P _{j} + beta where {} _{max} LEFT { F(P _{i} )-F(P _{i} - beta ) RIGHT } > _{min} LEFT { F(P _{j} + beta ) RIGHT .LEFT. -F(P _{j} ) RIGHT } ,`i != j,`beta =1.0,`0.1,`0.01,`0.001 . When applied to 3 different 15-generator cases, the proposed algorithm has consistently yielded optimized results compared to those of heuristic algorithms.

경제급전 최적화 문제를 해결하는 결정론적인 알고리즘에 존재하지 않아 지금까지는 비결정론적인 휴리스틱 알고리즘들이 제안되고 있다. 본 논문은 균형과 교환 방법을 도입하여 경제급전의 최적화 문제를 풀 수 있는 알고리즘을 제안하였다. 제안된 알고리즘은 초기치에 대해 성인걸음수와 아기걸음 수별로 발전량을 감소시켜 SIGMA P _{i} =P _{d}로 균형을 맞추고, 이 때 최소 발전비용을 가진 방법을 선택한다. 다음으로 선택된 방법에 대해 성인걸음-아기걸음 교환과 거인걸음 교환 방법으로 최적화한 값을 구하여 최소값 방법을 선택한다. 마지막으로 선택된 방법에 대해 P _{i} ± beta ,`( beta =0.1,`0.01,`0.001.`0.0001)의 교환을 수행하였다. 경제급전 문제의 시험사례로 빈번히 활용되고 있는 3개 데이터에 대해 제안된 알고리즘을 적용한 결과 2개 데이터에서는 성능을 향상시켰으며, 1개 데이터는 기존의 최적해와 동일한 결과를 얻었다. 제안된 알고리즘은 항상 동일한 결과를 얻을 수 있고, 모든 데이터에 적합하므로 경제급전 최적화 알고리즘으로 실제 적용이 가능하다.

In the absence of a deterministic algorithm for economic load dispatch optimization problem (ELDOP), existing algorithms proposed as solutions are inevitably non-deterministic heuristic algorithms. This paper, therefore, proposes a balance-and-swap algorithm to solve an ELDOP. Firstly, it balances the initial value to SIGMA P _{i} =P _{d} by subsequently reducing power generation for each adult-step and baby-step and selects the minimum cost-generating method. Subsequently, it selects afresh the minimum cost-generating method after an optimization of the previously selected value with adult-step baby-step swap and giant-step swap methods. Finally, we perform the P _{i} ± beta ,`( beta =0.1,`0.01,`0.001.`0.0001) swap. When applied to the 3 most prevalently used economic load dispatch problem data, the proposed algorithm has obtained improved results for two and a result identical to the existing one for the rest. This algorithm thus could be applied to ELDOP for it has proven to consistently yield identical results and to be applicable to all types of data.

경제급전 최적화 문제를 해결하는 결정론적인 알고리즘에 존재하지 않아 지금까지는 비결정론적인 휴리스틱 알고리즘들이 제안되고 있다. 이와 더불어 실시간 급전문제에 대한 연구는 거의 없는 실정이다. 본 논문은 발전정지 개 념을 도입하여 실시간 급전의 최적화 문제를 풀 수 있는 알고리즘을 제안하였다. 제안된 알고리즘은 단위 발전량당 최 대 비용이 소요되는 발전기는 발전을 중지시키는 기준을 적용하였다. 본 논문에서 제안한 발전정지 기준은 발전비용함 수에서 밸브효과에 따른 비선형 절대치 함수를 제외한 2차 함수만을 대상으로 하였다. 경제급전 문제의 시험사례로 빈 번히 활용되고 있는 데이터에 대해 제안된 알고리즘을 적용한 결과 기존 알고리즘들의 해를 크게 감소시킬 수 있었다.

In facing the lack of a deterministic algorithm for economic load dispatch optimization problem, only non-deterministic heuristic algorithms have been suggested. Worse still, there is a near deficiency of research devoted to real-time load dispatch optimization algorithm. In this paper, therefore, I devise a shut-off and swap algorithm to solve real-time load dispatch optimization problem. With this algorithm in place, generators with maximum cost-per-unit generation power are to be shut off. The proposed shut-off criteria use only quadratic function in power generation cost function without valve effect nonlinear absolute function. When applied to the most prevalent economic load dispatch benchmark data, the proposed algorithm is proven to largely reduce the power cost of known algorithms.

Cuckoo Search Algorithm (CSA), a new meta-heuristic algorithm based on natural phenomenon of Cuckoo species and Lévy flights random walk has been widely and successfully applied to several optimization problems so far. In the paper two modified versions of CSA, where new solutions are generated using two distributions including Gaussian and Cauchy distributions in addition to imposing bound by best solutions mechanism are proposed for solving economic load dispatch (ELD) problem with multiple fuel options. The advantages of CSA with Gaussian distribution (CSA-Gauss) and CSA with Cauchy distribution (CSA-Cauchy) over CSA with Lévy distribution and other meta-heuristic are fewer parameters. The proposed CSA methods are tested on two systems with several load cases and obtained results are compared to other methods. The result comparisons have shown that the proposed methods are highly effective for solving ELD problem with multiple fuel options and/nor valve point effect.

This paper presents a method to solve the optimal generation and dispatch of electrical power with multiple fuel options at different power levels. In general, the cost function for each generator is considered by a single quadratic function of power, for the optimal generation and dispatch problem. However, it is more realistic to represent the generation cost function for fossil fired plants by non smooth cost function i.e. the total generation cost function has non differentiable points such as segmented piece-wise quadratic function and valve point loading. Some generating units utilize multiple fuels sources (such as gas, coal, oil etc) are faced with the problem of determining which is the most economical fuel to be selected. This problem has been solved by Hota & Das [17] considering equal incremental cost. In this method the program has to be run for a number of times for fuel combinations for different units and then to select the best one out of all combinations. This is a complicated process. Also if some other type of nonlinear cost function is considered along with this, the method will not be applicable. In view of the above mentioned problems of economic load dispatch (ELD) of multiple fuel generation, a more general heuristic method known as Particle Swarm Optimization (PSO) is considered in this paper to solve the problem in which either type of nonlinearity in cost function can be considered. For solution of this problem dynamic PSO is proposed. It is dynamic in the sense that the velocity bound or limit is updated in each iteration. This proposed method is applicable to the generating units that can use multiple fuels through valve at different generation levels as well as other problems which results in multiple intersecting continuous and discontinuous cost curves for any unit. The advantage of the method is that the method is applicable to both continuous and discontinuous cost curves of systems. Also it does not require selection of unit and type of fuel to be used after computation for a number of combination which is a cumbersome task as used in ref [17]. The complete method is explained and validated by taking an example. The simulation is carried out using MATLAB software. It has been shown that the method is simple, direct, and practical. It can be used for real time implementation and operation.

In this paper, a Cuckoo Search Algorithm (CSA) is proposed for solving environmental economic load dispatch (EELD) problem with quadratic fuel function. Cuckoo Search is a new meta-heuristic algorithm inspired from the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds of other species for solving optimization problems with promising results. However, Cuckoo Search has not been applied to EELD problem so far. Therefore, the paper presents application of CSA to the problem. The effectiveness of the proposed method is tested on several cases of dispatch and loads. The obtained result including fuel cost, emission and computation time from CSA are compared to those from other method reported in the paper. The comparison result has indicated that the proposed CSA is a very efficient method for solving EELD problem.

This paper presents an Adaptive Cuckoo Search Algorithm (ACSA) for solving economic load dispatch problem where thermal units with multiple fuel options and valve point loading effect are taken into account. ACSA is first developed by improving the optimal solution search ability of conventional Cuckoo search algorithm. In ACSA, the initial eggs are evaluated and classified into two groups including good group and bad group. The updated step size in ACSA at the first new solution generation via Lévy is adaptive at each iteration and therefore the selection of the parameter is not an issue in the paper. The proposed ACSA method is tested on a ten-unit system considering multiple fuel options and valve point loading effect with different load cases. The comparisons of obtained results among the proposed method with others reported in the paper have indicated that ACSA is efficient for applying to the problem.

The optimal generation dispatch problem with only one fuel option for each generating unit has been solved for many recent years. However, it is more realistic to represent the fuel cost function for each fossil fired plant as a segmented piece-wise quadratic functions. This is because of development of technology in thermal plants to reach maximum fuel save. Those units are faced with the difficulty of determining which the most economical fuel to burn is. This paper presents two effective methods for solving economic load dispatch problem with multiple fuel options. An advantage of the methods is to formulate Lagrange mathematical function easily based on the Lagrange multiplier theory. The proposed methods are tested on one test system consisting of ten generating units with various load demands and compared to other methods. The simulation results show that the methods are very efficient for the optimal generation dispatch problem with multiple fuel options

This paper applies a novel hybrid rough set theory-particle swarm optimizer technique, namely rough particle swarm optimization (RPSO) algorithm, for solving non-convex economic load dispatch (NELD) problem. The RPSO algorithm is based on the notion of rough patterns that uses rough values defined with upper and lower intervals in which represent a set of values. This RPSO method is suggested to deal with the practical constraints such as valve point loading effect, generation limitation, ramp rate limits and prohibited operating zones in the NELD problems. Simulations were performed on four different power systems with 3, 6, 15 and 40 generating units and the results are compared with classical PSO and crazy PSO algorithms. The results of this study reveal that the proposed approach is able to find appreciable NELD solutions than those of previous algorithms.

In this paper, a Hopfield Lagrange network (HLN) is proposed for solving economic emission load dispatch (EELD) problem with multiple fuel options (MFO). Economic load dispatch (ELD) problem with MFO has been solved for recent years. However, it is more realistic to add CO2 emission to objective of ELD problem because generating units not only use fuels but also release emissions to the air. Consequently, ELD problem becomes EELD problem. HLN is a combination of Lagrange function and continuous Hopfield neural network where the Lagrange function is directly used as the energy function for the continuous Hopfield neural network. By using equivalent cost function and HLN, the paper proposed an effective method to solve EELD problem with MFO. The proposed method is tested on one test system consisting of ten generating units with various load demands and compared to other methods. In addition, the best compromise from the set of obtained solutions is found and compared to this from lamda-iteration (LI) method. The result comparisons have indicated that the proposed method is a highly effective method.

This paper presents a Bat Algorithm (BA) for solving economic emission dispatch (EELD) problem with quadratic fuel function. The BA is a new meta-heuristic algorithm which is a powerful optimal solution search algorithm owing easily selected control parameters and high successful rate as well as high ability for dealing with complex constraints. In addition to minimizing electricity generation fuel cost, emission released into the air from thermal plants is also another main objective needs to be minimized. In order to test the performance of the proposed BA one system with two load cases is employed. The obtained result by the BA compared to that from other methods has revealed that the proposed BA is a very promising meta-heuristic algorithm for solving economic emission load dispatch problem.

This paper proposes an Adaptive Cuckoo Search Algorithm (ACSA) for solving economic emission dispatch (EELD) problem with quadratic fuel function. The ACSA is developed by performing two adaptive updated step size parameters on conventional CSA in aim to enhance the convergence speed and quality solution of the conventional CSA. In addition to minimizing electricity generation fuel cost, emission released into the air from thermal plants is also another main objective needs to be minimized. In order to test the performance of the proposed ACSA two systems including a three unit system with one load case and a six unit system with three load cases are employed. The obtained result by the ACSA compared to that from other methods has revealed that the proposed ACSA is a very promising meata-heuristic algorithm for solving economic emission load dispatch problem.

본 논문은 경제급전 최적화 문제에 개별 발전회사별로 독립적으로 경제급전을 수행하는 방법에 비해 중앙에서 통합하여 경제급전 최적화를 수행하는 경우가 보다 경제적임을 보였다. 이 경우에 적용된 알고리즘으로 밸브지점으로 발전량을 조절하는 균형방법을 수행한 후 발전량을 감소시킬 때의 비용 감소분과 증가시킬 때의 비용 증가분 차이로 발전량을 상호 교환하는 방법으로 최적화를 수행하였다. 10대, 13대과 40대-발전기를 독립적으로 운영하는 경우와 통합된 63대-발전기를 경제급전하는 경우를 비교한 결과 통합운영 방법이 독립적 경제급전에 비해 발전비용을 획기적으로 감소시켜 경제적인 이익을 극대화 시킬 수 있음을 보였다.

This paper demonstrates that centralized economic load dispatch optimization is much more economical than independent optimization carried out by individual power generating companies. The algorithm applied here optimizes by balancing the generation power at the valve-point, then readjusting generation power by comparing incremental operating cost incurred by marginal increase in the generation power and decremental operating cost likewise incurred by marginal decrease in the generation power. Upon comparing 3 individual optimization cases of 10, 13, and 40 generators respectively with centralized optimization of 63 generators, centralized operation for economic load dispatch optimization has proven to maximize economic benefits by markedly reducing operation costs of individual optimization.

본 논문은 경제급전 최적화 문제에 균형-교환 방법을 제안하였다. 제안된 알고리즘은 모든 발전기를 가능한 한 밸브지점으로 운영한다고 가정한다. 초기치로 최대 발전량 로 설정하고, 각 발전기의 밸브지점 까지 발전량을 감소시켰을 때의 평균 발전단가 가 최대가 되는 발전기 의 발전량을 밸브지점 발전단가 로 감소시켰으며, 이면 의 발전기 발전량을 로 감소시켜 의 균형을 맞추었다. 다음으로, 의 범위에 대해 ”-10“ 간격으로 감소시키는 성인걸음법으로, 범위에 대해서는 ”-1“의 아기걸음법으로, 에 대한 이면 , 로 발전량을 교환하는 방법으로 최적화를 수행하였다. 다음으로 에 대해 미세한 교환을 수행하였다. 동적 경제급전 문제의 시험사례에 제안된 알고리즘을 적용한 결과 기존의 휴리스틱 알고리즘 최적화 발전비용을 크게 감소시켜 경제적인 이익을 극대화 시켰다.

This paper proposes a balance-swap method for the dynamic economic load dispatch problem. Based on the premise that all generators shall be operated at valve-points, the proposed algorithm initially sets the maximum generation power at . As for generator with , which is the maximum operating cost produced when the generation power of each generator is reduced to the valve-point , the algorithm reduces ’s generation power down to , the valve-point operating cost. When , it reduces the generation power of a generator with of to so as to restore the equilibrium . The algorithm subsequently optimizes by employing an adult-step method in which power in the range of is reduced by 10; a baby step method in which power in the range of is reduced by 1; and a swap method for of , in which power is swapped to , . It finally executes minute swap process for . When applied to various experimental cases of the dynamic economic load dispatch problems, the proposed algorithm has proved to maximize economic benefits by significantly reducing the optimal operating cost of the extant Heuristic algorithm.

본 논문은 복잡한 비평활 발전비용함수를 가진 경제급전의 최적화 문제를 풀기 위해 단순히 선형 근사함수를 이용하는 방법을 제안하였다. 제안된 알고리즘은 비평활 발전비용 함수를 선형으로 근사시키고, 요구량이 현재의 발전 량을 초과하는 경우 발전단가가 비싼 발전기의 가동을 중지시키고, 발전단가가 보다 큰 발전기의 발전량을 감소시켜 요구량과 발전량의 균형을 맞추는 개념을 도입하였다. 경제급전 문제의 시험사례로 빈번히 활용되고 있는 데이터에 대 해 제안된 알고리즘을 적용한 결과 기존의 휴리스틱 알고리즘의 최적화 해를 획기적으로 감소시킬 수 있었으며, 현재 실무적으로 적용되고 있는 2차 평활함수 근사법과 유사한 결과를 얻었다.

This paper proposes a simple linear function approximation method to solve an economic load dispatch problem with complex non-smooth generating cost function. This algorithm approximates a non-smooth power cost function to a linear approximate function and subsequently shuts down a generator with the highest operating cost and reduces the power of generator with more generating cost in order to balance the generating power and demands. When applied to the most prevalent benchmark economic load dispatch cases, the proposed algorithm is found to dramatically reduce the power cost than does heuristic algorithm. Moreover, it has successfully obtained results similar to those obtained through a quadratic approximate function method.

본 논문은 이차 발전비용 함수를 적용하는 경제급전의 최적화 문제에 대한 균형-교환 최적화 알고리즘을 제안하였다. 제안된 알고리즘은 초기치 , 에 대해 일 때까지 , 인 발전기 의 출력량을 로 균형과정을 수행하고, 교환과정은 에 대해 로 수행하였다. 제안된 방법을 15, 20과 38-발전기 사례에 적용한 결과 간단하면서도 항상 동일한 결과로 가장 좋은 결과를 나타내었다. 또한, 73-발전기를 통합하여 경제급전을 수행한 결과 독립적으로 운영하는 경우에 비해 발전비용을 현저히 절약할 수 있음을 보였다.

In this paper, I devise a balance-swap optimization (BSO) algorithm to solve economic load dispatch with a quadratic fuel cost function. This algorithm firstly sets initial values to , and subsequently entails two major processes: a balance process whereby a generator’s power of , is balanced by until ; and a swap process whereby is set at . When applied to 15, 20, and 38-generators benchmark data, this simple algorithm has proven to consistently yield the best possible results. Moreover, this algorithm has dramatically reduced the costs for a centralized operation of 73-generators – a sum of the three benchmark cases – which could otherwise have been impossible for independent operations.

비 볼록 발전비용함수에 대한 최적화 문제는 다항시간으로 해를 구하는 알고리즘이 알려져 있지 않아 전기 분야에서는 부득이 2차 함수만을 사용하고 있다. 본 논문은 비 볼록 발전비용함수의 경제급전 최적화 문제에 대한 밸브지점 최적화 알고리즘을 제안하였다. 제안된 알고리즘은 초기 치로 최대 발전량 Pi ←Pi max로 설정하고, 평균 발전단가가 max Ci 인 발전기 i의 발전량을 밸브지점 Pik로 감소시키는 방법을 적용하였다. 제안된 알고리즘을 13과 40-발전기 데이터에 적용한 결과 기존의 휴리스틱 알고리즘보다 좋은 성능을 보였다. 따라서 비 볼록 발전비용함수의 경제급전 문제 최적 해는 각 발전기의 밸브지점 발전량으로 수렴함을 보였다.

There is no polynomial-time algorithm that can be obtain the optimal solution for economic load dispatch problem with non-convex fuel cost functions. Therefore, electrical field uses quadratic fuel cost function unavoidably. This paper proposes a valve-point optimization (VPO) algorithm for economic load dispatch problem with non-convex fuel cost functions. This algorithm sets the initial values to maximum powers Pi ←Pi max for each generator. It then reduces the generation power of generator with an average power cost of max Ci to a valve point power Pik. The proposed algorithm has been found to perform better than the extant heuristic methods when applied to 13 and 40-generator benchmark data. This paper consequently proves that the optimal solution to economic load dispatch problem with non-convex fuel cost functions converges to the valve-point power of each generator.

This paper presents a new algorithm called Quantum-behaved Electromagnetism-like Mechanism Algorithm which is used to solve economic load dispatch of power system. Electromagnetism-like mechanism algorithm simulates attraction and repulsion mechanism for particles in the electromagnetic field. Every solution is a charged particle, and it move to optimum solution according to certain criteria. Quantum-behaved electromagnetism-like mechanism algorithm merges quantum computing theory with electromagnetism-like mechanism algorithm. Superposition characteristic of quantum methodology can make a single particle present several states, and the characteristic potentially increases population diversity. Probability representation of quantum methodology is to make particle state be presented according to a certain probability. And the quantum rotation gates are used to realize update operation of particles. The algorithm is tested for 13-generator system and 40-generator system, which validates it can effectively solve economic load dispatch problem. Through performance comparison, it is obvious the solution is superior to other optimization algorithm.

This paper proposes a Modified Particle Swarm Optimization with Time Varying Acceleration Coefficients (MPSO-TVAC) for solving economic load dispatch (ELD) problem. Due to prohibited operating zones (POZ) and ramp rate limits of the practical generators, the ELD problems become nonlinear and nonconvex optimization problem. Furthermore, the ELD problem may be more complicated if transmission losses are considered. Particle swarm optimization (PSO) is one of the famous heuristic methods for solving nonconvex problems. However, this method may suffer to trap at local minima especially for multimodal problem. To improve the solution quality and robustness of PSO algorithm, a new best neighbour particle called 'rbest' is proposed. The rbest provides extra information for each particle that is randomly selected from other best particles in order to diversify the movement of particle and avoid premature convergence. The effectiveness of MPSO-TVAC algorithm is tested on different power systems with POZ, ramp-rate limits and transmission loss constraints. To validate the performances of the proposed algorithm, comparative studies have been carried out in terms of convergence characteristic, solution quality, computation time and robustness. Simulation results found that the proposed MPSO-TVAC algorithm has good solution quality and more robust than other methods reported in previous work.

Optimal Power dispatch is the short-term decision of the optimal output of a number of power generation facilities, to meet the system demand, with the objective of Power dispatching at the lowest possible cost, subject to transmission lines power loss and operational constraints. The operational constraint includes power balance constraint, generator limit constraint, and emission dispatch constraint and valve point effects. In this paper, Opposition based Differential Evolution Algorithm (ODEA) has been proposed to handle the objective function and the operational constraints simultaneously. Furthermore, the valve point loading effects and transmission lines power loss are also considered for the efficient and effective Power dispatch. The ODEA has unique features such as self tuning of its control parameters, self acceleration and migration for searching. As a result, it requires very minimum executions compared with other searching strategies. The effectiveness of the algorithm has been validated through four standard test cases and compared with previous studies. The proposed method out performs the previous methods.