Non-convex Fuel Cost Functions Research Articles

Enhanced Colliding Bodies Optimization Towards the Solution of Economic Load Dispatch Problem

This paper proposes application of enhanced colliding bodies optimization algorithm (ECBO) for solving the economic load dispatch problem, which aims to get minimum generation cost through economic scheduling of generating unit. The algorithm is a novel efficient optimization algorithm, and its idea is attained from collisions, which are one-dimensional between two or more bodies. Each agent involved in this collision is shaped as a body having defined mass and defined velocity. To analyze the performance of the proposed method, it is tested on four different test systems consisting of 3, 5, 13, and 18 generating units. Both convex and non-convex fuel cost function have been considered. Numerical results have been compared with various well-known algorithms. A significant improvement in the results has been observed. It has been farther noted that an average 5.1% better results were provided by ECBO in terms of generation cost compared to other algorithms. Moreover, the simulation time is also improved. Besides this, Kruskal-Wallis non-parametric test has been performed.

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.

The application of an effective cuckoo search algorithm for optimal scheduling of hydrothermal system considering transmission constraints

This paper presents an implementation of an effective cuckoo search algorithm (ECSA) for solving hydrothermal scheduling (ST-FH-HTS) problems considering transmission power losses, nonconvex fuel cost function of thermal units, and transmission grid constraints such as the voltage of load buses, voltage of generator buses, capacity of transmission lines. The ECSA method has been developed based on the conventional cuckoo search algorithm (CCSA) which is a recently developed meta-heuristic algorithm inspired from the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other birds of other species for solving optimization problems. In the ECSA method, new eggs generated via Levy flights are replaced partially, and the newly generated eggs are then evaluated and ranked at once. Moreover, there is a boundary by the best solution technique proposed for replacing the invalid dimension in order to improve convergence rate and performance. The performance of ECSA has been investigated via comparisons with other methods by testing on eleven systems. In addition, the ECSA and other popular meta-heuristic algorithms such as CCSA, conventional particle swarm optimization, conventional differential evolution, and conventional Bat algorithm have been tested on a large-scale system with 50 thermal units and four hydrounits considering constraints from an IEEE 118-bus transmission line grid. The result comparisons from ECSA with other methods for 12 test systems have revealed that ECSA method is very efficient for solving ST-FH-HTS problems. Therefore, the ECSA can be a favorable method for solving the ST-FH-HTS problems.

Multi-gradient PSO algorithm for optimization of multimodal, discontinuous and non-convex fuel cost function of thermal generating units under various power constraints in smart power grid

Optimization of fuel cost function of large-scale thermal generating units under several constraints in smart power grid is a challenging problem. Because of these constraints, the fuel cost function becomes multimodal, discontinuous and non-convex. Although the global particle swarm optimization with inertia weight (GPSO-w) algorithm is a popular optimization technique, it is not capable of solving such complex problems satisfactory. In this paper, a novel multi-gradient PSO (MG-PSO) algorithm is proposed to solve such a challenging problem. In MG-PSO algorithm, two phases, called Exploration phase and Exploitation phase, are used. In the Exploration phase, the m particles are called Explorers and undergo multiple episodes. In each episode, the Explorers use a different negative gradient to explore new neighbourhood whereas in the Exploitation phase, the m particles are called Exploiters and they use one negative gradient that is less than that of the Exploration phase, to exploit a best neighborhood. This diversity in negative gradients provides a balance between global search and local search. The effectiveness of the MG-PSO algorithm is demonstrated using four (medium and large) power generation systems. Superior performance of the MG-PSO algorithm over several PSO variants in terms of several performance measures has been shown.

Adaptive Cuckoo Search Algorithm for Short-Term Fixed-Head Hydrothermal Scheduling Problem with Reservoir Volume Constraints

This paper present three versions of Cuckoo Search Algorithm (CSA) including conventional Cuckoo Search Algorithm (CSA), modified CSA (MCSA) and adaptive CSA (ACSA) for solving the fixed head short-term hydrothermal scheduling (ST-HTS) problem where the reservoir volume constraints and nonconvex fuel cost function of thermal unit as well as the power losses in transmission line are taken into account. Among the applied methods, ACSA is first developed in the study by performing two modifications on second new solution generation via the action of an alien egg to be abandoned. In the ACSA, all initial solutions or all solutions at the end of the previous iteration are evaluated and sorted into two kinds of solution, good solutions with lower fitness function and bad solutions with higher fitness function. The implementation of the first new solution generation first via Levy flights in the ACSA is carried out similarly to that in MCSA. However, at the second new solution generation the ACSA evaluates the current solutions to choose the best one and use the information of the best one with a random solution to generate the second new solutions via the action of an alien egg to be abandoned. In addition, the probability of an alien egg discovery is considered an adaptive variable, which is set to the largest value at the beginning and decreased as the iteration is increased. Due to the adaptive value of the parameter, the ACSA can search an optimal solution but the trial runs are significantly decreased compared to CSA and MCSA. The performance of the ACSA is validated by testing on two systems and comparing with CSA, MCSA and other existing methods available in the paper.

An efficient cuckoo bird inspired meta-heuristic algorithm for short-term combined economic emission hydrothermal scheduling

Abstract The short-term combined economic emission hydrothermal scheduling is a bi-objective problem minimizing both fuel cost and emission of a hydrothermal system while satisfying thermal, hydro and system constraints. In this paper, an efficient cuckoo bird-inspired meta-heuristic algorithm (CBIA) is first successfully applied for solving the bi-objective hydrothermal scheduling problem with nonlinear objectives and constraints. For implementation of the CBIA to the problem, the bi-objective problem is converted into single objective optimization by using the price penalty factor. The performance of the CBIA has been verified on two test systems with nonconvex fuel cost function of thermal units and a cascaded-reservoir system. The result comparisons between the proposed CBIA with other methods have shown that the CBIA is more efficient than many other methods. Therefore, the proposed CBIA can be a very powerful method for the bi-objective hydrothermal scheduling problems with cascaded reservoir system.

Cuckoo search algorithm for combined heat and power economic dispatch

This paper proposes a cuckoo search algorithm (CSA) for solving for the combined heat and power economic dispatch (CHPED) problem considering valve point loading effects on fuel cost function of pure power generation units and electrical power losses in transmission systems. The main objective of the CHPED problem is to minimize the total fuel cost for producing electricity and heat supplying to a load demand. The proposed CSA method inspired from the reproduction behavior of cuckoo birds has attracted many researchers implementing to engineering optimization problems since it has showed several advantages of few control parameters, high solution quality and fast computational time. The effectiveness and robustness of the proposed CSA have been validated on five different systems including three systems with quadratic fuel cost function of pure power units neglecting transmission losses and two systems with nonconvex fuel cost function of pure power units. The result comparisons between the CSA method and other methods for the test systems have revealed that the CSA method can obtain higher quality solution with faster computational time than many other methods. Therefore, the proposed CSA method can be a very efficient method for solving the CHPED problem.

Solving Short-Term Cascaded Hydrothermal Scheduling Problem Using Modified Cuckoo Search Algorithm

This paper presents a modified cuckoo search algorithm (MCSA) for solving shortterm cascaded hydrothermal scheduling (ST-CHTS) problem. The short-term cascaded hydrothermal scheduling is to determine the optimal operation for thermal plants and a cascaded reservoir system while satisfying all constraints including electrical constraints of both hydro and thermal plants and hydraulic constraints of reservoirs. The MCSA has been developed by modifying the search strategy via Levy flights to improve the performance of the conventional cuckoo search algorithm. The proposed method has been widely and successfully applied to many optimization problems in engineering fields; however, this is first time employed to search for the optimal solution of the ST-CHTS problem. The proposed method has been tested on two systems where thermal plants with nonconvex fuel cost function and a cascaded reservoir system are taken into account. The result comparison from the MCSA compared to other methods reported in the literature has revealed that the proposed MCSA is very efficient for solving the ST-CHTS problem.

비볼록 발전비용함수 경제급전문제의 개선된 밸브지점 최적화 알고리즘

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 for each generator. It then reduces the generation power of generator i with an average power cost of to a valve point power . 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.

The application of one rank cuckoo search algorithm for solving economic load dispatch problems

In this paper, a one rank cuckoo search algorithm (ORCSA) is proposed for solving economic load dispatch (ELD) problems. The main objective of the ELD problem is to minimize total cost of thermal generators while satisfying power balance constraint, prohibited operating zones, ramp rate constraints and operating limits of generators. Moreover, the generating units considered in this paper have different characteristics such as quadratic fuel cost function, nonconvex fuel cost function and multiple fuel options. The proposed ORCSA method has been developed by performing two modifications on the original cuckoo search algorithm (CSA) to improve optimal solution quality and computational time. The first modification is to merge new solution generated from both Lévy flights and replacement a fraction of egg together and to evaluate and rank the solutions at once only. A bound by best solution mechanism has been used in the second modification for properly handling the inequality constraints. The proposed ORCSA method has been tested on different systems with different characteristics of thermal units and constraints. The results obtained by ORCSA have been compared to those from other methods available in the literature and the result comparison has indicated that the ORCSA method can obtain better solution quality than many other methods. Therefore, the proposed ORCSA can be a very effective and efficient method for solving ELD problems.

Enhanced Particle Swarm Optimization for Short-Term Non-Convex Economic Scheduling of Hydrothermal Energy Systems

This paper presents an Enhanced Particle Swarm Optimization (EPSO) to solve short-term hydrothermal scheduling (STHS) problem with non-convex fuel cost function and a variety of operational constraints related to hydro and thermal units. The operators of the conventional PSO are dynamically controlled using exponential functions for better exploration and exploitation of the search space. The overall methodology efficiently regulates the velocity of particles during their flight and results in substantial improvement in the conventional PSO. The effectiveness of the proposed method has been tested for STHS of two standard test generating systems while considering several operational constraints like system power balance constraints, power generation limit constraints, reservoir storage volume limit constraints, water discharge rate limit constraints, water dynamic balance constraints, initial and end reservoir storage volume limit constraints, valve-point loading effect, etc. The application results show that the proposed EPSO method is capable to solve the hard combinatorial constraint optimization problems very efficiently.

Improved Immune Algorithm for Power Economic Dispatch Considering Units with Prohibited Operating Zones and Spinning Reserve

This study proposes an improved immune algorithm (IIA) with multiplier updating (MU) for power economic dispatch (PED) considering units with both of prohibited operating zones (POZ) and spinning reserve. The IIA equipped with an accelerated operation and a migration operation can efficiently search and actively explore solutions. The MU is introduced to handle the equality and inequality constraints of the power system. To show the advantages of the proposed algorithm, three realistic examples are investigated, and computational results of the proposed method are compared with that of previous methods. The proposed approach (IIA-MU) integrates the IIA and the MU, revealing that the proposed approach has the following merits - ease of implementation; applicability to non-convex fuel cost functions; better effectiveness than previous methods; better efficiency than immune algorithm with the MU (IA-MU), and the requirement for only a small population in applying the optimal PED problem of generators with POZ and spinning reserve.

Artificial Immune Algorithm for Practical Power Economic Dispatch Problems

This paper proposes an artificial immune algorithm with a multiplier updating method (AIA-MU) for practical power economic dispatch (PED) considering units with prohibited operating zones (POZ). The AIA equipped with a migration operation can efficiently search and actively explore solutions. The multiplier updating (MU) is introduced to handle the system constraints. To show the advantages of the proposed algorithm, two examples are investigated, and the computational results of the proposed method are compared with that of the previous methods. The proposed approach integrates the AIA and the MU, revealing that the proposed approach has the following merits - ease of implementation; applicability to non-convex fuel cost functions of the POZ; better effectiveness than previous methods, and the requirement for only a small population in applying the optimal PED problem of generators with POZ.

Improved Particle Swarm Optimization for Power Economic Dispatch with Prohibited Operating Zones

This study proposes an improved particle swarm optimization (IPSO) for power economic dispatch (PED) considering units with prohibited operating zones (POZ). The IPSO equipped with an accelerated operation and a migration operation can efficiently search and actively explore solutions. The multiplier updating (MU) is introduced to handle the equality and inequality constraints of the power system. To show the advantages of the proposed algorithm, one realistic example is investigated, and the computational results of the proposed method are compared with that of previous methods. The proposed approach integrates the IPSO and the MU, revealing that the proposed approach has the following merits ease of implementation; applicability to non-convex fuel cost functions; better effectiveness than previous methods; better efficiency than particle swarm optimization with the MU (PSO-MU), and the requirement for only a small population in applying the optimal PED problem of generators with POZ.

Power Economic Dispatch Problems Considering Prohibited Operating Zones and Spinning Reserve: A Hybrid Differential Evolution

This paper proposes a hybrid differential evolution (HDE) for power economic dispatch (PED) considering units with prohibited operating zones (POZ) and spinning reserve. The HDE equipped with an accelerated operation and a migration operation can efficiently search and actively explore solutions. The multiplier updating (MU) is introduced to handle the equality and inequality constraints of the system. To show the advantages of the proposed algorithm, one example is investigated, and the computational results of the proposed method are compared with that of the previous methods. The proposed approach integrates the HDE and the MU, revealing that the proposed approach has the following merits - ease of implementation; applicability to non-convex fuel cost functions; better effectiveness than previous methods; better efficiency than differential evolution with the MU (DE-MU), and the requirement for only a small population in applying the optimal PED problem of generators with POZ and spinning reserve.

A novel stochastic search method for the solution of economic dispatch problems with non-convex fuel cost functions

This paper develops a Novel Stochastic Search (NSS) method for the solution of economic dispatch problems with non-convex fuel cost functions. The NSS solution procedure consists of three steps, namely Direct Search (DS), Goal Neighborhood Approximation (GNA) and Marginal Cost Dispatch (MCD). The DS step identifies a set of feasible solutions in accordance with prescribed equality and inequality constraints. The GNA step processes those feasible solutions to identify an appropriate direction for searching the global optimal solution. Finally, in the MCD step, the marginal cost of each generating unit is regulated in order to establish the global optimal solution. The proposed NSS scheme is applied to solve three examples systems of increasing complexity. The results are compared to those obtained using the conventional Simulated Annealing (SA), Genetic Algorithm (GA), and Evolutionary Programming (EP) methods. The results demonstrate that the NSS method provides a fast, robust and highly effective scheme for the solution of economic dispatch.

Hybrid DE–SQP algorithm for non-convex short term hydrothermal scheduling problem

This paper proposes a hybrid method combining differential evolution (DE) and sequential quadratic programming (SQP) for solving short term hydrothermal scheduling problem with non-convex fuel cost function. In this paper, differential evolution (DE) is used as a global optimizer and sequential quadratic programming (SQP) method as a local optimizer to fine tune the solution. The proposed method has been tested on a multichain cascaded reservoir with an equivalent thermal test system and the simulation results are compared with existing methods reported in literatures. From the results, it clearly shows that the proposed method is giving better quality solutions than existing methods.

Nonconvex Economic Dispatch with Heuristic Load Patterns Using Improved Fast Evolutionary Program

This article presents an attempt to explore the feasibility of an improved fast evolutionary program (IFEP) search technique to solve extremely challenging nonconvex economic load dispatch (ELD) problem with transmission losses involving variations of consumer load patterns. The effectiveness of the proposed approach has been tested successfully on the standard 6-bus system, IEEE 14-bus system, and the IEEE-30 bus system with several heuristic load patterns. The numerical results reveal that the proposed approach can provide better optimal dispatch solutions than those of classical lambda-iteration method (CLIM). Aside from this, the computation time is reasonable even with nonconvex fuel cost functions where the gradient based search methods are inapplicable.