An effectively adaptive selective cuckoo search algorithm for solving three complicated short-term hydrothermal scheduling problems

Highlights

• A new ASCSA method is proposed by applying two new techniques on conventional CSA.

• Complicated constraints are properly handled in hydrothermal scheduling problems.

• The ASCSA reaches a high success rate and fast convergence to the optimal solution.

• The proposed ASCSA is more effective and robust than CSA and many other methods.

Abstract

This paper proposes an effectively adaptive selective cuckoo search algorithm (ASCSA) for solving short-term hydrothermal scheduling problems with available water constraint, reservoir volume constraints, and transmission network constraints. The proposed ASCSA is a newly improved version of the conventional cuckoo search algorithm to enhance the solution quality and reduce the maximum number of iterations based on two new techniques including the new ratio of the difference between the fitness function values and the integration of solutions into one group. The effectiveness of ASCSA has been validated via eight hydrothermal systems, in which the last two systems consisting of the IEEE 30-bus and IEEE 118-bus systems are considered with a set of constraints in the transmission network. To investigate the performance of ASCSA, several algorithms are also implemented in the paper such as conventional cuckoo search algorithm, modified cuckoo search algorithm, particle swarm optimization, global vision of particle swarm optimization with inertia weight, differential evolution, and improved differential evolution. From result comparisons of the test systems, the proposed ASCSA method has obtained lower total costs than other methods implemented for solving the problems. Therefore, the proposed ASCSA is a very efficient and favorable method for solving the considered hydrothermal scheduling problems.

Introduction

A hydrothermal system is composed of both thermal plants and hydropower plants supplying electricity to load via transmission lines. This system has become popular in power systems due to its significant contribution to power source since the renewable energy cannot afford enough power energy to the electric load. Therefore, the cooperation of the hydrothermal system has played an important role in power system operation to keep power system working stably and economically. Although hydropower plants have consumed no fuel cost due to the free water from natural rivers, the hydrothermal system is required to operate under the constraints of hydraulic reservoirs and water available for power generation in a given scheduled time. On the contrary to hydropower plants, the fuel cost for power generation of thermal power plants is the main objective during the operation but the set of constraints taken into account is simpler as only limitations on thermal generation and power balance are included. Therefore, the hydrothermal scheduling (HTS) aims to minimize the electricity generation fuel cost of thermal plants using fossil fuels while all constraints from thermal plants, hydropower plants, and the system must be exactly met [1]. In this problem, the thermal power plant constraints are easy to deal with since only limitations on generations are taken into consideration and other constraints such as fossil fuel constraint, fuel cost for start-up and shutdown process are neglected.

In the optimal scheduling of the fixed-head short-term hydrothermal systems, there are two types of problem with different hydraulic constraints consisting of the available amount of discharge water and reservoir volume limits. The water head of the reservoir in both types is considered to be a constant but the detailed constraints of them are different. Namely, the amount of water used to run the hydro turbines is constrained over the scheduled horizon in the first problem meanwhile the main constraints in the second problem such as reservoir volumes at the beginning and the end of the scheduled horizon and the limits of the reservoir volumes are also taken into account. The detailed synthesis of the solutions implemented in the literature for the two problems is given in Table 1.

As observed from the table, there are two groups of applied methods including conventional deterministic methods and artificial intelligence based methods. All methods in the first group and neural network based methods in the second group such as HNN and ALHN can find a single solution updated in every iteration and they mainly use some deterministic transition rules for approaching the optimum solution while the methods in the last group try to gradually discover the optimal solution among a population of potential searching directions. In the first group, a single path search line based on gradient-based techniques could quickly converge to an optimum solution. However, such deterministic methods and neural network based methods cannot deal with the non-convex problems where the objective functions or constraints of the problems are non-differentiable. Moreover, the conventional methods also suffer difficulties when dealing with large-scale problems with complicated constraints [7]. On the contrary, the meta-heuristic methods initialize a set of solutions at the beginning of the optimal solution search process. The solutions are newly generated in each iteration and the quality of these solutions is evaluated via a fitness function consisting of the objective function that needs to be minimized and penalty amount for constraint violation. Unlike the deterministic ones, many meta-heuristic methods stop the search process mainly based on the predetermined maximum number of iterations and the obtained solutions are capable of satisfying all constraints. The meta-heuristic algorithms are considered more powerful and effective than deterministic ones once they can deal with the problems with complex and non-convex objective function constraints.

For the implementation of GA method on short-term HTS problems, the diversification of the offspring in crossover operation and mutation operation must be performed to guarantee an optimal solution. This behavior is regarded as an advantage of the EP method as the main search operator generating new solutions is fulfilled by the mutation operation. Unlike GA, EP is slow for obtaining a near optimal solution. However, the convergence speed of EP and GA can be improved by adding other techniques. In fact, there have been several improvements applied to the method to enhance their searchability and speed up their convergence process [20]. The SA technique seemed to be better than GS via the result comparison reported in Ref. [29] for the second problem. However, it is a difficult task for a proper selection of control parameters for the SA method. Moreover, this method also suffers slow computation for obtaining the optimal solution. Therefore, this method is limited for implementation in practical problems. The conventional BFA has been successfully applied to several optimization problems. However, there is still a challenge for this method in implementation to large-scale systems [28]. In order to overcome this drawback, an adaptive run-length parameter has been used in the IBFA as the main factor to control its searchability and speed up the convergence.

The conventional cuckoo search algorithm (CCSA) is a meta-heuristic algorithm developed by Yang and Deb in 2009 [31] inspired from the cuckoo bird's reproduction behavior. CCSA was considered effective and robust for optimization problem by testing a set of benchmark functions and comparing with two other algorithms such as GA and PSO. However, there have been many studies, which have identified the disadvantages of this method and brought up several modifications on CCSA. These studies have tried to improve the performance of CCSA by fulfilling different modifications. Namely, a study in Ref. [32] has proposed a modification on global search via Lévy flights by decomposing all solutions into the better group and worse group and the method for generating new solutions for the different groups are also different. Also, a study in Ref. [33] has also focused on the global search improvement and proposed a mechanism to supervise updated step sizes. On the contrary, a study in Ref. [34] has applied NEH heuristic to generate a soliton of whole initial solutions so as to improving solution quality. Besides, such studies had the same point that they did not point out clear weak points of the method. As stated in Ref. [35], CCSA is comprised of three main factors including selection operation, exploration, and exploitation, which have resulted in the potential capacity for searching the global optimal solution for CCSA. The selection operation focuses on choosing a set of potential solutions and finds the best solution among the available solutions whilst the exploration and exploitation respectively aim to seek optimal solutions in the search space [35]. In CCSA, new solutions are generated two times by using Lévy flights and the action of alien eggs discovery in which the former is the global explorative random walk and the latter is the local exploitative random walk [35]. At the end of each iteration, the selection operation performs two stages to determine the current best solution. In these two stages, the first stage performs the comparison for each old and each new solution at the same nest to retain the better one and afterward the second stage carries out the comparison to choose the best solution with the lowest fitness function value among the stored solutions. Obviously, the conventional selection operation has missed several candidate solutions because abandoned solutions at nests can be better than the stored solutions at other nests. To overcome the drawback of CCSA, we propose an effectively adaptive selective cuckoo search algorithm (ASCSA) by constructing two new techniques on CCSA such as selective technique and adaptive technique. The selective technique keeps a set of dominant solutions and abandons solutions with higher fitness function value than the dominant solutions by integrating all old and new solutions into one group, ranking them in the descending order of fitness function value, and choosing a population of leading solutions. In the next iteration, the global explorative random walk will work based on the dominant solutions from ASCSA and the stored solutions from CCSA. Thus, the global search from ASCSA can work more effectively than that from CCSA because the dominant solutions at the end of the previous iteration of ASCSA have higher potential than the stored solutions of CCSA. In CCSA, the local exploitative random walk seeks a new solution to replace an old solution based on an increased value obtained by the difference between two other old solutions. The two points-based increased value will narrow the local search space when the search process comes to the last iterations because most solutions at this time are very close together and the difference between every two solutions are very small. As a result, CCSA may suffer from the local optimal solution. To overcome the restriction, we propose an adaptive technique to enhance the local exploitative random walk to avoid obtaining local optimal solutions. In the proposed technique, two new formulas are set up including a fitness difference ratio and a four points-based increased value. The technique enables the proposed ASCSA to choose a better way for updating new solutions by using either two points-based increased value or four points-based increased value. The improvement enables ASCSA to avoid a local optimum.

In the paper, ASCSA method is first applied for solving three HTS problems with available water constraint and reservoir volume constraints considering power losses in transmission lines and valve-point loading effects on thermal units. The performance of ASCSA has been verified via three HTS problems in which the first problem consists of four systems neglecting reservoir volume constraints, the second problem considers two systems considering the volume reservoir constraints and the third problem consists of the IEEE 30-bus and IEEE 118-bus system neglecting reservoir constraints but considering transmission power network constraints. On the other hand, valve-point loading effects on thermal units represented as a nonconvex fuel cost function are also taken into consideration for some systems in the three problems.

The organization of the remaining of the paper is as follows. The problem formulation is given in Section 2. A review of CCSA is provided in Section 3. ASCSA is proposed in Section 4. The numerical is presented in Section 5. Finally, the conclusion is given.