Combined heat and power economic dispatch using integrated civilized swarm optimization and Powell’s pattern search method
Graphical abstract
PSEUDO code of proposed technique.
Introduction
Society demands adequate and secure electricity at the cheapest possible price, so utility planners are trying to upgrade their operating strategies to improve conversion efficiency. Still, the conversion efficiency of primary fossil fuels to electricity is not more than 50–60%. In conventional thermal generating units, the significant fraction of heat energy is not converted into electric power, hence such heat produced can be used for industrial purpose. It is evident that the monetary value of the use of waste heat can enhance the economics of the thermal unit, increases fuel efficiency, and decreases the pollutant emission [1]. In a combined heat and power cogeneration system, simultaneous production of heat, as well as electric power, is possible from one fuel source [2]. The combined heat and power (CHP) generation are an emerging trend to improve conversion efficiency [3], [4]. Therefore, CHP generating units are widely used nowadays with proper scheduling for an economic operation.
Many approaches have been carried out from time to time to obtain an optimum solution for the CHP dispatch problem. Initially, traditional methods like dual and quadratic programming [5], gradient descent approaches [6], Lagrange relaxation technique [6] and variants of dynamic programming [7] were followed. These techniques are deterministic in nature and make a number of assumptions make the problem more tractable and lead to a local optimum solution. Jubril et al. [8] have applied semi-definite programming approach for solving CHP dispatch problem. Rong and Lahdelma [9] presented branch and bound algorithm for non-convex single-period CHP problem. Abdolmohammadi and Kazemi [10] introduced benders decomposition technique to solve CHP dispatch problem by decomposing the problem into a master and sub-problems. Dieu and Ogsakul [11] have proposed augmented Lagrange–Hopfield network for the CHP dispatch problem. Chen et al. [12] have applied modified direct search technique to solve CHP dispatch problem. Kim and Edgar [13] have presented the application of mixed-integer nonlinear programming approach for scheduling of a CHP dispatch problem. Hosseini et al. [14] have presented two level algorithms based on Lagrangian multipliers for the CHP scheduling problem.
The global search techniques have multi-point search strategies and having a number of exclusive advantages. Researchers have applied various global search techniques to solve CHP dispatch problem, such as, differential evolution (DE) [15], modified PSO [16], time varying acceleration coefficients PSO (TVAC-PSO) [17], bee colony optimization (BCO) algorithm [18], teaching learning-based optimization (TLBO) [19], group search optimization (GSO) [20], [21], opposition based GSO (OGSO) [22], with various degrees of success. Recently, researchers have explored some other optimization techniques. The gravitational search algorithm (GSA) has been applied to solve CHP dispatch problem [23]. Ghorbani [2] has applied exchanged market algorithm to solve CHP economic dispatch problem. Haghrah et al. [24] have applied real coded genetic algorithm with improved Mühlenbein mutation to search optimum solution for the CHP dispatch problem. Cuckoo search algorithm has been applied to solve CHP dispatch problem [1].
Each global search technique cannot surpass other optimization algorithms in all prospects, particularly when the problem is multi-dimensional and multi-modal. Recently, researchers have proposed various optimization techniques to solve numerical optimization problems. Wu [25] has proposed a population-based across neighborhood search (ANS) technique. In ANS technique, a collection of superior solutions found by individuals is maintained and updated dynamically. Javidy et al. [26] have proposed an optimization algorithm inspired by the ions motion in nature. The proposed algorithm mimics the attraction and repulsion of anions and captions to perform search. Mirjalili [27] has proposed a nature-inspired ant lion optimizer (ALO) optimization algorithm. The ALO algorithm mimics the hunting mechanism of antlions in nature. Mirjalili [28] has proposed a nature-inspired paradigm called Moth-Flame optimization (MFO) algorithm, inspired by the navigation method of moths in nature. Mirjalili and Lewis [29] have proposed an optimization algorithm, which mimics the social behavior of humpback whales, called a whale optimization algorithm. A population based algorithm called Sine Cosine algorithm has been proposed by Mirjalili [30]. In the proposed algorithm, the search is performed by mathematical model based on sine and cosine functions. Bansal et al. [31] have proposed an optimization algorithm by modeling the foraging behavior of spider monkeys called a spider monkey optimization algorithm. Thus, for large-scale systems, global search approaches independently may not obtain the solution satisfactorily. For complete justification to the problem, pros and cons of different methods are coordinated in a suitable manner, making the implementing strategy to be a meta-heuristic integrated algorithm. Selvakumar and Thanushkodi [32] introduced one such integrated technique, civilized swarm optimization (CSO). Civilized swarm optimization is an integration of PSO [33] and society civilization algorithm (SCA) [34] having different search procedures. The SCA carries out global search and PSO is responsible for performing local search operation. Society civilization algorithm is based on the communication terminology within the civilization. On the other hand, PSO brings the importance of personal best experiences and compare it with the best experience of other individuals in the swarm. Hence, the performance of CSO technique is enhanced by exploiting good qualities of SCA and PSO in a meaningful manner. The CSO has been successfully applied to economic load scheduling (ELD) [32] and multi-objective short-term hydrothermal scheduling problem [35].
Currently, optimization researchers are actively working to develop an algorithm, which is able to achieve a global optimal solution with high speed, better efficiency, and reliable convergence. In this attempt, Harman and McMinn [36] have illustrated results from a large empirical study that compares the behavior of both global and local search based optimization on real world problems, and suggested that a hybrid global-local search may be appropriate. Various researchers have proposed the appropriate integration of global and local search techniques to maintain the balance between exploration and exploitation capabilities of the algorithm to achieve good performance. Jia et al. [37] have proposed an effective memetic DE technique that utilizes a chaotic local search with a shrinking strategy. Piotrowski [38] has proposed memetic DE algorithm with global and local neighborhood-based mutation operators to solve numerical optimization problems. Kim and Liou [39] have developed an adaptive local search method for hybrid evolutionary multiobjective algorithms. Simon et al. [40] have proposed linearized version of biogeography-based optimization technique combined with periodic re-initialization and local search operators to search global optimal solution. Arab and Alfi [41] have proposed a memetic algorithm with a local search operator, namely, adaptive gradient descent, to improve accuracy and convergence speed simultaneously of the algorithm. Bao et al. [42] have investigated the behavior of memetic algorithm based on PSO and pattern search technique for support vector machine parameter optimization. Wu et al. [43] have incorporated local search techniques with superior solution guided PSO to improve the search capability of the algorithm. Palar et al. [44] have studied the impact of various local search methodologies for the multi-objective memetic algorithm.
Several integrated techniques are successfully applied to solve power system dispatch problems. Victoire and Jeyakumar [45] have applied an integrated technique based on PSO and local search technique, namely, sequential quadratic programming (SQP) to solve ELD problem. Basu [46] has applied integrated BCO-SQP technique for dynamic economic load dispatch (DELD) problem. Chang [47] has presented an approach of combined PSO and SQP to optimize the planning of large-scale passive harmonic filters. The DELD problem has been solved by a hybrid technique of evolutionary programming (EP) with SQP method [48], [49]. Titus and Jeyakumar [50] have presented hybrid EP, PSO and SQP methods to solve the DELD problem. Aydin and Özyön [51] have applied incremental artificial bee colony with a local search method to solve ELD problem. Narang et al. [52], [53] have integrated predator-prey optimization technique with local search technique, namely, Powell’s pattern search (PPS) method for the hydrothermal generation scheduling problem.
The CSO technique has a good ability to explore the search area effectively, but like other global search techniques, its exploitation ability is not competitive as compared to local search techniques. To improve the exploitation capability of the search algorithm, the CSO technique may be integrated with local search technique. The local search PPS method has been proven its ability to search nearby optimal solution in quick time. It is based on conjugate search direction and does not require the gradient or Hessian matrix of the function to be optimized. In the PPS method, search point is generated according to a pattern and accepts the point, which appears as an improvement over the prior search point [54]. The PPS method speeds up the search process and helps to avoid any possible stagnation of a local solution.
The intent of research work is to propose an integrated technique that embeds global and local search optimization techniques to solve CHP dispatch problem. The CSO is taken as a global search technique to ensure the search capability of the entire search space and the PPS method has been undertaken as local search technique, to improve the solution quality and avoids the stagnation. The proposed technique has been applied to solve CHP dispatch problem, considering the valve-point loading effect, transmission losses, and prohibited operating zones (POZs) constraint. To authenticate the applicability of the proposed technique, five CHP test systems have been undertaken which are of small, medium, and large size. Statistical analysis is also presented to validate the results obtained with proposed technique.
The paper is summarized in eight sections. Section 2 elaborates the formulation of the CHP dispatch problem having multiple interdependent constraints. Sections 3 and 4 presents CSO and PPS techniques, respectively. The constraint handling strategies have been discussed in Section 5. Section 6 elaborates the solution methodology. Test systems and results are presented in Section 7. Section 8 outlines the conclusions.
Section snippets
Combined heat and power dispatch problem
The objective of the CHP dispatch problem is to determine the unit power generation and heat production such that the production cost of heat and power generation is minimized while simultaneously satisfying various operational constraints related to heat production and power generation [2]. The cost function is mathematically expressed as follows:
Minimizewhere is total cost;, and are cost functions of
Civilized swarm optimization algorithm
The integration of SCA with PSO constitute CSO where SCA is a social-behavioral approach based on cultural evolution and PSO is a self-adaptive food searching strategy, indicating the activities of birds and fishes in their respective flocks and school [32]. In the SCA, the swarm is divided into societies and each society has its own leader, society leader (SL). The society leader is the best-performing particle of the society and other particles of society become society members (SMs).
Let
Powell’s pattern search
Powell’s pattern search method is based on the conjugate search direction method. By applying the PPS method, a nonlinear function can be optimized in finite steps with fast convergence. The PPS method does not need any information about the gradient and higher order gradients of the objective function and constraints. In the PPS method, the initial search is performed by moving the decision variable in the coordinate search direction (exploratory move) and afterwards the search is performed in
Constraint handling strategies
The objective of the CHP dispatch problem is to search decision variables, unit heat production, and power generation, so that the operating cost is minimized while satisfying all constraints. During the search process, decision variables may violate the constraints, hence there is a need to apply constraint-handling strategies to satisfy all constraints. There are many approaches those can be used to satisfy the constraints. In this work, the exterior penalty method approach is applied. The
Solution methodology
In this work, CSO and PPS methods are integrated where CSO is used to perform the search globally and PPS technique is applied to search locally. Initially, swarm particles are randomly generated between upper and lower limits of decision variables for heat only and conventional thermal only units. For cogeneration units, decision variables are initialized within FOR region. To upgrade the position of the particles, the selection of velocity limits is an important issue. The higher value of
Test systems and results
The proposed technique is implemented on five CHP test systems to search optimum power and heat dispatch. The valve-point loading effect and POZ constraint of conventional thermal units along with transmission losses are also considered. For all test systems, CSO, proposed technique, and proposed technique with MCH approach is applied 50 times and obtained results are compared with the results reported in the literature.
Conclusion
An integrated technique based on CSO with PPS method is proposed to search economic schedule for the CHP dispatch problem. The initial search is performed by CSO and to further exploit the promising search area, PPS technique is applied. The PPS method fine tune the civilized leader position obtained by CSO that helps to avoid convergence at any local optimal solution. The practical viability of the formulation is illustrated through five numerical examples and obtained results are compared
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