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Hydrothermal Systems Generation Scheduling Using Cultural Algorithm Xiaohui Yuan, Hao Nie, Yanbin Yuan, Anjun Su and Liang Wang

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Hydrothermal Systems Generation Scheduling Using Cultural Algorithm Xiaohui Yuan, Hao Nie, Yanbin Yuan, Anjun Su and Liang Wang 65 Q IWA Publishing 2009 Journal of Hydroinformatics | 11.1 | 2009 Hydrothermal systems generation scheduling using cultural algorithm Xiaohui Yuan, Hao Nie, Yanbin Yuan, Anjun Su and Liang Wang ABSTRACT This paper proposes an enhanced cultural algorithm to solve the short-term generation Xiaohui Yuan (corresponding author) Hao Nie scheduling of hydrothermal systems problem, in which differential evolution is embedded into a Anjun Su cultural algorithm and applies two knowledge sources to influence the variation operator of Liang Wang School of Hydropower and Information differential evolution and couples with simple selection criteria based on feasibility rules and Engineering, Huazhong University of Science and Technology, heuristic search strategies to handle constraints in the cultural algorithm effectively. A test 430074 Wuhan, China hydrothermal system is used to verify the feasibility and effectiveness of the proposed method. Tel.: +86 27 8800 8929 Results are compared with those of other optimization methods reported in the literature. It is Fax: +86 27 8754 3992 E-mail: [email protected] shown that the proposed method is capable of yielding higher quality solutions. Yanbin Yuan Key words | cultural algorithm, differential evolution, heuristic search, scheduling School of Resources and Environmental Engineering, Wuhan University of Technology, 430070 Wuhan, China INTRODUCTION include generator-load power balance equations and total water discharge constraint as the equality constraints and Short-term hydrothermal generation scheduling (SHGS) is reservoir storage limits and the operational limits of the one of the most important and challenging optimization hydro- as well as thermal generators as the inequality problems in the economic operation of power systems. In a constraints. Thus, the short-term hydrothermal generation hydrothermal power system, the water resources available scheduling problem becomes a typical large-scale, dynamic, for electrical generation are represented by the inflows to non-convex nonlinear constrained optimization problem. the hydropower plants and the water stored in their The importance of hydrothermal generation scheduling reservoirs. Thus, the available resources at each stage of is well recognized. An efficient generation schedule not the operation planning horizon depend on the previous use only reduces the production costs but also increases the of the water, which establishes a dynamic relationship system reliability and maximizes the energy capability of among the operation decisions made along the whole horizon. The purpose of short-term hydrothermal schedul- the reservoirs. Therefore, many methods have been devel- ing is to find the optimal amount of water release for hydro- oped to solve this problem over the past decades. The and thermal generation in the system to meet the load major methods include variational calculus (Grake & demands over a scheduling horizon of one day or a few Kirchmayer 1962), maximum principle (Papageorgiou days. As the sources for hydropower are natural water 1985), functional analysis (Soliman & Christensen 1986), resources with almost zero operational cost, the objective dynamic programming (Yang & Chen 1989; Yeh 1992; Tang of the short-term optimal hydrothermal scheduling problem & Peter 1995), network flow and mixed-integer linear essentially reduces to minimizing the fuel cost of thermal programming (Franco et al. 1994; Piekutowski 1994; plants over all the scheduling period of time while satisfying Oliveira & Soares 1995; Chang et al. 2001), nonlinear various constraints. The practical constraints to be satisfied programming (Guan & Peter 1995), progressive optimality doi: 10.2166/hydro.2009.056 Downloaded from http://iwaponline.com/jh/article-pdf/11/1/65/386327/65.pdf by guest on 01 October 2021 66 X. Yuan et al. | Hydrothermal systems generation scheduling using cultural algorithm Journal of Hydroinformatics | 11.1 | 2009 algorithm (Turgeon 1981; Lee 1989), Lagrangian relaxation This paper proposes a new enhanced cultural algorithm method (Tufegdzic 1996; Salam & Mohamed 1998) and (ECA) to solve the short-term generation scheduling of modern heuristics algorithms such as artificial neural hydrothermal systems problem. The proposed ECA method networks (Naresh & Sharma 1999), evolutionary algorithm is enhanced over the method of Yuan & Yuan (2006) by [17–20](Chen & Chang 1996; Yang & Yang 1996; Orero & simple feasibility-based selection comparison techniques Irving 1998; Werner & Verstege 1999), chaotic optimization and heuristic search strategies to handling constraints (Yuan & Yuan 2002), ant colony (Huang 2001), Tabu effectively, especially for the water dynamic balance search (Bai & Shahidehpour 1996) and simulated anneal- equation and reservoir end-volume equality constraints in ing (Wong & Wong 1994). But these methods have one or the SHGS problem. Moreover, differential evolution is used another drawback such as dimensionality difficulties, large as a population space to replace evolutionary programming memory requirement or an inability to handle nonlinear in the cultural algorithm, which was not considered in the characteristics, premature phenomena and trapping into previous work for solving the SHGS problem. Differential local optimum, taking too much computation time. Thus, evolution is a recently developed evolutionary algorithm, improving current optimization techniques and exploring which has been found to be a very robust optimization new methods to solve short-term hydrothermal generation technique. Differential evolution appeared a better choice scheduling has great significance so as to efficiently utilize for the population space than evolutionary programming water resources, which can be regarded as a renewable when dealing with constrained search spaces in cultural source of energy. algorithms. Finally, simulation results demonstrate the In recent years, a new optimization method known as feasibility and effectiveness of the proposed method for cultural algorithm proposed by Reynolds in 1994 (Rey- solving the SHGS problem in terms of convergence speed nolds 1994) has become a candidate for optimization and solution precision compared to those of the augmented applications due to its flexibility and efficiency. The Lagrange method, two-phase neural network, genetic cultural algorithm is a technique that incorporates domain algorithm and differential evolution. knowledge obtained during the evolutionary process so as This paper is organized as follows. The next section to make the search process more efficient. It has been provides the notation used in the paper along with the successfully applied to solve optimization problems and mathematical formulation of the short-term hydrothermal promises to overcome some shortcomings of the above scheduling problem. Then we introduce the basics of the optimization methods. cultural algorithm. The fourth section briefly describes In our earlier work, the optimal daily generation differential evolution. We then propose an enhanced scheduling of hydrothermal systems has been first solved cultural algorithm (ECA). The next section presents the by the cultural algorithm (CA) (Yuan & Yuan 2006). In the ECA for solving short-term hydrothermal generation sche- proposed CA method for solving the SHGS problem, belief duling problem. The seventh section gives a numerical cells with an n-dimensional regional-based schema are example, followed by our conclusions. adopted to support the acquisition, storage and integration of knowledge about constraints of the SHGS problem in CA. The belief space contains a set of these schemata, PROBLEM FORMULATIONS which can be used to guide the search in a direct way by pruning the infeasible regions and promoting the promising Notation regions. Meanwhile, evolutionary programming (EP) is To formulate the problem mathematically, the following used as a population space in cultural algorithms. The notation used in this paper is first introduced: exploration of the EP population serves to modify these belief-cell schemata, which are used to guide the pro- F composite fuel cost function t duction of new individuals in the population component Psi power generation of thermal plant i at time in return. interval Downloaded from http://iwaponline.com/jh/article-pdf/11/1/65/386327/65.pdf by guest on 01 October 2021 67 X. Yuan et al. | Hydrothermal systems generation scheduling using cultural algorithm Journal of Hydroinformatics | 11.1 | 2009 t fiðPsiÞ fuel cost of ith thermal plant at time interval t Objective function: XT XNs XT XNs Ns number of thermal plants t t t 2 min F ¼ fiðPsiÞ¼ {ai þ biPsi þ giðPsiÞ } ð1Þ Nh number of hydro plants t¼1 i¼1 t¼1 i¼1 ai,bi,gi thermal generation coefficients of ith plant Subject to the following constraints: T total time horizon System load balance: t time index, t ¼ 1,2, … ,T XNh XNs Pt þ Pt ¼ Pt t ¼ 1; 2; … ; T ð2Þ Pt system load demand at time interval t hi sj D D i¼1 j¼1 min Psi minimum power generation of thermal plant i Thermal plant power generation limits: Pmax maximum power generation of thermal plant i si Pmin # Pt # Pmax i ¼ ; ; ; N ; t ¼ ; ; ; T ð Þ t si si si 1 2 … s 1 2 … 3 Phi power generation of hydro plant i at time interval t Hydro plant power generation limits: min t max ; ; ; ; ; ; min Phi # Phi # Phi i ¼ 1 2 … Nh; t ¼ 1 2 … T ð4Þ Phi minimum power generation of hydro plant I max Hydro plant discharge limits: Phi maximum power generation of hydro plant i min t max ; ; ; ; ; ; t Qi # Qi # Qi i ¼ 1 2 … Nh; t ¼ 1 2 … T ð5Þ Vi water
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