The short-term hydrothermal scheduling is one of the most important and challenging optimization problems in the daily operation planning of power systems. The objective of the scheduling is to determine the optimal power output of both hydro and thermal plants in order to meet the required load demand at minimum operating cost while satisfying various constraints. With the insignificant operational cost of hydroelectric plants, the objective of minimizing the operational cost of a hydrothermal system reduces to minimizing the fuel cost of thermal plants. Due to increasing concern over environmental pollution caused by fossil fuel fired thermal power plants, harmful emission produced by the thermal plants must be minimized simultaneously. It is necessary for utilities to take not only the fuel cost but also emission into consideration. Thus, an alternative economic hydrothermal scheduling considering both the fuel cost and emission is required.
Up to now, several techniques have been developed to solve the hydrothermal scheduling problems. Various mathematical programming methods such as programming methodology (Tang and Peter, 1995 and Yang and Chen, 1989), lagrangian relaxation (Guan & Peter, 1998), non-linear network flow technique (Brannud, Bubenko, & Sjelvgren, 1986), and decomposition techniques (Pereira & Pinto, 1982) have been applied for solving the hydrothermal scheduling problems. As the hydrothermal scheduling problems are modeled with nonlinear and non-convex curves with prohibited operating regions, the conventional methods are not suitable for dealing with constraints of hydrothermal system. Thus, aside from the above methods, optimal hydrothermal scheduling problems have been solved by stochastic search algorithms like simulated annealing (Wong & Wong, 1994), genetic algorithm (Gil et al., 2003, Orero and Irving, 1998, Ramirez and Ontae, 2006 and Yuan and Yuan, 2002), artificial neural networks (Naresh & Sharma, 1999), evolutionary programming (Sinha, Chakrabarti, & Chattopadhyay, 2003), cultural algorithm (Yuan & Yuan, 2006), tabu search (Bai & Shahidehpour, 1996), differential evolution (Mandal & Chakraborty, 2008) and particle swarm optimization (Yu, Yuan, & Wang, 2007). Various heuristic methods such as heuristic search technique (Dhillon Jarnail, Dhillon, & Kothari, 2007), fuzzy satisfying evolutionary programming procedures (Basu, 2004) and fuzzy decision-making stochastic technique (Dhillon, Parti, & Kothari, 2002) have been applied to solve multi-objective short-term hydrothermal scheduling problems.
In most of these algorithms, penalty functions are used to handle the equality and inequality constraints where infeasible solutions are penalized depending on the amount of constraint violation. Despite the simplicity and ease of implementation of penalty functions, they require tedious process of choosing suitable penalty coefficients. In this work, heuristic rules are proposed to handle the water dynamic balance constraints and heuristic strategies based on priority list are employed to handle active power balance constraints. A feasibility-based selection technique is also suggested to handle the reservoir storage volumes constraints. These techniques can also help the evolutionary algorithm avoid premature convergence.
In the present work, a novel interactive bi-objective programming with valuable trade off approach was borrowed from Refs. Kuo, 2009a and Kuo, 2009b to solve the bi-objective hydrothermal scheduling problem. It can provide a valuable trade off solution for the bi-objective optimization problem. Quadratic approximation based differential evolution (QADE) is proposed to optimize the nonlinear hydrothermal scheduling problem. Differential Evolution (DE) developed by Storn and Price (1997), is one of the most promising evolution algorithms. It has been successfully applied to solve optimization problems particularly involving non-smooth objective functions. The quadratic approximation operator is a nonlinear operator which can accelerate the evolution process by generating a new solution vector lying at the point of minima of the quadratic curve passing through the three selected solution vectors (Deep & Das, 2008). The proposed approach applied for hydrothermal scheduling is evaluated on a sample test system with four cascaded hydro plants and three thermal plants (Basu, 2004). The results obtained with the proposed algorithm were analyzed and compared with the results of differential evolution (Mandal and Chakraborty, 2008) and interactive fuzzy satisfying method based on evolutionary programming (Basu, 2004) reported in the literature. The proposed algorithm is found to be quite encouraging as compared with the earlier reported approaches.
The rest of the paper is organized as follows. In Section 2, the formulation of the hydrothermal scheduling problem is introduced. Section 3 explains the proposed quadratic approximation based differential evolution with valuable trade off approach for hydrothermal scheduling. Section 4 describes heuristic strategies for preserving various constraints. Simulation results and comparison with other approaches are presented in Section 5. Finally in Section 6, the conclusions are derived.
In this paper, quadratic approximation based differential evolution with valuable trade off approach (QADEVT) has been successfully introduced to solve short-term hydrothermal scheduling with non-smooth fuel and emission cost functions. The major merits of the proposed approach are as follows: (1) Short-term combined economic emission scheduling (CEES) posed as a bi-objective optimization problem is converted into a single objective one by price penalty factor and valuable trade off ratio; (2) Quadratic approximation is combined into differential evolution to accelerate the evolution process; (3) Heuristic rules are proposed to handle water dynamic balance constraints and heuristic strategies based on priority list are employed to handle active power balance constraints; (4) The feasibility-based selection rules are developed to handle the reservoir storage volumes constraints. Additionally, comparing with the method reported in the literature (Mandal and Chakraborty, 2008), the fuel cost and pollutant emission using QADEVT method can be reduced about 63.29 $/h and 53.79 lb/h, respectively, for the compromising minimum fuel cost and pollutant emission case. Numerical experiments show that the proposed method can obtain better quality solutions with higher precision. Future work is to study the CEES problems with other constraints such as ramp rate limits and prohibited discharge zones.
