An improved binary particle swarm optimization for unit commitment problem

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Abstract

This paper proposes a new improved binary PSO (IBPSO) method to solve the unit commitment (UC) problem, which is integrated binary particle swarm optimization (BPSO) with lambda-iteration method. The IBPSO is improved by priority list based on the unit characteristics and heuristic search strategies to repair the spinning reserve and minimum up/down time constraints. To verify the advantages of the IBPSO method, the IBPSO is tested and compared to the other methods on the systems with the number of units in the range of 10–100. Numerical results demonstrate that the IBPSO is superior to other methods reported in the literature in terms of lower production cost and shorter computational time.

Introduction

Unit commitment (UC) is a very significant optimization task, which plays an important role in the operation planning of power systems. The unit commitment problem (UCP) in power systems refers to the optimization problem for determining the start-up and shut-down schedule of generating units over a scheduling period so that the total production cost is minimized while satisfying various of constraints. UCP can be considered as two linked optimization decision processes, namely the unit-scheduled problem that determines on/off status of generating units in each time period of planning horizon, and the economic load dispatch problem. Mathematically, the UCP has commonly been formulated as a complex nonlinear, mixed-integer combinational optimization problem with 0–1 variables that represents on/off status and continuous variables that represents unit power, and a series of prevailing equality and inequality constraints. However, the number of combinations of 0–1 variables grows exponentially as being a large-scale problem. Therefore, UCP is known as one of the problems which is the most difficult to be solved in power systems.

Many methods have been developed to solve the UCP in the past decades. The major methods include priority list method (PLM) (Senjyu, Miyagi, & Saber, 2006), dynamic programming (DP) (Su & Hsu, 1991), branch-and-bound methods (BBM) (Cohen & Yoshimura, 1983), integer and mixed-integer linear programming (MILP) (Khodaverdian et al., 1986, Samer and John, 2000), Langrangian relaxation (LR) (Feng and Liao, 2006, Ongsakul and Petcharaksm, 2004). Among these methods, PLM is simple and fast, but the quality of final solution is not guaranteed. DP is flexible but the disadvantage is the “curse of dimensionality”, which results it may leads to more mathematical complexity and increase in computation time if the constraints are taken into consideration. The shortcoming of BBM is the exponential growth in the execution time with the size of UCP. MILP adopts linear programming techniques to solve and check for an integer solution. MILP for solving UCP fail when the number of units increases because they require a large memory and suffer from great computational delay. These methods have only been applied to small UCP and have required major assumptions that limit the solution space. The main problem with the LR is the difficulty encountered in obtaining feasible solutions. Due to the non-convexity of the UCP, optimality of the dual problem does not guarantee feasibility of the primal UCP. Furthermore, solution quality of LR depends on the method to update Lagrange multipliers.

Aside from the above methods, meta-heuristic approaches such as artificial neural networks (ANN) (Dieu & Ongsakul, 2006), genetic algorithm (GA) (Damousis et al., 2004, Kazarlis and Bakirtzis, 1996, Senjyu et al., 2002), evolutionary programming (EP) (Juste, Kita, & Tanaka, 1999), memetic algorithm (MA) (Jorge & Smith, 2002), Tabu search (TS) (Mantawy, Abdel, & Selim, 1998), simulated annealing (SA) (Zhuang & Galiana, 1990), particle swarm optimization (PSO) (Lee and Chen, 2007, Ting et al., 2003) and greedy random adaptive search procedure (GRASP) (Viana, Sausa, & Matos, 2003) have also been used to solve UCP since the beginning of the last decade. These meta-heuristic methods optimization methods attract much attention, because of their ability to search not only local optimal solution but also global optimal solution and can easily deal with various difficult nonlinear constraints. However, these meta-heuristic methods require a considerable amount of computational time to find the near-global minimum especially for a large-scale UCP.

To reduce the search space in large-scale UCP, hybrid methods combining meta-heuristic approaches and deterministic optimization methods such as LR and GA (LRGA) (Cheng, Liu, & Liu, 2000), LR and MA (LRMA) (Jorge & Smith, 2002), and PSO combined with the LR (PSOLR) (Balci & Valenzuela, 2004) have been used to solve UCP. They are more efficient than the single methods due to less expensive production cost and a faster computational time. One main difficulty is their sensitivity to the choice of parameters. Thus, improving current optimization techniques and exploring new strategy to solve UCP has great significance.

PSO proposed by Kennedy and Eberhart in 1995 has become a candidate for optimization applications due to its flexibility and efficiency. In this paper, a new improved PSO (IBPSO) method has been proposed to solve UCP, which is integrated an improved discrete binary particle swarm optimization (BPSO) with the lambda-iteration method. The IBPSO is enhanced by priority list based on the unit characteristics and heuristic search strategies to repair the spinning reserve and minimum up/down time constraints. In the proposed IBPSO method, BPSO is used to solve the unit-scheduling problem and the lambda-iteration method is used to solve the economic load dispatch problem. In solving UCP, the BPSO and lambda-iteration method are run in parallel, adjusting their solutions in search of a better solution. Finally, the proposed IBPSO method is tested on the UCP systems with the number of units in the range of 10 to 100. Simulation results demonstrate the feasibility and effectiveness of the IBPSO method in terms of solution quality and computation time compared with those of other optimization methods reported in the literature.

This paper is organized as follows. Section 2 provides the mathematical formulation of the UCP. Section 3 briefly introduces the basics of PSO. Section 4 proposes an improved binary PSO (IBPSO) method for solving UCP. Section 5 gives the numerical example. Section 6 outlines the conclusions. Finally, acknowledgements are given.

Section snippets

Objective function

The objective of UCP is to find the generation scheduling over the scheduled time horizon such that the total production cost can be minimized while satisfied all kinds of constraints. The total cost F over the entire scheduling periods is the sum of the operating cost and start-up cost for all of the units. Thus, the objective function of the UC problem isminF=t=1Tt=1N[fi(Pit)+STit(1-uit-1)]uitwhere N is number of generators; T is total scheduling period; Pit is generation of unit i at time t

Overview of particle swarm optimization

Particle swarm optimization (PSO), first introduced by Kennedy and Eberhart, is one of the heuristic optimization algorithms (Clerc & Kennedy, 2002). A classical PSO maintains a swarm of particles that represent the potential solutions to the problem on hand. The classical PSO model consists of a swarm of particles moving in the D-dimensional space of possible problem solutions. Each particle embeds the relevant information regarding the D decision variables and is associated with a fitness

Improved binary particle swarm optimization (IBPSO) for UCP

The proposed IBPSO method for solving UCP consists of three stages. In the first stage, combination discrete binary particle swarm optimization with priority list is used to commit units to satisfy spinning reserve neglecting the minimum up and down time constraints. In the second stage, a heuristic search algorithm is applied to repair violations of the minimum up and down time constraints as well as decommit excessive spinning reserve units based on the unit schedule from the first stage. In

Numerical examples

In order to verify the feasibility and effectiveness of the proposed IBPSO method for solving UCP, the proposed IBPSO method is tested on different system sizes based on a basic system of 10 units from the literature (Juste et al., 1999). The scheduling time horizon T is chosen as one day with 24 intervals of one hour each. The spinning reserve requirement is set to be 10% of total load demand. For the systems of 20, 40, 60, 80 and 100 units, the basic 10-unit system is duplicated and total load

Conclusions

In this paper, we proposed an enhanced particle swarm optimization method (IBPSO) to solve unit commitment problem. The proposed IBPSO is a combination of binary particle swarm optimization with lambda-iteration method, which is enhanced by priority list to handle the spinning reserve constraints and a heuristic search algorithm to handle minimum up/down time constraints. The simulated results obtained after solving several UCP instances with the number of units in the range of 10–100 show that

Acknowledgements

The authors gratefully acknowledge the financial supports from National Natural Science Foundation of China under Grant Nos. 50779020 and 40572166.

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