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In this paper, the efficient population utilization strategy for particle swarm optimization (EPUSPSO) is proposed to solve the economic load dispatch (ELD) problem of power system. This algorithm improves the accuracy and the speed of its convergence by changing the number of particles effectively, and improving the velocity equation and position equation. In order to verify the effectiveness of the algorithm, this algorithm is tested in three different ELD cases of power system include IEEE 3-unit case, 13-unit case, and 40-unit case, and the obtained results are compared with those obtained from other algorithms using the same system parameters. The compared results show that the algorithm can find the optimal solution effectively and accurately, and avoid falling into the local optimal problem; meanwhile, faster speed can be ensured in the case.

Economic load dispatch is one of the important fundamental issues in power systems operation and planning. From solving that problem, fuel cost can be reduced and the reliability of the power system can be improved. The fundamental objective of the optimization problem is to minimize the cost of power generation by determine the output power of generating units with satisfying the load demand, operational requirement and other constraints. From past researches, we can know that due to the existing of valve point effect [

In recent years, various kinds of intelligent algorithms are widely used in ELD optimization because of their good global convergence and non-restricting by the object function, which include ant colony algorithm, simulated annealing algorithm, chaos algorithm and so on (Thesis [

Particle swarm optimization (PSO) was a biological evolution method which was proposed by Kennedy and Eberhart in 1995 [

In power systems, Economic load dispatch (ELD) is a method to schedule the power generator outputs with respect to the system constraints. Power system has to be optimized in such a way that it finally supplies all the loads with minimum fuel costs. Mathematically, the economic load dispatch problem function can be summarized as follows:

where,

F = total fuel cost;

In the unit “heat run” testing stage, the cost function of single generator can be obtained, which can be presented by a quadratic cost function as follows:

where

When the turbine admission valve at low level starts to open, the effect of wire drawing produce a rippling effect on the unit, this is called the valve point effect. If the valve point effect is neglected, there will be significantly affect the accuracy of the solution. When taking the ripples to valve point effect into consideration, the cost function becomes:

where,

Generator capacity should be satisfied:

where,

The power balance must meet the constraints as follows:

where,

In this paper, distribution network in the power system are concentrated, so the transmission losses can be neglected. As the system is a multi-unit power system with high dimension in this paper, the constraint can be simplified as follows:

Particle swarm optimization (PSO) is a biological evolution method inspired by the social behavior of animals (flock of birds), PSO is based on the concept of swarms and their intelligence and movement. A swam of particles represent a potential solution to the optimization problem. In the iterative process, each particle adjusts its position in term of its own experience, and the experience of its neighborhood particles at each time step. In other words, in PSO algorithm, particles can share information among individuals, start with a random population initialization and end with fast convergence to acceptable solution in the search space.

Efficient population utilization strategy for particle swarm optimization (EPUSPSO) [

where, x means the current position of individuals, and v means the current velocity of individuals. i and j mean the j-th dimension of the i-th particle. s and w are set as the current number of iterations and the weight parameter respectively, c is set as the learning factor and r is a random number between 0 and 1.

Equation (8) is the speed function, it means that each particle in the iterative process in accordance with two extreme values of the optimal solution to update their own pace. Equation (9) is the location function, by which particles update their location and got the optimal solution after numbers of iteration.

Although the PSO algorithm can solve multi-variable optimization problem, but it is easy to fall into local optima in solving problems of high complexity, usually in later iterations, ordinary particle swarm algorithm can’t obtain a faster convergence effect. But the EPUSPSO can finish good optimization without these problems.

EPUSPSO algorithm is an enhanced particle swarm algorithm using variable particles and the biggest improvement is to change the number of particles effectively by changing global optimization values. This method is called solution-sharing strategy. Content rules are as follows:

1) If the fitness value of global optimal solution has not been updated in two consecutive generations, then add a particle in the population.

Coefficient

2) If the fitness value of global optimal solution is updated in two consecutive generations. It means the existing particles are enough to handle the current solution searching procedures and then remove the particle from swarm population which has lower fitness value.

In addition, the algorithm adopts searching range sharing strategy and solution sharing strategy. In order to prevent the particles from falling into the local minimum, the activation probability of the search range sharing strategy could be defined as Pr(s) and the formula is as follows:

where, iteration in the formula means the maximum iteration.

The search range sharing strategy is used to reset all dimensions of individual particles’ current positions in the solution space. It can be divided into two different kinds of patterns according to the different searching space, that is the global pattern and the local pattern. In the global pattern, particles are restricted in the search boundary as (x_{min}, x_{max}) initially. In the local pattern, searching space boundary is restricted in past best solution (Pbest_{min}, Pbest_{max}). Selected from the individual optimal solution, Pbest_{max} is the maximum solution and the Pbest_{min} is the minimal solution.

At the same time, the singularity the existing particle velocity update process has improved by the setting of solution sharing strategy and the calculation formula is as follows:

In the formula, coefficient a represents a random serial number of a particle from the population and rand is a random number between 0 and 1. The formula Ps_{i} is as follows:

where, D is the dimension of the particle.

For evaluating the advantages and applicability of the proposed EPUSPSO method in dealing with different- dimensional, non-convex, non-conducting and multi-constrained optimization problems, tests are made through three different dimensions of typical ELD problems. Of all the test cases, valve-point loading effect of the ripple curve is considered and the transmission loss is neglected. For this work, MATLAB R2013a is used, all parameter of generators in cases below can be obtained in reference [_{max}), the minimum cost (F_{min}), and the average cost (F_{mean}) are acquired from the tests, and the results from other algorithms can be acquired from the references.

Case 1: The first test system consists of 3 thermal generating units with 6 bus bars by simulation. The expected load demand for this test case is 850 MW. The value of population size is 64. The dimension of the test is set to 3. Different methods are tested on the same system. Comparison of cost and the convergence characteristics are illustrated in

In

Case 2: The proposed method is also tested on the 13-unit system. The difference is that the expected load demand for this test case is 1800 MW, and the dimension is 13. Other parameters are the same as Case 1. The comparison of the convergence characteristics of various methods is demonstrated as follow, simulation results are shown in

It can be seen in

Case 3: In this section, EPUSPSO method is applied to a 40-generator system. The expected load demand for this test case is equal to 10,500 MW, and the dimension is 40. Other parameters are the same as Case 1.

The data in

From three cases above, we can see that EPUSPSO outperformed both accuracy and convergence speed of solution. No matter the dimension is lower or higher, EPUSPSO has good stability. With its advantages, the EPUSPSO algorithm is suitable for solving the optimization problems of ELD in large power systems.

Compared Item | |||
---|---|---|---|

PSO | 8389.37 | 8243.21 | 8457.12 |

IGA | 8245.01 | 8241.58 | 8361.86 |

MPSO | 8247.62 | 8234.07 | 8386.79 |

CEP | 8235.97 | 8234.07 | 8241.83 |

EPUSPSO | 8234.21 | 8229.57 | 8240.36 |

Compared Item | |||
---|---|---|---|

PSO | 18,415.29 | 18,217.64 | 18,620.43 |

IGA | 18,413.73 | 18,232.37 | 18,789.89 |

MPSO | 18,212.23 | 17,988.93 | 18,547.72 |

CEP | 18,190.32 | 18,048.21 | 18,404.04 |

EPUSPSO | 18,088.65 | 17,869.42 | 18,235.98 |

Compared Item | |||
---|---|---|---|

PSO | 129,546.83 | 127,614.75 | 137,650.24 |

IGA | 129,519.40 | 129,519.40 | 129,519.40 |

MPSO | 129,459.62 | 124,239.66 | 138,998.94 |

CEP | 124,793.48 | 123,488.29 | 126,902.89 |

EPUSPSO | 122,576.35 | 122,385.81 | 123,172.82 |

In this paper, we use EPUSPSO to solve different-dimensional and non-convex and nonlinear constraints problems in ELD. Compare with the traditional PSO, the algorithm overcomes the shortcomings of PSO algorithm that is easy to fall into local optima and the global optimal solution can be found in the optimization problem to a large extent. In the meantime, EPUSPSO has the faster speed of convergence, the higher accuracy and it is easy to achieve. Then we put this algorithm in an actual ELD problem to simulation which is considered the valve point effect, line capacity constraint and system stability constraints of generator units and the results demonstrate the feasibility and effectiveness of the proposed algorithm. We can also use this algorithm to other complex optimization problems in virtue of its universal property.

LeiWu,HaimingLi,ZhengyangWu,ChenbinWu, (2015) Economic Load Dispatch Based on Efficient Population Utilization Strategy for Particle Swarm Optimization. International Journal of Communications, Network and System Sciences,08,367-373. doi: 10.4236/ijcns.2015.89035