Closure to Discussion of “An Improved Particle Swarm Optimization for Nonconvex Economic Dispatch Problems”

The authors appreciate the discusser’s comments and observations. The following is offered in response to the questions in the discussion. We have proposed a heuristic constraint-handing technique for considering equality and inequality constraints of economic dispatch (ED) problems. The proposed constraint-handling technique is able to improve the solution quality even if it might spend more CPU time than any penalty factor approaches. The adjustment of power generation of a generator will make the change of the transmission network loss. Therefore, the should be recalculated after adjusting the generation of each generator in the proposed constraint-handling technique. If the difference between adjusted and previous generations of every generator is very small, the change of the can be ignored. That is, if , then stop the constraint-handling procedure; otherwise, recalculate the transmission network loss. Here, is the solution convergence tolerance. The crossover operation does not affect the original PSO trajectories. The position of a particle is updated by using the original PSO process. The crossover operation creates a trial vector, which is just used to update the and . The authors have found a numerical error that an incorrect input data of a generator has been used for test system 1. In [2], no load fuel-cost coefficient of the 7th generator is 287.71. However, we have used 278.71; thereby, slightly incorrect results for test system 1 were obtained. We have recalculated the fuel cost for test system 1 and revised the results in Tables I–V, respectively. Test system 2 consists of the 15-unit power system, which considers the prohibited operating zones, ramp rate limits, and transmission network losses. Therefore, test system 2 cannot be directly solved by mathematical programming approaches. The execution time does not depend on the chaotic sequences and/or the crossover operation. As shown in Fig. 3, the CCPSO converges early into final solution in comparison with COPSO. This means that COPSO spends more CPU time on adjusting power generation of each generator in the constraint-handling procedure than CCPSO.