Abstract: This paper takes the mathematical software Maple as the auxiliary tool to study the differential problems of four types of functions. We can obtain the infinite series forms of any order derivatives of these functions by using binomial theorem and differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order derivative values. On the other hand, we propose two examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple.

Abstract: Generation maintenance scheduling (GMS) is an important and effective part of Generation expansion planning (GEP) problem. Preventive-maintenance schedules need to be optimized to trade-off between two conflicting objectives, reducing the overall cost and improving the reliability. This paper presents a multi-objective binary gravitational search algorithm (BGSA) for solving GMS problem of generation systems as a sub-problem of the main GEP problem. In the proposed method, a fuzzy membership function is defined for each term in the objective function. There are three objective functions in this problem. The first objective function is leveling reserve capacity when unit maintenance outages are considered. The second and the third objectives which are also objectives of the main GEP problem, are to minimize the operation and maintenance (O&M) cost and the reliability index of Expected Energy Not Supplied (EENS). As GMS problem is a sub-problem of the main GEP problem, it is solved for a typical solution of the main GEP problem. The proposed method is applied to solve GMS problem for 4-bus test system from Grainger & Stevenson and IEEE-RTS 24-bus test system for a planning horizon of one year and two years, respectively. To verify the capability of the proposed BGSA based method, a binary genetic algorithm (BGA) method is also implemented to solve GMS problem and then the results are compared.