计及离散变量基于互补约束全光滑牛顿法的无功优化

Reactive Power Optimization With Discrete Variables Based on Complementarity Constraints Smooth Newton Method

针对电力系统无功优化确定性算法在处理离散变量时有困难及收敛域小的问题,提出把基于互补约束的全光滑牛顿算法用于含离散控制变量的电力系统无功优化。该方法使用光滑松弛函数,以避免海森(Hessian)矩阵的奇异性,将优化模型的1阶优化条件(Karush-Kuhn-Tucker,KKT)中的互补约束条件转化为光滑非线性方程,从而把非线性优化问题重构成一组非线性方程组的根求解问题,并用牛顿法进行求解。在此基础上,进一步提出以离散变量的2个边界构造其互补约束条件,并将约束条件直接嵌入到牛顿法中,实现离散变量在优化过程中的逐次逼近。算例表明:该无功优化方法具有大范围收敛性,突破了基于内点法等的无功优化技术要求系统初始点必须位于系统可行域之内的限制;采用互补约束条件处理离散变量,简单有效,能够可靠地同时得到连续变量及离散变量的最优解。

Focusing on the difficulty of processing the discrete variables and the small convergence region for the current deterministic methods of reactive power optimization, a new method based on complementarity constraint smooth Newton method was applied to reactive power optimization with discrete variables in this paper. In the method, the complementarity constraints sub-condtions in the Karush- Kuhn-Tucker （KKT） conditions were transformed to be a set of smooth nonlinear equations by introducing smooth relaxation function, which ensured the nonsingularity of Hessian matrix, and the nonlinear optimization problem was reconstructed as the solution of a set of smooth nonlinear equations, and solved by smooth Newton method. On this basis, the upper and lower integer bounds of the discrete variables are constructed to be complementarity constraints conditions, and embedded with smooth Newton method to approach its integer solution successively during the Newton iteration. The results of the samples demonstrate that the method proposed for reactive power optimization owns following advantages： firstly, it has large convergence region, which breakthroughs the requirement and restriction of reactive power optimization method based on the interior point algorithm that the initial point must be located in the feasible region of the system; secondly, the strategy of adopting complementarity constraints conditions to address discrete variables is simple and efficient, and can make the optimal solutions respectively for continuous variables and for discrete variables being reliably obtained at the same time.