【作者】 田君杨；

【导师】 韦化；

【作者基本信息】 广西大学 ， 电力系统及其自动化， 2014， 博士

【摘要】 优化理论在电网规划、运行等各方面得到广泛应用并发挥着越来越重要的作用,形成了各式各样的优化问题,其最优解的优劣直接影响电力系统的运行。几十年来,各种优化方法都被用于电力系统优化问题的求解,取得了许多有意义的成果。然而,由于电力系统优化问题具有非凸性,而传统优化方法难于确保其解的全局最优性,这使得电力系统全局最优解的求取面临着巨大的挑战。因此,研究新的全局优化理论,探究电力系统优化问题的全局最优解,具有重要的理论和现实意义。本文依据全局优化理论的最新突破性成果——矩量半定规划,开展电力系统全局优化算法的理论研究工作。借助概率领域的矩量理论将电力系统多项式优化问题转换为矩量表达,通过构造半正定的矩量矩阵推导出矩量空间的半定规划凸松弛模型,即矩量半定规划模型,该模型可通过增大矩量矩阵的阶次而逐渐逼近于原问题的全局最优解。并且,在求解时引入全局最优判定准则,以保证解的全局最优性。在电力系统优化问题中,{0,1}-经济调度和最优潮流问题是典型的非凸规划问题。其中{0,1}-经济调度问题属于混合整数规划问题,求解过程复杂,难于确保求得全局最优解,甚至得不到可行解；而最优潮流问题的全局最优解是学者们长期以来努力追求的目标,曾尝试采用半定规划凸松弛方法进行求取,但还是困难重重。本文采用矩量半定规划方法求解这两个问题,一般通过二阶松弛模型就能获得精确的全局最优解。主要研究成果如下：1)提出了{0,1}-经济调度的矩量半定规划模型,将{0,1}-经济调度问题中的整数约束表示为多项式互补约束形式,并将问题转换到矩量空间,通过引入半正定约束,建立相应的矩量半定规划松弛模型进行求解。计算结果表明,该模型不用对原问题分解,其最优解中能直接得到0/1变量的整数解,并满足全局最优判定准则。2)提出了求解{0,1}-经济调度问题多个全局最优解的矩量半定规划算法。{0,1}-经济调度属于组合优化问题,可能存在多个全局最优解,通过矩量半定规划的全局最优判定准则,可判断{0,1}-经济调度问题具有多少个全局最优解。当存在多个全局最优解时,所得矩量解为原问题的多个解在某取值概率下对应的矩量值,通过奇异值分解的特征值法可从矩量解中提取出{0,1}-经济调度问题的多个全局最优解。算例结果表明,该方法成功找到了多个有意义的全局最优解,这为电力系统组合优化问题的求解提供了有益的启示。3)提出了最优潮流的矩量半定规划模型,将最优潮流问题表示为不等式约束的多项式优化问题,同样采用矩量空间的半定松弛技术,建立相应的矩量半定规划松弛模型进行求解。对最优潮流的标准算例及常规半定规划方法求解时的反例均能求得秩1的矩量解,从而确保得到全局最优解。由此表明,该模型能够克服现有的半定规划方法求解最优潮流时不能得到秩1解的问题,具有更高的可靠性。4)提出了求解最优潮流问题的矩量半定规划全局优化算法。通过最优潮流矩量半定规划模型的秩1矩量解,可确定原问题的全局最优解是唯—的。此时,最优解的取值概率为狄拉克函数,则所得最优解的矩量值与原问题的全局最优解相等,因此最优潮流问题的全局最优解可从矩量解中直接获取。本文在国家自然科学基金(51167001)和国家重点基础研究发展规划项目(973项目)(2013CB228205)的资助下完成。更多还原

【Abstract】 Optimization theory has been widely applied in power system planning and operation, and it is playing an increasing significant role. This applications develop various optimization problems which solutions will affect the power system based on its merit. Over the past decades, all kinds of optimization are used with many positive results for the problems. However, power system optimization being nonconvex, it is an enormous challenge to solve the global optimum since traditional optimization methods can not assure the global optimality. Therefore, it has important theoretic and realistic meanings to study new global optimization theories and explore the global optimum for power system optimization problems.Based on moment semidefinite programming (MSDP), which is the brand-new achievement of global optimization, this thesis focuses on the theory study for the global optimization algorithm of power system. With moment theory in the field of probability, power system polynomial optimization problems can be transformed as moment expressions, and relaxed as semidefinite programming (SDP) models in the moment space through constructing semidefinite moment matrices. This kind of SDP model is called MSDP model, and its optimal value can gradually approach the original global optimum as increasing the order of moment matrix. Moreover, a judgment criterion of global optimum is introduced to guarantee the global optimality for the solution.In the power system optimization problems,{0,1}-economic dispatch (ED) and optimal power flow (OPF) are two typical nonconvex programming problems. The {0,l}-ED problem is a mixed integer program problem, its solution process is complex and is hard to get the global optimum, and even can not obtain a feasible solution. The global optimum of OPF problem is the goal of the long-term effort for the scholars, and it was attempted to be sovled by SDP convex relaxation method, but that is still difficult. This thesis employs MSDP method to solve these two problems, and obtains exact global optimum with2-order relaxation model in general. The main research achievements of this study are as follows.1) A MSDP model for {O,1}-ED problem is proposed. The integer constraints in the {O,1}-ED model are written as polynomial complementary form, and the problem is converted to moment space, then a corresponding MSDP model can be established by introducting the semidefinite constraints. The results show that, its optimal solutions can meet the criterion of global optimum, and the integer solutions of the0/1variables can be got without dividing the original problem.2) The MSDP algorithm of {0,1}-ED problem is proposed for solving multiple global optimal solutions. There may be one or more global optimal solutions for the {0,1}-ED since it belongs to the combined optimization problem. And the number of solutions of {0,1}-ED can be judged by the criterion of global optimum. When there are multiple global optimal solutions, the optimized result of MSDP is the moment of original solutions about a probability measure. And the multiple global optimal solutions of {0,1}-ED problem can be extracted from the MSDP solution by an eigenvalue method with singular value decomposition. An illustrative example showed that the method succeeded in finding multiple meaningful global optimal solutions. This process provides a useful insight into solving combinatorial optimization problems of power system.3) A MSDP model for OPF problem is proposed. The OPF model being written as polynomial optimization problem with inequality constraints, a corresponding MSDP model can be established by the semidefinite relaxation techniques in moment space. This MSDP model can be solved with rank-1moment solution for OPF standard cases and the counterexamples of existing SDP method, and the global optimum is guaranteed. It is showed that the MSDP model overcome the problem of no rank-1solution in solving OPF by existing SDP method.4) The MSDP global optimization algorithm of OPF problem is proposed. As the moment solution of MSDP-OPF model is rank-1, it ensures that the global optimal solution of OPF is unique. The probability measure of the solution is Dirac function now, and then the moment solution is equal to the original global optimum. Consequently, the global optimal solution of OPF can be got from moment solution directly.This thesis was supported in part by the National Natural Science Foundation of China (51167001) and by The National Basic Research Program of China (2013CB228205)更多还原