基于矩量理论的电力系统全局优化算法研究

发布时间：2018-07-20 09:02

【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。
[Abstract]:Optimization theory has been widely used in the planning and operation of power grid and plays a more and more important role, and has formed a variety of optimization problems. The optimal solution has a direct impact on the operation of power system. In the past few decades, various optimization methods have been used to solve the problem of power system optimization, and many meaningful results have been obtained. However, due to the non convexity of the power system optimization problem, the traditional optimization method is difficult to ensure the global optimality of the solution. This makes the global optimal solution of the power system face great challenges. Therefore, it is important to study the new global optimization theory and explore the global optimal solution of the power system optimization problem. Practical significance.
In this paper, based on the latest breakthrough of the global optimization theory, the moment semidefinite programming, the theoretical research work of the global optimization algorithm of power system is carried out. By means of the moment theory of probability domain, the polynomial optimization problem of power system is converted into moment expression, and a semi definite rule of moment space is derived by constructing a semi positive moment matrix. The convex relaxation model, that is, the moment semidefinite programming model, can be gradually forced to close to the global optimal solution of the original problem by increasing the order of the moment matrix, and the global optimal criterion is introduced to ensure the global optimality of the solution.
In the problem of power system optimization, the {0,1}- economic scheduling and the optimal power flow problem are typical non convex programming problems. Among them, the {0,1}- economic scheduling problem belongs to the mixed integer programming problem, and the solution process is complicated. It is difficult to ensure the global optimal solution and even not get the feasible solution. The global optimal solution of the optimal power flow problem is a long term for the scholars. Since the goal has been tried hard, the semi definite programming convex relaxation method has been tried, but it is still difficult. In this paper, the moment semidefinite programming method is used to solve these two problems, and the exact global optimal solution can be obtained by the two order relaxation model. The main research results are as follows:
1) a moment semidefinite programming model of {0,1}- economic dispatch is proposed. The integer constraints in the {0,1}- economic scheduling problem are expressed as polynomial complementary constraints, and the problem is converted to the moment space. By introducing the semi positive definite constraint, the corresponding moment semidefinite programming relaxation model is established. The results show that the model is not used for the original model. In the optimal solution, the integer solution of 0/1 variables can be directly obtained and the global optimal criteria can be satisfied.
2) a moment semidefinite programming algorithm for solving multiple global optimal solutions of {0,1}- economic scheduling problem.{0,1}- economic scheduling is a combinatorial optimization problem. There may be multiple global optimal solutions. Through the global optimal decision criteria of moment semidefinite programming, how many global optimal solutions of the {0,1}- economic scheduling problem can be judged. When the global optimal solution is given, the moment is solved as the moment value of the solution of the original problem in the probability of a value. By the eigenvalue method of singular value decomposition, the multiple global optimal solutions of the {0,1}- economic dispatch problem can be extracted from the moment solution. The results of the calculation show that the method has successfully found a number of meaningful global optimal solutions, which is the electric power. It provides a useful inspiration for solving system combinatorial optimization problems.
3) a moment semidefinite programming model for optimal power flow is proposed. The optimal power flow problem is expressed as a polynomial optimization problem with inequality constraints. A semi definite relaxation model of moments is established by using the semidefinite relaxation technique of moment space to solve the problem. The standard calculation example and the conventional semi definite programming method for solving the optimal power flow are inverse. The moment solution of rank 1 can be obtained and the global optimal solution can be ensured, which shows that the model can overcome the problem that the rank 1 solution can not be obtained when the existing semi definite programming method can solve the optimal power flow, and has higher reliability.
4) a moment semidefinite programming global optimization algorithm for solving the optimal power flow problem is proposed. Through the rank 1 moment solution of the optimal power flow moment semidefinite programming model, the global optimal solution of the original problem is determined only. At this time, the probability of the optimal solution is Dirac's function, and the moment value of the optimal solution is equal to the global optimal solution of the original problem. Therefore, the global optimal solution of the optimal power flow problem can be obtained directly from the moment solution.
This article is supported by the National Natural Science Foundation of China (51167001) and the national key basic research development project (973 project) (2013CB228205).
【学位授予单位】：广西大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TM73

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