【摘要】：優(yōu)化理論在電網(wǎng)規(guī)劃、運行等各方面得到廣泛應用并發(fā)揮著越來越重要的作用,形成了各式各樣的優(yōu)化問題,其最優(yōu)解的優(yōu)劣直接影響電力系統(tǒng)的運行。幾十年來,各種優(yōu)化方法都被用于電力系統(tǒng)優(yōu)化問題的求解,取得了許多有意義的成果。然而,由于電力系統(tǒng)優(yōu)化問題具有非凸性,而傳統(tǒng)優(yōu)化方法難于確保其解的全局最優(yōu)性,這使得電力系統(tǒng)全局最優(yōu)解的求取面臨著巨大的挑戰(zhàn)。因此,研究新的全局優(yōu)化理論,探究電力系統(tǒng)優(yōu)化問題的全局最優(yōu)解,具有重要的理論和現(xiàn)實意義。 本文依據(jù)全局優(yōu)化理論的最新突破性成果——矩量半定規(guī)劃,開展電力系統(tǒng)全局優(yōu)化算法的理論研究工作。借助概率領域的矩量理論將電力系統(tǒng)多項式優(yōu)化問題轉(zhuǎn)換為矩量表達,通過構造半正定的矩量矩陣推導出矩量空間的半定規(guī)劃凸松弛模型,即矩量半定規(guī)劃模型,該模型可通過增大矩量矩陣的階次而逐漸逼近于原問題的全局最優(yōu)解。并且,在求解時引入全局最優(yōu)判定準則,以保證解的全局最優(yōu)性。 在電力系統(tǒng)優(yōu)化問題中,{0,1}-經(jīng)濟調(diào)度和最優(yōu)潮流問題是典型的非凸規(guī)劃問題。其中{0,1}-經(jīng)濟調(diào)度問題屬于混合整數(shù)規(guī)劃問題,求解過程復雜,難于確保求得全局最優(yōu)解,甚至得不到可行解；而最優(yōu)潮流問題的全局最優(yōu)解是學者們長期以來努力追求的目標,曾嘗試采用半定規(guī)劃凸松弛方法進行求取,但還是困難重重。本文采用矩量半定規(guī)劃方法求解這兩個問題,一般通過二階松弛模型就能獲得精確的全局最優(yōu)解。主要研究成果如下： 1)提出了{0,1}-經(jīng)濟調(diào)度的矩量半定規(guī)劃模型,將{0,1}-經(jīng)濟調(diào)度問題中的整數(shù)約束表示為多項式互補約束形式,并將問題轉(zhuǎn)換到矩量空間,通過引入半正定約束,建立相應的矩量半定規(guī)劃松弛模型進行求解。計算結果表明,該模型不用對原問題分解,其最優(yōu)解中能直接得到0/1變量的整數(shù)解,并滿足全局最優(yōu)判定準則。 2)提出了求解{0,1}-經(jīng)濟調(diào)度問題多個全局最優(yōu)解的矩量半定規(guī)劃算法。{0,1}-經(jīng)濟調(diào)度屬于組合優(yōu)化問題,可能存在多個全局最優(yōu)解,通過矩量半定規(guī)劃的全局最優(yōu)判定準則,可判斷{0,1}-經(jīng)濟調(diào)度問題具有多少個全局最優(yōu)解。當存在多個全局最優(yōu)解時,所得矩量解為原問題的多個解在某取值概率下對應的矩量值,通過奇異值分解的特征值法可從矩量解中提取出{0,1}-經(jīng)濟調(diào)度問題的多個全局最優(yōu)解。算例結果表明,該方法成功找到了多個有意義的全局最優(yōu)解,這為電力系統(tǒng)組合優(yōu)化問題的求解提供了有益的啟示。 3)提出了最優(yōu)潮流的矩量半定規(guī)劃模型,將最優(yōu)潮流問題表示為不等式約束的多項式優(yōu)化問題,同樣采用矩量空間的半定松弛技術,建立相應的矩量半定規(guī)劃松弛模型進行求解。對最優(yōu)潮流的標準算例及常規(guī)半定規(guī)劃方法求解時的反例均能求得秩1的矩量解,從而確保得到全局最優(yōu)解。由此表明,該模型能夠克服現(xiàn)有的半定規(guī)劃方法求解最優(yōu)潮流時不能得到秩1解的問題,具有更高的可靠性。 4)提出了求解最優(yōu)潮流問題的矩量半定規(guī)劃全局優(yōu)化算法。通過最優(yōu)潮流矩量半定規(guī)劃模型的秩1矩量解,可確定原問題的全局最優(yōu)解是唯—的。此時,最優(yōu)解的取值概率為狄拉克函數(shù),則所得最優(yōu)解的矩量值與原問題的全局最優(yōu)解相等,因此最優(yōu)潮流問題的全局最優(yōu)解可從矩量解中直接獲取。 本文在國家自然科學基金(51167001)和國家重點基礎研究發(fā)展規(guī)劃項目(973項目)(2013CB228205)的資助下完成。
[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).
【學位授予單位】：廣西大學
【學位級別】：博士
【學位授予年份】：2014
【分類號】：TM73

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