基于特征值优化理论的小干扰稳定约束最优潮流研究
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摘要
现代电力系统互联规模日益扩大,系统间低频振荡时有发生,小干扰稳定问题日趋突出,对电力系统的安全稳定运行带来了致命的影响。电力系统稳定器PSS等阻尼控制装置虽可有效地抑制振荡,但不能确保低频振荡就此消失。因此在节能调度的同时,运行人员十分期望系统的运行预留一定的稳定裕度,紧急情况下,能通过调整机组出力的方式,消除低频振荡,避免系统瓦解。为此,小干扰稳定约束最优潮流应运而生,作为发电调度的有力工具它将系统的经济性和安全性有机地结合,是抑制低频振荡的重要补充,相关研究正引起各国研究人员的广泛兴趣。
由于系统状态矩阵谱横坐标函数的隐式特征和非李普希茨特性,小干扰稳定约束最优潮流的建模异常困难,实际上这一问题的数学本质是特征值优化问题。随着半定规划和现代内点理论的发展,特征值优化理论日趋成熟,已在多个领域有着成功的应用。
基于特征值优化理论,本文提出了一种小干扰稳定约束最优潮流的非线性半定规划模型和算法。所提方法是利用特征值优化理论,解决电力系统的小干扰稳定问题的有益尝试,具有重要的理论和实用价值。主要研究成果如下：
1.提出了一种小干扰稳定约束最优潮流的非线性半定规划模型,它借助于李雅谱诺夫定理引入正定约束来表达小干扰稳定,是原问题的精确表达,避免了满足小干扰稳定而带来的不必要经济代价。同时,将非对称矩阵的特征值优化问题转换为对称矩阵的半定规划问题,克服了灵敏度算法中由于谱横坐标函数的非李普希茨特性引起的迭代振荡问题。
2.提出了一种将半定约束等价转换为光滑、凹、非线性约束的方法。所提转换方法将半定规划中半定约束转换为非线性约束后,整个模型变为非线性规划模型,从而可以利用高效的现代内点算法求解。
3.通过选择控制器参数为变量,提出了阻尼控制器协调优化的非线性半定规划模型,它可以分别或同时对系统内PSS、FACTS参数进行协调优化。在建立模型时还可考虑多种运行方式,使模型解适应更大的运行范围,更具鲁棒性。
全文分为六章,归纳如下：
第一章,介绍了小干扰稳定约束最优潮流问题的研究背景,回顾了国内外相关研究现状,指出了各种方法的优缺点,阐述了建立小干扰稳定约束最优潮流精确模型的动机。
第二章,分别介绍了常规最优潮流问题和及其现代内点算法,回顾了电力系统多机动态模型及小干扰稳定分析问题的数学模型。分析了最优潮流问题和小干扰稳定分析问题的联系,指出研究小干扰稳定约束最优潮流问题的难点,为后续章节的展开奠定基础。
第三章,详细论述了特征值优化理论,包括其发展历史,介绍了线性半定规划模型、非线性半定规划模型以及求解算法,为后续章节提出的数学模型作理论准备。
第四章,讨论了小干扰稳定约束最优潮流问题,提出了这一问题的非线性半定规划模型,并采用模型转换的方法将非线性半定规划模型精确转换为非线性规划模型,用现代内点法求解。同时还对WSCC3机9节点系统、Kundur4机10节点系统和IEEE5机14节点系统进行了仿真计算和分析,验证了模型的有效性和算法的正确性及鲁棒性。
第五章,以第四章的理论成果为基础,提出了阻尼控制器协调优化的非线性半定规划模型,并采用相同的模型转换方法进行求解。通过两个系统的仿真结果,验证了模型的有效性。
第六章,概括总结了全文的主要研究工作及其创新点,指出了今后有待进一步开展的研究工作。
The interconnections of large regional power network bring more challenges for the security of modern power system. Due to small-signal instability, low frequency oscillations pose a lethal threat to power system. Although the damping controllers can enhance the small signal stability effectively, it cannot guarantee that no oscillations occur. Therefore, it requires dispatchers to schedule the system generation not only based on economic object with the aid of the powerful tool Optimal Power Flow (OPF), but also ensuring a security margin with respect to small signal stability. In this context, it has been proposed to include small signal stability constraints into the optimal power flow model, thus creating Small-Signal Stability Constrained OPF (SSSC-OPF) techniques. They have aroused more and more interest recently.
For the implicit and non-Lipschitz property of spectral abscissa of system state matrix, it is a big challenge to model the SSSC-OPF directly. In general, SSSC-OPF is an eigenvalue optimization problem naturally. This theory had not made much progress for a long time, until the dramatic breakthrough in the interior point methods for the semi-definite programming (SDP). Nowadays, the therory has been introduced to many fields, having a number of successful applications.
Based on the eigenvalue optimization, this dissertation presents a nonlinear semi-definite programming (NLSDP) model and algorithm for SSSC-OPF. The research has made important contribution to theory and practicality as well as a good attempt for enhancing small-signal stability using eigenvalue optimization. The main research achievements are as follows:
1. A nonlinear semi-definite programming (NLSDP) model for SSSC-OPF is proposed. According to the Lyapunov theorem, positive definite constraints are introduced to describe the small signal stability through an accurate equivalent expression. Thus, the SSSC-OPF will not spend more economic cost to meet the small-signal stability constraints. Furthermore, by transforming the eigenvalue optimization of a nonsymmetric matrix to SDP model over a symmetric matrix, the oscillation of the iterations of sensitivity-based algorithm because of the non-Lipschitz of spectral abscissa will not be suffered.
2. A transforming method which can formulate the positive definite constraints into smooth concave and nonlinear ones is proposed. Then the NLSDP model is transformed into a nonlinear programming (NLP), one which can be solved by the modern interior point method.
3. By choosing the controller parameters as variables, a NLSDP model for coordinated tuning of damping controller is proposed. It can coordinate the parameters of PSS and FACTS devices simultaneously or respectively. Besides, it can consider a wide range of operating conditions leading to a robust design.
There are six chapters in this dissertation. The main contents of the dissertation are arranged as follows:
In chapter1, the background of small-signal stability constrained OPF problem is introduced at first. The literature of the problem is also reviewed and the advantages and disadvantages of the reviewed methods are pointed out. Besides, the motivations of proposing an accurate model of SSSC-OPF based on eigenvalue optimization are analyzed in this chapter as well.
The conventional optimal power flow model and the modern interior point method are presented respectively in chapter2. Furthermore, multi-machine dynamic models and the small-signal stability analysis model are introduced. The relationships between the OPF problem and small-signal stability analysis are discussed. Meanwhile, the difficulty of SSSC-OPF is analyzed. This chapter is the foundation for the discussions of the coming chapters.
In chapter3, the eigenvalue optimization is introduced in details including its history, linear semi-definite programming model, nonlinear semi-definite programming model and their algorithm. This chapter provides the theoretical preparation for the coming chapters.
The small-signal stability constrained OPF problems are discussed in Chapter4, and the nonlinear semidefinite programming model of SSSC-OPF is proposed as well. By using the model transformation method, the NLSDP model is formulated to a NLP model accurately, and then solved by the modern interior point method. Extensive numerical simulations on WSCC3machine9bus system, Kundur4machine10bus system and IEEE5machine14bus system have shown that effectiveness of the model and correctness and robustness of the algorithm.
The chapter5extends the SSSC-OPF model in chapter and proposes a nonlinear semidefinite programming model for co-ordination of stabilizer parameter settings. It is solved by the same model transformation method. The feasibility is validated by the simulations of two systems.
The conclusions and remained questions worthy to be studied further are given in chapter6.
由于系统状态矩阵谱横坐标函数的隐式特征和非李普希茨特性,小干扰稳定约束最优潮流的建模异常困难,实际上这一问题的数学本质是特征值优化问题。随着半定规划和现代内点理论的发展,特征值优化理论日趋成熟,已在多个领域有着成功的应用。
基于特征值优化理论,本文提出了一种小干扰稳定约束最优潮流的非线性半定规划模型和算法。所提方法是利用特征值优化理论,解决电力系统的小干扰稳定问题的有益尝试,具有重要的理论和实用价值。主要研究成果如下：
1.提出了一种小干扰稳定约束最优潮流的非线性半定规划模型,它借助于李雅谱诺夫定理引入正定约束来表达小干扰稳定,是原问题的精确表达,避免了满足小干扰稳定而带来的不必要经济代价。同时,将非对称矩阵的特征值优化问题转换为对称矩阵的半定规划问题,克服了灵敏度算法中由于谱横坐标函数的非李普希茨特性引起的迭代振荡问题。
2.提出了一种将半定约束等价转换为光滑、凹、非线性约束的方法。所提转换方法将半定规划中半定约束转换为非线性约束后,整个模型变为非线性规划模型,从而可以利用高效的现代内点算法求解。
3.通过选择控制器参数为变量,提出了阻尼控制器协调优化的非线性半定规划模型,它可以分别或同时对系统内PSS、FACTS参数进行协调优化。在建立模型时还可考虑多种运行方式,使模型解适应更大的运行范围,更具鲁棒性。
全文分为六章,归纳如下：
第一章,介绍了小干扰稳定约束最优潮流问题的研究背景,回顾了国内外相关研究现状,指出了各种方法的优缺点,阐述了建立小干扰稳定约束最优潮流精确模型的动机。
第二章,分别介绍了常规最优潮流问题和及其现代内点算法,回顾了电力系统多机动态模型及小干扰稳定分析问题的数学模型。分析了最优潮流问题和小干扰稳定分析问题的联系,指出研究小干扰稳定约束最优潮流问题的难点,为后续章节的展开奠定基础。
第三章,详细论述了特征值优化理论,包括其发展历史,介绍了线性半定规划模型、非线性半定规划模型以及求解算法,为后续章节提出的数学模型作理论准备。
第四章,讨论了小干扰稳定约束最优潮流问题,提出了这一问题的非线性半定规划模型,并采用模型转换的方法将非线性半定规划模型精确转换为非线性规划模型,用现代内点法求解。同时还对WSCC3机9节点系统、Kundur4机10节点系统和IEEE5机14节点系统进行了仿真计算和分析,验证了模型的有效性和算法的正确性及鲁棒性。
第五章,以第四章的理论成果为基础,提出了阻尼控制器协调优化的非线性半定规划模型,并采用相同的模型转换方法进行求解。通过两个系统的仿真结果,验证了模型的有效性。
第六章,概括总结了全文的主要研究工作及其创新点,指出了今后有待进一步开展的研究工作。
The interconnections of large regional power network bring more challenges for the security of modern power system. Due to small-signal instability, low frequency oscillations pose a lethal threat to power system. Although the damping controllers can enhance the small signal stability effectively, it cannot guarantee that no oscillations occur. Therefore, it requires dispatchers to schedule the system generation not only based on economic object with the aid of the powerful tool Optimal Power Flow (OPF), but also ensuring a security margin with respect to small signal stability. In this context, it has been proposed to include small signal stability constraints into the optimal power flow model, thus creating Small-Signal Stability Constrained OPF (SSSC-OPF) techniques. They have aroused more and more interest recently.
For the implicit and non-Lipschitz property of spectral abscissa of system state matrix, it is a big challenge to model the SSSC-OPF directly. In general, SSSC-OPF is an eigenvalue optimization problem naturally. This theory had not made much progress for a long time, until the dramatic breakthrough in the interior point methods for the semi-definite programming (SDP). Nowadays, the therory has been introduced to many fields, having a number of successful applications.
Based on the eigenvalue optimization, this dissertation presents a nonlinear semi-definite programming (NLSDP) model and algorithm for SSSC-OPF. The research has made important contribution to theory and practicality as well as a good attempt for enhancing small-signal stability using eigenvalue optimization. The main research achievements are as follows:
1. A nonlinear semi-definite programming (NLSDP) model for SSSC-OPF is proposed. According to the Lyapunov theorem, positive definite constraints are introduced to describe the small signal stability through an accurate equivalent expression. Thus, the SSSC-OPF will not spend more economic cost to meet the small-signal stability constraints. Furthermore, by transforming the eigenvalue optimization of a nonsymmetric matrix to SDP model over a symmetric matrix, the oscillation of the iterations of sensitivity-based algorithm because of the non-Lipschitz of spectral abscissa will not be suffered.
2. A transforming method which can formulate the positive definite constraints into smooth concave and nonlinear ones is proposed. Then the NLSDP model is transformed into a nonlinear programming (NLP), one which can be solved by the modern interior point method.
3. By choosing the controller parameters as variables, a NLSDP model for coordinated tuning of damping controller is proposed. It can coordinate the parameters of PSS and FACTS devices simultaneously or respectively. Besides, it can consider a wide range of operating conditions leading to a robust design.
There are six chapters in this dissertation. The main contents of the dissertation are arranged as follows:
In chapter1, the background of small-signal stability constrained OPF problem is introduced at first. The literature of the problem is also reviewed and the advantages and disadvantages of the reviewed methods are pointed out. Besides, the motivations of proposing an accurate model of SSSC-OPF based on eigenvalue optimization are analyzed in this chapter as well.
The conventional optimal power flow model and the modern interior point method are presented respectively in chapter2. Furthermore, multi-machine dynamic models and the small-signal stability analysis model are introduced. The relationships between the OPF problem and small-signal stability analysis are discussed. Meanwhile, the difficulty of SSSC-OPF is analyzed. This chapter is the foundation for the discussions of the coming chapters.
In chapter3, the eigenvalue optimization is introduced in details including its history, linear semi-definite programming model, nonlinear semi-definite programming model and their algorithm. This chapter provides the theoretical preparation for the coming chapters.
The small-signal stability constrained OPF problems are discussed in Chapter4, and the nonlinear semidefinite programming model of SSSC-OPF is proposed as well. By using the model transformation method, the NLSDP model is formulated to a NLP model accurately, and then solved by the modern interior point method. Extensive numerical simulations on WSCC3machine9bus system, Kundur4machine10bus system and IEEE5machine14bus system have shown that effectiveness of the model and correctness and robustness of the algorithm.
The chapter5extends the SSSC-OPF model in chapter and proposes a nonlinear semidefinite programming model for co-ordination of stabilizer parameter settings. It is solved by the same model transformation method. The feasibility is validated by the simulations of two systems.
The conclusions and remained questions worthy to be studied further are given in chapter6.
引文
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