基于矩阵摄动理论的微电网小扰动稳定性分析
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摘要
将分布式电源以微电网形式接入电网并网运行，并与之互为支撑，是发挥分布式电源效能的有效方式。分布式电源渗透率的逐步提高对微电网的运行提出了更高的要求，微电网的小扰动稳定性是本文关注的重点。本文主要工作如下：
（1）建立了微电网中通过逆变器/发电机并网的两类典型的分布式电源的小扰动稳定性分析模型，并对逆变器三种典型控制方式下的小扰动分析模型进行了分析。在小扰动稳定性计算中，根据微电网小扰动分析模型的稀疏性和分块性，采用改进十字链表法构建了微电网系统的状态矩阵。在此基础上，阐述了微电网特征值分布的广域性和区域性，以及导致这种分区域现象的影响因素。
（2）针对特征值和特征向量灵敏度难以准确求解的问题，提出了基于矩阵摄动理论的微电网中特征值和特征向量灵敏度求解的方法。系统性地介绍了矩阵摄动理论，并对微电网系统的状态矩阵进行参数摄动分析。在此基础上分析了微电网中孤立特征值和和重特征值情况下特征值及其所对应特征向量对控制参数的灵敏度计算。此方法具有较高的计算精度，既避免了解析法求解时复杂灵敏度计算公式的推导，又避免了扰动法求解时系统状态矩阵的重复形成和特征值问题的反复求解。
（3）提出了基于矩阵摄动理论的微电网中逆变器下垂系数协调优化设计的方法，用于提高微电网的小扰动稳定性。从矩阵参数摄动的角度对逆变器下垂系数进行了详细深入的摄动分析，得出了下垂系数对微电网系统状态矩阵的影响形式和影响程度，分析了产生该种影响的不同因素。提出了下垂系数协调优化的目标函数和算法流程，其中目标函数采用考虑多运行场景的综合目标函数，同时涵盖系统小扰动稳定性、阻尼比、稳定裕度等方面的要求，算法流程中分析了矩阵摄动理论如何与序列二次规划算法相结合以实现参数协调优化。算例分别从参数优化分析、时域仿真验证和参数鲁邦性分析等方面验证了矩阵摄动理论在微电网参数优化设计中的有效性、所提参数优化目标函数的可行性和下垂系数协调优化的鲁棒适应性。
Microgrid has attracted increasing attention as an effective means of integratingdistributed generation units into the power systems. The increasing penetration ofdistributed generation units has put forward higher demand for the microgrid stableoperation. This thesis mainly focuses on the small signal stability analysis of amicrogrid. The major work can be summarized as follows:
(1) Small signal models of distributed generation units are analyzed based on twodifferent grid-connection types, namely inverter-interfaced distributed generationunits and generator-interfaced distributed generation units. Besides, a comparativeanalysis is made on the small signal models of three typical controls ininverter-interfaced distributed generation units. Taking into account the characteristicsof sparsity and partition of the small signal models, a sparse filling and addressingapproach for constructing state matrix based on improved cross chain table method isadopted in the small signal stability calculation. On these basises, the extensive andregional distribution characteristics of eigenvalues in a microgrid are then illustratedin detail, revealing the corresponding influence factors as well.
(2) A matrix perturbation based approach is proposed for the sensitivity solutionand analysis of eigen-solutions in a microgrid. Matrix perturbation theory isintroduced systematically, based on which parameter perturbations are utilized toanalyze the construction characteristics of the state matrix. The sensitivities ofeigenvalues and eigenvectors to control variables are then obtained and analyzedaccording to different spatial distribution characteristics of eigenvalues, namelydistinct eigenvalues and multiple eigenvalues. Through the proposed approach, thesensitivities of eigen-solutions can be calculated with satisfactory accuracy. Not onlyare the complex derivations of sensitivity formulas concerned in analytical methodavoided, but also repeated solutions of eigenvalue problem involved in numericaldisturbance method is no longer required.
(3) In order to improve the small signal stability of a microgrid, a novel approachbased on matrix perturbation theory is proposed for the coordinated optimization ofthe droop coefficients in inverter controllers of a microgrid. Rigorous parameterperturbation analysis is applied to the droop coefficients of inverter controllers toidentify the manner and degree of their influence on the system state matrix. Furthermore, the influence factors are investigated as well. An eigenvalue-basedobjective function is proposed, aimed at ensuring the stability of the system,enhancing the damping characteristics, and maintaining the stability margin for a widerange of operating conditions. In the parameter-optimization process, matrixperturbation theory combining with sequential quadratic programming is applied insolving the minimization problem with constraints. In the examples, theparameter-optimization process, the time-domain simulations, and the robustnessanalysis of the optimized coefficients are carried out and analyzed in detail.Theoretical analysis and evaluation results have confirmed the effectiveness of theoptimization approach, the feasibility of the objective function, and the robustness ofthe optimized parameters.
（1）建立了微电网中通过逆变器/发电机并网的两类典型的分布式电源的小扰动稳定性分析模型，并对逆变器三种典型控制方式下的小扰动分析模型进行了分析。在小扰动稳定性计算中，根据微电网小扰动分析模型的稀疏性和分块性，采用改进十字链表法构建了微电网系统的状态矩阵。在此基础上，阐述了微电网特征值分布的广域性和区域性，以及导致这种分区域现象的影响因素。
（2）针对特征值和特征向量灵敏度难以准确求解的问题，提出了基于矩阵摄动理论的微电网中特征值和特征向量灵敏度求解的方法。系统性地介绍了矩阵摄动理论，并对微电网系统的状态矩阵进行参数摄动分析。在此基础上分析了微电网中孤立特征值和和重特征值情况下特征值及其所对应特征向量对控制参数的灵敏度计算。此方法具有较高的计算精度，既避免了解析法求解时复杂灵敏度计算公式的推导，又避免了扰动法求解时系统状态矩阵的重复形成和特征值问题的反复求解。
（3）提出了基于矩阵摄动理论的微电网中逆变器下垂系数协调优化设计的方法，用于提高微电网的小扰动稳定性。从矩阵参数摄动的角度对逆变器下垂系数进行了详细深入的摄动分析，得出了下垂系数对微电网系统状态矩阵的影响形式和影响程度，分析了产生该种影响的不同因素。提出了下垂系数协调优化的目标函数和算法流程，其中目标函数采用考虑多运行场景的综合目标函数，同时涵盖系统小扰动稳定性、阻尼比、稳定裕度等方面的要求，算法流程中分析了矩阵摄动理论如何与序列二次规划算法相结合以实现参数协调优化。算例分别从参数优化分析、时域仿真验证和参数鲁邦性分析等方面验证了矩阵摄动理论在微电网参数优化设计中的有效性、所提参数优化目标函数的可行性和下垂系数协调优化的鲁棒适应性。
Microgrid has attracted increasing attention as an effective means of integratingdistributed generation units into the power systems. The increasing penetration ofdistributed generation units has put forward higher demand for the microgrid stableoperation. This thesis mainly focuses on the small signal stability analysis of amicrogrid. The major work can be summarized as follows:
(1) Small signal models of distributed generation units are analyzed based on twodifferent grid-connection types, namely inverter-interfaced distributed generationunits and generator-interfaced distributed generation units. Besides, a comparativeanalysis is made on the small signal models of three typical controls ininverter-interfaced distributed generation units. Taking into account the characteristicsof sparsity and partition of the small signal models, a sparse filling and addressingapproach for constructing state matrix based on improved cross chain table method isadopted in the small signal stability calculation. On these basises, the extensive andregional distribution characteristics of eigenvalues in a microgrid are then illustratedin detail, revealing the corresponding influence factors as well.
(2) A matrix perturbation based approach is proposed for the sensitivity solutionand analysis of eigen-solutions in a microgrid. Matrix perturbation theory isintroduced systematically, based on which parameter perturbations are utilized toanalyze the construction characteristics of the state matrix. The sensitivities ofeigenvalues and eigenvectors to control variables are then obtained and analyzedaccording to different spatial distribution characteristics of eigenvalues, namelydistinct eigenvalues and multiple eigenvalues. Through the proposed approach, thesensitivities of eigen-solutions can be calculated with satisfactory accuracy. Not onlyare the complex derivations of sensitivity formulas concerned in analytical methodavoided, but also repeated solutions of eigenvalue problem involved in numericaldisturbance method is no longer required.
(3) In order to improve the small signal stability of a microgrid, a novel approachbased on matrix perturbation theory is proposed for the coordinated optimization ofthe droop coefficients in inverter controllers of a microgrid. Rigorous parameterperturbation analysis is applied to the droop coefficients of inverter controllers toidentify the manner and degree of their influence on the system state matrix. Furthermore, the influence factors are investigated as well. An eigenvalue-basedobjective function is proposed, aimed at ensuring the stability of the system,enhancing the damping characteristics, and maintaining the stability margin for a widerange of operating conditions. In the parameter-optimization process, matrixperturbation theory combining with sequential quadratic programming is applied insolving the minimization problem with constraints. In the examples, theparameter-optimization process, the time-domain simulations, and the robustnessanalysis of the optimized coefficients are carried out and analyzed in detail.Theoretical analysis and evaluation results have confirmed the effectiveness of theoptimization approach, the feasibility of the objective function, and the robustness ofthe optimized parameters.
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