基于复杂系统理论的电网故障时空分布特性及结构脆弱性研究
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摘要
随着电网规模不断扩大和区域联网，电网发生连锁故障的风险不断增加。大量的故障统计数据表明，尽管电网发生连锁故障的概率较小，但其一旦发生，其后果是极其严重的。因此，认识电网连锁故障并揭示其形成机理具有重要的现实意义和理论价值。本文应用复杂系统理论分析电网故障的时空分布特性和电网结构的分形特性，试图从新的角度认识电网的自组织临界性，从而更细致地描述电网的脆弱性以及电网连锁故障的发生机理。本文重点在以下几个方面开展了较为深入的研究：
第一，发现电网故障在时间上具有强长程相关性，空间上具有幂律分布特性。经统计分析某省级电网12年零8个月的实际故障数据，得出电网故障时间间隔序列具有无标度特性；采用R/S和SWV模型计算Hurst指数，得到的Hurst指数均为0.86以上，说明电网故障序列具有强长程相关性，存在异常扩散；将故障序列按故障发生位置所属的设备分为五个子序列，各子序列仍然具有无标度性和长程相关性。通过对该省级电网故障的发生地点进行统计，得到电网故障的地点与其出现的频次呈幂律分布。
第二，发现电网拓扑结构具有相似性。通过对五个电网分别构建无权和加权网络模型，实证分析了电网的中心性和层次性，得到五个电网都具有小世界特性，并寻找到电网拓扑结构中的关键节点；发现无权电网具有异配性，度分布为指数分布，而加权电网具有同配性，点权分布有幂律尾现象。此外，详细讨论了在电网中起重要传输作用的三节点和四节点模体，得出由500kV和220kV线路构成的子图是电网模体的主要结构形式。这些结构子图也是电网的关键组元，并起着主要的传输作用。由于高电压等级的节点互连成为电网的模体，使得加权电网总体上表现为同配性质。
第三，发现电网拓扑结构的特征值谱具有多重分形特征。基于IEEE118节点系统和某省级电网系统，构建了以导纳为边权的加权拓扑模型，得到相应拓扑结构的邻接矩阵和拉普拉斯矩阵。采用MF-DFA方法对电网两类矩阵的特征值谱进行分析，发现这两类矩阵的特征值谱都表现出典型的多重分形特征，说明了电网结构在生长和演化过程中具有某种程度上的统计相似性，而且是一种内在的固有特性。虽然不同电网系统特征值谱的多重分形特性之间有些差异，但基于拉普拉斯矩阵特征值谱的多重分形特性在不同电网系统间的差异明显减少，表明拉普拉斯矩阵特征值谱体现了更一般的网络结构特征。
第四，提出分析电网线路故障传播的阈值模型。基于该阈值模型定量地分析了电网线路故障引起连锁故障的临界阈值，并发现线路的临界阈值越大，该线路对电网系统的脆弱性影响程度就越大。此阈值模型为电网寻找关键线路提供了新方法。通过两个不同规模的算例实证分析，证明了此方法的正确性和有效性。
The expansion and regional interconnection of power grid bring out an increasing risk ofgrid cascading failures. Large amount of statistical data of failures suggests that thoughcascading failures is less likely to occur, the consequence will be extremely severe once ithappens, since negligible common failures may lead to serious cascading failures. Therefore,understanding cascading failures of power grid and the mechanism of its formation holdsgreat practical significance and theoretical value. Numerous studies have proposed thatself-organized criticality can be used to describe the cascading failures of power grid.However, current confirmations of self-organized criticality exists in power grid are basedmostly on the power-law relationship between the scale of the grid failure and itscorresponding frequency. Few people have focused on studies of whether the grid system hasother feature of self-organized critical system. Thus in this paper, we apply complex systemtheory to the temporal and spatial correlation of grid faults’ behavior and the fractalcharacteristics of grid structure, which create a new interpretation of self-organized criticalityof power grid, bringing out a more detailed description of the vulnerability of power grid aswell as the mechanism of grid cascading failure.
First of all, we studied the long-range correlation of power grid’s temporal and spatialpatterns. The results imply not only that power grid’s failures are intermittent and paroxysmal,but also that they hold long-range correlation in time domain as well as power-lawdistribution in spatial domain, and fractal characteristics. Based on real faults data of certainprovincial power grid, several time interval series of power grid failures have beenconstructed. After applying KS test, the series present leptokurtic distribution. Statisticalanalysis on these time series based on three different time scales (minute/hour/day) showstheir nonlinear characteristics and power-law distribution. Strong long-range correlation ofthese time series has been discovered by calculating Hurst index with both R/S (rescaledrange analysis) and SWV (scaled windows variance methods) models, implying anomalousdiffusion of power grid's fault. After sorting these time series according to different types ofequipments of faults (The types of equipments of faults are classified into AC line, thermalpower, bus station, hydropower, DC line), we studied their long-range correlation respectivelyand found that the results remained the same. Through studying the spatial distribution onpower grid’s faults, we discover that its probability distribution obeys the power-law. Thelong-range correlation of power grid’s temporal and spatial patterns is one crucialmathematical characterization of self-organized criticality of power grid.
Further, the topology structure characteristics of five power grids have been compared,which indicates that topology structures of power grids from different places, different sizes,and different time share great similarities--all hold small world property. The main structuralforms of motifs of power grids have also been discovered. Through respectively constructingmodels for unweighted and weighted(taking transmission lines impedance or admittance asedge weight) power grids, we analyze their degree, excess average degree, node strength,excess average node strength, characteristic path length, betweeness, closeness, clusteringcoefficient, assortativity coefficient, motifs etc. The results show that power grids have largeclustering coefficient and short characteristic path length. Such small world property couldaccelerate the spread of faults. Moreover, critical nodes of power grids' topology have beendetermined and that each unweighted network of power grids is disassortative, while thecorresponding weighted power grid is assortative. Besides, three-node and four-node motifs,playing significant role in transmission of power grids, have been discussed with details andthe main structures of transmission motifs have been discovered.
Afterwards, multifractal characteristics of the topology have been discovered bystudying eigenvalue spectra of five power grids' topology. Through constructing adjacentmatrices of power grids as well as Laplace matrices, applying MF-DFA model to themultifractal analysis of eigenvalue spectra of both matrices, we found that both matricesshowed multifractality. In other words, spatial structures of power grid are multifractal. Later,it has been observed that the adjacent matrices show greater heterogeneity and strongermutifractal characteristics than those of the Laplace matrices. Fractal characteristic of powergrid topology is another crucial mathematical characterization of self-organized criticality ofpower grid.
Finally, in light of self-organized criticality, DC power flow model as well as parametersof power grids’ topology, and the instantaneous impact of power redistribution led bytransmission line faults, the threshold model is proposed which analyze transmission linefaults spreading. The critical threshold of cascading failures caused by transmission line faultshas been analyzed quantitatively based on the threshold model. The results show that thelarger the line critical threshold is, the greater it impacts on power system’s vulnerability.Accordingly, the line critical threshold can be used to find the key lines of the power gridswhich are also the vulnerable points or sources.
第一，发现电网故障在时间上具有强长程相关性，空间上具有幂律分布特性。经统计分析某省级电网12年零8个月的实际故障数据，得出电网故障时间间隔序列具有无标度特性；采用R/S和SWV模型计算Hurst指数，得到的Hurst指数均为0.86以上，说明电网故障序列具有强长程相关性，存在异常扩散；将故障序列按故障发生位置所属的设备分为五个子序列，各子序列仍然具有无标度性和长程相关性。通过对该省级电网故障的发生地点进行统计，得到电网故障的地点与其出现的频次呈幂律分布。
第二，发现电网拓扑结构具有相似性。通过对五个电网分别构建无权和加权网络模型，实证分析了电网的中心性和层次性，得到五个电网都具有小世界特性，并寻找到电网拓扑结构中的关键节点；发现无权电网具有异配性，度分布为指数分布，而加权电网具有同配性，点权分布有幂律尾现象。此外，详细讨论了在电网中起重要传输作用的三节点和四节点模体，得出由500kV和220kV线路构成的子图是电网模体的主要结构形式。这些结构子图也是电网的关键组元，并起着主要的传输作用。由于高电压等级的节点互连成为电网的模体，使得加权电网总体上表现为同配性质。
第三，发现电网拓扑结构的特征值谱具有多重分形特征。基于IEEE118节点系统和某省级电网系统，构建了以导纳为边权的加权拓扑模型，得到相应拓扑结构的邻接矩阵和拉普拉斯矩阵。采用MF-DFA方法对电网两类矩阵的特征值谱进行分析，发现这两类矩阵的特征值谱都表现出典型的多重分形特征，说明了电网结构在生长和演化过程中具有某种程度上的统计相似性，而且是一种内在的固有特性。虽然不同电网系统特征值谱的多重分形特性之间有些差异，但基于拉普拉斯矩阵特征值谱的多重分形特性在不同电网系统间的差异明显减少，表明拉普拉斯矩阵特征值谱体现了更一般的网络结构特征。
第四，提出分析电网线路故障传播的阈值模型。基于该阈值模型定量地分析了电网线路故障引起连锁故障的临界阈值，并发现线路的临界阈值越大，该线路对电网系统的脆弱性影响程度就越大。此阈值模型为电网寻找关键线路提供了新方法。通过两个不同规模的算例实证分析，证明了此方法的正确性和有效性。
The expansion and regional interconnection of power grid bring out an increasing risk ofgrid cascading failures. Large amount of statistical data of failures suggests that thoughcascading failures is less likely to occur, the consequence will be extremely severe once ithappens, since negligible common failures may lead to serious cascading failures. Therefore,understanding cascading failures of power grid and the mechanism of its formation holdsgreat practical significance and theoretical value. Numerous studies have proposed thatself-organized criticality can be used to describe the cascading failures of power grid.However, current confirmations of self-organized criticality exists in power grid are basedmostly on the power-law relationship between the scale of the grid failure and itscorresponding frequency. Few people have focused on studies of whether the grid system hasother feature of self-organized critical system. Thus in this paper, we apply complex systemtheory to the temporal and spatial correlation of grid faults’ behavior and the fractalcharacteristics of grid structure, which create a new interpretation of self-organized criticalityof power grid, bringing out a more detailed description of the vulnerability of power grid aswell as the mechanism of grid cascading failure.
First of all, we studied the long-range correlation of power grid’s temporal and spatialpatterns. The results imply not only that power grid’s failures are intermittent and paroxysmal,but also that they hold long-range correlation in time domain as well as power-lawdistribution in spatial domain, and fractal characteristics. Based on real faults data of certainprovincial power grid, several time interval series of power grid failures have beenconstructed. After applying KS test, the series present leptokurtic distribution. Statisticalanalysis on these time series based on three different time scales (minute/hour/day) showstheir nonlinear characteristics and power-law distribution. Strong long-range correlation ofthese time series has been discovered by calculating Hurst index with both R/S (rescaledrange analysis) and SWV (scaled windows variance methods) models, implying anomalousdiffusion of power grid's fault. After sorting these time series according to different types ofequipments of faults (The types of equipments of faults are classified into AC line, thermalpower, bus station, hydropower, DC line), we studied their long-range correlation respectivelyand found that the results remained the same. Through studying the spatial distribution onpower grid’s faults, we discover that its probability distribution obeys the power-law. Thelong-range correlation of power grid’s temporal and spatial patterns is one crucialmathematical characterization of self-organized criticality of power grid.
Further, the topology structure characteristics of five power grids have been compared,which indicates that topology structures of power grids from different places, different sizes,and different time share great similarities--all hold small world property. The main structuralforms of motifs of power grids have also been discovered. Through respectively constructingmodels for unweighted and weighted(taking transmission lines impedance or admittance asedge weight) power grids, we analyze their degree, excess average degree, node strength,excess average node strength, characteristic path length, betweeness, closeness, clusteringcoefficient, assortativity coefficient, motifs etc. The results show that power grids have largeclustering coefficient and short characteristic path length. Such small world property couldaccelerate the spread of faults. Moreover, critical nodes of power grids' topology have beendetermined and that each unweighted network of power grids is disassortative, while thecorresponding weighted power grid is assortative. Besides, three-node and four-node motifs,playing significant role in transmission of power grids, have been discussed with details andthe main structures of transmission motifs have been discovered.
Afterwards, multifractal characteristics of the topology have been discovered bystudying eigenvalue spectra of five power grids' topology. Through constructing adjacentmatrices of power grids as well as Laplace matrices, applying MF-DFA model to themultifractal analysis of eigenvalue spectra of both matrices, we found that both matricesshowed multifractality. In other words, spatial structures of power grid are multifractal. Later,it has been observed that the adjacent matrices show greater heterogeneity and strongermutifractal characteristics than those of the Laplace matrices. Fractal characteristic of powergrid topology is another crucial mathematical characterization of self-organized criticality ofpower grid.
Finally, in light of self-organized criticality, DC power flow model as well as parametersof power grids’ topology, and the instantaneous impact of power redistribution led bytransmission line faults, the threshold model is proposed which analyze transmission linefaults spreading. The critical threshold of cascading failures caused by transmission line faultshas been analyzed quantitatively based on the threshold model. The results show that thelarger the line critical threshold is, the greater it impacts on power system’s vulnerability.Accordingly, the line critical threshold can be used to find the key lines of the power gridswhich are also the vulnerable points or sources.
引文
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