Study On Fault’s Temporal-spatial Distribution Characteristics And Structural Vulnerability In Power Grid Based On Complex System Theory

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.