基于复杂网络的电网结构健壮性及Braess悖论现象研究

应用复杂网络理论，建立电力系统的改进导纳模型，结合电网拓扑特性和电气特性对电网的级联故障进行研究。通过随机移除传输线引发电网级联故障，研究网络的节点数、平均度、发电站数量以及发电站的分布状况对系统健壮性的影响，并对小世界电网级联故障过程中的布雷斯(Braess)悖论现象进行分析。研究表明：网络的健壮性与其拓扑结构密切相关，平均度较大时，最近邻耦合网络和小世界网络健壮性曲线存在多个分叉点；在小世界结构电网中，一般平均度和节点数越大，发电站的数量越多，电网健壮性越好；发电站分散分布比发电站集中分布的电网健壮性更好。对网络容量增加导致健壮性降低的布雷斯现象进行解释。     

Node-wise robustness against fluctuations of power consumption in power grids

We propose a new concept of node-wise robustness of power grids under variation of effective power in one load node using a mathematical model that takes into account the change in voltage and reactive power of load nodes. We employ the topology of the power grid in eastern Japan. We define the robustness as the threshold value of the effective power, above which the steady state loses its stability. We show that the robustness is highly heterogeneous among the load nodes. We find that the shortest path length from generators is most highly correlated with the robustness of the load nodes. We numerically demonstrate that the supply of reactive power enhances the robustness.     

An electrical betweenness approach for vulnerability assessment of power grids considering the capacity of generators and load

Most existing research on the vulnerability of power grids based on complex networks ignores the electrical characteristics and the capacity of generators and load. In this paper, the electrical betweenness is defined by considering the maximal demand of load and the capacity of generators in power grids. The loss of load, which reflects the ability of power grids to provide sufficient power to customers, is introduced to measure the vulnerability together with the size of the largest cluster. The simulation results of the IEEE-118 bus system and the Central China Power Grid show that the cumulative distributions of node electrical betweenness follow a power-law and that the nodes with high electrical betweenness play critical roles in both topological structure and power transmission of power grids. The results prove that the model proposed in this paper is effective for analyzing the vulnerability of power grids.     

Models for the modern power grid

This article reviews different kinds of models for the electric power grid that can be used to understand the modern power system, the smart grid. From the physical network to abstract energy markets, we identify in the literature different aspects that co-determine the spatio-temporal multilayer dynamics of power system. We start our review by showing how the generation, transmission and distribution characteristics of the traditional power grids are already subject to complex behaviour appearing as a result of the the interplay between dynamics of the nodes and topology, namely synchronisation and cascade effects. When dealing with smart grids, the system complexity increases even more: on top of the physical network of power lines and controllable sources of electricity, the modernisation brings information networks, renewable intermittent generation, market liberalisation, prosumers, among other aspects. In this case, we forecast a dynamical co-evolution of the smart grid and other kind of networked systems that cannot be understood isolated. This review compiles recent results that model electric power grids as complex systems, going beyond pure technological aspects. From this perspective, we then indicate possible ways to incorporate the diverse co-evolving systems into the smart grid model using, for example, network theory and multi-agent simulation.     

Random matrix theory models of electric grid topology

The random matrix theory is useful in the study of large systems such as electric grids. These transmission systems can be modeled as complex networks, with high-voltage lines the edges that connect nodes representing power plants and substations. We draw upon established literature of complex systems theory and introduce methods from nuclear and statistical physics to identify new characteristics of these networks. We show that most grids can be characterized by the Gaussian Orthogonal Ensemble, an indicator of chaos in many complex systems. Under certain circumstances, however, grids may be described by Poisson statistics, an indicator of regularity. We use the random matrix formalism to describe the interconnection of multiple grids and construct a simple model of a distributed grid.     

The Application of Fractal Transform and Entropy for Improving Fault Tolerance and Load Balancing in Grid Computing Environments

This paper applies the entropy-based fractal indexing scheme that enables the grid environment for fast indexing and querying. It addresses the issue of fault tolerance and load balancing-based fractal management to make computational grids more effective and reliable. A fractal dimension of a cloud of points gives an estimate of the intrinsic dimensionality of the data in that space. The main drawback of this technique is the long computing time. The main contribution of the suggested work is to investigate the effect of fractal transform by adding R-tree index structure-based entropy to existing grid computing models to obtain a balanced infrastructure with minimal fault. In this regard, the presented work is going to extend the commonly scheduling algorithms that are built based on the physical grid structure to a reduced logical network. The objective of this logical network is to reduce the searching in the grid paths according to arrival time rate and path’s bandwidth with respect to load balance and fault tolerance, respectively. Furthermore, an optimization searching technique is utilized to enhance the grid performance by investigating the optimum number of nodes extracted from the logical grid. The experimental results indicated that the proposed model has better execution time, throughput, makespan, latency, load balancing, and success rate.     

Optimal intentional islanding to enhance the robustness of power grid networks

Intentional islanding of a power system can be an emergency response for isolating failures that might propagate and lead to major disturbances. Some of the islanding techniques suggested previously do not consider the power flow model; others are designed to minimize load shedding only within the islands. Often these techniques are computationally expensive. We aim to find approaches to partition power grids into islands to minimize the load shedding not only in the region where the failures start, but also in the topological complement of the region. We propose a new constraint programming formulation for optimal islanding in power grid networks. This technique works efficiently for small networks but becomes expensive as size increases. To address the scalability problem, we propose two grid partitioning methods based on modularity, properly modified to take into account the power flow model. They are modifications of the Fast Greedy algorithm and the Bloom algorithm, and are polynomial in running time. We tested these methods on the available IEEE test systems. The Bloom type method is faster than the Fast Greedy type, and can potentially provide results in networks with thousands of nodes. Our methods provide solutions which retain at least 40–50% of the system load. Overall, our methods efficiently balance load shedding and scalability.     

Long-range response to transmission line disturbances in DC electricity grids

We consider a DC electricity grid composed of transmission lines connecting power generators and consumers at its nodes. The DC grid is described by nonlinear equations derived from Kirchhoff's law. For an initial distribution of consumed and generated power, and given transmission line conductances, we determine the geographical distribution of voltages at the nodes. Adjusting the generated power for the Joule heating losses, we then calculate the electrical power flow through the transmission lines. Next, we study the response of the grid to an additional transmission line between two sites of the grid and calculate the resulting change in the power flow distribution. This change is found to decay slowly in space, with a power of the distance from the additional line. We find the geographical distribution of the power transmission, when a link is added. With a finite probability the maximal load in the grid becomes larger when a transmission line is added, a phenomenon that is known as Braess’ paradox. We find that this phenomenon is more pronounced in a DC grid described by the nonlinear equations derived from Kirchhoff's law than in a linearised flow model studied previously in Ref. [1]. We observe furthermore that the increase in the load of the transmission lines due to an added line is of the same order of magnitude as Joule heating. Interestingly, for a fixed system size the load of the lines increases with the degree of disorder in the geographical distribution of consumers and producers.     

Building resilient power grids from integrated risk governance perspective: A lesson learned from china's 2008 Ice-Snow Storm disaster

In the past three decades, the electric energy industry made great contribution to support rapid social and economic development in China, and meanwhile has been grown at the highest rate in the human history owing to the economic reform. In its new national development plan, more investment has been put into installation of both electricity generating capacity and transmitting capacity in order to meet fast growing demand of electric energy. However, energy resources, both fossil fuel and renewable types, and energy consumption and load centers in China are not evenly distributed in both spatial and temporal dimensions. Moreover, dominated by coal as its primary energy source, the whole eastern China is now entering an environmental crisis in which pollutants emitted by coal power plants contribute a large part. To balance the regional differences in energy sources and energy consumption while meeting the steadily increasing demands for electric energy for the whole country, in addition to increase electric generating capacity, building large-scale, long-distance ultra high voltage power grids is the top priority for next five years. China is a country prone to almost all kinds of natural disasters due to its vast, complex geographical and climatic conditions. In recent years, frequent natural disasters, especially extreme weather and climate events, have threatened the safety, reliability and stability of electric energy system in China. Unfortunately, with fast growth rate but lacking of risk assessing and prevention mechanism, many infrastructure constructions, including national power grids, are facing integrated and complex economic, social, institutional and ecological risks. In this paper, based on a case analysis of the Great Ice Storm in southern China in January 2008, risks of building a resilient power grid to deal with increasing threats from extreme weathers are discussed. The paper recommends that a systematic approach based on the social-ecological system framework should be applied to assess the risk factors associated with the power grid, and the tools to deal with complex dynamic systems need to be applied to deal with constant changes in the whole social-ecological system.     

双激励全波长压电陶瓷超声换能器工作特性分析

对由两个半波长夹心式压电陶瓷换能器级联而成的双激励全波长压电陶瓷超声换能器进行了研究,给出了这种换能器优值的相关参量表达式,通过数值计算分析了该换能器的结构和材料参数对其特性的影响。研究结果表明,当两组压电陶瓷堆分别位于各自所在半波长换能器的位移节点,或者在压电陶瓷片数量确定情况下两组压电陶瓷堆的片数相同时,换能器均能得到最大的优值;在相当大的范围内增加压电陶瓷片的数量,换能器的优值稍有降低,但力因子迅速增大;金属块材料对换能器的优值影响甚微。可见双激励全波长换能器能在不明显降低换能器综合性能的情况下有效增加压电陶瓷的体积,因而可大幅提高换能器的功率容量和负载能力,更适用于大功率重负载的场合。      

基于子网划分的电网关键节点识别

结合电网拓扑结构和潮流追踪技术,提出一种基于子网划分的电网关键节点识别方法。首先,根据发电机节点的邻域信息和功率将发电机节点划分为不同的子集,然后根据电网的系数分配矩阵将负荷节点划分到为其提供最大功率的发电机节点子集中,完成子网划分。接着采用多属性决策法对每个子网的节点进行排序,进一步改进并计算每个子网的结构系数,作为衡量子网重要性的指标。根据子网重要性,从每个子网中提取特定比例的候选关键节点,对这些候选节点依据多属性决策法重新排序,得到关键节点的最终排序。以IEEE14、IEEE57和IEEE118三种节点系统为例进行分析,得到各个系统的子网划分结果和各个标准网络的重要节点排序结果。采用本文方法、PageRank法和多属性决策法分别进行关键节点排序,并对排序靠前的关键节点进行级联故障性能实验和网络效能实验。实验表明,本文算法选择的关键节点对整个网络的传播性能影响最大,优于其他两种关键节点识别方法。     

Operating characteristics analysis of a dual-excited full-wavelength piezoceramic ultrasonic transducer

A dual-excited full-wavelength piezoceramic ultrasonic transducer as a cascade of two half-wavelength sandwich piezoceramic transducers is studied.The relevant parameters' expressions of the figure of merit N for the transducer are derived,and the effects of the structure and material parameters of the transducer on its characteristics are further analyzed by numerical calculation.The results show that when the two piezoceramic stacks are respectively located at the displacement nodes of their own half-wavelength transducers,or the two piezoceramic stacks have the same number of pieces in the case of a certain number of piezoceramic pieces,the figure of merit of the transducer can reach a maximum.With increasing of the number of piezoceramic pieces in a fairly large range,the figure of merit of the transducer slightly decreases,but the force factor of the transducer increases rapidly.The metal materials of the transducer have little effect on its figure of merit.Thus it can be seen that the dual-excited full-wavelength transducer can effectively increase the volumes of the piezoceramic stacks in the case of that the transducer's comprehensive performance has only a little bit of degradation,so it's power capacity and load capability can be dramatically improved,which means the transducer is more suitable for high power and heavy load applications.     

The impact of model detail on power grid resilience measures

Extreme events are a challenge to natural as well as man-made systems. For critical infrastructure like power grids, we need to understand their resilience against large disturbances. Recently, new measures of the resilience of dynamical systems have been developed in the complex system literature. Basin stability and survivability respectively assess the asymptotic and transient behavior of a system when subjected to arbitrary, localized but large perturbations in frequency and phase. To employ these methods that assess power grid resilience, we need to choose a certain model detail of the power grid. For the grid topology we considered the Scandinavian grid and an ensemble of power grids generated with a random growth model. So far the most popular model that has been studied is the classical swing equation model for the frequency response of generators and motors. In this paper we study a more sophisticated model of synchronous machines that also takes voltage dynamics into account, and compare it to the previously studied model. This model has been found to give an accurate picture of the long term evolution of synchronous machines in the engineering literature for post fault studies. We find evidence that some stable fix points of the swing equation become unstable when we add voltage dynamics. If this occurs the asymptotic behavior of the system can be dramatically altered, and basin stability estimates obtained with the swing equation can be dramatically wrong. We also find that the survivability does not change significantly when taking the voltage dynamics into account. Further, the limit cycle type asymptotic behaviour is strongly correlated with transient voltages that violate typical operational voltage bounds. Thus, transient voltage bounds are dominated by transient frequency bounds and play no large role for realistic parameters.     

Understanding the cascading failures in Indian power grids with complex networks theory

Two huge blackouts, occurred separately on 30 and 31 July 2012 in India, spread over half the country when three of its five regional grids collapsed, leaving hundreds of millions of people without government-supplied electricity and ringing once again alarm bells with security problems in electric power grid systems. The first investigation reveals that the outage of the second (backup) 400 kV Bina–Gwalior–Agra line on 29 July 2012 led to the cascading failure through the grid, which can be simulated and explained from the perspective of the complex networks theory. In this paper, a new model of a power grid involving the active and reactive power loads is proposed and then used to analyze the cascading behavior of power grids, which is also used to explain the reason of the blackout happening in India. Furthermore, some strategic advices are given for improving the stability and security of power grids, especially Indian power grids.     

基于局部序参数的电网同步性能优化

采用类Kuramoto模型对电网中的节点进行建模,利用局部序参数描述节点的局部同步能力.研究发现相比小功率节点,大功率节点到其直接邻居节点更难达到同步.提出一种网络耦合强度的非均匀分配方法,在网络总耦合强度不变的情况下,增大大功率节点到其直接邻居节点的耦合强度以及相关节点对之间的连边耦合强度,减少其余节点对间的耦合强度.研究表明,这种方法可以在一定范围内降低电网的同步临界耦合强度,改善网络的同步性能;但当这种耦合强度的非均匀性过大时,网络的同步性能开始恶化.     

Osher Flux with Entropy Fix for Two-Dimensional Euler Equations

We compare in this paper the properties of Osher flux with O-variant and P-variant (Osher-O flux and Osher-P flux) in finite volume methods for the two-dimensional Euler equations and propose an entropy fix technique to improve their robustness. We consider both first-order and second-order reconstructions. For inviscid hypersonic flow past a circular cylinder, we observe different problems for different schemes: a first-order Osher-O scheme on quadrangular grids yields a carbuncle shock, while a first-order Osher-P scheme results in a dislocation shock for high Mach number cases. In addition, a second-order Osher scheme can also yield a carbuncle shock or be unstable. To improve the robustness of these schemes we propose an entropy fix technique, and then present numerical results to show the effectiveness of the proposed method. In addition, the influence of grid aspects ratio, relative shock position to the grid and Mach number on shock stability are tested. Viscous heating problem and double Mach reflection problem are simulated to test the influence of the entropy fix on contact resolution and boundary layer resolution.     

SVD–GFD scheme to simulate complex moving body problems in 3D space

The present paper presents a hybrid meshfree-and-Cartesian grid method for simulating moving body incompressible viscous flow problems in 3D space. The method combines the merits of cost-efficient and accurate conventional finite difference approximations on Cartesian grids with the geometric freedom of generalized finite difference (GFD) approximations on meshfree grids. Error minimization in GFD is carried out by singular value decomposition (SVD). The Arbitrary Lagrangian–Eulerian (ALE) form of the Navier–Stokes equations on convecting nodes is integrated by a fractional-step projection method. The present hybrid grid method employs a relatively simple mode of nodal administration. Nevertheless, it has the geometrical flexibility of unstructured mesh-based finite-volume and finite element methods. Boundary conditions are precisely implemented on boundary nodes without interpolation. The present scheme is validated by a moving patch consistency test as well as against published results for 3D moving body problems. Finally, the method is applied on low-Reynolds number flapping wing applications, where large boundary motions are involved. The present study demonstrates the potential of the present hybrid meshfree-and-Cartesian grid scheme for solving complex moving body problems in 3D.     

八叉树网格到非结构混合网格的转换

在数值模拟中, 非结构网格的优势是可以采用相同的数值格式统一处理任意复杂的计算区域, 但在网格生成过程中难度大, 并且不容易控制网格质量。树结构网格可以认为是介于结构网格和非结构网格之间的一种网格, 目前已经有相对成熟的方法快速在复杂区域内生成二维四叉树网格和三维八叉树网格。在实际应用中, 数值方法往往需要在连接协调的非结构网格上做离散, 树结构网格中不同尺寸的网格之间连接不是协调的, 在应用上会受到很多限制。文章实现了树结构网格到非结构混合网格的转换, 这种转换在二维情况下就是将四叉树网格转换为非结构三角形和四边形的混合网格, 三维情况下则将八叉树网格转换为非结构混合网格。这一转换过程的难点在于需要考虑数千种不同的八叉树单元, 并给出能实现连接协调的非结构混合网格划分。可以出现的网格单元包括六面体、三棱柱、金字塔和四面体这4种不同情况。通过特别的分类, 实现了程序的自动生成, 这种程序自动生成技术一方面可以避免人工编写大量程序时的失误, 另一方面也使得对数以千计的不同情况的处理成为可能。通过对几个简单网格的测试, 对网格数据转换方法做了初步的验证。      

Model for cascading failures with adaptive defense in complex networks

<正>This paper investigates cascading failures in networks by considering interplay between the flow dynamic and the network topology,where the fluxes exchanged between a pair of nodes can be adaptively adjusted depending on the changes of the shortest path lengths between them.The simulations on both an artificially created scale-free network and the real network structure of the power grid reveal that the adaptive adjustment of the fluxes can drastically enhance the robustness of complex networks against cascading failures.Particularly,there exists an optimal region where the propagation of the cascade is significantly suppressed and the fluxes supported by the network are maximal. With this understanding,a costless strategy of defense for preventing cascade breakdown is proposed.It is shown to be more effective for suppressing the propagation of the cascade than the recent proposed strategy of defense based on the intentional removal of nodes.     

Proactive robustness control of heterogeneously loaded networks

A proactive measure to increase the robustness of heterogeneously loaded networks against cascades of overload failures is proposed. It is based on load-dependent weights. Compared to simple hop weights, respective shortest flow paths turn a previously heterogeneous load distribution into a more homogeneous one for the nodes and links of the network. The use of these flow paths increases the networks robustness and at the same time reduces the investment costs into the networks capacity layout. These findings are of relevance for critical infrastructures like communication and transportation networks.