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Topological theory of resilience and failure spreading in flow networks
流网络中的韧性拓扑理论与故障扩散

Franz Kaiser ∘ ∘ ^(@){ }^{\circ} ) and Dirk Witthaut † † ^(†){ }^{\dagger}
Franz Kaiser ∘ ∘ ^(@){ }^{\circ} ) 和 Dirk Witthaut † † ^(†){ }^{\dagger} Forschungszentrum Jülich, Institute for Energy and Climate Research (IEK-STE), 52428 Jülich, Germany and Institute for Theoretical Physics, University of Cologne, 50937 Köln, Germany
德国尤利希研究中心，能源与气候研究所（IEK-STE），52428 尤利希，以及科隆大学理论物理研究所，50937 科隆

(Received 11 September 2020; accepted 7 May 2021; published 1 June 2021)
（收到日期：2020 年 9 月 11 日；接受日期：2021 年 5 月 7 日；发表日期：2021 年 6 月 1 日）

Abstract  摘要

Link failures in supply networks can have catastrophic consequences that can lead to a complete collapse of the network. Strategies to prevent failure spreading are thus heavily sought after. Here, we make use of a spanning tree formulation of link failures in linear flow networks to analyze topological structures that prevent failure spreading. In particular, we exploit a result obtained for resistor networks based on the matrix tree theorem to analyze failure spreading after link failures in power grids. Using a spanning tree formulation of link failures, we analyze three strategies based on the network topology that allow us to reduce the impact of single link failures. All our strategies either do not reduce the grid’s ability to transport flow or do in fact improve it-in contrast to traditional containment strategies based on lowering network connectivity. Our results also explain why certain connectivity features completely suppress any failure spreading as reported in recent publications.
供应链网络中的链路故障可能引发灾难性后果，导致网络完全崩溃。因此，防止故障扩散的策略备受关注。在此，我们利用线性流网络中链路故障的生成树公式来分析防止故障扩散的拓扑结构。特别地，我们基于矩阵树定理在电阻网络中获得的结果，分析了电网中链路故障后的故障扩散情况。通过链路故障的生成树公式，我们分析了三种基于网络拓扑的策略，这些策略使我们能够减少单个链路故障的影响。我们的所有策略要么不降低电网的输运能力，要么实际上提高了其输运能力——这与基于降低网络连通性的传统控制策略形成对比。我们的结果还解释了为什么某些连通性特征完全抑制了故障扩散，正如近期出版物所报道的那样。

DOI: 10.1103/PhysRevResearch.3.023161

I. INTRODUCTION  I. 引言

The theory of linear flow networks provides a powerful framework, allowing one to study systems ranging from water supply networks [1,2] and biological networks, such as leaf venation networks [3-6], to resistor networks [7-9], or ac power grids [10,11]. Failures of transportation links in these networks can have catastrophic consequences up to a complete collapse of the network. As a result, link failures in linear flow networks and strategies to limit their consequences are a field of active study [12-19].
线性流网络理论提供了一个强大的框架，使人们能够研究从供水网络[1,2]和生物网络（如叶脉网络[3-6]）到电阻网络[7-9]或交流电网[10,11]等各种系统。这些网络中交通连接的失效可能导致灾难性后果，甚至完全崩溃网络。因此，线性流网络中的连接失效及其后果限制策略是一个活跃的研究领域[12-19]。
The study of linear flow networks is intimately related to graph theory since most phenomena can be analyzed on purely topological grounds [7]. This connection dates back to work by Kirchhoff [8], who analyzed resistor networks and introduced several major tools that are now the basis of the theory of complex networks, such as the matrix tree theorem [7,8,20]. These tools can now serve as a basis for the analysis of failure spreading in ac power grids, which can be modeled as linear flow networks based on the dc approximation [11]. A substantial part of security analysis in power grids is dedicated to the study of transmission line outages since they can lead to cascading outages in a series of failures [21-23].
线性流网络的研究与图论密切相关，因为大多数现象都可以纯粹从拓扑学的角度进行分析[7]。这种联系可以追溯到 Kirchhoff[8]的工作，他分析了电阻网络并引入了几个现在成为复杂网络理论基础的重大工具，如矩阵树定理[7,8,20]。这些工具现在可以作为分析交流电网中故障扩散的基础，这些电网可以根据直流近似建模为线性流网络[11]。电网安全分析的大部分工作都致力于研究输电线路故障，因为它们可能导致一系列故障中的级联故障[21-23]。
The topological approach to failure spreading has been exploited to demonstrate that the strength of flow rerouting after link failures decays with distance to the failing link [12-15]. In particular, the so-called rerouting distance based
拓扑学方法已被用于证明，在链路故障后重新路由的强度随着与故障链路的距离而衰减[12-15]。特别是，所谓的重新路由距离基于
on cycles in the network has been found to predict flow rerouting very well [12]. However, the analysis of flow rerouting still lacks a theoretical foundation. Here, we demonstrate that these observations made for flow rerouting may be understood based on a formalism originally developed to study current flows in resistor networks that uses spanning trees (STs) of the underlying graph. Moreover, the formalism explains recent results regarding the shielding against failure spreading in complex networks.
在网络循环上的研究已被发现能很好地预测流量重定向[12]。然而，流量重定向的分析仍然缺乏理论基础。在这里，我们证明对流重定向所做这些观察可以通过一个最初用于研究电阻网络中电流的正式形式来理解，该形式使用基础图的生成树（ST）。此外，该形式解释了关于复杂网络中防止故障扩散的最新结果。
This paper is structured as follows. In Sec. II, we give an overview over the theory of linear flow networks and present an important lemma that relates the current flows in these networks to STs. In Sec. III, we demonstrate the analogy between such networks and ac power grids in the dc approximation and relate the ST formulation to line outages studied in power system security analysis. Finally, in Sec. IV we show how this formulation may be used to understand why certain connectivity features inhibit failure spreading extending on recent results [19].
本文的结构如下。在第二节中，我们概述了线性流网络的理论，并提出了一个重要引理，该引理将这些网络中的电流流与生成树联系起来。在第三节中，我们展示了这些网络与直流近似下的交流电网之间的类比，并将生成树公式与电力系统安全分析中研究的线路故障联系起来。最后，在第四节中，我们展示了如何使用这种公式来理解为什么某些连通性特征会抑制故障扩散，扩展了最近的结果[19]。

II. FUNDAMENTALS OF RESISTOR NETWORKS
第二节. 电阻网络基础

Resistor networks are a prime example of linear flow networks and have inspired research throughout centuries [7,8,24]. A resistor network can be described using a graph as follows. Let G = ( E , V ) G = ( E , V ) G=(E,V)G=(E, V) be a connected graph with vertex set V = { v 1 , … , v N } V = v 1 , … , v N V={v_(1),dots,v_(N)}V=\left\{v_{1}, \ldots, v_{N}\right\} and M M MM edges in the edge set E E EE. Then we assign a weight w k w k w_(k)w_{k} to each edge e k = ( a , b ) e k = ( a , b ) e_(k)=(a,b)e_{k}=(a, b) in the graph given by the inverse resistance w k = R k − 1 w k = R k − 1 w_(k)=R_(k)^(-1)w_{k}=R_{k}^{-1} between its terminal vertices a a aa and b b bb. If there is a potential difference v k = V a − V b v k = V a − V b v_(k)=V_(a)-V_(b)v_{k}=V_{a}-V_{b} between the terminal vertices of edge e k = ( a , b ) e k = ( a , b ) e_(k)=(a,b)e_{k}=(a, b), according to Ohm’s law there is a current flow i k i k i_(k)i_{k} between the two vertices given by
电阻网络是线性流网络的一个典型例子，并在数个世纪以来激发了相关研究[7,8,24]。电阻网络可以使用图来描述如下。设 G = ( E , V ) G = ( E , V ) G=(E,V)G=(E, V) 是一个连通图，其顶点集为 V = { v 1 , … , v N } V = v 1 , … , v N V={v_(1),dots,v_(N)}V=\left\{v_{1}, \ldots, v_{N}\right\} ，边集 E E EE 包含 M M MM 条边。然后我们为图中的每条边 e k = ( a , b ) e k = ( a , b ) e_(k)=(a,b)e_{k}=(a, b) 分配一个权重 w k w k w_(k)w_{k} ，该权重由其端点顶点 a a aa 和 b b bb 之间的电阻的倒数 w k = R k − 1 w k = R k − 1 w_(k)=R_(k)^(-1)w_{k}=R_{k}^{-1} 给出。如果边 e k = ( a , b ) e k = ( a , b ) e_(k)=(a,b)e_{k}=(a, b) 的端点之间存在电势差 v k = V a − V b v k = V a − V b v_(k)=V_(a)-V_(b)v_{k}=V_{a}-V_{b} ，根据欧姆定律，这两个顶点之间会有电流流 i k i k i_(k)i_{k} ，其表达式为
i k = v k R k = V a − V b R k i k = v k R k = V a − V b R k i_(k)=(v_(k))/(R_(k))=(V_(a)-V_(b))/(R_(k))i_{k}=\frac{v_{k}}{R_{k}}=\frac{V_{a}-V_{b}}{R_{k}}
In order to give a direction to the current flow, we assign an arbitrary orientation to each edge in the graph that is encoded by the graph’s edge-node-incidence matrix B ∈ R N × M B ∈ R N × M BinR^(N xx M)\mathbf{B} \in \mathbb{R}^{N \times M} defined as [7]
为了给电流流一个方向，我们为图中的每条边分配一个任意方向，该方向由图的边-节点关联矩阵 B ∈ R N × M B ∈ R N × M BinR^(N xx M)\mathbf{B} \in \mathbb{R}^{N \times M} 定义，如[7]所示
B n , ℓ = { 1 if line ℓ starts at node n − 1 if line ℓ ends at node n 0 otherwise B n , ℓ = 1  if line  ℓ  starts at node  n − 1  if line  ℓ  ends at node  n 0  otherwise  B_(n,ℓ)={[1" if line "ℓ" starts at node "n],[-1" if line "ℓ" ends at node "n],[0" otherwise "]:}B_{n, \ell}=\left\{\begin{aligned} 1 & \text { if line } \ell \text { starts at node } n \\ -1 & \text { if line } \ell \text { ends at node } n \\ 0 & \text { otherwise } \end{aligned}\right.
The current flows and voltages are then subject to Kirchhoff’s circuit laws [8]. The first of the laws, typically referred to as Kirchhoff’s current law, at an arbitrary node j ∈ V ( G ) j ∈ V ( G ) j in V(G)j \in V(G) reads as
电流流和电压需要满足基尔霍夫电路定律[8]。其中第一条定律，通常被称为基尔霍夫电流定律，在任意节点 j ∈ V ( G ) j ∈ V ( G ) j in V(G)j \in V(G) 上的表述为
∑ e k ∈ Λ ( j ) M i k = I j ∑ e k ∈ Λ ( j ) M   i k = I j sum_(e_(k)in Lambda(j))^(M)i_(k)=I_(j)\sum_{e_{k} \in \Lambda(j)}^{M} i_{k}=I_{j}
Here, I j ∈ R I j ∈ R I_(j)inRI_{j} \in \mathbb{R} is the current injected into or withdrawn from node j j jj, and Λ ( j ) ⊂ E ( G ) Λ ( j ) ⊂ E ( G ) Lambda(j)sub E(G)\Lambda(j) \subset E(G) is the set of all edges that connect to node j j jj respecting their orientation. The current law may be regarded as a continuity equation and thus states that the inflows and outflows at each node in the network have to balance with the current injections at the respective node. It may be written more compactly making use of the node-edgeincidence matrix
此处， I j ∈ R I j ∈ R I_(j)inRI_{j} \in \mathbb{R} 表示注入或从节点 j j jj 抽取的电流， Λ ( j ) ⊂ E ( G ) Λ ( j ) ⊂ E ( G ) Lambda(j)sub E(G)\Lambda(j) \subset E(G) 表示所有连接到节点 j j jj 并保持其方向的边的集合。电流定律可被视为连续性方程，因此它表明网络中每个节点的流入和流出必须与该节点的电流注入相平衡。利用节点-边关联矩阵，可以更简洁地表示：
B i = I B i = I Bi=I\mathbf{B i}=\mathbf{I}
where i = ( i 1 , … , i M ) ⊤ ∈ R M i = i 1 , … , i M ⊤ ∈ R M i=(i_(1),dots,i_(M))^(TT)inR^(M)\mathbf{i}=\left(i_{1}, \ldots, i_{M}\right)^{\top} \in \mathbb{R}^{M} is a vector of current flows and I = ( I 1 , … , I N ) ⊤ ∈ R N I = I 1 , … , I N ⊤ ∈ R N I=(I_(1),dots,I_(N))^(TT)inR^(N)\mathbf{I}=\left(I_{1}, \ldots, I_{N}\right)^{\top} \in \mathbb{R}^{N} is a vector of current injections. On the other hand, we can also introduce a more compact notation for Ohm’s law (1) by defining a vector of nodal voltage levels V = ( V 1 , … , V N ) ⊤ ∈ R N V = V 1 , … , V N ⊤ ∈ R N V=(V_(1),dots,V_(N))^(TT)inR^(N)\mathbf{V}=\left(V_{1}, \ldots, V_{N}\right)^{\top} \in \mathbb{R}^{N} and a diagonal matrix of edge resistances R = diag ( R 1 , … , R M ) ∈ R M × M R = diag R 1 , … , R M ∈ R M × M R=diag(R_(1),dots,R_(M))inR^(M xx M)\mathbf{R}=\operatorname{diag}\left(R_{1}, \ldots, R_{M}\right) \in \mathbb{R}^{M \times M} such that Ohm’s law reads as
其中 i = ( i 1 , … , i M ) ⊤ ∈ R M i = i 1 , … , i M ⊤ ∈ R M i=(i_(1),dots,i_(M))^(TT)inR^(M)\mathbf{i}=\left(i_{1}, \ldots, i_{M}\right)^{\top} \in \mathbb{R}^{M} 是电流流量的向量， I = ( I 1 , … , I N ) ⊤ ∈ R N I = I 1 , … , I N ⊤ ∈ R N I=(I_(1),dots,I_(N))^(TT)inR^(N)\mathbf{I}=\left(I_{1}, \ldots, I_{N}\right)^{\top} \in \mathbb{R}^{N} 是电流注入的向量。另一方面，我们也可以通过定义节点电压水平向量 V = ( V 1 , … , V N ) ⊤ ∈ R N V = V 1 , … , V N ⊤ ∈ R N V=(V_(1),dots,V_(N))^(TT)inR^(N)\mathbf{V}=\left(V_{1}, \ldots, V_{N}\right)^{\top} \in \mathbb{R}^{N} 和边电阻的对角矩阵 R = diag ( R 1 , … , R M ) ∈ R M × M R = diag R 1 , … , R M ∈ R M × M R=diag(R_(1),dots,R_(M))inR^(M xx M)\mathbf{R}=\operatorname{diag}\left(R_{1}, \ldots, R_{M}\right) \in \mathbb{R}^{M \times M} 来引入更简洁的欧姆定律（1）表示，使得欧姆定律表示为：
R i = B ⊤ V . R i = B ⊤ V . Ri=B^(TT)V.\mathbf{R i}=\mathbf{B}^{\top} \mathbf{V} .
Combining Ohm’s law with Kirchhoff’s current law, we arrive at the following relationship between nodal voltages V V V\mathbf{V} and nodal current injections I I I\mathbf{I} :
结合欧姆定律和基尔霍夫电流定律，我们得到节点电压 V V V\mathbf{V} 和节点电流注入 I I I\mathbf{I} 之间的关系：
I = B R − 1 B ⊤ V I = B R − 1 B ⊤ V I=BR^(-1)B^(TT)V\mathbf{I}=\mathbf{B R}^{-1} \mathbf{B}^{\top} \mathbf{V}
This Poisson-like equation has been analyzed in different contexts [7,12,25]. Note that Kirchhoff’s voltage law is automatically satisfied by virtue of Eq. (3), because the resulting vector of potential differences v = B T V v = B T V v=B^(T)V\mathbf{v}=\mathbf{B}^{T} \mathbf{V} vanishes along any closed cycle due to the duality between the graph’s cycle space and its cut space [7,26]. In addition to that, the potential at one node may be chosen freely without affecting the result.
这种类似泊松的方程在不同的背景下已被分析过[7,12,25]。请注意，由于公式(3)，基尔霍夫电压定律自动得到满足，因为由势差向量 v = B T V v = B T V v=B^(T)V\mathbf{v}=\mathbf{B}^{T} \mathbf{V} 在任意闭合回路中消失，这是由于图的回路空间与其割空间之间的对偶性[7,26]。此外，一个节点的电势可以自由选择而不影响结果。
The matrix connecting the two quantities is referred to as a weighted graph Laplacian or Kirchhoff matrix L = L = L=\mathbf{L}= B R − 1 B ⊤ ∈ R N × N B R − 1 B ⊤ ∈ R N × N BR^(-1)B^(TT)inR^(N xx N)\mathbf{B R}^{-1} \mathbf{B}^{\top} \in \mathbb{R}^{N \times N} and characterizes the underlying graph completely. It has the following entries [7]:
连接这两种量的矩阵被称为加权图拉普拉斯矩阵或基尔霍夫矩阵 L = L = L=\mathbf{L}= B R − 1 B ⊤ ∈ R N × N B R − 1 B ⊤ ∈ R N × N BR^(-1)B^(TT)inR^(N xx N)\mathbf{B R}^{-1} \mathbf{B}^{\top} \in \mathbb{R}^{N \times N} ，它完全表征了底层图。它具有以下条目[7]：
L m n = { ∑ ℓ ∈ Λ ( m ) w ℓ if m = n − w ℓ if m is connected to n by ℓ L m n = ∑ ℓ ∈ Λ ( m )   w ℓ       if  m = n − w ℓ       if  m  is connected to  n  by  ℓ L_(mn)={[sum_(ℓin Lambda(m))w_(ℓ)," if "m=n],[-w_(ℓ)," if "m" is connected to "n" by "ℓ]:}L_{m n}= \begin{cases}\sum_{\ell \in \Lambda(m)} w_{\ell} & \text { if } m=n \\ -w_{\ell} & \text { if } m \text { is connected to } n \text { by } \ell\end{cases}
Here, the weight of an edge ℓ ℓ ℓ\ell is again given by its inverse resistance w ℓ = R ℓ − 1 w ℓ = R ℓ − 1 w_(ℓ)=R_(ℓ)^(-1)w_{\ell}=R_{\ell}^{-1}. For a connected graph, this matrix has exactly one vanishing eigenvalue λ 1 = 0 λ 1 = 0 lambda_(1)=0\lambda_{1}=0 with corresponding unit eigenvector v 1 = 1 / N v 1 = 1 / N v_(1)=1//sqrtN\mathbf{v}_{1}=\mathbf{1} / \sqrt{N} such that L 1 = 0 L 1 = 0 L1=0\mathbf{L} \mathbf{1}=0. For this reason, the matrix is noninvertible. This is typically overcome by
在这里，边 ℓ ℓ ℓ\ell 的权重再次由其倒数电阻 w ℓ = R ℓ − 1 w ℓ = R ℓ − 1 w_(ℓ)=R_(ℓ)^(-1)w_{\ell}=R_{\ell}^{-1} 给出。对于连通图，该矩阵恰好有一个消失的特征值 λ 1 = 0 λ 1 = 0 lambda_(1)=0\lambda_{1}=0 ，具有相应的单位特征向量 v 1 = 1 / N v 1 = 1 / N v_(1)=1//sqrtN\mathbf{v}_{1}=\mathbf{1} / \sqrt{N} ，使得 L 1 = 0 L 1 = 0 L1=0\mathbf{L} \mathbf{1}=0 。因此，该矩阵是不可逆的。通常通过
making use of the graph’s Moore-Penrose-pseudoinverse L † L † L^(†)\mathbf{L}^{\dagger}, which has properties similar to the actual inverse [27].
利用图的莫尔-彭罗斯伪逆 L † L † L^(†)\mathbf{L}^{\dagger} 来克服这个问题，它具有与实际逆矩阵相似的特性[27]。
With this formalism at hand, we can in principle now determine the current on any edge given a particular injection pattern I and edge resistances R. As a start, consider the situation where each edge has a unit resistance R = diag ( 1 ) R = diag ( 1 ) R=diag(1)\mathbf{R}=\operatorname{diag}(1) and a unit current is injected into a particular vertex s s ss and withdrawn at another one t t tt such that I = e s − e t I = e s − e t I=e_(s)-e_(t)\mathbf{I}=\mathbf{e}_{s}-\mathbf{e}_{t}, where e i = e i = e_(i)=\mathbf{e}_{i}= ( 0 , … , 1 ⏟ i , … , 0 ) ⊤ ∈ { 0 , 1 } M ( 0 , … , 1 ⏟ i , … , 0 ) ⊤ ∈ { 0 , 1 } M (0,dots,ubrace(1ubrace)_(i),dots,0)^(TT)in{0,1}^(M)(0, \ldots, \underbrace{1}_{i}, \ldots, 0)^{\top} \in\{0,1\}^{M} are the unit vectors with entry one at position i i ii and zero otherwise. In this situation, the current across any edge in the graph ℓ = ( a , b ) ℓ = ( a , b ) ℓ=(a,b)\ell=(a, b) is given by the following lemma, which dates back to Kirchhoff [8,20] and has been popularized by Shapiro [7,28].
有了这个形式化方法，原则上我们可以根据特定的注入模式 I 和边电阻 R 来确定任何边的电流。首先，考虑这样一种情况：每条边都有一个单位电阻 R = diag ( 1 ) R = diag ( 1 ) R=diag(1)\mathbf{R}=\operatorname{diag}(1) ，并且一个单位电流注入到一个特定的顶点 s s ss ，并在另一个顶点 t t tt 撤出，使得 I = e s − e t I = e s − e t I=e_(s)-e_(t)\mathbf{I}=\mathbf{e}_{s}-\mathbf{e}_{t} ，其中 e i = e i = e_(i)=\mathbf{e}_{i}= ( 0 , … , 1 ⏟ i , … , 0 ) ⊤ ∈ { 0 , 1 } M ( 0 , … , 1 ⏟ i , … , 0 ) ⊤ ∈ { 0 , 1 } M (0,dots,ubrace(1ubrace)_(i),dots,0)^(TT)in{0,1}^(M)(0, \ldots, \underbrace{1}_{i}, \ldots, 0)^{\top} \in\{0,1\}^{M} 是单位向量，在位置 i i ii 处为 1，其余位置为 0。在这种情况下，图中任何边的电流由以下引理给出，该引理可追溯到基尔霍夫[8,20]，并由沙皮罗[7,28]推广。
Lemma 1. Put a 1-A current between the vertices s s ss and t t tt of a connected, unweighted graph G G GG such that I = e s − e t I = e s − e t I=e_(s)-e_(t)\mathbf{I}=\mathbf{e}_{s}-\mathbf{e}_{t}. Then the current on any other edge ( a , b a , b a,ba, b ) is given by
引理 1. 在一个连通的无权图 G G GG 的顶点 s s ss 和 t t tt 之间放置一个 1 安培电流，使得 I = e s − e t I = e s − e t I=e_(s)-e_(t)\mathbf{I}=\mathbf{e}_{s}-\mathbf{e}_{t} 。那么任何其他边（ a , b a , b a,ba, b ）上的电流由下式给出
i a b = N ( s , a → b , t ) − N ( s , b → a , t ) N i a b = N ( s , a → b , t ) − N ( s , b → a , t ) N i_(ab)=(N(s,a rarr b,t)-N(s,b rarr a,t))/(N)i_{a b}=\frac{\mathcal{N}(s, a \rightarrow b, t)-\mathcal{N}(s, b \rightarrow a, t)}{\mathcal{N}}
where N ( s , a → b , t ) N ( s , a → b , t ) N(s,a rarr b,t)\mathcal{N}(s, a \rightarrow b, t) is the number of STs that contain a path from s s ss to t t tt of the form s , … , a , b , … , t s , … , a , b , … , t s,dots,a,b,dots,ts, \ldots, a, b, \ldots, t and N N N\mathcal{N} is the total number of STs of the graph.
其中 N ( s , a → b , t ) N ( s , a → b , t ) N(s,a rarr b,t)\mathcal{N}(s, a \rightarrow b, t) 是包含从 s s ss 到 t t tt 的路径 s , … , a , b , … , t s , … , a , b , … , t s,dots,a,b,dots,ts, \ldots, a, b, \ldots, t 和 N N N\mathcal{N} 的 ST 的数量， N N N\mathcal{N} 是图中 ST 的总数。
Whereas this lemma only holds for graphs where all links have unit resistances, real-world resistor networks or other types of linear flow networks are typically weighted with nonhomogeneous resistances. However, the extension to weighted networks is straightforward as summarized in the following corollary (see, e.g., Theorem II. 2 in Ref. [7]).
尽管这个引理只适用于所有链路都具有单位电阻的图，但现实世界的电阻网络或其他类型的线性流网络通常使用非均匀电阻进行加权。然而，将其扩展到加权网络是直接的，如下面的推论（参见例如参考文献[7]中的定理 II. 2）所总结的。
Corollary 1. Put a 1-A current between the vertices s s ss and t t tt of a connected, weighted graph G G GG such that I = e s − e t I = e s − e t I=e_(s)-e_(t)\mathbf{I}=\mathbf{e}_{s}-\mathbf{e}_{t}. Then the current on any other edge ( a , b a , b a,ba, b ) is given by
推论 1. 在连通的加权图 2 的顶点 s s ss 和 t t tt 之间放置一个 1-A 电流，使得 I = e s − e t I = e s − e t I=e_(s)-e_(t)\mathbf{I}=\mathbf{e}_{s}-\mathbf{e}_{t} 。那么，任何其他边( a , b a , b a,ba, b )上的电流由下式给出
i a b = N ∗ ( s , a → b , t ) − N ∗ ( s , b → a , t ) N ∗ i a b = N ∗ ( s , a → b , t ) − N ∗ ( s , b → a , t ) N ∗ i_(ab)=(N^(**)(s,a rarr b,t)-N^(**)(s,b rarr a,t))/(N^(**))i_{a b}=\frac{\mathcal{N}^{*}(s, a \rightarrow b, t)-\mathcal{N}^{*}(s, b \rightarrow a, t)}{\mathcal{N}^{*}}
where N ∗ = ∑ T ∈ T ∏ e ∈ T w e N ∗ = ∑ T ∈ T   ∏ e ∈ T   w e N^(**)=sum_(T inT)prod_(e in T)w_(e)\mathcal{N}^{*}=\sum_{T \in \mathcal{T}} \prod_{e \in T} w_{e} is the sum over the products of the weights w e w e w_(e)w_{e} of all edges e ∈ T e ∈ T e in Te \in T that are part of the respective spanning tree T ; T T ; T T;TT ; \mathcal{T} is the set of all STs in the graph. Similarly, N ∗ ( s , a → b , t ) N ∗ ( s , a → b , t ) N^(**)(s,a rarr b,t)\mathcal{N}^{*}(s, a \rightarrow b, t) equals the sum over all STs that contain a path of the form s , … , a , b , … , t s , … , a , b , … , t s,dots,a,b,dots,ts, \ldots, a, b, \ldots, t, where each ST is weighted with the product of the weight w e w e w_(e)w_{e} of all edges that are part of it. We thus assign a weight to each ST given by the product of the weights of the edges on the ST and replace the unweighted STs in Lemma 1 by weighted STs.
其中 N ∗ = ∑ T ∈ T ∏ e ∈ T w e N ∗ = ∑ T ∈ T   ∏ e ∈ T   w e N^(**)=sum_(T inT)prod_(e in T)w_(e)\mathcal{N}^{*}=\sum_{T \in \mathcal{T}} \prod_{e \in T} w_{e} 是所有属于各自生成树 2 的边的权重 1 的乘积之和 T ; T T ; T T;TT ; \mathcal{T} 是图中所有生成树 ST 的集合。类似地， N ∗ ( s , a → b , t ) N ∗ ( s , a → b , t ) N^(**)(s,a rarr b,t)\mathcal{N}^{*}(s, a \rightarrow b, t) 等于所有包含形式为 s , … , a , b , … , t s , … , a , b , … , t s,dots,a,b,dots,ts, \ldots, a, b, \ldots, t 路径的生成树之和，其中每个生成树都根据其包含的所有边的权重 6 的乘积加权。因此，我们根据生成树上的边的权重乘积为每个生成树分配一个权重，并将引理 1 中的无权生成树替换为有权重的生成树。
We will demonstrate in the following sections how this lemma and corollary may be made use of to understand how failure spreading may be mitigated in linear flow networks such as ac power grids in the dc approximation.
我们将在以下几节中展示如何利用这个引理和推论来理解如何在直流近似下，如交流电网这样的线性流网络中如何缓解故障传播。

III. ANALOGY BETWEEN RESISTOR NETWORKS AND POWER FLOW IN ELECTRICAL GRIDS
III. 电阻网络与电力系统中的潮流的类比

Importantly, the theoretical framework developed in the last section may be applied not only to resistor networks but also to power grids. In this section, we demonstrate how these results may be used to gain insight into the mitigation of failure spreading in power grids.
重要的是，上一节中发展的理论框架不仅适用于电阻网络，也适用于电网。在本节中，我们展示了如何利用这些结果来深入了解电网中故障蔓延的缓解。
TABLE I. Analogy between resistor networks and ac power grids in the dc approximation.
表 I. 直流近似下电阻网络和交流电网的类比。
dc approximation  直流近似 Resistor network  电阻网络
Quantity  数量 Symbol  符号 Quantity  数量 Symbol  符号
Power injections  功率注入 P Nodal current  节点电流 I
Real power flow  实际功率流 F Current flow  电流流 i
Nodal phase angles  节点相角 ϑ ϑ vartheta\vartheta Nodal voltages  节点电压 V
Line susceptances  线路电导 b e b e b_(e)b_{e} Inverse edge resistance  逆边电阻 r e − 1 r e − 1 r_(e)^(-1)r_{e}^{-1}
dc approximation Resistor network Quantity Symbol Quantity Symbol Power injections P Nodal current I Real power flow F Current flow i Nodal phase angles vartheta Nodal voltages V Line susceptances b_(e) Inverse edge resistance r_(e)^(-1)| dc approximation | | Resistor network | | | :--- | :--- | :--- | :--- | | Quantity | Symbol | Quantity | Symbol | | Power injections | P | Nodal current | I | | Real power flow | F | Current flow | i | | Nodal phase angles | $\vartheta$ | Nodal voltages | V | | Line susceptances | $b_{e}$ | Inverse edge resistance | $r_{e}^{-1}$ |

A. Modeling power grids as linear flow networks
A. 将电网建模为线性流网络

Most electric power transmission grids are made up of ac transmission lines and are, as such, governed by the nonlinear ac power flow equations [11]. However, the real power flow over transmission lines can be simplified to a linear flow model in what is referred to as the dc approximation of the ac power flow. This approximation is based on the following assumptions [29].
大多数电力传输网络由交流输电线路组成，因此受非线性交流潮流方程的支配[11]。然而，输电线路上的实际功率流可以简化为线性流模型，这被称为交流潮流的直流近似。这种近似基于以下假设[29]。
(i) Nodal voltage magnitudes vary little.
(i) 节点电压幅值变化不大。
(ii) Transmission lines are purely inductive; that is, their resistance is negligible compared with their reactance r ℓ ≪ r ℓ ≪ r_(ℓ)≪r_{\ell} \ll x ℓ , ∀ ℓ ∈ E ( G ) x ℓ , ∀ ℓ ∈ E ( G ) x_(ℓ),AAℓin E(G)x_{\ell}, \forall \ell \in E(G).
(ii) 输电线路是纯电感性的；也就是说，与它们的感抗相比，它们的电阻可以忽略不计 r ℓ ≪ r ℓ ≪ r_(ℓ)≪r_{\ell} \ll x ℓ , ∀ ℓ ∈ E ( G ) x ℓ , ∀ ℓ ∈ E ( G ) x_(ℓ),AAℓin E(G)x_{\ell}, \forall \ell \in E(G) 。
(iii) Differences between nodal voltage angles ϑ n , n ∈ ϑ n , n ∈ vartheta_(n),n in\vartheta_{n}, n \in V ( G ) V ( G ) V(G)V(G), of neighboring nodes n , m n , m n,mn, m are small | ϑ n − ϑ m | ≪ 1 ϑ n − ϑ m ≪ 1 |vartheta_(n)-vartheta_(m)|≪1\left|\vartheta_{n}-\vartheta_{m}\right| \ll 1.
(iii) 相邻节点 n , m n , m n,mn, m 的节点电压相角 ϑ n , n ∈ ϑ n , n ∈ vartheta_(n),n in\vartheta_{n}, n \in 和 V ( G ) V ( G ) V(G)V(G) 之间的差异很小 | ϑ n − ϑ m | ≪ 1 ϑ n − ϑ m ≪ 1 |vartheta_(n)-vartheta_(m)|≪1\left|\vartheta_{n}-\vartheta_{m}\right| \ll 1 。
Typically, these assumptions are met if the power grid is not heavily loaded and if the power grid is modeled at the transmission level where line resistances are small [29]. As a result, the real power flow F ℓ F ℓ F_(ℓ)F_{\ell} along a transmission line e ℓ = e ℓ = e_(ℓ)=e_{\ell}= ( n , m ) ∈ E ( G ) ( n , m ) ∈ E ( G ) (n,m)in E(G)(n, m) \in E(G) in the dc approximation depends linearly on the nodal voltage phase angles ϑ n ϑ n vartheta_(n)\vartheta_{n} of neighboring nodes
通常情况下，当电网未处于重载状态，并且电网在输电层面进行建模（此时线路电阻较小）时，这些假设能够得到满足[29]。因此，在直流近似下，沿输电线路 e ℓ = e ℓ = e_(ℓ)=e_{\ell}= - ( n , m ) ∈ E ( G ) ( n , m ) ∈ E ( G ) (n,m)in E(G)(n, m) \in E(G) 的真实功率流 F ℓ F ℓ F_(ℓ)F_{\ell} 与相邻节点的节点电压相角 ϑ n ϑ n vartheta_(n)\vartheta_{n} 线性相关。
F ℓ = b ℓ ( ϑ n − ϑ m ) F ℓ = b ℓ ϑ n − ϑ m F_(ℓ)=b_(ℓ)(vartheta_(n)-vartheta_(m))F_{\ell}=b_{\ell}\left(\vartheta_{n}-\vartheta_{m}\right)
Here, b ℓ ≈ x ℓ − 1 b ℓ ≈ x ℓ − 1 b_(ℓ)~~x_(ℓ)^(-1)b_{\ell} \approx x_{\ell}^{-1} is the line susceptance of line ℓ ℓ ℓ\ell. Thus the vector of real power flow along the transmission lines in the power grid F = ( F 1 , … , F M ) ⊤ ∈ R M F = F 1 , … , F M ⊤ ∈ R M F=(F_(1),dots,F_(M))^(TT)inR^(M)\mathbf{F}=\left(F_{1}, \ldots, F_{M}\right)^{\top} \in \mathbb{R}^{M} takes the role of current flow vector in the case of resistor networks. On the other hand, the nodal voltage phase angles ϑ = ( ϑ 1 , … , ϑ N ) ⊤ ∈ R N ϑ = ϑ 1 , … , ϑ N ⊤ ∈ R N vartheta=(vartheta_(1),dots,vartheta_(N))^(TT)inR^(N)\boldsymbol{\vartheta}=\left(\vartheta_{1}, \ldots, \vartheta_{N}\right)^{\top} \in \mathbb{R}^{N} take the role of the nodal voltages V V V\mathbf{V}, and line weights are given by the line susceptances b k b k b_(k)b_{k} of an edge e k e k e_(k)e_{k} in correspondence with the inverse resistances R k − 1 R k − 1 R_(k)^(-1)R_{k}^{-1} in the case of resistor networks. Thus Ohm’s law (4) translates to power grids as
此处 b ℓ ≈ x ℓ − 1 b ℓ ≈ x ℓ − 1 b_(ℓ)~~x_(ℓ)^(-1)b_{\ell} \approx x_{\ell}^{-1} 为线路 ℓ ℓ ℓ\ell 的线路电纳。因此，在电力系统 F = ( F 1 , … , F M ) ⊤ ∈ R M F = F 1 , … , F M ⊤ ∈ R M F=(F_(1),dots,F_(M))^(TT)inR^(M)\mathbf{F}=\left(F_{1}, \ldots, F_{M}\right)^{\top} \in \mathbb{R}^{M} 中沿输电线路的真实功率流向量在电阻网络中充当电流流向量。另一方面，节点电压相角 ϑ = ( ϑ 1 , … , ϑ N ) ⊤ ∈ R N ϑ = ϑ 1 , … , ϑ N ⊤ ∈ R N vartheta=(vartheta_(1),dots,vartheta_(N))^(TT)inR^(N)\boldsymbol{\vartheta}=\left(\vartheta_{1}, \ldots, \vartheta_{N}\right)^{\top} \in \mathbb{R}^{N} 充当节点电压 V V V\mathbf{V} 的角色，线路权重由边 e k e k e_(k)e_{k} 的线路电纳 b k b k b_(k)b_{k} 给出，这与电阻网络中的倒数电阻 R k − 1 R k − 1 R_(k)^(-1)R_{k}^{-1} 相对应。因此，欧姆定律 (4) 转化为电网形式为
F = B d B ⊤ ϑ F = B d B ⊤ ϑ F=B_(d)B^(TT)vartheta\mathbf{F}=\mathbf{B}_{d} \mathbf{B}^{\top} \vartheta
Here, B d = diag ( b 1 , … , b M ) ∈ R M × M B d = diag b 1 , … , b M ∈ R M × M B_(d)=diag(b_(1),dots,b_(M))inR^(M xx M)\mathbf{B}_{d}=\operatorname{diag}\left(b_{1}, \ldots, b_{M}\right) \in \mathbb{R}^{M \times M} is the diagonal matrix of line susceptances. Again, Kirchhoff’s current law (3) holds, and we may express it using vector quantities as follows [11,12]:
此处 B d = diag ( b 1 , … , b M ) ∈ R M × M B d = diag b 1 , … , b M ∈ R M × M B_(d)=diag(b_(1),dots,b_(M))inR^(M xx M)\mathbf{B}_{d}=\operatorname{diag}\left(b_{1}, \ldots, b_{M}\right) \in \mathbb{R}^{M \times M} 为线路电纳的对角矩阵。同样，基尔霍夫电流定律 (3) 成立，我们可以用向量形式表示如下 [11,12]：
B F = P B F = P BF=P\mathbf{B F}=\mathbf{P}
Here, P = ( P 1 , … , P N ) ⊤ ∈ R N P = P 1 , … , P N ⊤ ∈ R N P=(P_(1),dots,P_(N))^(TT)inR^(N)\mathbf{P}=\left(P_{1}, \ldots, P_{N}\right)^{\top} \in \mathbb{R}^{N} is the vector of nodal power injections, which thus takes the role of nodal current injections I. We summarize these equivalences in Table I.
此处 P = ( P 1 , … , P N ) ⊤ ∈ R N P = P 1 , … , P N ⊤ ∈ R N P=(P_(1),dots,P_(N))^(TT)inR^(N)\mathbf{P}=\left(P_{1}, \ldots, P_{N}\right)^{\top} \in \mathbb{R}^{N} 为节点功率注入向量，因此充当节点电流注入 I 的角色。我们将这些等价关系总结在表 I 中。

B. Sensitivity factors in power grid security analysis
B. 电网安全分析中的敏感性因子

In power grid security analysis, linear sensitivity factors are used to study and prevent line overloads which could cause disturbances to power system operation and result in
在电网安全分析中，线性灵敏度因子被用于研究和预防线路过载，这可能导致电力系统运行扰动并造成
power outages [11]. One of these factors is the power transfer distribution factor (PTDF). The PTDF s , t , k s , t , k _(s,t,k)_{s, t, k} then quantifies the change in flow Δ F k Δ F k DeltaF_(k)\Delta F_{k} on line e k ∈ E ( G ) e k ∈ E ( G ) e_(k)in E(G)e_{k} \in E(G) if a power Δ P Δ P Delta P\Delta P is injected at node r r rr and withdrawn from node s s ss. It is calculated as [11]
停电[11]。这些因素之一是功率传输分布因子（PTDF）。PTDF s , t , k s , t , k _(s,t,k)_{s, t, k} 量化了当在节点 r r rr 注入功率 Δ P Δ P Delta P\Delta P 并从节点 s s ss 抽取时，线路 e k ∈ E ( G ) e k ∈ E ( G ) e_(k)in E(G)e_{k} \in E(G) 上的流量 Δ F k Δ F k DeltaF_(k)\Delta F_{k} 的变化。其计算公式为[11]
PTDF r , s , k = Δ F k Δ P PTDF r , s , k = Δ F k Δ P PTDF_(r,s,k)=(DeltaF_(k))/(Delta P)\mathrm{PTDF}_{r, s, k}=\frac{\Delta F_{k}}{\Delta P}
Now assume that a single line e m e m e_(m)e_{m} fails, for example, as a result of an overload, and is disconnected from the network. The change in power flow on a line e k e k e_(k)e_{k} may then be calculated by using the line outage distribution factor (LODF) [11]
现在假设线路 e m e m e_(m)e_{m} 发生故障，例如由于过载而与网络断开连接。然后可以使用线路故障分配因子（LODF）[11]计算线路 e k e k e_(k)e_{k} 上的功率变化。
LODF k , m = Δ F k F m ( 0 ) LODF k , m = Δ F k F m ( 0 ) LODF_(k,m)=(DeltaF_(k))/(F_(m)^((0)))\mathrm{LODF}_{k, m}=\frac{\Delta F_{k}}{F_{m}^{(0)}}
Here, F m ( 0 ) F m ( 0 ) F_(m)^((0))F_{m}^{(0)} is the flow on line e m e m e_(m)e_{m} before the outage. Mathematically, we can map the flow changes after a failure to the flow changes after changes in the injection patterns by considering power injections that effectively compensate for the flow on the link that is assumed to fail (see Refs. [11,12]). As a result, the two quantities are related as follows if e m = ( r , s ) e m = ( r , s ) e_(m)=(r,s)e_{m}=(r, s) is the failing link [11]:
此处， F m ( 0 ) F m ( 0 ) F_(m)^((0))F_{m}^{(0)} 表示线路 e m e m e_(m)e_{m} 在故障前的流量。从数学上讲，我们可以通过考虑能够有效补偿假设故障链路上的流量的功率注入，将故障后的流量变化映射到注入模式变化后的流量变化（参见参考文献 [11,12]）。因此，如果 e m = ( r , s ) e m = ( r , s ) e_(m)=(r,s)e_{m}=(r, s) 是故障链路 [11]，则这两个量之间的关系如下：
LODF k , m = PTDF r , s , k 1 − PTDF r , s , m LODF k , m = PTDF r , s , k 1 − PTDF r , s , m LODF_(k,m)=(PTDF_(r,s,k))/(1-PTDF_(r,s,m))\mathrm{LODF}_{k, m}=\frac{\mathrm{PTDF}_{r, s, k}}{1-\mathrm{PTDF}_{r, s, m}}
Note that the description of link failures using LODFs relies on the dc approximation of the nonlinear ac power flow equations. However, extended descriptions have been proposed that incorporate nonlinear terms [31]. Furthermore, the dc approximation and thus the LODF-based description of link failures are commonly used to model cascading failures in power grids, where a single link triggers the failure of other links [23,32,33]. A comparison of the effect of link failures in linear and nonlinear models of power flows can, for example, be found in Ref. [34].
请注意，使用 LODF 描述链路故障依赖于非线性交流功率流方程的 dc 近似。然而，已经提出了包含非线性项的扩展描述[31]。此外，dc 近似以及基于 LODF 的链路故障描述通常用于模拟电网中的级联故障，其中单个链路触发其他链路的故障[23,32,33]。例如，参考文献[34]中比较了线性模型和非线性模型中链路故障的影响。
On the basis of the analogy between electrical grids and resistor networks developed in the last sections, we will now show how the ST formula presented in Corollary 1 may be used for power system security analysis. In the language of power grids, the lemma yields the PTDF s , t , m s , t , m _(s,t,m)_{s, t, m} for an edge e m = e m = e_(m)=e_{m}= ( a , b ) ( a , b ) (a,b)(a, b) if a unit power Δ P Δ P Delta P\Delta P is injected at node r r rr and withdrawn from node s s ss. For this reason, the PTDF may be calculated as follows:
基于上一节中发展的电网与电阻网络的类比，我们将展示如何使用引理 1 中提出的 ST 公式进行电力系统安全分析。在电网的语言中，该引理给出了边 e m = e m = e_(m)=e_{m}= ( a , b ) ( a , b ) (a,b)(a, b) 的 PTDF s , t , m s , t , m _(s,t,m)_{s, t, m} ，如果单位功率 Δ P Δ P Delta P\Delta P 在节点 r r rr 注入并在节点 s s ss 撤出。因此，PTDF 可以按以下方式计算：
PTDF s , t , m = N ∗ ( s , a → b , t ) − N ∗ ( s , b → a , t ) N ∗ PTDF s , t , m = N ∗ ( s , a → b , t ) − N ∗ ( s , b → a , t ) N ∗ PTDF_(s,t,m)=(N^(**)(s,a rarr b,t)-N^(**)(s,b rarr a,t))/(N^(**))\operatorname{PTDF}_{s, t, m}=\frac{\mathcal{N}^{*}(s, a \rightarrow b, t)-\mathcal{N}^{*}(s, b \rightarrow a, t)}{\mathcal{N}^{*}}
Based on Eq. (11), which yields the LODF expressed in terms of the PTDF, we can make use of this expression to derive an equivalent expression for the LODF. If e k = ( r , s ) e k = ( r , s ) e_(k)=(r,s)e_{k}=(r, s) is the failing link and e m = ( a , b ) e m = ( a , b ) e_(m)=(a,b)e_{m}=(a, b) is the link where the flow changes are monitored, the expression based on Eq. (12) reads as
根据公式(11)，该公式以 PTDF 表示 LODF，我们可以利用此表达式推导出 LODF 的等效表达式。如果 e k = ( r , s ) e k = ( r , s ) e_(k)=(r,s)e_{k}=(r, s) 是失效链路，而 e m = ( a , b ) e m = ( a , b ) e_(m)=(a,b)e_{m}=(a, b) 是监测流量变化的链路，则基于公式(12)的表达式读作
LODF m , k = N ∗ ( r , a → b , s ) − N ∗ ( r , b → a , s ) N ∗ − [ N ∗ ( r , r → s , s ) − N ∗ ( r , s → r , s ) ] = N ∗ ( r , a → b , s ) − N ∗ ( r , b → a , s ) N ∗ − N ∗ ( r , r → s , s ) = N ∗ ( r , a → b , s ) − N ∗ ( r , b → a , s ) N ∖ { k } ∗ LODF m , k = N ∗ ( r , a → b , s ) − N ∗ ( r , b → a , s ) N ∗ − N ∗ ( r , r → s , s ) − N ∗ ( r , s → r , s ) = N ∗ ( r , a → b , s ) − N ∗ ( r , b → a , s ) N ∗ − N ∗ ( r , r → s , s ) = N ∗ ( r , a → b , s ) − N ∗ ( r , b → a , s ) N ∖ { k } ∗ {:[LODF_(m,k)=(N^(**)(r,a rarr b,s)-N^(**)(r,b rarr a,s))/(N^(**)-[N^(**)(r,r rarr s,s)-N^(**)(r,s rarr r,s)])],[=(N^(**)(r,a rarr b,s)-N^(**)(r,b rarr a,s))/(N^(**)-N^(**)(r,r rarr s,s))],[=(N^(**)(r,a rarr b,s)-N^(**)(r,b rarr a,s))/(N_(\\{k})^(**))]:}\begin{aligned} \mathrm{LODF}_{m, k} & =\frac{\mathcal{N}^{*}(r, a \rightarrow b, s)-\mathcal{N}^{*}(r, b \rightarrow a, s)}{\mathcal{N}^{*}-\left[\mathcal{N}^{*}(r, r \rightarrow s, s)-\mathcal{N}^{*}(r, s \rightarrow r, s)\right]} \\ & =\frac{\mathcal{N}^{*}(r, a \rightarrow b, s)-\mathcal{N}^{*}(r, b \rightarrow a, s)}{\mathcal{N}^{*}-\mathcal{N}^{*}(r, r \rightarrow s, s)} \\ & =\frac{\mathcal{N}^{*}(r, a \rightarrow b, s)-\mathcal{N}^{*}(r, b \rightarrow a, s)}{\mathcal{N}_{\backslash\{k\}}^{*}} \end{aligned}
FIG. 1. Different methods for mitigating failure spreading in linear flow networks. (a) The failure of a single link (red) with unit flow results in flow changes Δ F Δ F Delta F\Delta F (color scale) throughout the Scandinavian power grid. (b) Failure spreading to Finland may be reduced by strengthening a link that horizontally separates Sweden and Finland. © Adding nodes, thus increasing the length of the rerouting path, reduces failure spreading to Finland as well. (d) Adding two links to construct a network isolator results in a complete vanishing of flow changes in the other part of the grid. Grid topology was extracted from the open energy system model PyPSA-Eur [30].
图 1. 线性流网络中缓解失效蔓延的不同方法。(a) 单个链路（红色）的失效导致单位流量变化 Δ F Δ F Delta F\Delta F （颜色标度）在整个斯堪的纳维亚电网中发生。 (b) 通过加强水平分隔瑞典和芬兰的链路，可以减少对芬兰的失效蔓延。 (c) 通过增加节点，从而增加重路由路径的长度，同样可以减少对芬兰的失效蔓延。 (d) 通过添加两个链路构建网络隔离器，导致电网其他部分的流量变化完全消失。电网拓扑从公开能源系统模型 PyPSA-Eur [30]中提取。
Here, N ∖ { k } ∗ N ∖ { k } ∗ N_(\\{k})^(**)\mathcal{N}_{\backslash\{k\}}^{*} denotes the weight of all STs in the graph evaluated after removing the edge e k e k e_(k)e_{k} from the set of trees T T T\mathcal{T}. We thus found an expression for the LODFs that is based purely on certain STs in the graph. This equation is the basis of our analysis of subgraphs inhibiting failure spreading which we will perform in the following sections. Note that a similar expression for the LODFs based on spanning 2-forests has recently been derived by Guo et al. [16].
此处， N ∖ { k } ∗ N ∖ { k } ∗ N_(\\{k})^(**)\mathcal{N}_{\backslash\{k\}}^{*} 表示在从树集 T T T\mathcal{T} 中移除边 e k e k e_(k)e_{k} 后，图中所有 ST 的权重。因此，我们找到了一个基于图中某些特定 ST 的 LODF 表达式。该方程是我们分析抑制故障传播的子图的基础，我们将在以下几节中执行这一分析。请注意，基于生成 2-森林的类似 LODF 表达式最近已被 Guo 等人推导出来 [16]。

IV. MITIGATING FAILURE SPREADING
IV. 减缓故障传播

We have seen in the last section that the spreading of failures is studied using LODFs in power system security analysis. To prevent large flow changes on other links after the failure of a link e k e k e_(k)e_{k} which may potentially trigger dangerous cascades of failures, it is desirable for overall power system security to keep the LODFs small. A natural question to ask is thus the following: Can we design or alter the network topology in such a way that LODFs stay small? Based on Eq. (13) expressing the LODF in terms of STs, this question may be addressed in a purely topological manner. In particular, we deduce three strategies to reduce the effect of failure spreading.
在上一节中，我们看到故障传播是通过电力系统安全分析中的 LODF 来研究的。为了防止在边 e k e k e_(k)e_{k} 故障后，其他链路上的大流量变化可能引发危险的故障级联，保持 LODF 较小对于整体电力系统安全是有益的。因此，一个自然的问题是：我们能否设计或改变网络拓扑，使得 LODF 保持较小？基于表示 LODF 的 ST 方程 (13)，这个问题可以纯粹从拓扑学的角度来解决。特别是，我们推导出三种减少故障传播效应的策略。
(1) Fixing long paths between trigger link e k e k e_(k)e_{k} and monitoring link e l e l e_(l)e_{l} leaves only few degrees of freedom, which reduces the relative contribution of the numerator in Eq. (13).
(1) 固定触发链路 e k e k e_(k)e_{k} 与监控链路 e l e l e_(l)e_{l} 之间的长路径，仅剩下少量自由度，这降低了公式(13)中分子的相对贡献。
(2) Fixing specific paths between trigger link e k e k e_(k)e_{k} and monitoring link e l e l e_(l)e_{l} can force links of large weights to be not
(2) 固定触发链接 e k e k e_(k)e_{k} 和监控链接 e l e l e_(l)e_{l} 之间的特定路径可以迫使权重较大的链接
contained in the numerator, thus reducing its relative contribution to Eq. (13).
分子中的项，从而降低了其对公式(13)的相对贡献。
(3) Introducing symmetric elements between parts of the network may lead to a complete balancing between the two contributions in the numerator of Eq. (13).
(3) 在网络的各个部分之间引入对称元素可能导致式(13)分子中的两个贡献完全平衡。
In Fig. 1 we illustrate three possible ways to realize these strategies to mitigate the impact of the failure of a single link (red) in a real power grid. All three strategies provide significant relief to the right module of the Scandinavian power grid, which represents Finland, after a link failure occurred in the left module. Remarkably, all these strategies are intimately related to the graph’s topological properties as we will see in the following sections.
在图 1 中，我们展示了三种可能的方法来实现这些策略，以减轻实际电网中单个链路（红色）故障的影响。这三种策略在左模块发生链路故障后，为斯堪的纳维亚电网的右侧模块提供了显著缓解。值得注意的是，所有这些策略都与图的拓扑特性密切相关，正如我们将在以下几节中看到的那样。

A. The role of the rerouting distance
A. 重定向距离的作用

With Eq. (13) expressing LODFs using STs at hand it is intuitively clear that certain paths in the network should play an important role in predicting the overall effect of line outages. In particular, we can see immediately that for a given failing link e k e k e_(k)e_{k}, the numerator in Eq. (13) depends on the paths going through the link monitoring the flow changes e l e l e_(l)e_{l} whereas the denominator does not. Therefore we expect the flow changes to be smaller on another link e m e m e_(m)e_{m} that has a longer minimum path going through e m e m e_(m)e_{m} and e k e k e_(k)e_{k} compared with link e l e l e_(l)e_{l}. This is due to the fact that reducing the number of possible paths in the sum over all STs N ∗ ( r , a → b , s ) N ∗ ( r , a → b , s ) N^(**)(r,a rarr b,s)\mathcal{N}^{*}(r, a \rightarrow b, s) effectively reduces the number of STs by fixing a certain path.
通过公式(13)使用现有的流图（STs）表达 LODFs，可以直观地看出网络中某些路径应在预测线路故障的整体效应中发挥重要作用。特别是，我们可以立即看到对于给定的故障线路 e k e k e_(k)e_{k} ，公式(13)的分子取决于通过该线路监测的流量变化 e l e l e_(l)e_{l} ，而分母则不受影响。因此，我们预期在另一条具有更长的最小路径通过 e m e m e_(m)e_{m} 和 e k e k e_(k)e_{k} 的线路 e m e m e_(m)e_{m} 上，流量变化将比线路 e l e l e_(l)e_{l} 更小。这是因为减少所有 STs 求和中的可能路径数 N ∗ ( r , a → b , s ) N ∗ ( r , a → b , s ) N^(**)(r,a rarr b,s)\mathcal{N}^{*}(r, a \rightarrow b, s) ，通过固定某条路径，实际上减少了 STs 的数量。
FIG. 2. Flow changes decay exponentially with cyclic paths in different networks. (a) and (d) Number of spanning trees (STs) τ ( G / p ) τ ( G / p ) tau(G//p)\tau(G / p) in an Erdős-Rényi (ER) random graph G ( 200 , 300 ) G ( 200 , 300 ) G(200,300)G(200,300) with 300 edges and 200 vertices (a) and in the power flow test case “IEEE 118” [35] (d) that contain a randomly chosen cyclic path p p pp ( y y yy axis) plotted against the length of the path len( p p pp ) ( x x xx axis). The number of STs decays exponentially with the length of the path, thus appearing linear on a logarithmic y y yy scale. (b) and (e) The rerouting distance scales exponentially with the LODF evaluated here for a single trigger for both grids. © and (f) The exponential scaling is preserved when averaging over all possible trigger links. Shading indicates 0.25 and 0.75 quantiles, a line represents the median. In Figs. 8 and 9 in the Appendix we demonstrate that the scaling robustly occurs for ER random graphs by analyzing 20 random realizations.
图 2. 流量变化在不同网络中随循环路径呈指数衰减。(a)和(d) 展示了具有 300 条边和 200 个顶点的 Erdős-Rényi（ER）随机图 G ( 200 , 300 ) G ( 200 , 300 ) G(200,300)G(200,300) 中生成树（STs）的数量 τ ( G / p ) τ ( G / p ) tau(G//p)\tau(G / p) ，以及 IEEE 118[35]的功率流测试用例（包含随机选择的循环路径 p p pp （ y y yy 轴））中生成树的数量 τ ( G / p ) τ ( G / p ) tau(G//p)\tau(G / p) ，它们随路径长度 len( p p pp )（ x x xx 轴）的变化情况。生成树的数量随路径长度呈指数衰减，因此在对数 y y yy 尺度上呈现线性关系。(b)和(e) 展示了对于单个触发器，在两个电网中重新路由距离与所评估的 LODF 呈指数关系。©和(f) 当对所有可能的触发链进行平均时，指数关系仍然保持。阴影表示 0.25 和 0.75 分位数，实线表示中位数。在附录中的图 8 和图 9 中，我们通过分析 20 个随机实现，证明了对于 ER 随机图，这种缩放关系是稳健的。
This intuitive idea is demonstrated to hold also quantitatively in Figs. 2(a) and 2(d): We illustrate that the number of STs τ ( G / p ) τ ( G / p ) tau(G//p)\tau(G / p) scales approximately exponentially with the length of the cyclic path contained in the STs for an unweighted Erdős-Rényi (ER) random graph G ( 200 , 300 ) G ( 200 , 300 ) G(200,300)G(200,300) with 300 edges and 200 vertices [36] [Fig. 2(a)] and the power flow test case “IEEE 118” [35,37] [Fig. 2(d)]. To study this scaling, we contract a cyclic path p p pp between two arbitrarily chosen edges and quantify the number of STs using Kirchhoff’s matrix tree theorem [8]. The theorem states that the number of STs in a graph may be calculated using the determinant of the graph’s Laplacian matrix [7]
这一直观想法在图 2(a)和图 2(d)中也被定量地验证：我们展示了对于具有 300 条边和 200 个顶点的无权 Erdős-Rényi（ER）随机图 G ( 200 , 300 ) G ( 200 , 300 ) G(200,300)G(200,300) [36]（图 2(a)）以及电力潮流测试案例“IEEE 118” [35,37]（图 2(d)），包含在 ST 中的循环路径的长度与 ST 的数量 τ ( G / p ) τ ( G / p ) tau(G//p)\tau(G / p) 近似呈指数关系。为了研究这种标度关系，我们在任意选定的两条边之间收缩一个循环路径 p p pp ，并使用基尔霍夫矩阵树定理[8]量化 ST 的数量。该定理指出，可以通过计算图的拉普拉斯矩阵的行列式来得到图中的 ST 数量[7]。
τ ( G ) = det ( L u ) . τ ( G ) = det L u . tau(G)=det(L_(u)).\tau(G)=\operatorname{det}\left(L_{u}\right) .
Here, L u L u L_(u)L_{u} is the matrix obtained from the Laplacian matrix L L LL of G G GG obtained by removing the row and column corresponding to an arbitrarily chosen vertex u ∈ V ( G ) u ∈ V ( G ) u in V(G)u \in V(G). The number of STs τ ( G / p ) τ ( G / p ) tau(G//p)\tau(G / p) containing a path p p pp may be calculated by contracting the path in the graph and the Laplacian matrix and then taking the determinant of the resulting Laplacian. Taking the difference in the numerator of Eq. (13) between the path and a reversed path will in general not affect the exponential scaling since the difference of two exponential functions with different exponents or different prefactors will again scale exponentially. In Fig. 8 in the Appendix, we show
此处， L u L u L_(u)L_{u} 是从 G G GG 的拉普拉斯矩阵 L L LL 中删除对应任意选择的一个顶点 u ∈ V ( G ) u ∈ V ( G ) u in V(G)u \in V(G) 的行和列所得到的矩阵。包含路径 p p pp 的 STs τ ( G / p ) τ ( G / p ) tau(G//p)\tau(G / p) 的数量可以通过收缩图中的路径和拉普拉斯矩阵，然后取所得拉普拉斯矩阵的行列式来计算。在式 (13) 的分子中，将路径与一条反向路径的差值通常不会影响指数缩放，因为两个具有不同指数或不同前因子的指数函数的差值仍然会呈指数缩放。在附录中的图 8 中，我们展示了
that the same scaling robustly occurs in ER random graphs by analyzing it for 20 different random realizations of ER graphs.
通过分析 20 种 ER 图的随机实现，证明了相同比例的鲁棒性在 ER 随机图中发生。
We may thus expect an exponential decay of LODFs with the length of fixed, cyclic paths. This result complements recent progress made in the understanding of the role played by distance for failure spreading in linear flow networks. In Ref. [12], it was shown that flow changes after a link failure are not captured well by the ordinary graph distance between the failing link and the link monitoring flow changes. Instead, a different distance measure referred to as rerouting distance captures this effect much better. It is defined as follows:
因此，我们可以预期固定循环路径的长度与 LODFs 呈指数衰减。这一结果补充了最近在理解距离在线性流网络中故障传播作用方面取得的进展。在参考文献[12]中，表明链路故障后的流量变化不能很好地由故障链路与监控流量变化的普通图距离来捕捉。相反，一种称为重路由距离的不同距离度量能更好地捕捉这种效应。它定义如下：
Definition 1. A rerouting path from vertex r r rr to vertex s s ss via the edge ( m , n m , n m,nm, n ) is a path
定义 1. 从顶点 r r rr 通过边( m , n m , n m,nm, n )到顶点 s s ss 的重路由路径是一条路径
( v 0 = r , v 1 , … , v i = m , v i + 1 = n , v i + 2 , … , v k = s ) v 0 = r , v 1 , … , v i = m , v i + 1 = n , v i + 2 , … , v k = s (v_(0)=r,v_(1),dots,v_(i)=m,v_(i+1)=n,v_(i+2),dots,v_(k)=s)\left(v_{0}=r, v_{1}, \ldots, v_{i}=m, v_{i+1}=n, v_{i+2}, \ldots, v_{k}=s\right)
or  或者
( v 0 = r , v 1 , … , v i = n , v i + 1 = m , v i + 2 , … , v k = s ) v 0 = r , v 1 , … , v i = n , v i + 1 = m , v i + 2 , … , v k = s (v_(0)=r,v_(1),dots,v_(i)=n,v_(i+1)=m,v_(i+2),dots,v_(k)=s)\left(v_{0}=r, v_{1}, \ldots, v_{i}=n, v_{i+1}=m, v_{i+2}, \ldots, v_{k}=s\right)
where no vertex is visited twice. The rerouting distance between two edges ( r , s ) ( r , s ) (r,s)(r, s) and ( m , n ) ( m , n ) (m,n)(m, n) denoted by
其中没有顶点被访问两次。两个边 ( r , s ) ( r , s ) (r,s)(r, s) 和 ( m , n ) ( m , n ) (m,n)(m, n) 之间的重路由距离用
edist re [ ( r , s ) , ( m , n ) ] edist re [ ( r , s ) , ( m , n ) ] edist_(re)[(r,s),(m,n)]\operatorname{edist}_{\mathrm{re}}[(r, s),(m, n)]
is the length of the shortest rerouting path from r r rr to s s ss via ( m , n ) ( m , n ) (m,n)(m, n) plus the length of edge ( r , s ) ( r , s ) (r,s)(r, s). Equivalently, it is the length of the shortest cycle crossing both edges ( r , s ) ( r , s ) (r,s)(r, s) and
其长度是最短重路由路径从 r r rr 到 s s ss 通过 ( m , n ) ( m , n ) (m,n)(m, n) 的长度加上边 ( r , s ) ( r , s ) (r,s)(r, s) 的长度。等价地，它是跨越边 ( r , s ) ( r , s ) (r,s)(r, s) 和
FIG. 3. Spanning trees (STs) may be used to explain the shielding effect of certain connectivity structures between different parts of a network. (a) and (b) A square grid is divided into two parts by either weakening the links connecting two parts [(a), blue, w e = 0.1 ] w e = 0.1 {:w_(e)=0.1]\left.w_{e}=0.1\right] or strengthening the links perpendicularly separating the two parts [(b), blue, w e = 10 ] w e = 10 {:w_(e)=10]\left.w_{e}=10\right]. © and (d) For both divisions, the failure of a single link with unit flow (red) significantly reduces failure spreading to the other part of the network. (e)-(h) Different STs (black) that contain specific paths of the form ( v 0 = r , v 1 , … , v i = m , v i + 1 = n , v i + 2 , … , v k = s v 0 = r , v 1 , … , v i = m , v i + 1 = n , v i + 2 , … , v k = s v_(0)=r,v_(1),dots,v_(i)=m,v_(i+1)=n,v_(i+2),dots,v_(k)=sv_{0}=r, v_{1}, \ldots, v_{i}=m, v_{i+1}=n, v_{i+2}, \ldots, v_{k}=s ) used to calculate the flow changes on link ( m , n m , n m,nm, n ) for a failure of link ( r , s r , s r,sr, s ) by virtue of Eq. (7). (e) and (f) For the weakly connected network shown in (a) and ©, a monitoring link in the same part (e) may lead to STs that contain only one weak link (blue shading). Thus the contribution of this ST to the sum over all STs is much stronger than for a monitoring link in the other part, where STs have to contain at least two weak links [(f), blue shading]. (g) and (h) For the strongly connected network shown in (g) and (h), the STs with the highest contribution are the ones containing all edges with strong weights [(g), blue shading]. (h) If links ( m , n ) ( m , n ) (m,n)(m, n) and ( r , s ) ( r , s ) (r,s)(r, s) are in different parts, no ST may contain all edges with strong weights (blue shading), thus reducing failure spreading in this case.
图 3. 覆盖树（STs）可用于解释网络不同部分之间某些连通性结构所起的屏蔽效应。（a）和（b）一个正方形网格被分割成两部分：通过减弱连接两部分的链接 [(a)，蓝色， w e = 0.1 ] w e = 0.1 {:w_(e)=0.1]\left.w_{e}=0.1\right] ] 或加强垂直分隔两部分的链接 [(b)，蓝色， w e = 10 ] w e = 10 {:w_(e)=10]\left.w_{e}=10\right] ]。（c）和（d）对于这两种分割方式，单个单位流量的链接（红色）的失效会显著减少失效向网络其他部分的蔓延。（e）-（h）包含特定形式（ v 0 = r , v 1 , … , v i = m , v i + 1 = n , v i + 2 , … , v k = s v 0 = r , v 1 , … , v i = m , v i + 1 = n , v i + 2 , … , v k = s v_(0)=r,v_(1),dots,v_(i)=m,v_(i+1)=n,v_(i+2),dots,v_(k)=sv_{0}=r, v_{1}, \ldots, v_{i}=m, v_{i+1}=n, v_{i+2}, \ldots, v_{k}=s ）路径的不同 STs（黑色），这些路径用于通过公式（7）计算链接（ m , n m , n m,nm, n ）在链接（ r , s r , s r,sr, s ）失效时的流量变化。（e）和（f）对于（a）和（c）中显示的弱连通网络，同一部分的监控链接（e）可能导致只包含一个弱链接（蓝色阴影）的 STs。因此，这种 ST 对所有 ST 求和的贡献远强于其他部分的监控链接，后者必须至少包含两个弱链接 [(f)，蓝色阴影]。 (g)和(h)对于图(g)和(h)中强连通网络，贡献最大的流连通集(ST)包含所有具有强权重的边[(g)，蓝色阴影]。(h)如果链接 ( m , n ) ( m , n ) (m,n)(m, n) 和 ( r , s ) ( r , s ) (r,s)(r, s) 位于不同部分，则没有任何 ST 可能包含所有具有强权重的边（蓝色阴影），从而在这种情况下减少失效传播。
( m , n m , n m,nm, n ). If no such path exists, the rerouting distance is defined to be ∞ ∞ oo\infty.
( m , n m , n m,nm, n )。如果不存在这样的路径，则重路由距离定义为 ∞ ∞ oo\infty 。
Note that we include the weight of the edge ( r , s ) ( r , s ) (r,s)(r, s) to make sure the rerouting distance is symmetric. The rerouting distance defined this way is a proper distance metric as shown in Ref. [12]. With the arguments made before at hand it is intuitively clear why the rerouting distance performs very well in predicting the effects of line outages. Indeed, we observe an exponential scaling of the LODFs for a given trigger link in the ER random graph [Fig. 2(b)] and in the test case "IEEE 118 " [Fig. 2(e)]. The same observation still holds if we average over all monitoring links located at a fixed rerouting distance to the possible trigger links over which we average thereafter [Figs. 2© and 2(f)]. In Fig. 9 we show that the observed scaling robustly appears by comparing it for 20 different realizations of ER random graphs.
请注意，我们包含了边 ( r , s ) ( r , s ) (r,s)(r, s) 的权重以确保重路由距离的对称性。以这种方式定义的重路由距离是一种适当的距离度量，如参考文献[12]所示。基于之前提出的论点，直观上很明显为什么重路由距离在预测线路故障的影响方面表现非常出色。实际上，我们观察到在 ER 随机图[图 2(b)]和"IEEE 118"测试用例[图 2(e)]中，给定触发链路的 LODFs 呈现出指数缩放。如果我们对所有位于固定重路由距离的可能触发链路处的监控链路进行平均，这一观察结果仍然成立[图 2©和图 2(f)]。在图 9 中，我们通过比较 ER 随机图的 20 种不同实现，展示了观察到的缩放现象稳健地出现。

B. The role of strong and weak network connectivity
B. 强网络连通性和弱网络连通性的作用

Our second strategy to reduce failure spreading after link failures is based on fixing specific paths in the network in such a way that they cannot contain certain links with large weights. This way, the numerator in Eq. (13) does not contain the contribution of the links with large weights whereas the denominator does, thereby reducing the overall impact of the link failure. Note that in contrast to the last section, the fixed
我们减少链路故障后故障传播的第二种策略是基于在网络中固定特定路径，以使它们不能包含具有较大权重的某些链路。这样，等式(13)中的分子不包含具有较大权重的链路的贡献，而分母包含，从而减少了链路故障的整体影响。请注意，与上一节相比，固定
paths do not necessarily have to be long to prevent failure spreading. We will demonstrate this strategy for two cases: First, we use this reasoning to demonstrate that weakening the links between two parts of the network-thus effectively dividing it into communities-may reduce failure spreading between them. This is expected as weakly connected networks generally suppress failure spreading from one part to the other one, but this also limits the possibility of power flow between the parts. This is no longer true for the second strategy: We illustrate why also strengthening the links that separate two parts of the network perpendicularly to the community boundary reduces the impact of link failures.
路径不必一定很长才能防止故障蔓延。我们将通过两种情况来展示这一策略：首先，我们利用这一推理证明，削弱网络两个部分之间的连接——从而实际上将其划分为社区——可能会减少它们之间的故障蔓延。这符合预期，因为弱连接网络通常抑制故障从一个部分蔓延到另一个部分，但这也会限制各部分之间的功率流动。对于第二种策略，情况就不再是这样了：我们说明，为什么垂直于社区边界加强分离网络两个部分的连接也能减少链路故障的影响。
The two strategies are illustrated for a simple 3 × 6 3 × 6 3xx63 \times 6 square grid in Fig. 3. We divide the square grid into two parts by either weakening the links that separate the parts [Fig. 3(a)] or strengthening the links perpendicular to these links [Fig. 3(b)]. We then monitor the flow changes (color scale) after the failure of a single link (red) in both cases [Figs. 3© and 3(d)]. For weak connectivity, the failure of link e k = ( r , s ) e k = ( r , s ) e_(k)=(r,s)e_{k}=(r, s) (dashed orange line) leads to a different contribution of the numerator in Eq. (13) if the monitoring link e ℓ = ( m , n ) e ℓ = ( m , n ) e_(ℓ)=(m,n)e_{\ell}=(m, n) (green line) is contained in the same part [Fig. 3(e)] as compared with a different, weakly connected part [Fig. 3(f)] in an otherwise symmetrical situation. Note that the distance between monitoring link and trigger link is also the same in both Fig. 3(e) and Fig. 3(f). For a link in the same part, the numerator also
在图 3 中，通过一个简单的 3 × 6 3 × 6 3xx63 \times 6 方形网格说明了这两种策略。我们将方形网格分为两部分，通过削弱分隔这两部分的连接[图 3(a)]或加强垂直于这些连接的连接[图 3(b)]。然后，在两种情况下，我们监测单个连接（红色）失效后的流量变化（颜色标度）[图 3©和图 3(d)]。对于弱连通性，如果监测连接 e ℓ = ( m , n ) e ℓ = ( m , n ) e_(ℓ)=(m,n)e_{\ell}=(m, n) （绿线）包含在相同的部分中，连接 e k = ( r , s ) e k = ( r , s ) e_(k)=(r,s)e_{k}=(r, s) （虚线橙色）的失效会导致方程(13)中分子有不同的贡献，与在一个对称情况下不同、弱连通的部分[图 3(f)]相比[图 3(e)]。请注意，在图 3(e)和图 3(f)中，监测连接和触发连接之间的距离也是相同的。对于同一部分的连接，分子也
FIG. 4. Network isolators that lead to a complete vanishing of LODFs are created using certain symmetric paths in the network. (a) STs that contain a path starting at node r r rr and terminating at node s s ss and containing the edge ( m , n m , n m,nm, n ) (blue) or ( n , m n , m n,mn, m ) (red) have to cross the subgraph consisting of dotted, colored edges in the center. Since each path can contain each vertex and edge only once, each ST passing through the subgraph in one way (blue) has a counterpart passing through the subgraph in the other way (red). (b) Failure of a link (red) results in vanishing LODFs (color scale) in the part connected by a network isolator as predicted using the ST formulation of link failures.
图 4. 通过网络中特定的对称路径创建的网络隔离器会导致 LODFS 完全消失。(a) 包含从节点 r r rr 开始、以节点 s s ss 结束且包含边( m , n m , n m,nm, n )（蓝色）或边( n , m n , m n,mn, m )（红色）的 ST 必须穿过中心由虚线彩色边组成的子图。由于每条路径只能包含每个顶点和边一次，每个通过子图的一种方式（蓝色）的 ST 都有一个通过子图的对应方式（红色）。(b) 链路（红色）的失效导致由网络隔离器连接的部分 LODFS 消失（颜色标度），正如通过链路失效的 ST 公式预测的那样。
contributes with STs containing only one weak link (thin line, blue shading). For a trigger link located in the other part, each ST connecting trigger link and monitoring link has to contain at least two weak links (shaded blue). Since the contribution in the numerator is proportional to the product of all weights along the ST and the situation is otherwise symmetric, we expect a weaker LODF and thus a shielding effect if the two links are contained in different, weakly connected parts.
仅包含一个弱链（细线，蓝色阴影）的 ST 做出贡献。对于位于其他部分的触发链，每个连接触发链和监测链的 ST 必须至少包含两个弱链（蓝色阴影）。由于分子中的贡献与 ST 沿线的所有权重乘积成正比，且情况是对称的，如果两条链位于不同的弱连接部分，我们预期 LODFS 较弱，从而产生屏蔽效应。
A similar observation holds in the case of strong connectivity: If the monitoring link e ℓ = ( m , n ) e ℓ = ( m , n ) e_(ℓ)=(m,n)e_{\ell}=(m, n) is contained in the same part of the network as the trigger link e k = ( r , s ) e k = ( r , s ) e_(k)=(r,s)e_{k}=(r, s) [Fig. 3(g)], now separated through strong connections, spanning trees connecting the two links may contain two-or generally, all-strong links. For a trigger link in the other part of the network, the spanning tree connecting them can contain maximally one-or generally, all minus one-strong links. Again, the term in the numerator scales with the link weights contained in the spanning trees. Therefore we expect the effect of link failures to be stronger for links located in the same part as compared with links contained in the other part, which is confirmed when simulating the failure of a single link in Fig. 3(d).
在强连通性的情况下，也有类似的观察结果：如果监控链路 e ℓ = ( m , n ) e ℓ = ( m , n ) e_(ℓ)=(m,n)e_{\ell}=(m, n) 与触发链路 e k = ( r , s ) e k = ( r , s ) e_(k)=(r,s)e_{k}=(r, s) 位于网络的同一部分中 [图 3(g)]，现在通过强连接分离，连接这两个链路的生成树可能包含两个或一般地，所有强链路。对于位于网络其他部分的触发链路，连接它们的生成树最多可以包含一个或一般地，所有减去一个的强链路。同样，分子中的项与生成树中包含的链路权重成正比。因此，我们预期与位于其他部分相比，位于同一部分的链路失效效应更强，这在模拟图 3(d) 中单个链路失效时得到了证实。

C. The role of symmetry
C. 对称性的作用

As a third strategy for reducing failure spreading, we suggest building networks in such a way that the terms in the numerator of Eq. (13) balance. In this case, failure spreading reduces to zero for the respective links. In order to balance the terms in the numerator of Eq. (13), we need the spanning trees passing through the monitoring link e ℓ = ( a , b ) e ℓ = ( a , b ) e_(ℓ)=(a,b)e_{\ell}=(a, b) in both directions to have exactly the same weight
作为减少故障蔓延的第三种策略，我们建议构建网络，使得式(13)分子中的项相互平衡。在这种情况下，对于相应的链路，故障蔓延减少到零。为了平衡式(13)分子中的项，我们需要通过监控链路 e ℓ = ( a , b ) e ℓ = ( a , b ) e_(ℓ)=(a,b)e_{\ell}=(a, b) 的双向生成树具有完全相同的权重。
N ∗ ( r , m → n , s ) = N ∗ ( r , n → m , s ) ⇒ ∑ T ∈ T ( r , m → n , s ) ∏ e ∈ T w e = ∑ T ∈ T ( r , n → m , s ) ∏ e ∈ T w e . N ∗ ( r , m → n , s ) = N ∗ ( r , n → m , s ) ⇒ ∑ T ∈ T ( r , m → n , s )   ∏ e ∈ T   w e = ∑ T ∈ T ( r , n → m , s )   ∏ e ∈ T   w e . {:[N^(**)(r","m rarr n","s)=N^(**)(r","n rarr m","s)],[=>quadsum_(T inT(r,m rarr n,s))prod_(e in T)w_(e)=sum_(T inT(r,n rarr m,s))prod_(e in T)w_(e).]:}\begin{aligned} \mathcal{N}^{*}(r, m \rightarrow n, s) & =\mathcal{N}^{*}(r, n \rightarrow m, s) \\ \Rightarrow \quad \sum_{T \in \mathcal{T}(r, m \rightarrow n, s)} \prod_{e \in T} w_{e} & =\sum_{T \in \mathcal{T}(r, n \rightarrow m, s)} \prod_{e \in T} w_{e} . \end{aligned}
Here, T ( r , m → n , s ) T ( r , m → n , s ) T(r,m rarr n,s)\mathcal{T}(r, m \rightarrow n, s) is the set of all spanning trees containing a path of the form ( r , … , m , n , … , s ) ( r , … , m , n , … , s ) (r,dots,m,n,dots,s)(r, \ldots, m, n, \ldots, s). This equality is, for example, fulfilled if for each tree T ∈ T ( r , m → n , s ) T ∈ T ( r , m → n , s ) T inT(r,m rarr n,s)T \in \mathcal{T}(r, m \rightarrow n, s) there is a counterpart T ∈ T ( r , n → m , s ) T ∈ T ( r , n → m , s ) T inT(r,n rarr m,s)T \in \mathcal{T}(r, n \rightarrow m, s) of the same weight. This may be accomplished by introducing certain symmetric elements, referred to as network isolators [19], into the
此处， T ( r , m → n , s ) T ( r , m → n , s ) T(r,m rarr n,s)\mathcal{T}(r, m \rightarrow n, s) 是包含形式为 ( r , … , m , n , … , s ) ( r , … , m , n , … , s ) (r,dots,m,n,dots,s)(r, \ldots, m, n, \ldots, s) 路径的所有生成树的集合。这种等式，例如，如果对于每个树 T ∈ T ( r , m → n , s ) T ∈ T ( r , m → n , s ) T inT(r,m rarr n,s)T \in \mathcal{T}(r, m \rightarrow n, s) 都存在一个相同权重的对应树 T ∈ T ( r , n → m , s ) T ∈ T ( r , n → m , s ) T inT(r,n rarr m,s)T \in \mathcal{T}(r, n \rightarrow m, s) ，则得到满足。这可以通过引入某些对称元素，称为网络隔离器[19]，来实现
graph as demonstrated in Fig. 4: For each ST connecting trigger link e k = ( r , s ) e k = ( r , s ) e_(k)=(r,s)e_{k}=(r, s) and monitoring link e ℓ = ( m , n ) e ℓ = ( m , n ) e_(ℓ)=(m,n)e_{\ell}=(m, n) and containing a path of the form ( r , … , m , n , … , s ) ( r , … , m , n , … , s ) (r,dots,m,n,dots,s)(r, \ldots, m, n, \ldots, s) (gray and blue lines) there is an ST containing a path of the form ( r , … , n , m , … , s ) ( r , … , n , m , … , s ) (r,dots,n,m,dots,s)(r, \ldots, n, m, \ldots, s) (gray and red lines). If we compare the product of weights for a single tree T 0 ∈ T ( r , m → n , s ) T 0 ∈ T ( r , m → n , s ) T_(0)inT(r,m rarr n,s)T_{0} \in \mathcal{T}(r, m \rightarrow n, s) and its counterpart T 0 ∗ ∈ T ( r , n → m , s ) T 0 ∗ ∈ T ( r , n → m , s ) T_(0)^(**)inT(r,n rarr m,s)T_{0}^{*} \in \mathcal{T}(r, n \rightarrow m, s), such that both contain exactly the same edges except for the edges connecting the two parts, i.e., the links marked as blue and red arrows in Fig. 4(a), we can see that these products are equal except for the links r 1 r 1 r_(1)r_{1} and r 2 r 2 r_(2)r_{2} (red links) being contained only in T 0 T 0 T_(0)T_{0}, and b 1 b 1 b_(1)b_{1} and b 2 b 2 b_(2)b_{2} (blue links) being contained only in T 0 ∗ T 0 ∗ T_(0)^(**)T_{0}^{*}. We can thus conclude that the above equality is fulfilled, i.e., the product of weights is equal for both trees T 0 T 0 T_(0)T_{0} and T 0 ∗ T 0 ∗ T_(0)^(**)T_{0}^{*}, if
图如 4 所示：对于每个连接触发链路 e k = ( r , s ) e k = ( r , s ) e_(k)=(r,s)e_{k}=(r, s) 和监控链路 e ℓ = ( m , n ) e ℓ = ( m , n ) e_(ℓ)=(m,n)e_{\ell}=(m, n) 且包含形式为 ( r , … , m , n , … , s ) ( r , … , m , n , … , s ) (r,dots,m,n,dots,s)(r, \ldots, m, n, \ldots, s) （灰色和蓝色线）的路径的 ST，存在一个包含形式为 ( r , … , n , m , … , s ) ( r , … , n , m , … , s ) (r,dots,n,m,dots,s)(r, \ldots, n, m, \ldots, s) （灰色和红色线）的路径的 ST。如果我们比较单个树 T 0 ∈ T ( r , m → n , s ) T 0 ∈ T ( r , m → n , s ) T_(0)inT(r,m rarr n,s)T_{0} \in \mathcal{T}(r, m \rightarrow n, s) 与其对应树 T 0 ∗ ∈ T ( r , n → m , s ) T 0 ∗ ∈ T ( r , n → m , s ) T_(0)^(**)inT(r,n rarr m,s)T_{0}^{*} \in \mathcal{T}(r, n \rightarrow m, s) 的权重乘积，两者除了连接两部分的边（即图 4(a)中标记为蓝色和红色箭头的链路）外包含完全相同的边，我们可以看到这些乘积除了链路 r 1 r 1 r_(1)r_{1} 和 r 2 r 2 r_(2)r_{2} （红色链路）仅包含在 T 0 T 0 T_(0)T_{0} 中，以及链路 b 1 b 1 b_(1)b_{1} 和 b 2 b 2 b_(2)b_{2} （蓝色链路）仅包含在 T 0 ∗ T 0 ∗ T_(0)^(**)T_{0}^{*} 中外，其余部分相等。因此我们可以得出结论，上述等式成立，即树 T 0 T 0 T_(0)T_{0} 和 T 0 ∗ T 0 ∗ T_(0)^(**)T_{0}^{*} 的权重乘积相等，如果
b 1 ⋅ b 2 = r 1 ⋅ r 2 . b 1 ⋅ b 2 = r 1 ⋅ r 2 . b_(1)*b_(2)=r_(1)*r_(2).b_{1} \cdot b_{2}=r_{1} \cdot r_{2} .
In this case, a failure of link e k = ( r , s ) e k = ( r , s ) e_(k)=(r,s)e_{k}=(r, s) does not result in any flow changes on link e ℓ = ( m , n ) e ℓ = ( m , n ) e_(ℓ)=(m,n)e_{\ell}=(m, n) at all. This reasoning has been generalized recently, where the concept was termed network isolators [19]. We also note that similar arguments were put forward by Guo et al. [16]. On general grounds, network isolators are defined as follows [19].
在这种情况下，链路 e k = ( r , s ) e k = ( r , s ) e_(k)=(r,s)e_{k}=(r, s) 的故障不会导致链路 e ℓ = ( m , n ) e ℓ = ( m , n ) e_(ℓ)=(m,n)e_{\ell}=(m, n) 上的任何流量变化。这一推理最近已被推广，其中该概念被称为网络隔离器[19]。我们还注意到，郭等人[16]也提出了类似的论点。从一般的角度来看，网络隔离器被定义为[19]如下。
Lemma 2. Consider a linear flow network consisting of two parts with vertex sets V 1 V 1 V_(1)V_{1} and V 2 V 2 V_(2)V_{2} and assume that a single link in
引理 2。考虑一个由两部分组成的线性流网络，其顶点集为 V 1 V 1 V_(1)V_{1} 和 V 2 V 2 V_(2)V_{2} ，假设其中有一条链。
FIG. 5. Sign reversal of LODFs by symmetric subgraphs. (a) and (b) Modifying the subgraph connecting two graphs from the two parallel lines to the two crossing lines leads to a sign reversal of the LODFs in the connecting subgraphs (shades of gray). This is in line with the compensatory effect of the symmetric subgraphs used to create the network isolator in Fig. 4.
图 5. 对称子图对 LODFs 符号的翻转。(a)和(b)将连接两条平行线的子图修改为连接两条交叉线的子图，导致连接子图中的 LODFs 符号发生翻转（灰色阴影）。这与图 4 中用于创建网络隔离器的对称子图的补偿效应一致。
the induced subgraph G ( V 1 ) G V 1 G(V_(1))G\left(V_{1}\right) fails, i.e., a link ( r , s r , s r,sr, s ) with r , s ∈ r , s ∈ r,s inr, s \in V 1 V 1 V_(1)V_{1}. If the adjacency matrix of the mutual connections has unit rank rank ( A 12 ) = 1 rank rank A 12 = 1 rank rank(A_(12))=1\operatorname{rank} \operatorname{rank}\left(\mathbf{A}_{12}\right)=1, then the flows on all links in the induced subgraph G ( V 2 ) G V 2 G(V_(2))G\left(V_{2}\right) are not affected by the failure; that is,
诱导子图 G ( V 1 ) G V 1 G(V_(1))G\left(V_{1}\right) 发生故障，即一条边 ( r , s r , s r,sr, s ) 具有 r , s ∈ r , s ∈ r,s inr, s \in V 1 V 1 V_(1)V_{1} 。如果相互连接的邻接矩阵的单位 rank rank ( A 12 ) = 1 rank rank A 12 = 1 rank rank(A_(12))=1\operatorname{rank} \operatorname{rank}\left(\mathbf{A}_{12}\right)=1 ，那么在诱导子图 G ( V 2 ) G V 2 G(V_(2))G\left(V_{2}\right) 上的所有边的流量不受故障影响；也就是说，
Δ F m , n ≡ 0 ∀ m , n ∈ V 2 . Δ F m , n ≡ 0 ∀ m , n ∈ V 2 . DeltaF_(m,n)-=0quad AA m,n inV_(2).\Delta F_{m, n} \equiv 0 \quad \forall m, n \in V_{2} .
The subgraph corresponding to the mutual interactions is referred to as a network isolator.
对应于相互作用的子图被称为网络隔离器。
Note that network isolators of arbitrary size may be understood using the same reasoning as presented above for a network isolator consisting of only four links.
请注意，任意规模的网络隔离器都可以使用上述仅由四个连接组成的网络隔离器的推理方法来理解。

1. Sign reversal of flow changes
1. 流量变化的符号反转

Based on the symmetric elements-the network isolators-introduced in Sec. IV C, we can demonstrate yet another application of the ST formulation to link failures: We can modify the grid in such a way that the LODFs and thus the flow changes change their sign. This is again based on the symmetry of LODFs in terms of the paths ( r , … , m , n , … , s ) ( r , … , m , n , … , s ) (r,dots,m,n,dots,s)(r, \ldots, m, n, \ldots, s) and ( r , … , n , m , … , s ) ( r , … , n , m , … , s ) (r,dots,n,m,dots,s)(r, \ldots, n, m, \ldots, s). If we apply a symmetric modification such that paths of the first form are replaced by parts of the latter one, we can reverse the sign of the resulting flow changes in the other part. In particular, if we interchange the two terms appearing in the nominator of Eq. (13) for a subset of edges, we can change the sign of the LODF for these edges
基于第四节 C 中引入的对称元素——网络隔离器，我们可以展示 ST 公式的另一个应用实例于连接故障：我们可以修改网格，使得 LODFS 和因此流量变化改变它们的符号。这再次基于 LODFS 在路径 ( r , … , m , n , … , s ) ( r , … , m , n , … , s ) (r,dots,m,n,dots,s)(r, \ldots, m, n, \ldots, s) 和 ( r , … , n , m , … , s ) ( r , … , n , m , … , s ) (r,dots,n,m,dots,s)(r, \ldots, n, m, \ldots, s) 方面的对称性。如果我们应用一个对称修改，使得第一种形式的路径被后者的部分所替换，我们可以反转其他部分的流量变化的符号。特别是，如果我们交换出现在方程(13)分子中的两个项于一个边缘子集，我们可以改变这些边缘的 LODF 的符号
N ∗ ( r , m → n , s ) → N ∗ ( r , n → m , s ) N ∗ ( r , n → m , s ) → N ∗ ( r , m → n , s ) ⇒ LODF ℓ , k → − LODF ℓ , k N ∗ ( r , m → n , s ) → N ∗ ( r , n → m , s ) N ∗ ( r , n → m , s ) → N ∗ ( r , m → n , s ) ⇒ LODF ℓ , k → − LODF ℓ , k {:[N^(**)(r","m rarr n","s) rarrN^(**)(r","n rarr m","s)],[N^(**)(r","n rarr m","s) rarrN^(**)(r","m rarr n","s)],[=>LODF_(ℓ,k) rarr-LODF_(ℓ,k)]:}\begin{aligned} \mathcal{N}^{*}(r, m \rightarrow n, s) & \rightarrow \mathcal{N}^{*}(r, n \rightarrow m, s) \\ \mathcal{N}^{*}(r, n \rightarrow m, s) & \rightarrow \mathcal{N}^{*}(r, m \rightarrow n, s) \\ \Rightarrow \operatorname{LODF}_{\ell, k} & \rightarrow-\operatorname{LODF}_{\ell, k} \end{aligned}
This can be achieved using a modification similar to the one shown in Fig. 4(a): If the initial network contains the subgraph indicated by blue dashed arrows in the center, we can revert the sign of the LODF ℓ , k LODF ℓ , k LODF_(ℓ,k)\mathrm{LODF}_{\ell, k} by changing this subgraph to the one indicated by red dashed arrows. This is demonstrated in Fig. 5: Changing the subgraph in the center connecting the two graphs from the “x”-shaped subgraph [Fig. 5(a)] to the " = = == "-shaped subgraph [Fig. 5(b)] leads to a sign reversal of the LODFs in the second graph (shades of gray), while the magnitude of LODFs is the same in both panels. This modification thus allows us to simultaneously change the sign of all LODFs in a subgraph, which may prevent overloads that are caused by flows going in a particular direction.
这可以通过类似于图 4(a)中所示的一种修改来实现：如果初始网络包含中心处由蓝色虚线箭头指示的子图，我们可以通过将该子图更改为由红色虚线箭头指示的子图来改变 LODF ℓ , k LODF ℓ , k LODF_(ℓ,k)\mathrm{LODF}_{\ell, k} 的符号。这在图 5 中得到了证明：将连接两个图的中心子图从“x”形状的子图[图 5(a)]更改为“ = = == ”形状的子图[图 5(b)]，导致第二个图中 LODFS 的符号反转（灰色阴影），而两个面板中 LODFS 的幅度相同。因此，这种修改使我们能够同时改变子图中所有 LODFS 的符号，这可能防止由特定方向流动引起的过载。

D. Comparison of strategies for mitigating failure spreading
D. 缓解故障蔓延的策略比较

Our theoretical analysis has led to three different strategies to mitigate failure spreading by optimizing the network topology. We will now quantify to what extent these modifications in topology improve the overall network resilience in terms of the impact of a single line failure.
我们的理论分析导致了三种通过优化网络拓扑来缓解故障蔓延的不同策略。现在，我们将量化这些拓扑修改在多大程度上提高了网络的整体韧性，特别是在单线故障的影响方面。
To begin with, we quantify the suppression of failure spreading between two preselected parts of the network. As an indicator we use the ratio of the LODFs evaluated at a given distance d d dd to the failing link m m mm suggested in Ref. [19]
首先，我们量化网络中两个预先选定的部分之间故障传播的抑制程度。作为指标，我们使用在给定距离 d d dd 处评估的 LODFs 与参考文献[19]中建议的故障链路 m m mm 的比值。
R ( m , d ) = ⟨ | LODF k , m | ⟩ d e k ∈ O ⟨ | LODF k , m | ⟩ d e k ∈ S R ( m , d ) = ⟨ | LODF k , m | ⟩ d e k ∈ O ⟨ | LODF k , m | ⟩ d e k ∈ S R(m,d)=((:|LODF_(k,m)|:)_(d)^(e_(k)in O))/((:|LODF_(k,m)|:)_(d)^(e_(k)in S))R(m, d)=\frac{\langle | \mathrm{LODF}_{k, m}| \rangle_{d}^{e_{k} \in O}}{\langle | \mathrm{LODF}_{k, m}| \rangle_{d}^{e_{k} \in S}}
FIG. 6. Failure spreading between Finland and the rest of Scandinavia is suppressed for all three strategies. We evaluate the ratio of LODFs, R ¯ ( m ) := ⟨ R ( m , d ) ⟩ d R ¯ ( m ) := ⟨ R ( m , d ) ⟩ d bar(R)(m):=(:R(m,d):)_(d)\bar{R}(m):=\langle R(m, d)\rangle_{d}, averaged over distance d d dd [Eq. (14)] between the right part of the grid, i.e., Finland, and its left part, i.e., the remainder of Scandinavia (see Fig. 1). We average the ratio over all distances d d dd for a given trigger link m m mm and sort the values by magnitude for the initial Scandinavian power grid (dark blue circles). We then analyze the ratio for the three strategies outlined in Sec. IV B and shown in Fig. 1. We observe that all strategies consistently yield reduced failure spreading between the two parts. Strengthening a specific link [blue triangles; cf. Fig. 1(b)] inhibits failure spreading more than increasing the length of the rerouting path [light blue squares; cf. Fig. 1©], while adding a network isolator [light blue diamonds; cf. Fig. 1(d)] completely suppresses failure spreading.
图 6. 对于所有三种策略，芬兰与其他斯堪的纳维亚地区的故障传播均受到抑制。我们评估了网格右侧部分，即芬兰，与其左侧部分，即斯堪的纳维亚地区剩余部分（见图 1）之间，距离 d d dd 平均的 LODFs R ¯ ( m ) := ⟨ R ( m , d ) ⟩ d R ¯ ( m ) := ⟨ R ( m , d ) ⟩ d bar(R)(m):=(:R(m,d):)_(d)\bar{R}(m):=\langle R(m, d)\rangle_{d} 比值[公式(14)]。我们对给定触发链路 m m mm 的所有距离 d d dd 的比值进行平均，并按初始斯堪的纳维亚电网的幅度排序（深蓝色圆圈）。然后，我们分析了第四节 B 中概述的三种策略（见图 1）的比值。我们观察到所有策略都一致地减少了这两个部分之间的故障传播。加强特定链路[蓝色三角形；参见图 1(b)]比增加重路由路径的长度[浅蓝色正方形；参见图 1©]更能抑制故障传播，而添加网络隔离器[浅蓝色菱形；参见图 1(d)]则完全抑制了故障传播。
Here, O O OO and S S SS are the two preselected parts of the network that are supposed to be protected against each other in terms of failure spreading, m ∈ S m ∈ S m in Sm \in S is the failing link located in part S S SS, and d d dd is the unweighted edge distance between trigger link m m mm and monitoring link k k kk. We average the absolute LODF over all links k k kk located in the other ( O O OO, numerator) and the same ( S S SS, denominator) part located at the fixed distance d d dd. The ratio assumes values between R ≈ 1 R ≈ 1 R~~1R \approx 1 if LODFs in both parts assume similar values and R ≈ 0 R ≈ 0 R~~0R \approx 0 if failure spreading to the other part O O OO is suppressed completely.
此处， O O OO 和 S S SS 是网络中预先选定的两个部分，它们在失效传播方面应相互保护， m ∈ S m ∈ S m in Sm \in S 是位于部分 S S SS 中的失效链路，而 d d dd 是触发链路 m m mm 和监控链路 k k kk 之间的无权边距离。我们对所有位于其他（ O O OO ，分子）和相同（ S S SS ，分母）部分且距离固定为 d d dd 的链路上的绝对 LODF 进行平均。该比率在两个部分的 LODF 值相似时取值范围为 R ≈ 1 R ≈ 1 R~~1R \approx 1 ，在完全抑制对其他部分 O O OO 的失效传播时取值范围为 R ≈ 0 R ≈ 0 R~~0R \approx 0 。
In Fig. 6 we analyze to what extent the three strategies shown in Fig. 1 are able to reduce failure spreading between Finland and the remainder of Scandinavia. We analyze the LODF ratio for all possible trigger links m m mm that are present in both the modified and the initial grid and compare the ratio for a given link by averaging the ratio over the distance d d dd. Thereby, we are able to compare to what extent failure spreading caused by the failure of a given link is reduced in each grid modification scenario. We observe that all three strategies consistently suppress failure spreading to the other part as measured by a reduction in the LODF ratio. Whereas strengthening a single link [Fig. 1(b)] suppresses spreading more strongly than an increase in rerouting [Fig. 1©], adding a network isolator [Fig. 1(d)] provides the strongest reduction in failure spreading by setting the LODF ratio to zero.
在图 6 中，我们分析了图 1 所示的三种策略在多大程度上能够减少芬兰与斯堪的纳维亚其余地区之间的故障传播。我们分析了存在于修改后和初始网格中的所有可能的触发链 m m mm 的 LODF 比率，并通过将给定链的比率在距离 d d dd 上平均来比较该比率。通过这种方式，我们能够比较在每个网格修改场景中，由给定链的故障引起的故障传播在多大程度上得到减少。我们观察到，所有三种策略都一致地通过 LODF 比率的降低来抑制故障传播到其他部分。而加强单个链[图 1(b)]的抑制效果强于重新路由的增加[图 1©]，而添加网络隔离器[图 1(d)]通过将 LODF 比率设置为零，提供了最强的故障传播减少效果。
While all three strategies suppress failure spreading between the two parts, we did not yet consider their overall impact on the entire network, i.e., including their impact on the same part where the trigger link is located. To quantify the
虽然所有三种策略都抑制了两个部分之间的故障传播，但我们尚未考虑它们对整个网络的整体影响，即包括对触发链所在相同部分的影响。为了量化
FIG. 7. Systematic analysis of the overall impact of a given strategy for mitigating failure spreading. We compare the impact of each of the strategies shown in Fig. 1 on the overall grid resilience measured by the LODF ratio R ― ( m ) := ⟨ R ( m , d ) ⟩ d R ¯ ( m ) := ⟨ R ( m , d ) ⟩ d bar(R)(m):=(:R(m,d):)_(d)\overline{\mathcal{R}}(m):=\langle\mathcal{R}(m, d)\rangle_{d} averaged over distance d d dd [Eq. (15)], which expresses to what extent the impact of the failure of a given link m m mm on the network differs from its impact in the grid without the modification. We average the ratio over all distances to calculate a link-based measure of grid resilience. (a) We observe that strengthening a single link has an overall positive impact on grid resilience and reduces the LODF ratio up to tenfold (dark blue links), with only a few links showing an increase (red). (b) An increase in rerouting as shown in Fig. 1© improves resilience in most links as well; the effect is, however, less pronounced than in the previous case. © Adding a network isolator strongly improves resilience in Finland, while slightly weakening it in the rest of Scandinavia. Thus all three strategies consistently have a positive impact on link-based resilience in Finland.
图 7. 对缓解失效传播的给定策略的整体影响进行系统性分析。我们比较图 1 中所示每种策略对整体网格弹性（通过 LODF 比率 R ― ( m ) := ⟨ R ( m , d ) ⟩ d R ¯ ( m ) := ⟨ R ( m , d ) ⟩ d bar(R)(m):=(:R(m,d):)_(d)\overline{\mathcal{R}}(m):=\langle\mathcal{R}(m, d)\rangle_{d} 在距离 d d dd 上平均[公式(15)]衡量）的影响，该比率表示给定链路 m m mm 的失效对网络的影响与未修改网格中的影响之间的差异程度。我们将该比率在所有距离上平均，以计算基于链路的网格弹性度量。（a）我们观察到，加强单个链路对网格弹性具有整体积极影响，并将 LODF 比率降低高达十倍（深蓝色链路），只有少数链路出现增加（红色）。（b）如图 1©所示的重路由增加，也提高了大多数链路的弹性；然而，其效果不如前一种情况明显。© 添加网络隔离器显著提高了芬兰的弹性，而略微削弱了斯堪的纳维亚其他地区的弹性。因此，所有三种策略在芬兰的链路弹性上均具有积极影响。
overall impact, we now consider the ratio of LODFs before and after the grid modification
在考虑整体影响后，我们现在考虑网格修改前后的 LODF 比率
R ( m , d ) = ⟨ | LODF k , m | ⟩ d e k ∈ G ′ ⟨ | LODF k , m | ⟩ d e k ∈ G R ( m , d ) = ⟨ | LODF k , m | ⟩ d e k ∈ G ′ ⟨ | LODF k , m | ⟩ d e k ∈ G R(m,d)=((:|LODF_(k,m)|:)_(d)^(e_(k)inG^(')))/((:|LODF_(k,m)|:)_(d)^(e_(k)in G))\mathcal{R}(m, d)=\frac{\langle | \mathrm{LODF}_{k, m}| \rangle_{d}^{e_{k} \in G^{\prime}}}{\langle | \mathrm{LODF}_{k, m}| \rangle_{d}^{e_{k} \in G}}
Here, G G GG is the initial network, and G ′ G ′ G^(')G^{\prime} is the network after the topology has been modified according to a chosen strategy. As before, m m mm denotes the failing link, and the magnitudes of the LODFs are averaged over all links k k kk at a given distance d d dd to the trigger link m m mm. Only links which are present in both G G GG and G ′ G ′ G^(')G^{\prime} are considered as trigger links. While being defined similarly to the ratio of LODFs in Eq. (14), the main difference between the two quantities is the following: The ratio considered here compares the impact of a link failure in two different networks, while the ratio in Eq. (14) compares the impact on two different parts of the same network. The ratio defined here thus quantifies whether a given modification leads to lower average LODFs in the entire grid or whether it increases the vulnerability of some links. It assumes values of unity, R ( m , d ) ≈ 1 R ( m , d ) ≈ 1 R(m,d)~~1\mathcal{R}(m, d) \approx 1, if the impact of the failure on the entire grid is approximately the same in the initial and the modified grid and deviates from unity if the impact of a failure of the given link m m mm on links at a distance d d dd is reduced [ R ( m , d ) < 1 ] [ R ( m , d ) < 1 ] [R(m,d) < 1][\mathcal{R}(m, d)<1] or increased [ R ( m , d ) > 1 ] [ R ( m , d ) > 1 ] [R(m,d) > 1][\mathcal{R}(m, d)>1].
此处， G G GG 表示初始网络， G ′ G ′ G^(')G^{\prime} 表示根据所选策略修改拓扑后的网络。与前文相同， m m mm 表示失效链路，LODFs 的大小在距离触发链路 m m mm 的给定距离 d d dd 的所有链路 k k kk 上进行平均。仅考虑同时存在于 G G GG 和 G ′ G ′ G^(')G^{\prime} 中的链路作为触发链路。虽然这两个量与公式 (14) 中的 LODF 比率定义相似，但它们的主要区别如下：此处考虑的比率比较了两个不同网络中链路失效的影响，而公式 (14) 中的比率比较了同一网络中两个不同部分的影响。因此，此处定义的比率量化了给定修改是否会导致整个网格的平均 LODF 降低，或者是否会增加某些链路的脆弱性。当初始网格和修改后网格中失效对整个网格的影响大致相同时，该比率取值为 1 ( R ( m , d ) ≈ 1 R ( m , d ) ≈ 1 R(m,d)~~1\mathcal{R}(m, d) \approx 1 )；当给定链路 m m mm 的失效对距离为 d d dd 的链路的影响减少 [ R ( m , d ) < 1 ] [ R ( m , d ) < 1 ] [R(m,d) < 1][\mathcal{R}(m, d)<1] 或增加 [ R ( m , d ) > 1 ] [ R ( m , d ) > 1 ] [R(m,d) > 1][\mathcal{R}(m, d)>1] 时，该比率偏离 1。
In Fig. 7, we analyze this ratio for the Scandinavian grid for each strategy and the resulting grid modification at the border between Finland and the remainder of Scandinavia shown in Fig. 1. To be precise, we evaluate the distance-averaged LODF ratio R ― ( m ) := ⟨ R ( m , d ) ⟩ d R ¯ ( m ) := ⟨ R ( m , d ) ⟩ d bar(R)(m):=(:R(m,d):)_(d)\overline{\mathcal{R}}(m):=\langle\mathcal{R}(m, d)\rangle_{d} for all possible trigger links m m mm. For all three strategies, we observe a reduction in failure spreading, i.e., R ― ( m ) < 1 R ¯ ( m ) < 1 bar(R)(m) < 1\overline{\mathcal{R}}(m)<1, if the trigger link m m mm is located in the bulk of Finland or in western Norway. The benefits are strongest if a network isolator is added [Fig. 7©] and weakest if rerouting distance is increased [Fig. 7(b)]. For a trigger link m m mm located in the central part of the grid, i.e., in Sweden, the addition of an isolator has a slightly negative effect such that R ― ( m ) > 1 R ¯ ( m ) > 1 bar(R)(m) > 1\overline{\mathcal{R}}(m)>1, which is, however, much weaker than the positive effects on the other parts of the grid. The
在图 7 中，我们分析了斯堪的纳维亚电网针对每种策略的该比率，以及芬兰与斯堪的纳维亚其余地区边界处产生的电网修改情况，如图 1 所示。具体而言，我们评估了所有可能的触发链路 m m mm 的距离平均 LODF 比率 R ― ( m ) := ⟨ R ( m , d ) ⟩ d R ¯ ( m ) := ⟨ R ( m , d ) ⟩ d bar(R)(m):=(:R(m,d):)_(d)\overline{\mathcal{R}}(m):=\langle\mathcal{R}(m, d)\rangle_{d} 。对于所有三种策略，如果触发链路 m m mm 位于芬兰主体区域或挪威西部，则观察到故障扩散减少，即 R ― ( m ) < 1 R ¯ ( m ) < 1 bar(R)(m) < 1\overline{\mathcal{R}}(m)<1 。当添加网络隔离器时，效益最强[图 7©]，而当重路由距离增加时，效益最弱[图 7(b)]。对于位于电网中心部分的触发链路 m m mm ，即瑞典，添加隔离器会产生轻微的负面影响，即 R ― ( m ) > 1 R ¯ ( m ) > 1 bar(R)(m) > 1\overline{\mathcal{R}}(m)>1 ，然而这种负面影响远弱于对电网其他部分产生的正面效应。
two other strategies have a negligible impact if m m mm is in this part of the grid. In all cases, the ratio indicates an increase in failure spreading for a few trigger links that are located in the vicinity of the topology modification. To conclude, we observe that the choice of a favorable strategy depends on the goal to be achieved. If trigger links in Sweden or in the vicinity of the border between Sweden and Finland have been identified as links that are likely to fail, none of the strategies will strongly increase grid resilience or will even deplete it. If, on the other hand, the goal is to protect the grid against likely link failures that emerge in Finland, all three strategies consistently provide a certain benefit to grid resilience which is also confirmed by the results in Fig. 6. In this case, adding a network isolator most likely provides the best results.
如果 m m mm 位于该网格的这一部分，则另外两种策略的影响可以忽略不计。在所有情况下，该比率表明，对于位于拓扑修改附近的一些触发链路，故障传播有所增加。总之，我们观察到，选择有利的策略取决于要实现的目标。如果瑞典或瑞典与芬兰边境附近的触发链路已被确定为可能发生故障的链路，则没有任何策略会显著提高电网弹性，甚至可能会消耗弹性。另一方面，如果目标是保护电网免受可能出现在芬兰的链路故障的影响，则三种策略始终为电网弹性提供一定的益处，这也由图 6 的结果所证实。在这种情况下，添加网络隔离器最有可能提供最佳结果。
In total, the LODF ratios R ( m , d ) R ( m , d ) R(m,d)R(m, d) and R ( m , d ) R ( m , d ) R(m,d)\mathcal{R}(m, d) provide a complementary view on the different strategies by measuring the extent to which failures are suppressed between the two parts of a network, on the one hand, and the impact of a strategy on the network as a whole on the other one. For this reason, they can be used to balance the pros and cons of a grid modification and thus allow one to find which strategy performs best for the given grid or even allow one to study the impact of a combination of different strategies.
总而言之，LODF 比率 R ( m , d ) R ( m , d ) R(m,d)R(m, d) 和 R ( m , d ) R ( m , d ) R(m,d)\mathcal{R}(m, d) 从不同角度提供了互补的视角，一方面测量网络两个部分之间故障抑制的程度，另一方面评估一种策略对整个网络的影响。正因如此，它们可用于权衡网格修改的利弊，从而帮助人们找到最适合给定网格的策略，甚至可以研究不同策略组合的影响。

V. CONCLUSION  V. 结论

We demonstrated how a spanning tree formulation of link failures may be used to understand which topological patterns aid the mitigation of failure spreading in power grids and other types of linear flow networks. In particular, we derived and explained three strategies for reducing the effect of link failures in linear flow networks based on spanning trees. Our results offer an understanding of previous strategies used to inhibit failure spreading in power grids and may thus help to increase power grid security.
我们展示了如何通过链路失效的生成树公式来理解哪些拓扑模式有助于缓解电网和其他类型的线性流网络中的失效传播。特别是，我们基于生成树推导并解释了三种减少线性流网络中链路失效影响的方法。我们的结果为理解先前用于抑制电网失效传播的策略提供了认识，从而可能有助于提高电网的安全性。
All strategies analyzed here for reducing failure spreading are based on extending-or at least not reducing-the network’s ability to transport flows. This is in contrast to typical containment strategies in power grid security which are based
这里分析的所有减少失效传播的策略都是基于扩展——至少不减少——网络传输流的能力。这与电网安全中典型的控制策略形成对比，后者基于
FIG. 8. Exponential decay of the number of spanning trees (STs) τ ( G / p ) τ ( G / p ) tau(G//p)\tau(G / p) in Erdős-Rényi (ER) random graphs, with length of randomly chosen cyclic path len ( p ) ( p ) (p)(p), occurs robustly. Each panel shows the number of STs in a different, random realization of an ER graph G ( 200 , 300 ) G ( 200 , 300 ) G(200,300)G(200,300) with 300 edges and 200 vertices after collapsing a randomly chosen cyclic path. We analyze 200 randomly chosen cyclic paths for each ER graph (dots) and perform a least-squares fit of an exponential function on the semilog scale (dashed lines). The number of STs decays exponentially with the length of the path, thus appearing linear on a logarithmic y y yy scale.
图 8. 在 Erdős-Rényi（ER）随机图中，跨越树（STs）数量 τ ( G / p ) τ ( G / p ) tau(G//p)\tau(G / p) 呈指数衰减，当随机选择的循环路径长度 len ( p ) ( p ) (p)(p) 时，这种现象稳健发生。每个面板展示了在 ER 图 G ( 200 , 300 ) G ( 200 , 300 ) G(200,300)G(200,300) （具有 300 条边和 200 个顶点）的随机实现中，在折叠一条随机选择的循环路径后，跨越树的数量。我们对每个 ER 图分析 200 条随机选择的循环路径（点），并在半对数尺度上对指数函数进行最小二乘拟合（虚线）。跨越树的数量随路径长度的增加呈指数衰减，因此在对数 y y yy 尺度上表现为线性。
on islanding the power grid, i.e., reducing the connectivity for the sake of security. We illustrated how to exploit the intimate connection to graph theory to find and analyze subgraphs that allow for improving both power grid resilience and efficiency at the same time.
在电力系统中实现离网，即为了安全而降低连通性。我们展示了如何利用与图论的紧密联系，寻找和分析子图，从而同时提高电力系统的弹性和效率。
Our results offer a new understanding on a graphtheoretical level of network structures that have been found to inhibit or enhance failure spreading. We illustrated the fruitful approach of analyzing failure spreading in power grids by using spanning trees for several subgraphs but are confident that other subgraphs for enhancing or inhibiting failure spreading may be unveiled using this formalism.
我们的研究结果为图论层面上的网络结构提供了新的理解，这些结构已被发现能够抑制或增强故障传播。我们通过使用生成树分析了几个子图的故障传播，展示了这一富有成效的方法，但我们也相信，使用这种形式化方法可能会揭示出其他能够增强或抑制故障传播的子图。
Finally, the question arises regarding to what extent our theoretical results are relevant for the stability of real-world power grids, in particular, the stability to large-scale blackouts. In fact, a power grid blackout is typically triggered by
最后，问题在于我们的理论结果对现实世界电网的稳定性有多大相关性，特别是对大规模停电的稳定性。事实上，电网停电通常是由
the outage of a single transmission element, more rarely a single generation element [21]. When such a transmission line outage occurs, power flow is redistributed to parallel transmission paths, which may cause secondary overloads. Hence the scenario considered in this paper is of high practical relevance.
单个输电元件的故障，更罕见的是单个发电元件[21]。当发生此类输电线路故障时，电力将重新分配到并联输电路径，这可能导致次级过载。因此，本文所考虑的场景具有很高的实际相关性。
Our results have been derived for the linearized dc approximation; hence they will hold only approximately for scenarios where the dc approximation is no longer valid. In particular, there is no longer an exact analogy between resistor networks and ac power grids when flows are calculated nonlinearly using ac power flow models. However, the impact of line failures in high-voltage grids is typically well described by the linearized dc approximation [19,34]. Deviations occur mainly for high-loading scenarios, but even then the dc approximation usually gives a reasonable first-order estimate of the flow redistribution. It must be noted that the assumptions leading
我们的结果是在线性化直流近似下推导出来的；因此，它们仅在线性化直流近似不再有效的情况下近似成立。特别是，当使用交流潮流模型非线性地计算流量时，电阻网络和交流电网之间不再存在精确的类比关系。然而，高压电网中线路故障的影响通常可以用线性化直流近似很好地描述[19,34]。偏差主要发生在高负载场景中，即便如此，直流近似通常也能给出流量重新分配的一阶合理估计。必须指出的是，导致
FIG. 9. Decay of averaged LODFs with rerouting distance to the trigger link is robust throughout 20 different realizations of Erdős-Rényi (ER) random graphs. Each panel shows the decay of LODFs for a different, random realization of an ER graph G ( 200 , 300 ) G ( 200 , 300 ) G(200,300)G(200,300) with 300 edges and 200 vertices. We observe an approximately exponential scaling of LODFs with rerouting distance when averaging over all possible links located at a fixed rerouting distance to the trigger link. Shading indicates 0.25 and 0.75 quantiles; a line represents the median.
图 9. 平均 LODF 随触发链重路由距离衰减在 20 种不同的 Erdős-Rényi（ER）随机图中表现出鲁棒性。每个面板显示了具有 300 条边和 200 个顶点的 ER 图 G ( 200 , 300 ) G ( 200 , 300 ) G(200,300)G(200,300) 的不同随机实现中 LODF 的衰减情况。当对所有位于触发链固定重路由距离的所有可能链进行平均时，观察到 LODF 与重路由距离近似呈指数缩放关系。阴影表示 0.25 和 0.75 分位数；线表示中位数。
to the dc approximation are not necessarily violated during the initial stages of a cascade. Secondary overloads occur when the current or real power flow exceeds a threshold. If the reactance x ℓ x ℓ x_(ℓ)x_{\ell} is not too large, this will happen well before the angle difference becomes large. During the final stages of a cascade, nonlinear and dynamical effects must be taken into account.
到直流近似的约束在级联的初始阶段不一定被违反。次级过载发生在电流或实际功率流超过阈值时。如果电抗 x ℓ x ℓ x_(ℓ)x_{\ell} 不是太大，这将在角度差变得很大之前就会发生。在级联的最终阶段，必须考虑非线性和动态效应。
Nevertheless, the focus of our study is on flow networks where flow distribution and redistribution after failures are governed by Kirchhoff’s laws. Further studies are necessary to assess whether parts of our results may in some sense be transferred to topological models where flows are routed along shortest paths [38-40].
然而，我们的研究重点在于遵循基尔霍夫定律的流网络，其中流量的分布和故障后的重新分配受到基尔霍夫定律的支配。进一步的研究是必要的，以评估我们的部分结果是否可以在某种程度上转移到流量沿最短路径路由的拓扑模型中[38-40]。

ACKNOWLEDGMENTS  致谢

We gratefully acknowledge support from the German Federal Ministry of Education and Research (Grant No. 03EK3055B) and the Helmholtz Association (via the joint initiative “Energy System 2050 - A Contribution of the Research Field Energy” and Grant No. VH-NG-1025).
我们衷心感谢德国联邦教育与研究部（项目编号 03EK3055B）和亥姆霍兹协会（通过“能源系统 2050——能源领域研究的贡献”联合倡议和项目编号 VH-NG-1025）的支持。

APPENDIX  附录

In this Appendix, we demonstrate that the scaling robustly occurs for ER random graphs by analyzing 20 random realizations (Figs. 8 and 9).
在本附录中，我们通过分析 20 个随机实现（图 8 和图 9）证明了 ER 随机图具有稳健的缩放特性。
  1. *f.kaiser@fz-juelich.de
    † † ^(†){ }^{\dagger} d.witthaut@fz-juelich.de
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