Identification of Critical Transmission Lines in Complex Power Networks

Abstract

Growing load demands, complex operating conditions, and the increased use of intermittent renewable energy pose great challenges to power systems. Serious consequences can occur when the system suffers various disturbances or attacks, especially those that might initiate cascading failures. Accurate and rapid identification of critical transmission lines is helpful in assessing the system vulnerability. This can realize rational planning and ensure reliable security pre-warning to avoid large-scale accidents. In this study, an integrated “betweenness” based identification method is introduced, considering the line’s role in power transmission and the impact when it is removed from a power system. At the same time, the sensitive regions of each line are located by a cyclic addition algorithm (CAA), which can reduce the calculation time and improve the engineering value of the betweenness, especially in large-scale power systems. The simulation result verifies the effectiveness and the feasibility of the identification method.

1. Introduction

Due to the expansion of power grids and increases in load demand, the power systems gradually approach their operating limits. This increases the uncertainty of the system dynamic behavior and the risk of large blackouts [1]. Many power outages have occurred in recent years [2,3], for instance, the western North America blackouts in 1996 [4], the North American blackouts in 2003 [5], and the India blackouts in 2012 [6]. Frequent accidents expose the potential problems of current power system analysis methods, especially concerning system vulnerability.

In a blackout, critical lines often trigger or promote the failure propagation. These lines usually play important roles in the network structure, and their disconnections greatly increase power system vulnerability and can even cause serious consequences. Accurately identifying the critical transmission lines can realize rational planning and provide reliable security pre-warning. It also helps to ensure control measures are carried out effectively to avoid the occurrence of blackouts.

At present, researches on the identification of critical or vulnerable transmission lines are usually based on three main factors: the network topology [7,8,9,10,11,12,13,14,15,16], the electrical characteristics [17,18,19,20,21,22,23], and the failure mechanism [24,25,26,27]. Power grids can be conceptually described as networks, so that topology-based theory provides a feasible way to study network vulnerability [7] and locate the critical transmission lines [8,9]. Some indices, such as betweenness [10,11,12,13], are proposed to describe the importance of different components. In topology-based methods, the betweenness of an edge is defined as the number of the shortest paths through it. Reference [14,15,16] indicated that the nodes and lines with high betweenness and high degrees play key roles in guaranteeing the connectivity and stability of the grid. However, pure topology-based methods ignore the electrical characteristics and often cannot completely reflect the complexity of operations [17]. Some researchers try to locate important substations and lines by using the electrical parameters [18,19,20,21,22]. An electrical betweenness was proposed in Reference [19] to quantify the role of each line in the entire grid by using the sum of absolute value of power in different directions. Reference [20] improved the electrical betweenness by considering the maximal load demand and the capacity of generators in power grids. A concept of network centrality for power systems based on system responses and topological features was presented in Reference [21]. It aims to identify critical components with an equivalent electrical model. Chen et al. [22] calculated electrical efficiency based on the admittance matrix, but assumed that power was transmitted through the most effective paths. In reality, power flow is not limited to transfer along the lines with the smallest impedance, but in all possible ways. In Reference [23], a hybrid flow betweenness (HFB) based identification method was proposed. The new betweenness covers the direction of power flow, the maximum transmission capacity, the electrical coupling degree between different lines, and outage transfer distribution factor with a more comprehensible physical background. From the perspective of the development mechanism of large-scale blackouts, Reference [24] searched for sensitive transmission lines on the basis of the criticality of system cascading collapses. Wang et al. [25] used the fault chain theory to perform cascading failures initiated by faults on different transmission lines. The number of disconnected lines is basically used to derive the vulnerability index for identifying the sensitive initial lines. This type of method [24,25,26,27] is usually based on numerous simulations, and cannot reflect the actual power flow properties and operational states well.

It is reasonable to assess network vulnerability and identify critical transmission lines of power systems with electrical and topological characteristics. However, the existing methods are mainly based on the inherent parameters of the whole power system, without enough ability to take real-time data into account. In this study, a new integrated betweenness is proposed to comprehensively consider the function of a line and the impact when it is removed from a power system. Compared with previous studies, the betweenness evaluates the role of the transmission line from the perspective of the power transmission distance, improves the impact analysis of fault flow distribution, and considers the system security with the tolerance of the remaining network. At the same time, a new searching algorithm, named cyclic addition algorithm (CAA), is proposed to locate the sensitive regions that are significantly affected by a removed line. Thus the calculation time of the betweenness can be reduced and the new identification method is more valuable in engineering applications.

The rest of the paper is structured as follows: Section 2 introduces the definition and the calculation method of the integrated betweenness of a transmission line, which consists of the indices of the power transmission, the fault flow distribution, and the influence on system security. Section 3 presents the detailed identification process with integrated betweenness and CAA. The simulation results with the IEEE 39-bus system are provided in Section 4, which prove the accuracy and effectiveness of the identification method. Finally, conclusions are given in Section 5.

2. Integrated Betweenness to Identify Critical Lines

“Vulnerability” or its opposite concept “robustness” is often used to evaluate the ability of a power grid to provide key services or functions in normal operations, with random failures or under intentional attacks [18]. A critical transmission line plays an important role in maintaining the security and stability of the power system. When it is removed due to external attacks or its own failure, the system may suffer from a serious impact, and even have a high probability of cascading trips. Therefore, the integrated betweenness proposed in this study comprehensively considers the role of a transmission line in a power system, the redistribution of power flow caused by a fault on the line, and the corresponding influence on system security.

Taking line i as an example, D_i, H_i and C_i are respectively the indices of the power transmission, the fault flow distribution, and the influence on system security. The integrated betweenness W_i of line i can be calculated by:

$$\{\begin{array}{c}{W}_i=D_i^{\prime }+H_i^{\prime }+C_i^{\prime }\\ D_i=D_{Pi}-D_{P0}\\ H_i=\frac{{\displaystyle \sum _{m\in \psi }{Z}_i^m}}{B_i}\\ C_i=\frac{1}{2}(C_i^{\psi }+C_i^{\max })\end{array}$$

(1)

where D_i′, H_i′, and C_i′ are normalized absolute values of D_i, H_i, and C_i, as can be obtained by a linear normalization method. The values of D_i′, H_i′ and C_i′ are all between 0 and 1. D_i is the index of power transmission, and is used to reflect the role of line i in the power transmission network. It can be calculated by the difference of the transmission distances of the whole power grid with and without line i. D_P0 is the transmission distance of active power in the grid inclusive of line i, while D_Pi is the transmission distance after the removal of line i; H_i is the index of fault flow distribution. It reflects the impact caused by a fault on line i. H_i can be obtained by the ratio of the amount and the distribution entropy B_i of transferred power. Z_i^m is the power change on line m when a fault happens on line i. Ψ is the set of lines with increased power. The greater the transferred power is, the larger the impacts on the other lines are. The smaller the distribution entropy B_i is, the more concentrated the power flow transfer is. A substantial increase in the transmission power can bring about harmful effects, such as the exposure of hidden failures. C_i is the index of influence on system security. C_i^ψ and C_i^max are respectively the global and the maximum influence considering the tolerance of the remaining network. The calculation method of D_i, H_i, and C_i is described in Section 2.1, Section 2.2 and Section 2.3.

2.1. The Index of Power Transmission

The importance of a line depends on its role in power transmission. The transmission distance of active power is defined as:

$$D_P={\displaystyle \sum _{l\in T}{\displaystyle \sum _{g\in G,d\in L}\left|x_l{p}_l(g,d)\right|}}$$

(2)

where x_l is the electrical length of line l. It can be expressed by the reactance value, x_l = x_ab, where x_ab is the element at row a and column b in the reactance matrix of a power grid; a and b are the two ends of line l; p_l(g, d) is the active power on line l when power is transmitted from generator node g to load node d; G is the set of generators, L is the set of loads and T is the set of lines.

The above equation can be simplified to:

$$D_P={\displaystyle \sum _{l\in T}\left|x_l\right|\left|p_l\right|}$$

(3)

where p_l is the total active power on line l, as calculated by a DC power flow model [28]:

$$p_l=p_{ab}=\frac{\theta (a)-\theta (b)}{x_{ab}}$$

(4)

where θ(a) and θ(b) are the phase angles of a and b. The index of power transmission of line i is [29]:

$$D_i=D_{Pi}-D_{P0}$$

(5)

where D_P0 and D_Pi are the transmission distances with and without transmission line i. D_i can express the change of the transmission distance of active power directly. From Equations (3)–(5), D_i can be expressed as:

$$D_i={\displaystyle \sum _{l\in T_i}\left|\mathrm{\Delta }{\theta }_l\right|}-{\displaystyle \sum _{l\in T_0}\left|\mathrm{\Delta }{\theta }_l\right|}$$

(6)

where ∆θ_l is the phase difference of the two ends of line l; T₀ and T_i are the sets of lines in the system before and after the removal of line i. Based on DC power flow model, ∆θ can be calculated by:

$$\mathrm{\Delta }\mathit{\theta }={\mathit{A}}^{\mathrm{T}}\mathit{\theta }={\mathit{A}}^{\mathrm{T}}{\mathit{B}}^{-1}{\mathit{P}}^{ing}$$

(7)

where ∆θ is the matrix of phase difference; A is the node-branch incident matrix; B is the electrical susceptance matrix; and, P^ing is the vector of injected power. Equation (7) can be used to calculate ∆θ₀ and ∆θ_i, the matrixes of transmission distances with and without line i. Equation (5) can then be expressed as:

$$D_i={\displaystyle \sum _{a=1}^{r_i}\left|\mathrm{\Delta }{\theta }_{ia}\right|}-{\displaystyle \sum _{a=1}^{r_0}\left|\mathrm{\Delta }{\theta }_{0a}\right|}$$

(8)

where r₀ and r_i are the numbers of lines in the system before and after the removal of line i; ∆θ_0a and ∆θ_ia are the elements at row a in matrix ∆θ₀ and ∆θ_i.

2.2. The Index of Fault Flow Distribution

F_i^m(g, d) is defined as the change of power on line m, as related to the power transferred from generator g to load d, when a fault happens on line i. It can reflect the impact on line m caused by the removal of faulty line i. F_i^m(g, d) can be calculated by the line outage distribution factor (LODF) [30]:

$$F_i^m(g,d)=P_{gd}\times {\mathrm{LODF}}_i^m(g,d)$$

(9)

where LODF_i^m(g, d) is the power increment of line m when unit power is transferred between generator g and load d, and a fault happens on line i; P_gd is the power flow from generator g to load d, as calculated by the power flow tracing method [31].

Considering the reverse overload, the change of power can be calculated as:

$$Z_i^m=\left|{\displaystyle \sum _{g\in G,d\in L}{P}_{0m}(g,d)}+{\displaystyle \sum _{g\in G,d\in L}{F}_i^m(g,d)}\right|-\left|{\displaystyle \sum _{g\in G,d\in L}{P}_{0m}(g,d)}\right|$$

(10)

where P_0m(g, d) is the power flow on line m in normal operating condition, and Z_i^m is the change of power on line m when line i is disconnected. In the DC power flow model, Z_i^m can also be obtained using the calculation result of Equation (4). For the lines with increased power, the sharing ratio of power flow transfer on each line is:

$${\delta }_i^m=\frac{Z_i^m}{{\displaystyle \sum _{m\in \psi }{Z}_i^m}}$$

(11)

where δ_i^m is the sharing ratio of line m when line i is disconnected; Ψ is the set of lines with increased power. The entropy of the fault flow distribution is:

$$B_i=-{\displaystyle \sum _{m\in \psi }{\delta }_i^m\ln {\delta }_i^m}$$

(12)

The entropy B_i is used to describe the distribution of Z_iⁿ, n = 1, 2, 3, …, m, … When the power is uniformly distributed, each component will shoulder the burden equally. The sharing ratio of each path is δ = 1/N, where N is the number of transferable paths. B_i is the maximum, as is lnN. Similarly, the smaller the value of B_i is, the more concentrated the transferred power flow is [32]. It means that the disconnection of line i will have significant influence on a few lines, which may overload these lines and even trigger a cascading failure. Thus, the index H_i can be used to show the distribution of fault flow, i.e.,

$$H_i=\frac{{\displaystyle \sum _{m\in \psi }{Z}_i^m}}{B_i}$$

(13)

2.3. The Index of Influence on System Security

Although the disconnections of some lines can cause a significant transfer of power flow, detrimental consequences may not be caused if other lines have large capacities and can successfully share the transferred power. Sometimes, the influence is not great, but the operating condition is close to the safety limits, and the disconnection of a line may force the system into a dangerous operating condition. Hence it is important to consider the redistribution of power flow and the tolerance of the remaining network when evaluating the importance of a line in the whole system. In this paper, the global influence and the influence on each line caused by the removal of line i are analyzed. The index of the influence on line m is:

$$C_i^m=\frac{Z_i^m}{S_m}$$

(14)

where S_m is the remaining available channel of line m before the removal of line i. The global influence can by calculated by:

$$C_i^{\psi }=\frac{1}{r_0-1}{\displaystyle \sum _{m\in \psi }{C}_i^m}$$

(15)

where r₀ is the number of lines in the power system, and C_i^Ψ is the average value of C_i^m. The influence on system security is:

$$C_i=\frac{1}{2}(C_i^{\psi }+C_i^{\max })$$

(16)

where C_i^max is the maximum value of C_i^m, and C_i is the average value of the global influence and the maximal influence.

2.4. The Threshold of the Integrated Betweenness

If the removal of line i causes the overload of any other line in the power grid, line i can be considered a critical component [33]. In order to better assess whether the line is critical, three thresholds D_th, H_th and C_th are defined next:

(1) D_th:

$$D_{th}=\underset{a\in T_0}{\min }\left|x_a\right|\left|p_{cpa}\right|-\underset{b\in T_0}{\min }\left|x_b\right|\left|p_b\right|$$

(17)

where x_a is the electrical length of line a; p_cpa is the remaining available channel of line a; x_b is the electrical length of line b; p_b is the total active power on line b; and, T₀ is the set of lines in the power system. For line i, D_i = D_Pi − D_P0 and D_Pi has already subtracted $\left|x_i\right|\left|p_i\right|$, so min $\left|x_b\right|\left|p_b\right|$ is also subtracted from D_th.

(2) H_th: H_i can reflect the impact caused by a fault on line i. To avoid missing critical lines, H_th should be the minimum H_i when one remaining line is overloaded after the fault. To get the minimum H_i, the entropy B_i of the fault flow distribution should be as large as possible and ${\displaystyle \sum _{m\in \psi }{Z}_i^m}$ in Equation (13) should be as small as possible. Let us assume the smallest remaining available channel of a transmission line is p_mincp, and p_mincp = minp_cpa, a $\in$ T₀. B_i takes the maximum value, i.e., ln(r₀ − 1), and ${\displaystyle \sum _{m\in \psi }{Z}_i^m}$ takes p_mincp ignoring the complex conditions. The threshold of H is:

$$H_{th}=\frac{p_{\min cp}}{\ln (r_0-1)}$$

(18)

(3) C_th: When only one remaining line is overloaded and other lines are not affected, C_i^max = 1 and C_i^ψ =1/(r₀ − 1). The threshold of C is:

$$C_{th}=\frac{1}{2}(1+\frac{1}{r_0-1})$$

(19)

W_th equals to the sum of normalized D_th, H_th and C_th, i.e.,

$$W_{th}=D_{th}^{\prime }+H_{th}^{\prime }+C_{th}^{\prime }$$

(20)

where D_th′, H_th′ and C_th′ are normalized absolute values of D_th, H_th and C_th. The transmission line, the integrated betweenness of which is greater than W_th, can be regarded as a critical component. The mathematical formulae for D_th, H_th, and C_th give out small values. Such small threshold values prevent missing the detection of critical lines. After the above process, these lines can be sorted according to W.

3. Identification Process with Integrated Betweenness and CAA

The integrated betweenness of each line is affected by the changes in the grid’s structure or the operating mode. The calculation time is large when the approach is applied to the data of the whole power network. In fact, D_i, H_i, and C_i are all related to the changes of power flow before and after line i is removed, where the changes mainly occur on particularly sensitive lines. Therefore, a cyclic addition algorithm (CAA) is proposed to locate the sensitive regions which consist of the transmission lines with large variations of power. The impact on the sensitive regions is analyzed rather than the impact on the whole power network. Thus, the calculation time of the betweenness can be significantly reduced and the proposed identification method is more suited to the real-time analysis.

The process to identify the critical transmission lines is shown in Figure 1. First, the basic data of the grid, such as the structural parameters and the information of power flow, are obtained. The data can be collected by phasor measurement units (PMUs). When the network structure or operating mode changes, the identification process of critical lines in this state will start. Possible changes of the network structure or operating mode include the disconnection/failure of primary equipment (generators, transformers, or lines), the connection/reconnection of primary equipment, and significant redistribution of power flow within the grid. Such changes can be caused by (1) faults, e.g., line trip, generator trip, and line failure; (2) planned adjustments, e.g., the operation of new generators/lines and planned load shedding. Then W_th and the integrated betweenness of each line are calculated: sensitive regions can be located by CAA, and the remaining regions are regarded as non-sensitive regions in this study. Since the influence of the non-sensitive regions on the calculation results is quite small, those regions can be simplified as sources and loads. The number of lines and substations in the system is greatly reduced after the simplification, so that the integrated betweenness W can be calculated more quickly. With the calculation method introduced in Section 2, W of each line can be obtained. Finally, the critical lines can be identified.

Location of Sensitive Regions with CAA

The power flow is transferred on all feasible paths, but the transferred power on each path is quite different. As can be seen from previous studies [26,27,28,29,30,31], when a transmission line is disconnected or removed from the power network, the power is usually transferred to other paths connected with the two-end nodes of the removed line, and mainly along the paths with short electrical distances, as is shown in Figure 2.

The topological analysis-based method can provide a quick and effective way to identify the network structure and find propagation paths of power flow transfer [34]. Some new approaches, such as the compressive sensing-based approach [35] and graphical learning-based approach [36,37] were proposed in recent years. Based on Dijkstra’s algorithm [38], CAA is proposed in this study. The sensitive regions can be obtained through the cyclic addition of different lines. The following is a detailed description of the process.

Supposing there are a substations and b lines, all substations in the power grid can be simplified as nodes and form V according to graph theory, as is V = {v₁, v₂, ..., v_a}. All lines are represented by the edge set E, E = {e₁, e₂, e₃, …, e_b}. V and E constitute G(V, E), the topology graph of the power grid. All different paths connecting v_i and v_j in graph G form set D(G, v_i, v_j), and the length of path d is l(d). The line reactance is used as the weight of an edge, and l(d) is the sum of the weights of all edges in path d.

Theoretical principle 1: According to graph theory, if G cannot be fully connected when node v is removed, v is called the cut point in G, such as point v₄ in Figure 3. Any edge and the paths connected with its two ends can only exist in the same part divided by cut points.

Theoretical principle 2: According to the conclusion in Reference [39], any path between two nodes in graph G(V, E) must be the shortest path between the same nodes in G_g (V_g, E_g), where G_g is a sub-graph of G with a few (including b) edges. It can be described as:

$$d\in D(\mathrm{G},v_p,v_q)\leftrightarrow d=d_{\mathrm{shortest}}({\mathrm{G}}_g,v_p,v_q)$$

(21)

where d_shortest (G_g, v_p, v_q) is the shortest path between v_p and v_q in graph G_g. Therefore, the path between two nodes in G(V, E) can be transformed into the shortest path between the same nodes in a sub-graph.

In this study, line e₀ is defined as the removed line, while v_p and v_q are its two-end nodes. Vector L is built to store the shortest distance between any two nodes which has been found currently. For the graph with a nodes, there are C_a² elements in L and the elements can be described as L(1–2), L(1–3), L(1–4), …, L(1–a), L(2–3), L(2–4), …, L((a − 1)–a), where L(i–j) is the shortest distance between v_i and v_j, and i and j can be interchanged. Vector HJ is built to store the successor nodes along the shortest path. Similarly, there are C_a² elements in HJ and the elements can be described as HJ(1–2), HJ(1–3), HJ(1–4), …, HJ(1–n), HJ(2–3), HJ(2–4), …, HJ((a − 1)–a). Each sub-graph has corresponding values for L and HJ.

Step 1: Find cut points and divide the graph into different parts. Find the parts G’ where e₀ is located. Suppose that there are m lines and n nodes in G’. Delete e₀ and get all the other edges in graph G’. Let k = 1.

Step 2: Select k edges, and there are C_m−1^k different selection results. Let c = 1.

Step 3: Deal with the selection result c, where k edges and their connected nodes constitute a sub-graph G’_kc. If v_p or v_q is an isolated node without any connected lines, or v_p and v_q are located in different islands in G′_kc, go to step 8, i.e., the sub-graph can be ignored directly; otherwise go to step 4.

Step 4: Take away and rank k edges. Let f = 1 and initialize L_kc and HJ_kc. When there is no edge in graph G’_kc, the values of the elements in L_kc are equal to an infinite value, as is L_kc = [∞, ∞, ∞, …, ∞]. All elements in HJ_kc are Ø, showing there is no path currently.

Step 5: Put edge f back. The two-end nodes of f are v_u and v_v, v_u and v_v $\in$ V′_kc. Next apply the following judgment: If L_kc(u − v) < l(f), go directly to step 7. Otherwise, let L_kc(u − v) = l(f), HJ_kc(u − v) = v, and go to step 6.

Step 6: Deal with L_kc and HJ_kc:

For w (1 ≦ w ≦ n, w ≠ u, w ≠ v):

If L_kc(u − w) > L_kc(u − v) + L_kc(v − w), change the value of L_kc(u − w) into L_kc(u − v) + L_kc(v − w), and change HJ_kc(u − w) into v; If L_kc(v − w) > L_kc(u − v) + L_kc(u − w), change the value of L_kc(v − w) into L_kc(u − v) + L_kc(u − w), and change HJ_kc(v − w) into u;

If L_kc(u − w) = L_kc(u − v) + L_kc(v − w) ≠ ∞, insert v into HJ_kc(u − w); If L_kc(v − w) = L_kc(u − v) + L_kc(u − w) ≠ ∞, insert u into HJ_kc(v − w); then go to step 7.

Step 7: If f = k, go to step 8; otherwise f = f + 1, then go back to step 5.

Step 8: If c = C_m−1^k, go to step 9; otherwise c = c + 1, then go back to step 3.

Step 9: if k = m−1, go to step 10; otherwise k = k + 1, then go back to step 2.

Step 10: All the paths between v_p and v_q and their lengths can be got. Remove duplicate paths.

Supposing that there are y paths between v_p and v_q, the lengths of all paths (l(1), l(2), ..., l(y)) can be derived by the above process. According to the principle of electric circuits, the transferred power flow on path d changes inversely with the length l(d). Thus, the power sharing coefficient of edge i is:

$${\xi }_i={\displaystyle \sum _{o\in \mu }\frac{1}{l(o){\displaystyle \sum _{s=1}^y\frac{1}{l(s)}}}}$$

(22)

where μ is the set of paths which contain line i. Judge whether the power sharing coefficient can satisfy:

$${\xi }_i\ge {\xi }_{th}$$

(23)

where ξ_th is the threshold value, ξ_th ≥ 0. If (23) is satisfied, line i will be regarded as a sensitive line, as is closely related to line e₀. The sensitive lines should be selected effectively and the sensitive regions which consist of sensitive lines should be fully connected. According to the definition of sensitive lines or transmission sections in Reference [40], the threshold value usually varies between 0.2 and 0.3. Based on the above analysis, the threshold ξ_th can be obtained by the following iterative process: ξ_th is initialized with 0.3. If the sensitive regions are not fully connected, ξ_th will be decreased with a fixed step size, until the requirement of full connection can be satisfied or the value of ξ_th is 0.

To improve the accuracy of integrated betweenness and to avoid missing sensitive lines, ξ_th can be taken as 0 directly in an extreme situation.

Sensitive regions can be obtained by the above process. For G′ with m lines and n nodes, the calculation amount of the cyclic process in CAA in the worst condition is:

$$P={\displaystyle \sum _{k=1}^{m-1}{\mathrm{C}}_{m-1}^k}\cdot k\cdot (4n-6)$$

(24)

where P = 1 means that one element in L or HJ is updated once. For G′ with m lines and n nodes. C_m−1^k is the number of sub-graphs with k lines. In the worst case, all sub-graphs can satisfy the requirement in step 3, and none will be ignored. 2(n − 2) + 1 elements in L and 2(n − 2) + 1 elements in HJ are updated when one transmission line is put back through steps 5 and 6. The process of CAA only contains some simple comparisons based on a pure topology model. Therefore, the calculation time of the identification method can be significantly reduced. At the same time, it should be noticed that CAA is effective for most lines in power systems. For the lines which do not have other paths connected with their two ends, the removal of the lines can result in the islanding phenomenon. In these cases, the injected power of sources and absorbed power of loads in each island should be balanced first. After that, the calculation of the integrated betweenness can be carried out using the analysis of the whole power network.

4. Test Cases

In this section, IEEE 39-bus system is used as the test system to demonstrate the effectiveness of the proposed method.

4.1. Location of Sensitive Regions

4.1.1. Location of Sensitive Regions with CAA

Line e₃₃ is taken as an example and used to show the detailed process to locate the sensitive regions.

Firstly, all substations and lines are simplified as nodes and edges. Line e₃₃ is the removed line, while v₂₆ and v₂₉ are its two-end nodes. Divide the graph into different parts with cut points v₂₆ and v₁₆ in Figure 4. G’ which contains v₂₆, v₂₈, v₂₉, e₃₂, e₃₃ and e₃₄ is found, and then e₃₃ is deleted. The edges e₃₂ and e₃₃ in graph G’ are taken away. Let k = 1. Select 1 edge, and it can be seen that there is no path between v₂₆ and v₂₉ in all the sub-graphs with one edge and two nodes. Let k = 2. Taking sub-graph G’₂₁ as an example, the sub-graph is shown in Figure 5. Since the topology of G′₂₁ is simple, the power sharing coefficient of each edge can be obtained directly. To better illustrate the application of CAA, the calculation process is introduced in detail.

With the analysis of step 3, the sub-graph is kept. Take away and rank the 2 edges. Let f = 1, and initialize L₂₁ and HJ₂₁. The elements in L₂₁ are [L₂₁(26–28), L₂₁(26–29), L₂₁(28–29)]. The elements in HJ₂₁ are [HJ₂₁(26–28), HJ₂₁(26–29), HJ₂₁(28–29)]. L₂₁ = [∞, ∞, ∞] and HJ₂₁ = [Ø, Ø, Ø].

Put edge e₃₄ back. The weight of the edge l(e₃₄) is 0.0151, which is smaller than L₂₁(28–29). Therefore, let L₂₁(28–29) = 0.0151. After the step 5, L₂₁ = [∞, ∞, 0.0151] and HJ₂₁ = [Ø, Ø, 29]. Because L₂₁(26–28) = L₂₁(28–29) + L₂₁(26–29) = ∞ and L₂₁(26–29) = L₂₁(28–29) + L₂₁(26–28) = ∞, do nothing to L₂₁ and HJ₂₁ according to step 6. Go to step 7, i.e., f = f + 1 = 2. Then go back to step 5.

Similarly, put edge e₃₂ back. L₂₁ = [0.0474, 0.0625, 0.0151] and HJ₂₁ = [28, 28, 29].

The length of the shortest path between nodes v₂₆ and v₂₉ in graph G′₂₁ is 0.0625 according to L₂₁(26–29). The successor node of v₂₆ is v₂₈, and the successor node of v₂₈ is v₂₉, which can be got by HJ₂₁(26–29) and HJ₂₁(28–29). Then the shortest path v₂₆–v₂₈–v₂₉ can be obtained. The power sharing coefficients of e₃₂ and e₃₄ are 1. ξ_th is set to 0.3. The sensitive region, which includes line e₃₂ and e₃₄ is now derived.

By two simple comparisons, the analysis of all lines in the whole network can be simplified as the analysis of two lines. The order of the matrixes, such as A and B in Equation (7), can be reduced from 39 to 3 during the calculation of the integrated betweenness.

4.1.2. Comparison of CAA and Depth-First-Search Method

For the purpose of comparing the performance of CAA and Depth-First-Search (DFS) [41] algorithm, line e₃₃ is used as an example. Both algorithms are used separately to find the paths from v₂₆ to v₂₉ after line e₃₃ is removed.

The CAA proposed in the study uses the topological method to search for all possible paths connecting the two-end nodes of the removed line and obtain the power sharing coefficients of different edges. The DFS algorithm explores a graph by starting at a node and going as deep as possible. Such DFS traversal is a type of backtracking method, and therefore exhibits poor performance.

With CAA, the relevant sub-graph can be located at first, which effectively cleans up all useless edges and reduces the search scope. Then, the path and its length can be obtained directly after two comparisons. With DFS, v₂₆ is set as the starting node. v₂₅, v₂₇ and v₂₈ can be reached in the next step. The paths through v₂₅ or v₂₇ are not feasible by several explorations and backtracking. The only feasible path is v₂₆–v₂₈–v₂₉.

As can be seen from Section 4.1.1 and the above analysis, the proposed CAA can delete useless lines and sub-graphs to avoid invalid search directions. In this case, the search results with CAA and DFS are identical, but the computation time with CAA is 78.6% shorter than that with DFS.

4.1.3. Accuracy Analysis of the Location Method

(1) To show the validity of the location method, branch e₃₃ is removed at 0.5 s. The simulation result based on PSASP is shown in

Table 1. Simulation result when line e₃₃ is removed.

	Line	Power Change (MW)		Line	Power Change (MW)		Line	Power Change (MW)
1	e34	189.7265	12	e13	0.584	23	e2	−0.6828
2	e32	186.0633	13	e10	0.103	24	e14	−0.6843
3	e9	2.514	14	e28	0.0099	25	e6	−0.8824
4	e7	2.4037	15	e29	0.0051	26	e25	−0.8841
5	e26	2.1787	16	e24	−0.0082	27	e16	−1.2161
6	e30	1.964	17	e27	−0.0305	28	e12	−1.3197
7	e19	1.3781	18	e23	−0.0328	29	e20	−1.3791
8	e18	1.3198	19	e22	−0.0565	30	e3	−1.4682
9	e21	1.2928	20	e8	−0.0709	31	e4	−2.1515
10	e17	1.2161	21	e15	−0.6759	32	e31	−2.2252
11	e11	0.5896	22	e1	−0.6828	33	e5	−2.3553

. As can be seen from the simulation results, there is a large increase in the power transferred on the transmission lines in the sensitive regions, while the increases on un-sensitive lines are much smaller.

(2) Line e₂₀ is removed at 0.2 s. The simulation result is shown in

Table 2. Simulation result when line e₂₀ is removed.

	Line	Power Change (MW)		Line	Power Change (MW)		Line	Power Change (MW)
1	e19	289.8958	12	e31	−63.5244	23	e10	−20.1339
2	e21	288.6156	13	e26	63.2027	24	e3	15.7069
3	e5	237.9716	14	e4	61.5607	25	e22	0.1593
4	e25	224.116	15	e9	−48.6027	26	e23	0.0991
5	e6	222.5013	16	e1	45.4289	27	e27	0.0972
6	e8	−208.162	17	e2	45.4289	28	e28	−0.033
7	e18	81.8674	18	e15	45.2833	29	e33	0.0229
8	e12	−81.6569	19	e14	45.1093	30	e32	−0.0216
9	e17	75.1401	20	e7	−28.4259	31	e34	−0.0211
10	e16	−75.1401	21	e11	−25.1939	32	e24	0.0187
11	e30	−63.6899	22	e13	−25.0956	33	e29	−0.0182

. When ξ_th is 0.3, there are e₄, e₅, e₆, e₈, e₁₂, e₁₆, e₁₇, e₁₈, e₁₉, e₂₁, e₂₅, e₂₆, e₃₀, and e₃₁ in sensitive regions. The first 14 branches with large power variations are included.

4.2. Analysis of Identification Results

4.2.1. Identification Result with Integrated Betweenness and CAA

ξ_th is set as 0 to insure the accuracy of the identification. The capacity of each line is 5 times the initial active power. The sensitive regions can be located by CAA, and the non-sensitive regions are simplified as sources and loads. Based on that, the integrated betweenness of each line can be calculated. For the lines whose removal can result in islanding phenomenon, the injected power of sources and the absorbed power of loads in each island should be balanced. In this study, the absorbed power of each load and the injected power of each source are reduced in proportion to their initial power and according to the minimum load shedding method as described in Reference [42].

During the identification process, the shortest computation time of W for a single transmission line is 0.515 ms, and the longest time for a single line is 8.416 ms. The computation time of the whole process is 0.061 s. Without CAA, the computation time is 0.286 s when the approach is applied to the data of the whole power network. The results prove that the CAA is effective in reducing the computation time.

The final results of the indices and the betweenness are shown in

Table 3. Calculation result of transmission lines.

	Line	Index			W		Line	Index			W
		D	H	C				D	H	C
1	e27	0.7439	1	0.7975	2.5415	18	e18	0.0749	0.3855	0.0893	0.5497
2	e3	1	0.4868	0.8282	2.3150	19	e17	0.0845	0.4046	0.0590	0.5482
3	e20	0.6249	0.3874	0.9302	1.9426	20	e1	0.0454	0.1499	0.2332	0.4286
4	e31	0.5744	0.3437	1	1.9182	21	e2	0.0454	0.1499	0.2332	0.4286
5	e4	0.4285	0.3214	0.8308	1.5809	22	e13	0.0473	0.2791	0.0370	0.3636
6	e23	0.2676	0.5499	0.4379	1.2555	23	e30	0.0042	0.0887	0.2409	0.3339
7	e34	0.67893	0.3823	0.0890	1.1502	24	e5	0.0681	0.1215	0.0919	0.2816
8	e12	0.3574	0.4769	0.1919	1.0263	25	e33	0	0.2001	0.0626	0.2628
9	e21	0.1928	0.2595	0.5214	0.9738	26	e7	0.0397	0.1927	0.0276	0.2601
10	e29	0.0523	0.5406	0.3566	0.9495	27	e32	0	0.1421	0.0322	0.1743
11	e11	0.1849	0.6203	0.0867	0.8919	28	e6	0.0517	0.0467	0.0556	0.1541
12	e16	0.1981	0.4950	0.1285	0.8218	29	e24	0.0334	0.0832	0.0055	0.1222
13	e22	0.1972	0.5363	0.0419	0.7755	30	e28	0.0329	0.0760	0.0049	0.1139
14	e8	0.1445	0.3247	0.2794	0.7488	31	e14	0.0698	0.0247	0.0047	0.0993
15	e9	0.0232	0.6239	0.0892	0.7364	32	e15	0.0698	0.0247	0.0047	0.0993
16	e10	0.0957	0.4873	0.0881	0.6711	33	e19	0.0363	0.0231	0.0013	0.0607
17	e25	0.0434	0.2404	0.3199	0.6038	34	e26	0.0014	0.0190	0.0149	0.0354

. W_th = 0.2546. e₆, e₁₄, e₁₅, e₁₉, e₂₄, e₂₆, and e₂₈ cannot be regarded as critical lines. Compared with the results calculated by the global network, the average error rate of index H with the data of sensitive regions is 2.8%, and the average error rate of index C is 1.02%. The average error rate of integrated betweenness W is 2.24%. The accurate identification result can still be obtained through the process of simplification.

4.2.2. Comparison of Lines with Different Integrated Betweenness

Taking lines e₂₇, e₂₀ and e₂₉ as examples, the increased power of transmission lines at 1 s in IEEE 39-bus system is shown in Figure 6, when e₂₇, e₂₀ and e₂₉ are removed individually at 0.5 s.

It can be seen clearly from Figure 6a that the total amount of power flow transfer is quite large and it is mainly concentrated on lines e₂₄, e₂₈, and e₂₉ when line e₂₇ is removed. It means that the disconnection of line e₂₇ has significant influence on the power system. The comparison of Figure 6b,c shows that the transferred power is more concentrated when line e₂₉ is removed. But the total transferred power and the global impact is less compared with the removal of line e₂₀. The above phenomena are consistent with the values of D, H, and C, which proves the validity and rationality of the indices.

4.3. Comparison of Identification Results

To verify the effectiveness of the identification method with integrated betweenness, the top 10 lines are attacked. When compared with the attacks performed according to the hybrid flow betweenness (HFB) as introduced in Reference [23], the number of islands, the total value of the loads, and the average vulnerability of the remaining network are shown in Figure 7. The loss of loads can be determined according to the minimum load shedding method [42]. The average vulnerability of the transmission lines is evaluated using the probability of incorrect tripping caused by the current increment. The linear function of the probability can be obtained according to Reference [43]. In this study, the probability of incorrect tripping is set to 0 when there is no current increment on the transmission line, and the probability is set to 1 when the current is greater than 1.4 times the rated value.

On the basis of the direction and distribution of power flow, HFB considers the electrical coupling between the lines and the influence caused by a single line fault. Compared with HFB, the integrated betweenness proposed in this study improves the impact analysis of fault flow distribution and considers the tolerance of the remaining network. The top 10 lines are e₂₇, e₃, e₂₀, e₃₁, e₄, e₂₃, e₃₄, e₁₂, e₂₁, and e₂₉ according to the proposed integrated betweenness and e₁₁, e₂₀, e₃, e₁₆, e₂₉, e₁₂, e₁, e₁₉, e₇, and e₅ as according to the HFB. The lines are attacked and disconnected in turn, and the impact of each disconnection on the remaining transmission lines is analyzed.

(1) Integrated betweenness attack

According to the identification results of the proposed method, line e₂₇ with high D, H and C will be disconnected first. The current increments on lines e₂₄, e₂₈ and e₂₉ are 5.268 p.u., 5.8586 p.u. and 5.4645 p.u. These lines exhibit high current increments and will therefore have a high risk of tripping. Then, line e₃ is disconnected. It will have impact on several transmission lines, namely e₁, e₂, e₆, e₇, e₁₄, e₁₅, e₂₅, e₃₀, and e₃₁. At the same time, the current on e₂₄, e₂₈, and e₂₉ continues to rise and the system risk will be further increased. Following the disconnections of lines e₂₀ and e₃₁, the power flow is redistributed to lines e₄, e₅, e₁₇, e₁₈, and e₁₉.

Islanding phenomenon occurs when e₄ is disconnected. The network is split into a 6-node isolated island and a 33-node system. Although the network is forced to be divided into two parts, the total amount of loads is reduced during the balance of the injected power of sources, and the absorbed power of loads in each island. Therefore, the pressure on the transmission system and the overloads of e₁, e₂, e₇, e₁₄, e₁₅, e₁₇, e₁₈, e₁₉, and e₃₀ are relieved. Subsequently, e₂₃ is disconnected. The node v₂₁ is separated from the network, and the current on e₂₄, e₂₈, and e₂₉ is slightly decreased. After the attacks on e₃₄, e₁₂, e₂₁, and e₂₉, some nodes fall off, but the average vulnerability of the remaining lines keeps increasing.

(2) HFB attack

When line e₁₁ is disconnected, e₉, e₁₀, e₁₄, and e₁₅ will have a high risk of tripping. Then e₂₀, e₃, e₁₆, e₂₉, e₁₂, and e₁ are attacked one by one. Some transmission lines, such as e₂, e₆, e₁₇, e₁₈, e₁₉, e₂₁, e₂₅, e₂₆, and e₃₀, are affected by the power flow transfer. Most of the current increments on these lines are between 0.2 and 0.3 p.u. and the maximum current increment is 4.4196 p.u. Subsequently, line e₁₉ is disconnected and the node v₁₅ is isolated from the network. Finally, e₇ and e₅ are attacked and the average probability of incorrect tripping is approximately 0.7.

It can be seen from Figure 7 that the islanding phenomenon is more obvious and the loss of loads is greater under the integrated betweenness attack performed on the IEEE 39-bus system. The lines with high betweenness play critical roles in the power grid. When nearly 30% of the lines are removed under integrated betweenness attack, the loss of load remains at 24%, and the vulnerability of the remaining network is also greatly improved.

4.4. Extreme Cases and Real-Time Behavior

4.4.1. Extreme Cases

IEEE 39-bus system is used as the test system in this section. The injected and absorbed power is changed to present some extreme cases.

(1) All the injected power of generators and the absorbed power of loads are reduced to 0.05 times the initial value. The calculation result is shown in

Table 4. Calculation result of transmission lines.

Line	W	Line	W	Line	W	Line	W	Line	W
e1	0.2489	e8	0.5680	e15	0.0955	e22	1.2320	e29	0.6616
e2	0.2489	e9	0.6824	e16	1.0743	e23	0.9019	e30	0.1394
e3	1.6463	e10	0.6000	e17	0.5979	e24	0.1177	e31	1.1108
e4	0.9140	e11	0.8219	e18	0.4032	e25	0.3507	e32	0.1483
e5	0.2073	e12	0.9458	e19	0.0597	e26	0.0234	e33	0.2122
e6	0.1092	e13	0.3337	e20	1.1916	e27	1.8976	e34	1.0783
e7	0.2291	e14	0.0955	e21	0.5660	e28	0.1099

. Based on the identification method, W_th = 2.3149, and no line is selected to be a critical transmission line.

(2) All of the injected power of generators and the absorbed power of loads are increased 4.99 times in respect to their initial values, to bring the system close to its stability limit. The disconnection of any line will result in a serious consequence. In this case, W_th = 0.00822. The calculation result is shown in

Table 5. Calculation result of transmission lines.

Line	W	Line	W	Line	W	Line	W	Line	W
e1	0.4372	e8	0.7937	e15	0.0993	e22	1.2659	e29	0.9495
e2	0.4372	e9	0.7544	e16	1.1781	e23	1.2555	e30	0.3339
e3	2.3150	e10	0.6711	e17	0.6456	e24	0.1222	e31	1.9182
e4	1.5848	e11	0.8919	e18	0.4753	e25	0.6090	e32	0.1743
e5	0.2816	e12	1.1008	e19	0.0607	e26	0.03546	e33	0.2628
e6	0.1541	e13	0.3636	e20	1.9426	e27	2.5415	e34	1.1502
e7	0.2514	e14	0.0993	e21	0.9870	e28	0.1139

. All lines are regarded as critical transmission lines.

4.4.2. Real-Time Behavior of Power Grids with Different Scales

The computation times of test systems with various scales are statistically analyzed. The systems include IEEE 39-bus system, IEEE118-bus system, and several large-scale systems, which are randomly synthesized by IEEE standard 30-bus and 118-bus system [44]. The standard systems are connected with high-tension transmission lines. Each system is calculated ten times. The average computation time of each test system is shown in

Table 6. Computation time of different test systems.

Test System	Number of Buses	Computation Time (s)
IEEE39	39	0.061
IEEE118	118	0.319
SYN472	472	1.285
SYN1062	1062	2.891
SYN3000	3000	7.639
SYN7680	7680	32.268
SYN12000	12,000	73.512

.

Based on the above data, the computation time can be qualitatively analyzed: the computation time of the system with 39–5000 buses can be of the order of seconds, and the computation time of the system with 5000–10,000+ buses can be of the order of minutes.

The implementation of “real-time identification of critical lines” depends on two important parts: (1) measurement data acquisition; (2) data processing and analysis. The measurement data can be obtained by wide area measurement system (WAMS). According to Reference [45], WAMS is required to complete the measurement and transmission of PMU data in 30–50 ms for dynamic monitoring and real-time decision-making purposes. Based on the data in Table 6, the time scale of the identification process can be of the order of minutes.

5. Conclusions

In order to implement accurate and fast identification of critical lines, an integrated betweenness is proposed which considers the topological and electrical characteristics of the power network. The indices of power transmission, fault flow distribution, and the influence on system security are used to assess the importance of different lines with respect to the functionality of themselves, relevance with other transmission lines, and system security. Although the three indices are different, they are all related to the changes in the power system before and after the removal of a line. Based on this, the sensitive regions containing lines with large variations of power are located by CAA. The analysis of the whole network is replaced by the analysis of sensitive regions, which reduces the calculation time effectively and provides the possibility for real-time analysis in large-scale power systems.

This study focuses on the identification of critical lines which play important roles in power transmission and whose disconnections have a great influence on system vulnerability. Together with the transmission lines which are heavily overloaded or have high risks of failures, the identification results can help formulate plans to avoid blackouts. The simulation results show that the proposed method is effective and practical.