Vulnerability assessment of power system based on Theil entropy of branch potential energy

Reference [21] ignored the fault propagation of the power grid in the construction process of correlation networks. Reference [22] only considers the change of the active power after the branch fault in the calculation of the correlation network weight, which ignores the reactive power transmitted in the power grid and node voltage state. A large number of branch potential energy will be produced and distributed on each branch when a major disturbance happened in power system. And the system stability can be reflected by the distribution of the branch potential energy in the network. The branch potential energy is composed of two parts corresponding to the active and reactive power transmitted on the branches [23].

By integrating the branch voltage difference and phase angle difference, a branch energy function model can be established [24], and the expression is given as follows:

$$\begin{eqnarray}&&\begin{array}{l}{E}_{ij}=\displaystyle {\int }_{{\delta }_{ij}^{s}}^{{\delta }_{ij}}\left[{U}_{i}^{2}{G}_{ij}-{U}_{i}{U}_{j}\left({G}_{ij}\,\cos \,{\delta }_{ij}+{B}_{ij}\,\sin \,{\delta }_{ij}\right)-{P}_{ij}^{s}\right]d{\delta }_{ij}\\ \,+\,\displaystyle {\int }_{{U}_{ij}^{s}}^{{U}_{ij}}\left[\displaystyle \frac{-{U}_{i}^{2}{B}_{ij}+{U}_{i}{U}_{j}({B}_{ij}\,\cos \,{\delta }_{ij}+{G}_{ij}\,\sin \,{\delta }_{ij})-{Q}_{ij}^{s}}{{U}_{ij}}\right]d{U}_{ij}\end{array}\end{eqnarray} \tag{ 1 }$$

where i and j are the node numbers of branch m. E_ij is the branch energy function of branch m(i-j), ${\delta }_{ij}^{s},$ ${U}_{ij}^{s},$ ${P}_{ij}^{s}$ and ${Q}_{ij}^{s}$ represent the initial state values of the phase angle difference ${\delta }_{ij},$ voltage difference ${U}_{ij},$ active power ${P}_{ij}$ and reactive power ${Q}_{ij}.$ ${G}_{ij}$ of branch m(i-j). is the conductance of branch m(i-j). ${B}_{ij}$ is the susceptance of branch m(i-j). ${U}_{i}$ and ${U}_{j}$ are the voltage amplitudes of node i and node j respectively. $\cos \,{\delta }_{ij}$ and $\sin \,{\delta }_{ij}$ are the cosine and sine values of branch phase angle.

Due to the absorption of the branch potential energy, the global energy distribution becomes unbalanced, which results in the vulnerability of the branch and power grid. The greater the impact of branch potential energy is, the worse the stability of the system and the higher the vulnerability is [25]. The branch potential energy is mainly affected by the complex power on the branch. The active power transmitted by the branch is mainly determined by the voltage phase angle difference between the branch nodes, while the reactive power mainly depends on the voltage amplitude difference between the branch nodes. When the system is attacked, the system state will transit from the initial state to the fault state. The transmission power of the branch will change, which will result in the accumulation of the branch potential energy. The uneven distribution of branch potential energy in the network leads to different vulnerability of each branch. The higher the branch potential energy is, the higher the vulnerability is. The branch potential energy can represent the state characteristics of the power system, so the branch potential energy can be used as the weight of the state transition network to reflect the power grid state comprehensively.

Based on the above analysis, the construction of the state transition correlation network based on the branch potential energy function is the primary work to evaluate the vulnerability of the power grid. To construct the state transition correlation network, each branch is taken as an initial fault. Moreover, the correlation between branches is analyzed based on the branch potential energy. Finally, a weighted state transition correlation network is established which takes the branch potential energy variation as the weight. The specific process is shown in Algorithm 1.

Algorithm1: Construction of the two-level state transition network
1	Input grid parameters and calculate the power flow and capacity margin of each branch.
2	According to the rule of "power grid branch corresponds to the unweighted network node", map the null graph.
3	For I  = 1: L (L is the number of grid branches)
4	Remove the branch $i$ and calculate the state changes of the remaining branches in the power grid: active power variation ${\rm{\Delta }}{p}_{ij},$ reactive power variation ${\rm{\Delta }}{q}_{ij},$ phase angle variation ${\rm{\Delta }}{\theta }_{ij}$ and voltage amplitude variation ${\rm{\Delta }}{v}_{ij}.$
5	For j = 1: L
6	If the active power on branch j is overload
7	Continue to remove the branch j and recalculate the state changes of the remaining branches in the grid.
8	End if
9	End for
10	For j = 1: L
11	According to Formula (1), calculate the transient energy variation of branch j;
12	If the disconnection of branch i causes the potential energy change of branch j
	Establish an edge between nodes i and j in the state correlation networks, and assign the variation of the branch potential energy to edge (i-j) as weights.
	End if
13	End for
14	End for

The branch potential energy comprehensively reflects the operation state changes in the power grid, and its distribution on the branches can reflect the ability to withstand the impact. Therefore, the proposed state transition network accords with the power transfer and coupling relationship between transmission lines in the actual operation of the power grid. Taking figure 1 as an example, the process of generating a two-level state transition network by power system mapping is illustrated. There are 9 branches in the system, so the two-level state transition network has 9 nodes. If there are interactions among the 9 nodes, bidirectional edges will be generated to represent their correlations. Edge weight△E1–2 represents the potential energy change of branch L2 after the disconnection of branch L1. It should be noted that the influence of the fault of the branch i on the potential energy of branch j is different from that of the fault of branch j on branch i. Therefore, the branches in the secondary state transition network are bidirectional and have two weights, that is, one direction corresponds to one weight.

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According to the above analysis, the state transition network structure is a bidirectional weighted correlation complex network. In the correlation network, the out-degree of nodes reflects the influence of the node on the adjacent nodes and corresponds to the fault propagation characteristics of the branch. The higher the number of the out-degree of nodes is, the wider the influence range of the node is. And the larger the weight of the out-degree of nodes is, the greater the impact of the node on the corresponding node is. Therefore, this paper only studies the vulnerability of power system under the influence of the out-degree in the correlation network. Considering that the connectivity of a node is an important criterion to measure its criticality, we need to establish the connectivity index of the correlation network. This index can comprehensively reflect the total connectivity strength of nodes and their distribution in the related edges. Therefore, this paper constructs a branch potential energy entropy index based on the Theil entropy theory, which is shown in Formula (2).

$$\begin{eqnarray}&&PEE{D}_{u}={S}_{u}\times {T}_{u}\end{eqnarray} \tag{ 2 }$$

where $PEE{D}_{u}$ is the branch potential energy entropy degree after the interruption of node u. ${S}_{u}$ is the weight of the node u. ${T}_{u}$ is the Theil entropy of the node u in the correlation network, and u is the node number in the correlation network.

Theil entropy standard is an index to calculate the income inequality based on the information entropy, which is the difference between the maximum value of information entropy and the actual entropy [26]. It is shown in Formula (3).

$$\begin{eqnarray}&&{T}_{u}=\,\mathrm{log}\,n-H\left(x\right)=\displaystyle \sum _{i=1}^{n}{x}_{i}\times \,\mathrm{log}\,n{x}_{i}\end{eqnarray} \tag{ 3 }$$

where, $\mathrm{log}\,n$ is the maximum information entropy when n uncertain events appear with the same probability. H (x) is the actual entropy of the system, and ${x}_{i}$ is the probability of the occurrence of the uncertain event i.

If the weight of edge ${e}_{uv}$ in state correlation network is defined as ${E}_{uv},$ then the weight of node u is:

$$\begin{eqnarray}&&{S}_{u}=\displaystyle \displaystyle \sum _{v\in {{\rm{\Gamma }}}_{u}}{E}_{uv}\end{eqnarray} \tag{ 4 }$$

Where, $v$ is the adjacent node of node $u$ in the state in correlation network. ${{\rm{\Gamma }}}_{u}$ is the set of adjacent nodes of node $u.$

According to Formula (4), the connection probability ${x}_{uv}$ between node $v$ and node u in the correlation network is:

$$\begin{eqnarray}&&{x}_{uv}=\displaystyle \frac{{E}_{uv}}{{S}_{u}}\end{eqnarray} \tag{ 5 }$$

Combined with Formula (3) to (5), the entropy index of branch potential energy is obtained as follows:

$$\begin{eqnarray}&&PEE{D}_{u}={S}_{u}\left(\displaystyle {\sum }_{v\in {{\rm{\Gamma }}}_{u}}{x}_{uv}\,\mathrm{ln}\left({n}_{u}{x}_{uv}\right)\right)\end{eqnarray} \tag{ 6 }$$

where, ${n}_{u}$ is the degree of the node u in the state correlation network.

According to Formula (6), the greater the branch potential energy entropy degree $PEE{D}_{u}$ is, the more unbalanced the distribution of branch potential energy is. The branch potential energy is mainly accumulated on some lines in the system, which is more likely to cause the branch overload and even cascading failures. On the contrary, the smaller the branch potential energy entropy is, the more balanced the distribution of the system potential energy is. The branch with large capacity bears more power flow impact, while the branch with small capacity bears smaller power flow impact, and the system is in a relatively balanced state. This index can effectively reflect the influence of the active power and reactive power on the branch potential energy, which is closer to the actual operation of the power system. The vulnerability of branches can be reflected by the branch potential energy entropy degree

The fault of a line can propagate through the network and trigger other failures, and even lead to a blackout. This propagation, also called the cascading failures, is well known as the mechanism of large blackouts [27]. In the traditional branch vulnerability studies, a branch is selectively removed and the changes of system performance are usually taken as the verification indicators, such as network efficiency, load loss rate under AC cascading failure model and the outage loss index VaR, CVaR, etc. In this paper, the AC cascading failure model [28] is borrowed to perform the static intentional attacks according to the transmission lines sequence, and the load loss rate of the system after the fault is calculated. Thus, the cascade process of one day is as follows. Select a line and attack (or remove) it as an initial fault of cascading failures, and then judge whether there exists the branch overload or not. If yes, continue to remove the overload branches until there is no branch overload, and then calculate the system load loss. Then the average load loss rate after the attack of each line after N days simulation is:

$$\begin{eqnarray}&&{P}_{loss}\left({\rm{i}}\right)=\displaystyle \frac{\displaystyle {\sum }_{{\rm{n}}=1}^{N}{P}_{loss}\left({\rm{i}}\right)}{N}\end{eqnarray} \tag{ 7 }$$

where, ${P}_{loss}(i)$ represents the average load loss rate of the system after the attack of line i. Obviously, the larger ${P}_{loss}(i)$ is, the more vulnerable the branch is. The sequence of vulnerable lines derived from ${P}_{loss}$ is called cascading failures simulation (CFS) sequencing [29].

In this paper, the AC cascading fault model is used in the static intentional attacks according to transmission lines sequence. Then the system load loss rate is calculated, and the branch potential energy entropy algorithm proposed in this paper is verified.

The traditional power flow entropy model only considers the power flow value, and ignores the difference of the ability of branches to withstand power flow impact. That's to say, the same power flow impact may cause the overload of small capacity branches, but a little impact on the large capacity branches. The branch potential energy Theil entropy method can be used to identify the importance of nodes in complex networks. Thus, this paper establishes a comprehensive vulnerability evaluation model of the branches based on the Theil entropy theory, in which the importance of the branch fault to the system is measured by the influence of the two-level cascading failures on the system. Considering the change and accumulation of the branch potential energy caused by the branch disconnection, the evaluation method is closer to the actual operation of power grid.

The specific model is shown in figure 2. Firstly, according to the corresponding principle of branches mapped as nodes and the correlation between branches mapped as edges, a null graph is generated. Secondly, according to the power flow data of the initial operation state of the system and the two-level cascading failures, the potential energy changes of the remaining branches caused by a branch fault are calculated. Then, the potential energy variation of the branch is given to the branches of the null graph as weights to generate a bidirectional two-level state transition network. Finally, the branch potential energy entropy degree (PEED) of the nodes in the state transition network is calculated by using the branch potential energy Theil entropy method, and the PEED order in the two-level state transition network is equivalent to the power system branch vulnerability order, and the power system vulnerability assessment is completed.

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In this paper, the effectiveness and correctness of the proposed branch vulnerability assessment model based on the branch potential energy Theil entropy is verified by the simulations of the standard IEEE-39 bus system. The structure of the IEEE-39 bus system is shown in figure 3. There are 39 nodes, 46 branches, 10 generators, 21 load nodes.

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In this paper, the simulation days N are set to 10000 days in the AC cascading failures model (CFS). The effectiveness of the proposed method is verified by comparing the accuracy K/M identified by the first m branches between the proposed method and the CFS sequence. K is the number of the matched branches in the two sequences.

According to Formula (6), the first 17 vulnerable branches in the IEEE-39 bus system are shown in table 1. According to the analysis of figure 3, among the top 10 most vulnerable branches, branches 14, 37, 33, 20, 39, 46 and 34 (7 lines in total) are the only paths for generators 31, 35, 33, 32, 36, 38 and 34 to deliver power to the grid. Branch 27 is the only way for generators 33 and 34 to transmit power to the system, and its fault will cause serious shortage of the active power in the system, or even system disconnection. Branches 35 and 42 are located on the important transmission lines, whose disconnection will lead to a large-scale transfer of power flow so that the power flow equilibrium will be reduced. For example, the fault of branch 42 will lead to a power imbalance between node 26 and node 27, which may lead to a series of problems such as power angle instability. Therefore, from the perspective of the network structure and function, it can be found that the PEED method can identify the vulnerable branches of the system.

In order to further verify the identification effect of the proposed method, the CFS sequence is used as an index to be compared with the existing methods: the maximum flow method (MF) [30], the improved maximum flow method (IMF) [31], PageRank algorithm (PRA) [32] and Cascading index (CEI) [33]. Table 2 lists the comparison results of the first 12 vulnerable lines by the different methods. Firstly, although the methods have different perspectives on vulnerable branch evaluation, the overall distribution trend of the branch vulnerability obtained by this method is similar to those obtained by the above methods. Moreover, in the first 12 branches, there are 5 to 8 branches consistent with those got by other methods. It shows that this method can find out the vulnerable branches of the power system comprehensively and evaluate the vulnerability of power system branches more accurately. From the perspective of the comparison accuracy with the CFS sequence, the PEED sequence used in this paper has higher identification accuracy than other methods. It should be noted that if the first 10 lines are selected to be compared, the PEED still has better recognition effect and accuracy.

Figure 4. shows the comparison results of the potential energy entropy degrees of the nodes in the state transition network of the IEEE-39 node system, and that is, the comparison results in the power grid considering the two-level cascading failures. Table 3 shows the distributions of the fault branches and the number of the power system islands when the primary and secondary faults occur. In the process of two-level cascading failures simulations, there are 17 lines out of operation which will cause the exits of the rest of the lines from the system due to their overloads. Among the 17 most vulnerable lines identified by the branch PEED sequence, 16 lines are consistent with it, only branches 32 and 3 are exceptions. It can be seen from figure 4. that branch 32, which can induce the two-level cascading failures, ranks 18 in the entropy sequence of branch potential energy, and also belongs to the lines with high vulnerability. Although branch 3 can't induce the two-level cascading failures, it exits from the operation frequently due to an overload in the secondary fault, which indicates that branch 3 has a weak ability to resist the power flow impact. So the branch sequence induced by the N-2 cascading failures and identified by the branch potential energy entropy degree PEED sequence are consistent with each other, which shows the proposed method are effective.

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This section takes a real-life regional power grid of China as an example for analysis. The regional power grid system has 32 nodes, 42 branches, 9 generators, 20 load nodes. For simplicity, it is called the NE power grid later, and figure 5. is the schematic diagram of the NE power system.

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Table 4 shows the comparison between the proposed method and the CFS sequence in the NE power grid. The simulation days in the AC cascading failures model are still set to 10000 days. In the first 15 vulnerable branches, 12 of them are consistent with the results in the verification index CFS sequence, and the accuracy is 80%. The three unidentified branches are Branch 1, branch 13 and branch 29 respectively. Branch 1 and branch 29 are the only paths for load node 32 (load 702MW) and load node 30 (load 250MW) to absorb power from the system respectively. Although the faults of these two branches will cause great losses of loads, the rest of the system will not be greatly impacted. Therefore, branch 1 and branch 29 rank higher in the CFS sequence considering the load loss, but lower in the PEED sequence considering the system state change.

According to the PEED sequence, branches 35, 36, 39, 40, 34, 41, 42, 37 and 38 (9 branches in total) are the only paths for 9 generators 6, 9, 2, 3, 5, 4, 8, 1 and 7 to transmit power to the grid. Branch 5 is the only path for generators 2 and 3, load nodes 13 and 17 connected with the system. Branches 30, 2 and 16 play the same role with branch 5 in the system. Branch 30 is the only path for generators 1 and 7, load node 23 to connect with the system. Branch 2 is also the only path for generators 2 and 3, load nodes 10, 13 and 17 to connect with the system. Branch 16 is the only path for generator 5 and load node 22 to connect with the system. The above branch faults will cause serious shortage of the active power in the system, resulting in islands and great losses of loads. Branch 15 and branch 31 are the branches of the high voltage loop network of the system, which are located on the important transmission lines. Their faults will lead to a decrease of the system connectivity and a wide range transfer of power flows.

Due to the high voltage level of the NE power grid, most of the important branches are double circuit lines, and the short circuit capacity of the system is large, so the secondary fault of the remaining branches doesn't occur after the first level fault of the branches.

The results are compared with the other methods mentioned in section B. Taking the CFS sequence as index, table 5 shows the comparison of the identification results of the first 12 branches by different methods. It can be seen that compared with the other methods, the branch potential energy entropy method still has higher identification accuracy in the NE power grid. Select the first six lines from results by the above methods and remove them from the system in order. Once a line is removed, we calculate the proportion of the residual load to the total load for each line removed using the AC power flow model, that is, the residual load rate of system.

The residual load rate is shown in figure 6. According to the results of the PEED, the residual load of the system is less than 40% after attacking the six lines continuously, which can cause an extremely serious impact on the power grid. The simulation results show that compared with the other methods, this method is more sensitive to identifying vulnerable lines.

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Table 6 shows the time consumed by the algorithm in this paper. In the two test networks in this paper, the time consumed by the branch potential energy entropy method is much less than that of CFS. If the server with better performance is used, the computing time will be shorter and the online application is expected to be realized.