Topology reconstructure for power systems based on transient dynamics: A sparse recovery perspective

Abstract

Aiming at the traditional topology identification based on steady-state operation, a topology identification method considering power system transient data is proposed. Firstly, the power system is dynamically modeled. Through theoretical derivation, the feature vectors that can reflect the topology information are extracted, and the topology identification problem is transformed into a sparse vector recovery problem. Based on compressive sensing theory, the orthogonal matching pursuit algorithm is adopted to solve the sparse recovery problem. Since the identification process is bidirectional, there may be some identification conflicts. For this consideration, an optimization strategy is introduced to improve the original algorithm. The influence of each algorithm parameter on the topology identification performance is then studied. By considering the transient process, a large amount of effective identification data was obtained in only a few processes. Finally, a simulation test on the proposed algorithm on the IEEE standard 22-bus power distribution system is conducted. The results show that the improved algorithm has outperformed the traditional algorithm.

Keywords

Power distribution network topology reconstructure sparse recovery transient data

1. Introduction

With the continuous development of power technology, power transmission management is playing an increasingly important role. A good management of the power distribution system is closely related to the real-time monitoring of its status, while the reasonable monitoring of the power distribution system requires obtaining its network topology structure in advance. However, the topology of power distribution system may change due to the aging of the line, the action of the switch and etc., which brings obstacles to the real-time monitoring of the system. In addition, the topology identification of the power distribution system is of great significance to its fault line identification, power flow calculation, power distribution and stability control [1, 2, 3], which further implies the research value of this subject.

The topology structure of the power distribution system resembles many similarities with complex networks. For example, the movement of nodes follows similar dynamic equation constraints, the coupling state of the network is similar, and the network structures both have sparse characteristics [4, 5]. Therefore, many methods of researching complex networks can be used for reference in the study of power distribution systems. At present, many scholars have conducted related researches on the topology identification of complex networks. There are two most commonly used methods for this type of research. One is based on network synchronization, and the other is based on statistical theory. Guanrong Chen and etc. studied the global synchronization and asymptotic stability of complex dynamical networks. Based on the reference state, they obtained sufficient conditions for global synchronization and stability. On this basis, Shuna Zhang, Xiaoqun Wu and etc., realize the restoration of the complex network topology by adopting a method of constructing external network synchronization [6, 7]. This type of method requires constructing an external network as the same size of the original network, and the external network is completely consistent with the target network through design and drive. Therefore, this kind of method is easier to implement in small-scale networks. But when the network size is too large, the convergence speed of the algorithm will get slow. In addition, if the target network falls into internal self-synchronization during the synchronization process, it will cause the external network to fail to synchronize with the target network [8]. Scholars such as EM Lourenco researched the power grid using a Bayesian method, and developed a Bayesian-based hypothesis testing program. The program can be used by combining standardized Lagrangian multipliers with topology error handling. This method is used to replace the need to run the repeated state estimator for hypothesis evaluation, which reduces the possibility of topology error identification [9]; the method based on Bayesian estimation requires statistical methods to obtain prior information about the network topology, but In many cases, the prior information of the network topology is not available, especially for the power distribution system with constantly changing topology.

In recent years, a topology identification method based on sparse recovery theory has appeared in the field of topology identification research. The method fully utilizes the sparse characteristics of the structure of the power distribution system, and does not require prior information of the network. Also, the convergence speed of the algorithm is very impressive. Babakmehr Mohammad and other scholars combined the concepts of compressed sensing and graph theory, and proposed a smart grid topology identification and monitoring solution [10, 11]. But their research is based on the steady-state operation data of the power distribution system. While in the actual application process, a steady state of the power distribution system may last for a long period of time, which will result in small differences in the data collected during this period. Thus, there is little effective identification data that can be obtained. In other words, it takes a long period of time to obtain sufficient data for topology identification, and it is necessary to ensure that the network topology does not change during the acquisition of these data, which greatly limits the real-time and practicality of this method.

In this context, the data collection process is greatly shortened due to the sampling from transient data and thus the efficiency of topology identification will be notably improved. Apart from promoting topology identification efficiency, utilizing transient data can also raise the identification performance. This is because the transient operation process of the power distribution system will produce data with large differences, that is, valid data. And with the development of phasor measurement unit (PMU) technology, the accuracy and real-time performance of recording data also make it possible to accurately and efficiently record and save these transient data [12]. In addition, the PMU also comes with its own communication function, which can timely transmit the collected effective data back to the data processing center for real-time identification of the topology of the power distribution system. The transient data not only improves the efficiency of topology identification, but also improves the success rate of topology identification to a certain extent.

2. Problem description and modeling

A power distribution system with $n$ generators or loads and $l$ transmission lines can be represented by a graph set $G=(S_{n},S_{l})$ with $n$ nodes and $l$ edges, where $S_{n}=S_{ng}\cup S_{nl}$ , $S_{ng}$ represents a set composed of generator nodes, $S_{nl}$ represents a set composed of load nodes; $S_{nl}=\{l_{i,j},i,j\in S_{n}\}$ represents a set composed of $l$ edges, corresponding to $l$ transmission lines of the power distribution system. Because when the entire power distribution system is operating normally, whether it is always running in a steady state or with small fluctuations on the load side, the voltage phase angle difference of all nodes in the network is relatively small. In other words, the value of $\theta_{i}-\theta_{j}$ in the network is relatively small, thus, $\sin(\theta_{i}-\theta_{j})\approx\theta_{i}-\theta_{j},\cos(\theta_{i}-\theta% _{j})\approx 1$ . Therefore, the DC model can describe the dynamics of the entire system well. The corresponding dynamic equation of each node of the power grid is as follows [13, 14]:

$\displaystyle\left\{{\begin{array}[]{l}\left\{{\begin{array}[]{l}M_{i}\ddot{% \delta}_{i}+D_{i}\dot{\delta}_{i}=P_{i}^{g}-\frac{\left|{E_{i}}\right|\left|{V% _{i}}\right|}{X_{di}}\sin(\delta_{i}-\theta_{i})\\ \dot{\theta}_{i}=\frac{1}{T_{i}}P_{i}^{d}+\frac{\left|{E_{i}}\right|\left|{V_{% i}}\right|}{X_{di}}\sin(\delta_{i}-\theta_{i}){\kern 1.0pt}{\kern 1.0pt}-\frac% {1}{T_{i}}\sum\limits_{j=1}^{n}{\left|{V_{i}}\right|\left|{V_{j}}\right|B_{ij}% (\theta_{i}}-\theta_{j})\\ \end{array}}\right.,i,j\in S_{n_{g}}\\ \dot{\theta}_{i}=\frac{1}{T_{i}}P_{i}^{d}-\frac{1}{T_{i}}\sum\limits_{j=1}^{n}% {\left|{V_{i}}\right|\left|{V_{j}}\right|B_{ij}(\theta_{i}}-\theta_{j}),i,j\in S% _{n_{l}}\\ \end{array}}\right.$ (1)

Where $M_{i},D_{i},P_{i}^{g},\delta_{i},X_{di},\left|{E_{i}}\right|$ represent the inertia coefficient, damping coefficient, injected power, rotor angle, transient reactance and electromotance of the $i$ -th generator node, respectively; $\left|{V_{i}}\right|$ is the voltage amplitude of the $i$ -th node, $\theta_{i}$ is the relative phase angle of the $i$ -th node that compared to the reference node, $P_{i}^{d}$ is the positive active power consumption of the $i$ -th node, and $T_{i}$ is the frequency power factor of the $i$ -th node. $B_{ij}$ denotes the $(i,j)$ -th entry of matrix $B$ , and the definition of matrix $B$ is as follows:

$\displaystyle B_{ij}=\left\{{\begin{array}[]{ll}\sum\limits_{i\neq j}^{n}b_{ij% },&i=j\\ -b_{ij},&i\neq j,l_{i,j}\in S_{l}\\ 0,&i\neq j,l_{i,j}\notin S_{l}\\ \end{array}}\right.$ (2)

Where $b_{ij}$ denotes the susceptance between node $i$ and node $j$ of the power distribution system. Transform the above formula to get the following equation:

$\displaystyle y_{i}=A_{i}x_{i}+\eta_{i}$ (3)

Where $y_{i}=[y_{i}(t_{1}),y_{i}(t_{2}),\ldots,y_{i}(t_{S})]^{T}\subset R^{S\times 1}$ . $t_{s}$ and $S$ represent the sampling time point and the maximum number of sampling respectively. $A_{i}=[\phi_{i}(t_{1}),\phi_{i}(t_{2}),\ldots,\phi_{1}(t_{S})]\subset R^{S% \times(n-1)}$ is the perception matrix, and $x_{i}=[B_{i1},\ldots,B_{i(i-1)},B_{i(i+1)},\ldots,B_{in}]^{T}\subset R^{(n-1)% \times 1}$ is the sparse vector containing network topology information to be estimated. It is worth noting that the element $B_{ii}$ was removed in the researched problem. because it is of no value in obtaining the power grid topology and its modulus value is relatively large compared to other elements, which may affect the algorithm performance. $\eta_{i}$ is the Gaussian white noise caused by measurement error. The definition of $y_{i}(t_{s}),\phi_{i}(t_{s}),(s=1,2,\ldots,S)$ is as follows:

$\displaystyle\left\{{\begin{array}[]{ll}y_{i}(t_{s})=\frac{P_{i}^{d}}{\left|{V% _{i}}\right|}+\frac{\left|{E_{i}}\right|}{X_{di}}\sin(\delta_{i}-\theta_{i})-% \frac{T_{i}}{\left|{V_{i}}\right|}\dot{\theta}_{i},&i\in S_{ng}\\ y_{i}(t_{s})=\frac{P_{i}^{d}}{\left|{V_{i}}\right|}-\frac{T_{i}}{\left|{V_{i}}% \right|}\dot{\theta}_{i},&i\in S_{nl}\\ \phi_{i}(t_{s})=[f_{i}^{\theta_{1}},\cdots,f_{i}^{\theta_{i-1}},f_{i}^{\theta_% {i+1}},\cdots,f_{i}^{\theta_{n}}],&i\in S_{n}\\ \end{array}}\right.$ (4)

Where $f_{i}^{\theta_{j}}=|{V_{j}}|(\theta_{i}-\theta_{j})$ , for convenience of expression, the time scale $t_{s}$ on the right side of the above equation has been omitted. The derivative $\dot{\theta}_{i}$ of the nodal phase angle adopts first-order forward difference as an approximation, that is, $\dot{\theta}_{i}={[\theta_{i}(t_{s}+\Delta t)-\theta_{i}(t_{s})]}/{\Delta t}$ . The time interval of the difference should be neither too large nor too small, otherwise a large approximation error will be introduced. How to determine the appropriate $\Delta t$ value will be discussed later. The above problem can be transformed into a recovery problem of sparse vector $x_{i}$ , which is equivalent to solving the following optimization problem:

$\displaystyle\hat{x}_{i}=\arg\min\left\|{y_{i}-A_{i}x_{i}}\right\|_{2}^{2}$ (5)

Where $\hat{x}_{i}$ is the vector containing network topology information to be estimated.

If sufficient time scale data can be obtained, the above problem can be settled by a sparse recovery method. In general, data obtained from the power distribution system comes from its steady state or transient state, as shown in Fig. 1. During steady-state operation, the data of all nodes remain basically stable, with only small fluctuations within a certain range. The difference between such data is very small, and the effective data that can be used for topology identification is also very limited. In the process of transient operation, the data generated by each node of the power distribution system is quite different, so there are more effective data that can be used for topology identification. Despite the fact that the power distribution system is under steady-state operation most of the time, occasional load fluctuations on the load side may redistribute the power flow of the power distribution system, and the power distribution system will then enter a transient process, and then gradually return to stability. Assuming that a large load fluctuation suddenly occurs at time $t_{2}$ , then the power distribution system will immediately enter a transient process, which will produce a series of data with large differences. Effective data that can be used for topology identification are abundant. After identifying and judging that the system enters the transient process during the operation of the power distribution system, the sampling frequency can be appropriately increased to obtain as much efficient data as possible, which not only reduces invalid data, but also improves the efficiency of topology identification. To a certain extent, the identification success rate can also be improved.

Figure 1.

Dynamical process of $\theta$ and the valid sample points in power distribution network.

In general, because the data difference is not large in the steady state, the PMU will sample the data at a smaller frequency. Due to the large difference between the time scale data difference before and after the transient process, a threshold can be set in advance. If the difference between the two sampled data exceeds this threshold, it is considered that the system has entered a transient process, and the control center immediately sends instructions to the PMU to increase the sampling frequency. In other words, the sampling frequency is a function related to the difference between the sampled data before and after. Taking the phase angle of the node as the reference judgment value, then,

$\displaystyle f_{i}=\left\{{\begin{array}[]{l}f_{1},\Delta\theta_{i}\leqslant% \delta_{i}\\ f_{2},\Delta\theta_{i}>\delta_{i}\\ \end{array}}\right.$ (6)

Where $f_{i}$ represents the sampling frequency of the PMU of the $i$ -th node. $f_{1},f_{2}$ represent the sampling frequency of steady-state operation and transient operation, respectively. $\Delta\theta_{i}=\theta_{i}(t_{s+1})-\theta_{i}(t_{s})$ represents the difference between the sampling and measurement values of the $i$ -th node at time $t_{s+1}$ and time $t_{s}$ , and $\delta_{i}$ is the judgment threshold set for node $i$ in advance.

3. Improved orthogonal matching pursuit algorithm based on weighted average strategy

After collecting enough effective data of $A_{i},y_{i}$ , the above optimization problem can be solved. Traditional methods include least squares method, ridge regression algorithm, etc. [15]. Although these methods can also solve the problem very well in general, they all have one thing in common. That is, the above problem is a sparse vector reconstruction problem, which is not used. For the reconstruction of sparsevectors, the greedy algorithm can effectively reduce the running times of the algorithm. In this paper, the classic orthogonal matching pursuit algorithm is adopted. Compared with the least square method and ridge regression algorithm, this algorithm has faster execution speed and smaller identification error. And because the orthogonal matching pursuit algorithm stops the calculation when it reaches the iterative output threshold. The possibility of algorithm misjudgment is greatly reduced.

Assuming that $P_{i}^{g},P_{i}^{d},\delta_{i},\left|{V_{i}}\right|$ and $\theta_{i}$ can be measured by the PMU and stored in a specific location, then the corresponding matrix $A_{i}$ and vector $x_{i}$ can be obtained according to Eq. (4), and then $y_{i}$ can be estimated using the orthogonal matching pursuit algorithm. The specific execution steps of the orthogonal matching pursuit algorithm are shown in Table 1.

Table 1
Process of orthogonal matching pursuit algorithm

Algorithm 1: Orthogonal matching pursuit algorithm (OMP)

Input: matrix

A_{i}

, vector

y_{i}

i=1,2,\ldots,n

. Output: Estimated value

B

of the power distribution system topology. When

i\leqslant n

, repeat all the following steps: Step 1: Initialize the residual

r_{0}=y_{i}

, the reconstruction signal

x_{i0}=

0, the index set

\Lambda_{0}=\varphi

, the number of iterations

n=S

, and the counter

k=

0. Step 2: Calculate the projection coefficient (inner product value) of each column of the residual and perception matrix

A_{i}

c_{k}=A_{i}^{T}r_{k-1}

. Step 3: Find the largest element

c_{k}^{\ast}=\max\{c_{k}\}

c_{k}

and the corresponding position pos. Step 4: Update the index set

\Lambda_{k}=\Lambda_{k-1}\cup\{\textit{pos}\}

, and the atomic set

A_{\Lambda_{k}}=A_{\Lambda_{k}}\cup\{A_{i}(:,\textit{pos})\}

. Step 5: Use least squares to find the approximate solution

x_{k}=(A_{\Lambda_{k}}^{T}A_{\Lambda_{k-1}})^{-1}A_{\Lambda_{k}}^{T}y_{i}

. Step 6: Update the residual

r_{k}=y_{i}-A_{i}x_{k}

. Step 7: Determine whether the iteration meets the stopping condition

\left|{r_{k}}\right|<\varepsilon

k\geqslant P

. If it is satisfied, it will stop and output

\hat{x}_{i},r

i=i+1

. Otherwise, go to step 1. End: Construct matrix

B

according to estimated

n

vectors

\hat{x}_{i}

In each iteration of the orthogonal matching pursuit algorithm, the column most relevant to $x_{i}$ in $A_{i}$ will be selected to calculate the corresponding elements in $\hat{x}_{i}$ , until the stop condition is reached, the iteration will be stopped and $\hat{x}_{i}$ will be output. It can be seen from the logic of the algorithm that for each node, the algorithm detects its connectivity with other nodes. In other words, each edge is estimated twice.

Assuming that $\hat{x}_{i}$ and $\hat{x}_{j}$ are two already solved column vectors, there will be a $B_{ij}$ and $B_{ji}$ corresponding to them. Ideally, these two elements should be exactly the same, but due to measurement noise and calculation errors, there may be differences in the results of the two calculations, that is, one element is 0 and the other is not 0. Therefore, whether this edge exists or not requires further judgment. In this paper, a weighted average strategy is designed for this concern. From the logic of the algorithm execution, the accuracy of $\hat{x}_{i}$ estimation is positively correlated with the number of non-zero elements of the vector, so a vector with fewer non-zero elements has greater “decision power”.

Assuming that the obtained transient data is sufficient, for each node, $\Psi$ groups $A_{i},y_{i}$ can be obtained to estimate the vector $x_{i}$ $\Psi$ times. The final result obtained for the $\psi$ -th time is denoted as $B^{\psi}$ ( $\psi=1,2,\ldots,\Psi)$ . For all $B_{ij}^{\psi}$ and $B_{ji}^{\psi}$ , the positive times are respectively denoted as $\rho_{ij}$ and $\rho_{ji}$ . For each $\hat{x}_{i}$ and $\hat{x}_{j}$ , the average sparsity (also known as the average degree of nodes) of the identification results for $\Psi$ times is recorded as $\Gamma_{i}$ and $\Gamma_{j}$ respectively. So as long as there is a difference in sign between $B_{ij}^{\psi}$ and $B_{ji}^{\psi}$ , a weight coefficient is introduced as follows:

$\displaystyle\varpi_{ij}=\left(1-\text{sgn}(\Gamma_{i}-\Gamma_{j})\frac{\Gamma% _{i}}{\Gamma_{i}+\Gamma_{j}}\right)\rho_{ij}+\left(1-\text{sgn}(\Gamma_{j}-% \Gamma_{i})\frac{\Gamma_{j}}{\Gamma_{i}+\Gamma_{j}}\right)\rho_{ji}$ (7)

Where $\varpi_{ij}$ represents the number of edges between nodes $i$ and $j$ in $\Psi$ estimations. As long as this coefficient is not less than an empirical threshold, then it can be considered that there are eventually edges between nodes $i$ and $j$ . The algorithm flow design of the weighted average strategy is shown in Table 2.

Table 2

Process of improved orthogonal matching pursuit algorithm with an optimized weighted strategy

Algorithm 2: Improved orthogonal matching pursuit algorithm with an optimized weighted strategy (OMPOW)

Input: Sensing matrix

A_{i}^{\psi}

, measurement vector

y_{i}^{\psi}

\psi=1,2,\ldots,\Psi

. Output: Estimated value

B

of the power distribution system topology. Step 1: Get all

B^{\psi}

from Algorithm 1. Step 2: When there is any

\textit{sgn}(B_{ji}^{\psi})\neq\textit{sgn}(B_{ji}^{\psi})

, calculate

\omega_{ij}

according to Eq. (7), if

\omega_{ij}\geqslant\alpha\Psi

, go to step 3, otherwise go to Step 4. Step 3: Calculate the arithmetic mean of all non-zero

B_{ji}^{\psi}

and

B_{ji}^{\psi}

, then record it as

\bar{B}_{ij}

B_{ij}^{\textit{new}}=B_{ji}^{\textit{new}}=\bar{B}_{ij}

. Step 4:

B_{ij}^{\textit{new}}=B_{ji}^{\textit{new}}=

0. Step 5:

B=B^{\textit{new}}

, and output

B

4. Numerical simulation and analysis

In this part, this paper verifies the proposed algorithm on MATPOWER’s IEEE standard 22-bus power distribution test system [16]. In the simulation process, the assumption that the test noise $\xi_{i}$ of each node data obeys a Gaussian random distribution with zero mean, that is, $\xi_{i}=\zeta\times\eta_{0}\times\theta_{i}$ , $\zeta\sim N(0,1)$ , while $\eta_{0}$ denotes the noise intensity scaled at an 0.5% interval from 0 to 5%. Unless otherwise specified, the maximum sampling times $S$ for simulation is 18, and the default noise level is 1%. The transient reactance vector of one generator node of the IEEE standard 22-bus power distribution system is $X_{d}=$ 0.2. Power and voltage data (such as $P_{i}^{g}$ and $V_{i}$ ) are obtained through MATPOWER’s power flow calculation. Since the changes of $P_{i}^{g}$ and $P_{i}^{d}$ during the entire transient process are small enough, it can be assumed that they are constants throughout the entire transient process.

The change of the topology recovery success rate at different noise intensity and sampling time intervals is shown in Fig. 2, where the topology recovery success rate is defined as the ratio between the number of edges that are correctly identified and the total number of network edges throughout the whole simulation process. It can be seen from the figure that when the measurement noise is 0, the topology recovery success rate is almost 100% under the condition of small sampling interval, but when the sampling interval is too large (such as $\Delta t\geqslant 0.7$ ), the success rate of topology identification decreases slightly. Although a larger sampling interval will introduce an error, this error will be greatly reduced by the small proportionality coefficient ${T_{i}}/{\left|{V_{i}}\right|}$ in Eq. (4), which will eventually make the error small relative to $y_{i}(t_{m})$ on the right side of the equation. So, it will not have a great impact on the final identification result. On the other hand, a small $\Delta t$ will greatly amplify the error caused by measurement noise. It can be seen from the figure that when there is no noise, the smaller the sampling interval, the smaller the approximate error of the phase angle derivative function. Therefore, a very small $\Delta t$ can make the identification success rate 100%. However, as long as a very small noise (such as 0.25%) occurs, it will be amplified many times, leading to a rapid decline in the identification success rate. According to the color block diagram, the identification effect is the best when the sampling time interval $\Delta t$ is 0.3, so the following simulations all set $\Delta t=$ 0.3.

Figure 2.

Topology recovery performance vs. sample interval and noise level over 500 realizations.

The influence of the iteration stop threshold $\varepsilon$ on the topology reconstructure results is also studied in this paper. The graph of the identification success rate changing with the iteration stopping threshold $\varepsilon$ is shown in Fig. 3.

As can be seen from the figure, the smaller the iteration stopping threshold is, the higher the success rate of topology identification will be. However, the smaller iteration termination threshold means that the algorithm takes longer to execute, which will reduce the efficiency of topology identification and increase the calculation cost in the identification process. Therefore, under the premise of ensuring the highest topology identification success rate, selecting a maximum iteration stopping threshold $\varepsilon$ can not only ensure a better identification effect, but also greatly reduce the running time of the algorithm, thereby improving the efficiency of topology identification. According to Fig. 3, the value of $\varepsilon$ is between 0.02 and 0.04. Therefore, in the subsequent simulations, we set $\varepsilon$ to 0.03. See Table 3 for other simulation parameters.

Table 3

System partial parameter list

Simulation parameters	Values
Motor inertia coefficient $M_{i}$	1
Motor damping coefficient $D_{i}$	1
Motor frequency power factor $T_{i}$	0.02
Threshold coefficient $\alpha$	0.5
Maximum iteration number $P$	22

Figure 3.

Topology recovery performance vs. iteration stopping threshold over 500 realizations.

Figure 4.

Comparison of susceptance matrix recovery performance in one single test.

Figure 4 shows the network topology color map solved by the orthogonal matching pursuit algorithm and the solution result after adding the designed weighted average strategy. Among them, the deep blue spots indicate that there is no connection between the two nodes. The slight blue dots indicate that there is a connection between the two nodes. And the green spots indicate that there is actually an edge between the two nodes but the algorithm identification result shows that there is no connection (false positive). The slight yellow spots indicate that there is actually no edge between the two nodes but the algorithm identification result shows that there is (false negative). And the yellow spots indicate the negative element of the diagonal of the susceptance matrix. It can be seen from the figure that due to the existence of identification conflicts, the topology matrix directly output is not symmetric, which is obviously not in line with reality. Under the given simulation parameters, the orthogonal matching pursuit algorithm only has false positive results, no false negative result. And after processing with the given weighted average strategy, the false positive results are well eliminated.

The optimized weighted strategy introduced by the algorithm is designed based on the negatively correlated relationship between the identification accuracy rate and the sparsity of the column vector corresponding to each node. As shown in Fig. 5, node 2 requires more sampling times than node 1 when the same identification accuracy is achieved. This is because the sparsity of the estimated vector $\hat{x}_{2}^{\psi}$ corresponding to node 2 is basically around 3, while the sparsity of the vector $\hat{x}_{1}^{\psi}$ corresponding to node 1 is only around 1. By comparing Fig. 5A and B, it can be found that the new algorithm with the weighted average strategy not only reduces the number of sampling size required to achieve the same identification success rate, but also improves the success rate of topology identification under the same sampling size.

Figure 5.

Topology recovery performance vs. sample size $S$ for each node over 500 realizations.

Figure 6.

Topology recovery performance vs. sample size $S$ for the whole grid over 500 realizations.

Figure 7.

Topology recovery performance vs. noise level $\eta_{0}$ for the whole grid over 500 realizations.

In the simulation process, the maximum sampling size $S$ and noise level $\eta_{0}$ have a great influence on the result of topology identification. In order to further study their influence on the identification result, a series of simulation tests are carried out in this paper. It can be seen from Fig. 6 that it takes at least 5 measurements to make it possible to restore the topology of the IEEE standard 22-bus distribution system. Under the condition that the topology restoration can be carried out, the restoration effect of the proposed optimization algorithm is better than that of the original orthogonal matching pursuit algorithm. In addition, when the noise level varies from 0 to 5%, the algorithm with weighted average strategy also has higher success rate of topology identification than the original algorithm (see Fig. 7). As mentioned in the previous section, when there is no noise at all, the identification success rate of both algorithms is relatively high. When noise is first introduced, the identification success rate suffers a relatively large drop, and the subsequent change trend is relatively slow. It is because the introduced noise signal is greatly reduced by the small coefficient before $\dot{\theta}_{i}$ . In other words, the algorithm proposed in this paper is robust within the noise level tested. Although the noise level reaches 5%, the proposed algorithm can still guarantee the identification accuracy of more than 90%.

5. Conclusion

By considering the use of transient operating data in the distribution network, this paper greatly improves the speed of traditional topology reconstructure methods in power networks. In addition, an optimized weighted strategy based on sparse recovery algorithm is introduced to address the conflict issue caused by topology bidirectional identification in power system. Our proposed algorithm can well address topology identification problems in power systems providing the topology remains unchanged in the sampling process. The simulation results based on the standard IEEE standard 22-bus power distribution system in MATPOWER prove the feasibility of the algorithm. By collecting and storing transient data in the PMU, the power network topology can be identified quickly and efficiently, which is of great significance for state estimation as well as line outage detection in practical engineering.

References

Alam

et al., PMU based line outage identification using comparison of current phasor measurement technique, International Journal of Electrical Power and Energy Systems, 115 (2020), 105501–105501.

Seg

, Power flow control capability of the power transistor-assisted Sen transformer and the unified power flow controller: a close comparison, IET Generation, Transmission and Distribution 14(15) (2020), 3033–3041.

Shu

Y.B.

and Tang

, Analysis and recommendations for the adaptability of China’s power system security and stability relevant standards, CSEE Journal of Power and Energy Systems 3(4) (2017), 334–339.

Arianos

et al., Power grid vulnerability: A complex network approach, Chaos: An Interdisciplinary Journal of Nonlinear Science 19(1) (2009), 175.

Chen

Z.H.

et al., Robustness of interdependent power grids and communication networks: A complex network perspective, IEEE Transactions on Circuits and Systems II: Express Briefs 65(1) (2017), 115–119.

, and Chen

G.R.

, Global synchronization and asymptotic stability of complex dynamical networks, IEEE Transactions on Circuits and Systems II: Express Briefs 53(1) (2006), 28–33.

Zhang

S.N.

X.Q.

et al., Recovering structures of complex dynamical networks based on generalized outer synchronization, IEEE Transactions on Circuits and Systems I: Regular Papers 61(11) (2014), 3216–3224.

Chen

, and Chi

K.T.

, Synchronization: an obstacle to identification of network topology, IEEE Transactions on Circuits and Systems II: Express Briefs 56(4) (2009), 310–314.

Lourenco

E.M.

Costa

A.S.

and Clements

K.A.

, Bayesian-based hypothesis testing for topology error identification in generalized state estimation, IEEE Transactions on Power Systems 19(2) (2004), 1206–1215.

10.

Babakmehr

et al., Smart-grid topology identification using sparse recovery, IEEE Transactions on Industry Applications 52(5) (2016), 4375–4384.

11.

Babakmehr

et al., Compressive sensing-based topology identification for smart grids, IEEE Transactions on Industrial Informatics 12(2) (2016), 532–543.

12.

Dusabimana

and Yoon

S.G.

, A survey on the micro-phasor measurement unit in distribution networks, Electronics 9(2) (2020), 305.

13.

Nishikawa

and Motter

A.E.

, Comparative analysis of existing models for power-grid synchronization, New Journal of Physics 17(1) (2015), 015012.

14.

Stott

Jardim

and Alsac

, DC power flow revisited, IEEE Transactions on Power Systems 24(3) (2009), 1290–1300.

15.

Tibshirani

, Regression shrinkage and selection via the lasso: a retrospective, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73(3) (2011), 273–282.

16.

Zimmerman

R.D.

et al., MATPOWER: Steady-state operations, planning, and analysis tools for power systems research and education, IEEE Transactions on Power Systems 26(1) (2010), 12–19.