Power Grid Topology Classification Based on Time Domain Analysis Jia He, Maggie X. Cheng and Mariesa L. Crow, Fellow, IEEE

Abstract—Power system monitoring is significantly im- all depend on the correct network topology information. proved with the use of Phasor Measurement Units (PMUs). Topology change, no matter what caused it, must be The availability of PMU and recent advances in ma- immediately updated. An unattended topology error may chine learning together enable advanced data analysis that lead to cascading power outages and cause large scale is critical for real-time fault detection and diagnosis. This blackout ( [2], [3]). In this paper, we demonstrate that paper focuses on the problem of power line outage. We use a machine learning framework and consider the problem using a machine learning approach and real-time mea- of line outage identification as a classification problem in surement data we can timely and accurately identify the machine learning. The method is data-driven and does outage location. We leverage PMU data and consider not rely on the results of other analysis such as power the task of identifying the location of line outage as flow analysis and solving power system equations. Since it a classification problem. The methods can be applied does not involve using power system parameters to solve in real-time. Although training a classifier can be time- equations, it is applicable even when such parameters are consuming, the prediction of power line status can be unavailable. The proposed method uses only voltage phasor done in real-time. angles obtained from PMUs. It first extracts Compared to previous works on line outage detec- domain features from the PMU data streams, and then trains a classifier to learn the network topology based on tion and identification, this work does not rely on the the features. The proposed method is results of other critical tasks such as state estimation tested by using simulated data from PSAT. It is shown ( [4]) or load-flow analysis ( [5]). These tasks require that the frequency domain features are robust against not only network connectivity information but also line dynamic load changes and noise in data. The prediction parameters and power injections. In this paper, we use performance is comparable to previous works that require measurement data only to determine the status of power detailed system information such as line parameters. lines, thus there is no interlocking with other tasks Index Terms—Line Outage, Machine Learning, Classi- such as state estimation. It is paramount that the line fication, Network Topology outage identification can be performed independently and prior to the tasks that rely on network topology I.INTRODUCTION information to work. The data-driven approach compared to the previous works that involve solving accurate state In recent years topics on anomaly detection in power equations is a fundamental breakthrough. systems and subsequent fault localization are heavily Compared to earlier works that use machine learning investigated. They received much attention due to the approaches, the proposed work distinguishes itself in importance of the topics and the challenges to address several ways: 1) It is not based on a particular flow them. We focus on one type of anomaly in this paper: model (DC or AC). The input to the classifiers is derived power line outage. information from PMU measurements, and the classifiers Power line outages caused by overgrown trees, wild do not use model-specific information; 2) We do not use animals, and inclement weather conditions account for voltage magnitudes and power injections. We only use most power disruptions ( [1]). It is critical to develop phasor angles. Instead of using phasor angles directly, real-time procedures that can quickly detect and identify we use the difference between the phasor angles from the outage location since other important tasks such as two adjacent buses to build . We then use the flow analysis, state estimation, and contingency analysis spectrum features extracted from the time series as input This work was supported in part by the National Science Founda- for the classifiers. These features more accurately capture tion of USA under grants 1854077 and 1854078. the correlation with the network topology than those used J. He and M. Cheng are with Illinois Institute of Technology, in previous work; 3) The performance of the classifiers Chicago, IL 60616 USA (e-mail: [email protected]). M. Crow is with Missouri University of Science and Technology, are reported by using precision and recall as metrics, Rolla, MO 65401 USA (e-mail: [email protected]). instead of an overall accuracy or error rate as in the past 2 work. The overall accuracy is high mainly because the method is developed to solve the line outage identifi- classifiers perform well for the non-case, i.e., it predicts cation problem by using the DC model. An important the network has no outage when there is no outage. For assumption is that line outage is sparse in the power grid. rare event detection, the non-case is the majority case The proposed method also achieves low-complexity, but in the dataset. Even though a classifier performs poorly different from [5], the proposed method is completely for the ”yes” case, the overall accuracy is still high. data-driven, and does not need the line parameters (e.g., However, we think it is a reasonable argument that to reactance). detect the ”yes” case accurately is more important. We [9] solved outage detection and state estimation as therefore emphasize how to improve the detection rate coupled problems. It is also not rare to address line (recall) in this paper; 4) Our work does not depend on the outage detection and identification as one coupled prob- sparsity assumption as in [5]. Since each line is predicted lem (see [10], [11]). Both [10] and [11] use incremental separately, the number of line outages that it can predict changes from measurements between two time instances is unlimited; 5) Using our feature extraction method, as random variables to build time series. [10] uses the classifiers do not require PMU data from all buses. phasor angles and [11] uses voltage magnitudes. In [10] results show that it requires only a subset change point detection is performed based on the time of PMU data. We further show which PMU location series directly, and line outage is detected from the is dispensable and which is not, and the conclusion first series that reported the change. In [11] detection is specific to the line we try to predict. This implies and identification are based on changes on conditional that accurate topology information of the network is not correlation. There are also several works using message essential to the machine learning algorithms. However, passing algorithms for network topology inference, e.g., knowing which pair of buses is connected by a line [12], [13]. can reduce the feature space. The knowledge of the full In recent years, there are a handful of works using topology is a plus, without which the algorithms still machine learning approaches for power line outage de- work although with longer training time; 6) Finally, to tection. To name a few, in [14], a classical support vector our knowledge, this is the first work to predict power machine (SVM) approach is developed for single line line outage in a very large system (1354-bus with 1991 outage detection. In [15] and [16] neural networks are lines) at high detection rates. developed to predict single and double line outages, and The rest of the paper is organized as follows: in both use only a single hidden layer. In [15], the input Section II, we review some related work on the topics of to the neural network includes phasor angles and power power line outage in modern power systems; in Section injections; in [16] the input is voltage magnitudes and III, we provide an overview of the methodology, the phasor angles. [16] addressed partial data inference and analysis behind the proposed method, and then detailed [17] discussed optimal locations for PMUs. It was solved description of the feature extraction method and descrip- as an optimization problem to maximize the system’s tion of the classifiers; Section IV provides extensive ability to detect single-line outages. simulation results on line outage identification; Section V concludes the paper with a plan for future work. III.PROPOSED METHODS II.RELATED WORK A. Overview We briefly review some representative works on power This paper focuses on the identification of power line line outage detection and identification. In [6], a proce- outage, i.e., to locate which lines are out. We use the dure is developed to detect a single line outage based measurement data from buses for this task. Throughout on line connectivity information and PMU data. Line the implementation of the method, we assume no prior outage is identified based on pre-outage flow analysis. knowledge about the admittance matrix or state equa- To perform flow analysis, the system admittance matrix tions of the system. is needed. In [7], the method is extended to address First, we need to identify the measurement that shows double line outages. It is pointed out in [7] that missing high correlation with power line topology. PMU data may lead to indistinguishable outages. This Recall that the active power transfer is a function of conclusion is consistent with our findings in this paper. phasor angles. Let θi be the phasor angle at bus i, and Multiple line outage detection and identification using Pij be the active power transfer from bus i to bus j. We PMU data are addressed in recent works. In [8], maxi- have mum likelihood estimation is used to estimate a linear multinomial regression model. In [5], a low-complexity Pij = |Vi||Vj| (Gijcos(θi − θj) + Bijsin(θi − θj)) (1) 3

where |Vi| is the voltage magnitude at bus i, and Gij Although the plots look similar in shape, the differences and Bij are line parameters between bus i and bus j. in amplitudes across rows are more significant than If the power line impedance z = r + jx, then the differences across columns, which suggests that the 1 r jx admittance y = z = r2+x2 − r2+x2 = g + jb. Usually amplitudes of the dominant frequencies under different r  x, so the real part in y is close to zero. Therefore, line outage scenarios are very different, more so than the real part of the admittance matrix will be close to the differences caused by varying loads under the same zero, i.e., Gij → 0, thus the power transfer equation can topology. be transformed to: . Pij = |Vi||Vj| (Bijsin(θi − θj)) (2) It is also observed that the difference between the phasor angles at two buses i and j connected by a circuit is usually very small, so (θi − θj) is usually a very small number (usually less than 0.262 in radians). Therefore we use (θi −θj) to approximate sin(θi −θj). The power transfer equation can be further simplified as: . Pij = |Vi||Vj| (Bij(θi − θj)) (3) In the per-unit system, the numerical values of voltage magnitudes |Vi| and |Vj| are between 0.95 to 1.05, so it incurs very little error if we assume them to be 1.0. Therefore in the normal operation when two buses are connected by a transmission line, the power transfer amount is loosely proportional to the angle difference between the two buses. It is almost a perfect dependent relationship. When the line between two buses is taken out, the phasor angles at two buses are no longer con- strained by this relationship and become conditionally Fig. 1: Observed time series for θ1 − θ39 when a line independent. We expect that a line outage will disrupt outage happens. X-axis is time, Y-axis is the angle the phasor angles between two buses. At the difference in radians. Top row: when line (1, 2) is out; of line outage, we observe a transition from a relatively middle row: when line (1, 39) is out, bottom row: when stable signal to a more dramatically changing signal. line (2, 30) is out. The three columns correspond to three Based on the analysis above, the only information used different load settings. in the classification problem is the phasor angles at buses. For a fair comparison among different machine learning methods, all methods are provided with the same data. B. Features Extraction Assume there are angle measurements from m buses, Phasor angles in the form of time series cannot serve the data contains m-dimensional time series {θ }, where t as predictors, since the length of the time series, n, θ ∈ Rm, and t = 1 . . . n. t can be very long or very short. The dimension of the The phasor angle measurements are first smoothed to predictors should not depend on the length of the time avoid angle oscillation between [0, 2π), otherwise the series. periodic change from 2π − δ to 0 even for a small δ In this paper, we use Fast to con- can totally undermine the classifier. When a line outage vert a time series to frequency domain features. Fast occurs at a branch, we observe a change in the difference Fourier Transform (FFT) takes a discrete signal in the of angles between the two buses that the branch connects, time domain and transforms that signal into its discrete as well as from other pairs of buses. Although the change frequency domain representation, in the magnitude is small, the change in the dynamic n−1 feature is significant — a high frequency component X −i2πkj/n occurred at the time of the abrupt change. Xk = xje Fig. 1 shows the angle difference between bus 1 and j=0 bus 39 when line outages happen in the IEEE 39-bus where Xk is a and corresponds to the system. Fig. 2 shows the frequency domain features. frequency of k cycles per n samples. 4

(l) where |Xk | is the amplitude corresponding to frequency (l) k for line l. The predictor variables include Xk for each k and each l. W is the total number of lines that are observed. Two well-known classifiers are tested for identifying line outages: • • Random Forest 1) Logistic Regression: In logistic regression, Y is the binary outcome variable, X = (x1, . . . , xm) is the set of predictor variables. Let p be the probability of Y being 1. The logit model is given by ( [19]):  p  log = β + β x + β x + . . . β x = β · X e 1 − p 0 1 1 2 2 m m

where β1, . . . , βm are the regression coefficients indicat- ing the relative effect of each particular predictor variable on the outcome, and β0 is the intercept. Training data are used to estimate the coefficients β. Fig. 2: Frequency domain features. X-axis is the fre- A separate logit model is estimated for each line quency index, and Y-axis is the amplitude. outage. If there are L lines that might have outage, there are L models, i.e. L sets of β. We use β(l) to denote {X } k = the coefficients for the line l. For line outage detection, FFT results in a sequence of k , with (l) (l) 0, . . . , n−1. We consider the dominant non-zero frequen- we have a vector of p = Prob(y = 1) indicating the probability of line l being broken. Given the coefficient cies ( i.e., the frequencies with the largest amplitudes) (l) as predictor variables for the classification problem, so β and X from the new data point, the probability of that the dynamic features of the time series are fully outage on line l is computed as follows: captured. 1 p(l) = The dominant frequencies are selected by using the 1 + e−β(l)·X 2 following methods: Let Mn = max |Xk| . Mn follows 1≤k≤n The logistic regression method is used for both sin- Gumbel distribution asymptotically. It has been proved gle line outage and multiple line outage. The process (l) (l) in [18] that there exists two sequences (an) and (bn) involves first estimating β and then predicting p such that for the new data, for l = 1,...,L. The computational Mn − an  −e−t complexity does not increase with the number of outages. lim P ≤ t = e . n→∞ bn 2) Random Forest: Random Forest is a tree-based With the level of significance α (e.g., α = 0.05), we have method [20]. Different from regression models, Random 1 − α = e−e−t . Solving it for t, we get threshold t = Forest uses decision trees as building blocks to construct − log(− log(1 − α)). Then we can select the dominant prediction models. frequencies using this threshold. A classification tree here is used to predict a qual- A more convenient approach is to sort the amplitudes itative response (Yes or No). Using decision trees for and then select the top K (e.g., K = 3) dominant classification essentially involves two steps: In the first non-zero frequencies. Although a fixed value K may step — training, we divide the predictor space defined correspond to a different significance level α, when it by the set of all possible values for the m-dimensional comes to line outage identification, the results from the predictor variables into J distinct and non-overlapping two approaches are very close and non-distinguishable. high-dimensional regions {R1,...,RJ }, then in the sec- ond step— prediction, for a new observation that falls into the region R , we assign it to the most commonly C. Classification Algorithms j occurring class of training observations in Rj. In the We use classification algorithms to identify which training step, the regions are constructed to minimize the lines are out. The response variable Y is a binary vari- classification error rate, i.e., the fraction of the training  (l)  able, and the predictors are a collection of |Xk |,... , observations in that region that do not belong to the most 5 common class. In practice, either cross-entropy or Gini the incidence matrix and the PMU measurements from Index is used to evaluate the classification error rate. buses. Random forest involves producing multiple trees The incidence matrix provides the network topology, which are then combined to yield a single consensus but does not have line impedance information. We use prediction by averaging all the predictions. When build- this information to estimate the models for line outage ing individual decision trees, each time a split of the identification. In the simulation, we consider single line predictor space is considered, a random sample of p outages, as well as simultaneous double line and three predictors are randomly chosen as split candidates out line outages. The simultaneous outages for four or more of the full set of m predictors. The split then uses only lines are not simulated for this small network as some p p one of√ the predictors. is typically an integer smaller buses will be disconnected from the rest of the network, than m. and bus voltages will have no variation after the outage, Using random forest for the classification of line thus the problem of identifying broken lines becomes too outage can be carried out in two ways: trivial. 1) Binary outcome. For each possible line outage, We load the 39-bus system into PSAT, and run the time we train a separate model to decide if the line is domain analysis for a total of T seconds. For the 39-bus broken, and then apply each of the models to the system, we have used T = 200. Data are collected at new observation. If multiple models yield ”Yes”, the interval of 8 data points per second. For line outage then there are multiple line outage. simulation, a number t between 1 to T is chosen, and 2) Multi-categorical outcome. We build one training line outage at time t is inserted in the simulation. The model for all possible classes. If there are L dataset has 16, 215 data points, with 50% for the training possible line outages, there are L + 1 classes, set and 50% for the validation set. or L + 1 possible outcomes, with outcome= 0 The PSAT simulation gives measurements at PMUs. indicating no line is broken, outcome= l indicating We first use PMU measurements from all buses for line l is broken. This method can only be used training and testing, then we test on the situation where for single line outage prediction, as in the case of not all buses have PMUs installed, and therefore we multiple line outages, the number of classes grows have to deal with . In addition, random exponentially with the number of lines, i.e., there measurement errors are added, and the robustness of the are a total of 2L outcomes. algorithms against noise is tested. on single line outage detection indicate Two prediction models are tested, i.e., Logistic Re- that multi-categorical Random Forest underperforms bi- gression (LR) and Random Forest with binary outcome nary Random Forest by at least 20% in detection rate. (RF). For multiple line outage detection, the multi-categorical version cannot even detect 50% of the line outages due to B. Performance Metrics having too many categories. Having too many categories with limited training data is detrimental to the method. We use recall and precision to assess the performance Therefore, in the following we will refer to the binary of the algorithms. Precision is defined as the fraction of version whenever Random Forest is mentioned. true outages that are detected among the total detected outages, while recall (i.e., detection rate) is the fraction of true outages that are detected among all true outages. IV. RESULTS

A. Experiment Set-up True Y The tests are implemented on IEEE standard 39-bus Positive Negative Positive True Positive (TP) False Positive (FP) test system. Time domain simulation of the system is Yˆ implemented in Power System Analysis Toolbox (PSAT) Negative False Negative (FN) True Negative (TN) [21]. PSAT is a free open source package equipped with modules for solving power flow (PF), optimal power TP TP flow(OPF), continuation power flow (CPF) and time Precision = , Recall = domain simulation (TDS). In this paper, we do not TP+FP TP+FN use power flow analysis, since we assume neither the Previous work (e.g., [5], [10], [11], [14]–[17]) used admittance matrix nor the Jacobian matrix is available. overall accuracy as performance metrics. We use the time domain analysis part of PSAT, and the TP+TN only information needed for the proposed methods is Accuracy = TP+TN+FP+FN 6

An algorithm with low true positive rate (TP) can still Gaussian distribution. To test the robustness of the classi- have high accuracy if negative cases are dominant in fiers to noise, we add Gaussian noise t to voltage phasor the dataset and the algorithm has a perfect true negative angle difference between two buses i and j, i.e., the rate (TN). For rare event detection, it is important to measured difference D˜t = Dt + t, where Dt = θit − θjt 2 achieve high detection rate, and therefore recall is a is the true signal, and t ∼ N (0, σ ) is the noise term, T better indicator than overall accuracy. α P with σ = T Dt. α is a parameter controlling the noise t=1 C. Line Outage in IEEE 39-Bus System level. We choose α ∈ [0, 0.3]. Results show that both The results for single line outage are summarized in classifiers are robust to noise in data measurements. Even Table I. In this experiment, the dataset contains cases with a high noise-to-signal ratio, recall is still nearly corresponding to either no outage or there is a single 90%, and precision is above 95% (Fig. 3). line outage. Both precision and recall are high for both 1.00 algorithms. 1.00 0.99

TABLE I: Single Line Outage Detection Results for 39- 0.95 0.98 0.97 Bus System. 0.90 Recall Precision 0.96

Method Precision Recall Accuracy 0.85 0.95

Logistic Regression Logistic Regression Random Forest Random Forest Logistic Regression 0.990 0.981 0.999 0.80 0.94 0 0.01 0.02 0.05 0.1 0.2 0.3 0 0.01 0.02 0.05 0.1 0.2 0.3 Random Forest 0.997 0.922 0.998 alpha alpha (a) Recall (b) Precision The results for double line outages are summarized in Table II. In this experiment, the dataset contains Fig. 3: Single line outage prediction with noisy data in cases corresponding to no outage, a single line outage, the 39-bus system. or double line outages. The true positive in column “Single” represents when there is single line outage and it is successfully detected, in column “Double” it E. Robustness to Missing Data represents when there are two simultaneous outages and both outages are successfully detected. Both precision This experiment aims to study the influence of missing and recall are high for Random Forest, but Logistic measurements on the classification results. The exper- Regression shows performance degradation from the iments are performed on the 39-bus system, and the previous experiment. This is because the size of the results are the average of 5 random runs. We use the training set increases and the “yes” case and non-case outage of Line(1,2) as an example, and examine the become more imbalanced. By using a higher weight on results when the measurements from one bus is missing, the “yes” case, the recall of the Logistic Regression can which is equivalent to the case where no PMU is be improved as shown in Table II. LR-W and RF-W are installed on that bus. weighted versions of LR and RF, respectively. We distinguish two cases, 1) training data does not include this bus and therefore the model is estimated TABLE II: Double Line Outage Detection Results for without using the features related to this bus; and 2) 39-Bus System. training data include this bus and therefore the model is Precision Recall Accuracy trained with full features but the new data point missed Method Single Double Single Double Single Double these features. LR 0.974 0.637 0.758 0.757 0.994 0.979 Table III shows that when the the model is trained RF 0.989 0.998 1 0.981 0.999 0.999 with full features, but missing data occurs in the test data on a bus directly connected to the open line, its LR-W 0.967 0.531 0.858 0.865 0.996 0.974 effect on prediction is detrimentally. But if the missing RF-W 0.989 0.998 1 0.982 0.999 0.999 data occurs on buses at least one hop away, prediction is only slightly affected. On the other hand, if the model is trained without using features from a particular bus, D. Robustness to Noise prediction is almost not affected or only slightly affected. In real life, meters often introduce random measure- Experiments on other lines show similar results with the ment errors. Measurement errors are assumed to follow consistent conclusion. 7

TABLE III: Detection result on line(1,2) with different For downsampling, we sample only 30% of non-case missing measurement. The first row is for case 1), and data. For weighting the rare class, we use a simple ratio- the second row is for case 2). based method where the rare class takes a class weight Logistic Regression Random Forest in reverse proportion to the ratio of ”yes” case in the Missing data bus # Precision Recall Precision Recall dataset. Both methods effectively improve the detection 1 1 1 0.86 rate as shown in Table IV. No missing data 1 1 1 0.86 Downsampling not only improves the detection rate of the ”yes” case, but also significantly reduces the training 0.18 0.11 1 0.51 1 (connected) 0 0 0 0 time. Fig. 4 shows the precision, recall, and training time of two classifiers Logistic Regression and Random 1 1 1 0.83 2 (connected) 0 0 0 0 Forest as we decrease the rate. The plots show the results of outage detection at Line #23 connecting 1 1 1 0.86 39 (one-hop) 0.14 0.97 1 0.86 buses 356 and 505. In the experiment, the non-case is downsampled with a sampling rate p, where p takes value 1 1 1 0.86 3 (one-hop) 1 1 1 0.86 in [1.0, 0.8, 0.7, 0.6, 0.5]. 1 1 1 0.86

25 (one-hop) 1.0 1.0 1 1 1 0.86 0.8 0.8 1 1 1 0.86 16 (four-hop) 0.6 0.6 Recall

1 1 1 0.86 Precision 0.4 0.4 0.2 0.2

LR LR RF RF 0.0 0.0

1 0.8 0.7 0.6 0.5 1 0.8 0.7 0.6 0.5 F. Large Systems with Imbalanced Dataset p p To validate the line outage detection algorithms, we (a) Recall (b) Precision

LR applied the algorithms on a larger system, the 1354-bus RF 140 portion of European transmission system [22], [23]. The 120 100

network contains 1354 buses and 1991 branches. We 80 60 run the time domain simulation in PSAT and generate TrainingTime(seconds) 40 time series for voltage measurements at buses. For each 20

1 0.8 0.7 0.6 0.5 network topology, we run the simulation under 10 differ- p ent settings of loads, so the dataset size is 19, 910 data (c) Training Time points. Table IV shows the result. Fig. 4: The performance of both classifiers improves TABLE IV: Single Line Outage Detection Results for when we down sample the non-cases, and less training the 1354-bus System. time is needed (especially for LR). Plots show results for Line #23(356,505). Method Precision Recall Accuracy Original LR 0.875 0.695 0.992 For large systems, we also obtained results for locating N+ = 1 RF 0.998 0.719 0.994 N− 1990 multiple simultaneous outages. We randomly chose three Downsampled LR 0.847 0.758 0.993 lines to have outage among 70 randomly sampled lines. N+ = 1 RF 0.998 0.779 0.995 N− 599 When the outages on Line #12, #24, #69 happened, both Downsampled, weighted LR 0.714 0.764 0.992 algorithms show high probabilities of outage on these N+ 1 w− 1 N− = 599 , w+ = 599 RF 0.996 0.793 0.995 three lines (see Fig. 5).

With this dataset, the ”yes” case (with line outage) V. CONCLUSIONAND OUTLOOK is significantly rarer than the non-case (without line In this paper, the problem of line outage identification outages), and the sample ratio is 1:1990. The ”yes” has been thoroughly studied. Under the proposed ma- case tends to be underestimated, i.e., the detection rate chine learning framework, two classifiers are considered: or recall will be low. In this case, the classifiers are Logistic Regression and Random Forest. biased towards the non-case class as it is the dominant The proposed algorithms have been tested via simu- class. To overcome the ”rare event” issue, we take two lation. When applied to single and multiple line outage countermeasures: downsampling the non-case, or adding identification, the proposed algorithms achieved remark- larger weight to the ”yes” case. able detection rates. The IEEE 39-bus standard test 8

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