I Introduction
Anomaly in power systems is common, which can easily expand and cause power failures or system blackout if can’t be detected in time. The most famous example is the blackout in North East America on the 14th August 2003. In recent years, there have been increasing deployments of phasor measurement units (PMUs), which constitute wide area measurement system (WAMS) [1] for power systems. Compared with traditional supervisory control and data acquisition (SCADA) system, WAMS can provide synchrophasor measurements with higher sampling rates, which makes it possible for detecting eventualities in a timely manner. For example, the root mean square data of voltage and current can be sampled at the rate of Hz or Hz in the WAMS, while they are obtained only at the rate of Hz or Hz in the SCADA system [2].
The high sampling rate synchrophasor measurements contain rich information on the operation state of the system, which stimulates the research of advanced data analytics. For example, in [3]
, the voltage phasors are used to compute the Lyapunov components to estimate the shortterm voltage stability. In
[4], the dimensionality of the phasormeasurementunit (PMU) data is analyzed, and an PCAbased data dimensionality reduction algorithm is proposed for early event detection. In [5], a parallel detrended fluctuation analysis approach is proposed for fast event detection on massive volumes of PMU data. In [6], a densitybased detection algorithm is proposed to detect local outliers, which can differentiate highquality synchrophasor data from the lowquality one during system physical disturbance. In
[7], a waveletbased algorithm is proposed for detecting the occurrence of significant changes in timesynchronized voltage and frequency measurements.Due to the significant deployments of PMUs in WAMS, highdimensional synchrophasor measurements with fast sampling rate are collected, and the demand for theories capable of processing highdimensional data has grown dramatically. Random matrix theory (RMT), introduced by Wishart in 1928
[8], is an important mathematical tool for statistical analysis of highdimensional data. As for highdimensional random matrices, the importance of RMT for statistics comes from the fact that it may be used to correct traditional tests or estimators which fail in the ‘large , large ’ setting, where is the number of parameters (dimensions) andis the sample size. RMT starts with asymptotic theorems on the distribution of eigenvalues or singular values of random matrices with certain assumptions, and eventually gives macroscopic quantity to indicate the data behavior. The theorems ensure convergence of the empirical eigenvalue distributions to deterministic functions as the matrices grow large, which makes RMT naturally suitable for highdimensional data analysis. Nowadays, RMT has been widely used in wireless communication
[9], finance [10], quantum information [11], etc. In recent years, some work that makes substantial use of results in the RMT has emerged in the power field. For example, in [12], an architecture with the application of RMT into smart grid is proposed. In [13], based on RMT, a datadriven approach to reveal the correlations between various factors and the power system status is proposed. In [14] and [15], RMT is used for power system transient analysis and steadystate analysis, respectively.In this paper, based on RMT, a datadriven approach is proposed for anomaly detection in power systems. It tracks the data behavior by moving a window at continuously sampling times. For each moving data window, the empirical eigenvalue distribution (EED) in the complex plane is analyzed and compared with the theoretical limits, and the mean spectral radius (MSR) of the eigenvalues is calculated to indicate the system state in macroscopic. In order to realize anomaly declare automatically in realtime analysis, an anomaly indicator based on the MSR is designed and the corresponding confidence level for each sampling time is calculated. The main contribution of this paper can be summarized as follows: 1) The approach is purely datadriven and does not require making assumptions or simplifications on the complex power systems. 2) The approach is sensitive to the variation of the data behavior, which makes it possible for detecting the anomaly in an early phase. 3) It is theoretically and experimentally justified that the approach is robust against random fluctuations and measuring errors of the data. 4) In the approach, the product of multiplicative data matrices can reinforce the anomaly signals, which makes it much easier for anomaly detection. 5) The proposed approach has fast computing speed, which ensures its application for realtime online analysis.
The rest of this paper is organized as follows. Section II conducts anomaly analysis based on the asymptotic theorem in RMT, and the MSR is introduced as the macroscopic indicator to indicate the system states. In Section III, spatiotemporal data set is formulated by arranging highdimensional synchrophasor measurements in chronological order and specific steps of RMT for anomaly detection are given. The advantages of our approach are systematically analyzed in this section. The synthetic data generated from IEEE 300bus, 118bus and 57bus test systems are used to validate the effectiveness and advantages of our approach in Section IV. Conclusions are presented in Section V.
Ii Anomaly Analysis Based on Random Matrix Theory
In this section, anomaly analysis has been conducted based on RMT. First, asymptotic theorem in RMT is used to analyze the empirical spectral density (ESD) of highdimensional ‘signal+noise’ matrices, which illustrates the differences of ESD when a system operates in normal and abnormal states. Then the MSR is introduced as the macroscopic indicator to indicate the system states in quantity.
Iia Asymptotic Theorem for ‘Signal+Noise’ Matrix
Let be a nonHermitian random matrix with i.i.d. entries . The mean
and the variance
. The product of nonHermitian random matrices can be defined as(1) 
where is the singular value equivalent [16] of . As but , according to the Ring law [17], the ESD of
converges to the limit with probability density function (PDF)
(2) 
1) : We first consider the case in the Ring law. Let be a highdimensional data matrix and we apply the Ring law for . In normal state, is considered as a standard random matrix scaled by (i.e., , where is a random matrix with the mean and the variance , and satisfying the singular value equivalent hypothesis), and in equation (1). Then the ESD of converges to the limiting spectral density , which is shown in Figure 1(a). In the complex plane, the blue dots represent the eigenvalues of , the radius of the inner circle of the ring is and the radius of the outer circle of the ring is unity. However, what will happen in abnormal state? Here, “abnormal” means signals occur in and the correlations among the entries have been changed. Then is considered of the type , where is a standard random matrix, which represents random noise or small fluctuations, and is a lowrank matrix which represents anomaly signals. Figure 1(b) shows the ESD of does not converge to the Ring law and the extreme eigenvalues (outliers) caused by anomaly signals are out of the outer circle of the ring. More surprising facts are that the outliers are in the neighborhood of the eigenvalues of , which has been proved in [18].
2) : Furthermore, we explore the case in the Ring law. Let be a highdimensional data matrix and we apply the Ring law for the product of . In normal state, is considered as a standard random matrix scaled by (i.e., , where is a random matrix with the mean and the variance , and satisfying the singular value equivalent hypothesis), and according to equation (1). Then the ESD of converges to the limiting spectral density , which is shown in Figure 2(a). In the complex plane, the blue dots represent the eigenvalues of , the radius of the inner circle of the ring is and the radius of the outer circle of the ring is unity.
In abnormal state, is considered of the type , where is a standard random matrix satisfying the singular value equivalent hypothesis, which represents random noise or small fluctuations, and is a lowrank matrix which represents anomaly signals. Then the product of is calculated as
(3) 
where is the random term, is the deterministic term, and represents the ‘mixed’ terms, each containing at least one random factor and one deterministic factor. Figure 2(b) shows the ESD of does not converge to the Ring law and some extreme eigenvalues (outliers) caused by anomaly signals are out of the outer circle of the ring. More surprising facts are that the outliers are only determined by the deterministic term and the ‘mixed’ terms do not affect the asymptotic location of the outliers, which are shown in Figure 3. Similar conclusions for the Circular law can be found in [19]. Thus, the product of multiple ‘signal+noise’ matrices can reinforce the signals, which makes it much easier to detect the anomaly.
IiB Macroscopic Indicator
Based on the analysis in Section IIA, it can be concluded that the ESDs are different for a highdimensional random matrix with or without anomaly signals, which inspires us to investigate the statistics regarding the empirical eigenvalues in the complex plane to indicate the data behavior in macroscopic quantity. The mean spectral radius (MSR) is the mean distribution radius of the eigenvalues of the product in equation (1), which is defined as
(4) 
where are the eigenvalues of and is the radius of in the complex plane. It can be used to measure the distribution of the eigenvalues in macroscopic quantity. For example, the MSR of the eigenvalues of the standard (i.e., for each row of ) under both normal and abnormal system states corresponding to in Section IIA are shown in Figure 4. It can be concluded that the MSR decreases when the system state changes from normal to abnormal.
Iii Random Matrix Theory for Anomaly Detection
Based on the analysis in Section II, an RMTbased anomaly detection approach is proposed in this section. First, spatiotemporal data set is formulated by arranging highdimensional synchrophasor measurements in chronological order in power systems. Then, details on the RMT for realtime data analysis are presented, in which a moving data window method is used. Finally, steps of the RMT for anomaly detection are given, and we systematically analyze the advantages of the approach.
Iiia SpatioTemporal Data Formulation
Assume there are dimensional measurement variables (such as dimensional voltage measurements from PMUs installed in a power system) . At the sampling time , the
dimensional measurements can be formulated as a column vector
. For a series of time , a spatiotemporal data set is formulated by arranging these vectors in chronological order. To be mentioned is that, by stacking the measurements in a series of time together, the spatiotemporal data contains the most information on the operating states of the system.IiiB RealTime Data Analysis for Anomaly Detection
In realtime analysis, we can move a window on at continuous sampling times and the last sampling time is the current time, which enables us to track the data behavior in realtime. For example, at the sampling time , the obtained data window is formulated as
(5) 
where for is the sampling data at time .
For the data matrix , we convert it into the standard form by
(6) 
where , , and . The singular value equivalent of is introduced as
(7) 
where
is a Haar unitary matrix, and
.According to equation (1), the product of (such as for the sampling time) nonHermitian matrices is obtained by
(8) 
Then is converted to the standard form by
(9) 
where ,
is the standard deviation of
, and thus . Then the eigenvalues of can be obtained and the MSR in equation (4) can be calculated. Since is a complicated matrix function of the product, it can be considered as a random variable.
In practice, for a series of time , is generated for each sampling time with continuously moving windows and the last sampling time is considered as the current time. In order to realize anomaly declare automatically in realtime analysis, an anomaly indicator is defined based on the generated . For example, at the sampling time , the anomaly indicator is calculated as
(10) 
where and is the absolute value function. Here, for the sampling times is considered to follow a student tdistribution with degree of freedom. Now we convert for the last sampling time into the standard form by
(11) 
where and are the mean and standard deviation of for the sampling times, and follows the standard tdistribution with degree of freedom. We can obtain the confidence level of once and are calculated. For example, let and , then the confidence level for the last sampling time is . Thus, an anomaly can be declared automatically by comparing with the threshold defined empirically.
IiiC Anomaly Detection Approach and Its Advantages
Based on the research mentioned above, an RMTbased approach is proposed for anomaly detection in power systems. The specific steps are given in Table IIIC.
Steps of the RMTbased Anomaly Detection Approach 

1: A spatiotemporal data set is formulated by arranging 
synchrophasor measurements in a series of time in chronological 
order. 
2: For the sampling time : 
2a) Obtain the data matrix as in equation (5) by using a 
() window on ; 
2b) Convert into the standard form matrix through 
equation (6); 
2c) The singular value equivalent of is introduced as in 
equation (7); 
2d) The product of nonHermitian matrices is calculated 
through equation (8); 
2e) Convert into the standard form through equation (9); 
2f) Calculate the eigenvalues of and 
compare the ESD with the theoretical Ring law; 
2g) Calculate through equation (4) and the corresponding 
in equation (10); 
3: Draw the and curves for a series of time . 
4: Calculate the confidence level of for each sampling time and 
declare anomalies where . 
Step 1 is conducted for data set formulation. For each sampling time, obtain a moving data window on the data set and the last sampling time is considered as the current time, which is shown in Step 2a. In Step 2b2f, the Ring law in RMT is used for the data window analysis. The MSR and corresponding anomaly indicator for each sampling time are calculated in Step 2g. The and curves are drawn in Step 3 to indicate the data behavior. In Step 4, the confidence level of for each sampling time is calculated and compared with the threshold to realize anomaly declare automatically.
The RMTbased anomaly detection approach is purely datadriven and suitable for highdimensional data analysis, such as tens, hundreds or more. It is sensitive to the variation of the data behavior, which makes it possible for detecting the early anomalies. The steps above involve no mechanism models, thus avoiding the errors brought by assumptions and simplifications. For each sampling time, a spatiotemporal data window instead of just the current sampling data is analyzed in the approach. The average result makes it robust against random fluctuations and measuring errors in the data. What’s more, our approach is practical for realtime analysis for the fast computing speed.
Iv Case Studies
In this section, the effectiveness of our approach is validated with the synthetic data generated from IEEE 300bus, 118bus and 57bus test systems [20]. Detailed information about IEEE 300bus, 118bus and 57bus test systems can be found in case300.m, case118.m and case57.m in Matpower6.0 package [21]. And the simulation environment is MATLAB2016. Three cases in different scenarios were designed: 1) The first case, leveraging the synthetic data from IEEE 300bus test system, tested the effectiveness of our approach with for anomaly detection, in which an anomaly signal was set by a sudden increase of impedance. 2) In the second case, we tested the effectiveness of our approach with for anomaly detection and compared that with the case . In this case, the synthetic data was generated from IEEE 118bus test system and an increasing anomaly signal was set by gradually increasing the active load. 3) We validated the advantages of our approach in anomaly detection by comparing it with other existing techniques in the last case. In this case, the synthetic data was generated from IEEE 57bus test system and an increasing anomaly signal was set by gradually increasing the active load.
Iva Case Study on
In this case, the synthetic data generated from IEEE 300bus test system contained voltage measurement variables with sampling times. For the generated data set
, a little white noise
() was introduced to represent random fluctuations and measuring errors, i.e., . The scale of the added white noise is , where represents the trace function and is the signalnoiserate. In order to test the effectiveness of our approach with in equation (8), an assumed anomaly signal was set by a sudden increase of impedance from bus to and others stayed unchanged, which was shown in Table I. The generated data was shown in Figure 5. In the experiments, the size of the moving window was set as and the signalnoiserate was set to be . The experiments were conducted for times and the results were averaged.fBus  tBus  Sampling Time  Impedance(p.u.) 

33  34  0.2  
20  
Others  Others  Unchanged 
The anomaly detection results are shown in Figure 6. The anomaly indicator in Figure 6(b) is normalized into . It is noted that the curve begins at , because the initial moving window includes 499 times of historical sampling and the present sampling data. And the curve begins at for is calculated through and in equation (10). In the realtime calculation of for each data point on the curve, a series of sampling times (i.e., times of historical sampling times and the current sampling time) are considered to follow a student tdistribution. The threshold is set to be . The anomaly detection processes are shown as follows:
I. During , and remain almost constant and the corresponding values of are small, which indicates no anomalies occur and the system operates in normal state. For example, at , the calculated value of is . As is shown in Figure 7(a), the ESD converges almost surely to the theoretical Ring law.
II. At , and begin to change rapidly and the corresponding is , which indicates an anomaly is detected and the system operates in abnormal state. As is shown in Figure 7(b), the ESD does not converge to the Ring law. It is noted that, from , the curve is almost Ushaped, because the delay lag of the anomaly signal to is equal to the moving window’s width.
III. At , returns to normal and remains constant afterwards, which indicates the anomaly signal disappears and the system returns to normal state. It is noted that is large and the corresponding is , which is caused by the change of the system state.
IvB Case Study on
In this case, we test the effectiveness of our approach with and compare that with the case . The synthetic data generated from IEEE 118bus test system contained voltage measurement variables with sampling times and a little white noise was introduced to represent random fluctuations and measuring errors. In the simulation, an increasing anomaly signal was set by a gradual increase of active load at bus and others stayed unchanged, which was shown in Table II. The generated data was shown in Figure 8. In the experiments, the size of the moving window was set as and the signalnoiserate was set to be .. The experiments were conducted for times and the results were averaged.
Bus  Sampling Time  Active Power(MW) 

20  
Others  Unchanged 
The anomaly detection results of our approach with are shown in Figure 9, where and are normalized into . It is noted that the curves corresponding to begin at , because the initial moving window contains times of sampling data and the initial is calculated through the product of consecutive moving data windows. And the curves corresponding to begin at , because the initial is calculated through and in equation (10). In realtime analysis, the parameter and the threshold are set the same as in Case A. The anomaly detection processes are shown as follows:
I. During , and remain almost constant and the corresponding of are small random values, which indicates no anomalies occur and the system operates in normal state. For example, at , the calculated of corresponding to are , respectively. As is shown in Figure 10, the ESD converges almost surely to the theoretical Ring law.
II. At , and change rapidly and the corresponding of for are , which indicates an anomaly is detected and the system operates in abnormal state. As is shown in Figure 11, the ESD does not converge to the Ring law. It is noted that, with the increase of the number of multiplicative data windows, the values of decrease more rapidly from , which makes it easier for detecting the anomaly signal. It validates our claim in Section IIA that the product of multiple ‘signal+noise’ matrices can reinforce the signals.
III. From , and remain almost constant for the increasing signal is contained throughout the moving window afterwards.
IvC Case Study on Comparison with Other Approaches
In this case, we compare our spectrum analysis (SA) approach with oneclass support vector machines (SVMs)
[22], structured autoencoders (AEs)
[23]and long short term memory (LSTM) networks
[24] to illustrate the advantages of our approach for anomaly detection, i.e., more sensitive to the variation of the data behavior and robust against random fluctuations and measuring errors. The IEEE 57bus test system is used to generate the synthetic data. In the simulation, an increasing anomaly signal was set by a gradual increase of active load at bus and others stayed unchanged, which was shown in Table III. The generated data contained voltage measurement variables with sampling times, which was shown in Figure 12. In the experiment, the signalnoiserate was set as . For SVMs, AEs and LSTM, we trained the detection models only using a normal data sequence during and computed the testing errors for the remaining sequence during , in which one times of sampling was used as a training/testing sample. For our approach, both the case of and were tested. The parameters involved in the detection approaches are summarized as in Table IV.Bus  Sampling Time  Active Power(MW) 

20  
Others  Unchanged 
Approaches  Parameter Settings 

SVMs  the upper bound on the fraction of training errors : 0.03; 
the kernel function: ;  
AEs  the model depth: ; 
the number of neurons in each layer of encoder: ; 

the number of neurons in each layer of decoder: ;  
the initial learning rate: ;  
the activation function: ; 

the minimum reconstruction error: ;  
the optimizer: .  
LSTM  the time steps: ; 
the model depth: ;  
the number of neurons in each layer: ;  
the initial learning rate: ;  
the activation function: ;  
the minimum loss: ;  
the optimizer: .  
SA()  the moving window’s size: ; 
the number of multiplicative data windows: .  
SA()  the moving window’s size: ; 
the number of multiplicative data windows: . 
The anomaly detection results of different approaches are normalized into , which are shown in Figure 13
. For SVMs, the normalization result of signed distance to the separating hyperplane is plotted; for AEs and LSTM, the normalized values of testing errors are plotted; for SA (
) and SA (), is plotted, where is the normalized value of . It can be obtained that:I. SA () and SA () are able to detect the anomaly signal much earlier (i.e., ) than other approaches, which validates our approach is more sensitive to the variation of the data behavior by exploring the data correlations and robust against random fluctuations and measuring errors. The reason is that, in our approach, a data window instead of just the current sampling data is analyzed for each sampling time. The average result makes our approach more robust against random fluctuations and measuring errors of the data.
II. SA () outperforms SA () in anomaly detection, which indicates the product of multiple moving data windows can reinforce the anomaly signal so that it can be detected much easier.
III. The detection curves in our SA approach increase rapidly when the voltage collapses from , while that of other approaches vibrates with the voltage values. It indicates our approach can reflect the system state more accurately in macroscopic.
Approaches  SVMs  AEs  LSTM  SA()  SA() 

(unit: s) 
Furthermore, in order to illustrate the efficiency of different detection approaches, the for each sampling time was counted. For SVMs, AEs and LSTM, the for each testing sample was counted, which does not include the model training time. The experiments were conducted on a server with GHz central processing unit (CPU) and GB random access memory (RAM). The for SVMs, AEs, LSTM, SA () and SA () are shown in Table V. Considering our approach is an unsupervised approach without any training, it can be concluded that our approach has competitive performance in detection efficiency.
V Conclusion
Based on random matrix theory, a datadriven approach is proposed for anomaly detection in power systems. It is capable of detecting the anomaly in an early phase by exploring the variation of the data behavior. The mean spectral radius gives insight into the system states from a macroscopic perspective, which is used to indicate the data behavior in our approach. An anomaly indicator based on the MSR is designed and the corresponding confidence level is calculated to realize anomaly declare automatically. Our approach is purely datadriven without making assumptions and simplifications on the complex power systems. It is robust against random fluctuations and measuring errors, and it has fast computing speed. Case studies on synthetic data corroborate the effectiveness and advantages of our approach, which indicates our approach can be served as a primitive for realtime data analysis in power systems.
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