Among challenges for big data analytics towards grid modernization, data-driven approach and data utilization are of great significance in power system operation in smart grids . Current power systems are huge in size and complex in topology. Model-based methods can not always meet the real-life needs when assumptions and simplifications are prerequisites for these mechanism models. Massive datasets are accessible, however, when the operation of the power system is monitored by a large number of sensors such as phase measuurment units (PMUs). For instance, China has deployed 1717 PMUs as of 2013  and there are about 500 PMUs installed by July 2012 in America .
The synchronized measurements of a large power grid enable the joint modeling of temporal statistical properties across the spatial nodes. The spatial-temporal couplings pose opportunities and challenges. Towards this goal, random matrix theory (RMT) is used for data analysis first in 
. RMT starts in the early 20th century. Due to the increase in the dimensionality of collected datasets, RMT has been commonly used for considering problems regarding the behavior of eigenvalues of large dimensional random matrices in physics, finance, wireless communication, etc[5, 6].
Due to the large size of datasets, randomness or uncertainty is at the heart of data modeling and analysis in a complex, large power gird when rapid fluctuations in voltages and currents are ubiquitous. Often, these fluctuations exhibit some certain (or deterministic) distribution properties . Our approach exploits the massive datasets across the large grid that are distributed in both spatially and temporally. Random matrix theory (RMT) appears very natural for the problem at hand. In a random matrix of size we use variables to represent the spatial nodes. For the -th node where , there are observations to represent the temporal samples When the number of nodes and data samples are large, very unique mathematical phenomenon occurs such that power mathematical tools such as free probability  can be exploited to develop big data analytics for joint spatial-temporal datasets. This is the central purpose of this paper.
Free probability is a powerful tool for solving random matrix problems, such as additive and multiplicative free convolution. Based on free probability theory, asymptotic limits of the testing functions (free self-adjoint matrix polynomials), can be obtained numerically through certain algorithms. Closed form expressions exist only for some simple matrix polynomials. The obtained asymptotic limits provide the rigorous bounds in mathematics which can help distinguish signals and noise in grid data. Theanomaly detection is conducted through hypothesis testing and an indicator for fault location is designed using some mathematical tricks.
I-a Contributions of Our Paper
in the last several years. Motivated for machine learning from massive datasets, our line of research is based on the modern high-dimensional statistics where RMT is central to this paradigm. The contributions of this paper can be summarized as follows:
The aim of this paper is to exploit the polynomials of large random matrices in the context of big data analytics for a large power grid. To our best knowledge, this attempt is for the first time. This analysis is made possible by a recent breakthrough [11, 12] in the literature of mathematics. Our result represents one of the first applications of these algorithms in engineering.
Based on certain new algorithms developed in this paper, an indicator for fault location is proposed and validated to be valid by simulations and real-world cases.
I-B Related Work
There are numerous researches on data driven methods for modeling and analysing large power systems. Le, Chen and Kumar  propose a linearized analysis method for early event detection using partial least squares estimation. Lim and DeMarco [7, 14]
propose a singular value decomposition (SVD)-based voltage stability assessment from principal component analysis (PCA). Both methods in the above are PCA related and the selection of the eigenvalues has a crucial influence. Lim et al select the largest eigenvalue and Xie et al
adopt the method of the threshold of cumulative variance proportion. Their common disadvantage is that the pre-defined threshold depends on the experience without the consideration of the statistical characteristic of the grid data, so the redundancy or loss of information is unavoidable.
This paper is organized as follows. Section II establishes the random matrix model for the power grid as the basis of this paper. Section III proposes the anomaly detection method from the way of hypothesis testing and an indicator for fault location. Section IV provides a brief introduction of the algorithm for obtaining the asymptotic spectral distribution of free self-adjoint polynomial which is essential for our analytical framework. In Section V, numerical case studies validate our methods with simulation data and real-world data. In Section VI, the estimation of signal strength is discussed under the linear assumption. Conclusion and further direction of this research are given in Section VII. For the sake of simplicity, some details and the supplementary materials are deferred to the Appendix.
Ii Random Matrix Model and Data processing for power grid
Ii-a Random Matrix Model for Power Grid
where denotes the active power, the reactive power, the voltage phase angle and the voltage amplitude respectively.
To characterize the role of each block of the Jacobian matrix, denote:
Therefore, under the situation that is relatively constant, the model between and is obtained as:
random vectors observed at timea random matrix is formed as follows:
It is worth noting that only voltage magnitude of PMU data is used. The voltage magnitude are more sensitive to topology change than phase angle and they remain relatively stable in normal operating condition . Without dramatic topology changes, rich statistical empirical evidence indicates that the Jacobian matrix keeps nearly constant, so does . Thus (5) is rewritten as:
where , and . Here and are random matrices. To model the fast time scale stochastic variation in a load, we assume that
is a random matrix with Gaussian random variables as its entries, following[4, 9].
Ii-B Data Processing Method
The sampling data matrix of real power grid is always non-Gaussian, so a normalization procedure in  is adopted to conduct data preprocessing. Meanwhile, a Monte Carlo method is employed to estimate the empirical spectral distribution (ESD) of raw grid data according to the asymptotic property theory.
The data processing procedure above is organized as following steps in Algorithm 1. The parameter denotes the number of buses and denotes the sampling period. Note that is extremely small, e.g. and is set to 10 in our simulation cases.
Ii-C Validation of Proposed Model
Marchenko-Pastur Law (M-P Law) , a basic theorem in random matrix theory, is introduced to verify the random matrix model for power grid.
Theorem II.1 (M-P Law ).
Let be a random matrix whose entries with the mean and the variance , are independent identically distributed (i.i.d). As with the ratio .
is the corresponding sample covariance matrix. Then, the asymptotic spectral distribution of is given by:
where , . Here, is called Wishart matrix.
According to Algorithm 1, we obtain the ESD of the sample covariance matrix of real-world datasets for 34 PMUs. The is collected in normal operation.
As illustrated in Fig 1, the histogram of the ESD of coincides with the M-P Law. Although the asymptotic convergence is considered under infinite dimensions, i.e., the asymptotic results are fairly accurate for moderate matrix sizes such as s. It effectively explains why RMT is practical for the real-world datasets in a power grid.
Iii Anomaly Detection and Fault Location
Iii-a Hypothesis Testing For Anomaly Detection
Based on the random matrix model for power gird in II-A, the problem of anomaly detection is formulated in terms of the hypothesis testing :
where is the sample covariance matrix of the grid data collected in normal operation and is the sample covariance matrix of the grid data for abnormal operation. This problem is a matrix hypothesis testing [5, 6]
. Test statistics are central to hypothesis testing.
In this paper is adopted as test statistics. Here, is a self-adjoint polynomial of large random matrices, i.e. The measures the difference between two sample covariance matrices.
Theorem III.1 (The self-adjoint matrix polynomial of large Hermitian random matrices ).
Let be a family of independent, normalized Wishart matrices. Assume that for every Hermitian matrix of the form
where P is a free self-adjoint matrix polynomial, we have with probability one that: 1. The empirical spectral distribution of a free self-adjoint matrix polynomial converges weakly to a compactly supported on the real line as goes to infinity.
2. For any , almost surely there exits such that for all , , where ’Sp’ means the spectrum and ’Supp’ means the support.
Theorem III.1 implies that if is true, the ESD of will coincide with a theoretical curve111This curve can be calculated by a certain algorithm. See Section III for details., i.e. the asymptotic spectral distribution (ASD) of . Besides, no eigenvalue exits outside of the support of the theoretical curve.
In this paper, the eigenvalues outside of the support are called outliers.
According to Theorem III.1, the proposed detection method is summarized as follows:
Calculate and from the sample data with the preprocessing method stated in Algorithm 1.
Compare the theoretical curves corresponding with the ESDs of different matrix polynomials .
Anomaly detection is conducted: if outlier exists, will be rejected, i.e. signals exist in the system.
Based on the hypothesis testing (9), we propose a statistic indicator denoted by
The function of is similar to the signal-to-noise ratio.
Notice that our proposed detection method is quite sensitive to the signal even if the signal is extremely weak . In order to reduce the false alarm probability, it is necessary for the values of of the normal and the abnormal load variation to be different. So the choice of the polynomial functions is crucial.
In this paper, we study two typical self-adjoint matrix polynomials. The first one is the multivariate linear polynomial:
The second one is the multivariate nonlinear polynomial:
Here, both and are the sample covariance matrices. The simulation results in Section V will show that the performance of the nonlinear polynomial is much better than the linear one.
Iii-B Fault Location
In this subsection, we investigate the fault location based on the proposed anomaly detection method in III-A. Since the selected polynomials are real and symmetric, the following equations
hold. Here, ,
denote the left and right eigenvector matrix;is an eigenvalue of and it indicates the energy of the corresponding eigenvector .
For the element in , the derivative of (15) leads to the following:
Left multiply (16) by . Note that and and we have
Let . Obviously, only and other elements of equal to zero. So (17) is simplified as:
Finally, the contribution of the -th row to the eigenvalue is obtained by:
In the work of Lim et al , the singular vector correspondingto the largest singular value is used to conduct fault location. The simulation results in  show that the singular vector tells which buses are contributing to the corresponding singular value.
For the hypothesis testing in III-A, not only the largest eigenvalue of the covairance matrix but also the outliers are viewed as the “signals”. This observation inspires us to improve Lim’s method by studying those eigenvectors corresponding to outliers. In particular, we design a new location indicator denoted by
to quantify each bus’s contribution to the anomaly. Since that and , obviously,
Thus, and . From the above, is a reasonable indicator that measures the correlation between the -th bus and the load variation. The location (denoted as ) of the most sensitive bus can be expressed as
All the eigenvalues outside the support of the theoretical curve are considered. The functions of
play a role of machine learning that classify between the noise and signals.Outliers are remained as the useful signals for anomaly detection and fault location.
Iv The asymptotic spectral distribution of free selfadjiont polynomial
In this section, we study the asymptotic spectral distribution (ASD) of and , on the premise that both and are Wishart matrices.
Iv-a The ASD of
For obtaining the ASD of , we introduce the operator-valued setting  briefly. Let be a unital algebra and be a subalgebra containing the unit. A linear map is a conditional expectation. For a random variable , we define the operator-valued Cauchy transform: for which is invertible in . Let . In the following theorem , we will use the notation .
Theorem IV.1 ().
Let and be self-adjoint operator-valued random variables free over . Then there exists a Frechet analytic map so that
for all ,
Moreover, if , then is the unique fixed point of the map. and for any , where means the n-fold composition of with itself. Same statements hold for , with replaced by
Theorem IV.2 (Stieltjes inversion formula ).
For any open interval , such that neither a nor b are atoms for the probability measure , the inversion formula
Theorem IV.1 provides an iterative algorithm to compute the operator-valued Cauchy transform of . Then, the ASD of is easily obtained through the Stieltjes inversion formula.
Iv-B The ASD of
The ASD of is obtained by linearizing the nonlinear polynomial. Through Anderson’s linearation trick , we have a procedure that leads finally to an operator:
In the case of ,
Therefore, can be easily written in the form of , where
Theorem IV.3 ().
Consider that that has a self-adjoint linearization
Then, for each and all small enough , the operators and are both invertible and
V Case Studies
The proposed method is tested with the simulated data generated from IEEE 118-bus and a Polish 2383-bus system, respectively. Detailed information of the system is referred to the case118.m and case118.m in Matpower package and Matpower 4.1 User’s Manual .
In subsection V-A, V-B and V-C, the proposed method is tested with simulated data in the standard IEEE 118-bus system, as shown in Fig. 37 in Appendix B. The results generated form the Polish 2383-bus system are deferred to the Appendix C.
In subsection V-D, the fault location method is validated by real-world 34-PMU data.
For Cases 1-3, set the sample dimension, i.e. the number of buses, as . The spectral density distribution of free adjoint polynomial can be obtained through the algorithm in Section III as long as . Meanwhile, the sample dimension is required to be large enough to guarantee the accuracy of results in the proposed asymptotic theory of eigenvalue distributions. Therefore, in Cases 1-3 , we set the sample length to be equal to , i.e. , and select six sample voltage matrices presented in Tab. I. The load variation is shown in Fig. 38 in Appendix B.
|Cross Section (s)||Sampling (s)||Descripiton|
|Reference, no signal|
|Existence of a step signal for Bus 22|
|Steady load growth for Bus 22|
|Steady load growth for Bus 52|
|Chaos due to voltage collapse|
*We choose the temporal end edge of the sampling matrix as the marked time for the cross section. E.g., for , the temporal label is 217 which belong to .
Power grid operates with only white noises during 0 s to 900 s; we choose sampling matrix as the reference. Similarly, we mark other kinds of system operation status as –, and choose their relevant sampling matrix – for the test.
V-a Case 1: Anomaly Detection with the Multivariate Linear Polynomial
We conduct anomaly detection using and the reference matrix through the proposed hypothesis testing (9). Note that covariance matrix is generated from by the preprocessing in II-B. Then, we choose the multivariate linear polynomial
to conduct this detection. Simulation results are shown in Fig. 7. The red curve represents the ASD of obtained by Theorem IV.1. The ESD histogram of is plotted by using Algorithm 1. Outliers are highlighted by ellipses. The values of defined in (11) in – are presented in Tab. II.
1) For Fig. 7(a)-(d), the actual histograms agree with the theoretical curve very well except a few spikes (called outliers); (e): for the white noise case, there are no outliers, as expected by the theory.
2) Sort by the values of : .
From result 1), we observe that this method distinguishes the signals from the white noise successfully. Result 2) implies that the size of outliers indicates the strength of the signals.
However, the discrimination between the ramp signals and the step signals are not very obvious. See Table II for details. To deal with this disadvantage, a direct approach is to increase the sensitivity for a given signal-to-noise ratio. In particular, a nonlinear polynomial is adopted.
|Existence of a step signal||0.1096|
|Steady load growth||0.0357|
|Steady load growth||0.0609|
|Chaos due to voltage collapse||0.4219|
V-B Case 2: Anomaly Detection with the Multivariate Nonlinear Polynomial
In Case 2, the process is similar to V-A except that the test statistic function is replaced with
which is a multivariate nonlinear polynomial in two matrices. This is the simplest second-order polynomial. Of course, we can study higher orders that will be left for the work in a next paper.
1) For Fig. (e)e, the histogram agrees perfectly with the theoretical curve when in the absence of spikes; for other figures (a)-(d), there exist outliers.
2) Sort by the values of : .
Result 2) show that the step signals and the ramp signals are remarkably distinguished by the size of outliers. This implies that the nonlinear polynomial is more sensitive to outliers for the same signal-to-noise ratio, compared with its linear case. Furthermore, these results indicate that the anomaly’s influence on the grid can be estimated quantitatively by the sizes of outliers.
Compared with linearity, nonlinearity is more flexible in problem modeling and closer to the reality. Some other multivariate nonlinear polynomials may be more effective for the power grid with special load characteristics. The search for such an optimal nonlinear polynomial is beyond the scope of this paper.
|Existence of a step signal||0.9027|
|Steady load growth||0.1152|
|Steady load growth||0.2783|
|Chaos due to voltage collapse||5.2332|
V-C Case 3: Fault Location with Simulation Data
In Case 3, the indicator defined in (20) is used to conduct fault locations. As introduced in III-B, the proposed fault location method defined in (22) is based on the hypothesis testing. Case 1 and Case 2 validate our proposed detection method and show that the nonlinear polynomial is more effective than the linear one Thus, is selected as the test statistics in this case.
Fig. 18 is the 3D Plot for the time series of the indicator Figures (a)a, (c)c and (c)c show that the proposed indicator captures the bus information that is most affected by the topology change, and the location results are consistent with the the event description in Table I. The fault location fails in Fig. (d)d due to voltage collapse. This result is close to the real fact.
Fig. 23 is the result obtained by Lim’s method in . The location results are close to our method at most time points. However, in some points, it is not easy to determine which bus is the most vulnerable to the load variation, especially in Fig. (a)a and (c)c. The reason is that the peaks in other buses are not independent of the statistic information corresponding to the largest singular value.
The reason why our method performs better is given by studying the distribution of the corresponding eigenvectors. We select the cross section as an example. Let denote the -th component of the eigenvector corresponding to the eigenvalue . Then, we normalize it such that . For a fixed , the distribution of is denoted by . The distribution of is plotted in Fig. 26 by dashed lines. The red solid line represents the standard normal distribution
standard normal distribution. As shown in Fig. (a)a, the for four randomly selected eigenvalues well inside the support fits extremely well with the standard normal distribution. In some sense, there is no signal contained in these eigenvectors. On the contrary, the for outliers is markedly different from the standard normal distribution in Fig. (b)b. This means that not only the largest eigenvalue but also all other outliers contain the most statistical information about the signal. In other words, all outliers matter!
V-D Case 4: Fault Location with Real 34-PMU Data
In this subsection, we evaluate the fault location indicator with real-world 34-PMU data. The real power data is a chain-reaction fault that happened in 2013 in one large power grid in China. The sample rate is 50 Hz and the total sample time is seconds (s). Fig. LABEL:34-PMU_powerflow and Fig. LABEL:34-PMU_chain-reaction illustrate the three-dimensional power flow at the whole time and the fault time respectively. The chain-reaction fault starts at s.
Similarly to the data processing in simulation Case 3, set the sample dimension and the sample length . The location of the most sensitive bus can be determined using defined in (20), using the method of (22). The result shown in Fig. 30 illustrates that the 18-th PMU () is the most sensitive one which is in agreement with the actual accident situation. This case validates the proposed method in real-life grid.
Vi The estimation of the signal strength
Here we estimate the signal strength under the linear assumption:
where is the grid data matrix, is the noise matrix and represents the signal. Let denotes the -th sampling matrix and . Define the matrix product as
It is natural to wonder how close the eigenvalues of are to those of .
Theorem VI.1 ().
Let be an integer, and assume are complex-valued random variables. For each , let be an i.i.d random matrix with atom variable , respectively. In addition, for each , let be a deterministic matrix with rank and operator norm . Define the products
and . Let , and suppose that for all sufficiently large , there are no eigenvalues of in the band , and there are eigenvalues of the product in the region , and after labeling these eigenvalues properly,
as for each .
Theorem VI.1 reveals two main points:
when the sizes of matrices are large, the cross terms in (25) can be negligible outliers exit if signals exist in the system.
the combined strength of the signals can be bounded according to (26).
The outliers of are asymptotically close to the outliers of the product defined in (25). This implies that the combined signal strength can be estimated by calculating the eigenvalues of directly. In practice, the can be obtained by measurements but the is difficult to know.
The results in Fig 36 show an interesting phenomenon of products of random matrices: outliers appear only in the presence of the signals.
Built upon random matrix theory (RMT), we obtain new statistical models using massive datasets across the power grid. In this paper, we take advantage of a breakthrough by [12, 11] in free probability to calculate the asymptotic spectrum distribution of the self-adjoint matrix polynomials. Our problem of anomaly detection is formulated in terms of hypothesis testing. Fault location is also conducted. The new approach has advantages over previous ones. The results generated from the 2383-bus system agree with the asymptotic theory much better than those the 118-bus system.
As a starting point, this paper considers only two simplest examples of the self-adjoint matrix polynomials: and where and are the large-dimensional sample covariance matrices, respectively, for the null and alternative hypotheses. This problem is related to the difference between two mixed quantum states (e.g., Wishart matrices with fixed trace) [24, 25, 26] where analysis is conducted by free probability. Specifically, if we define the mixed quantum states as we can study the difference following .
Different trace functions as done in  can be considered in the future. Specifically, we consider the linear eigenvalue statistics (LES) of the difference of the two quantum states where is an arbitrary function of some certain smooth properties and is the th eigenvalue of the difference This is the extension of the LES for one single random matrix in . What is the optimal function ?
The algorithm of [12, 11] is very general. The techniques in quantum information theory, e.g., [24, 25, 26], have more explicit expressions to give use more transparent solutions to our problems at hand. All these papers are unified within the paradigm of free probability, a fast growing branch of random matrix theory. Note that the first use of RMT in a large power grid was by the same authors of this paper in . The whole paradigm of using RMT allows one to exploit the theory of asymptotically large random matrices. The whole framework lies in the empirical observation that the asymptotic limits are very close to that finite-size random matrices, even for moderate sizes! The empirical success of using the asymptotic theory of [12, 11] in finite-size cases of our studies in a power grid will pave the way for studies of big data data analytics using other massive datasets collected in such as internet of things (IOT).
In this paper, the simple white noise representations are adopted to model the stochastic variation in load; some recent studies have adopted Ornstein-Uhlenbeck process  - Validation of the Ornstein-Uhlenbeck process for load modeling based on PMU measurements. These inspire us to adopt some achievements about Ornstein-Uhlenbeck Process based on RMT in the future work.
Appendix A The Specific Procedures of calculating The ASD of
The following steps give the precise statement of the algorithm in IV-B .
Compute the linearization of
through Anderson’s linearization trick.
Compute the Cauchy transform through the scalar-valued Cauchy transforms :
Calculate the Cauchy transform of
by applying Theorem IV.1. The Cauchy transform of is then given by
According to Corollary IV.3, the scalar-valued Cauchy transform of is obtained by
Compute the distribution of via the Stieltjes inversion formula.
Appendix B The standard IEEE 118-bus system and The load variation
Appendix C The 2383-Bus case
For simplicity, the signals for each bus are shown in Tab. IV. The detection results, the values of location results are shown in Fig.42, Tab.V and Fig. 46 respectively. The results generated from the 2383-bus system are the same as the 118-bus system.
|59||3100 3200||Steady load growth|
|5100||Existence of a step signal|
|6000 6100||Chaos due to voltage collapse|
|Others||1 10000||No signal|
|3100 3200||Steady load growth||0.2538|
|5100||Existence of a step signal||2.045|
|6000 6100||Chaos due to voltage collapse||9.668|
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