SPECIAL SECTION ON DATA-DRIVEN MONITORING, FAULT DIAGNOSIS AND CONTROL OFCYBER-PHYSICAL SYSTEMS

Received November 7, 2017, accepted December 9, 2017, date of publication December 18, 2017, date of current version March 9, 2018.

Digital Object Identifier 10.1109/ACCESS.2017.2784841

Big Data Modeling and Analysis for PowerTransmission Equipment: A Novel RandomMatrix Theoretical ApproachYINGJIE YAN 1, GEHAO SHENG1, (Member, IEEE), ROBERT CAIMING QIU 2,3, (Fellow, IEEE),AND XIUCHEN JIANG1,(Member, IEEE)1Department of Electrical Engineering, Shanghai Jiaotong University, Shanghai 200240, China2Department of Electrical and Computer Engineering, Tennessee Technological University, Cookeville, TN 38505, USA3Big Data Research Center, Department of Electrical Engineering, Shanghai Jiaotong University, Shanghai 200240, China

Corresponding author: Yingjie Yan (yanyingjie@sjtu.edu.cn)

This work was supported in part by the National Natural Science Foundation of China under Grant 51477100 and Grant 61571296, in partby the National High Technology Research and Development Program of China (863 Program) under Grant 2015AA050204, in part by theChina State Grid Corp Science and Technology Project, and in part by the National Science Foundation, Division of Computer andNetwork Systems under Grant NSF CNS-1619250.

ABSTRACT This paper explores a novel idea for power equipment monitoring and finds that randommatrixtheory is suitable for modeling the massive data sets in this situation. Big data analytics are mined fromthose data. We extract the statistical correlation between key states and those parameters. In particular,the (empirical) eigenvalue spectrum distribution and the (theoretical) single ring law are derived from large-dimensional random matrices whose entries are modeled as time series. The radii of the single ring laware used as statistical analytics to characterize the measured data. The evaluation of key state and anomalydetection are accomplished through the comparison of those statistical analytics.

INDEX TERMS Big data analytics, big data model, power transmission equipment, key state, largedimensional random matrix, the single ring law.

I. INTRODUCTIONWith the development of Smart Grid and Energy Internet,the modern power system is gradually evolving into a systemthat aggregates large scale data. The real-time data acqui-sition, transmission and storage of the grid, together withthe fast analysis of massive data, have become the basis forthe reliable operation of the grid. With the development ofthe state monitoring technology, and information applicationsystems, such as SCADA system, production managementsystem and EMS system, the volume of the transmissionequipment state data has been growing exponentially [1]–[3].It is attractive to apply big data analytics in this con-text [4], [5].

Big data analytics in power systems are of interestin the fields of data acquisition, storage, analysis andvisualization [6], [7], but there is little attention paid tothe characterization of the transmission equipment state.Power transformers faces lots of difficulties, in the areasof anomaly detection and fault diagnosis. The functionalrelations between different status parameters and anomaly

patterns are too complicated to be described with explicitfunctions or equations. In [8], the equivalent icing thicknessis calculated by formulas involving wire tension, wire incli-nation and wind speed, but the parameters in the formulas aredifferent between wires, which makes the calculating resultincorrect. In [9], pattern recognition is realized using neuralnetwork method. The accuracy of anomaly detection cannotbe ensured with association between anomaly pattern andseveral state variables alone.

A. CONTRIBUTIONWe are motivated for exploiting big data analytics in powerequipment monitoring. To our best knowledge, the ran-dom matrix theory is, for the first time, introduced intothis context here in this paper. In particular, we study thecondition evaluation of power transformers. The frame-work of modeling large-scale data with random matriceshas been systematically pursued by one co-author (Qiu)in wireless network [10], sensor network [11], and SmartGrid [6], [12], [14]–[16]. The paper is a new contribution to

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this framework. More details are following:

1) The measured data for power transformers have strongstatistical correlations. This fact demands a funda-mentally different analysis from that of our previouspapers [14]–[16] that are based on i.i.d. assumptions.It is well known that correlated random variables aremuch harder to deal with than i.i.d. ones. This diffi-culty deserves a special note in the context of high-dimensionality in our problem at hand. We deal withmatrix-valued random variables which are noncommu-tative random variables. The topic is very difficult inmathematical literature. We are lucky to find a theoret-ical model that fits our data very well.

2) The latest development of random matrix theory dealswith two basic problems [13]: finite matrix sizeand non-Gaussian matrix entries. Our work may bereviewed as a new contribution to these problems in thatexperimental results are agreements with the theoreti-cal predictions.

3) The limiting spectrum distribution of a random matrixwith ARMA entries are compared with the empiricalspectral distribution, to enable the anomaly detectionof the transformer state.

4) A data model is proposed for the status data of powerequipment, solving problems such as diversity, hetero-geneity, and sampling synchronization in data sources.

5) Our paper is built upon our previous works [14]–[16].We pursue a completely different direction in thispaper, however, in the context of power equipmentrather than power system grids.

B. RELATED WORKRecent work on anomaly detection of power transformerare generally based on analyzing these changes during thetransformer operation process [17]. Reference [18] uses anovel Cumulative Sum chart to detect change points in thedata flow without repair them. Solutions are proposed todetect and repair X-outliers in load curve data in powersystem [19], [20]. Using the mapping between the faultmodes and the data sources, the huge information will bereduced by the massive data rough set information entropymethod [21], [22]. Although it solves the problem of missingdata, the integrity of the information is damaged. C-meansclustering method is used to separate the noise data in dealingwith the training set of support vector machine [23], [24].However, this clustering method directly removes isolatednoise data and destroys the state quantity data link continuity.Some works are designed to study the unique characteristicsof matrix and distribution of data from the perspective ofprobability and statistics [25], [26], resulting in event detec-tion or condition evaluation [27].

II. BIG DATA MODELLING FOR POWER EQUIPMENTIn this section, a large amount of historical test dataand fault samples during equipment faults and defects are

TABLE 1. Data of status parameters in substation.

collected from PMS system of the electric power company ofa province and literature [28], [29]. The correlation betweenstate parameters and fault types is mined through associationrules. A novel algorithm for the key state evaluation of equip-ment is proposed by applying the large dimensional randommatrix theory to the on-line monitoring data stream.

A. DATA OF STATUS PARAMETERS MEASUREDIN SUBSTATIONWith the development of power system, data arising fromon-line monitoring, production management and meteoro-logical environment are gradually integrated into a unifiedinformation platform. The storage of status data is usuallyin the form of time series, so that the data of each sta-tus parameter can be converted to stationary time seriesthat are modeled by ARMA. Besides, in the analysis oflarge-dimensional data, ARMA models are studied in powerequipment [17], [30], [31].

Since different state data have different sampling peri-ods and different sampling time points, the large dimen-sional matrix of the ARMA model must be constructedfrom the ARMA time sequences after pre-processing, accord-ing to the types of state variables. Taking substation A asan example, there are six converter transformers and sixreactors in the substation. An on-line monitoring device isinstalled in each piece of equipment (12 in total). Monitoringparameters include oil chromatogram, winding temperature,grounding current, bushing dielectric loss, vibration, micro-meteorological and so on. In Table I, the second columnshows the sampling period of monitoring variables. Thethird column shows the original data matrix after the pre-processing. The raw data are shown in Table I, with threefootnotes for winding temperature,1 bushing dielectric loss2

and oil gas.3

B. MODELING STATUS PARAMETERS IN LARGEDIMENSIONAL RANDOM MATRICESWe need to adjust the elements in rows and columns to getthe ratio of row number to column number c.

1The status parameters of winding temperature include two types ofparameters: top oil temperature and bottom oil temperature.

2The status parameters of bushing dielectric loss include three types ofparameters: current,capacity and dielectric loss.

3The status parameters of oil gas include 18 types of parameters. Thereare 9 gases (H2, C2H2, CH4, C2H4, C2H6, CO, CO2, C, water) and theirpercent in the whole gases.

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TABLE 2. The large dimension matrices of key state evaluation and thematrix size.

Assume there are N observation sites and, from eachsite, the observed time series is xi ∈ CT×1, where i =1, 2, · · · ,N . From theseN vectors, the original data matrixXis formed as

X =

xT1xT2...

xTN

N×T

∈ CN×T . (1)

If N is relatively small compared with T , we can split eachxi into k segments such that xi = (x1i , x

2i , · · · , x

ki ), x

ki ∈

C(T/k)×1. k is decided to make a moderate ratio c = Nk2T , for

reasons that will be clear later in following context of randommatrix theory. So the new matrix Y is written as

Y =(x11 , x

21 , · · · , x

k1 , · · · , x

1N , x

2N , · · · , x

kN

)T. (2)

As shown in (2), the original matrixX of size N ×T is turnedintoY of size (kN )×(T/k). The result is shown in column 4 ofTable I.

The key states are extracted according to the correspondingrelationships of state variables from the documents, standardsand related references in China State Grid which can reflectthe load, insulation and mechanical capacity of the trans-formers in substations. And the five key states are shownin column 1 of Table II. For example, the key state loadcapability includes load, winding temperature and ambienttemperature. So the large dimensional matrix X1 is formedby stacking the matrix row-by-row as following.

X1 =

loadwinding tempambient temp

= [load]168×144[winding temp]168×144[ambient temp]168×144

(3)

X2,X3, · · · ,X5 are constructed in the same way as X1, asshown in Table II. Those matrices characterize the key stateof power equipment.

III. LARGE DIMENSIONAL MATRIX OF ARMA MODELIn this section, the stationary and invertible ARMA(p, q)model are combined with the large-dimensional samplecovariance matrix. The relationship between the limitingspectral density function and the coefficients of ARMA

FIGURE 1. Ring circle of Gaussian random matrix with i .i .d andmeasured data.

model is established in order to deduce the single ring lawof large-dimensional matrix [4], [25]–[27]. The key stateevaluation and anomaly detection can be accomplished, basedon the limiting spectral density function and the single ringlaw.

A. RANDOM MATRIX THEORY AND THE SINGLE RING OFMEASURED DATALet X = {xi,j} be a p × n random matrix whose entriesare independent identically distributed (i.i.d.) with the meanµ(x) = 0 and the variance σ 2(x) < ∞. As p, n → ∞

with the ratio c = p/n ∈ (0, 1], the ESD of the corre-sponding sample covariance matrix Sn = (1/N )XXH

∈

CN×N converges to the distribution of M-P law with densityfunction [10].

fMP(x) =

√(b− x)(x − a)2πxcσ 2 a ≤ x ≤ b

0 otherwise(4)

where a = σ 2(1−√c)2, b = σ 2(1+

√c)2. So the single

ring of Gaussian random matrix is showed as the red linesin the Fig. 1. Take the random matrix with size 168×432 forexample. The eigenvalues are restricted into the ring singletheoretically to obey the M-P law.

Consider the the single ring obtained from measured dataof status parameters in substation. According to Section 2.B,the large dimensional matrixXp×n is constructed to representthe key state of power equipment. The measured data isX = (x1, x2, . . . , xn), where x1, x2, . . . , xn are independentcolumn variables of each status parameter. The eigenvaluesof measured data are beyond the limits of the ring as the redlines in Fig.1. In fact, as shown in Fig. 1, nearly 30 percent ofeigenvalues of measured data are within the inner circle pre-dicted by the randommatrix of i.i.d. entries. The phenomenonshows that the measured data of ARMA series do not obeythe M-P law. The reason is that the measured data of statusparameters do not follow i.i.d. assumptions, and there arestrong relationship among the parameters. So this discoverymotivates us to address this problem using the matrix ofARMA entries in the next section, to fit the measured databetter.

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B. LIMITING SPECTRAL DENSITY FUNCTION AND RINGLAW OF LARGE-DIMENSIONAL SAMPLE COVARIANCEMATRIX OF ARMA MODELAccording to the measured data, we assume column variablext is ARMA(p, q) process with φ(B)xt = θ (B)εt , and φ(B) =1− ϕ1B− . . .− ϕpBp, θ (B) = 1+ θ1B+ . . .+ θqBq, whereB is the delay operator with Bjxt = xt−j, εt ⊂ N (0, σ 2). Thepower spectral density function of an ARMA(p, q) process isdefined as [4]:

8(ω) =σ 2

2π|θ (e−iω)|2

|φ(e−iω)|2−π ≤ ω ≤ π (5)

Take the AR(1) model as example [32], [33] so that assumingeach column variable ofX is AR(1) process. The k-th columnvariable of X is xk = (x1, x2, . . . , xp), and its AR(1) processis assumed as

xt = ϕxt−1 + εt , t = 1, 2, . . . , p,εt ⊂ N (0, σ 2) (6)

where ϕ ∈ (−1, 1). According to (5) and (6), the powerspectral density function of an AR(1) process is defined as

8(ω) =12π

(σ 2

1+ ϕ2 − 2ϕ cosω)−π ≤ ω ≤ π (7)

The sample covariance matrix of X is Sn = 1n

n∑i=1

xixi =

1nXX

H . The empirical spectral density function [4] of Sn is

FSn (x) =1p

p∑t=1

I (λt < x), (8)

where I (·) is the indicator function, λ1, λ2, · · · , λp are theeigenvalues of Sn, t ∈ (1, p). Only if p → ∞, n → ∞ andp/n = c, the empirical spectral density function tends to bea non-random probability distribution F(x), which representsthe limit spectral distribution function.Moreover, the Stieltjestransform s(z) of F(x) (as a mapping from C+ into C+)satisfies the equation [4], [33]

s(z) = sFSn =

∫1

x − zdFSn (x) =

1ptr(Sn − zI)−1 (9)

where I is the unitary matrix. In the same way, the ESD ofS̄ = 1

nX′X is F S̄n (x) and the Stieltjes transform is s̄(z) = sF S̄n

by (9). So the relationship of s(z) and s̄(z) can be indicated asfollows [37].

s(z) = s̄(z)/c+ (1− c)/(cz) (10)

According to the association between ESD and limit spec-tral distribution function of Sn [33], [37] shown by (11),the reverse s-transform function of z satisfies the equa-tion (12). The s(z) represent the s-transform of ESD of Sn andg(z) represents the reverse s-transform of the limit spectraldistribution function of Sn.

z = −1s(z)+

12π

∫1

cs(z)+ |2π8(ω)|−1dω (11)

g(z) =12π

∫1

cs(z)+ (2π8(ω))−1dω (12)

The limit spectral density function (LSD) is shown by (13)when the imaginary part of z turns to 0 (ν → 0).

fX(x) =1π

limν→0

Im(g(z)) (13)

The LSD of sample covariance matrix of large dimen-sional matrix X based on AR(1) model is indicated by (14)[38], [39], according to the combination of equations(7),(10),(12) and equation (13).

fX(x) =

12πx

√R+ 2r + 2

√w2

R x ∈ �

0 otherwise(14)

where R, r,w, � is related to parameters ϕ, σ, c and thedetailed expressions are shown in Appendix to be morereadable.

The ring law can be deduced by the LSD. Considering thatthe data in status parameter matricesXp×n in Section 2 are allreal, the sample covariance matrix ofX is replaced by the sin-gular value equivalent Xu = U

√XXH . U denotes the Haar-

unitary matrix and the eigenvalue distribution of Xu spreadsin the complex plane. Consider the non-Hermitian randommatrix X with eigenvalue decomposition Xu = Un3nVn,where 3n = diag(s1, s2, . . . , sn), Un and Vn denote therandom Haar-unitary matrix independent with 3. The ESDof Xu converges to the interval as (15) [4].

{z ∈ C : a ≤ |z| ≤ b} (15)

where a = (∫x−2v(dx))−1/2 and b = (

∫x2v(dx))−1/2. Take

the large dimensional matrix of AR(1) model as example,the s-transform s(z) of limit spectral distribution functionand the LSD fX(x) of Xu can be calculated by (9) and (15).So the eigenvalues of singular equivalent matrix Xu convergeto the limit spectral distribution function which means thatthe distribution of eigenvalues in the complex plane is closeto a ring circle with the inner radius a and outer radius b.

C. DATA ANOMALY DETECTION USING RING CIRCLE ANDRECONSTRUCTION CURVETheKPCA (Kernel Principal Component Analysis)method iswidely used in image processing [34]–[36]. The eigenvaluesof Xu are spread in a ring circle as known in Section B. Theradius of circle can be calculated from the complex modelparameters ϕ, σ, c. But in the real application, lots of timeshould be taken to calculate these parameters only to get theaccurate radius which shows less necessary. Thus, we use thisKPCAmethod to obtain a circle that will lie within the singlering of the scattered eigenvalues and we can quickly get theeffective distance of eigenvalues to the origin (similar to theaverage distance but more accurate).Let Y2×M = {y1, . . . , yM } denote a set of data and{y1, . . . , yM } is the 2-dimensional array of the real part andimaginary part of eigenvalues ofXu. Through theKernel PCAmethod, the data are mapped into feature space F by a non-linear function φ. Define the kernel matrix K whose entriesare Ki,j = k(yi, yj) = φ(yi)Tφ(yj). One popular kernel

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function is a Gaussian function k(yi, yj) = exp(− |yi−yj|2

d ),where d is the kernel parameter. Through eigen decomposi-tion of K, λ1, λ2, . . . , λk are the nonzero eigenvalues of Kand υ1, υ2, . . . , υk are the eigenvectors. For each y, its h-th

principal component can be calculated as th =M∑i=1υhi k(yi, y).

Then the image of y from its projections onto the firstH (≤ h)principal components in space F can be reconstructed by theprojection function [28]:

PHφ(y) =H∑h=1

thvh =H∑h=1

th(1/√λhM )φ(Y)υh (16)

Hence, for the data {y1, . . . , yM }, the reconstruction error is

ρ =M∑i=1‖ PHφ(yi)− φ(yi) ‖. Denote w as the approximate

reconstruction of data y, the gradient of ρ should be 0 as

1ρ =M∑i=1βik(w, yi)(w − yi) = 0. The w can be calculated

by (17), where βi =H∑h=1

thυhi . Finally, we obtain

w =M∑i=1

βik(w, yi)yi/M∑i=1

βik(w, yi) (17)

So the KPCA reconstruction curve can be obtained by W ={w1,w2, . . . ,wH }. The reciprocal of the minimum distancefrom the curve to the origin is defined as the scatter distanceC = min(‖ wi ‖), where i = 1, 2, . . . ,H . And Chistory is thehistorical scatter distance calculated based on the historicaldata. The deviation percentage of the historical density isdefined as the key state evaluation value P

P = 1− C/Chistory (18)

The key statematrix is judged as anomalouswhen P is smallerthan the threshold. The anomaly state and anomaly timepoint are calculated according to the normality of the residualsequence matrix using the method in Section 3.C.

D. MODEL VALIDATIONMeasured data for loads at two different operations are usedfor model validation. As shown in Fig.2, the real spectraldistribution is consistent with the theoretical spectral dis-tribution. The inner circle together with the KPCA curvechanges according to different parameters of AR(1) model.When the load data is more dynamic, ϕ and σ will be smaller.This phenomenon can be used to distinguish measured dataof transformers under different states. Without knowing thedetailed coefficients of times series or calculating the theoret-ical inner radius, just through the KPCA scattering distanceof eigenvalues, we can detect the difference. The applicationsteps are shown in next section.

IV. MODEL APPLICATIONS FOR KEY STATE EVALUATIONAND EARLY WARNINGThe correlation between the temporal part and the spatial partof state variables should be considered in the key states of

FIGURE 2. Model Validation of measured data Measured data ofload A and B (average value removed) according to (6). Load A: ϕ = 0.82,σ = 0.60 Load B: ϕ = 0.58, σ = 0.32 Load A is more dynamic than Bwhich makes the smaller inner radius and KPCA scatter distance.

FIGURE 3. Steps of key state evaluation.

transmission equipment. The procedures are shown in Fig 3.Each blockWeekn,. . . ,Weekn−1 is a large dimensional matrixcomposed of state variables and corresponds to Table I. Eachblock represents a key state matrix of one week.

According to Fig. 3, the steps of key state evaluation are asfollows:

1) After data collection and preprocessing, large dimen-sional matrices of different state variables are acquiredand combined to form the key state matrices X1 ∼ X5according to Table I.

2) The eigenvalues and eigenvectors of sample covariancematrices of these key state matrices weekly are com-puted. The corresponding rings and spectral distribu-tions are calculated.

3) The rings and spectral distributions of the current dataand historical data are compared using the approach inSection 3.A. The key state evaluation value P is calcu-lated. The key state matrix is judged to be anomalouswhen P is larger than the threshold.

4) If the key state matrix is anomalous, the large dimen-sional matrix of residual sequence is constructed byfitting the anomalous matrix to the AMRA model.

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FIGURE 4. Comparison of the rings and spectral functions ofWeek I and II. Week II’s spectral distribution function is narrower thanthat of Week I. The reconstruction curve of KPCA shows that the scatterdensity of Week I is C1 = 0.32, far smaller than the scatter density ofWeek II C2 = 0.7.

V. CASE STUDIESTo verify the anomaly detection method in this article,the load data, on-line monitoring data, and meteorologicaldata of a certain period of time at a substation are studied.

A. CASE STUDY 1Choose all the monitoring data of transformers of one substa-tion from June to July in 2012. Both the load and the windingtemperature are changing in a larger scale in Week II (fromJune 15 to June 21) than Week I (from June 8 to June 14)and Week III (from June 22 to June 28). But the data ofambient temperature in the three weeks are nearly the same.To indentify the key state load capacity of the transformers inthe substation, we construct three matricesXload1, . . . ,Xload3denoting the key state matrix of Week I-III according to themethod in Section II. Then the eigenvalues of the samplecovariance matrices, the corresponding rings and the spectraldistributions are calculated. The result is shown in Fig. 4.

According to the scatter density of Week II, the key stateload capacity evaluation value Pload2 = 0.55, less thanthe threshold 0.8.The result indicates that the key state loadcapacity is abnormal and the abnormal status parameters needto be detected. Fitting the ARMA model with the week II’swinding temperature and load data, high dimensional matri-cesXres and its covariance matricesRres are constructed fromthe residual sequences. According to Section 3.C, the resultof normality testing of the corresponding eigenvectors ofRresis shown in Fig. 5.

In the same way, the spectral distribution and ring cir-cles of key state insulation capacity, mechanical capacity areobtained. But there is no anomaly detected in these key stateswhich means only load capacity is abnormal. The above anal-ysis shows that the abnormal key state load capacity may becaused by the increased oil temperature which is influencedby the load. It means that the anomaly on oil temperatureis caused by load, and the transformer does not necessar-ily have overheating defect. We can make conclusions thatearly warning is not necessarily and we only need to payclose attention to the operation situation of the transformer.

FIGURE 5. The eigenvector corresponding to the maximum eigenvalueand its normality test. The load is abnormal on time t = 376(June 28, 2012), and the winding temperature is abnormal on t = 380.

FIGURE 6. Density of points in circle ring. The scatter distance ofinsulation capacity (overheating) and insulation capacity (discharge)reduces slowly and the ring of insulation moisture, mechanical propertiesis almost unchanged.

As shown in operation records of the substation, the actualsituation was that the transformer ran above the nameplaterating with planned overload factor 1.1∼1.2 in June 28, dueto the dispatching needs, and it returned to normal load threehours later. This situation shows that the transformer wasindeed running under abnormal state in a period of time withno overheat defect and then returned to normal operationsituation, which is the same as the above conclusions.

B. CASE STUDY 2Lightning happens frequently from April to August in thearea around the transmission lines which are connected withthe substation. The data of oil gas(CO, CO2, H2, CH4), oiltemperature and partial discharge from April to June in 2013are used to evaluate the state of substation. The data of H2and CH4 increase horizontally and the rising rates of CO/CO2speed up quickly in June 16. Additionally, oil temperatureand partial discharge are obviously higher than normal valuefrom June 14 to June 20. The key state matrices of equipmentare constructed using all the data of 12 weeks from April toJune. The scattered point intensity of each weekąŕs circlering is calculated as shown in Fig. 6. The slow change ofthe ring characteristics shows that the insulation capacity

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FIGURE 7. Comparison of circle ring representing insulation of dischargeand overheat. The inner circle of insulation capacity (overheating) circlering in the week from June 14 to June 20 increases significantly, whichshows that the insulation capacity (overheating) degrades significantly inthe week. The insulation capacity (discharge) also degrades significantly.

TABLE 3. The abnormal parameters related to abnormal key states.

(overheating) and the insulation capacity (discharge) aregradually degraded.

Using the data of Week I (from April 2 to April 8),Week II(from June 7 to June 13) andWeek III(from June 14 toJune 20), the circle ring of insulation capacity (overheating)and insulation capacity (discharge) are constructed in Fig.7.

These steps show that serious anomalies appear in the keystates of equipments from June 14 to June 20. It is necessaryto detect the anomaly state variables and anomaly time pointin the key state matrices. The result is shown in Tab. III.

According to Tab. III, there was an anomalous oil tem-perature in the transformer in June 15, which made thetransformer overheating that last several days. Significantanomalies appeared in the data of gases such as H2, CO, CO2from June to August. The anomaly of H2 shows that thereis a slight discharge phenomenon. The anomaly of CO andCO2 shows that there exists deterioration in solid insulation.According to above steps, key state evaluation results areshowed in Fig. 8. We can make conclusions as follows:A latent fault is highly likely to exist in the transformer dueto the internal deterioration in insulation capacity (overheat-ing) and insulation capacity (discharge) of the transformer,which is aggravated in mid June. The insulation deteriorationis caused by slight discharge and involved solid insulationjudged by the state variables of deterioration. Thus, we shouldclosely track the changes in the chromatographic and arrangea power-off test timely.

Information can be acquired as follows by the operationrecords of the substation and off-line test reports: (1) Therewas a phase to earth fault caused by lightning in the linestwo thousand kilometers away from the station. There was

FIGURE 8. Comparison of key state evaluation results.

a successful coincidence brake after 0.3 seconds. (2) Slightdischarge phenomenon was found in the transformer in thepartial discharge off-power test in July 2. These recordsshowed that the phase to earth fault near the substation led toa short term large current impulse of the transformer, whichwas quite likely to cause the internal insulation deteriorationof the transformer and a discharge happened. The recordsshowed that the conclusions of the anomaly detection arebasically the same as the actual situation.

The above two examples show that the anomaly detectionusing large dimensional matrices is valid and feasible. Theevaluation conclusion is the same as the actual situation.Compared with the traditional threshold comparison method,the method using large dimensional matrices combines therelation between the temporal and spatial parts of the statevariables and mines the latent developing trend of the datawith high accuracy.

VI. CONCLUSIONThe measured data for power transformers have strong sta-tistical correlations. This fact demands a fundamentally dif-ferent analysis from that of our previous papers that arebased on i.i.d. assumptions. It is well known that correlatedrandom variables are much harder to deal with than i.i.d.ones. This difficulty deserves a special note in the contextof high-dimensionality in our problem at hand. We deal withmatrix-valued random variables which are noncommutativerandom variables. The topic is very difficult in mathematicalliterature. We are lucky to find that experimental results arein agreement with the theoretical predictions. In particular,the limiting spectrum distribution of a random matrix withARMA entries are compared with the empirical spectral dis-tribution, to enable the anomaly detection of the transformerstate. A data model is proposed for the status data of powerequipment, solving problems such as diversity, heterogeneity,and sampling synchronization in data sources. To our bestknowledge, this paper represents the first attempt to userandommatrix theory for power equipment monitoring. Also,from the systematical development of our framework, thiswork is remarkable in many ways. As pointed out above,

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now we deal with large random matrices whose entries arecorrelated random variables, rather than much simpler i.i.d.cases.

In the future, more data collections are needed to validatethis new framework. Future challenges include the limitedaccess of power monitoring devices, lack of standard pro-cedures, and real-time evaluation. Transmission lines andgas insulation systems can be studied, as we have done fortransformers in this paper. On the other hand, from theoreticalpoint of view, we can study more complicated functionsof random matrices. More accurate finite-size results fromrandom matrix theory can be used. Also, free probability isrelevant in this context by studying the polynomial of many(matrix-valued) random variables [4], [11].

APPENDIXThe formula in Appendix is the explanation for the equation(13) in Section 3.A.

u = max((1+ ϕ)2/σ 2, (1− ϕ)2/σ 2) (19)

v = min((1+ ϕ)2/σ 2, (1− ϕ)2/σ 2) (20)

r = −1− (3(1− ux)2 + 3(1− vx)2

− 2(1− ux)(1− vx))/(8c2) (21)

w = (2− (u+ v)x)(4y2 + (u− v)2x2)/(8c3)− 2/c (22)

e = −(2− (u+ v)x)2(3(u− v)2x2 − 4(1− ux)(1− vx)

+ 16c2)/(256c4)− (u+ v)x/(2c2) (23)

� = {x : (1−√c)2/u ≤ x ≤ (1+

√c)2/v,

×(2p3 + 27q2 − 72pr)2 > 4(p2 + 12r)3} (24)

R = −√(4(−r2 − 12e)3+(2r3+27w2 − 72re)2))1/3/1.26

+ 2r3 + 27w2− 72re+ (2r3 + 27w2

− 72re

+

√4(−r2 − 12e)3 + (2r3 + 27w2 − 72re)2)− 2r/3

(25)

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YINGJIE YAN received the B.E. degree fromShanghai Jiaotong University, Shanghai, China,in 2010, where he is currently pursuing the Ph.D.degree from the Department of Electrical Engi-neering. His current research interests includeelectric power equipment online monitoring, bigdata methods for fault diagnosis, and conditionevaluation of power transmission equipments.

GEHAO SHENG (M’08) was born in Hunan,China. He received the B.E., M.S., and Ph.D.degrees in electric power system and automationfrom the Huazhong University of Science andTechnology, Wuhan, China, in 1996, 1999, and2003, respectively. From 2003 to 2005, he was aPost-Doctoral Researcher with the Department ofElectrical Engineering, Shanghai Jiao Tong Uni-versity, Shanghai, China, where he is currently anAssociate Professor. His current research interests

include the condition monitoring of power apparatus.

ROBERT CAIMING QIU (S’93–M’96–SM’01–F’14) received the Ph.D. degree in electrical engi-neering fromNewYorkUniversity, NewYork, NY,USA. He was with Verizon, Waltham, MA, USA,and Bell Labs, Lucent, Whippany, NJ, USA. Hehas involved in wireless communications and net-work, machine learning, smart grid, digital signalprocessing, EM scattering, composite absorbingmaterials, RF microelectronics, UWB, underwateracoustics, and fiber optics. He was the Founder-

CEO and a President of Wiscom Technologies, Inc., manufacturing and mar-keting WCDMA chipsets. Wiscom was sold to Intel in 2003. He is currentlya Professor with the Center for Manufacturing Research, Department ofElectrical and Computer Engineering, Tennessee Technological University,Cookeville, TN,USA,where hewas anAssociate Professor in 2003 before hebecame a Professor in 2008. Since 2015, he has also been with the ResearchCenter for Big Data Engineering and Technologies, State Energy Smart GridCenter, Department of Electrical Engineering, Shanghai Jiaotong University.His current interest is in wireless communication and networking, machinelearning, and the smart grid technologies. He holds over six patents andauthored over 70 journal papers/book chapters and 90 conference papers.He has 15 contributions to 3GPP and the IEEE standards bodies. In 1988,he developed the first three courses on 3G for Bell Laboratories researchers.He served as an Adjunct Professor with Polytechnic University, Brooklyn,NY, USA. He serves as an Associate Editor for the IEEE TRANSACTIONS

ON VEHICULAR TECHNOLOGY and other international journals. He co-authoredthe books Cognitive Radio Communication and Networking and authoredthe book Big Data and Smart Grid. He is a Guest Editor for the bookUltra-WidebandWireless Communications and three special issues on UWB,including the IEEE JOURNAL ON SELECTEDAREAS INCOMMUNICATIONS, the IEEETRANSACTIONS ON VEHICULAR TECHNOLOGY, and the IEEE TRANSACTIONS ON

SMART GRID. He serves as a member of TPC for GLOBECOM, ICC, WCNC,MILCOM, and ICUWB. He served on the Advisory Board of the New JerseyCenter for Wireless Telecommunications. He is included in Marquis WhosWho in America.

XIUCHEN JIANG was born in Shandong, China.He received the B.E. degree in high voltageand insulation technology from Shanghai JiaotongUniversity, Shanghai, China, in 1987, the M.S.degree in high voltage and insulation technologyfrom Tsinghua University in 1992, and the Ph.D.degree in electric power system and automationfrom Shanghai Jiaotong University in 2001.He iscurrently a Professor with the Department of Elec-trical Engineering, Shanghai Jiaotong University.

His research interests include electrical equipment on-line monitoring, andcondition-based maintenance and automation.

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