Inferring Power System Frequency Oscillations using Gaussian Processes

Mana Jalali, Vassilis Kekatos, Siddharth Bhela, and Hao Zhu

Abstract— Synchronized data provide unprecedented oppor-tunities for inferring voltage frequencies and rates of changeof frequencies (ROCOFs) across the buses of a power system.Aligned to this goal, this work puts forth a novel framework forlearning dynamics after small-signal disturbances by leveragingthe tool of Gaussian processes (GPs). We extend results oninferring the input and output of a linear time-invariantsystem using GPs to the multi-input multi-output setup byexploiting power system swing dynamics. This physics-awarelearning technique captures time derivatives in continuous time,accommodates data streams sampled potentially at differentrates, and can cope with missing data and heterogeneous levelsof accuracy. While Kalman filter-based approaches requireknowing all system inputs, the proposed framework handlesreadings of system inputs, outputs, their derivatives, andcombinations thereof on an arbitrary subset of buses. Relyingon minimal system information, it further provides uncertaintyquantification in addition to point estimates for dynamic gridsignals. The required spatiotemporal covariances are obtainedby exploring the statistical properties of approximate swingdynamics driven by ambient disturbances. Numerical testsverify that this technique can infer frequencies and ROCOFsat non-metered buses under (non)-ambient disturbances for alinearized dynamic model of the IEEE 300-bus benchmark.

I. INTRODUCTION

Maintaining the stability of power systems requires know-ing frequency oscillations across the grid. Due to the highpenetration of distributed and renewable energy resources,power systems may experience increased oscillations. On theother hand, modern devices (such as the frequency distur-bance recorders (FDRs) comprising the frequency monitor-ing network (FNET) [1], [2]) can collect time-synchronizedpower grid measurements (frequency, voltage magnitude,voltage phase angle) across the grid and relay them over theInternet for processing on a centralized server. Since FDRsare installed at the distribution level, their voltage phasereadings are susceptible to local injection fluctuations andarbitrary shifts caused by transformers, yet frequency read-ings could help predict and control system instabilities [3].Reliable estimates of frequency excursions and their acceler-ations are also instrumental towards realizing load frequencycontrol [4]. Nonetheless, frequency measurements are notavailable at all buses, while the available data streams maybe sampled at different rates or even include missing entriesdue to communication failures, and computing ROCOFs astime differences is sensitive to measurement noise.

M. Jalali and V. Kekatos are with the Bradley Dept. of ECE,Virginia Tech, VA, USA. Emails: {manaj2,kekatos}@vt.edu.S. Bhela is with Siemens Technology, Princeton, NJ, USA. Email:siddharth.bhela@siemens.com. H. Zhu is with the Dept. of ECE,the University of Texas at Austin, Austin, TX 78712, USA. Email:haozhu@utexas.edu. The work was supported by the US NSF grants1751085 and 1923221.

The existing literature on inferring power system dynamicscan be broadly classified into data- and model-based meth-ods. Data-based methods typically engage synchrophasormeasurements to learn the system’s dynamic states. Forexample, missing entries from PMU data streams can berecovered via the low-rank plus sparse decomposition of thePMU data matrix pursued in [5]. If all PMU data is lost forone or more consecutive times, a robust matrix completionapproach stacking data in a Hankel matrix shows promiseto estimate the lost signal [6], [7]. Arranging synchrophasorreadings in higher-order tensors rather than matrices couldpotentially impute data over prolonged communication fail-ures via tensor factorization [8]. Nonetheless, data-basedtechniques cannot extrapolate on non-metered buses andignore any information on the dynamic system model.

Dynamic state estimation (DSE) aims at inferring thepower system states using both a system model and mea-surements processed typically through a Kalman filter (KF);see [9] for a comprehensive review. Plain KFs are optimalestimators that adopt a linear system model, while nonlinearpower system dynamics can be handled through KF variants,such as the extended [10], [11]; the unscented [12], [13];and ensemble KFs [14], [15]. By and large, KF-based DSEsolutions operate on a localized fashion assuming eachbus is connected to an infinite bus, and thus ignore thedynamic correlation among generators and loads [16], [17].KFs operate on data collected over uniformly sampled timeintervals, which renders them vulnerable to missing data.DSE approaches approximate continuous differential equa-tions with discrete finite differences. Reference [18] suggestsa function basis expansion to predict the local frequencyunder again a localized single-bus state-space model usingexpectation maximization and prediction error minimizationalgorithms. The work in [19] uses physics-informed deepneural networks to find frequencies for a single-machineinfinite bus model.

Synchrophasor readings can be used to infer more coarsedynamical system information, such as locating the sourcesof oscillations. The latter task can be accomplished by com-paring the arrival time of traveling waves [20]; by measuringthe dissipated energy of power flows [21], [22]; or via robustprincipal component analysis [23]. Yet these methods requireall buses to be monitored.

In a nutshell, existing methods for inferring power systemoscillations cannot extrapolate to other buses, presume uni-form sampling rates, and cannot naturally deal with missingdata. In this paper, we aim at inferring the voltage frequenciesand acceleration across buses and time instances using thepowerful toolbox of GPs. We consider a centralized esti-

mation scheme in which system operators collect frequencyand/or ROCOF measurements on a subset of buses andtimes possibly at different sampling rates. Depending onwhich dynamic grid quantities are measured and which arewanted, one can envision various application scenarios, suchas a1) Estimating system frequencies and/or ROCOFs at non-metered buses to ensure stability; a2) Predicting frequenciesand ROCOFs at monitored buses; or a3) Imputing missingvalues from an FDR data stream.

The contribution of this work is twofold. First, introduceGP modeling to learning power system frequencies andROCOFs. This provides a comprehensive learning paradigmfor processing data streams with different sampling rates andmissing data, extrapolating to non-metered buses, obviatingthe need for approximating ROCOFs from finite differences,and providing a complete pdf description in addition to pointestimates. Second, extend the idea of GP modeling lineardynamical systems to the MIMO setting by exploring anapproximate homogeneous model for swing dynamics drivenby ambient disturbances and utilizing its statistical propertiesto build physically meaningful covariances for the GP model.

Notation: Column vectors (matrices) are denoted by lower-(upper-) case boldface letters. Operator dg(x) returns adiagonal matrix with x on its main diagonal. Symbol (·)>stands for transposition, matrix IN is the N × N identitymatrix, x = dx

dt denotes time differentiation, and E is theexpectation operator. Notation x ∼ N (µ,Σ) means x isdrawn from a multivariate Gaussian distribution with meanµ and covariance Σ.

II. PROBLEM STATEMENT AND PRELIMINARIES

Consider a power system with N buses comprising setN . The dynamic behavior of the power system is capturedby a set of nonlinear differential-algebraic equations. Un-der small-signal analysis and certain simplifications, powersystem dynamics can be approximated by a second-ordermulti-input multi-output (MIMO) linear time-invariant (LTI)system described by the swing equation [24, Ch. 3]

Mω + Dω + Lθ = p (1)

where vectors θ := [θ1 . . . θN ]>, ω = θ, and p :=[p1 . . . pN ]> collect respectively the deviations of nodalvoltage angles, voltage frequencies, and active power injec-tions from their nominal steady-state values. We henceforthdrop the term deviations for brevity. Vector ω collects therelated ROCOFs. Matrices M := dg({Mn}Nn=1) and D :=dg({Dn}Nn=1) carry the per-generator inertia and dampingcoefficients. Matrix L is the (reduced) negative Jacobianmatrix of active power injections with respect to voltage an-gles evaluated at the current operating point. Within standardapproximations, matrix L is assumed symmetric and positivesemi-definite (psd); see [25]. Note that the dynamic modelin (1) involves only buses hosting generators as load buseshave been eliminated via Kron reduction [26].

The task of inferring oscillations dealt with in this workcan be abstractly posed as: Knowing model (1) and givenmeasurements of frequencies and/or ROCOFs at different

buses and times, estimate frequencies and ROCOFs at non-metered buses and times. The proposed learning frameworkcan accommodate different power systems monitoring tasks.For example, one can infer oscillations at non-metered buses;whereas for metered buses, one can impute missing fre-quency data, predict future values, or provide more reliableestimates of ROCOFs from frequency measurements.

The proposed framework is built upon a physics-aware sta-tistical model for power system oscillations using Gaussianprocesses (GPs). Gaussian processes have previously beenadopted for learning in single-input single-output (SISO) LTIsystems [27], or differential equations evolving across timeand/or continuous spatial dimensions [28]. Leveraging theparticular properties of power systems oscillations, we putforth a GP-based Bayesian model for learning under theMIMO LTI system setup of (1). Before presenting our model,let us briefly review GPs.

A GP is a random process with the additional propertythat any collection of a finite number of its samples forms aGaussian random vector [29, Ch.1]. Take for instance a timeseries z(t) and two sets of time indices T1 and T2. If z(t) isa GP, the two vectors z1 and z2 collecting the samples overT1 and T2 are jointly Gaussian or

z =

[z1z2

]∼ N

([µ1

µ2

],

[Σ11 Σ>21Σ21 Σ22

]). (2)

Now suppose one may wish to estimate z2 given an imple-mentation of z1. From (2), the conditional pdf of z2 givenz1 is Gaussian with mean and covariance [30, Ch. 6.4]

E[z2|z1] = µ2 + Σ21Σ−111 (z1 − µ1) (3a)

Cov[z2|z1] = Σ22 −Σ21Σ−111 Σ>21. (3b)

Equation (3a) provides the minimum mean square error(MMSE) estimate of z2 given z1, while the covariancein (3b) quantifies its uncertainty. Judiciously selecting theparameters for the Gaussian distribution of z is obviouslyimportant. The mean vector is usually set to zero, whilethe covariance matrix takes the parametric form of a kernelfunction as E[z(t)z(t′)] = k(t, t′); see [30]. The Gaussiankernel k(t, t′) = e−β(t−t

′)2 for β > 0 is a popular choicefor learning processes z(t) that are known to be smooth.

Granted a kernel function, applying GP learning topower system oscillations is straight-forward: Stack all fre-quency/ROCOF measurements in z1, gather the unknownquantities in z2, and apply (3) to obtain point estimates andconfidence metrics. These steps are depicted in Figure 1. Thekey challenge here is in selecting meaningful covariances orequivalently kernel functions. In doing so, two observationsare in order: First, we seek spatiotemporal kernels as theprocesses at hand evolve both across time and the spatialdimension of finitely many buses. Second, one should exploitthe fact that the involved processes are related through time-differentiation and the swing dynamics. We pursue these twoobjectives by considering an approximate model for ambientdynamics as detailed next.

Bayesian inference

constructing

covariances

PMUFDR

detrending

and filtering

preprocessing inferenceinputs

point and

uncertainty

estimates

Fig. 1. Workflow for inferring frequencies/ROCOFs.

III. DESIGNING COVARIANCES UNDER HOMOGENEOUSAMBIENT DYNAMICS

To design the covariances required in modeling frequencyoscillations as GPs, we resort to the particular setting ofswing dynamics driven by ambient disturbances and overbuses with a homogeneous inertia-damping model. We com-mence with the latter by postulating the next assumption.

Assumption 1. The inertia and damping coefficients arehomogeneous across buses, i.e., D = γM for a given γ > 0.

This approximation is reasonable since such coefficientsscale roughly proportionally to the power rating of eachgenerator [31]. Assumption 1 has been frequently adoptedfor studying power system dynamics [32], [33]. From As-sumption 1, the MIMO LTI system of (1) decomposes into asystem of N separable SISO LTI eigen-systems. The novelidea here is to first model each SISO eigen-system usingGPs, and then linearly combine the eigen-states to arrive ata statistical model for the original MIMO system of (1).Starting with the decomposition, we paraphrase the nextclaim from [25] and [33] for completeness.

Proposition 1. Consider matrix LM := M−1/2LM−1/2

and its eigenvalue decomposition LM = VΛV>. UnderAssumption 1, the swing dynamics of (1) are equivalent to

y + γy + Λy = x (4)

under the state and input transformations

y = V>M1/2θ and x = V>M−1/2p. (5)

Proof. Matrix LM is symmetric psd since L is symmetricpsd. Therefore, its eigenvectors are orthonormal or V>V =VV> = IN . Moreover, its eigenvalues {λi}Ni=1, placed inthe main diagonal of Λ, satisfy 0 = λ1 < λ2 ≤ . . . ≤ λN .This is because L is known to be a weighted Laplacian ofthe power grid graph [26]. If the grid is connected, then Lhas a single zero eigenvalue and the remaining eigenvaluesare positive [32]. The claim carries over to LM due to thecongruence transformation LM = M−1/2LM−1/2. If D =γM, the equivalence of (1) with (4) follows by substitutingθ = M−1/2Vy and p = M1/2Vx provided by (5) in (1),and premultiplying by V>M−1/2.

Because Λ is diagonal, the MIMO LTI system of (4) isdecomposed into the N SISO eigen-systems

yi + γyi + λiyi = xi, i = 1, . . . , N (6)

where λi ≥ 0 is the i-th eigenvalue of LM . We term yian eigen-frequency or eigen-state and xi an eigen-input. Asdiscussed in [25], [34], and [33], the transfer function ofeigen-system i with respect to state yi(t) is

Hi(s) =s

s2 + γs+ λi. (7)

The first eigen-system corresponding to λ1 = 0 is first-order, and the remaining ones are second order. It is noteasy to verify they are all stable. Heed that if rather thanyi, we select yi as the state, the first eigen-system becomesmarginally unstable. This is an additional reason we chose(eigen)-frequencies as the system states.

To apply a GP model on eigen-states, we need to describethe spatiotemporal covariance E[yi(t+ τ)yj(t)] for all i andj ∈ N , and t + τ and t ∈ T . To do so, we focus onambient oscillations occurring when the swing equation iscontinuously driven by small-magnitude random variationsin generation and demand at all buses. Ambient oscillationsoccur under normal conditions and thus correspond to themajority of observed frequency data. They are typicallyattributed a white noise profile across time and buses withvariances scaling proportionally to generator ratings as de-scribed next; see [25] and references therein.

Assumption 2. Ambient disturbances are zero-mean randomprocesses with covariance E[p(t + τ)p>(t)] = εMδ(τ),where δ(τ) is the Dirac delta function and ε > 0 is a givenconstant.

Under this statistical model for ambient disturbances, wenext compute the spatiotemporal covariances of eigen-statesby adopting a result from [25]. It is worth noticing thatAssumption 2 can be waived as demonstrated in [35], whichgeneralizes the current work. Excluding λ1 = 0, we assumed4λi > γ for all i = 2, . . . , N , although over-damped eigen-systems can be handled similarly.

Proposition 2. Under Assumption 2, the covariance ofeigen-states is

E[y(t+ τ)y>(t)] = dg ({ki(τ)}) (8)

where k1(τ) = 12γ e−γ|τ | and for i = 2, . . . , N

ki(τ) = Bie−γ|τ |/2 cos(2πfi|τ |+ ψi) (9)

with fi =√|4λi − γ2|/(4π); Bi := ε

2

√1

(4πfi)2+ 1

γ2 ; and

ψi = arctan(

γ4πfi

).

Proof. Under Assumption 2, the covariance of the eigen-inputs can be easily derived as

E[x(t+ τ)x>(t)] = V>M−1/2E[p(t+ τ)p>(t)]M−1/2V

= εV>M−1/2MM−1/2Vδ(τ)

= εIδ(τ). (10)

Because the eigen-inputs are uncorrelated and the eigen-systems are decoupled, the eigen-states are uncorrelated witheach other. Moreover, since each eigen-system i is driven

by zero-mean white noise, its output yi(t) is wide-sensestationary with mean zero and covariance ki(τ) = E[yi(t+τ)yi(t)] that can be computed as the convolution

ki(τ) = ε hi(τ) ∗ hi(−τ) ∗ δ(τ).

Here hi(τ) is the impulse response of eigen-system i com-puted as the inverse Laplace transform of Hi(s) in (7)

hi(t) = Aie−γt/2 cos(2πfit+ ψi)u(t)

where Ai :=√1 + γ2

|4λi−γ2| and u(t) being the unit stepfunction. Interestingly, the convolution ε hi(τ) ∗ hi(−τ)yields (9) as shown in [25], [34].

Having captured the first- and second-order statistics ofeigen-states, we can now transition to the original spaceof power system frequencies via the transformation ω =M−1/2Vy. Frequency oscillations are apparently zero-meanand their covariance can be easily computed from (8) as

E[ω(t+ τ)ω>(t)] = M−1/2V dg ({ki(τ)})V>M−1/2.(11)

This covariance provides an explicit form for the kernelfunction k ((n, t1), (m, t2)) = E[ωn(t1)ωm(t2)], which canbe evaluated as the (m,n)-th entry of the matrix on the right-hand side (RHS) of (11) upon setting τ = t1− t2. Note thatthe kernel function ki exhibits shift-invariance in the sensethat ki(t1, t2) = ki(t1− t2). This is why we have oftentimesabused notation and denoted ki as taking a single-argument.The covariance of (11) applies to noiseless readings ofsystem frequencies. In practice, the collected frequency dataare corrupted by zero-mean Gaussian measurement noisewith variance σ2

n per bus. Then, the RHS of (11) shouldbe appended a dg({σ2

n}Nn=1)δ(τ) term.It should be emphasized that if ambient inputs are Gaus-

sian distributed, eigen-states and hence original states (fre-quencies) are Gaussian as well. Then GPs become the naturalchoice. A GP model however can still be used when swingdynamics are driven by non-Gaussian ambient disturbancesor non-ambient disturbances (e.g., deterministic inputs ofimpulse- or step-type caused by generator or line trips).1 Weelaborate on this in the next section, which reviews how GPsextend to the time derivatives of random processes.

IV. COVARIANCES UPON TIME DIFFERENTIATION

So far, we have presented a statistical model assumingthat both vector z1 of observations and vector z2 of wantedquantities in (2)–(3) involve only frequencies. An appealingproperty of GPs is that time differentiation of a GP yields aGP. Therefore, our GP model for frequencies can be easilymodified to yield a GP model for ROCOFs. This is importantas it enables including available ROCOF measurements in z1and/or estimating ROCOFs by having them in z2.

To explain this key feature of GPs, we adopt the alternativeinterpretation of GPs provided in [27]. One can postulate

1Our numerical tests considered ambient and non-ambient disturbances,both under a non-homogeneous ineertia/damping swing model.

that each eigenstate yi(t) can be expressed as a linearcombination of K fixed basis functions {φki (t)}Kk=1 as [30]

yi(t) =

K∑k=1

wki φki (t) = w>i φi(t), i = 1, . . . , N (12)

where φi(t) := [φ1i (t) · · · φKi (t)]> is a mapping from t toRK and wi is a vector of unknown coefficients. Following aBayesian approach, we further postulate that wi is randomwith prior distribution wi ∼ N (0, IK). We are interestedin the joint probability density function (pdf) for samples ofyi(t) collected at times T := {t1, . . . , tT }, and stacked inyi := [yi(t1) · · · yi(tT )]>. From (12), it follows that yi =Φiwi for the T × K matrix Φi := [φi(t1) · · · φi(tT )]>.Because wi is Gaussian, vector yi becomes Gaussian too.It is not hard to verify that yi ∼ N (0,Ki) where Ki =ΦiΦ

>i � 0. Since the latter holds for any collection of

sampling instances T , signal yi(t) is a GP. If we assumeE[wiw

>j ] = 0 for j 6= i and that matrix Ki is Toeplitz with

entries dictated by (9), then (12) agrees with the modelingof Section III. Observe also that its (`, k)-th is

[Ki]`,k = ki(t` − tk) = φ>i (t`)φi(tk) (13)

with the kernel function ki(t` − tk) defined in (9).If yi(t) is provided by (12), its time derivative is yi(t) =

w>i φi(t). And because wi is Gaussian, the random processyi(t) is also a GP. If vector yi collects the samples of yi(t)over T , its covariance matrix is

E[yiy>i ] = ΦiE[wiw>i ]Φ

>i = ΦiΦ

>i = Ki (14)

where Φi is the element-wise time derivative of Φi. Inter-estingly, matrix Ki := ΦiΦ

>i relates to Ki as

[Ki]`,k = φ>i (t`)φi(tk)

=∂2

∂t`∂tkφ>i (t`)φi(tk)

=∂2ki(t`, tk)

∂t`∂tk.

The first equality stems from the definition of Ki, the secondfrom the definition of time differentiation, and the third onebecause ki(t`, tk) = φ>i (t`)φi(tk). The cross-covariancesE[yiy>i ] can be computed likewise as E[yiy>i ] = ΦiΦ

>i =

Ki, whose (`, k)-th entry would be ∂ki(t`,tk)∂t`

.In a nutshell, eigen-ROCOFs can be modeled as zero-

mean GPs with covariances obtained by differentiating therelated kernel functions. Based on these GP models, it isstraightforward to arrive at a GP model for the originalROCOFs ω = M−1/2Vy similar to (11). Having modeledfrequencies and ROCOFs as GPs, we can utilize the Bayesianinference of (3) to estimate frequencies and ROCOFs at non-metered buses and times.

In some applications, one might be interested in a par-ticular frequency band of power system frequencies. Recallthat the term power system frequencies {ωn(t)}Nn=1 refers infact to the time-domain signal of interest here. For example,inter-area oscillations typically correspond to the frequency

Fig. 2. Frequency estimate at bus 36 under ambient disturbances.

content of ωn(t) in the range of 0.1-0.8Hz [24, Ch. 12]. Asindicated by (7), each eigen-system i exhibits either a low-pass or band-pass behavior around its resonance frequencyfi. If the focus lies on a particular band, the eigen-systemswith out-of-band resonant frequencies can be ignored and theobserved signals can be passed through bandpass filters. Inother words, filtered frequency measurements can be approx-imated as ω = M−1/2VDyD +n, where n is measurementand modeling noise; yD is the subvector of y collecting theeigen-states with in-band resonant frequencies; and matrixVD samples the associated eigenvectors (columns of V).To reduce the computational complexity of the proposedlearning scheme, matrix V should be replaced by VD

in (11). Another advantage of such filtering is that having toinfer fewer eigen-states means that learning can be conductedusing fewer metered buses.

V. NUMERICAL TESTS

The novel GP-based learning framework was evaluatedusing synthesized data on the IEEE 300-bus power systembenchmark, which was Kron-reduced to a 69-bus systemupon eliminating load buses [26]. Generator parameters wereobtained from [36], so that Assumption 1 on homogeneitywas actually not met. To build the postulated covariances,we need to know (M, γ,L). We chose γ = 1>D1

1>M1and

L was set to the B-matrix obtained from MATPOWER’sfunction makeJac. Although covariances were derived pre-suming homogeneous inertia-damping coefficients, powersystems dynamics data were generated using the non-homogeneous (M,D) parameters from the benchmark. Weconverted the swing equation to its state-space representationusing MATLAB’s ss function, and solved given input p(t)using the ode45 command. Dynamics were simulated ata time resolution of 1 ms, but sampled and processedat the reporting rate of 15 samples per second accordingto the IEEE C37.118.1-2011 standard [37]. Synthesizedfrequency readings were corrupted by zero-mean additivewhite Gaussian noise (AWGN) with standard deviation of0.01 rad/s to comply with the maximum frequency errorsallowed by [37]. Benchmark data and the developed codecan be found at https://github.com/manajalali/GP4GridDynamics.

Fig. 3. ROCOF estimate at bus 34 under ambient disturbances.

Fig. 4. Frequency estimate for bus 36 under ambient disturbances withmetered buses selected at random.

We first evaluated the proposed GP framework under am-bient disturbances with covariance E[p(t)p>(t)] = 0.01 ·M.Focusing on inter-area oscillations, we filtered frequencymeasurements using MATLAB’s filtfilt function. Thefilter was designed to maintain the K = 5 eigen-stateswith resonant frequencies fi’s falling within 0.1-0.8Hz. Weassumed frequencies were being collected at M = 20 outof the 69 buses. Metered buses were selected based on theplacement scheme proposed in [38], which aims at maxi-mizing the minimum eigenvalue of the posterior covariancematrix when estimating yD by row-sampling the linear-Gaussian model ω = M−1/2VDyD + n. The non-meteredfrequency deviations and ROCOFs were estimated using (3).The ±σ uncertainty interval was computed by taking thesquare root of the diagonal entries of (3b). Figure 2 and 3depict the estimation results respectively for the frequencyand ROCOF at a non-metered bus. The results confirmthat the GP framework can successfully infer oscillationsunder ambient disturbances even when Assumption 1 oninertia/damping homogeneity is waived. To study the effectof meter placement, we repeated the previous test by select-ing M = 20 metered buses at random. Figure 4 shows thefrequency estimation outcome. The results demonstrate thatthe sensor placement does not significantly affect the pointestimation of the GP paradigm. However, the uncertaintyinterval was increased.

We also analyzed the capability of the GP paradigm topredict system states. Unlike the previous tests, we did notfocus on inter-area oscillations but used the raw frequency

Fig. 5. Frequency prediction for bus 66 under ambient disturbance withrandomly sampled measurements.

Fig. 6. Estimated frequencies at buses 1 and 63 of the 69-bus system undera non-ambient disturbance (a fault near bus 1).

measurements from 60 randomly selected buses across thegrid. We used the frequencies observed from these busesduring 7 seconds to predict the frequencies at non-meteredbuses for the next 7 seconds. Figure 5 demonstrates howuncertainty increases as we advance further in the future.

In this final test, we evaluated the proposed scheme undernon-ambient dynamics, thus waiving both assumptions underwhich covariances were derived. To generate a non-ambientsetup, we simulated a generator trip on bus 1 using animpulse function. Further, we assumed that bus 1 at whichthe generator trip has occurred is not metered. Figure 6depicts the performance of the GP paradigm for estimatingfrequencies at buses 1 and 63. The results confirm thatalthough the proposed GP framework was formulated forambient disturbances, it applies also to other settings.

VI. CONCLUSIONS AND FUTURE WORK

A novel method for learning power system frequenciesand ROCOFs using GPs has been put forth. The requiredspatiotemporal covariances have been derived upon leverag-ing the statistical properties of ambient dynamics under anapproximate rendition of the swing equation. The proposedframework can accommodate different sampling rates, miss-ing data, extrapolate dynamics at non-metered buses, andestimates ROCOFs in a parametric fashion without resortingto finite differences. The point estimates are endowed witha complete pdf characterization, which can be instrumentalin detecting and identifying bad, erroneous, and/or attacked

data. Numerical tests on a non-homogeneous power systembenchmark corroborated the efficacy of the proposed schemein extrapolating and predicting frequencies and ROCOFsunder ambient and non-ambient disturbances. This work setsthe foundations for interesting research directions, includingreal-time implementations, parameter estimation, and modalanalysis. Utilizing the GP estimation paradigm to infervoltage angles and power deviations would be useful forfault identification. Finally, exploring more detailed gener-ator models and using other measurements, such as fieldvoltages, could improve observability and enhance frequencyestimation.

REFERENCES

[1] Y. Zhang, P. Markham, T. Xia, L. Chen, Y. Ye, Z. Wu, Z. Yuan,L. Wang, J. Bank, J. Burgett, R. W. Conners, and Y. Liu, “Wide-areafrequency monitoring network (FNET) architecture and applications,”IEEE Trans. Smart Grid, vol. 1, no. 2, pp. 159–167, Sep. 2010.

[2] Z. Zhong, C. Xu, B. J. Billian, L. Zhang, S. S. Tsai, R. W. Conners,V. A. Centeno, A. G. Phadke, and Y. Liu, “Power system frequencymonitoring network (FNET) implementation,” IEEE Trans. PowerSyst., vol. 20, no. 4, pp. 1914–1921, Nov. 2005.

[3] W. Yu, W. Yao, and Y. Liu, “Definition of system angle referencefor distribution level synchronized angle measurement applications,”IEEE Trans. Power Syst., vol. 34, no. 1, pp. 818–820, Oct. 2019.

[4] C. Zhao and S. H. Low, “Optimal decentralized primary frequencycontrol in power networks,” in Proc. IEEE Conf. on Decision andControl, Los Angeles, CA, Dec. 2014, pp. 2467–2473.

[5] P. Gao, M. Wang, S. G. Ghiocel, and J. H. Chow, “Modeless re-construction of missing synchrophasor measurements,” in Proc. IEEEPower & Energy Society General Meeting, National Harbor, MD, Jul.2014, pp. 1–5.

[6] S. Zhang and M. Wang, “Correction of corrupted columns throughfast robust Hankel matrix completion,” IEEE Trans. Signal Process.,vol. 67, no. 10, pp. 2580–2594, May 2019.

[7] S. Zhang, Y. Hao, M. Wang, and J. H. Chow, “Multichannel Hankelmatrix completion through nonconvex optimization,” IEEE J. Sel.Topics Signal Process., vol. 12, no. 4, pp. 617–632, Aug. 2018.

[8] D. Osipov and J. H. Chow, “PMU missing data recovery using tensordecomposition,” IEEE Trans. Power Syst., vol. 35, no. 6, pp. 4554–4563, May 2020.

[9] J. Zhao, A. Gomez-Exposito, M. Netto, L. Mili, A. Abur, V. Terzija,I. Kamwa, B. Pal, A. K. Singh, J. Qi, Z. Huang, and A. P. S.Meliopoulos, “Power system dynamic state estimation: Motivations,definitions, methodologies, and future work,” IEEE Trans. Power Syst.,vol. 34, no. 4, pp. 3188–3198, Jul. 2019.

[10] Zhenyu Huang, K. Schneider, and J. Nieplocha, “Feasibility studiesof applying Kalman filter techniques to power system dynamic stateestimation,” in Intl. Power Engineering Conf., Singapore, Dec. 2007,pp. 376–382.

[11] Kuang-Rong Shih and Shyh-Jier Huang, “Application of a robustalgorithm for dynamic state estimation of a power system,” IEEETrans. Power Syst., vol. 17, no. 1, pp. 141–147, Aug. 2002.

[12] S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinearestimation,” Proc. IEEE, vol. 92, no. 3, pp. 401–422, Nov. 2004.

[13] S. Wang, W. Gao, and A. P. S. Meliopoulos, “An alternative method forpower system dynamic state estimation based on unscented transform,”IEEE Trans. Power Syst., vol. 27, no. 2, pp. 942–950, May 2012.

[14] W. S. Rosenthal, A. M. Tartakovsky, and Z. Huang, “Ensemble Kalmanfilter for dynamic state estimation of power grids stochastically drivenby time-correlated mechanical input power,” IEEE Trans. Power Syst.,vol. 33, no. 4, pp. 3701–3710, Jul. 2018.

[15] N. Zhou, D. Meng, and S. Lu, “Estimation of the dynamic states ofsynchronous machines using an extended particle filter,” IEEE Trans.Power Syst., vol. 28, no. 4, pp. 4152–4161, Nov. 2013.

[16] A. K. Singh and B. C. Pal, “Decentralized dynamic state estimation inpower systems using unscented transformation,” IEEE Trans. PowerSyst., vol. 29, no. 2, pp. 794–804, Mar. 2014.

[17] J. Zhao and L. Mili, “Power system robust decentralized dynamic stateestimation based on multiple hypothesis testing,” IEEE Trans. PowerSyst., vol. 33, no. 4, pp. 4553–4562, Jul. 2018.

[18] J. Dong, X. Ma, S. M. Djouadi, H. Li, and Y. Liu, “Frequencyprediction of power systems in FNET based on state-space approachand uncertain basis functions,” IEEE Trans. Power Syst., vol. 29, no. 6,pp. 2602–2612, May 2014.

[19] G. S. Misyris, A. Venzke, and S. Chatzivasileiadis, “Physics-informedneural networks for power systems,” in Proc. IEEE Power & EnergySociety General Meeting, Montreal, Canada, Aug. 2020, pp. 1–5.

[20] P. N. Markham and Y. Liu, “Electromechanical speed map develop-ment using FNET/GridEye frequency measurements,” in Proc. IEEEPower & Energy Society General Meeting, National Harbor, MD, Jul.2014, pp. 1–5.

[21] X. Deng, D. Bian, D. Shi, W. Yao, Z. Jiang, and Y. Liu, “Line outagedetection and localization via synchrophasor measurement,” in Proc.IEEE Conf. on Innovative Smart Grid Technologies, Chengdu, China,May 2019, pp. 3373–3378.

[22] S. N. Nuthalapati, Power System Grid Operation Using SynchrophasorTechnology. Cham, Switzerland: Springer, 2019.

[23] T. Huang, N. M. Freris, P. R. Kumar, and L. Xie, “A synchrophasordata-driven method for forced oscillation localization under resonanceconditions,” IEEE Trans. Power Syst., pp. 1–1, Mar. 2020.

[24] P. Kundur, Power System Stability and Control. New York, NY:McGraw-Hill, 1994.

[25] P. Huynh, H. Zhu, Q. Chen, and A. E. Elbanna, “Data-driven esti-mation of frequency response from ambient synchrophasor measure-ments,” IEEE Trans. Power Syst., vol. 33, no. 6, pp. 6590–6599, Nov.2018.

[26] T. Ishizaki, A. Chakrabortty, and J. Imura, “Graph-theoretic analysisof power systems,” Proc. IEEE, vol. 106, no. 5, pp. 931–952, May2018.

[27] T. Graepel, “Solving noisy linear operator equations by Gaussianprocesses: Application to ordinary and partial differential equations,”in Intl. Conf. on Machine Learning, Washington, DC, Aug. 2003, p.234–241.

[28] M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Inferring solutionsof differential equations using noisy multi-fidelity data,” Journal ofComputational Physics, vol. 335, pp. 736–746, Apr. 2017.

[29] C. Rasmussen and C. Williams, Gaussian Processes for MachineLearning. Cambridge, MA: MIT Press, 2006.

[30] C. M. Bishop, Pattern Recognition and Machine Learning. NewYork, NY: Springer, 2006.

[31] B. K. Poolla, S. Bolognani, and F. Dorfler, “Optimal placement ofvirtual inertia in power grids,” IEEE Trans. Autom. Contr., vol. 62,no. 12, pp. 6209–6220, Dec. 2017.

[32] L. Guo, C. Zhao, and S. H. Low, “Graph Laplacian spectrum andprimary frequency regulation,” in Proc. IEEE Conf. on Decision andControl, Miami Beach, FL, Dec. 2018, pp. 158–165.

[33] F. Paganini and E. Mallada, “Global analysis of synchronizationperformance for power systems: Bridging the theory-practice gap,”IEEE Trans. Autom. Contr., vol. 65, no. 7, pp. 3007–3022, Sep. 2020.

[34] S. Liu, H. Zhu, and V. Kekatos, “A dynamic response recoveryframework using ambient synchrophasor data,” IEEE Trans. PowerSyst., Jul. 2021, (submitted). [Online]. Available: https://arxiv.org/abs/2104.05614

[35] M. Jalali, V. Kekatos, S. Bhela, H. Zhu, and V. Centeno, “Inferringpower system dynamics from synchrophasor data using Gaussianprocesses,” IEEE Trans. Power Syst., May 2021, (submitted). [Online].Available: https://arxiv.org/abs/2105.12608

[36] P. Demetriou, M. Asprou, J. Quiros-Tortos, and E. Kyriakides, “Dy-namic IEEE test systems for transient analysis,” IEEE Systems Journal,vol. 11, no. 4, pp. 2108–2117, Jul. 2017.

[37] IEEE C37.118.1-2011 Std. for Synchrophasor Measurements forPower Systems, IEEE Std., 2011.

[38] R. Ramakrishna and A. Scaglione, “Grid-graph signal processing(Grid-GSP): A graph signal processing framework for the power grid,”IEEE Trans. Signal Process., vol. 69, pp. 2725–2739, 2021.