Online Learning of Power Transmission Dynamics

Transcription

1 Online Learning of Power Transmission Dynamics ndrey Y. Lokhov, Marc Vuffray, Dmiry Shemeov, Deepjyoi Deka and Michael Cherkov Theoreical Division, Los lamos Naional Laboraory, Los lamos, NM Cener for Nonlinear Sudies, Los lamos Naional Laboraory, Los lamos, NM Complexiy Sciences Cener, Universiy of California a Davis, Davis, C Skolkovo Insiue of Science and Technology, Moscow, Russia arxiv: v1 [cs.sy] 27 Oc 2017 bsrac We consider he problem of reconsrucing he dynamic sae marix of ransmission power grids from imesamped PMU measuremens in he regime of ambien flucuaions. Using a maximum likelihood based approach, we consruc a family of convex esimaors ha adap o he srucure of he problem depending on he available prior informaion. The proposed mehod is fully daa-driven and does no assume any knowledge of sysem parameers. I can be implemened in near real-ime and requires a small amoun of daa. Our learning algorihms can be used for model validaion and calibraion, and can also be applied o relaed problems of sysem sabiliy, deecion of forced oscillaions, generaion re-dispach, as well as o he esimaion of he sysem sae. Index Terms Transmission grid dynamics, Swing equaions, Parameer learning, Phasor measuremen unis, Reconsrucion algorihm, Synchronous measuremens I. INTRODUCTION Ensuring sable, secure and reliable operaions of he power grid is a primary concern for sysem operaors [1]. Securiy assessmen and conrol acions heavily rely on he accuracy of he assumed power sysem model and is parameers and of he esimaed sae [2]. Thus, inaccuracies in sae esimaion daa or in he neworked dynamic model can impac he assessmen of he sysem sabiliy and he efficacy of he corresponding conrol measures. In his paper, we explore he possibiliy o leverage he proliferaion of Phasor Measuremen Unis (PMUs) ha collec ime synchronous daa in a disribued way, for validaing he assumed power sysem model and he curren sysem sae. In paricular, our goal is o develop a daa-efficien learning framework for performing an online reconsrucion of he dynamic model using he minimal number of assumpions and exclusively relying on he PMU measuremens. number of recen works showed promising resuls in aacking his problem [3], [4], [5], [6], [7], [8], [9]. Here, we propose o exend he scope of exising works o he problem of exracing he dynamic sae marix from PMU measuremens in a purely daa-driven way, wihou assuming any knowledge of model parameers. We ake advanage of he separaion of scales ha exiss in he regime of ambien flucuaions around he seady sae leading o power sysem dynamics excied by sochasic load variaions. Under quie The work was suppored by funding from he U.S. DOE/OE as par of he DOE Grid Modernizaion Iniiaive. general and widely acceped assumpions in his ambien regime, we develop a provably consisen maximum likelihood based mehod ha recovers he dynamic sae marix wih a low number of observaions. Imporanly, he proposed mehodology can be naurally exended o cases of unknown nework opology and parial observaions, and has a low compuaional complexiy which is conducive for real-ime esimaions. n accurae esimaion of he dynamic sae marix has a large number of applicaions ha have been well explored in he lieraure [10], [11], including model validaion and parameer calibraion [3], [4], probing he proximiy o insabiliy and helping in design of he corresponding emergency conrol acions [12], [13], opimizaion and resource allocaion [14], [15], as well as idenificaion and analysis of forced oscillaions in he sysem [16]. The poenial abiliy o use he learned dynamic parameers o simulaneously perform a purely measuremen-based sae esimaion of deviaions in power consumpion from nominal values represens anoher aracive feaure of our framework. validaed sae esimaion can improve resource allocaion for generaion re-dispach. The paper is organized as follows: in Secion II we formulae he model and he reconsrucion problem; in Secion III we sae our learning mehod and discuss he convergence properies of he proposed algorihm; in Secion IV we illusrae our approach on a es sysem, and provide an empirical assessmen of he performance of our algorihms; finally, in Secion V we discuss possible exensions of our mehod and sae some open problems. II. PROBLEM FORMULTION We model he power nework by a graph G = (V, E) wih a se of N nodes (buses) V and a se of edges (ransmission lines) E V V. We consider he regime of ambien oscillaions around he seady sae ha is governed by he dynamics of generaor angles. I is common o model ambien dynamics wih a classical equivalen model of aggregaed generaors [17] ha corresponds o a nework-reduced power sysem where passive loads are eliminaed via Kron reducion [18]. lhough his modeling choice is no necessary for our analysis, i faciliaes a uniform mahemaical descripion where we can assume ha every node i in V essenially corresponds o a generaor wih non-zero ineria M i and

2 damping D i coefficiens wih emporal evoluion governed by he swing equaions [1]: M i θi + D i ( θ i ω (0) ) = P (m) i P (e) i, (1) where ω (0) is he synchronous frequency (60 Hz in U.S..); θ i and θ i (= θ i / ) respecively correspond o he generaor roor angles and speeds; P (m) i is he ne power injecion (e.g. he generaor mechanical power inpu); and P (e) i is he elecrical power oupu. P (e) i can be furher expressed as a sum of power flows ou of node i: P (e) i = (ij) E P ij, wih P ij = V i V j (g ij cos(θ i θ j ) + b ij sin(θ i θ j )), (2) where conducance g ij > 0 and suscepance b ij > 0 correspond o he real and imaginary pars of he complex admiance y ij = g ij ĵb ij (ĵ 2 = ) associaed wih each line (i, j) E in he nework. In he viciniy of he synchronous sae, he difference of roor angles is ypically small, so ha θ i θ j is close o zero for every pair (i, j) E. Therefore, in he regime of moderae ambien flucuaions i is sandard [1] o linearize he expression (2) around he curren operaing poin using cos(θ i θ j ) 1 and sin(θ i θ j ) (θ i θ j ). Over he period of ime where he volage magniudes can be approximaely considered as consan, line admiances are characerized by effecive suscepances β ij = V i V j b ij and conducances γ ij = V i V j g ij, ha absorb consan volage magniudes by definiion. Given hese simplificaions due o DC linearizaion, we assume ha he following relaion is valid in expecaion in he seady sae regime: EP (m) i = EP (e) i = ( ) γ ij + β ij (θ (0) i θ (0) j ), (3) where θ (0) i (ij) E denoe he mean seady sae values of roor angles, and deviaions of phase from hese values δ i = θ i θ (0) i are small. Noe ha conducances g ij are ypically negligible for power ransmission lines and hence γ ij is usually omied in he expression (3) under he assumpion of purely inducive lines [1]. Finally, he resuling dynamic model ha we consider in his paper akes he following form: M i ω i + D i ω i = β ij (δ i δ j ) + δp i, (4) (i,j) E where δp i = P (m) i P (e) i represens he effec of exogenous power deviaions, and ω i = δ i denoes he relaive generaor roor speed measured wih respec o he reference synchronous frequency ω (0). From (4), equaions for he whole sysem can be wrien in he marix form as [ δ ω ] [ 0N N I = N N M L M D ] [ [ δ + ω] 0 M δp ], (5) where y = [y 1,..., y N ] T denoes a generic N-componen vecor and y can refer o θ, θ, ω, ω and δp ; 0 is a N- dimensional vecor; and 0 N N and I N N denoe N N zero and ideniy marices, respecively. L is he suscepanceweighed Laplacian marix defined as L ij = β ij for (ij) E; L ii = (ik) E β ik; and L ij = 0 oherwise. Finally, M and D respecively represen diagonal ineria and damping marices paramerized by M i D i. For compacness, le us rewrie he sysem (5) as Ẋ = d X + ξ, (6) where X is a shorhand noaion for he sysem sae vecor [δ, ω] a ime, and ξ is he δp dependen vecor. cycle (60Hz) Learning βij= Vi Vj bij Mi, Di minues 10xminues hours Figure 1. Separaion of scales in our esimaion problem in he seing of ambien flucuaions. Learning procedure should be sample-efficien and only use he amoun of daa observed during he ime period which is no longer han minues or ens of minues, he ypical scale over which d is sable. In his paper, our goal is o esimae he dynamic sae marix d from PMU daa providing ime series measuremens of dynamic variables X. In principle, d can be compued if all parameers enering (1) are known. s moivaed above, here we deliberaely pursue a purely daa-driven approach ha will provide a characerizaion of he sysem dynamic based on he soluion of he inverse problem using he observed ime series, and hence would allow o validae he assumed dynamic model. However, he dynamic sae marix can be assumed o be consan only a ime scales for which he parameers enering (5) remain unchanged. In he ambien flucuaions seing, here exiss a naural separaion of ime scales in parameer variaions, see Figure 1. The parameers of generaors, such as ineria M i or damping coefficiens D i can be regarded as sable on he scale of several hours, wih a poenial slow drif due o droop or local feedback conrol [2]. lhough infrequen changes in he sysem may provoke a change in V i, overall volage magniudes may be considered sable on he ime inerval of he order of ens of minues [1]. The same conclusion holds for he values b ij ha may flucuae due o variaions in emperaures ha rarely happen on shorer scales compared o ens of minues. On he oher hand, we assume ha ambien flucuaions hemselves are caused by flucuaions of loads Pi m and Pi e around base nominal values or generaor noise [9], and herefore occur a very shor scales of he order of he frequency cycle of 60 Hz. Because of his reason and due o he aggregaed naure of loads, he oal power deviaions δp i are commonly modeled as random Gaussian variaions [19]. Therefore, learning mus ake place using measuremens obained during he inerval of ime ha happens on he scale of minues, below he

3 scale on which volage magniudes migh change, and above he scale of load flucuaions δp i. We model δp i as a zero mean Gaussian noise erm wih sandard deviaion σ Pi ha incorporaes he aggregaed ambien flucuaions in power injecion and consumpion. PMU measuremens are discree in naure, and arrive as ime-separaed daa samples wih a ypical frequency of several cycles. Therefore, he observed daa poins approximaely follow he dynamics represening he discreizaion of (6) wih a cerain sep. Using he Euler-Maruyama discreizaion scheme, we ge o he firs order in : X +1 = X + Bξ, (7) wih = ( d + I 2N 2N ), ξ N (0, I 2N 2N ) is he sandard mulivariae normally disribued noise and B summarizes he resuling scale of flucuaions. I is reasonable o assume ha load flucuaions are spaially independen across nodes so ha B is diagonal; however, variance a individual buses can be differen, so ha B ii = M i σ Pi for i [N +1, 2N] has a meaning of he noise sandard deviaion a node i and B ii = 0 for i [1, N]. s we will see below, is an imporan parameer ha drives he reconsrucion procedure. Indeed, can no be smaller han he resoluion of he PMU daa, and should no be oo small so ha ξ could be convenienly inerpreed as uncorrelaed whie noise across ime. he same ime, can also no be oo large because in his case X would be essenially independen across ime, meaning ha he dynamic sae marix d could no be recovered in principle, and one could only hope o ge he esimaion of he seady sae covariance [20]. In order o faciliae he learning ask, i is advanageous o selec in such a way ha i saisfies he rade-off beween he amoun of observaions used and he accuracy achieved. In wha follows and unless saed oherwise, we will assume ha ξ is independen and idenically disribued for he purpose of reconsrucion of he dynamic sae marix d. III. ESTIMTORS. Maximum likelihood formulaion In his secion, we presen our esimaors for he dynamic sae marix from T discree observaions of he sysem {X },...,T. Consider he empirical cross-correlaion marices wih and wihou displacemen ha respecively read Σ 1 = 1 X +1 X, (8) Σ 0 = 1 T X X. (9) Based on Eq. (7) we inroduce he following esimaor for he marix which exiss whenever he cross-correlaion marix in Eq. (9) is inverible:  = Σ 1 Σ 0. (10) We refer o his esimaor as o he Maximum Likelihood (ML) esimaor for he. Noice ha he 2N 2N marices Σ 0 and Σ 1 are a mos of rank as expressions (8) and (8) are sums of rank one marices. This implies ha he marix Σ 0 is no inverible for T 2N. When T 2N +2, he marix is inverible wih probabiliy one since rank { XT,..., } { } X 1 = rank ξ T 2,..., ξ1, X1 2N and ξ are independen normally disribued vecors. Bounds on he expeced reconsrucion error for he ML esimaor are given by he following heorem. Theorem 1 (Reconsrucion error for discree dynamics): Le ɛ > 0 and T > 2N + 2. The ML esimaor (10) reconsrucs he marix wih probabiliy a leas 1 ɛ wihin a Frobenius norm error bounded by  F B 2 ɛ E [Tr(Σ 0 )] E [ Σ 0 F] 2. (11) Proof: The proof is given in ppendix. The bound in Theorem 1 is valid wihou any assumpions on he marix. For a sable sysem wih a seady sae dynamics one expecs ha Σ 0 concenraes o is expecaion as T. In his case Theorem 1 shows ha he error on he ML esimaor decreases as he inverse square-roo of he number of observaions and increases linearly wih he noise inensiy. The dynamic sae marix describing he coninuous dynamics in Eq. (6) is esimaed from he discree dynamic marix using he relaion  d =  I 2N 2N. (12) Guaranees on he reconsrucion error for he coninuous dynamic marix follow from Theorem 1. Corollary 2: Le ɛ > 0 and T > 2N + 2. The error on he reconsrucion of d is wih probabiliy a leas 1 ɛ upperbounded by σ 2 P i N Âd d F ɛ i=1 M 2 i () E [Tr(Σ 0 )] E [ Σ 0 2 F]. (13) Proof: I is a direc applicaion of Theorem 1 wih B saisfying B ii = M i σ Pi for i [N + 1, 2N]. The main implicaion of Corollary 2 is ha he error on d decreases wih respec o he produc T. This produc corresponds o he oal observaion ime of he sysem obs. Therefore, if he number of daa samples T is large enough for Σ 0 o concenrae o is average, he reconsrucion error only depends on obs = T hrough is inverse square roo. Noice ha for a fixed observaion ime, he error does no depend on he discreizaion, see [21] for an exended discussion on his propery. s menioned earlier, (10) can be inerpreed as he maximum likelihood esimaor for he dynamic marix. This means ha he esimaor  can be seen as he oucome of some minimizaion procedure. While obaining  hrough an opimizaion problem migh seem unnecessarily more complicaed, formulaing he esimaor as a minimizaion procedure renders possible he incorporaion of exra informaion in he

4 esimaion. This exra informaion can ake he form of a prior on he ineria, damping parameers or line suscepances in he sysem, or i can serve as a prior on he locaion of zero elemens in. s i is crucial o keep he learning ime below he ypical ime for which sysem parameers drif (see Figure 1), adding exra informaion in he reconsrucion help in accomplishing his ask by increasing he learning accuracy for a fixed observaion period. The precise minimizaion procedure is given by he following proposiion. Proposiion 3 (Maximum-Likelihood esimaor): Given T observaions {X },...,T resuling from he discree linear dynamics (7), he maximum likelihood esimaor of represens he soluion of he following leas-squares regression  = argmin X +1 X 2 2. (14) Moreover, for T 2N + 2, he minimum of he leas-square regression is achieved by  = Σ 1Σ 0. Proof: The proof is given in ppendix B. Noice ha he firs N rows of he dynamic sae marix d in Eq. (5) are [0 N N I N N ]. Even hough i seems naural o incorporae his informaion in he esimaion procedure, in fac i appears o be unnecessary as hese rows are no direcly affeced by noise (given ha B ii = 0 for i [1, N]) and are always reconsruced perfecly. However he siuaion is differen for he diagonal lower-righ block of, [ M D ] ha is subjec o noise. Thanks o he opimizaion formulaion in Eq. (14), we can resric he regression o marices ha have a diagonal lower-righ N N block o obain he following consrained esimaor,  = argmin X +1 X 2 2 s.. ij = 0 i, j [N + 1, 2N] and i j. (15) While he consrained esimaor (15) provides a more accurae reconsrucion han is unconsrained counerpar (10), i is compuaionally more expensive as i requires o solve an opimizaion problem. This rade-off beween compuaional power and accuracy can be crucial for pracical applicaions. B. Exensions Oher exensions of he ML esimaor can be considered based on prior informaion ha we can incorporae in he leas-squares regression (14). For insance, when he sysem parameers are expeced o drif slowly i is beneficial o incorporae a prior on he marix. This prior can ake he form of a Gaussian prior cenered around a previous reconsrucion. This ranslaes ino a leas-squared regression wih a Tikhonov regularizaion [22], i.e. ) ( T  = argmin X +1 X ν Âprev 2 2, (16) where ν is a parameer indicaing our degree of confidence ha he curren marix is close o is previous esimae Âprev. There exiss insances when he sysem parameers migh no have been previously esimaed, for example when he opology has been modified due o ouages, lines ripping or conroller changes [9], bu he opology of he observed grid is known o be sufficienly sparse. In his case i would be beneficial o promoe he sparsiy of he marix wih a Laplace prior P () e λ 1. This prior leads o he following LSSO esimaor  = argmin ( T ) X +1 X λ 1. (17) The LSSO esimaor proves o be much more efficien han he unconsrained leas-squares (14) when he marix is sparse, rendering he sample requiremen very weakly dependen of he size of he problem N [21]. Finally in cases where he sae of only a subse of buses is observed [20], i is sill possible o rerieve par of he dynamic sae marix corresponding o he visible par of he sysem. This can be done wih he so-called sparse plus lowrank heurisic producing  and L, he minimizers of X +1 ( + L)X λ 1 + η L, (18) where L summarizes he effecs of he unobserved measuremens and he nuclear norm penaly represens a convex surrogae for he low rank consrain, see [23] for more deails. IV. CSE STUDY We illusrae he performance of our learning algorihms on he IEEE 39-bus 10-generaor es sysem [24]; he opology of his sysem is shown in Figure 2 (). We assume ha PMUs are locaed a he generaor buses. Firs, we perform Kron reducion and eliminae passive loads, obaining he parameers of he sysem (5) obeyed by he generaors [9]. Given he esablished parameers of he reduced nework and hence he ground ruh d, we use he discreized represenaion (7) o simulae he dynamics and produce he ime series on δ i and ω i for each generaor i V using he smalles resoluion = 1/60 sec (1 cycle) for which our model (4) is sill valid; in Figure 2 (B), we show one example of such run wih he simulaed daa over 10 minues. In all simulaions, he load variaion is fixed a he level of σ Pi = 0.01 p.u. [9]. We use daa obained in his way in all reconsrucion experimens repored below. Noice ha PMUs migh have a lower ime resoluion, subsampling hese daa poins wih a differen ime sep, for insance once every k cycles (for many measuring devices, he maximum sampling rae corresponds o k = 2 or k = 3). Moreover, because of he daa processing reasons, one migh wish o deliberaely sample daa a a lower frequency for he reconsrucion purposes. Therefore, i is insrucive o check he sensiiviy of he algorihmic performance o he chosen subsampling sep. Bu prior o ha, we need o esablish he measure of performance for he wo esimaors inroduced in his paper: he Unconsrained Maximum Likelihood (UML)

5 1, 1 5, 5 8, 8 Gen 1 Gen 8 Gen 10 Gen 9 B Gen 2 Gen 3 Gen 5 Gen 4 Gen 7 3, 3 6, 6 9, 9 4, 4 7, 7 10, 10 Gen 6 Figure 2. () Topology of he IEEE 39-bus sysem wih 10 generaors [24]. (B) Synheic PMU-measured daa generaed using he parameers of he Kron-reduced sysem and sampled a 60 Hz frequency. 10 minues of daa is presened for every generaor excep generaor 2 ha serves as he reference slack bus. The ime couning he number of sample poins is expressed in erms of he number of cycles. Noe ha he ime series for he generaor 1 has a disinc appearance due o a significanly larger ineria coefficien M 1. esimaor (10) and he Consrained Maximum Likelihood (CML) esimaor (15), where he word consrained means ha he suppor paern of d has been explicily enforced. We use boh algorihms o produce an esimae Âd of he dynamic sae marix d. For a given ime separaion beween measuremen samples {X }, firs he discree marix  a he corresponding scale (7) is esimaed, and hen he dynamic sae marix is recovered using he linear approximaion (12). In order o accoun for he addiional sparsiy srucure presen in d, we supplemen he applicaion of (10) in he UML esimaor wih a pos-esimaion hresholding of marix elemens ha are known o be zero, in paricular in he block corresponding o he diagonal submarix M D in (5). We quanify he qualiy of he esimaion using he relaive Frobenius error ε, defined as ε = Âd d F d F, (19) where Âd is he recovered marix and d is he ground ruh dynamic sae marix used o produce daa (a finer discreizaion = 1/60 sec). The dependence on is shown in Figure 3 (). In his figure, he observaion window obs is fixed o 10 min, and herefore he number of samples T seen by he algorihms decreases wih as T = obs /. ccording o he Corollary 2, he reconsrucion error ε should say consan in T = obs. We see ha for boh algorihms we don see any clear plaeau in ε( ) dependence even for small. This is due o he increasing wih error in he firs-order approximaion (7) of he fine-grained dynamics and hence o he level of validiy of (12). Neverheless, we see ha for boh esimaors ε is growing slowly wih, and he error seen for = 3 cycles is very close o he one realized when he fines possible discreizaion = 1 is aken, alhough he algorihms use hree imes less samples in he former case. Moreover, given ha = 3 cycles represens a normal sampling rae for many measuremen devices, we use his sampling rae in he subsequen experimens. The final hing ha we observe is ha he algorihm CML ha explois he srucure of d sysemaically yields a lower error compared o UML even wih he added heurisic pos-reconsrucion hresholding; on he oher hand, i should be noed ha CML is slower as one needs o solve he opimizaion problem (15) insead of jus invering he marix (in he presen experimens, he opimizaion was carried ou using he Ipop solver). Sill, boh algorihms run in seconds for his es case, which represens a premise for an online real-ime implemenaion. B UML CML 2.5% [cycles] 2.5% 1.75% T Δ = obs [minues] UML CML Figure 3. Performance of he esimaors as a funcion of () ime discreizaion and (B) oal observaion ime obs = T. Each daa poin is averaged over 50 independen realizaions of he dynamics for reducing he noise due o saisics. In Figure 3 (B), we sudy he performance of our esimaor as a funcion of he number of samples T for a fixed sampling sep = 3 cycles (and hence for growing observaion ime

6 obs from 1 o 20 minues). We see ha he experimen confirms he conclusion of he Corollary 2: he esimaed dynamic sae marix Âd quickly converges o he ground ruh marix d, wih CML algorihm achieving he relaive error of 2.5% by obs = 10 min and 1.75% by obs = 20 min. This fac shows ha i is possible o esimae he dynamic sae marix o an impressive accuracy under he ime consrains oulined in Figure 1. Besides he accurae predicion of he dynamic sae marix d per se, a desirable feaure of he esimaors would consis in an accurae predicion of he properies of his marix, in paricular including he specral properies [9]. Indeed, he criical eigenvalue is known o serve as measures of proximiy o he insabiliy [10], [12], [13], while he associaed criical eigenvecor migh provide useful informaion on he sysem response and faciliae he design of conrol acions such as he generaion re-dispach [12], [16]. In Figure 4, we es he accuracy of he criical eigenvalues predicion using he samples obained wihin he obs = 10 min observaion inerval a he sampling rae = 3 cycles. Is is apparen ha he criical eigenvalues are prediced o a good accuracy, which shows ha our learning procedure can be used for he online monioring of he sysem sabiliy. Im( d ) rue d UML d CML d λ criical Re( d ) Figure 4. Qualiy of predicion of he criical eigenvalues of he dynamic sae marix d wih obs = 10 min and = 3 cycles. V. DISCUSSION ND CONCLUSIONS In his work, we explored he maximum likelihood based approach o he problem of esimaion of he dynamic sae marix from PMU measuremens. In paricular, we consruced and esed wo leas-squares esimaors, one based on a fas inversion of he empirical covariance marix, and anoher one based on he soluion of a convex quadraic regression aking full advanage of he problem srucure and leading o a more accurae reconsrucion, bu a an expense of a slighly higher compuaional ime. These wo esimaors realize a common rade-off in pracical applicaions beween he accuracy and he speed of compuaions. In his conribuion, we have verified he properies of our algorihms on synheic daa from sandard IEEE es case; in fuure sudies, i would be insrucive o es heir performance on daa colleced from real-world PMUs. s clarified in secion III, our framework is very broad and can naurally accommodae he regularized online learning by incorporaing he previously learned models as a prior hus poenially decreasing he compuaional ime even furher, as well as exensions o he case of sparse nework opologies and o realisic scenario of incomplee observaions due o a parial PMU coverage [20]. I would be useful o perform a heoreical finie-sample analysis of hese exensions in he fuure similarly o he analysis presened in his paper, as well as o provide an empirical assessmen of he algorihm performance in realisic problems. s we commened while moivaing he dynamic learning problem, esimaed parameers can be useful in a number of asks relaed o opimizaion, conrol and securiy of he ransmission grid. noher aracive feaure of our mehodology consiss in an abiliy o perform daa-driven sae esimaion of power flucuaions hrough esimaion of he marix B relaed o power flucuaions (7), as explained in he proof of he Proposiion 3 in ppendix B. I is imporan o quanify he poenial advanage of using he learning-based mehod in hese applicaions. Some of relevan open quesions ha we did no address in his work include he consrucion of opimal esimaors in he cases of more general noise disribuions han he Gaussian case (for example, he power-law disribued noise), as well as for noise correlaed in space and ime. We anicipae ha in he regime of weak spaial and emporal coupling he esimaors inroduced in his paper should be sufficienly robus and perform reasonably well, bu his saemen should be horoughly checked on realisic es cases. Finally, an ineresing and naural direcion for fuure exploraion consiss in exending our scheme o non-saionariy and o higher-order dynamic models. PPENDIX PROOF OF THEOREM 1 Firs muliply boh sides of Eq. (7) by X sum over o obain Σ 1 = Σ 0 + R, where R ij = 1 B ii ξ,i X,j. and perform a (.20) (.21) Since T 2N + 2, he cross-correlaion marix Σ 0 is inverible. Muliplying boh sides of Eq. (.20) by Σ 0 gives  = RΣ 0. In expecaion he difference in Frobenius norm beween he ML esimaor and he marix is upperbounded by he following expression [ ] E  F = E [ RΣ 0 F], (.22) E [ R F Σ 0 F], (.23) E [ R 2 F ] E [ Σ 0 F] 2, (.24)

7 where in wo las lines we have used Cauchy-Schwarz inequaliy for he Frobenius norm and for he expecaion, respecively. In order o compue he expeced Frobenius norm of R, we firs evaluae he expecaions of is elemens squared E [ ( ) Rij 2 ] 2 T Bii = = 1=1 2=1 E [ξ 1,iX 1,jξ 2,iX 2,j], (.25) ( ) 2 T Bii E [ X 2,j], (.26) where we used ha E [ ξ 2,i] = 1 and E [ξ1,ix 1,jξ 2,iX 2,j] = 0 if 1 2. I is now easy o compue he expeced Frobenius norm of he marix R, E [ R 2 ] B 2 F = 2 E [Tr (Σ 0)]. (.27) The final sep of he proof consiss in combining Eq. (.24), Eq. (.27) and Markov inequaliy. PPENDIX B PROOF OF PROPOSITION 3 The likelihood funcion is he probabiliy densiy funcion (PDF) of he observaion given he parameers, B ha define he model. Since ξ = B (X +1 X ) and he noise is idenically and independenly disribued, we can relaed he PDF of X = X 1,..., X T for a given, B o he PDF of a single ξ, i.e. ρ X (X) = ρ X 1 (X 1 ) de(b) T ρ ξ (B (X +1 X )). (B.28) The maximum likelihood esimaor is given by he argmax of he likelihood in Eq. (B.28) wih respec o, B. Equivalenly one can minimize he opposie of he logarihm of he likelihood o obain [ ] (Â, B) = argmin,b ln de B ln ρ ξ (B (X +1 X )) (B.29) fer replacing he normally disribued PDF for ξ, we arrive a he following opimizaion problem [ ] (Â, B) N = argmin ln B ii + (2B2 ii ) (X +1 X ) 2 i,b i=1 (B.30) Noice ha in Eq. (B.30), he opimizaion over is independen of he value of B and can be performed separaely, yielding he opimizaion problem (14) ha afer some algebra can be equivalenly represened as  = argmin Tr( Σ 0 2 Σ 1 ). (B.31) Whenever Σ 0 is inverible, he minimizaion in Eq. (B.31) can be done analyically and gives  = Σ 1Σ 0. The esimae B can hen be obained by solving (B.30). REFERENCES [1] P. Kundur, N. J. Balu, and M. G. Lauby, Power sysem sabiliy and conrol. McGraw-hill New York, 1994, vol. 7. [2] P. W. Sauer, M.. Pai, and J. H. 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