Reconstructing the power grid dynamic model from sparse measurements

Transcription

1 Reconsrucing he power grid dynamic model from sparse measuremens Andrey Lokhov wih Michael Cherkov, Deepjyoi Deka, Sidhan Misra, Marc Vuffray Los Alamos Naional Laboraory Banff, Canada

2 Moivaion: learning from correlaed samples, or ime series Assume linear dynamics: Ẋ () = AX () + ξ(), wih ξ() Gaussian noise Given N ime series, is i possible o reconsruc he srucure and parameers of A? x 1 1 x 2 x N. 2 A 12 N

3 Moivaion: learning from correlaed samples, or ime series Assume linear dynamics: Ẋ () = AX () + ξ(), wih ξ() Gaussian noise Given N ime series, is i possible o reconsruc he srucure and parameers of A? x 1 1 x 2 x N. 2 A 12 N Wha happens if only N O < N ime series are observed? X = X O X H

4 Example: graph-based anomaly deecion in cyber-physical sysems Task: deec and localize aacks on CPS using physical measuremens Smar Facories & Indusry Criical Infrasrucures & Smar Grid SCADA Aacked nodes Self-Driving Cars and Avionics Complex Transporaion Approach: assuming linearized dynamics, learn he normal graph and monior changes Normal operaion Seing: srucure unknown, usually no hidden nodes Faul or aack

5 Example: reconsrucing he power grid dynamics Sae esimaion and parameer learning in dynamics of he ransmission power grid θ i = f i, M i ḟ i + τ i f i = p i j i β ij (θ i θ j ) + ξ i (). Task: reconsruc parameers of generaors and lines (evolve slowly, hours) and injecions and consumpions (evolve rapidly, minues) from sensor measuremens θ 1 θ 3 θ 4 θ 2 Seing: srucure known, bu hidden observaions (sparsely locaed PMUs)

6 Reducion o he saic problem? For sable sysems: explore Lyapunov equaion for he saionary covariance marix AΣ + ΣA + I = 0 [Wang, Bialek, Turisyn 2015], [Zare, Jovanović, Georgiou 2016] Disadvanages: requires knowledge of some par of A, hard o generalize o hidden case Subsampling independen samples: use saic Gaussian graphical model learning Disadvanages: only saionary regime, wasing samples (desiring log N samples), Σ has less informaion (supp(a) supp(σ 1 )) 0.0 i.i.d. samples 0.1 (b) 10:00 10:30 11:00 11:30 12:00 0 (c) 10:00 10:30 11:00 11:30 12:00 0

7 In wha follows For simpliciy, consider discree-ime dynamics: X +1 = AX + ξ, wih ξ whie noise! Complee observaions on all nodes (a) Known graph srucure: leas-squares objecive (b) Unknown graph: l 1 and l 0 regularizaions! Parially observed sysem (a) Known graph: convex formulaion, incomplee soluion (b) Unknown graph: sparsiy and low-rank regularizaions (c) Non-convex EM-ype algorihm Remark: Inuiively and rigorously [Beno e al., 2010], in he case of coninuous equaions, here exiss an opimal discreizaion sep

8 Complee observaions: known graph srucure Assuming he uniform prior on A, P(A X, ξ) exp( T 1 =1 T 1 Â MMSE = Â MAP = argmin X +1 AX 2 A =1 X +1 AX 2 /2σ 2 ) For a sufficien number of samples M N, Â = ( T 1 =1 X +1 X ) ( T X X =1 ) = Σ,+1 (Σ, ) A = Âij

9 Unknown graph and high-dimensional regime Regularized leas-squares: [Beno, Ibrahimi, Monanari 2010] Â = argmin A ( T 1 =1 X +1 AX 2 + λ A 1 ) Reconsrucs graph srucure wih M log N samples under incoherence condiion and assumpions on (λ min, λ max ) of covariance marix Open quesion: similarly o he Gaussian GM selecion, assumpions-free algorihm? Candidae: non-convex l 0 sparsiy consrain [Misra, Vuffray, AL, Cherkov 2017]

10 Parial observaions: convex formulaion Likelihood of observaions: P(Ã O X, ξ) = dx H P(A X, ξ), X H Leads o a convex Lasso ype formulaion for small H : (Â O, ˆL) = argmin A O,L [ T 1 =1 Ã O = A O A OH A 1 H A HO A O + L X O +1 (A O + L)X O 2 + λ 1 A O 1 + λ 2 L ] Adapaion of [Giraud and Tsybakov 2012], [Jalali, Sanghavi 2012] M log N under incoherence assumpion. If he graph is known, one could furher aemp o decompose he marix L ino sparse facors, see e.g. [Wien, Tibshirani, Hasie 2009]. Open quesion: is i possible o devise assumpions-free algorihm? Candidae: non-convex explici rank consrain rank(l) H [Yuan & Laurizen, Meinshausen 2012] ogeher wih an l 0 sparsiy consrain [Misra, Vuffray, AL, Cherkov 2017]

11 Parial observaions: convex formulaion Likelihood of observaions: P(Ã O X, ξ) = dx H P(A X, ξ), X H Leads o a convex Lasso ype formulaion for small H : (Â O, ˆL) = argmin A O,L [ T 1 =1 Ã O = A O A OH A 1 H A HO A O + L X O +1 (A O + L)X O 2 + λ 1 A O 1 + λ 2 L ] Adapaion of [Giraud and Tsybakov 2012], [Jalali, Sanghavi 2012] M log N under incoherence assumpion. If he graph is known, one could furher aemp o decompose he marix L ino sparse facors, see e.g. [Wien, Tibshirani, Hasie 2009]. Open quesion: is i possible o devise assumpions-free algorihm? Candidae: non-convex explici rank consrain rank(l) H [Yuan & Laurizen, Meinshausen 2012] ogeher wih an l 0 sparsiy consrain [Misra, Vuffray, AL, Cherkov 2017]

12 Parial observaions: convex formulaion Likelihood of observaions: P(Ã O X, ξ) = dx H P(A X, ξ), X H Leads o a convex Lasso ype formulaion for small H : (Â O, ˆL) = argmin A O,L [ T 1 =1 Ã O = A O A OH A 1 H A HO A O + L X O +1 (A O + L)X O 2 + λ 1 A O 1 + λ 2 L ] Adapaion of [Giraud and Tsybakov 2012], [Jalali, Sanghavi 2012] M log N under incoherence assumpion. If he graph is known, one could furher aemp o decompose he marix L ino sparse facors, see e.g. [Wien, Tibshirani, Hasie 2009]. Open quesion: is i possible o devise assumpions-free algorihm? Candidae: non-convex explici rank consrain rank(l) H [Yuan & Laurizen, Meinshausen 2012] ogeher wih an l 0 sparsiy consrain [Misra, Vuffray, AL, Cherkov 2017]

13 Parial observaions: alernaive convex formulaion Likelihood P(A X, ξ) can be rewrien in he saic form over he rajecories: P(A X, ξ) ( de B exp X BX ), X [X=1,..., X =T ], A A B = A A 1 + A A A A 1 + A A A A A A A 1 Likelihood of observaions: P( B O X, ξ) = dx H P(B X, ξ), X H B O = B O B OH B 1 H B HO Leads o a Graph Lasso ype convex formulaion for B O L 0 and L 0: [ ] (ˆB O, ˆL) = argmin r(σ(b O L)) log de(b O L) + λ 1 B O 1 + λ 2 r(l) B O,L Adapaion of [Chandrasekaran, Parillo and Willsky 2012]

14 Parial observaions: alernaive convex formulaion Likelihood P(A X, ξ) can be rewrien in he saic form over he rajecories: P(A X, ξ) ( de B exp X BX ), X [X=1,..., X =T ], A A B = A A 1 + A A A A 1 + A A A A A A A 1 Likelihood of observaions: P( B O X, ξ) = dx H P(B X, ξ), X H B O = B O B OH B 1 H B HO Leads o a Graph Lasso ype convex formulaion for B O L 0 and L 0: [ ] (ˆB O, ˆL) = argmin r(σ(b O L)) log de(b O L) + λ 1 B O 1 + λ 2 r(l) B O,L Adapaion of [Chandrasekaran, Parillo and Willsky 2012]

15 Parial observaions: Expecaion-Maximizaion approach Given iniial guess A (s=0), ierae unil convergence: Expecaion: compue Q(A, A (s) ) = E [ P(A X O X H, ξ) X O, A (s)] Maximizaion: updae A (s+1) = argmax Q(A, A (s) ) A The closes reference [Shumway, Soffer 1982] No widely considered (hard o analyse), bu naural choice if he graph is known

16 Pah forward! Theoreical analysis of he algorihms! Esablishing bes algorihms in pracice (using modern solvers, EM)! Applicaion o he power grid and cyberphysical daa ses