Learning through Power Distribution Grid Probing

Transcription

1 Learig through Power Distributio Grid Probig Sid Bhela, Vassilis Kekatos, ad Harsha Veeramachaei INFORMS 2017 October 23 rd, 2017 Housto, TX Ackowledgemets:

2 Learig loads i distributio grids Reduced observability due to sheer extet ad limited meterig ifrastructure However, load estimates eeded for grid optimizatio, billig, ad eergy theft detectio Leverage smart meter data ad smart iverters 2

3 Problem statemet ad prior work Load learig: Give readigs from a set of metered (cotrollable) buses M, recover the system state ad hece the power ijectios at o-metered buses O. Collected readigs - smart meter data (voltage magitudes, power ijectios) - sychrophasor data (bus voltage ad curret phasors) For sufficietly rich (pseudo)measuremet sets, solve as D-PSSE [Dzafic-Jabr-Pal et al 13], [Klauber-Zhu 15], [Gomez-Exposito et al 15] Liear estimator if grid is equipped with micro-pmus [A. vo Meier 15] Meter placemet i distributio grids [Lui-Poci 14] Probig trasmissio grids for estimatig oscillatio modes [Trudowski-Pierre 09] System idetificatio i DC microgrids [Agjelichioski-Scaglioe 17] 3

4 Liearized distributio flow model Sigle-phase radial grid with N+1 odes ad N lies u m u r` + jx` p m,q m p,q Approximate LDF model [Bara-Wu 89], [Bologai-Dorfler 15], [Deka et al 17] u ' Rp + Xq ' Xp Rq odal voltage phasor u := V 2 V 0 2 V = V e j The iverses of (R,X) are reduced graph Laplacia matrices 4

5 Passive load learig Partitio data ito metered ad o-metered buses um u O = RM,M R M,O R O,M R O,O pm p O + XM,M X O,M X O,O q O X M,O qm + Model for smart meter data u M R M,M p M X M,M q M = [R M,O X M,O ] po q O + M Model for sychrophasor data um R M,M p M X M,M q M M X M,M p M + R M,M q M = RM,O X M,O R M,O q O X M,O po + No-metered ijectios ca be uiquely recovered via least-squares if regressio matrices are full colum-rak 5

6 Idetifiability with passive load learig Propositio 1: Give smart meter data o M = M 1 [ M 2, the ijectios at O are idetifiable if every bus i O ca be coected to uique buses i M 1 ad M 2. Propositio 2: Give PMU data o M, the ijectios at O are idetifiable if every bus i O ca be coected to a uique bus i M. o-metered buses O metered buses M S. Bhela, V. Kekatos, ad S. Veeramachaei, Power distributio system observability with smart meter data, IEEE GlobalSIP, Motreal, Caada, Nov

7 Key ideas Partitio reduced (resistive) Laplacia L := R 1 = LM,M L > M,O L M,O L O,O Rak of Schur complemet rk(l) =rk(l M,M )+rk(l O,O L > M,OL 1 M,M L M,O) Matrix iversio lemma ad Sylvester's iequality L M,O R M,O = L 1 M,M L M,O(L O,O L > M,OL 1 M,M L M,O) 1 Matrix is geerically ivertible iff there exists a perfect matchig i the bipartite graph defied by its sparsity patter [Tutte 47] L M,O = O Perfect matchig easily foud as max flow problem M W. T. Tutte, The factorizatio of liear graphs, Joural of the Lodo Mathematical Society, 1947 Ford ad Fulkerso, Maximal flow through a etwork, Caadia J. of Mathematics,

8 Idetifiable setups with smart meter data IEEE 123-bus etwork with O =4 Frequecy Coditio umber IEEE 123-bus etwork with O = 10 Frequecy Coditio umber 8

9 Idetifiable setups with PMUs IEEE 123-bus etwork with O =4 100 Frequecy Coditio umber IEEE 123-bus etwork with O = 10 Frequecy Coditio umber 9

10 Load learig through grid probig Time t =1: Record data associated with state v 1 {u 1,p 1,q 1 } 2M Time t =2 Probe oliear physical system by perturbig power ijectios at cotrollable buses [how?] Record data {u 2,p 2,q} 2 2M state v 2 6= v 1 associated with Repeat say every secod for t =1,...,T Exploit the fact that {p t,q} t 2O ivariat durig probig remai o-metered buses O metered buses M 10

11 Coupled power flow (CPF) metered buses o-metered bus (u 0, 0 ) (p 1,q 1,u 1 ) (p 2,q 2,u 2 ) (p 3,q 3 ) t =1 (p 1,q 1 ) (p 2,q 2 ) (p 3,q 3 ) ecessary coditio M 2 O T (u 0, 0 ) (p 0 1,q 0 1,u 0 1) (p 0 2,q 0 2,u 0 2) (p 3,q 3 ) t =2 11

12 Idetifiability through grid probig Give smart iverter data o metered odes ad assumig ivariat o-metered loads, fid the states {v t } T t=1 ad hece the o-metered loads. p (v t )=ˆp t q (v t )=ˆq t u (v t )=û t 8 2 M 8 2 M 8 2 M p (v t )=p (v t+1 ) q (v t )=q (v t+1 ) 8 2 O 8 2 O couplig equatios t =1,...,T t =1,...,T 1 Theorem: If O ca be partitioed ito disjoit sets {Ōk} T/2 k=1 such that each oe of them ca be matched to M, the states {v t } T t=1 ad the ijectios at O are locally idetifiable. T =1: passive scheme with smart meter data T =2: passive scheme usig PMU data O! M Checkig if matchig exist is easy (for ay T) S. Bhela, V. Kekatos, ad H. Veeramachamei, Power Grid Probig for Load Learig: Idetifiability over Multiple Time Istaces, i Proc. IEEE CAMSAP, Curacao, December

13 Idetifiable setups with probig Frequecy Coditio umber SCE 34-bus feeder: T =4, O = 18 coditio umbers of CPF Jacobias for radom setups 13

14 Idetifiable setups with probig (cot d) Frequecy Coditio umber SCE 34-bus feeder: T =6, O = 21 coditio umbers of CPF Jacobias for radom setups 14

15 CPF as pealized SDR Lift states {v t } to psd ad rak-oe matrices {V t = v t vt H } T t=1 Solve CPF as relaxed pealized SDP mi TX Tr(MV t ) t=1 s.to Tr(M k V t )=ŝ t k, k 2 M t,t=1,...,t Tr(M k V t )=Tr(M k V t+1 ), k 2 O,t=1,...,T 1 V t 0, t =1,...,T rk(v t )=1, t =1,...,T M k : matrices depedet o bus admittace matrix Rak-oe solutios guarateed for M = B ad lightly loaded system R. Madai, M. Ashpraphijuo, J. Lavaei, ad R. Baldick, Power system state estimatio with a limited umber of measuremets, IEEE CDC, Las Vegas, NV, Dec Zhag, Madai, Lavaei, Power system state estimatio with lie measuremets, IEEE CDC

16 CPSSE as pealized SDR Data are oisy ad o-metered loads may vary durig probig mi TX Tr(MV t )+ t=1 TX t=1 s.to Tr(M k V t )+ t k =ŝ t k, X k2m t f k ( t k)+ TX 1 t=1 X f k ( t k) k2o k 2 M t,t=1,...,t Tr(M k V t )=Tr(M k V t+1 )+ t k, k 2 O,t=1,...,T 1 V t 0, t =1,...,T Data-fittig optios: - weighted least-squares (WLS) f k ( t k)= ( t k )2 2 k - weighted least-absolute value (WLAV) f k ( t k)= t k k S.Bhela, V. Kekatos, ad H. Veeramachamei, Ehacig observability i distributio grids usig smart meter data, IEEE Tras. o Smart Grid, (early access)

17 Numerical tests with sythetic data Probability of Success β kv t v t+1 k 2 Probability of correct rak-1 CPF solutio Probig works better for sufficietly RMSE differet states (v t, v t+1 ) WLAV WLS 0.04 State RMSE for CPSSE (T = 2) SNR 17

18 Numerical tests with real data WLAV WLS RMSE :00 a.m. 12:00 p.m. 2:00 p.m. 4:00 p.m. Time Actual load/solar data (Peca Str project) RMSE o system state over oe day Probig by chagig power factors i smart iverters 18

19 Coclusios Passive ijectio learig Smart meter ad sychrophasor data Idetifiability for sigle-phase radial grids Simple LS solver usig LDF model No time couplig; timescale depeds o data Meshed ad polyphase grids? Optimal meter selectio? Active ijectio learig Iverter data collected via itetioal probig Idetifiability for possibly meshed ad polyphase grids Improvemets with icreasig T SDP-/SOCP-based solvers for CPF ad CPPSE Optimal probig desig? PMU data? Lie flow data? Probig for topology idetificatio? Thak You! 19