Acknowledgements:

Learning through Power Distribution Grid Probing Sid Bhela, Vassilis Kekatos, and Harsha Veeramachaneni�

INFORMS 2017 October 23rd, 2017

Houston, TX

Learning loads in distribution grids

n  Reduced observability due to sheer extent and limited metering infrastructure

2

n  Leverage smart meter data and smart inverters

n  However, load estimates needed for grid optimization, billing, and energy theft detection

Problem statement and prior work

n  Collected readings

-  smart meter data (voltage magnitudes, power injections)

-  synchrophasor data (bus voltage and current phasors)

3

Load learning: Given readings from a set of metered (controllable) buses recover the system state and hence the power injections at non-metered buses

M,

O.

n  For sufficiently rich (pseudo)measurement sets, solve as D-PSSE

[Dzafic-Jabr-Pal et al’13], [Klauber-Zhu’15], [Gomez-Exposito et al’15]

n  Linear estimator if grid is equipped with micro-PMUs [A. von Meier’15]

n  Meter placement in distribution grids [Lui-Ponci’14]

n  Probing transmission grids for estimating oscillation modes [Trudnowski-Pierre’09]

n  System identification in DC microgrids [Angjelichinoski-Scaglione ’17]

Linearized distribution flow model

4

n  Single-phase radial grid with N+1 nodes and N lines

n  Approximate LDF model [Baran-Wu’89], [Bolognani-Dorfler’15], [Deka et al’17]

nodal voltage phasor u ' Rp+Xq

✓ ' Xp�Rq

Vn = |Vn|ej✓n

um

pm, qm pn, qn

un

r` + jx`

un := |Vn|2 � |V0|2

n  The inverses of (R,X) are reduced graph Laplacian matrices

Passive load learning

n  Model for synchrophasor data

5

n  Partition data into metered and non-metered buses

n  Model for smart meter data

n  Non-metered injections can be uniquely recovered via least-squares if regression matrices are full column-rank

uM �RM,MpM �XM,MqM

✓M �XM,MpM +RM,MqM

�=

RM,O XM,O

XM,O �RM,O

� pO

qO

�+ ✏

uM

uO

�=

RM,M RM,O

RO,M RO,O

� pM

pO

�+

XM,M XM,O

XO,M XO,O

� qM

qO

�+ ⌘

uM �RM,MpM �XM,MqM = [RM,O XM,O]

pO

qO

�+ ⌘M

Identifiability with passive load learning

6 S. Bhela, V. Kekatos, and S. Veeramachaneni, “Power distribution system observability with smart meter data,” IEEE GlobalSIP, Montreal, Canada, Nov. 2017.

metered buses non-metered buses O

M

Proposition 1: Given smart meter data on the injections at  are identifiable if every bus in can be connected to unique buses in and M1 M2.

OM = M1 [M2,

O

Proposition 2: Given PMU data on , the injections at are identifiable if every bus in can be connected to a unique bus in

M O

O M.

Key ideas

7

n  Partition reduced (resistive) Laplacian

n  Rank of Schur complement

n  Matrix inversion lemma and Sylvester's inequality

RM,O = �L�1M,MLM,O(LO,O � L>

M,OL�1M,MLM,O)

�1

L := R�1 =

LM,M LM,O

L>

M,O LO,O

�

rk(L) = rk(LM,M) + rk(LO,O � L>

M,OL�1M,MLM,O)

n  Matrix is generically invertible iff there exists a perfect matching in the bipartite graph defined by its sparsity pattern [Tutte’47]

LM,O

n  Perfect matching easily found as max flow problem

LM,O =

2

66664

0 ⇤ ⇤ ⇤⇤ 0 ⇤ 00 ⇤ 0 ⇤0 0 0 ⇤0 0 ⇤ 0

3

77775

O M

W. T. Tutte, “The factorization of linear graphs,” Journal of the London Mathematical Society, 1947 Ford and Fulkerson, “Maximal flow through a network,” Canadian J. of Mathematics, 1956

Identifiable setups with smart meter data

8

100 105 1010 1015 1020

Condition number

0

5

10

15

20

25

30

35

40

45

Freq

uenc

y

100 105 1010 1015 1020

Condition number

0

2

4

6

8

10

12

Freq

uenc

y

IEEE 123-bus network with |O| = 10

IEEE 123-bus network with |O| = 4

Identifiable setups with PMUs

9

100 101 102 103 104

Condition number

0

20

40

60

80

100

120

140

Freq

uenc

y

100 105 1010 1015 1020

Condition number

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Freq

uenc

y

IEEE 123-bus network with |O| = 10

IEEE 123-bus network with |O| = 4

Load learning through grid probing

10

n  Probe nonlinear physical system by perturbing power injections at controllable buses [how?]

n  Record data associated with state

n  Exploit the fact that remain invariant during probing

metered buses non-metered buses O

M

v1

Time t = 2

Time Record data associated with state

{u1n, p

1n, q

1n}n2Mt = 1 :

{u2n, p

2n, q

2n}n2M

v2 6= v1

Repeat say every second for t = 1, . . . , T

{ptn, qtn}n2O

Coupled power flow (CPF)

11

non-metered bus metered buses

(p2, q2, u2)(u0, ✓0) (p1, q1, u1) (p3, q3)t = 1

necessary condition

|M| �2|O|

T

(p1, q1) (p2, q2) (p3, q3)

(u0, ✓0)

t = 2

(p3, q3)(p02, q02, u

02)(p01, q

01, u

01)

Identifiability through grid probing

12

Given smart inverter data on metered nodes and assuming invariant non-metered loads, find the states and hence the non-metered loads.

coupling equations

{vt}Tt=1

pn(vt) = ptn 8n 2 Mqn(vt) = qtn 8n 2 Mun(vt) = ut

n 8n 2 M

t = 1, . . . , T

pn(vt) = pn(vt+1) 8n 2 O

qn(vt) = qn(vt+1) 8n 2 O

t = 1, . . . , T � 1

Theorem:  If   can be partitioned into disjoint sets such that each one of them can be matched to , the states and the injections at are locally identifiable.

{Ok}T/2k=1

{vt}Tt=1 O

O

M

n  passive scheme with smart meter data T = 1 :

n  passive scheme using PMU data T = 2 : O ! M

S. Bhela, V. Kekatos, and H. Veeramachameni, “Power Grid Probing for Load Learning: Identifiability over Multiple Time Instances,” in Proc. IEEE CAMSAP, Curacao, December 2017.

n  Checking if matching exist is easy (for any T)

Identifiable setups with probing

13

105 1010 1015 1020

Condition number

0

50

100

150

200

250

300

350

400

450

500

Freq

uenc

y

SCE 34-bus feeder: T = 4, |O| = 18

condition numbers of CPF Jacobians for random setups

Identifiable setups with probing (cont’d)

14

105 1010 1015 1020

Condition number

0

50

100

150

200

250

300

Freq

uenc

y

SCE 34-bus feeder: T = 6, |O| = 21

condition numbers of CPF Jacobians for random setups

CPF as penalized SDR

15

n  Solve CPF as relaxed penalized SDP

R. Madani, M. Ashpraphijuo, J. Lavaei, and R. Baldick, ‘Power system state estimation with a limited number of measurements,’ IEEE CDC, Las Vegas, NV, Dec. 2016.

Zhang, Madani, Lavaei, ‘Power system state estimation with line measurements,’ IEEE CDC 2016.

n  Lift states to psd and rank-one matrices {vt} {Vt = vtvH

t}Tt=1

n  Rank-one solutions guaranteed for and lightly loaded system M = �B

matrices dependent on bus admittance matrix Mk :

minTX

t=1

Tr(MVt)

s.to Tr(MkVt) = stk, k 2 Mt, t = 1, . . . , T

Tr(MkVt) = Tr(MkVt+1), k 2 O, t = 1, . . . , T � 1

Vt ⌫ 0, t = 1, . . . , T

rk(Vt) = 1, t = 1, . . . , T

CPSSE as penalized SDR

16

n  Data are noisy and non-metered loads may vary during probing

n  Data-fitting options:

- weighted least-squares (WLS)

- weighted least-absolute value (WLAV)

fk(✏tk) =

(✏tk)2

�2k

fk(✏tk) =

|✏tk|�k

S.Bhela, V. Kekatos, and H. Veeramachameni, “Enhancing observability in distribution grids using smart meter data,” IEEE Trans. on Smart Grid, (early access) 2018.

min �TX

t=1

Tr(MVt) +TX

t=1

X

k2Mt

fk(✏tk) +

T�1X

t=1

X

k2O

fk(✏tk)

s.to Tr(MkVt) + ✏tk = stk, k 2 Mt, t = 1, . . . , T

Tr(MkVt) = Tr(MkVt+1) + ✏tk, k 2 O, t = 1, . . . , T � 1

Vt ⌫ 0, t = 1, . . . , T

Numerical tests with synthetic data

17

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.10.20.30.40.50.60.70.80.9

11.1

β

Prob

abilit

y of

Suc

cess

n  Probability of correct rank-1 CPF solution

n  Probing works better for sufficiently different states

5 10 15 20 25 30 35 400

0.02

0.04

0.06

0.08

0.1

0.12

SNR

RM

SE

WLAVWLS

n  State RMSE for CPSSE

kvt � vt+1k2

(vt,vt+1)

(T = 2)

Numerical tests with real data

18

10:00 a.m. 12:00 p.m. 2:00 p.m. 4:00 p.m.Time

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

RM

SE

WLAVWLS

n  RMSE on system state over one day

n  Probing by changing power factors in smart inverters

n  Actual load/solar data (Pecan Str project)

Conclusions

19

n  Inverter data collected via intentional probing

n  Identifiability for possibly meshed and polyphase grids

n  Improvements with increasing T

n  SDP-/SOCP-based solvers for CPF and CPPSE

n  Optimal probing design? PMU data? Line flow data? Probing for topology identification?

Passive injection learning

n  Smart meter and synchrophasor data

n  Identifiability for single-phase radial grids

n  Simple LS solver using LDF model

n  No time coupling; timescale depends on data

n  Meshed and polyphase grids? Optimal meter selection?

Active injection learning

Thank You!