sid bhela, vassilis kekatos, and harsha veeramachaneni · learning through power distribution grid...
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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!