L1 minimization method for estimating sparsely represented errors
in electric energy system:
Simulation based investigation of IEEE 300 bus test case with a linearized model
Euiseok Hwang
Advisors: Rohit Negi. Vijayakumar Bhagavatula
Mar. 10, 2010
*Acknowledgements : Zhijian Liu, Marija Ilić
Outline
ƒ Motivation
ƒ L1 minimization based state estimation for a linear
measurement model
ƒ Sparsely represented error simulations
ƒ Sparse errors in node voltage angle measurements
ƒ Summary
2
Motivation
ƒ State estimation of electric energy systems
ƒ Estimate electrical states of the network from noisy measurements
based on given network topology and parameters
ƒ State estimation for sparsely represented measurement error
ƒ i.e., arbitrarily large values for a few elements whose sensor may
have failed, while zero errors for others
ƒ L0-min. method may be efficient, but L0-min. is an NP-hard problem
ƒ L1-min., polynomial-time, is equivalent to L0-min. when the
measurement matrix satisfies a restricted isometric prop. (RIP)
ƒ Investigation of L1-min. methods using numerical simulation
3
State Estimation based on a Linear Measurement Model
ƒ General state estimation problem*
ƒ Nonlinear measurement model, yj = hj(x) +ej, ej : mean 0, variance σj2
ƒ Weighted least square (WLS) state estimation
min J ( x ) = ∑ j =1 ( y j − h j ( x ) ) σ 2j
2
m
x
s.t. gi ( x ) = 0; i = 1,..., ng
ci ( x ) ≤ 0; i = 1,..., nc
m measurements, n state variables, n < m
gi(·) and ci(·) are equality and inequality constraints
ƒ State estimation for a linear measurement model**
(Linearized Model between node active powers, P, & voltage angles, δ)
Pi = −∑ j =1,≠i Yij (δ j − δ i )
⎡P ⎤
y = Ax + e,
x=⎢ ⎥
⎣δ ⎦
n
→ P = Wδ
min ( y − Ax ) C−1 ( y − Ax ) ,
T
x
C = diag (σ 2j )
0 ⎤
⎡I
s.t. ⎢
⎥x = 0
0
−
W
⎣
⎦
*A. Monticelli, Electric Power System State Estimation, Proc. IEEE, vol. 88, no.2 (2000)
**P. Schavemaker, and L. v. d. Sluis, Electrical Power System Essentials, Wiley
4
Estimation of Sparsely Represented Measurement Errors
ƒ Problem statement
ƒ Linear measurements
y = Ax + e,
e is 2n×1 and sparse, near zero values except k elements, k << 2n
A is a m×2n matrix, m < 2n
ƒ Estimate e from y
ƒ L1 minimization method for estimating a sparse error
ƒ Sparse errors in node voltage angle measurements and noiseless
node active power measurements
⎡ P ⎤ ⎡I
y = ⎢ s ⎥ = ⎢ s
⎣δ ⎦ ⎣0
min Δδ 1
Δδ
0⎤ ⎡ P ⎤ ⎡ 0 ⎤
+
,
I ⎥⎦ ⎢⎣ δ ⎥⎦ ⎢⎣ Δδ ⎥⎦
⎡I
A =⎢ s
⎣0
0⎤
⎡0⎤
e
=
,
⎢ Δδ ⎥
I ⎥⎦
⎣ ⎦
s.t. Ws Δδ = b,
b = Ws δ − Ps
Is is a m×n matrix, m < n,
where Ps is a subset of P, Ps= Is P
Δδ is n×1 and sparse
Ws = I s W
2
Δδ 2
* Least square method : min
Δδ
s.t. Ws Δδ = b
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Estimation of Sparsely Represented Measurement Errors
ƒ Estimating sparsely represented error, Δδ, from
min Δδ 1
Δδ
s.t. Ws Δδ = b
"
×
=
%
#
#
b : m ×1, m < n
Ws : m × n, m < n
Δδ : n × 1, Δδ 0 << n
6
Simulation Results
ƒ Sparse errors in bus voltage angle meas. (IEEE 300 test case)
ƒ 290 power measurements (noiseless)
ƒ 32 voltage angle errors, Δδj ~ N ( 0, 0.252 (=var(δ)))
Phase estimation error (290 power meas., 32 phase error)
1000 rmse=0.011 (avg for 20 test sets)
-0.05
0
0.05
0.1
Occurrence
min ||Δδ||1
0.1
0
-0.1
-0.2
1000 rmse=0.008 (avg for 20 test sets)
500 avg no. of err/test=0.70
^
x ^l1min
Δδˆ L,1 ||xl1min||0=299
0.2
2000
1500
= 32
^
x ^mmse
Δδˆ LS, ||xmmse||0=299
0.3
Δδˆ LS − Δδ
500 avg no. of err/test=0.65
0
-0.1
0
0.4
[rad]
Occurrence
min ||Δδ||2
1500
0
-0.1
x,Δδ
||x||
=32
, 0Δδ
0.5
2000
Δδˆ L1 − Δδ
-0.3
-0.4
-0.05
0
0.05
Phase estimation error (rad)
0.1
Histogram of voltage angle estimation error
(290 power meas., 32 voltage angle errors, …)
0
50
100
150
bus (δt)
200
250
300
Injected and estimated voltage angle errors
7
Simulation Results
ƒ Sparse errors in bus voltage angle meas. (IEEE 300 test case)
ƒ 4, 8, 16 and 32 errors, degrees of voltage angle error columns of
W≥4
0
10
290 P meas. (out of 299), deg(W(:,Δδ~=0)) ≥ 4
random phase error, Δδ ~ N(0,0.252)
Estimation failure rate
fail if max(Δδ^-Δδ)>0.05, σ =0.25
δ
-1
10
min || Δδ ||2
min || Δδ ||1
-2
10
0
4
8
16
No. of phase errors
32
Estimation failure rates as a function of the number of voltage angle errors
8
Summary
ƒ Evaluation of IEEE 300 test case with a linearized model
ƒ Sparse voltage angle errors can be better estimated
by L1 minimization method than LS method
(290 noiseless power meas. and 4~32 voltage angle errors)
ƒ Further investigations
ƒ Sparse errors in node voltage angle measurements and
Gaussian noise in node active power measurements
ƒ Subset of node voltage angle measurements with sparsely
represented errors
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