11 th Int. Workshop on EPCC, May -5, 011, Altea, Spain A Robust State Estimator Based on Maimum Eponential Square (MES) Wenchuan Wu Ye Guo Boming Zhang, et al Dept. of Electrical Eng. Tsinghua University Beijing, China
Introduction How to suppress bad data in SE? Largest normalized residual (LNR) approach Residual smearing problem? Robust estimator M-estimators (Such as WLAV, QC, QL, SHGM) leverage bad data? Calculation speed?
Introduction() The Proposed Maimum Eponential Square (MES) Estimator Differentiable objective function Avoid leverage point problem Strong ability to suppress bad data Similar implementation with WLS estimator Fast calculation speed (approaches the speed of FDSE+LNR)
Maimum Eponential Square (MES) Model For measurement equation z h() e A maimization problem with eponential square objective function m ma J ( ) wi ep( rsi ) i 1 r si = z i h( ) i
Mathematical Characteristics Larger residual has less impact on objective function 1. ep( r si ) 1 0. 8 0. 6 0. 4 BD 0. r si 0-6 -5-4 -3 - -1 0 1 3 4 5 6
Eplained by information theory Parzen window method with Gaussian kernel: The estimated pdf for random variable f e γ m 1 1 z ep i zi n (1) m i 1 σ: The width of Parzen window A nonparametric estimation method No prior knowledge about the random variables distribution type is needed, so gross errors can be treated in Parzen window method T
Eplained by information theory() If the information loss of the estimator is measured by Renyi s quadratic entropy as follows H f d e log e e e Maimizing (1), as done in MES method, is just to minimize (), the information loss of the estimator
MES vs WLS If no gross error eists, the MES estimator can be epanded into a Taylor series such as i. e. m m 4 i si i si si i 1 i 1 ma J ( ) w ep r w 1 r o r min m i 1 wr i si MES..WLS Which is similar with WLS estimator
Solution Method -Optimization condition: J i m 1 i T H W( )( z h( )) 0 f( ) 0 i ( ) wi ep( rsi ) Wii ( ) i ( ) / Hessian matri: J T ( ) H W( ) I diag{ z h } H @ Q Newton Iteration: Q k -H T W( k )( z h( k )) f k 1 k k f f ( )
Leverage Point Problem The relationship between residual r and measurement error e r = Ke K: residual sensitivity matri For WLS estimator: WLS T -1-1 T -1 K = I - H H R H H R The K WLS is less relevant with measurement values K ii 0 for a leverage point
Leverage Point Problem() For MES estimator: The K WLS is relevant with measurement value For the bad measurement i with a large residual, ( ) w ep( r ) the corresponding MES -1 T K = I - HQ H W W ( ) ( ) / 0 ii i MES K ii 1 i i si MES estimator can successfully identify BD even at conventional leverage point.
Leverage Point Test Measurement 1 is a leverage point in WLS estimator (due to the small reactance of branch 1-) Gross error is added to measurement 1 MES estimator did not suffer leverage point problem. 1 3 4 =0.1 p.u. 5 =1.0 p.u. 1 3 V 1 1.045 30 6 =1.0 p.u. V 3 1.0 V 1.05 35
Global Optimal Point MES method is similar to Parzen window method. In the Parzen window method, the average value of pdf in the Parzen window is assumed as the value of pdf at the window s centre point. If is smaller, the assumption is correct. If is larger, optimal solution may deviate from true value. should be adjusted from larger to smaller so as for the MES estimator to reach the true optimal solution
Global Optimal Point() 1.05 (1.05) 1.0-30 Bus1 Bus r+j=0.+j One BD J() P 1 +jq 1 =0.695+j0.0696 (-0.673+j0.0684) (0.663+j0.0677) smaller J() P 1 +jq 1 =-0.554+j0.0709 (-0.543+j0.0715) (0.51-j0.071) larger
How to get global optimal point adjust σ step by step from a larger to a smaller. The converge domain of Newton method: Keep Hessian matri f negative definite at each adjust step f ( ) f k 1 f k f( ) J( ) 0 ˆ k 1 ˆ 0 1 k Converge interval of in Newton method ˆ 1
Numerical Results 9-bus system Conforming errors at 3 rd and 4 th TR#1 Bus1 BusA Bus Gen Gen1 1 BusB BusC TR# 3 4 TR#3 Bus3 Gen3
Numerical Results() Bad Data wrongly estimated Correctly estimated Measurement Number True value 1 Meas. value FDSE+LNR method Estimated Meas. 3 Meas. est. Error 1-3 Estimated Meas. 4 MES method Meas. est. error 1-4 1 st 71.64 71.64 51.61 0.03 71.71-0.07 1.0000 nd -71.64-71.64-51.61-0.03-71.71 0.07 1.0000 3 rd 30.70 10.70 11.3 19.38 30.8-0.1 8.07 10-6 ( ) 4 th -30.57-10.57-11.41-0.00-30.53-0.04 8.951 10-6 LNR method fails to identify these conforming errors MES estimator estimates accurate results directly
Objective function ωi() for measurement i values Numerical Results(3) ( ) 1. 1st nd 3r d 4t h Good data 1 0. 8 0. 6 0. 4 0. 0 1 3 4 5 6 7 8 9 10 11 1 13 Times of σ tunning i ( ) wi ep( rsi ) Bad data J(). 5 1. 5 1 0. 5 0 19. 5 19 1 3 4 5 6 7 8 9 10 11 1 13 Times of σ tunning m J ( ) ( ) From down to 0 i 1 i
Numerical tests 118-bus system Totally 11 different bad data percentages ranging from 0% to 10% are generated For each bad data percentage, 30 different cases are randomly produced and calculated the averages of mean absolute estimation errors of bus voltages and angles under each bad data percentage are recorded
Numerical tests() (Average est. err.) Mean est. error of V, p.u. Mean est. error of θ, rad 0. 0016 0. 0014 0. 001 0.001 0. 0008 0. 0006 0. 0004 0. 0045 0.004 0. 0035 0.003 0. 005 0.00 0. 0015 0.001 0. 0005 V θ FDSE+LNR MES 0 1 3 4 5 6 7 8 9 10 Bad data percentage, % FDSE+LNR MES 0 1 3 4 5 6 7 8 9 10 Bad data percentage,% FDSE+LNR MES FDSE+LNR MES
Practical Applications A provincial power system in Central-China 546 buses, 737 branches, daily peak load 8GW Measurement acceptance rate inde : n m 100% n : the number of measurements whose residual is less than a threshold. m : the number of whole measurements
Practical Application() η inde η (%) CPU t i me( s) 95 94 93 9 91 90 89 88 87 6 5 4 3 1 0 FDSE+LNR MES 1 3 4 5 6 7 8 9 10 Case ID CPU time FDSE+LNR MES 1 3 4 5 6 7 8 9 10 Case I D MES(93-94%) FDSE+LNR (90-9%) MES(5 sec) FDSE+LNR(4 sec)
Practical Application(3) Residual distribution: MES estimator separates more residuals to a small region and to a significant large region. Number of measurements 1600 1400 100 1000 800 600 400 00 0 Clearly good 1358 166 MES 397 47 FDSE+LNR MES Lacking clarity 187 11 11 7 5 43 39 31 35 6 18138 [0,1) [1,) [,3) [3,4) [4,5) [5,6) [6,7) [7, ) Weighted residual region Clearly bad
Thank You!