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A Measurement-based Framework for Modeling, Analysis and Control of Large-Scale Power Systems using Synchrophasors Aranya Chakrabortty Electrical & Computer Engineering Department North Carolina State University LCCC Workshop, Lund University 18 th May, 2011

Wide Area Measurements (WAMS) • 2003 blackout in the Eastern Interconnection EIPP (Eastern Interconnection Phasor Project) NASPI (North American Synchrophasor Initiative) Power System Research Consortium (PSRC, 2006-present) Industry Members • Rensselaer (Joe Chow, Murat Arcak) • Virginia Tech (Yilu Liu) • Univ. of Wyoming (John Pierre) • Montana Tech (Dan Trudnowski) • Technical Research (RPI) 1. Model Identification of large-scale power systems 2. Post-disturbance data Analysis 3. Controller and observer designs, robustness, optimization 2

Main trigger: 2003 Northeast Blackout NYC before blackout New England Ohio Power flow INTER-AREA INTER-AREA STABLE UNSTABLE NYC after blackout Hauer, Zhou & Trudnowsky, 2004 Kosterev & Martins, 2004 Lesson learnt: 1. Wide-Area Dynamic Monitoring is important 2. Clustering and aggregation is imperative 3

Model Aggregation using distributed PMU data 6-machine, 30 bus, 3 areas Problem Formulation: 1. Model Reduction • How to form an aggregate model from the large system PMU PMU PMU • Chakrabortty & Chow (2008, 2009, 2010), Chakrabortty & Salazar (2009, 2010) 4

Model Aggregation using distributed PMU data 6-machine, 30 bus, 3 areas Problem Formulation: 1. Model Reduction • How to form an aggregate model from the large system PMU Area 1 PMU PMU Aggregate Transmission Network Area 3 Area 2 • Chakrabortty & Chow (2008, 2009, 2010), Chakrabortty & Salazar (2009, 2010) 4

Two-area Model Estimation ~    I I I PMU PMU     E E 2 2 1 1 H 1 H 2 ~ ~ x 2 x 1 x e  V    V   V V 1 1 1 2 2 2 5

Two-area Model Estimation ~    I I I PMU PMU     E E 2 2 1 1 H 1 H 2 ~ ~ x 2 x 1 x e  V    V   V V 1 1 1 2 2 2     Problem: How to estimate all parameters?  H H H P H P E E     1 2  2 1 1 2 1 2 m m 2 sin     ( H H H H x x x 1 2 ) e 1 2 1 2 x 1 , x 2 , H 1 , H 2 Swing Equation 5

IME: Method ( Reactance Extrapolation) • Key idea : Amplitude of voltage oscillation at any point is a function of its electrical distance from the two fixed voltage sources. 1 2  2   0 E E 1 ~ ~ ~ I V V 1 2 x ~ x        ( ) [ ( 1 ) cos( ) ] sin( ), V x E a E a j E a a 2 1 1  e  x x x 1 2 ~ • Voltage magnitude : 2 2     2     2 2 | ( ) | 2 ( ) cos( ) , ( 1 ) V V x c E E a a c a E a E 1 2 2 1 6

IME: Method ( Reactance Extrapolation) • Key idea : Amplitude of voltage oscillation at any point is a function of its electrical distance from the two fixed voltage sources. 1 2  2   0 E E 1 ~ ~ ~ I V V 1 2 x ~ x        ( ) [ ( 1 ) cos( ) ] sin( ), V x E a E a j E a a 2 1 1  e  x x x 1 2 ~ • Voltage magnitude : 2 2     2     2 2 | ( ) | 2 ( ) cos( ) , ( 1 ) V V x c E E a a c a E a E 1 2 2 1 • Assume the system is initially in an equilibrium ( δ 0 , ω 0 = 0, V ss ) :      ( ) ( , ) V x J a 0    ( , ) V a E E     2  0 1 2 ( , ) : ( ) sin( ) J a a a 0 0    ( , ) V a    0 0 6

Reactance Extrapolation         2 ( t ) ( , ) ( , ) sin( ) ( ) V x t V a E E a a 0 1 2 0 can be computed A from measurements at x     2 ( t ) ( , ) ( ) V n x t A a a Note: Spatial and temporal dependence are separated     2 • Fix time: t=t* ( , *) ( ) ( *) V n x t A a a t How can we use this relation to solve our problem? 7

Reactance Extrapolation     2 ( , *) ( ) ( *) V n x t A a a t PMU PMU 1 2 2    0 E E 1 x 2 x e + x 2 x 2      2 At Bus 2, a ( ) ( *) V A a a t 2 n , Bus 2 2 2  e  x x x  1 2 ( 1 ) V a a , 2 2 2 n Bus    x x ( 1 ) V a a e 2  , 1 1 1 At Bus 1,     n Bus 2 a ( ) ( *) 1 V A a a t n , Bus 1 1 1   x x x 1 2 e • Need one more equation - hence, need one more measurement at a known distance 8

Reactance Extrapolation     2 ( , *) ( ) ( *) V n x t A a a t 3 PMU PMU 1 2 2    0 E E 1 PMU x 2 x e x  2 2 x e + x 2 x 2      2 At Bus 2, a ( ) ( *) V A a a t 2 n , Bus 2 2 2  e  x x x  1 2 ( 1 ) V a a , 2 2 2 n Bus    x x ( 1 ) V a a e 2  , 1 1 1 At Bus 1,     n Bus 2 a ( ) ( *) 1 V A a a t n , Bus 1 1 1   x x x 1 2 e • Need one more equation  ( 1 ) V a a n , Bus 3 3 3  - hence, need one more measurement at a known distance  ( 1 ) V a a , 1 1 1 n Bus 8

IME: Method ( Inertia Estimation) • From linearized model   cos( ) 1 E E  1 2 0 f s    2 2 ( ) H x x x 1 2 e H H  where f s is the measured swing frequency and 1 2 H  H H 1 2 • For a second equation in H 1 and H 2 , use law of conservation of angular momentum                 2 2 2 ( ) ( ) 0 H H H H dt P P P P dt 1 1 2 2 1 1 2 2 1 1 2 2 m e m e  • Reminiscent of Zaborsky’s result H   1 2   H   H 1 2 2 1  H 2 1 • However, ω 1 and ω 2 are not available from PMU data, Estimate ω 1 and ω 2 from the measured frequencies ξ 1 and ξ 2 at Buses 1 and 2 9

IME: Method ( Inertia Estimation) • Express voltage angle θ as a function of δ , and differentiate wrt time to obtain a relation between the machine speeds and bus frequencies: where,           ( ) cos( ) a b c   1 1 1 1 2 1 2 1 2 1      2   2   2 cos( ) a b c ( 1 ) , ( 1 ), a E r b E E r r 1 1 1 2 1 1 1 2 i i i i i 2 2  c E r           ( ) cos( ) a b c 2 i i   2 1 2 1 2 1 2 2 2 2      2 cos( ) a b c 0.6 2 2 1 2 2 0.4 • ξ 1 and ξ 2 are measured, and a i , b i , c i Frequency (r/s) are known from reactance extrapolation. 0.2 x e x 1 x 2 • Hence, we calculate ω 1 / ω 2 to solve 0 for H 1 and H 2 . -0.2 -0.4 0 0.2 0.4 0.6 0.8 1 Normalized Reactance r 10

Illustration: 2-Machine Example • Illustrate IME on classical 2-machine model ( r e = 0) • Disturbance is applied to the system and the response simulated in MATLAB    0 . 0292 0 . 0316 0 . 0371 V V V 1 m 2 m 3 m    1 . 0320 1 . 0317 1 . 0136 V V V 1 ss 2 ss 3 ss    0 . 0301 0 . 0326 0 . 0376 V V V 1 2 3 n n n IME Algorithm Exact values: x 1 = 0.3382 pu x 1 = 0.34 pu, Voltage oscillations at 3 buses x 2 = 0.3880 pu x 2 = 0.39 pu H 1 = 6.48 pu s IME  ( ) G s H 2 = 9.49 pu  1 sT Exact values: H 1 = 6.5 pu, H 2 = 9.5 pu Bus angle oscillations Bus frequency oscillations 11

Application to WECC Data • 2000 gens • 11,000 lines • 22 areas, 6500 loads Colstrip Grand Coulee Malin SVC Vincent 12

Application to WECC Data 1.1 1.08 • 2000 gens • 11,000 lines 1.06 • 22 areas, 6500 loads Bus Voltage (pu) Bus 1 1.04 Bus 2 1.02 Midpoint 1 0.98 Colstrip Grand Coulee 0.96 0.94 0 50 100 150 200 250 300 Time (sec) Malin SVC Needs processing to get usable data • Sudden change/jump • Oscillations Vincent • Slowly varying steady-state (governer effects) 12

WECC Data 1.1 1.08 1.06 Bus Voltage (pu) Bus 1 1.04 Band-pass Bus 2 1.02 Midpoint Filter 1 0.98 Choose pass-band 0.96 covering typical swing mode range 0.94 0 50 100 150 200 250 300 Time (sec) Quasi-steady State Oscillations 1.15 0.02 Bus 1 Bus 2 0.015 1.1 Midpoint Fast Oscillations (pu) 0.01 Slow Voltage (pu) 1.05 0.005 + 0 1 -0.005 -0.01 0.95 -0.015 -0.02 0.9 0 50 100 150 200 250 300 60 65 70 75 80 85 90 Time (sec) Time (sec) 13

WECC Data Oscillations Interarea Oscillations 0.02 0.02 Bus 1 Bus 1 Bus 2 0.015 0.015 Bus 2 Midpoint Interarea Oscillations (pu) Midpoint Fast Oscillations (pu) 0.01 0.01 0.005 0.005 ERA 0 0 -0.005 -0.005 -0.01 -0.01 -0.015 -0.015 -0.02 -0.02 60 65 70 75 80 85 90 0 5 10 15 20 Time (sec) Time (sec) • Can use modal identification methods such as: ERA, Prony, Steiglitz-McBride -3 x 10 20 Bus 3 15 Jacobian Curve 10 Bus 1 x e x e 5 2 2 Bus 2 0 x 2 x 1 0 0.2 0.4 0.6 0.8 1 Normalized Reactance r 14