1 How to Study and Distinguish Forced Oscillations Lei CHEN PhD, Associate Professor Department of Electrical Engineering Tsinghua University chenlei08@tsinghua.edu.cn
2 Natural and Forced Oscillations Two types of oscillations are widely observed Natural/Free oscillation - Oscillations due to undamped system modes Forced oscillation - Oscillations from periodic sources external to the system x Ax n x( t) e i i1 t ic i Natural x Ax f () t f() t is periodic Forced
3 Oscillation Type Distinguishing Why should we distinguish oscillation type? Different control measures for different oscillation types Natural oscillation: Increase the damping ratio of the critical mode and the oscillation will decay Forced oscillation: Remove the external disturbance Why is this problem difficult? The approach should be measurement-based, or it is not online applicable Both oscillation types show sustained oscillation with constant amplitude in steady state Actual oscillation waveforms are complicated
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5 How to Study Forced Oscillation Natural oscillation Determined by system features Eigenvalue analysis or modal analysis Time-domain simulation Forced oscillation Determined by both external disturbances and system features Time-domain simulation method is applicable, but not capable of analytic and quantitative analysis Extended modal analysis
6 Extended Modal Analysis Linearized system Modal transformation Uncoupled system x Ax Bu x Φz 1 z Λz Bu 150 120 90 0.015 0.01 0.005 60 Gen 1 Gen2 Gen3 Gen430 180 0 Natural oscillation u=0 rt T zr zr0e r x0e rt * T zr zr0e r x0e t r * r t 210 240 270 300 330 n1 rt x () t z e e e i ir r0 r1 n1 r1 j( t ) j( t ) dr ir r dr ir r 2 z e cos( t ) rt ir r0 dr ir r ir ir Mode shape
7 Extended Modal Analysis Forced oscillation z Λ z z Λ z 1 Bu Assumption: only one sinusoidal disturbance Bu = PT PTm sint Focus on steady state response P T P e j t Tm Complex phasor representation z z r r T r PTm j r T r P j Tm r e e jt jt n n r r r r r1 r1 x() t z z j ( ) ( ) n T T T T r r r r r r r r r r r1 (j r)(j r) P j t Tme
8 Steady State Response of Forced Oscillation The rth mode in the ith state variable xi, r ( t) Bi, r sin( t i, r ) B ( a ) ( v b ) 2 2 2 nr r nr i, r 2 2 2 (1 vr ) (2 rvr ) p l 2 2 rvra (1 vr ) vrbnr ir, arctan 2 2 a(1 vr ) 2rvr bnr a ( ) b r ir lr r ir lr ir lr ir lr r Damping ratio nr Natural frequency r v r / nr Frequency ratio Frequency ratio inducing the largest amplitude v a a a 1 2(1 2 )( ) ( ) ( ) 2 2 2 4 2 r r b nr b nr b nr 0 v 1 r Poorly damped r Resonance
9 Resonance - When the frequency of the external disturbance is close to the frequency of a poorly damped mode, the system oscillates with a large amplitude, which is often larger than that of the disturbance Easy to Understand
10 Steady State Response at Resonance When resonance with a poorly damped mode, the mode dominates the response p ir lr l ( ) sin t x t i nr ir lr r nr r ir j ir ir e lr e j lr lr Damping ratio Right eigenvector Left eigenvector i State variable l Location of disturbance
11 p Oscillation Amplitude ir lr l ( ) sin t x t i nr ir lr r nr Oscillation amplitude is affected by Amplitude of disturbance Location of disturbance Damping ratio of mode p l lr r Larger amplitude of disturbance induces larger amplitude of oscillation lr Larger, which means the location of disturbance has stronger controllability on the mode, induces larger amplitude of oscillation Smaller damping ratio induces larger amplitude of oscillation
12 p Mode Shape ir lr l ( ) sin t x t i nr ir lr r nr Relative amplitude and phase of different state variables is also determined by the right eigenvector ir ir Same as natural oscillation Not affected by the location of disturbance At resonance, the mode shape of forced oscillation converges to system mode shape, which brings difficulties in oscillation type distinguishing and source location
13 Simulations 1 G1 G3 3 Inter-area mode Right eigenvector 5 6 7 8 9 10 11 C7 C9 2 L7 L9 4 G2 G4 2.15%
14 Simulations Disturbance on G1 Disturbance on G2 Disturbance on G3 Disturbance on G4
15 How to Distinguish Forced Oscillation Distinguishing natural and forced oscillations based on system measurements is critical for control measure decisions In steady state, both natural and forced oscillations show sustained oscillations with nearly constant amplitude A damped oscillation is obviously natural oscillation Not Critical Oscillation with negative damping will converge to constant-amplitude oscillation due to nonlinearities such as saturations and limits in actual systems Mode shapes are similar when resonance Not easy to distinguish
16 Influence factor Fundamental Differences Natural oscillation - system features Forced oscillation - system features + external disturbances The steady state waveform of natural oscillation is mainly sinusoidal If the external disturbance is non-sinusoid, the forced oscillation waveform will also deviate a lot from sinusoid An obvious non-sinusoidal waveform is a sufficient but unnecessary indicator of forced oscillation Intrinsic system damping Natural oscillation - zero or negative Forced oscillation positive How to obtain the intrinsic damping from its outward performance?
17 1. Harmonic Content of Steady State Waveform Obvious non-sinusoidal waveform in steady state is a sufficient but unnecessary condition of forced oscillation Harmonic content Harmonic content higher than a given threshold is an indicator of nonsinusoidal waveform, and forced oscillation h m m m 2 3 1 h m m 1 Amplitude of fundamental m i Amplitude of ith harmonic A recommended value of the threshold is 0.11, which is the harmonic content index for a triangle wave Simple but Practical
18 Examples h=0.34>0.11 Forced oscillation WECC FO, 2015 From Dan
19 2. Features of Start Up Waveform Different intrinsic system dampings result in different features of start up waveform Start up: the stage when the amplitude increases
20 Features of Start Up Waveform The envelope of start up waveform Ae t B Natural oscillation A 0, B 0, 0 Forced oscillation A 0, B A, 0
21 Features of Start Up Waveform The envelope of start up waveform the difference is σ Steps Peak-peak value increment of peak-peak value logarithm X i Linear fitting to get the slope Ae S>ε: natural oscillation, S<-ε: forced oscillation t B Yi X i X i 1 A(e 1)e T Z ln Y (ln( A(e 1)) T )+ T i i i S T T ( i 1) T
22 Examples Measured S=0.14>0, Natural Simulated S=-0.06<0, Forced
23 3. Spectral Methods (From Ruichao and Dan) The actual system response contains three components The intrinsic damping is contained in the ambient component
24 Natural (undamped) Steady State Response y r t = 2 c r u n v n x 0 cos ω n t + c r u n v n x 0 Transient M + l=1 q l t 2 c r u n v n b 2l cos ω n t + c r u n v n b 2l M + l=1 q l t N i=1 i n,n c r u i v i b 2l e λ it Noise 1: Random Sinusoidal Noise Noise 2: Colored Noise Forced y r t = m=1 N c r u i v i b 1 e λ it A m cos mω 0 t + A m Forced i=1 M + q l t N c r u i v i b 2l e λ it Colored Noise l=1 i=1
25 Signal-Noise Separation First separate the whole response into signal and noise Power spectral density (PSD) of natural oscillation S yr y r ω n = S ysr y Sr ω n + S ynr y Nr ω n S xsr x Sr ω n = 2πc r u n v n x 0 2 δ 0 2 PSD of signal/transient component 2 2 S ynr y Nr ω n 2πc r u n v n b 2l Sql q l ω δ 0 2 l=1 For two different measurements M PSD of noise, dominated by sinusoidal noise ω n 2 α S = S y S1 y S1 = c 1u n S ys2 y S2 ω 2 α N = S y N1 y ω N1 n = c 1u n Equal n c 2 u S n yn2 y N2 ω 2 n c 2 u n 2
26 Signal-Noise Separation Power spectral density (PSD) of forced oscillation S yr y r ω 0 = S ysr y Sr ω 0 + S ynr y Nr ω 0 N 2πA1 c r u i v i b 1 S yrf y rf ω 0 = jω 0 λ i i=1 2 δ 0 2 PSD of signal/forced component M S yrn y rn ω 0 = l=1 N cr u i v i b 2l jω 0 λ i i=1 2 S ql q l ω 0 PSD of noise For two different measurements ω n α S = S y S1 y S1 α S ys2 y S2 ω N = S y N1 y N1 ω n NOT Equal n S yn2 y N2 ω n
27 Natural Examples α S = 3.80 and α N = 3.85 Forced α S = 7.28 and α N = 0.91 Hz Hz Hz 60.02 60 59.98 0 20 40 60 80 100 120 Time (sec.) 60.1 60 59.9 0 20 40 60 80 100 120 Time (sec.) 60.1 60 Noise Response Transient Response Noise + Transient Response Gen 7 spd 59.9 0 20 40 60 80 100 120 Time (sec.) Hz Hz Hz 60.02 60 60.1 60 60.1 60 Noise Response Forced Response Noise + Forced Response Gen 7 spd 59.98 0 20 40 60 80 100 120 Time (sec.) 59.9 0 20 40 60 80 100 120 Time (sec.) 59.9 0 20 40 60 80 100 120 Time (sec.) Hz 60.02 60 Noise Response Gen 22 spd Hz 60.02 60 Noise Response Gen 22 spd Hz Hz 59.98 0 20 40 60 80 100 120 Time (sec.) 60.1 60 59.9 0 20 40 60 80 100 120 Time (sec.) 60.1 60 Transient Response Noise + Transient Response 59.9 0 20 40 60 80 100 120 Time (sec.) Hz Hz 59.98 0 20 40 60 80 100 120 Time (sec.) 60.1 60 59.9 0 20 40 60 80 100 120 Time (sec.) 60.1 60 Forced Response Noise + Forced Response 59.9 0 20 40 60 80 100 120 Time (sec.)
28 Cross-Spectrum Difference Function To detect the existence of the random sinusoid component cross-spectrum difference function S r Ω Y rw1 Y rw2 Y rw2 Y rw3 Y rwi the scaled DFT of the signal over window i Cross-spectrum index E S r Ω S g Ω 2 C rg Ω E S r Ω S r Ω E S g Ω S g Ω r, g: different measurement channels Criterion C rg Ω = 1 C rg Ω < 1 Natural Forced
29 Examples Natural Forced 1.028 1.028 1.0275 1.0275 Voltage amplitude (p.u) 1.027 1.0265 1.026 Voltage amplitude (p.u) 1.027 1.0265 1.026 1.0255 1.0255 1.025 0 10 20 30 40 50 60 70 80 90 100 Time (s) 1.025 0 10 20 30 40 50 60 70 80 90 100 Time (s) 1 1 0.9 0.9 0.8 0.8 Mean with Standard Deviation 0.7 0.6 0.5 0.4 0.3 Mean with Standard Deviation 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.1 0.1 0 0 2 4 6 8 10 12 14 16 18 20 Channel 0 0 2 4 6 8 10 12 14 16 18 20 Channel
30 Examples An actual forced oscillation incident in the western North American power system 1 0.9 0.8 0.7 0.6 C rg [ o ] 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 Pair
Conclusions Some methods for online distinguishing natural and forced oscillations are proposed The problem is NOT well solved Due to the complexity of actual oscillation curves, many methods, though have solid theoretical foundations and perform well with simulation results, do not perform well with actual records More practical approaches are still needed
32 Thanks! Lei CHEN PhD, Associate Professor Department of Electrical Engineering Tsinghua University chenlei08@tsinghua.edu.cn