Estimation of electromechanical modes in power systems using system identification techniques Vedran S. Peric, Luigi Vanfretti, X. Bombois E-mail: vperic@kth.se, luigiv@kth.se, xavier.bombois@ec-lyon.fr KTH Stockholm, Sweden Statnett SF Oslo, Norway Laboratoire Ampere UMR CNRS 5005 Ecole Centrale de Lyon, France Seminar Massachusetts Institute of Technology (MIT) February 26, 2015.
Acknowledgment The work presented here is part of work package 3, on off-line model validation, of the EU funded FP7 itesla project: http://www.iteslaproject.eu/ The work has been carried in collaboration between: KTH: Vedran Peric, L. Vanfretti Laboratoire Ampere UMR CNRS 5005: Xavier Bombois STATNETT SF: L. Vanfretti
Outline Electric power systems Electromechanical oscillations (modes) Basic principles of mode estimation Optimal probing signal design for mode estimation Optimal signal selection for ambient mode estimation Practical implementation 3/23
Electric power systems The largest and most complex machine ever built by humankind Generation (Thermal and hydro power plants, renewables) Transmission (High voltage grid electric power highways) Distribution (Medium and low voltage grid local roads) Consumers 4/23
Electromechanical oscillations Generators or rotating masses Transmission grid or elastic shaft 5/23
f [Hz] Need for the oscillation monitoring 50.15 50.1 20110219_0755-0825 February 19th 2011 North-South Inter-Area Oscillation PMU Data 50.05 50 49.95 49.9 Oscillations if lightly damped can lead to a system black-out 49.85 08:00:00 08:05:00 08:10:00 08:15:00 Oscillations occupy transmission capacities and increase losses Freq. Mettlen Freq. Brindisi Freq. Kassoe Continuously monitor frequency and damping 6
Types of power system dynamic responses Input Output 1 0.5 0 10 5 0 Permanent low power disturbances 100 120 140 160 Switching/Faults 100 120 140 160 H (s) Power Syst em Unknown Dynamics + + y(t) Out put µ(t) Measurement Noise 4 2 0 20 10 0-10 Ambient Data 100 120 140 160 Transient Ringdown 100 105 110 115 ( s z )( s z )...( s z ) ( )( )...( ) 1 2 m H () s k s p 1 s p 2 s p n p i (frequency and damping ratio) 7
Algorithms for measurement based mode estimation Mode estimation based on measured signals Transient response analysis Ambient analysis Probing Transient response: Prony, ERA, Pencil Matrix Well established Good accuracy Not suitable for real-time monitoring Ambient and probing : SysID and signal processing algorithms Suitable for real-time monitoring Lower accuracy 8/23
Ambient based mode estimation Assumptions: The system operates in quasi steady state period Behavior of the system is modeled by a linear model System is excited only by small load changes modeled as white noise Load Changes Power System H(jω) Measurements The model of measured stochastic signal is determined by a model set and N set of parameters. Most common model set is: H z = B z A z = q k=0 p b k z k 1 + k=0 a k z k 1 min ε t, θ 2 θ N t=1 ε(t, θ) = y(t) y(t t 1 9/23
Probing based mode estimation Inputs (load noise) Power system dx/dt=ax+bu y=cx+du Outputs (PMUs) Deterministic signal Probing signals FACTS devices AVR Turbine governors ARMA model estimation Exactly known excitation brings new information that can be used for improved mode identification 10/23
Model of the system determined as a solution of the optimization problem N 1 min θ N t=1 In case of probing, the model is: ARMAX Box Jenkins Difference between ambient and probing based mode estimation ε t, θ 2 B( z, ) C( z, ) y( t) u( t) e( t) A( z, ) D( z, ) ε(t, θ) = y(t) y (t t 1 e(t) random load H(θ,z) u(t) - designed input signal G(θ,z) y(t) measured signal 11/23
The goal is to identify the critical damping ratio of G(z) Parameter covariance matrix u(t) - designed input signal G(θ,z) The critical damping ratio is parameter of G(z) (element of θ) e(t) H(θ,z) y(t) random measured load signal 1 N 1 * N * P F (, ) F (, ) ( ) d F (, ) F (, ) d 2 u 0 u 0 u e 0 e 0 2 2 How should the probing signal look like??? 1) Length 2) Frequency spectrum 3) Time domain 12/23
Global algorithm (two steps) Spectrum calculation (LMI optimization) Time domain signal realization FIR filter Sample autocorrelation optimization Multi-sine input signal max var(ζ) white noise - e(t) Probing Φ u (ω) calculation LMI ACF (r d ) Multisine Signal realization FIR filter min( r-r d 2 ) min(u 2 peak /u 2 rms ) u(t) u(t) u(t) 13/23
Spectrum calculation Requirements : 1) Control effort 2) System disturbance 3) Accuracy Opt. criterion: Spectrum parameterization: 1) ( ) u Constraint: var( ) Keeping in mind: k k 2 1 2 u 2 2 u min J ( ) d G (s) ( ) d u(t) Input Output (frequency deviation) m M jr 2 2 ce r u Ar r Ar r rm 2 r 1 2) ( ) ( ) ( ) e P e r r - tolerance T i i i 1 N 1 * N * P F (, ) F (, ) ( ) d F (, ) F (, ) d 2 u 0 u 0 u e 0 e 0 2 2 14/23
Time domain signal realization Optimization based signal realization with constrained amplitude Multisine realization with crest factor minimization FIR filter signal realization (constant crest factor) max var(ζ) white noise - e(t) Probing Φ u (ω) calculation LMI ACF (r d ) Multisine Signal realization FIR filter min( r-r d 2 ) min(u 2 peak /u 2 rms ) u(t) u(t) u(t) 15/23
Power spectrum Sample autocorrelation ACF Optimization u( k) k MK 0 Efficient recursive algorithm Sample by sample Signal realization with constrained i 1 k 1 ( ) u( i) u( i ) k ( ) ( ) 2 min ACF ACF k ( ) rm Every sample result of a simple optimization problem u des m magnitude ce r jr Fiting Criterion Autocorrelation 10 5 10 4 10 3 Criterion Value Crest Factor 1 10 2 40 50 60 70 80 90 Applied Signal Limit 2000 Limit=30 1500 Limit=40 1000 Limit=50 Limit=60 500 Desired 0-500 -1000 2.5 2 1.5-1500 0 5 10 15 20 Time-lag () 16/23
Multisine signal realization Multisine signal: Two variables that describe multisine Amplitude Phase Crest factor: u( t) A cos( t ) r 1 max{ u( t)} max{ u( t)} max{ u( t)} Cr{ u( t)} rms{ u( t)} 1 var{ u( t)} ud 2 Crest factor minimization algorithm M r r r Initial random phases Cut the peaks FFT Take the phases Construct the signal Crest improved Yes END 17/23
y(t)=h(t)*e(t) where e(t) is a white nose Convenient realization- only random number generator required FIR filter determined by spectral factorization from autocorrelation coefficients FIR signal realization FIR filter y(n) b x(n)... b x(n k) 0 n e(t) white noise u ( ) m rm ce r jr c r Spectral Factorization FIR filter Autocorrelation Optimal input signal sequence B.D.O. Anderson, K. Hitz and N. Diem, Recursive algorithm for spectral factorization, IEEE Trans. Circuits and Systems, vol.21, no.6, pp.742,750, Nov. 1974. 18/23
Optimal probing signal design results 15000 4 x 105 Power spectrum 10000 5000 Power spectrum 3 2 1 0 0 0.5 1 1.5 2 2.5 Frequency [Hz] Minimized input 0 0 0.5 1 1.5 2 2.5 Frequency [Hz] Minimized output Input spectrum parameterization White noise Multi-sine FIR filter var{u(t)} 10410.0 1441.58 1933.55 var{y(t)} 1.6761 1.598 1.5515 var{uy(t)} 6881.10 2318.81 2518.24 19/23
Optimal signal selection for mode estimation Optimality criterion is the variance of the estimate N 1 * N * P 1 F (, ) F (, ) ( ) d F (, ) F (, ) d 2 u 0 u 0 u e 0 e 0 F (, ) F 2 * (, ) d e 0 e 0 2 2 Compute asymptotic variance for each measured signal Rank the signals Select the top ones 20/23
Optimal signal selection results 1 2 3 4 5 6 7 8 9 10 0.23 0.22 0.21 0.2 0.19 0.18 0.17 0.16 Voltage magnitudes 0.15 37 26 36 32 34 39 38 43 22 21 1 2 3 4 5 6 7 8 9 10 0.15 0.14 0.13 0.12 0.11 0.1 Voltage angles 0.09 49 48 50 17 40 31 18 42 44 47 21/23
KTH Wide Area Measurements System Real system PMU 1 Data in IEEE C37.118 Protocol Real time simulator PMU 2 PMU n Communication Network PDC SDK User application Source Measurements Comm. Infrastructure Decoder Power system or real-time simulator Phasor measurement units (PMUs). Phasor data concentrator (PDC) - aggregator. Communication Network: composed by routers/switches, fiber optic links (or other medium) Software Development Kit (SDK) 22/23
Application example - ambient mode estimator Real-life data The critical mode is at 0.39 Hz with damping ratio 9 % (in average) Other modes are observable at 0.2 Hz, 1 Hz and 1.4 Hz 0.5 Hz mode sporadically appears as a poorly damped mode 23/23
Conclusions Monitoring or electromechanical modes is important Staged experiments (probing) can provide good accuracy Probing signal has to be carefully designed In case of ambient mode estimation, signals used have to be carefully selected Testing of the mode estimation tools need to include others components of the system 24/23
Thank you! Questions? Vedran Perić email:vperic@kth.se