Least costly probing signal design for power system mode estimation

Transcription

1 1 Least costly probing signal design for power system mode estimation Vedran S. Perić, Xavier Bombois, Luigi Vanfretti KTH Royal Institute of Technology, Stockholm, Sweden NASPI Meeting, March 23, 2015.

2 Outline Motivation Probing-based mode estimation Optimization problem formulation Results Conclusions Q&A 2/13

3 Oscillation monitoring _ February 19th 2011 North-South Inter-Area Oscillation 50.1 PMU Data f [Hz] Oscillations if lightly damped can lead to a system black-out :00:00 08:05:00 08:10:00 08:15:00 Oscillations occupy transmission capacities and increase losses Frequency and damping ratio monitoring 3/13

4 Algorithms for measurement based mode estimation Mode estimation using 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 4/13

5 Probing based mode estimation Inputs (load noise) Power system dx/dt=ax+bu+fz y=cx+du+gz Outputs (PMUs) Deterministic signal Probing signals FACTS devices AVR Turbine governors Measured signals Exactly known excitation brings new information that can be used for improved mode identification 5/13

6 Model of the system e(t) random load Mathematical description of the probing-based mode estimation H(θ,z) u(t) - designed input signal G(θ,z) y(t) measured signal In case of probing, the model is: Bz (, θ) Cz (, θ) yt () = ut () + et () Az (, θ) Dz (, θ) Box Jenkins Mode estimation as an optimization problem 1 min θθ NN NN εε tt, θθ 2 tt=1 εε(tt, θθ) = yy(tt) yy (tt tt 1 ARMAX 6/13

7 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 = ωθ ωθ ω ω + ωθ ωθ ω θ * σ Φ N * P F (, ) F (, ) ( ) d F (, ) F (, ) d 2 u 0 u 0 u e 0 e How should the probing signal look like??? 1) Length 2) Frequency spectrum 3) Time domain 7/13

8 Spectrum calculation Requirements : 1) Control effort 2) System disturbance 3) Accuracy Opt. criterion: k k ω ω ω ω u 2 2 u min J = Φ ( ) d + G (s) Φ ( ) d u(t) Input Output (frequency deviation) Constraint: var( ) Keeping in mind: ζ = e Pe < r r - tolerance T i i θ i 1 N 1 * N * P = F (, ) F (, ) ( ) d F (, ) F (, ) d θ ωθ ωθ Φ ω ω + ωθ ωθ ω 2 u 0 u 0 u e 0 e 0 σ 2 2 8/13

9 Global algorithm (two steps) Spectrum calculation (LMI optimization) Time domain signal realization FIR filter Sample autocorrelation optimization Multi-sine input signal [1] 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) [1] J.W. Pierre, et al. Probing signal design for power system identification, IEEE Trans. Power Syst., vol.25, no.2, pp , May /13

10 Power spectrum Sample autocorrelation ACF Optimization u( k) k M+ K τ = 0 Efficient recursive algorithm Signal realization with constrained i= τ + 1 Φ k 1 ( τ) = uiui () ( τ) k ( ( τ) ( τ) ) 2 min ACF ACF k ( ω) = r = m Sample by sample Every sample result of a simple optimization problem u des m magnitude ce r jωr Fiting Criterion Autocorrelation Criterion Value Crest Factor Applied Signal Limit 2000 Limit= Limit= Limit=50 Limit= Desired Time-lag (τ) 10/13 2 Crest Factor

11 Optimal probing signal design results Minimized input 4 x 105 Minimized output Power spectrum Power spectrum Frequency [Hz] Frequency [Hz] Input spectrum parameterization White noise Multi-sine FIR filter var{u(t)} var{y(t)} var{uy(t)} Optimal probing allows us to reduce probing power and/or system disturbance while maintaining desired accuracy 11/13

12 Conclusions Monitoring of electromechanical modes is important Staged probing tests can provide a good accuracy Shape of the probing signal affects estimation accuracy Several considerations can be taken into account during design process Optimal probing allows us to reduce probing power and/or system disturbance while maintaining accuracy The proposed method is easy to implement 12/13

13 Thank you! Questions? Vedran Perić