New Approaches for the Design of Synchrophasor Estimation Systems

Transcription

1 New Approaches for the Design of Synchrophasor Estimation Systems Francisco Messina, Pablo Marchi, Leonardo Rey Vega, and Cecilia G. Galarza Facultad de Ingeniería, UBA, Argentina Centro de Simulación Computacional, CONICET, Argentina Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

2 Algorithm Requirements of the IEEE Std. The algorithms must be compliant with the IEEE C Std. and its recent amendment C a The IEEE Std. defines limits for both stationary and dynamic tests. Error Metrics TVE(t) = X (t) X (t), FE(t) = f (t) f (t), X (t) RFE(t) = f (t) f (t), where: X (t): synchrophasor. f (t): frequency. f (t): ROCOF. X (t): synchrophasor estimation. f (t): frequency estimation. f (t): ROCOF estimation. Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

3 Brief Review of the State of the Art Stationary Algorithms DFT recursive algorithm. Windowed and interpolated variations of the DFT: WDFT, IpDFT. Problems: poor filtering and dynamic behavior. Dynamic Algorithms Taylor based algorithms: 4PM, 6PM, TWLS, TFT-WLS, IpD 2 FT. PLL based algorithms: SRF-PLL, DDSRF-PLL, EPLL. Problems: filtering still poor. Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

4 A Modular Approach Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

5 A Modular Positive-Sequence Estimation Algorithm We have proposed the following system [1]: x a [n] x b [n] x c [n] 3PD x d [n] x q [n] PF PF y d [n] y q [n] DSS-PLL â[n] φ[n] ω[n] α[n] NCA ã[n] φ[n] ω[n] α[n] Why this particular structure? [1] F. Messina, P. Marchi, L. Rey Vega, C. G. Galarza, and H. Laiz, A Novel Modular Positive-Sequence Synchrophasor Estimation Algorithm for PMUs, accepted for publication in the IEEE Transactions on Instrumentation and Measurement. Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

6 Three-Phase Demodulation (3PD) The 3PD is related to the Park or abc-dq transformation. For the fundamental component, it produces: [ ] [ ] cos(φ1 [n]) cos(2ω0 nt + φ x dq [n] = a 1 [n] + a sin(φ 1 [n]) 2 [n] 2 [n]). sin(2ω 0 nt + φ 2 [n]) Unlike a 1PD, it avoids the creation of double-frequency terms for balanced input signals. It separates the positive- and negative- sequence components and completely eliminates the zero-sequence component. The interference power frequency distribution is changed similarly. Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

7 The Pre-Filters (PFs) The PFs are used to relax the PLL filtering requirements. Linear-phase is a strong requirement: TVE max µ φ sup η Ω p τ PF (η) 1%, Thus, we use symmetrical FIR PFs. The IFIR method can be used to significantly reduce the computational cost: C 10 log 10(δ p δ s /2) M ft + M 10 log 10(δ p δ s /2) (1 M(f s + f p )T ), Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

8 The Digital State-Space PLL (DSS-PLL) The DSS-PLL extends the standard SRF-PLL by introducing a state-space model for amplitude, phase, frequency, and ROCOF. y d [n] RT z d [n] APD u a [n] H a (z) â[n] φ[n] y q [n] z q [n] u φ [n] ω[n] G φ (z) T + α[n] T 2 /2 φ p [n] = φ[n n 1] z 1 φ[n + 1 n] Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

9 The Narrowband Compensation Algorithm (NCA) The NCA can be summarized as follows [2]: α[n] = α[n + m], ω[n] = ω[n + m] + δ α[n], φ[n] = φ[n + m] + δ ω[n] + δ2 2 ã[n] = â[n + m]/g( ω[n]), α[n] θ( ω[n]), τ = mt + δ: total PMU delay. φ[n], φ[n]: uncompensated and compensated phase estimate. ω[n], ω[n]: uncompensated and compensated frequency estimate. α[n], α[n]: uncompensated and compensated ROCOF estimate. â[n], ã[n]: uncompensated and compensated amplitude estimate. [2] F. Messina, L. Rey Vega, C. G. Galarza, and H. Laiz, An Accurate Phase Compensation Algorithm for PMUs, in Conference on Precision Electromagnetic Measurements (CPEM), Ottawa, Canada, Jul Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

10 Dynamic Tests: Amplitude & Phase Modulation Test The modulation frequency is set to 2 Hz, amplitude modulation factor to 0.1, and phase modulation factor to 0.1 rad Result Upper bound 0.25 Result Upper bound TVE [%] TVE [%] t [s] t [s] Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

11 Dynamic Tests: Frequency Ramp Test After 5 s of a pure sinusoidal signal, the ramp rate is set to 1 Hz/s, being the initial frequency 48 Hz and the final frequency 52 Hz. RFE [Hz/s] Result Std. limit Exclusion interval RFE [Hz/s] Result Std. limit Exclusion interval t [s] t [s] Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

12 Conclusions and Future Work We have decoupled the synchrophasor estimation problem into demodulation, filtering, dynamic tracking, and compensation. It is worth to emphasize that all requirements of the IEEE Std. were met with this structure. As one possible improvement, one could use quasi-linear-phase IIR pre-filters. As another one, one could design independently the three loop filters. Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

13 An Optimization Based Approach Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

14 Synchrophasor Estimation System General model for a synchrophasor estimation system [3]: U[n] X [n] + A(ν) Û[n] I[n] U[n]: signal phasor. X [n]: input phasor. A(ν): zero-phase filter. I[n]: interference phasor. Û[n]: estimated phasor. [3] F. Messina, P. Marchi, L. Rey Vega, and C. G. Galarza, Design of Synchrophasor Estimation Systems with Convex Semi-Infinite Programming, in IEEE Electrical Power and Energy Conference (EPEC), Ottawa, Canada, Oct Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

15 Problem and Objective Function Problem The problem to be addressed is the design of the filter A(ν) = e T (ν)a, where e(ν) = (1,..., cos(νn/2)) T, and a R N/2+1. Objective Function The general goal is the minimization of the TVE or Û[n] U[n] 2. Obviously, this metric depends on the unknown signal phasor U[n]. Applying the Cauchy-Schwarz inequality to Û[n] U[n] 2 gives: f (a) = νsb ν sb A(ν) 1 2 dν = a T Pa + q T a + r This is a convex quadratic function of a, which should be minimized. Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

16 Constraints The restrictions imposed by the designer should be included in the optimization problem formulation. General Constraint Formulation Let X θ [n] = U θ [n] + I θ [n] be the input phasor associated with a particular test, θ Θ the parameter, and n N Z the time index of interest. The associated TVE is TVE θ [n] so that the constraint reads: sup TVE 2 θ [n] ε2, (n,θ) N Θ where ε is the maximum TVE as specified by the designer. Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

17 Constraints: Examples Interference Rejection Signal: X θ [n] = a s e j(νsn+φs) + a i e j(ν i n+φ i ), N = Z, θ = (ν s, ν i ) T, Θ = Ω s Ω i. Constraint: A(ν s ) 1 + a i a s A(ν i ) TVE INT, θ Θ. Frequency Ramp Signal: X [n] = a s e jφs e jγn2, n = 0,..., N f 1, N = {n 1,..., n 2 }. Approximation: θ = γn, Θ = [γn 1, γn 2 ]. Constraint: a T P(θ)a + q T (θ)a + 1 TVE 2 FR, θ Θ, P(θ) = b(θ)b H (θ), q(θ) = 2 Re{b(θ)}, (b(θ)) k = cos(2θk)e jγk2. Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

18 Filter Design Formulation CSIP Problem The filter design problem is a convex semi-infinite program (CSIP): min a f (a) subject to g k (a, θ k ) 0, θ k Θ k, k = 1,..., p, a R N/2+1 : optimization variables. Θ k R m k : infinite sets. f : R N/2+1 R: objective (convex) function. g k : R N/2+1 Θ k R: constraint (convex in a for each θ k ) functions. Property: Any local minimum is also a global solution. Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

19 A Design Example We have designed a half latency (τ = 20 ms, N = 40) P class PMU filter with F s = 50 fps, f 0 = 50 Hz, T = 1 ms, f p = 2 Hz, f s = 46 Hz. Problem Specifications TVE STA 0.25% TVE INT 0.25% TVE FR 0.25% TVE AM 0.75% TVE PM 0.75% AO max 2.5% PO max 2.5% Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

20 Frequency and Impulse Responses Also shown are standard equiripple (EQU) and least-squares (LS) designs. Magnitude (db) Magnitude Response (db) Frequency (Hz) Impulse Response CSIP EQU LS CSIP EQU LS Amplitude Samples Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

21 Dynamic Tests: Amplitude Modulation Test Specifications Modulation frequency is 2 Hz and amplitude modulation factor CSIP EQU LS TVE [%] t [s] Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

22 Dynamic Tests: Frequency Ramp Test Specifications Ramp rate is 1 Hz/s, initial frequency 48 Hz and final frequency 52 Hz CSIP EQU LS TVE [%] t [s] Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

23 Conclusions and Future Work The synchrophasor estimation problem was posed as a convex semi-infinite program for guaranteed optimal performance. This provides a systematic and flexible design tool. Customized designs can be obtained for both frequency and time constraints. The chosen objective function tends to maximize the filter bandwidth. We are working on generalizations to provide frequency and ROCOF estimations with specific constraints. Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24

24 Thank you for your attention! Messina et al. (FI-UBA,CSC-CONICET) SimPMU 2016 October 27 th / 24