performance of synchrophasor estimation algorithms - · pdf fileperformance of synchrophasor...

Performance of Synchrophasor

Estimation Algorithms

G. Barchi1, D. Belega2, D. Fontanelli1, D. Macii1, D. Petri1

1University of Trento, Trento, Italy 2“Politehnica” University of Timisoara, Timisoara, Romania

Workshop on Synchrophasor Estimation Processes for Phasor Measurement Units:

Algorithms and Metrological Characterization

Lausanne, Switzerland, December 9, 2014

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Phasor Measurement Units (PMUs) • PMU: an instrument that measures

• Amplitude

• Phase

• Frequency

• Rate of change of frequency (ROCOF)

of voltage or current waveforms synchronized to the UTC

• Originally used at the transmission level, PMUs have recently become more and more interesting in distribution networks

1. Topology and/or fault detection

2. Reverse power flow detection

3. Protection relaying

4. Volt-Var optimization

5. Grid state estimation

Workshop on Synchrophasor Estimation Processes for PMUs – Lausanne, Switzerland, December 9, 2014 2

Traditional PMU

Micro PMU

General block diagram


Antialiasing

filter ADC

Analog

front-end

Synchronized

Clock to UTC

x(t) x[n] Embedded

computing device

tr

|𝑝 𝑟|

Arg{𝑝 𝑟}

GPS RX

or PTP

𝑓 𝑟

𝑅𝑂𝐶𝑂𝐹 𝑟

Estimation

algorithms

over

adjustable

observation

intervals

Modems From/to PDCs

Signal model - 1


ttttftXtx nhpa cos 121 0

nominal frequency

off-nominal (static) fractional freq. offset

average amplitude

amplitude fluctuations

phase fluctuations initial phase

Narrow-band disturbances: harmonics, out-of-band interharmonics

Wideband disturbances: noise, DC decaying offset

Sync N collected samples (e.g. odd)

r-th observation interval

…

time reference tR

t

Synchronized data acquisition (Hp. central reference time)

Sampling time: fs=Mf0

≈C nominal waveform cycles in an observation interval

Signal model - 2


2

12

1

2

0 2

NN

nh

nM

j

rr nnnenpnxrr

,...,,...Re

Discrete-time electrical waveform in the rth observation interval

rrp

r

nj

ar enX

np

12

Discrete-time synchrophasor

sraa nTtnr

srpp nTtnr

rr tf 12 0

rp rpArg ˆ

Synchrophasor estimation Frequency estimation

rf

Instantaneous frequency

sr

r

nTt

p

rdt

dfnf

2

110

rOFCRO ˆ

sr

r

nTt

p

rdt

dnROCOF

2

2

2

1

Instantaneous ROCOF

ROCOF estimation

Initial phase at tr

(Some) performance parameters Measured quantity

Performance parameters Notes

Phasor

Global accuracy in steady-state. Sometimes separate magnitude/phase

accuracy can be more interesting

Responsiveness in transient conditions

Frequency

Frequency accuracy in steady-state


ROCOF

ROCOF accuracy in steady-state



TVErt

TVErt

TVEs TTVETTVEt

infsup

r

rr

rp

ppTVE

ˆ

rrr ffFE

ˆ

rrr ROCOFOFCRORFE

ˆ

FErt

FErt

FEs TFETFEt

infsup

RFErt

RFErt

RFEs TRFETRFEt

infsup

We mainly focus on this

The synchrophasor estimation model in IEEE

Standards C37.118.1 and C37.118.1a


Antialiasing

filter ADC

Analog

front-end

Synchronized

Clock to UTC Digital quadrature

oscillator (f0)

sin(2π/M·n) cos(2π/M·n)

x(t)

x[n]

LP

filter

tr

tr

𝑅𝑒{𝑝 𝑟}

Im{𝑝 𝑟}

The Standards do not recommend any specific estimators. However,

Annex C reports an example for synchrophasor measurement only!

Static vs. dynamic estimators

• Classic phasor estimators

• εa(·) and εp(·) assumed to be generally negligible within an observation interval

• If they are not, they are usually regarded as disturbances

• Dynamic phasor estimators

• εa(·) and εp(·) assumed to be generally significant within an observation interval

• They are regarded as part of the measurand


Windowed DFT

IpDFT

4PM/6PM estimator

IpD2FT

Taylor-WLS estimator

Several effective algorithm have been proposed.

…

…

DCR + LP filtering

The windowed DFT phasor estimator


2

1

2

1

22

N

Nn

nCN

j

rr enwnxN

p

ˆ

rr nhdrrr EECWpWpp 2*ˆ

nominal waveform cycles

even is if

odd is if

MCC

MCC

M

1

Window scalloping loss in case of off-nominal frequency deviation

Mainlobe frequency shift

Image term interference

Contribution due to dynamic fluctuations

εa(·) and εp(·), if present

Harmonic and Inter-harmonic interference

Wideband noise contribution

2

1

2

1

2

0

1

N

Nn

nN

j

enwNW

W

-10 -8 -6 -4 -2 0 2 4 6 8 100

1

2

3

4

5

6

7

8

[%]

Max.

TV

E [%

]

A classic example


Rectangular window case with C=1

Scalloping loss due to off-nominal frequency deviation + image infiltration

…+ AM with amplitude = 10% of

fundamental , + PM with

amplitude = 10° and f = 5 Hz

…+AM+PM + 2nd harmonic = 1% of

fundamental

…+AM+PM + 50 harmonics = 1% of

fundamental + noise with SNR=50 dB

WDFT phasor estimator accuracy: summary

Uncertainty contribution

Notes Possible solutions

Scalloping loss due to

frequency offset

• Strongly depends on window • Error grows with observation

interval length C

• Flat-top windows • Can be compensated (if δ known

or through IpDFT – see later)


• Decreases when C increases • Windows for image rejection • Increase observation interval

length C

Dynamic fluctuations

• If AM or PM, TVE grows almost linearly with modulating amplitude and frequency

-

Harmonics and inter-harmonics

• 2nd, 3rd harmonics and OoB inter-harmonics very critical because of spectral interference due to mainlobe overlapping

• If a B-term cosine-class window is used, we should set C>B+1

• Use windows with a lower sidelobes

Wideband noise • Significant when SNR<40 dB • Longer observation intervals


Max. image tone rejection (MIR) window


2

1

2

121

NNnn

N

banw

B

b

k ,..., cos 0

18

14

18

42

2

12

2

0

C

Ca

C

Ca

B=2

Off-nominal frequency deviations only

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

[%]

TV

E [

%]

With scalloping loss compensation - C=1

Hanning

Min. sidelobe

Optimal

The case of angle estimation • Since w(n) has an even symmetry with respect to tr, its real-valued

transform W(·) has no impact on phase estimates

• If image influence is made negligible and if εa(·)≈εp(·) ≈ 0 the angle estimates based on DFT estimators can be very accurate, i.e.


rr pp ArgˆArg

The IpDFT phasor estimator

• Basic idea: estimate the static frequency deviation in

the r-th observation interval and use it to compensate

for the scalloping loss


2

1

2

1

22

N

Nn

nkN

j

rw enwnxN

kXr

ig rr ,ˆ

r

w

r

W

CXp r

ˆ

iW

iW

iCX

iCX

r

r

w

w

r11

If the window makes image interference negligible (e.g. MIR)

111

110

CXCX

CXCXi

rr

rr

ww

ww

if

if where

1

1

r

rr

iBiB

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 -150

-125

-100

-75

-50

-25

0

COSINE WINDOW (J=4)

Frequency (bin)

dB

DTFT DFT

X(l)

X[k] d

If for instance, a B-term MSD window is used

IpDFT phasor estimator accuracy: summary





frequency offset

• Almost completely compensated . Residual uncertainty depends on δr

estimation accuracy

-


• Depends on window type and observation interval length

• MIR windows for C=1 or C=2 • MSD windows for C>2


• Cause significant TVE increments • TVE and similar to WDFT if C

and window are the same.

• Phasor dynamic model (IpD2FT – see later)


• 2nd, 3rd harmonics and OoB inter-harmonics very critical because of spectral interference due to mainlobe overlapping

• If a B-term cosine-class window is used, we should set C>B+1

• Use window with lower sidelobes

Wideband noise • Significant when SNR<40 dB • Longer observation intervals

TVEst

Performance analysis example Test type

Max.

TVE [%]

Windows

Hann Hamming MIR1 MIR2

Freq. dev. 10% - 2.27 1.03 0.19 1.73

Freq. dev. 10% + noise with SNR=50 dB

mean 2.12 1.00 0.15 1.63

mean + std. 2.22 1.05 0.21 1.71 Freq. dev. 10% + 1% 2nd-order harmonic

- 2.77 1.56 0.74 2.20

Freq. dev. 10% + 1% 3rd-order harmonic

- 2.35 1.15 0.26 1.83

Freq. dev. 10% +

AM + PM - 2.61 1.29 0.45 2.01


Test type Max.

TVE [%]

Windows


Freq. dev. 10% - 0.38 0.42 1.00 0.12


mean 0.25 0.35 0.77 0.09


- 0.50 0.48 1.04 0.20


- 0.38 0.43 1.01 0.12

Freq. dev. 10% +

AM + PM - 1.05 1.22 1.93 0.74

1 c

ycle

(C

=1)

2 c

ycle

s (C

=2)

The Taylor’s dynamic phasor model

• Basic idea: estimate not only the phasor, but also its derivatives w.r.t. time to track the amplitude and phase fluctuations.

• Derivatives can be also used to estimate frequency and ROCOF!


srsr

tj

ar TN

ttTN

tetX

tp rrp

r 2

1

2

11

2

r

KrK

rrrrrrrKrr tttt

K

tpt

tpttptptptp ,

!!

'''

, 2

2

reference time time shift

observation interval

Taylor’s series order Typically, K = 2, 3

Significant amplitude and phase changes (part of the measurand)

The 4PM and 6PM phasor estimators


Xr-N Xr Xr-2N

tr tr-1 tr-2

NN

XXjXp rr

rPM

r2

2

14

sin

ˆˆˆˆ

**

2

2

212

211

221

123

6

22

2

24

21

22

2

NN

XXXXXX

NN

XXXjXp

rrrNrrrNrrrr

PMr

sin

ˆˆˆcosˆˆˆ

sin

ˆˆˆˆˆ

*********

Dynamic change of the

phasor in one cycle (1-order

derivative estimator)

Dynamic variation of the phasor in two cycles

(1- and 2-order derivative estimators)

1-cycle DFT (with rectangular window)

• Phasor derivatives are approximated by finite differences over 2 or 3 consecutive 1-cycle long intervals

4-parameter (4 PM, K=1)

6-parameter (6 PM, K=2)

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

[%]

Max T

VE

[%

]

6PM

4PM

Performance comparison overview


Scalloping loss due to off-nominal frequency deviation + image infiltration

…+ AM and PM modulation

…+AM+PM + 2nd harmonic = 1% of

fundamental

…+AM+PM + 50 harmonics = 1% of

fundamental + noise with SNR=50 dB

4PM/6PM phasor estimator accuracy: summary





frequency offset

• Derivative terms track off-nominal frequency changes, thus partially compensating for the scalloping loss. However, worse than IpDFT.

• Different windows ?


• Effect not very clear. On average, it looks comparable to WDFT.


• TVE generally better than WDFT (for 6PM). • Results can be better or worse than IpDFT

over C=1 or C=2, depending on window type. • Response times longer than WDFT/IpDFT for

the same C.

Harmonic and inter-harmonic

• TVE increments comparable with those of DFT for the same C.

Wideband noise • TVE std. dev. comparable with WDFT for the

same C. TVE bias could be slightly larger.

The Taylor-WLS phasor estimator - 1


TNNrr

Nr

Nrr xxxxx

21

21

21

21 101 ,...,,x

Vector of data in the observation interval

rHH

KKHH

KKr BBB xp , 1

2

where is the estimate of

TKrKrrrrrKrKrKr pppppppp ,,,,*,

*,

*,

*,, ,,,,,...,,p 110011

BK: N 2(K + 1) matrix including the coefficients of the linear system that relate xr with pr,K and

ks

tt

k

rk

kr Tdt

pd

Kp

r

!

,

1

Least square fitting based on phasor Taylor’s series expansion

Kr ,p

Hermitian operator

2

12

12

1 1 NNN wwwdiag ,...,,WLS = LS if a rectangular window is used (Ω=IN)

The Taylor-WLS phasor estimator - 2

• The Taylor-WLS can estimate easily all quantities of interest in one shot

• At least two variants/enhancements exist of the T-WLS: • The Taylor-Fourier Transform (TFT) (De la O Serna – 2011): includes the harmonics till

order H in the signal model to better compensate their effect; considerable computational complexity

• The T-WLS-DC (Petri – 2014): includes the decaying DC offset in the signal model, which is disregarded in synchrophasor measurement standards.


0,ˆ

rr pp Phasor estimator:

2

0

01

00

21

,

*,,

ˆ

ˆÎmˆ

r

rrsr

p

pp

f

fff

Frequency estimator:

ROCOF estimator:

2

0

0101022

0

2

,

*,,

*,,*

,,

^

ˆ

ˆÎmˆˆReˆÎm

ˆr

rrrrrr

sr

p

pppppp

p

fROCOF

For K=2

T-WLS phasor estimator accuracy: summary




Scalloping loss due to frequency offset

• Compensation through phasor tracking in WLS sense

• Combination with IpDFT? • Iterative application of

WLS


• It is included in the model and estimated accordingly. Negligible for any window shape and length

-

Dynamic fluctuations • Better than 4PM/6PM and IpDFT in

both accuracy and responsiveness -


• 2nd harmonic and OoB inter-harmonics still critical.

• Generally worse than IpDFT when the same window and length

• Increase observation interval length C

Wideband noise • Optimal for white noise reduction • Large impact on frequency and

ROCOF

• Increase observation interval length C

Performance analysis example


Test type Max.

TVE [%]

Windows


Freq. dev. 10% - 0.20 0.17 0.17 0.19


mean 0.33 0.18 0.15 0.23


- 4.06 2.93 2.64 3.43


- 8.07 3.77 2.99 5.25

Freq. dev. 10% +

AM + PM - 0.31 0.27 0.26 0.29

Test type Max.

TVE [%]

Windows


Freq. dev. 10% - 0.08 0.04 0.03 0.06


mean 0.09 0.07 0.06 0.08


- 0.93 0.76 0.66 0.86


- 0.11 0.07 0.06 0.10

Freq. dev. 10% +

AM + PM - 0.27 0.13 0.07 0.06

1 c

ycle

(C

=1)

2 c

ycle

s (C

=2)

The IpD2FT phasor estimator - 1


2

12

1

0

12

12

02

2

NN

nh

K

k

nM

j

krk

nM

j

krk

r nnnepnepnnxrr

,...,,...,*,,

rrr nh

*rrw ΕEppX δWδW IP

WDFT of xr[n] with K=2 and window w[n] Disturbances

If disturbances are negligible and δ is known, by inverting the system

Real-valued 3x3 matrix of the normalized DTFT of nkw(n) at bins –δ, 1 –δ, –1–δ

Real-valued 3x3 matrix of the normalized DTFT of nkw(n) at bins 2C+δ, 2C+1 +δ, 2C-1+δ

r

r

w

w

Xp

Xp

ImδWδWIm

ReδWδWRe

IPr

IPr

1

1

Estimate

Estimate

The IpD2FT phasor estimator - 2


rf^

rROCOFrp

Initialize

Compute

r

r

w

w

Xˆˆp

Xˆˆp

ImδWδWIm

ReδWδWRe

rIrPr

rIrPr

1

1

Estimate frequency deviation

2

0

01

2,

*,,

ˆ

ˆˆImˆ

r

rrr

p

ppM

Does result change significantly?

Compute

YES

NO

even is if

odd is if

MCC

MCC

Mr

r

rr

ˆˆ

ˆˆ

1

0rδ

IpD2FTphasor estimator accuracy: summary




Scalloping loss due to frequency offset

• Almost completely compensated in 2 or 3 iterations

• Combination with IpDFT could avoid iterations


• It is estimated and compensated. • Generally negligible for any window

shape and length • MIR windows if relevant


• TVE, FE, RFE and step response times grow with C and decrease with the window spectrum sidelobe width

-


• 2nd harmonics and OoB inter-harmonics still critical.

• Results better than T-WLS (if window spectrum mainlobes not overlapped)

• If a B-term cosine-class window is used, C>B+1 (probably > B+2)

• Increase C

Wideband noise • Significant when SNR<40 dB • Large impact on frequency and ROCOF

• Longer observation intervals

Performance analysis example


TVE [%] FE [mHz] RFE [Hz/s]

Test type B=2 B=3 B=4 B=2 B=3 B=4 B=2 B=3 B=4

Freq. dev. only (±10%) 0.00 0.00 0.00 0.1 0.02 <0.01 10-4 <10-4 <10-4

Freq. dev.+ noise

(60 dB)

bias 0.00 0.00 0.00 0.05 0.1 0.15 9·10-3 1.3·10-2 2.6·10-2

std. 0.01 0.01 0.01 0.50 0.65 0.75 4.3·10-2 6.6·10-2 12·10-2

Freq. dev.+ 2nd harm (10%) 0.12 0.50 1.4 15 15 52.5 2.5 12 41

Freq. dev.+ 3rd harm (10%) 0.10 0.01 0.00 5.8 0.5 0.01 2.2 0.24 <0.01

Freq. dev.+ AM 0.09 0.09 0.06 6 2.5 2.0 3.6 2.1 1.5

Freq. dev.+ AM + PM 0.24 0.12 0.09 45 33 27 4.5 2.7 2.4

A good tradeoff: C=4 with Minimum Sidelobe Level (MSL) windows

Amplitude step (±10%) Phase step (±10 degrees)

Window type MSL (B=2) MSL (B=4) MSL (B=2) MSL (B=4)

TTVE [%] 1.0 1.2 0.9 1.3 1.0

TFE [mHz] 5 3.2 2.7 3.5 3.1

TRFE [mHz/s] 10 4.0 3.9 4.0 3.9

Response times for different thresholds (in nominal cycles)

Conclusions • Phasor estimators based on the Taylor’s model offer global better performance

in the presence of dynamic changes

• IpDFT/IpD2FT exhibit good/excellent performance in removing scalloping loss,

image interference and harmonics

• T-WLS is better under the effect of modulations and wideband noise.

• Dynamic estimators over four-cycle observation intervals with a suitable

window can ensure compliance to stricter M-class specifications except in the

case OoB inter-harmonics interference

• OoB inter-harmonics rejection requires longer observation intervals (about 10

cycles) with the best techniques considered.

• WDFT, IpDFT, 4PM/6PM and T-WLS, exhibit a comparable computational

complexity (grows approx. linearly with N). The IpD2FT instead has a larger

complexity which depends also on the number of iterations.


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