Development and validation of a least squares algorithm for static and dynamic synchrophasor tests
Luciano Domínguez Pose Instituto Nacional de Tecnología Industrial (INTI)
lucianod@inti.gob.ar Abstract: INTI had developed a least squares fit algorithm based on an iterative matrix resolution method, covering all tests of the IEEE C37.118.1 standard. We present the method, results, validation and conclusions of this work.
Keywords: Synchrophasor-PMU- Calibration-Algorithm –Least square.
1. INTRODUCTION
INTI is part of a public private consortium 1 which aim is to develop a measuring system to monitor the electrical grid, based on synchrophasors technology. A work packaged in this project includes the development of a reference system for PMU calibration. In this work we present an algorithm developed with that purpose, both for static and dynamic tests based on [1,2,3,4]
In Fig. 1 we can appreciate the block diagram of INTI PMU calibration system. A 120V and 5 A signal is applied to the Device under test (DUT) and to the digitalization stage (DAQ) via current and voltage transducers. The digitalized output signals from the DAQ are fitting using an algorithm based on the least square method (LS) for obtain the reference synchrophasors, and thus be able to compare them with those produced by the PMU under test
and is the DC component. The ̅ vector represents the time of each sample.
The polynomial used by the fiting algorithm to estimate the reference signal can be seen in eq. (2) and eq. (3). Amplitude, frequency, phase and DC component are estimated using polynomials for each of them ranging from degree zero to degree two.
0( )̅ = 0 + 1 . ̅ + 2 . ̅ 2
(2)
( ̅) = 0 + 1 . ̅
(3)
The algorithm estimates 0, 1 , 2, 0, 1, 0, .
2. LS FITTING ALGORITHM
The signal injected by the reference signal source can be modeled as ( ̅) = 0( )̅ . (2 ̅ . ( )̅ + 0) + (1) where 0( ̅) is the amplitude of the signal, ( )̅ is the frequency of the system, 0 is the phase angle
Figure 1. Scheme of INTI PMU calibration system.
We use expansion in Taylor series over ( ̅) in function of frequency ( ̅) . To estimate the digitalized signal in the sample vector ̅ , we use:
1 FONARSEC Project is financed by Ministerio de Ciencia, Tecnología e Innovación Productiva.
1
̅ ≅ ( ̅ , ̅′) .
(4)
where ′̅ are the seed values of ( ̅),
( ̅ , ̅′) =
= [ ( ̅ , ′̅ ) ( ̅ , ′̅ ) ̅ . ( ̅ , ′̅ ) .̅ ( ̅ , ′̅ )
̅2. ( ̅ , ′̅ ) ̅2 . ( ̅ , ′̅ ) 1 ]
(5)
where
( ̅ , ′̅ ) = (2 ̅ ( 0′ + 1′ ̅ ))
(6)
( ̅ , ′̅ ) = (2 ̅ ( 0′ + 1′ ̅ ))
(7)
and is the fit coefficients vector
= [ 0 0 1 1 2 2 3 ]
(8)
that minimizes the cuadratic vector error,
‖ ̅ − ( ̅ , ̅′) . ‖22
(9)
and we obtain the solution through the following equation:
( ̅ , ′̅ ) = [ ( ̅ , ̅′) . ( ̅ , ̅′)]−1. ( ̅ , ̅′) . ̅ (10)
To solve (10) we use an iterative method, starting with an approximate initial value of 0′ y 1′. Then,
∆ 0 = ( 0 1 − 0 1)/(2 02)
(11)
∆ 1 = ( 0 2 − 0 2)/(2 02)
(12)
0 ≅ 0′ + ∆ 0
(13)
1 ≅ 1′ + ∆ 1
(14)
Once we obtain 0 y 1, we directly calculate
0 = √ 02 + 02
(15)
= ( 0 − 0 )/ 0
(16)
0
=
( 0)
0
(17)
= 3
(18)
3. COVERED TESTS
The developed method allows to perform all the tests that requires the cited standard of synchrophasors, either in voltage and current magnitudes. In all tests are calculated TVE (Total Vector Error), FE (Frequency Error) and RFE (ROCOF Frequency Error) [5]
Stationary Tests: a. Magnitude and phase. b. Harmonic distortion. c. Interarmonic distortion.
Dynamic Tests: d. Magnitude and phase modulation. e. Frequency ramp. f. Magnitude and phase step.
4. VALIDATION AND RESULTS The validation of the algorithm was done through a comparison with METAS. The validation method consisted in the comparison of the results of the algorithms developed by both institutes, for stationary and dynamic tests. In this paper only results for dynamic tests are presented.
4.1 FREQUENCY RAMP The test consists of the application of a frequency ramp signal to the system that can be model as
= [ 0 + 2]
(19)
where 0 is the nominal frequency of the system,
and
=
is the ramp slope in Hz/s. For the
test we used frequencies from 45 to 55 Hz with a
rising of = 1 / , sampling frequency =
18 and 0 = 50 . Table 1 and Figure 2
shows the results for TVE, FE y RFE
DYNAMIC FREQ RAMP TEST
max
mean
min
max C37.118
passed
TVE (%) 0,08 0,04 0,001
1
FE (Hz) 0,004 0,0036 0,0031 0,005
RFE (Hz/s) 3x10-6 3x10-6 3x10-6 0,1
Table 1. Results of comparison between METAS and INTI for frequency ramp tests.
2
Figure 2. Frequency ramp TVE results.
Figure 3. Amplitude modulation TVE results.
4.2 AMPLITUDE AND PHASE MODULATION
The test consists of the application of a modulation signal to the system that can be model as,
= [1 + ( )]. cos[ 0 +
( − )]
(20)
where is the amplitude of the signal, 0 is the nominal frequency of the system, is the
modulation frequency, is the amplitude modulation factor and is the phase modulation
factor.
4.2.1 AMPLITUDE MODULATION
For amplitude modulation test we used the following configuration: =0.1 to 4.9 Hz with gaps of 0.2 Hz. = 0.1 , = 0, sampling frequency = 18 kHz and 0 = 50 Hz. In table 2 and figures 3 and 4 we can see the results
for TVE, FE y RFE.
DYNAMIC AMPLITUDE MODULATION TEST
max mean
min
max C37.118
passed
TVE (%)
0,075 0,033
0,001
3
FE (Hz) 0,015 0,006 0,0001 0,06
RFE (Hz/s)
0,33
0,19
0,002
3
Table 2. Results of comparison between METAS and INTI for amplitude modulation tests.
Figure 4. Amplitude modulation FE results.
4.2.2 PHASE MODULATION For phase modulation a signals with the following configuration was applied: =0.1 to 4.9 Hz with steps of 0.2 Hz. = 0.1, = 0 In table 3 we can see the results for TVE, FE y RFE.
TVE (%)
FE (Hz)
RFE (Hz/s)
DYNAMIC PHASE MODULATION TEST
max mean
min
max C37.118
passed
0,073 0,057 0,043
3
0,05 0,03 0,0001 0,06
2,61 1,63 0,02
3
Table 3. Results of comparison between METAS and INTI for phase modulation tests.
4.3 PHASE STEP The mathematical representation of the applied signal is:
= [1 + 1( ) ]. cos[ 0 + 1( ) ]
(21)
where is the amplitude of the signal, 0 is the nominal frequency of the system, 1( ) is a unit step function, is the magnitude step size and the phase step size. The sampling frequency
= 18 kHz.
3
The results presented below have been performed
under the following conditions: number of tests:
180.
Phase
increase
per
test:
90
srtaedp.
=1,
=
90
rad step
.
Synchrophasor
report
rate:
50
frames s
.
Step instant: 0.5
Response time (s) Delay time (s) Overshoot (%)
DYNAMIC PHASE STEP TEST
max
max C37.118
passed
0,038 0,199
0,0023 0,005
1,8
5
Table 4. Results of comparison between METAS and INTI for Phase step tests.
5. CONCLUSIONS Iterative matrix adjustment algorithms have been developed, using the Taylor series expansion method. The algorithms have been validated and the obtaining results in all tests were below the required limits by the synchrophasor standard IEEE C37.118.1. We can concluded that the algorithm developed at INTI can be used in the PMU reference system
We are currently working on the development of algorithms based on other methods to compare and optimize each tests.
ACKNOWLEDGMENTS We are grateful to J. P. Braun and S. Siegenthaler of METAS, for the valuable help provided and the shared data for the validation of the algorithms.
REFERENCES
[1] G. Stenbakken, M. Zhou, "Dynamic Phasor Measurement Unit Test System", Power Engineering Society Meeting, IEEE, 2007.
Figure 5. INTI estimation algorithm phase step fitting.
[2] G. Stenbakken, J. Ren, M. Kezunovic, "Dynamic Characterization of PMUs Using Step Signals", IEEE Power & Energy Society General Meeting, 2009.
[3] J. P. Braun, S. Siegenthaler, "Calibration of PMUs with a Reference Grade Calibrator", Precision Electromagnetic Measurement, (CPEM), 2014.
[4] J. P. Braun, C. Mester, "Reference grade
Calibrator for the testing of the dynamic
a)
behavior of phasor measurement units",
Precision Electromagnetic Measurement,
(CPEM), 2012.
b) Figure 6. Phase step results referred to: a) Response time.
b) Delay.
[5] IEEE C37.118.1-2011, Standard for Synchrophasor Measurements for Power Systems, 2011.
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