![Page 1: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/1.jpg)
Numerical algorithms for power system protection
Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj HavelkaUniversity of Zagreb
Faculty of Electrical Engineering and [email protected], [email protected]
2010/2011.
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Introduction
Quality of digital relays depends on: Numerical algorithm quality (software) Hardware quality General digital relay characteristics:
selectivity, stability, satisfactory trip time and sensitivity
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Lecture parts First part
Types of signals Sampling theory Sampling and A/D circuits Numerical methods: Interpolation formulas
numerical integration and differentiation, curve fitting, Fourier analysis and digital filtering
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Lecture parts Second part
Sinus wave based algorithms Fourier based algorithms Least squares based algorithms Differential equation based algorithms
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Lecture parts Third part: Real time algorithm testing
50 Hz Signal Simulated short circuit Real short circuit
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Signal classification hierarchy Digital signals
1-0 (On Off or TTL) signal Pulse train (counters, timers)
Analog signals DC signal (slow) Signal in time domain (fast) Signal in frequency domain
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Signali
Digitalni
Analogni
1-0 (On-Off)
Pulsirajući signal
DC
Vremensko područje
Frekvencijsko područje
TTL
Brojači i uklopnoisklopni satovi
ADC/DAC(Sporo)
ADC/DAC(Brzo)
ADC(Brzo)+Analiza
1
0
t
t
t
t
f
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Basic elements of digital protection
AD converter resolution Nyquist’s theorem Analog filters Transducers Sample and hold circuit
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Let us assume that numerical values of some function x(t) are given at equally spaced intervals every t seconds.
1/t is then called sampling frequency.
Signal can then be represented by discrete set of samples:
[x(0), x(t), x(2t), …, x(kt),…]
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AD converter resolution Every sample of analog signal is
converted in to digital value with final number of bits
Conversion is preformed in AD converter
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3 bit resolution; 23=8 combinations, which means 8 discrete divisions that analog signal can be represented with
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Nyquist’s theorem
Sampling frequency (how often is AD conversion preformed)
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Nyquist’s theorem To avoid signal alias sampling
frequency must be at least two times higher then maximum frequency component in analog signal
For accurate waveform representation sampling frequency should be at least 5 to 10 times higher then maximum frequency component in analog signal
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Analog filtering
Izla
z fil
tera
Izla
z fil
tera
frekvencija
Prijelaznopodručje
f1 f2f1
a) b)
R
L
C
R
U1 U2
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Transducers and surge protection circuits Reduce voltages and currents (10 V
and 20 to 40 mA) to suit hardware requirements
Protect hardware from overvoltages Signal distortion is the problem (current
transducers saturation)
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Sample and Hold circuit
Tc T
Vc
C
Sf(t)
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transducer
Surge protective
circuit
LP Filter
AMUX
Signal conditioning subsystem
Sample andHold circuit AD conv.
DMUX
Conversion subsystem
Digital processing relay subsystem
Basic components of digital relay
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Numerical differentiation
Derivatives in point k is
kkk fh
xff
32
3
1
2
11)(
12
1
kkk
kkk
fff
fff
111
kn
kn
kn fff
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Numerical integration
n
i
n
jij
n
in
jij
jii
n
jj
iji
jin
xxxx
xx
yxx
xxyxLxg
00
0
0
0
n
i
b
a
n
iiin
jij
jii
n
ji
i
b
a
n
in
jij
jii
n
ji
i
b
a
Aydxxxxx
xx
ydxxxxx
xx
ydxxf0 0
0
0
0
0
0
Lagrange interpolation formula
Trapezoidal formula
b
a
x
x
nn
n
yyyyyh
dxxfdxxf0
1210 2222
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Curve fitting Linear fit:
Exponential fit:
General polynomial fit:
General linear fit:
Levenberg-Marquardt fit:
ii xaay 10
ixai eay 1
0
iii xaxaay 2210
)()( 22110 iii xfaxfaay
),,,,( 210 aaaxfy ii
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Least square method
vertikalna udaljenost izmeđuizmjerene točke i točke nakrivulji dobivene metodom
najmanjih kvadrata
krivulja dobivena metodomnajmanjih kvadrata
x
y
(xn, yn) )( iii xuy
N
i
mimiii
N
ii xaxaxaayS
1
22210
1
2
0
ka
S(k=0, 1, 2, ..., m)
imi
ii
i
mm
i
mi
mi
mi
miiii
m
iii
yx
yx
y
a
a
a
xxxx
xxxx
xxxN
1
0
221
132
2
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Fourier analysis
Fourier series
Fourier transform
1 1
000 )sin()cos(2
)(n n
nn tnbtnaa
tf
dtetfF tj )()(
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Discrete Fourier transform DFT Samples of signals from AD: time domain No need for curve fitting Use DFT: frequency domain
1
0
21 N
k
ikN
jekX
Nix
1
0
2N
i
ikN
jeixkX
k=0…N-1
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Smoothing Windowsperiod T
diskontinuiteti
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Smoothing Windows
N
nnw
2cos46.054.0)(
n=0, 1, …, N-1
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Digital filters Input signal is discrete They are software programmable They are stable and predictable They do not drift with temperature or humidity
and do not require precision components They have superior performance to cost ratio They do not age
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Digital filters
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Signal generator
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Control loop
ALGORITHMtext ALGORITHMte
xttext
Measure Calculate Measure Calculate Measure
Δt=1/fs
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Sine wave based algorithms Waveform is assumed to be sinusoidal They predict amplitude at every moment They can be used for impedance calculation Six are presented:
Sample and first derivative with two points Sample and first derivative with three points First and second derivative Two sample technique Three sample technique R i X calculation with three sample technique
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Sample and first derivative with two points
)sin()( 011 tIts
)cos()( 0101 tIts
kk ih
i
32
3
1
2
11
)(1
1 kkk iih
i 2
122
2
tf
iiiI
o
kkkk
2
0
121
21
)()(
ts
tsI
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Sample and first derivative with two points
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Sample and first derivative with three points
21432
1
kkkk iii
ti
2
2122
4
43
tf
iiiiI
o
kkkkk
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Sample and first derivative with three points
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First and second derivative
tVtv 0sin)(
tVv 00 cos
tVv 020 sin
2
20
11
2
1120
2 2
2
1
t
vvv
t
vvV kkkkk
2
0
2
20
2 1
v
vV
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First and second derivative
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Two sample technique
kk tIi 0sin
)(sinsin 0101 ttItIi kkk
ttIttIi kokk 0001 sincoscossin
20
0121
22
)(sin
cos2
t
tiiiiI kkkk
2
0
011111
sin
coscos
tIV
tvivivivi kkkkkkkk
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R i X calculation with three sample technique
kk tVv 0sin
ttVv kk 001 sin
ttVv kk 002 2sin
kk tIi 0sin
ttIi kk 001 sin
ttIi kk 001 2sin
222
2211
2
2
kkk
kkkkkkf iii
ivivivR t
iii
ivivX
kkk
kkkk
0
221
1221 sin
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R i X calculation with three sample technique
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Fourier algorithms Waveform does not have to be pure sine The basic assumption is that the waveform is periodic The principle of work is moving frame Moving frame is constant in size which means that it
always contains the same number of points
t4t1 t2 t3
Am
plit
ud
a
Vrijeme
t
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Fourier series with whole period
1 1
000 )sin()cos(2
)(n n
nn tnbtnaa
tv
Tt
t
n dttntvT
a0
0
)cos()(2
0
Tt
t
n dttntvT
b0
0
)sin()(2
0
ttdtvT
aVTt
t
x
0
0
01 cos)(2 NN yyyyy
xI 2222
2 1210
NNNNx tvtvtvtvt
tNV 0101101000 coscos2cos2cos
2
2
jN
tjT
t j 2
cos2
coscos 0 j=0 do N
If fs is 600 Hz then in one period of 20 ms there are 12 samples.Weighting factors are calculated in advance for fixed samplin frequency.
84111751026
120
2
3
2
1
226
1vvvvvvvvv
vvVx
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Fourier series with whole period
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Fourier series with whole period – third harmonic
n=3
ttdtvT
VTt
t
x
0
0
03, 3cos)(2 ttdtv
TV
Tt
t
y
0
0
03, 3sin)(2
1
1,3
03, 22
2 N
jjxj
Nx Wv
vv
NV
1
1,33,
2 N
jjyjy Wv
NV
za fs=600 Hz
108642
1203 226
1vvvvv
vvVx 11975313 6
1vvvvvvVy
23
233 yx VVV
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Fourier series with whole period – third harmonic
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Fourier series with half period
4152
60
2 2
3
2
1
223
1vvvv
vvV x
42513
2 2
3
2
1
3
1vvvvvV y
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FFT algorithm
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Least squares based algorithms All components of measured waveform must be
predicted in mathematical model. After curve fitting data about amplitude, harmonics,
angle, etc. are obtained Downside is large number of calculations They are complex Four of them are presented:
Algorithm with general polynomial fit LSQ 1, 3 multivariable algorithm LSQ 1, 3, 5 multivariable algorithm LSQ 1, 3, 5 ,7 multivariable algorithm
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General polynomial fit algorithm
)sin()( 0 tKtv
sincoscossin)( 00 tKtKtv
!3
)(sin
30
00
ttt
!2
)(1cos
20
0
tt
30
200 )(
6
cos)(
2
sin)(cossin)( t
Kt
KtKKtv
33
2210)( xaxaxaaxf
3
2
3
2
1
0
6543
5432
432
32
ii
ii
ii
i
iiii
iiii
iiii
iii
xv
xv
xv
v
a
a
a
a
xxxx
xxxx
xxxx
xxxN
ii tx 0 tit i if
f
fix
ssi 0
0
21
t4t1 t2 t3A
mp
litu
da
Vrijeme
t
21
20 aaK
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General polynomial fit algorithm
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LSQ 1, 3 multivariable algorithm 303101 3sinsin tKtKtv
10331033101110111 3cossin3sincoscossinsincos tKtKtKtKtv
41431321211111 xaxaxaxatvS
4
3
2
1
4
3
2
1
44434241
34333231
24232221
14131211
tv
tv
tv
tv
x
x
x
x
aaaa
aaaa
aaaa
aaaa
111 cosKx
112 sinKx
333 cosKx
334 sinKx
2021 sin ta
2022 cos ta
2023 3sin ta
2024 3cos ta
t4t1 t2 t3
Am
plit
ud
a
Vrijeme
t
01 t tt 2 tt 23 tt 34
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LSQ 1, 3 multivariable algorithm
![Page 52: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/52.jpg)
LSQ 1, 3, 5 multivariable algorithm
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![Page 53: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/53.jpg)
LSQ 1, 3, 5 multivariable algorithm
![Page 54: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/54.jpg)
LSQ 1, 3, 5, 7 multivariable algorithm
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![Page 55: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/55.jpg)
LSQ 1, 3, 5, 7 multivariable algorithm
![Page 56: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/56.jpg)
Differential equation based algorithm There is no need to assume that the waveform is sine The fundamental approach is based on the fact that
all protected equipment can be represented by differential equations of first or second order.
The methods are described by reference to transmission line
Three algorithms are presented: Integration algorithm Third harmonic filtration algorithm Differential algorithm
![Page 57: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/57.jpg)
Integration algorithmR L
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![Page 58: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/58.jpg)
Integration algorithm
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Third harmonic filtration algorithm
Elimination of m and n harmonics:
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0
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m
n
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dttidttidtti
For fs=600 Hz :
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![Page 60: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/60.jpg)
Third harmonic filtration algorithm
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Differential algorithm
dt
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![Page 62: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/62.jpg)
Differential algorithm
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Differential algorithm
![Page 64: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/64.jpg)
Algorithms and Real-Time operation System is operating in real-time if it can
guarantee fulfillment of various tasks in specific time
OS in real-time Hardware and software in real time
Control loop time
![Page 65: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/65.jpg)
Algorithms and Real-Time operation
regulacijskitransformator
napona od 0 domaksimalno
10V
ulazni podaciizlazni podaci memorija CPU 2 GHz
DA pretvornik
proradnisignal
sustav za podešenje iprilagodbu signala
sustav za pretvorbuanalognog signala u
digitalni
sustav za digitalnu obradusignala (računalo)
Izvor napona od 220 Vefektivno i frekvencije 50
Hz
DAQ PločicaPCI-MIO-16E-4
Proizvođač:National
Instruments
![Page 66: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/66.jpg)
Power system signal
![Page 67: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/67.jpg)
Sine wave based algorithms
![Page 68: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/68.jpg)
Fourier based algorithms
![Page 69: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/69.jpg)
Least squares based algorithms
![Page 70: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/70.jpg)
Differential equation based algorithms
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Short circuit
text
text
text
text
text
text
P
380 V 380 V
Z=R+jX
N1:N2=2
SC
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Sinus wave based algorithms
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Fourier based algorithms
![Page 74: Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering](https://reader030.vdocuments.us/reader030/viewer/2022032802/56649de55503460f94addb5d/html5/thumbnails/74.jpg)
Least squares based algorithms
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Differential equation based algorithms