Numerical algorithms for power system protection

Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj HavelkaUniversity of Zagreb

Faculty of Electrical Engineering and Computingante.marusic@fer.hr, juraj.havelka@fer.hr

2010/2011.

Introduction

Quality of digital relays depends on: Numerical algorithm quality (software) Hardware quality General digital relay characteristics:

selectivity, stability, satisfactory trip time and sensitivity

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

Lecture parts Second part

Sinus wave based algorithms Fourier based algorithms Least squares based algorithms Differential equation based algorithms

Lecture parts Third part: Real time algorithm testing

50 Hz Signal Simulated short circuit Real short circuit

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

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

Basic elements of digital protection

AD converter resolution Nyquist’s theorem Analog filters Transducers Sample and hold circuit

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),…]

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

3 bit resolution; 23=8 combinations, which means 8 discrete divisions that analog signal can be represented with

Nyquist’s theorem

Sampling frequency (how often is AD conversion preformed)

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

Analog filtering

Izla

z fil

tera

Izla

z fil

tera

frekvencija

Prijelaznopodručje

f1 f2f1

a) b)

R

L

C

R

U1 U2

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)

Sample and Hold circuit

Tc T

Vc

C

Sf(t)

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

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

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

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

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

Fourier analysis

Fourier series

Fourier transform

1 1

000 )sin()cos(2

)(n n

nn tnbtnaa

tf

dtetfF tj )()(

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

Smoothing Windowsperiod T

diskontinuiteti

Smoothing Windows

N

nnw

2cos46.054.0)(

n=0, 1, …, N-1

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

Digital filters

Signal generator

Control loop

ALGORITHMtext ALGORITHMte

xttext

Measure Calculate Measure Calculate Measure

Δt=1/fs

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

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

Sample and first derivative with two points

Sample and first derivative with three points

21432

1

kkkk iii

ti

2

2122

4

43

tf

iiiiI

o

kkkkk

Sample and first derivative with three points

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

First and second derivative

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

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

R i X calculation with three sample technique

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

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

Fourier series with whole period

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

Fourier series with whole period – third harmonic

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

FFT algorithm

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

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

General polynomial fit algorithm

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

LSQ 1, 3 multivariable algorithm

LSQ 1, 3, 5 multivariable algorithm

505303101 5sin3sinsin tKtKtKtv

61651541431321211111 xaxaxaxaxaxatvS

6

5

4

3

2

1

6

5

4

3

2

1

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

tv

tv

tv

tv

tv

tv

x

x

x

x

x

x

aaaaaa

aaaaaa

aaaaaa

aaaaaa

aaaaaa

aaaaaa

26

255 xxK

5

615 tan

x

x

LSQ 1, 3, 5 multivariable algorithm

LSQ 1, 3, 5, 7 multivariable algorithm

10331033101110111 3cossin3sincoscossinsincos tKtKtKtKtv

1077107710551055 7cossin7sincos5cossin5sincos tKtKtKtK

8

7

6

5

4

3

2

1

8

7

6

5

4

3

2

1

8887868584838281

7877767574737271

6867666564636261

5857565554535251

4847464544434241

3837363534333231

2827262524232221

1817161514131211

tv

tv

tv

tv

tv

tv

tv

tv

x

x

x

x

x

x

x

x

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

LSQ 1, 3, 5, 7 multivariable algorithm

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

Integration algorithmR L

v(t)

i(t)

dt

diLRiv y

x

2

1

2

1

2

1

t

t

t

t

t

t

dttvdiLdttiR 4

3

4

3

4

3

t

t

t

t

t

t

dttvdiLdttiR

tvv

iiLtii

R

2212

1212 t

vviiLt

iiR

22

3434

43

43123421

43123421

iiiiiiii

vviiiivvR

21344312

43214321

2 iiiiiiii

vviiiivvtL

Integration algorithm

Third harmonic filtration algorithm

Elimination of m and n harmonics:

0

0

0 0

00

m

m

n

n

dttidttidtti

For fs=600 Hz :

63

3

6

0

63

3

6

0

63

3

6

0

dttvdttvdidiLdttidttiR

5432230145233210

5432230145233210

iiiiiiiiiiiiiiii

vvvviiiiiiiivvvvR

3210452354322301

5432321054323210

2 iiiiiiiiiiiiiiii

vvvviiiiiiiivvvvtL

Third harmonic filtration algorithm

Differential algorithm

dt

tdiLtRitv

t

ii

dt

di kk

211

t

iiLRiv kk

kk

211

t

iiLRiv kk

kk

2

211

121

11

2

2k

k

kkk

kkk

v

v

L

R

t

iii

t

iii

31122

31122

kkkkkk

kkkkkk

iiiiii

iiviivR

31122

21122

kkkkkk

kkkk

iiiiii

vivitL

Differential algorithm

Differential algorithm

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

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

Power system signal

Sine wave based algorithms

Fourier based algorithms

Least squares based algorithms

Differential equation based algorithms

Short circuit

text

text

text

text

text

text

P

380 V 380 V

Z=R+jX

N1:N2=2

SC

Sinus wave based algorithms

Fourier based algorithms

Least squares based algorithms

Differential equation based algorithms