- Email: [email protected]
Z-function convolution in EHV power network electromagnetic transient analysis
Z.function convolution in EHV power network electromagnetic
transient analysis W D Humpage, K P Wong and T T Nguyen Department of Electrical and Electronic Engineering, University of Western Australia, Australia
Time-convolution in the forms arising in transformation from the frequency-domain to the time-doinain has been widely used in the earlier development o f electromagnetic transient methods in power systems. Independent work has recently led to new methods based on the z-transform, and the present paper develops methods based on discrete convolution arising from transformation to the time-domain now from the z-plane. The recursive solution sequences to which this leads combines high accuracy with low computing time requirements. Checks and controls in the synthesis of transmission line forward impulse response and surge impedance functions in the z-plane ensure that these are always stable system functions, and that numerical solution procedures which include them have very high inherent stability. The formulations developed are applied to transmission line energization from an equivalent source model, and the electromagnetic transients in short-circuit fault operation. Close comparisons are made between representative solutions from standard time-convolution analysis and from the methods o f the present paper. Keywords: electric power transmission, electromagnetic transients, z-plane function convolution
I. Introduction Following the earlier development H ° of several different forms of electromagnetic transient analysis in high-voltage power networks, recent work has reported power transmission line transient models which are based on digital £tlters xx. One of the applications of fitter models is in the derivation of computational algorithms, and recent work has produced solution sequences derived from that basis. These require the explicit structures of filter networks and they are formed in terms of the delay functions in their forward and return signal-flow paths. To reflect filter concepts formally into transient analysis leads to methods which involve the convolution of z-plane functions. Principal among the methods developed so far have been those in which transformation from the frequency-domain to the time-domain leads to time-convolution, but no paper so far has referred to formulations which derive from the convoluReceived: 19 October 1982
Vol 5 No 1 January 1983
tion of z-plane functions. In principle, the z-plane is used as an intermediate transform step a2 between the frequencydomain and the time-domain. For any given boundary conditions, a complete analysis formulation is developed almost entirely in the z-plane. Only the final group of equations required in the solution are transformed from the z-plane to the time-domain in terms of discrete-function convolution. Surprisingly concise sequences provide both fully comprehensive electromagnetic transient analysis formulations and also the individual and explicit steps of recursive evaluation. The present paper develops this logical extension of recent investigations of power transmission line electromagnetic transient modelling by digital filters. II. Principal symbols transmission line forward impulse response FQ(z) transmission line surge-impedance function Zo(z) source impedance function z~(z) matrices of source resistance and inductance coefficients v(z), i(z) vectors of voltage and current variables in z-plane v(n), i (n) vectors of voltage and current variables in recursive solution sequences F(z), F(n) forward characteristics B(z), B(n) backward characteristics vector of intermediate variables w(n) modal transformation matrices Ca, C2 At sample interval in time-domain ratio of wave transit time to sampling interval m The subscripts s and r identify variables at the switching and remote ends of the transmission line, respectively. The subscript f identifies variables at a point of short-circuit fault discontinuity. The superscript p distinguishes phasevariables from modal variables.
III. Z-plane response functions Multi-product forms of transmission line response functions in the z-plane lead directly to the cascade f'flter structures of earlier work al, but the z-function convolution procedures of the present paper are approached more directly when numerator and denominator functions have open poly-
0142-0615/83/010031-09 $03.00 © 1983 Butterworth & Co (Publishers) Ltd
31
nomial forms. It is also preferable first to consider the functions for one mode so that the subsequent working is in scalar form. The steps of the working apply identically to each mode, and, when these are complete, modes re-group to a vector form. On this basis, the forward impulse response F~ (z) and the surge impedance function Zo(z) are expressed in the forms:
m 1
DQ(z)= ~
bkz -k
(10)
k=l
M2
No(z)=
CkZ- k - ~ k=l
dkz -k
(11)
k=l
m2
Do(z) = ~, dk z-k F~(z) = - -
(12)
k=l
k=l
(1)
M~
Ns(z ) = (e -- 1) z-1
1 + ~" bk z-k
(13)
k=l
[ M2
Z~ 1 + E
IV. Z-function convolution
CkZ-
k=l
Zo(z) =
(2)
M, 1 + ~ dkz -k
IV. 1 Convolution involving the forward impulse response
In considering z-function convolution involving the forward impulse response, F~ (z), input and output functions, g~ (z) and f~(z), are introduced so that it is required to evaluate:
g=l
Owing to the term z -m in F~(z), only previous-step values arise in operations involving F~ (z) following transformation to the time domain. By comparison, the form of the surge impedance function does not ensure this, and it is then necessary to be able to separate the present-step component from all previous-step components. In the z-plane, this corresponds to a separation in the surge impedance function so that: (3)
Zo(z) = & + Zd(Z)
f~ (z) = F~ (z) g~ (z)
(14)
On using F~ (z) from equation (6): j~ (z) -
Fz -m [1 + N~(z)]
1 +D~(z)
g~ (z)
(15)
An intermediate function W~(z) is now formed using:
z- mg~ (z) W~(z) -
From equation (2), this is achieved by:
(16)
] + D~ (z)
M~
Z~ ~ CgZ- k - Z Q ~ dgz -k k=l
Zo(z) = Z~ +
k=l
M, 1 + ~ dk z-k
In terms of this, the output function f~(z) is found from: (4)
f~(z) = F [ 1 +N~(z)] W~(z)
k=l
On transforming to the time-domain:
Where a lumped-parameter source model, of the kind considered in earlier work n arises, the z-plane source-impedance function, Zs(z), is given by:
f~(n) = F[1 +N~(n)]* W~(n)
(e -- 1) z -a Z , ( z ) : z~ + Zc
1 +z -1
(s)
1 +&(z) Z~No(z) 1 + Do(z)
Zo(z) = Z~ 4
ZcN,(z) Zs(z) = z c -~
1 + z -1
(6)
(7)
(8)
W~(n) = g~ (n -- m) -- D~ (n)* W~(n)
(20)
In summation form, and for FQ (z) one-sided:
F~(n)=F[ W~(n)+ ~' NQ(k)W~(n--kt
(2])
k=l
ml
where:
WQ(n)=g~(n--m)-- ~
k=l
Dg.(k) W~(n--k)
(23)
k=l
m1
N~.(z) = ~
32
(19)
and then transforms to:
Fz -m [1 +NQ(z)] F~ (z) =
(18)
where * denotes discrete-sequence convolution and n the sampling point in the time-domain respectively. The expression for W~(z) first rearranges to:
WQ(z) = z-mg~ (z) -- D~ (z) W~(z)
In preparation for the subsequent derivations, equations (1), (4) and (5) are arranged in the forms:
(17)
ak z-k
(9) In these forms in the time-domain N~ (k) and D e (k) are
Electrical Power & Energy Systems
Circuit breaker
the kth terms in the z-plane numerator and denominator polynomials so that: N~ (k) = ak D~ (k)
/
,~czl i ~p~z~,zglz ~
"1
]
o
(24)
= bk
IV.2 Convolution involving the surge impedance function For input and output functions g0(z) and fo(z), and the surge impedance function of equation (7), it is required to evaluate in this case:
)
vp(z) m
Transmission line model
Source model
Z~No(z) ]
fo(z) = [ Z £ + i ~ ] g o ( z )
(251
On defining an intermediate function Wo(Z)then: re(z) = Z~go(z) + Z ~ g o ( z ) We(z)
(26)
We(Z) = go(z) -- Do(z) We(Z)
(27)
Figure 1. Single line equivalent of switching configuration in line energization circuit, the equations of the transmission line in modal variables in the z-plane are: Vs(Z) -- Z0(z) is(z) = F£(z) Vr(Z)
(35)
vr(z ) = F~(z) [vs(z) + Zo(z) is(Z)]
(36)
Transforming to the time-domain now gives: M2
re(n) = Z2go(n) + Z~ ~,
No(k) Wo(n -- k)
(28)
Phase- and modal-variables are inter-related by:
k=l
M~ We(n) = go(n) -- ~ Do(k) Wo(n -- k)
(29)
k=l for No(k) = c k -- d k
(30)
Do(k) = dk
(31)
IV.3 Convolution involving the source impedance function The steps here are closely similar to those of section (IV.2). Following transformation to the time-domain, the expression fs (z) = Z s (z) gs (z) is evaluated from: fs(n) = Zcgs(n ) + Z e ( e -- 1) Ws(n -- 1)
(32)
Ws(n) = g s ( n ) - Ws(n -- 1)
(33)
V. Energization of an open-circuited transmission line from an equivalent source model Switching over-voltage evaluations in power transmission line energization are often based on an equivalent representation of the network to which the incoming circuit is switched, as in Figure 1. When the equivalent is in a lumpedparameter form, transmission line voltages at the switchingend of the line in vector vsP(z) are related to the specified source voltages of the system equivalent in vector VsP(z) by:
vf(z) = Vf(z) -Z~(z) ~(z)
(34)
As in Figure 1, line conductor currents at the switching-end are in vector isP(z), and ZsP(z) is the source-model impedance matrix. In equation (34), variables are in phase co-ordinates. When the end of the transmission line remote from that at which circuit-breaker switching takes place is on open-
Vol 5 No 1 January 1983
vP(z) = C1 v(z),
v(z) = C;lvP(z)
(37)
iV(z) = C2i(z),
i(z) = C~1iV(z)
(38)
Previous work TM14 has considered the effect of the frequency- dependence of modal transformation matrices following which C1 and C2 identify the real parts of the modal transformation matrices evaluated at a selected frequency. On re-expressing the transmission line relationships of equations (35) and (36) in phase-variable form: vP(z) -- ZOO(z)isP(z) = F~'vP (z)
(39)
vP(z) = F~P(z) [vsP(z) isP(z) + Zo(z) is(z)]
(40)
where: Zg(z) = C,Zo(z) C; 1
(41)
F¢(Z) = C1F£(z ) C11
(42)
On using equation (34) to eliminate vsP(z) from equation (39): VsP(z) -- [ZOO(z)+ ZsP(z)] iV(z) : FP(z) vrP(z)
(43)
In solving for isP(Z) using equation (43), Z°o(z) and ZPs(Z) are separated into the forms: ZOO(z) = Z p + ZVoa(z)
(44)
z,v(z) = z~ + z~(z)
(45)
With these separations, and using Ye = [Z~' + Zcp]-I: isP(Z) = rc{VPs(z) - - F g ( z ) VrP(Z) --
[ZoPa(z) + Z~(z)l iV(z)}
(46)
33
Whereas combining source model and transmission line equations requires transformation between modal- and phase-variables, it is preferable in reducing the numerical work of subsequent convolution evaluations as much as possible to use the diagonal forms ofF~ (z) and Zo(z). Transforming equation (46) to achieve this then gives:
iAz) = c ; 1 YA vP(z) - C,FQ(z) vAz) --
C1Zoct(z ) is(Z) -- Zp(z) isP(z)}
(47)
This represents the first step in the development. The second one is to substitute for vs(z ) in equation (35) into (36) to give: vr(z) = FQ(z) [F~(z) vr(z ) + 2Zo(z ) is(z)]
(48)
Given is(z ) from equation (47), (48) provides a solution for vr(z ). Having achieved this in the z-plane, we now transform equations (47) and (48) to the time-domain to give: vr(n ) = F~(n)* [FQ(n)* Vr(n ) + 2Zo(n)* is(n)l
(49)
is(n) = C2-' Ye [VsP(n) -- C,F~(n)* Vr(n ) --ClZoa(n)* is(n ) - Z P ( n ) * iP(n)]
(50)
The separate convolutions to be evaluated in the numerical solution of equations (49) and (50) for vr(n) are now identified by: fl(n) = FQ(n)* Vr(n)
(5l)
f~(n) = Zoa(n)* is(n)
(52)
fs(n) = Z P ( n ) * isP(n)
(53)
f4(n) = fi(n) + 2Z~is(n) + 2f2(n} Vr(n ) = F [W~2(n ) + alW~2(n -- 1) + a2W22(n -- 2)1 W~2 (n) = f4(n --m) --blW2(n -- 1) --b2W~2{n -- 2) This fully defined sequence now provides a complete recursive solution. Sequential pole-closing in circuit-breakers is represented by a time-control of the diagonal elements of the source impedance matrix and which is in turn included in Yc. A representative solution for a 160 km 400 kV transmission line, for which the principal data is collected together in the Appendix, is given in Figure 2. Here, there is trapped charge on the conductors of the transmission line immediately prior to energization. In Figure 2, solutions from the sequence are compared with those of a standard time-convolution formulation ls.
V l . Short-circuit fault constraints A representation of short-circuit fault conditions is required in several areas of power system electromagnetic transient analysis, including the specialist area of transmission line protection system evaluations and research into signalprocessing methods for fault monitoring and real-time protection purposes 16' 17. A basic primary system for a high-speed protection system configuration is shown in Figure 3. A single point-to-point interconnection has fault infeeds at both ends. It is required that any given type of short-circuit fault can be represented at any location on the interconnection from any given pre-fault operating condition. Here the fault location sub-divides the line into two sections. The principal variables in each section are indicated in Figure 3. The source model and line equations for the solution are given as follows: Source model at end-s
The vector of switching-end line currents is then found from:
v~ (z) = V f ( z ) - Z f ( z )
in ( z )
(56)
is(n) = C21 Ye [VsP(n) --Clfl(n) -- C,f2(n) -- f3(n)] (54) Section 1 Remote-end voltages in vrP(n) are then available from: vrP(n ) = C,F~(n)* [f,(n) + 2f2(n) + 2ZQis(n)]
Subject to the definition of the different convolutions, equations (54) and (55) provide a complete solution in the time-domain. It is of interest to give the explicit recursive sequences of the numerical solution for a particular form of F~(z) and Zo(z). When these are second-order representations, the complete sequence is as follows: fl(n) = F
[WQ1(n)
+ al W~I (n -- 1) + a2W~l (n -- 2)]
WQl(n) = vr (n -- m) -- b 1W~l(n -- 1) -- b2 W~l (n -- 2) f2(n)
= Z~(Cl--all)
Wo(n -- 1) + Z~(c2--d2)Wo(n--2)
(57)
vf(z) -- Zo(z ) irl (,7,) = F£1 (z) [Vsl (,7,)+ Zo(z ) isl (z)]
(58)
Section 2 vf(z)--Zo(z)is,2(z) = F~2(z) [v,,2(z) +Zo(z)ir2(Z)]
(59)
vr2 (z) --Zo(z) iy2(z) = F~2(z) [vf(z) + Zo(z) is2 (z)]
(60)
Source at end-r Vr~ (z) = Vrp (z) -- Zrp (z) ir~ (z)
(61)
Fault location
Wo(n) = is(n) -- d~Wo(n -- 1) -- d2Wo(n -- 2)
i~(=) = - (i~ (z) + iV(z))
(62)
f3(n) = Zc(e -- 1) Ws(n -- 1)
if(z) = ayv (z)
(63)
Ws(n ) = isP(n) -- Ws(n -- 1)
JAn) = C~'Y~ [VP(n) - Clfi(n) -C~ f2(n) -fs(n)]
34
vsl(z)--Zo(z)is1(z)=F~l(z)[vf(z)+ Zo(z)irl(Z)] (55)
In equation (63), G~ is a matrix of fault-path conductances. In developing a solution procedure for equations (56)-(63),
Electrical Power & Energy Systems
960
>
7201
~"
480
""
o
240
L o
0
,i,
3
U
~o -240 c-
4
5 Time,
f
6 ms
'\'
8
9
O u
-480 a
r" 13._
-720 -960
590
--
>
480
--
o~
370
o >
260
C
0
150
:J
,
r"
o U
40
43
-
70
I
I
I
I
1
2
3
4 Time
m o
ca.
-180 -290
I 5
6
7
8
ms
-
?
600 F
42ol>
240 60
' /''
1 "6
-120
>
o
",
9
-300
7 -480 o
2
3
#'\' 4
5
• Timems
'f'
6
I
7
W
U
- 660
a -840
c Q_
Figure 2. Switching overvoltage transients in 400 k V system; pole scatter closing times: a = 0.0 s, b = 500 #s, c = 1 ms; trapped charge d i s t r i b u t i o n : a = - - 1.0 p.u., b = 0.5 p.u., c = - - 0.5 p.u., switching at peak voltage in phase a f r o m 10 000 M V A f a u l t level 160 km, 4 0 0 k V single-circuit transmission line; - - - standard c o n v o l u t i o n , - recursive discrete c o n v o l u t i o n
V o l 5 No 1 January 1983
35
vs, (z)
vf (z)
E
Vrz (z)
i i
I
I
'r '(z)~, s2 (z)
~r2(Z)l
p
Ft,(z),
s
Zo(Z )
i
node
F&(z), Zo(z )
--~ ~ 'f (Z)
r node
/
//,,I//
Figure 3. Model for short-circuit fault analysis we first adopt line and source impedance functions in the
From equations (73), (74) and (78):
form:
vf(z) = K f [F~I (z) Fsl(Z) + FQ2(z) Fez (z) Zo = Z~ + Zou(Z)
(64)
Z p = ZPx + Z~(z)
(65)
Z f = ZPrc+ ZP(z)
(66)
It is also convenient to group variables into the characteristics:
(79)
for K r = £ i ' [C~ + C;' Y~Cj
(80)
This completes the z-plane working. On transforming to the time-domain:
Frl (z) = vf(z) + Z0(z) irl (z)
(67)
is1 (n) = C~' Ysp [VsP(n) -- Clfl(n) -- C,f2(n) -- f3(n)]
Fs2(Z) = vf(z) + Zo(z) is2(z)
(68)
irl (n) = Y~ [Vy(Z) -- f4(n) -- fs(n)]
Equations (56) and (57) now combine to give:
is2(n) = Y~ Ivy(z) -- f6(n) -- fv(n)]
isl (z) = C21Ysp [VsP(z) - CIF~I (z) Frl (z) -- C~Zoa(Z) isl(Z ) - - Z P ( z ) i P (z)]
iez (z) = C ; I y p [V~P(n) -- G f s ( n ) -- C,fg(n) -- fm(n)] (69)
Similarly, at the other end of the transmission line, using equations (60) and (61):
--
C,Zoa(Z) it2 (z) -- ZP(z)
i~2(z)l
vf(n) = Kf [f4(n) + fs(n) + f6(n) + fT(n)] Vsl (n) = fl(n) + f2(n) + Z~isj (n) Fez(n) = fs(n) + fg(n) + Z~in(n)
iez(Z) : C 2 ' Y r p [VrP(z) -- C1FQ2(z ) V$2(z )
(70)
for:
Fsl (n) = Vsl (n) + f2(n) + Z~isl (n) VH(n ) = vf(n) + fs(n) + Z{~irl(n)
Ysp = l Zg + Z ~ ] - '
(71)
Vrp = [Z[ + ZPre]-'
(72)
In combining short-circuit fault constraints at the point of fault with the line equations at that point, ira (z), and iez(z) are formed from equations (58) and (59) to give:
Fs2(n ) = v[(n) + G(n) + ZQis2(n) Fr2(n ) = Fez(n) + f9(n) + Z~iez(n) For the separate convolutions: fl(n) = f~l(n)* Frl(n)
Zoa(n)*
(73)
f2(n) =
is2(Z) = Y~ [vy(z) -- F~2(z) Fez(z) -- Zod(Z) is2(z)]
(74)
f3(n) = ZP(n) * iP(n)
fs(n) = F~2(n)* Fg2(n)
f4(n) = FQ1 (n)* Fsl(n)
fg(n) = Zod(n)* iez(n)
Fsl(Z) = Vsl (z) + Zo(z) i,1 (z)
(75)
Fez(z) = Fez(z) + Zo(z)iez(z)
(76)
and YQ = Z~ 1
fs(n) = Zod(n)* irl(n)
fT(n) = Zoa(n)* i~(n)
flo(n) = ZP(n) * i~(n)
The selection of the fault type is made solely through the G~ matrix. For a single-phase-to-earth fault on phase 'a' through fault-path conductance G#, for example, the fault selector matrix is given by:
(77)
using: Gfv~(z) = -- (ir~ (z) + iP(z))
isl (n)
fo(n) = F~2(n)* Fez(n)
it1 (z) = YQ [v[(z) -- F~I (z) Fs] (z) -- Zoo(Z) irl(Z)]
where:
36
+ Zod(Z) it1 (z) + Zoa(Z) ise(z)]
~= (78)
o
(81)
0
Electrical Power & Energy Systems
4o°I
>
3O0 200 5
>
IO0
I_
o
15
0
cO U
¥
I=
-100
u3
Z
I
f~.
1 Time,
,A
I
7I/
8
I ~9
10
/
ms
-200
c-
13,_ - 30C
400 34O > IGI ID
280
-
220
-
160
-
.5>
o L
U D "O cO U
100 I
~o
l,~i
-20~
,,,,Q © G) ID c13._
I
I 4
2
I 5 Time,
1
I
I
I
6
7
8
9
ms
-80 -140
>
300
-
200
-
100
-
o
rz~
I 1
0
2
3
4
5
e
9
%
-100
> L 0
15 - 2 o 0 "ID cO U
-300
U
¢,
-
400
~_ - 5 0 0 -
_
l
600
Figure 4.
V o l t a g e t r a n s i e n t s in e v o l v i n g f a u l t sequence; - - -
Vol 5 No 1 January 1983
standard convolut!on, --
recursive discrete c o n v o l u t i o n
37
For a phase-to-phase fault between phases 'b' and 'c' through conductance Gbc:
0]
G; =
Gbc
--Gbc
--G bc
G bc
(82)
Initial-condition values from which electromagnetic transient analysis begins are found directly from a steady-state phasor analysis of the primary system configuration of Figure 3. In particular, the analysis methods completely avoids the pre-fault convolutions which are required in continuous convolution formulations 14 when transforming directly from the frequency-domain to the time-domain. Solutions for an evolving fault sequence in which singlephase-to-earth, double-phase.to-earth, and three-phase-toearth fault constraints are required in succession, are shown in Figure 4. In Figure 4, the phase-a-to-earth fault at 0.0 s evolves to a double-phase-to-earth fault at 1.0 ms and to a 3-phase-to-earth fault at 3.0 ms. The initial fault on-set coincides with a voltage peak in phase a. There is a fault level of 10 000 MVA at each line termination and the fault location is 128 km from the termination to which the voltage transients relate. The line is a 160 km, 400 kV single-circuit transmission line. The principal line data is given in the Appendix. The fault location is 80% of the line length from end s in Figure 3, and the transient waveforms of Figure 4 are those at the s-end of the line.
Conclusions The forms of convolution which arise following transformation from the frequency-domain to the time-domain have been widely applied in electromagnetic transient analysis in power systems, but no previous work has referred to formulations based on the convolution of z-plane functions. These extend the recent work on power transmission line modelling by digital filters and provide a formal counterpart to it. The solution methods to which they lead combine high accuracy with very high inherent stability and low computing time requirements. On a CDC Cyber 72/73, the computing time requirement for the solution of Figure 2 is 150 ms, and that for the solution in Figure 4 is 200 ms. In addition, z-plane convolution methods are particularly direct to derive and apply. Although they are developed mainly in the present paper in the context of a single transmission line, they easily extend to modelling multi-node networks 18, and, where required, to include the nonlinear magnetization characteristics and the principal features of surge arresters 19in them. The methods appear to be particularly well suited to cross-bonded cable system analysis2°. Here, the detailed working in representing sheath earthing and transpositions can be considerable, but the present paper shows that this can now largely be achieved in the z-plane. VII.
Acknowledgements The authors are grateful to the Australian Research Grants Committee for financial support and to the West Australian Rcgional Computing Centre for running their programs.
VIII.
38
The generous support of power systems research in tile Department of Electrical and Electronic Engineering at the University of Western Australia by Professor A R Billings, Head of Department, together with his professional cooperation at all times, is gratefully acknowledged. T T Nguyen gratefully acknowledges the award of a University Research Scholarship. IX. References 1 Bickford, J P and Doepel, P S 'Calculation of switching transients with particular reference to line energisation' Proc. Inst. Electr. Eng. Vol 114 No 4 (1967) pp 465477 Battisson, M J e t al. 'Calculation of switching phenomena in power systems' Proc. Inst.. Electr. Eng. Vol 114 No 4 (1967) pp 478-486 Dommel, H W 'Digital computer solution of electromagnetic transients in single- and multi-phase networks'
IEEE Trans. Power Appar. & Syst. Vol PAS-88 No 4 (1969) pp 388-396
4 Bickford, J P, Mullineux, N and Reed, J R 'Computation of power system transients' lEE Monograph 18 (Peter Peregrinus 1976) p 75
Budner, A 'Introduction of frequency-dependent line parameters into an electromagnetic transient program' IEEE Trans. Power Appar. & Syst. Vol PAS-89 No 1 (1970) pp 88-97 Snelson, J K 'Propagation of travelling waves on transmission lines: frequency-dependent parameters' IEEE Trans. PowerAppar. & Syst. Vol PAS-91 No 1 (1972) pp 85-91 Semlyen, A and Dabuleanu, A 'Fast and accurate switching transient calculations on transmission lines with ground return using recursive convolutions' IEEE Trans. Power Appar. & Syst. Vol PAS-94 No 2 (1975) pp 561-569 Carroll, D P and Nozari, F 'An efficient computer method for simulating transients in transmission lines with frequency-dependent parameters' IEEE Trans. PowerAppar. & Syst. Vol PAS-94 No 4 (1975) pp 1167-1174
Semlyen, A and Roth, A 'Calculations of exponential propagation step responses' IEEE Trans. Power Appar. & Syst. Vol PAS-96 No 3 (1977) pp 667-671 10 Ametani, A 'A highly efficient method of calculating transmission line transients' IEEE Trans. Power Appar. & Syst. Vol PAS-95 (1976) pp 1145-1149 11 Humpage, W D, Wong, K P and Nguyen, T T 'Digital filter model of power transmission line' Int. J. Electr. Power & Energy Syst. Vol 3 No 4 (1981) pp 197-207
12 Humpage, W D et al. 'Z-transform electromagnetic
Electrical Power & Energy Systems
transient analysis in power systems' Proc. Inst. Electr. Eng. Part C: Gen. Trans. & Distr. Vol 127 No 6 (1980) pp 370-378
•--,,-6.78m ~
0 .
13 Magnusson, P C 'Travelling waves on multiconductor open-wire lines: a numerical survey of the effects of frequency-dependence of modal composition' IEEE Trans. Power Appar. & Syst. Vol PAS-92 (1973) pp 999-1008
~-- 9.98 m
magnetic transient analysis in power systems' Proc. Inst. Electr. Eng. Part C: Gen. Trans. & Distr. Vol 127 No 6 (1980) pp 386-394 16 Humpage, W D et al. 'Dynamic simulation of high speed protection'Proc. Inst. Electr, Eng. Vol 121 No 6 (1974) pp 474-480 17 Humpage, W D and Wong, K P 'Some aspects of the dynamic response of distance protection' Trans, I. E. Aust, Vol EEl5 No 3 (1979) pp 122-129 18 Humpage,W D, Wong, K P and Nguyen, T T 'Development of z-transform electromagnetic transient analysis methods for multi-node networks', Proc. Inst. Electr. Eng. Part C: Gen. Trans. & Distr. Vol 127 No 6 (1980) pp 279-385 19 Humpage, W D, Wong, K P and Nguyen, T T 'Surge diverter and transformer nonlinearities in z-transform electromagnetic transient analysis in power systems' Proc. Inst. Electr. Eng. Part C: Gen. Trans. & Distr. Vol 128 No 2 (1981) pp 63-69 20 Humpage, W D, Wong, K P and Nguyen, T T 'Z-transform electromagnetic transient analysis of cross-bonded cable transmission systems' Proc. Inst. Electr. Eng. Part C: Gen. Trans. & Distr. Vol 128 No 2 (1981) pp 55-62
Appendix: Transmission line data The tower configuration and conductor spacing of the
Vol 5 No 1 January 1983
.
.
.
.
.
--~ 0
0
4153m -,--- 8.15m - ~ O0
14 Bergmann, R Ch G and Ponsioen,P J M 'Calculation of electrical transients in power systems: untransposed transmission line with frequency dependent parameters' Proc. Inst. Electr. Eng. Vol 126 No 8 (1979) pp 764-770
15 Humpage,W D, Wong, K P and Nguyen, T T 'Timeconvolution and z-transform methods of electro-
.
31.24m
1204m /
1
/)//////////////,
Figure 5. 400 kV transmission line spacings
400 kV transmission line to which the over-voltage evaluations of Section V and the short circuit fault analysis studies of Section VI relate is shown in Figure 5. Other basic line data is collected together in Table 1.
Table 1. Basic transmission line data Number of circuits Number of conductors per phase Number of earth-wires Conductor position symmetry Conductor resistivity, £Zm Earth-wire resistivity, ~ m Conductor strand diameter, cm E~'rth-wire strand diameter, cm Geometric mean diameter for 4-conductor bundle, cm Outer diameter of earth-wire, cm Number of effective strands in phase conductors Number of effective strands in earth wire Earth resistivity, ~2m
1 4 1 none 3.2 x 10-8 2.69 x 10-s 0.32 0.32 30.94 2.86 54 54 20.0
39
transient analysis W D Humpage, K P Wong and T T Nguyen Department of Electrical and Electronic Engineering, University of Western Australia, Australia
Time-convolution in the forms arising in transformation from the frequency-domain to the time-doinain has been widely used in the earlier development o f electromagnetic transient methods in power systems. Independent work has recently led to new methods based on the z-transform, and the present paper develops methods based on discrete convolution arising from transformation to the time-domain now from the z-plane. The recursive solution sequences to which this leads combines high accuracy with low computing time requirements. Checks and controls in the synthesis of transmission line forward impulse response and surge impedance functions in the z-plane ensure that these are always stable system functions, and that numerical solution procedures which include them have very high inherent stability. The formulations developed are applied to transmission line energization from an equivalent source model, and the electromagnetic transients in short-circuit fault operation. Close comparisons are made between representative solutions from standard time-convolution analysis and from the methods o f the present paper. Keywords: electric power transmission, electromagnetic transients, z-plane function convolution
I. Introduction Following the earlier development H ° of several different forms of electromagnetic transient analysis in high-voltage power networks, recent work has reported power transmission line transient models which are based on digital £tlters xx. One of the applications of fitter models is in the derivation of computational algorithms, and recent work has produced solution sequences derived from that basis. These require the explicit structures of filter networks and they are formed in terms of the delay functions in their forward and return signal-flow paths. To reflect filter concepts formally into transient analysis leads to methods which involve the convolution of z-plane functions. Principal among the methods developed so far have been those in which transformation from the frequency-domain to the time-domain leads to time-convolution, but no paper so far has referred to formulations which derive from the convoluReceived: 19 October 1982
Vol 5 No 1 January 1983
tion of z-plane functions. In principle, the z-plane is used as an intermediate transform step a2 between the frequencydomain and the time-domain. For any given boundary conditions, a complete analysis formulation is developed almost entirely in the z-plane. Only the final group of equations required in the solution are transformed from the z-plane to the time-domain in terms of discrete-function convolution. Surprisingly concise sequences provide both fully comprehensive electromagnetic transient analysis formulations and also the individual and explicit steps of recursive evaluation. The present paper develops this logical extension of recent investigations of power transmission line electromagnetic transient modelling by digital filters. II. Principal symbols transmission line forward impulse response FQ(z) transmission line surge-impedance function Zo(z) source impedance function z~(z) matrices of source resistance and inductance coefficients v(z), i(z) vectors of voltage and current variables in z-plane v(n), i (n) vectors of voltage and current variables in recursive solution sequences F(z), F(n) forward characteristics B(z), B(n) backward characteristics vector of intermediate variables w(n) modal transformation matrices Ca, C2 At sample interval in time-domain ratio of wave transit time to sampling interval m The subscripts s and r identify variables at the switching and remote ends of the transmission line, respectively. The subscript f identifies variables at a point of short-circuit fault discontinuity. The superscript p distinguishes phasevariables from modal variables.
III. Z-plane response functions Multi-product forms of transmission line response functions in the z-plane lead directly to the cascade f'flter structures of earlier work al, but the z-function convolution procedures of the present paper are approached more directly when numerator and denominator functions have open poly-
0142-0615/83/010031-09 $03.00 © 1983 Butterworth & Co (Publishers) Ltd
31
nomial forms. It is also preferable first to consider the functions for one mode so that the subsequent working is in scalar form. The steps of the working apply identically to each mode, and, when these are complete, modes re-group to a vector form. On this basis, the forward impulse response F~ (z) and the surge impedance function Zo(z) are expressed in the forms:
m 1
DQ(z)= ~
bkz -k
(10)
k=l
M2
No(z)=
CkZ- k - ~ k=l
dkz -k
(11)
k=l
m2
Do(z) = ~, dk z-k F~(z) = - -
(12)
k=l
k=l
(1)
M~
Ns(z ) = (e -- 1) z-1
1 + ~" bk z-k
(13)
k=l
[ M2
Z~ 1 + E
IV. Z-function convolution
CkZ-
k=l
Zo(z) =
(2)
M, 1 + ~ dkz -k
IV. 1 Convolution involving the forward impulse response
In considering z-function convolution involving the forward impulse response, F~ (z), input and output functions, g~ (z) and f~(z), are introduced so that it is required to evaluate:
g=l
Owing to the term z -m in F~(z), only previous-step values arise in operations involving F~ (z) following transformation to the time domain. By comparison, the form of the surge impedance function does not ensure this, and it is then necessary to be able to separate the present-step component from all previous-step components. In the z-plane, this corresponds to a separation in the surge impedance function so that: (3)
Zo(z) = & + Zd(Z)
f~ (z) = F~ (z) g~ (z)
(14)
On using F~ (z) from equation (6): j~ (z) -
Fz -m [1 + N~(z)]
1 +D~(z)
g~ (z)
(15)
An intermediate function W~(z) is now formed using:
z- mg~ (z) W~(z) -
From equation (2), this is achieved by:
(16)
] + D~ (z)
M~
Z~ ~ CgZ- k - Z Q ~ dgz -k k=l
Zo(z) = Z~ +
k=l
M, 1 + ~ dk z-k
In terms of this, the output function f~(z) is found from: (4)
f~(z) = F [ 1 +N~(z)] W~(z)
k=l
On transforming to the time-domain:
Where a lumped-parameter source model, of the kind considered in earlier work n arises, the z-plane source-impedance function, Zs(z), is given by:
f~(n) = F[1 +N~(n)]* W~(n)
(e -- 1) z -a Z , ( z ) : z~ + Zc
1 +z -1
(s)
1 +&(z) Z~No(z) 1 + Do(z)
Zo(z) = Z~ 4
ZcN,(z) Zs(z) = z c -~
1 + z -1
(6)
(7)
(8)
W~(n) = g~ (n -- m) -- D~ (n)* W~(n)
(20)
In summation form, and for FQ (z) one-sided:
F~(n)=F[ W~(n)+ ~' NQ(k)W~(n--kt
(2])
k=l
ml
where:
WQ(n)=g~(n--m)-- ~
k=l
Dg.(k) W~(n--k)
(23)
k=l
m1
N~.(z) = ~
32
(19)
and then transforms to:
Fz -m [1 +NQ(z)] F~ (z) =
(18)
where * denotes discrete-sequence convolution and n the sampling point in the time-domain respectively. The expression for W~(z) first rearranges to:
WQ(z) = z-mg~ (z) -- D~ (z) W~(z)
In preparation for the subsequent derivations, equations (1), (4) and (5) are arranged in the forms:
(17)
ak z-k
(9) In these forms in the time-domain N~ (k) and D e (k) are
Electrical Power & Energy Systems
Circuit breaker
the kth terms in the z-plane numerator and denominator polynomials so that: N~ (k) = ak D~ (k)
/
,~czl i ~p~z~,zglz ~
"1
]
o
(24)
= bk
IV.2 Convolution involving the surge impedance function For input and output functions g0(z) and fo(z), and the surge impedance function of equation (7), it is required to evaluate in this case:
)
vp(z) m
Transmission line model
Source model
Z~No(z) ]
fo(z) = [ Z £ + i ~ ] g o ( z )
(251
On defining an intermediate function Wo(Z)then: re(z) = Z~go(z) + Z ~ g o ( z ) We(z)
(26)
We(Z) = go(z) -- Do(z) We(Z)
(27)
Figure 1. Single line equivalent of switching configuration in line energization circuit, the equations of the transmission line in modal variables in the z-plane are: Vs(Z) -- Z0(z) is(z) = F£(z) Vr(Z)
(35)
vr(z ) = F~(z) [vs(z) + Zo(z) is(Z)]
(36)
Transforming to the time-domain now gives: M2
re(n) = Z2go(n) + Z~ ~,
No(k) Wo(n -- k)
(28)
Phase- and modal-variables are inter-related by:
k=l
M~ We(n) = go(n) -- ~ Do(k) Wo(n -- k)
(29)
k=l for No(k) = c k -- d k
(30)
Do(k) = dk
(31)
IV.3 Convolution involving the source impedance function The steps here are closely similar to those of section (IV.2). Following transformation to the time-domain, the expression fs (z) = Z s (z) gs (z) is evaluated from: fs(n) = Zcgs(n ) + Z e ( e -- 1) Ws(n -- 1)
(32)
Ws(n) = g s ( n ) - Ws(n -- 1)
(33)
V. Energization of an open-circuited transmission line from an equivalent source model Switching over-voltage evaluations in power transmission line energization are often based on an equivalent representation of the network to which the incoming circuit is switched, as in Figure 1. When the equivalent is in a lumpedparameter form, transmission line voltages at the switchingend of the line in vector vsP(z) are related to the specified source voltages of the system equivalent in vector VsP(z) by:
vf(z) = Vf(z) -Z~(z) ~(z)
(34)
As in Figure 1, line conductor currents at the switching-end are in vector isP(z), and ZsP(z) is the source-model impedance matrix. In equation (34), variables are in phase co-ordinates. When the end of the transmission line remote from that at which circuit-breaker switching takes place is on open-
Vol 5 No 1 January 1983
vP(z) = C1 v(z),
v(z) = C;lvP(z)
(37)
iV(z) = C2i(z),
i(z) = C~1iV(z)
(38)
Previous work TM14 has considered the effect of the frequency- dependence of modal transformation matrices following which C1 and C2 identify the real parts of the modal transformation matrices evaluated at a selected frequency. On re-expressing the transmission line relationships of equations (35) and (36) in phase-variable form: vP(z) -- ZOO(z)isP(z) = F~'vP (z)
(39)
vP(z) = F~P(z) [vsP(z) isP(z) + Zo(z) is(z)]
(40)
where: Zg(z) = C,Zo(z) C; 1
(41)
F¢(Z) = C1F£(z ) C11
(42)
On using equation (34) to eliminate vsP(z) from equation (39): VsP(z) -- [ZOO(z)+ ZsP(z)] iV(z) : FP(z) vrP(z)
(43)
In solving for isP(Z) using equation (43), Z°o(z) and ZPs(Z) are separated into the forms: ZOO(z) = Z p + ZVoa(z)
(44)
z,v(z) = z~ + z~(z)
(45)
With these separations, and using Ye = [Z~' + Zcp]-I: isP(Z) = rc{VPs(z) - - F g ( z ) VrP(Z) --
[ZoPa(z) + Z~(z)l iV(z)}
(46)
33
Whereas combining source model and transmission line equations requires transformation between modal- and phase-variables, it is preferable in reducing the numerical work of subsequent convolution evaluations as much as possible to use the diagonal forms ofF~ (z) and Zo(z). Transforming equation (46) to achieve this then gives:
iAz) = c ; 1 YA vP(z) - C,FQ(z) vAz) --
C1Zoct(z ) is(Z) -- Zp(z) isP(z)}
(47)
This represents the first step in the development. The second one is to substitute for vs(z ) in equation (35) into (36) to give: vr(z) = FQ(z) [F~(z) vr(z ) + 2Zo(z ) is(z)]
(48)
Given is(z ) from equation (47), (48) provides a solution for vr(z ). Having achieved this in the z-plane, we now transform equations (47) and (48) to the time-domain to give: vr(n ) = F~(n)* [FQ(n)* Vr(n ) + 2Zo(n)* is(n)l
(49)
is(n) = C2-' Ye [VsP(n) -- C,F~(n)* Vr(n ) --ClZoa(n)* is(n ) - Z P ( n ) * iP(n)]
(50)
The separate convolutions to be evaluated in the numerical solution of equations (49) and (50) for vr(n) are now identified by: fl(n) = FQ(n)* Vr(n)
(5l)
f~(n) = Zoa(n)* is(n)
(52)
fs(n) = Z P ( n ) * isP(n)
(53)
f4(n) = fi(n) + 2Z~is(n) + 2f2(n} Vr(n ) = F [W~2(n ) + alW~2(n -- 1) + a2W22(n -- 2)1 W~2 (n) = f4(n --m) --blW2(n -- 1) --b2W~2{n -- 2) This fully defined sequence now provides a complete recursive solution. Sequential pole-closing in circuit-breakers is represented by a time-control of the diagonal elements of the source impedance matrix and which is in turn included in Yc. A representative solution for a 160 km 400 kV transmission line, for which the principal data is collected together in the Appendix, is given in Figure 2. Here, there is trapped charge on the conductors of the transmission line immediately prior to energization. In Figure 2, solutions from the sequence are compared with those of a standard time-convolution formulation ls.
V l . Short-circuit fault constraints A representation of short-circuit fault conditions is required in several areas of power system electromagnetic transient analysis, including the specialist area of transmission line protection system evaluations and research into signalprocessing methods for fault monitoring and real-time protection purposes 16' 17. A basic primary system for a high-speed protection system configuration is shown in Figure 3. A single point-to-point interconnection has fault infeeds at both ends. It is required that any given type of short-circuit fault can be represented at any location on the interconnection from any given pre-fault operating condition. Here the fault location sub-divides the line into two sections. The principal variables in each section are indicated in Figure 3. The source model and line equations for the solution are given as follows: Source model at end-s
The vector of switching-end line currents is then found from:
v~ (z) = V f ( z ) - Z f ( z )
in ( z )
(56)
is(n) = C21 Ye [VsP(n) --Clfl(n) -- C,f2(n) -- f3(n)] (54) Section 1 Remote-end voltages in vrP(n) are then available from: vrP(n ) = C,F~(n)* [f,(n) + 2f2(n) + 2ZQis(n)]
Subject to the definition of the different convolutions, equations (54) and (55) provide a complete solution in the time-domain. It is of interest to give the explicit recursive sequences of the numerical solution for a particular form of F~(z) and Zo(z). When these are second-order representations, the complete sequence is as follows: fl(n) = F
[WQ1(n)
+ al W~I (n -- 1) + a2W~l (n -- 2)]
WQl(n) = vr (n -- m) -- b 1W~l(n -- 1) -- b2 W~l (n -- 2) f2(n)
= Z~(Cl--all)
Wo(n -- 1) + Z~(c2--d2)Wo(n--2)
(57)
vf(z) -- Zo(z ) irl (,7,) = F£1 (z) [Vsl (,7,)+ Zo(z ) isl (z)]
(58)
Section 2 vf(z)--Zo(z)is,2(z) = F~2(z) [v,,2(z) +Zo(z)ir2(Z)]
(59)
vr2 (z) --Zo(z) iy2(z) = F~2(z) [vf(z) + Zo(z) is2 (z)]
(60)
Source at end-r Vr~ (z) = Vrp (z) -- Zrp (z) ir~ (z)
(61)
Fault location
Wo(n) = is(n) -- d~Wo(n -- 1) -- d2Wo(n -- 2)
i~(=) = - (i~ (z) + iV(z))
(62)
f3(n) = Zc(e -- 1) Ws(n -- 1)
if(z) = ayv (z)
(63)
Ws(n ) = isP(n) -- Ws(n -- 1)
JAn) = C~'Y~ [VP(n) - Clfi(n) -C~ f2(n) -fs(n)]
34
vsl(z)--Zo(z)is1(z)=F~l(z)[vf(z)+ Zo(z)irl(Z)] (55)
In equation (63), G~ is a matrix of fault-path conductances. In developing a solution procedure for equations (56)-(63),
Electrical Power & Energy Systems
960
>
7201
~"
480
""
o
240
L o
0
,i,
3
U
~o -240 c-
4
5 Time,
f
6 ms
'\'
8
9
O u
-480 a
r" 13._
-720 -960
590
--
>
480
--
o~
370
o >
260
C
0
150
:J
,
r"
o U
40
43
-
70
I
I
I
I
1
2
3
4 Time
m o
ca.
-180 -290
I 5
6
7
8
ms
-
?
600 F
42ol>
240 60
' /''
1 "6
-120
>
o
",
9
-300
7 -480 o
2
3
#'\' 4
5
• Timems
'f'
6
I
7
W
U
- 660
a -840
c Q_
Figure 2. Switching overvoltage transients in 400 k V system; pole scatter closing times: a = 0.0 s, b = 500 #s, c = 1 ms; trapped charge d i s t r i b u t i o n : a = - - 1.0 p.u., b = 0.5 p.u., c = - - 0.5 p.u., switching at peak voltage in phase a f r o m 10 000 M V A f a u l t level 160 km, 4 0 0 k V single-circuit transmission line; - - - standard c o n v o l u t i o n , - recursive discrete c o n v o l u t i o n
V o l 5 No 1 January 1983
35
vs, (z)
vf (z)
E
Vrz (z)
i i
I
I
'r '(z)~, s2 (z)
~r2(Z)l
p
Ft,(z),
s
Zo(Z )
i
node
F&(z), Zo(z )
--~ ~ 'f (Z)
r node
/
//,,I//
Figure 3. Model for short-circuit fault analysis we first adopt line and source impedance functions in the
From equations (73), (74) and (78):
form:
vf(z) = K f [F~I (z) Fsl(Z) + FQ2(z) Fez (z) Zo = Z~ + Zou(Z)
(64)
Z p = ZPx + Z~(z)
(65)
Z f = ZPrc+ ZP(z)
(66)
It is also convenient to group variables into the characteristics:
(79)
for K r = £ i ' [C~ + C;' Y~Cj
(80)
This completes the z-plane working. On transforming to the time-domain:
Frl (z) = vf(z) + Z0(z) irl (z)
(67)
is1 (n) = C~' Ysp [VsP(n) -- Clfl(n) -- C,f2(n) -- f3(n)]
Fs2(Z) = vf(z) + Zo(z) is2(z)
(68)
irl (n) = Y~ [Vy(Z) -- f4(n) -- fs(n)]
Equations (56) and (57) now combine to give:
is2(n) = Y~ Ivy(z) -- f6(n) -- fv(n)]
isl (z) = C21Ysp [VsP(z) - CIF~I (z) Frl (z) -- C~Zoa(Z) isl(Z ) - - Z P ( z ) i P (z)]
iez (z) = C ; I y p [V~P(n) -- G f s ( n ) -- C,fg(n) -- fm(n)] (69)
Similarly, at the other end of the transmission line, using equations (60) and (61):
--
C,Zoa(Z) it2 (z) -- ZP(z)
i~2(z)l
vf(n) = Kf [f4(n) + fs(n) + f6(n) + fT(n)] Vsl (n) = fl(n) + f2(n) + Z~isj (n) Fez(n) = fs(n) + fg(n) + Z~in(n)
iez(Z) : C 2 ' Y r p [VrP(z) -- C1FQ2(z ) V$2(z )
(70)
for:
Fsl (n) = Vsl (n) + f2(n) + Z~isl (n) VH(n ) = vf(n) + fs(n) + Z{~irl(n)
Ysp = l Zg + Z ~ ] - '
(71)
Vrp = [Z[ + ZPre]-'
(72)
In combining short-circuit fault constraints at the point of fault with the line equations at that point, ira (z), and iez(z) are formed from equations (58) and (59) to give:
Fs2(n ) = v[(n) + G(n) + ZQis2(n) Fr2(n ) = Fez(n) + f9(n) + Z~iez(n) For the separate convolutions: fl(n) = f~l(n)* Frl(n)
Zoa(n)*
(73)
f2(n) =
is2(Z) = Y~ [vy(z) -- F~2(z) Fez(z) -- Zod(Z) is2(z)]
(74)
f3(n) = ZP(n) * iP(n)
fs(n) = F~2(n)* Fg2(n)
f4(n) = FQ1 (n)* Fsl(n)
fg(n) = Zod(n)* iez(n)
Fsl(Z) = Vsl (z) + Zo(z) i,1 (z)
(75)
Fez(z) = Fez(z) + Zo(z)iez(z)
(76)
and YQ = Z~ 1
fs(n) = Zod(n)* irl(n)
fT(n) = Zoa(n)* i~(n)
flo(n) = ZP(n) * i~(n)
The selection of the fault type is made solely through the G~ matrix. For a single-phase-to-earth fault on phase 'a' through fault-path conductance G#, for example, the fault selector matrix is given by:
(77)
using: Gfv~(z) = -- (ir~ (z) + iP(z))
isl (n)
fo(n) = F~2(n)* Fez(n)
it1 (z) = YQ [v[(z) -- F~I (z) Fs] (z) -- Zoo(Z) irl(Z)]
where:
36
+ Zod(Z) it1 (z) + Zoa(Z) ise(z)]
~= (78)
o
(81)
0
Electrical Power & Energy Systems
4o°I
>
3O0 200 5
>
IO0
I_
o
15
0
cO U
¥
I=
-100
u3
Z
I
f~.
1 Time,
,A
I
7I/
8
I ~9
10
/
ms
-200
c-
13,_ - 30C
400 34O > IGI ID
280
-
220
-
160
-
.5>
o L
U D "O cO U
100 I
~o
l,~i
-20~
,,,,Q © G) ID c13._
I
I 4
2
I 5 Time,
1
I
I
I
6
7
8
9
ms
-80 -140
>
300
-
200
-
100
-
o
rz~
I 1
0
2
3
4
5
e
9
%
-100
> L 0
15 - 2 o 0 "ID cO U
-300
U
¢,
-
400
~_ - 5 0 0 -
_
l
600
Figure 4.
V o l t a g e t r a n s i e n t s in e v o l v i n g f a u l t sequence; - - -
Vol 5 No 1 January 1983
standard convolut!on, --
recursive discrete c o n v o l u t i o n
37
For a phase-to-phase fault between phases 'b' and 'c' through conductance Gbc:
0]
G; =
Gbc
--Gbc
--G bc
G bc
(82)
Initial-condition values from which electromagnetic transient analysis begins are found directly from a steady-state phasor analysis of the primary system configuration of Figure 3. In particular, the analysis methods completely avoids the pre-fault convolutions which are required in continuous convolution formulations 14 when transforming directly from the frequency-domain to the time-domain. Solutions for an evolving fault sequence in which singlephase-to-earth, double-phase.to-earth, and three-phase-toearth fault constraints are required in succession, are shown in Figure 4. In Figure 4, the phase-a-to-earth fault at 0.0 s evolves to a double-phase-to-earth fault at 1.0 ms and to a 3-phase-to-earth fault at 3.0 ms. The initial fault on-set coincides with a voltage peak in phase a. There is a fault level of 10 000 MVA at each line termination and the fault location is 128 km from the termination to which the voltage transients relate. The line is a 160 km, 400 kV single-circuit transmission line. The principal line data is given in the Appendix. The fault location is 80% of the line length from end s in Figure 3, and the transient waveforms of Figure 4 are those at the s-end of the line.
Conclusions The forms of convolution which arise following transformation from the frequency-domain to the time-domain have been widely applied in electromagnetic transient analysis in power systems, but no previous work has referred to formulations based on the convolution of z-plane functions. These extend the recent work on power transmission line modelling by digital filters and provide a formal counterpart to it. The solution methods to which they lead combine high accuracy with very high inherent stability and low computing time requirements. On a CDC Cyber 72/73, the computing time requirement for the solution of Figure 2 is 150 ms, and that for the solution in Figure 4 is 200 ms. In addition, z-plane convolution methods are particularly direct to derive and apply. Although they are developed mainly in the present paper in the context of a single transmission line, they easily extend to modelling multi-node networks 18, and, where required, to include the nonlinear magnetization characteristics and the principal features of surge arresters 19in them. The methods appear to be particularly well suited to cross-bonded cable system analysis2°. Here, the detailed working in representing sheath earthing and transpositions can be considerable, but the present paper shows that this can now largely be achieved in the z-plane. VII.
Acknowledgements The authors are grateful to the Australian Research Grants Committee for financial support and to the West Australian Rcgional Computing Centre for running their programs.
VIII.
38
The generous support of power systems research in tile Department of Electrical and Electronic Engineering at the University of Western Australia by Professor A R Billings, Head of Department, together with his professional cooperation at all times, is gratefully acknowledged. T T Nguyen gratefully acknowledges the award of a University Research Scholarship. IX. References 1 Bickford, J P and Doepel, P S 'Calculation of switching transients with particular reference to line energisation' Proc. Inst. Electr. Eng. Vol 114 No 4 (1967) pp 465477 Battisson, M J e t al. 'Calculation of switching phenomena in power systems' Proc. Inst.. Electr. Eng. Vol 114 No 4 (1967) pp 478-486 Dommel, H W 'Digital computer solution of electromagnetic transients in single- and multi-phase networks'
IEEE Trans. Power Appar. & Syst. Vol PAS-88 No 4 (1969) pp 388-396
4 Bickford, J P, Mullineux, N and Reed, J R 'Computation of power system transients' lEE Monograph 18 (Peter Peregrinus 1976) p 75
Budner, A 'Introduction of frequency-dependent line parameters into an electromagnetic transient program' IEEE Trans. Power Appar. & Syst. Vol PAS-89 No 1 (1970) pp 88-97 Snelson, J K 'Propagation of travelling waves on transmission lines: frequency-dependent parameters' IEEE Trans. PowerAppar. & Syst. Vol PAS-91 No 1 (1972) pp 85-91 Semlyen, A and Dabuleanu, A 'Fast and accurate switching transient calculations on transmission lines with ground return using recursive convolutions' IEEE Trans. Power Appar. & Syst. Vol PAS-94 No 2 (1975) pp 561-569 Carroll, D P and Nozari, F 'An efficient computer method for simulating transients in transmission lines with frequency-dependent parameters' IEEE Trans. PowerAppar. & Syst. Vol PAS-94 No 4 (1975) pp 1167-1174
Semlyen, A and Roth, A 'Calculations of exponential propagation step responses' IEEE Trans. Power Appar. & Syst. Vol PAS-96 No 3 (1977) pp 667-671 10 Ametani, A 'A highly efficient method of calculating transmission line transients' IEEE Trans. Power Appar. & Syst. Vol PAS-95 (1976) pp 1145-1149 11 Humpage, W D, Wong, K P and Nguyen, T T 'Digital filter model of power transmission line' Int. J. Electr. Power & Energy Syst. Vol 3 No 4 (1981) pp 197-207
12 Humpage, W D et al. 'Z-transform electromagnetic
Electrical Power & Energy Systems
transient analysis in power systems' Proc. Inst. Electr. Eng. Part C: Gen. Trans. & Distr. Vol 127 No 6 (1980) pp 370-378
•--,,-6.78m ~
0 .
13 Magnusson, P C 'Travelling waves on multiconductor open-wire lines: a numerical survey of the effects of frequency-dependence of modal composition' IEEE Trans. Power Appar. & Syst. Vol PAS-92 (1973) pp 999-1008
~-- 9.98 m
magnetic transient analysis in power systems' Proc. Inst. Electr. Eng. Part C: Gen. Trans. & Distr. Vol 127 No 6 (1980) pp 386-394 16 Humpage, W D et al. 'Dynamic simulation of high speed protection'Proc. Inst. Electr, Eng. Vol 121 No 6 (1974) pp 474-480 17 Humpage, W D and Wong, K P 'Some aspects of the dynamic response of distance protection' Trans, I. E. Aust, Vol EEl5 No 3 (1979) pp 122-129 18 Humpage,W D, Wong, K P and Nguyen, T T 'Development of z-transform electromagnetic transient analysis methods for multi-node networks', Proc. Inst. Electr. Eng. Part C: Gen. Trans. & Distr. Vol 127 No 6 (1980) pp 279-385 19 Humpage, W D, Wong, K P and Nguyen, T T 'Surge diverter and transformer nonlinearities in z-transform electromagnetic transient analysis in power systems' Proc. Inst. Electr. Eng. Part C: Gen. Trans. & Distr. Vol 128 No 2 (1981) pp 63-69 20 Humpage, W D, Wong, K P and Nguyen, T T 'Z-transform electromagnetic transient analysis of cross-bonded cable transmission systems' Proc. Inst. Electr. Eng. Part C: Gen. Trans. & Distr. Vol 128 No 2 (1981) pp 55-62
Appendix: Transmission line data The tower configuration and conductor spacing of the
Vol 5 No 1 January 1983
.
.
.
.
.
--~ 0
0
4153m -,--- 8.15m - ~ O0
14 Bergmann, R Ch G and Ponsioen,P J M 'Calculation of electrical transients in power systems: untransposed transmission line with frequency dependent parameters' Proc. Inst. Electr. Eng. Vol 126 No 8 (1979) pp 764-770
15 Humpage,W D, Wong, K P and Nguyen, T T 'Timeconvolution and z-transform methods of electro-
.
31.24m
1204m /
1
/)//////////////,
Figure 5. 400 kV transmission line spacings
400 kV transmission line to which the over-voltage evaluations of Section V and the short circuit fault analysis studies of Section VI relate is shown in Figure 5. Other basic line data is collected together in Table 1.
Table 1. Basic transmission line data Number of circuits Number of conductors per phase Number of earth-wires Conductor position symmetry Conductor resistivity, £Zm Earth-wire resistivity, ~ m Conductor strand diameter, cm E~'rth-wire strand diameter, cm Geometric mean diameter for 4-conductor bundle, cm Outer diameter of earth-wire, cm Number of effective strands in phase conductors Number of effective strands in earth wire Earth resistivity, ~2m
1 4 1 none 3.2 x 10-8 2.69 x 10-s 0.32 0.32 30.94 2.86 54 54 20.0
39