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Phase-domain multiphase transmission line models
ELSEVIER
Electrical Power & Energy Systems, Vol. 19, No. 4, pp. 241-248, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved PII: S0142-0615(96)00039-7 o142-0615/97/$17.oo + 0.oo
Phase-domain multiphase transmission line models F Castellanos, J R M a r t i and F M a r c a n o Department of Electrical Engineering, The University of British Columbia, Vancouver, B.C., Canada V6T 1 Z4
models avoid the use of transformation matrices to relate the decoupled eigenmode propagation in the modal domain to the coupled-mode propagation in the phase domain. Since in their final form the equations are solved entirely in phase coordinates, these models can be regarded as phase-domain line models. They differ, however, from other recently proposed phase-domain models (e.g., [5-8]) in that the frequency dependent wave propagation functions are still first synthesized in the frequency domain, as in [3] and [4].
In general, multiphase t,,ansmission line and cable models for transient programs, such as the EMTP, base their solutions on transformations between the phase and modal domains. The main difficulty with these models is the synthesis of the frequency dependent transformation matrices. This paper presents two line models that circumvent this problem by writing the propagation functions directly in the phase domain and thus avoid the use of modal transformation matrices. The first model (developed in connection with our work on corona modelling) avoids the use of modal transformation matrices by separating the ideal-line travelling effect from the loss effects (resistance and internal inductance). The second proposed model is a full frequency dependent distributed parameter model based on idempotent decomposition. By using idempotents instead of eigenvectors a,: the basisfor modal decomposition, the problems associated with the indeterminacy of the frequency dependent eigenvector functions are eliminated. © 1997 Elsevier Science Ltd. All rights reserved
1.1 Lumped line model (z-line) The first proposed phase-domain line model (z-line, for 'z-lumped line') has been developed in connection with our work on a wide-band corona model [9]. This line model follows the basic ideas advanced in [10], except that it is formulated entirely in phase coordinates, thus eliminating the problems associated with modal transformation matrices. The premise here is that if the nonlinear corona model has to be lumped between relatively short line sections then advantage can be taken of the space discretization. This space discretization can be used to 'simplify' the modelling of the distortion part of the propagation function ([e-7(")x]). That is, the distortion due to the resistance and the internal inductance can be grouped into one lumped frequency dependent impedance matrix in phase coordinates [Z loss (0J)]while ideal propagation (at the speed of light in the medium) due to the external magnetic field [Lext] and capacitance [C] can be represented by an ideal-line segment. Since for ideal propagation all modes travel at the speed of light, the equations for this part of the model (ideal line) can be formulated directly in phase coordinates. This model is similar to the constantparameter line model in the EMTP (cp-line) where the resistance is lumped in the middle and at the end points of the line. As opposed to the cp-line model, however, the proposed z-line model can fully take into account the frequency dependence of the line parameters (in [Z l°ss(w)]) and exactly takes into account the asymmetry of any arbitrary line configuration without the need for a transformation matrix. The fd-line (frequencydependent) model in the EMTP works in modal coordinates and assumes that the transformation matrix relating mode and phase is constant and real and, therefore,
Keywords: transmission line modelling, EMTP simulation, phase-domain modelling, idempotent analysis
I. I n t r o d u c t i o n Overhead transmission lines and underground cables are the basic transmission links across a power system. Wave propagation and distortion along lines and cables affects the waveshapes reaching the system equipment and the associated stresses upon the equipment. Despite their importance in transient studies, the accurate modelling of transmission lines in time domain simulations is still not fully satisfactorily resolved. Good time-domain frequency dependent transmission line and cable models, based on Wedepohl and Hedman's theory [1, 2] of the modal decomposition of natural propagation modes (eigenmodes) are available in the EMTP program [3, 4]. The main problem with these models is the frequency dependence of the transformation matrices (matrices of eigenvectors) that connect the modal and phase domains. This paper introduces two new line models being developed at the University of British Columbia. These
241
Phase-domain transmission line models." F. Castellanos et al.
242 Corrections for losses and internal flux ;
7
Ideal line, only external flux "
;
where [ZY] or [¥Z] are full matrices that couple the propagation of voltage and current waves in all the phases. The series impedance matrix [Z] is a full matrix with elements of the form Zi j .~ Rii q-Ja3(Lij • ext + ALij )
[z g)]
(2)
where Rij is the resistance of the conductor plus the correction for the ground return effect, w is the frequency, Lext U is the inductance related to the external flux, and A L O. is the inductance related to the internal flux inside the conductor plus the correction for ground return. Matrix [Z] can thus be rewritten as
[Lext] [c]
[Z] = [Z l°ss] + j w [ L ext]
(3)
[Z] = [Z l°ss] -~-[Z ext]
(4)
Figure 1. Separation of basic effects in the z-line model it cannot correctly simulate strongly asymmetrical configurations. The only limitation of the proposed z-model is that the losses are assumed to be lumped and, therefore, a certain number of line sections are needed to simulate the actual distributed nature of these parameters. 1.2 Idempotent line model (id-line) The second proposed line model (id-line, for 'idempotentline') is a fully frequency dependent distributed-parameter line model. This model is conceptually similar to the current frequency dependent line models in the EMTP (fd-line and q-line, [3] and [4]), except that it overcomes the problems of these models with respect to the frequency dependence of the modal transformation matrices. In general, the main problem in synthesizing the elements of the transformation matrices as functions of frequency is that the eigenvectors that make up these matrices are only defined up to a complex constant factor in the case of distinct eigenvalues or are multiple-valued in the case of repeated eigenvalues. This means, for example, that different scalings ('normalizations') of the eigenvectors will result in different frequency functions for the elements of the transformation matrix. The question of which one is the 'right' normalization is very difficult to answer. The problem of multiple-choice eigenvector functions is aggravated in the case of repeated eigenvalues, which occurs in certain regions of the frequency spectrum. The eigenvectors for N repeated eigenvalues can lie anywhere in a given N-dimensional plane. The proposed id-line model solves these problems by avoiding the use of eigenvectors as base functions for modal decomposition. Instead, the phase-domain propagation function [e -Tphx] (matrix function) is expressed as a linear combination of the natural propagation modes e -7'x (scalar functions) with idempotent coefficient matrices. The idempotent matrices [11] result from the column-row products of the matrix of eigenvectors (modal transformation matrix) and its inverse, and are uniquely defined, independent of the particular scaling of the eigenvectors or of the multiplicity of the eigenvalues.
II. Proposed lumped line model (z-line) The modelling of transmission lines is based on the travelling wave equations, which can be expressed, for a given frequency ~ as, d2V
[ZY]V and d2I [YZ]I 2=
(1)
with Z!OSS tj = Ri j q_ju;/kLij
(5)
The shunt admittance matrix [Y] can be written as (assuming conductance G ~ 0) [Y] =jw[C],
with
[C] = [p]-i
(6)
[P] is the Maxwell coefficients matrix. The [Y] matrix does not require any corrections for ground return and contains only elements that depend on the capacitances (geometry) of the system. With equations (3)-(6), we can rewrite [YZ] as [YZ] = jw[C]([Z l°ss] + jw[Lext])
(7)
[VZ] = j w [ C ] [Z l°ss] - co2 [C] [L ext]
(8)
Equation (8) shows how the product [YZ] can be expressed as the combination of two main components: the first one related to the losses and ground effect corrections and the second one related to the ideal propagation. If we assume that the effect of the losses term is small compared with the effect of the ideal component, then we can lump the losses together and obtain the model shown in Figure l. This condition is met when the section length is small. In the general case of frequency dependent parameters the term [zl°ss(~)] is a function of both the frequency and the section length. I1.1 Modelling of the multiphase ideal line Once the impedance corrections ([Z loss (w)]) are extracted
from the complete model, the ideal transmission line can be easily modelled in the phase domain. Because the travelling time, r, is the same for all modes and the solution of the system of travelling wave equations is decoupled, Dommel's ideal line equation [12] for the time-domain single-phase case can be rewritten for the multiphase case, as illustrated in Figure 2. Using circuit theory, the terminal voltage vectors vk(t) and Vm(t) can be expressed as functions of the terminal current vectors ik(t ) and im(t), the characteristic impedance matrix [Zc], and the history source vectors ekh(t ) and emh(t). The history source vectors are calculated at each time step as functions of past values, through the expressions ekh(t) = Vm(t -- r) + [Zc]im(t -- r)
(9)
emh(t) : vk(t -- r) + [Zc]ik(t -- 7-)
(10)
Phase-domain transmission line models: F. Castellanos et al.
ik(t)
IZcl
kv~(t)~,
lZcl
ira(t)
ekll(t) ~ ~ + ) (~
m Vm(t)
emh(tt-~
Figure 2. Multiphase ideal transmission line model in the time domain
(where s = j ~ ) , which corresponds to a parallel R - L impedance in the continuous time domain. . Each element of [Zl°Ss(~)] is expressed as a sum of several of these blocks (total number of blocks = m), plus an additional resistance term, Riio, in the diagonal elements to satisfy the dc condition (Rii o -----Rii @ dc). With the subscriptfindicating a 'fitted' function:
m l°ssw ~ sKii(l) Diagonal elements (12) Z~ii ( ) = Rii o -'l- ~ s q- Pii(l)
with [Zc] = [y]--1([y] [zext]} 1/2
m ~loss, , = ~ sKij(1) Off-diagonal elements
(11)
In this case, the matrix [Z¢] represents the coupled system in the phase domain and is a full matrix. The implementation of this model in the EMTP is similar, with small modifications, to the standard coupled R - L model [13]. Like in the case of the single-phase ideal line model in the EMTP, the multiphase ideal line model is independent of the integration rule, and represents an exact solution of the line equations. 11.2 Modelling of the multiphase lumped loss section The elements of the matrix [zl°ss(w)] are frequency dependent functions, and they can be synthesized by rational function approximations, as in the frequency dependent line models in the EMTP [3, 4]. Mathematically, however, one has to be very careful when synthesizing coupled systems (as opposed to modal-domain decoupled systems). From network synthesis theory, there are a number of conditions regarding the relationship among poles and zeros of the elements of the matrix that need to be met to assure the stability of the resulting synthesis. In general, these conditions will not be met if each element of the matrix is synthesized independently of the other elements. A new synthesis procedure is used to achieve a coordinated synthesis considering all elements of the matrix together. This procedure preserves the stability of the full fitted matrix at all frequencies. The main concepts of the synthesis procedure used for the new z-line model are the following: . The basic fitting block is the function s K / ( s + p )
243
/% t~)
~'~' s + Pij0)
(13)
e The same number of blocks (m) is used for all the [zl°ss(w)] elements. e The Ks and p s are calculated in a coordinated way, which preserves the positive definitive condition of the [Zl°S~(w)] matrix with the fitted values. e A minimum least squares technique is used to optimize the fitting. Once the [zl°s~(w)] functions are represented in the form of equations (12) and (13) the system of voltagedrop equations is solved in the discrete time domain using the trapezoidal rule as the integration method. Figure 3 shows the fitting of some of the elements of [Z l°s~(~)] for a typical double-circuit transmission line. Nine blocks per function were used for these fits. 11.3 Short-section transmission line solution The phase domain solutions of the ideal line section and the [Zl°~ (w)] section can be combined to simulate a short section of a transmission line. For the short-section model, the part corresponding to [ZJ°~(~)] is divided into two sections. Each half section of [ZL°~(~)] is added at the ends of the ideal line section, as shown in Figure 4. A set of simulations was done to study the relation between the section length of the z-line model and the frequency of the study. The transmission line used for the testing was a typical flat-configuration three-phase 230 kV line. The base case is shown in Figure 5a and consists of one sinusoidal source attached to a node with
10
4
1 0 - -
original/ 120
0
=9
~10
1
[loo v
.~ 101 original
1(~ 0
m=9 -5
10o Frequency
(a)
(Hz)
10 s
0"l
1 0° 1 05 Frequency (Hz)
1
(b)
Figure 3. Typical elements of the lumped loss matrix and their fittings. (a) Diagonal element and (b) off-diagonal element
244
Phase-domain transmission line models. F. Castellanos et al.
[Zl°~))]/2
and [C]
[ZJ°~s)]/2
Figure 4. Proposed line-section for the z-line model different resistance terminations at the other ends. The source frequency was set equal to the frequency used for the calculation of the line parameters. The time-domain response of the z-line model was compared against the solution obtained with the constant-parameter model of the EMTP. By repeating this process for several frequencies an empirical relation between the length of the linesection and the frequency of the analysis was established. The results are shown in Figure 5b and they give the recommended maximum length of the z-line section that could be used for a study at a given dominant frequency.
11.4 Full-length transmission line solution and test cases
The solution for a complete transmission line is accomplished by connecting several of the z-line short-section models in cascade. The implementation of the full-length z-line model was done in a stand-alone program written in ADA-95 and several cases were tested. The results of these cases were compared against the frequency dependent (fd-line) model in the E M T P [3] and the F D T P (Frequency Domain Transients Program) of Wedepohl [11, 14]. Since the F D T P program solves the system entirely in the frequency domain (using an inverse Fourier transform to obtain the time-domain waveforms), its accuracy is not affected by issues of the frequency dependence of the line parameters. Figure 6 shows the general system and line configuration used for one of the test cases. Circuit 1 is energized at the beginning of the simulation (t = 0) with a three-phase sinusoidal voltage source with a peak value 1 p.u. The receiving end o f circuit 1 is connected to a three-phase Y-connected resistive load of 500 ~2/phase. Circuit 2 is connected to ground at the sending end through a small three-phase Y-connected resistive path of 10 ~/phase. The transmission line was represented using 40 sections
of 2.5 km each, for a bandwidth, according to Figure 5b of about 8 Khz (The graphic in Figure 5b was obtained allowing an average error of about 3 %). The time step A t of the simulation was chosen as 8.339 #s to exactly match the travelling time of the ideal line sections. For the L and C elements of the network (including the L s in [Zl°Ss(~)], a A t = 8.339 #s gives a bandwidth of about 12 K H z for a distortion of about 3% with the trapezoidal rule of The sectionalization integration ( f = ½fNyquist : ~ )1. requirements of the model (number of sections and size of At : 7-section) to achieve a certain bandwidth of accurate modelling are, therefore, not too different from those imposed by the distortion of the integration rule on the L and C components: 8 K H z for sectionalization versus 12 K H z for distortion in this case. Nine R - L blocks were used to fit the [Zl°SS(~v)] functions over an extended frequency range. A pole truncation algorithm was then applied to discard those poles close to and beyond the Nyquist frequency. This truncation procedure reduced the number of R L blocks used in the transient simulation to only four per function, with no sacrifice of the accuracy of the synthesis. The simulation results for this case are shown in Figure 7. The asymmetry in this case emerges from the line configuration and the untransposed characteristic of the system. The main goal of the simulation is to observe the induced voltages in parallel circuit 2. Both transient and steady state results are presented. The voltage at the receiving end of conductor 5 is shown in Figure 7 for simulations performed with the F D T P (considered as the reference solution), the frequency dependent fd-line model in the EMTP, and the proposed z-line model. The results with the constant transformation matrix fd-line model show a noticeable difference in magnitude and phase angle as compared to the full frequency dependent F D T P and z-line models. Figure 7b gives a magnification of the initial transient which emphasizes the differences. For the case presented the z-line model was about seven times slower in terms of computer time than the fd-line model. In general, we have found the model to be about N times slower than the fd-line, where N is the number of phases of the line (i.e., N = 3 for a single-circuit line, N = 6 for a double-circuit line, etc.). The current code, 2 10
a •
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................................. 102 103 104
105
106
Frequency(Hz)
(a).
(b).
Figure 5. Analysis of the z-line model frequency-length response. (a) Base case for analysis and,(b) z-line maximum section length vs maximum frequency of interest
Phase-domain transmission line models." F. Castellanos et al.
245
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Circuit 1
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Unlransposed coupled double
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Circuit 2
~au=~ 5~.0ASCP.(E~)
(a).
(an_~m_ hnms)
(b).
Figure 6. Case study. (a) One-line diagram of the study case and (b) line configuration
f
0.02
0.1
Jf
0.01
005
IlllWI1VV'
(
o_
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(a).
0s
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Time(sec)
i
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(b).
Figure 7. Voltage at the receiving end of conductor 5. Comparison between FDTP model (w), fd-line model (f) and z-line model (z). (a) Receiving voltage and (b) initial transient however, is development code and, therefore, not yet optimized. III. I d e m p o t e n t
line model
(id-line)
The frequency dependent line and cable models of [3, 4] are based on solving the line travelling wave equations in the 'modal domain', u,;ing diagonalizing transformation matrices. That is, the original full-matrix propagation equations in phase coordinates,
dzVph dx 2
--
d2Iph [ZphYph] Yph and ~ = [YphZph] Iph
(14)
are transformed into diagonal-matrix equations in modal coordinates
d2Vm -- [ZmVm]Vm and d2Im = [YmZm]Im
(15)
by the transformations [P] = eigenvectors of [ZphYph] and
(16)
[Q] = eigenvectors of [YphZph]
(17)
The line model of [3] makes the simplification of assuming constant (frequency independent) eigenvector matrices [P] and [Q], while the cable model of [4] synthesizes the frequency-dependent elements of these matrices with rational functions. A fundamental problem of the eigenvector description (modal transformation matrices) is that the eigenvectors are not uniquely defined. In the case of distinct eigenvalues individual eigenvectors can be multiplied by an arbitrary complex-number scaling factor and the new [P] or [Q] matrices will still be valid diagonalizing matrices. In the case of N repeated eigenvalues the corresponding eigenvectors can be anywhere in an N-dimensional space. This freedom in the location of eigenvectors may lead to serious mathematical complications in formulations based on synthesis by continuous frequency-domain functions (for example, by rational-function approximations). The problems related to the indeterminacy of the eigenvector solution can be avoided if the line modelling problem is formulated not in terms ofeigenvector decomposition but in terms of idempotent decomposition. A full idempotent-based transmission line model is
246
Phase-domain transmission line models. F. Castellanos et al.
I ,
N
(21) leads to convolutions in the time domain. To avoid these numerically very expensive convolutions, reference [4] synthesizes the elements of [Q(w)] by rational functions. Difficulties encountered with this approach in the modelling of overhead lines seem to be related to the indeterminacy of the frequency functions that constitute [Q(w)], particularly in frequency regions with repeated eigenvalues. (An interesting theoretical analysis of problems related to repeated eigenvalues can be found in [15].) As indicated earlier, indeterminacy problems can be eliminated when the formulation is established in terms of idempotents instead of eigenvectors. The breaking up of equation (21) in terms of idempotents can be done as follows. (A three-phase line is used to explain the procedure, which applies equally to an N-phase line.) (1) Write [Q] in terms of its constituent columns:
=
Diagonal elements
0.8
0.6 ® "-0I g:
I[
=o.4 0.2
[Q] = [CIC2C3] 10 -2
10 0
10 2
10 4
10 6
10 a
(22)
where C i are the eigenvectors. (2) Write [Q]-I in terms of its constituent rows:
Frequency (Hz) Figure 8.
Fitting of the phase propagation matrix
[Q]-l =
elements
currently under development by our research group. Due to space limitations, however, only the fundamental principles of this model will be introduced in this paper. II1.1 I d e m p o t e n t - b a s e d p r o p a g a t i o n e q u a t i o n s The concepts and properties of the idempotents used here are taken from reference [11]. Basically an idempotent matrix [/,.] is a matrix that has the property [I2] = [Ii]
(19)
where ~-i is the time-delay (phase velocity) of component mode i. In the frequency dependent line and cable models of [3, 4], the propagation functions (distortion and travelling time) e -y'' are synthesized by a minimum-phase rational function pi(w) (distortion) plus a time delay (travelling time) Ai(oJ ) = pi(co)e -Jw~'i
(20)
Equation (20) defines (in a synthesis form suitable for modelling) each natural propagation mode of the line. In the conventional approach, the component modes are combined into a coupled-mode propagation matrix in the phase domain using the transformation matrix of eigenveetors [Q]:
[Aph ] = [Q] [Am][O] -1
(23)
• Even though it is not necessary in the derivation, it is interesting to mention here that when the eigenvectors of[P] and [Q] in equations (16) and (17) are normalized according to the Euclidean norm (vector length equal to one), then [Q]-I = [p]t and the row vectors in (23) correspond to the eigenvectors of [P]. (3) Write equation (21) in terms of the column and row partitions in (22) and (23)
[Aph ] = [Q] [Am][Q]-1
(lS)
and, of course, [Ii]" = [Ii] for any integer n. From this apparently innocuous property one can intuit that a matrix function response expressed as a power expansion based on idempotents will bring out the basic constituent responses of the system (eigenmodes). The application of this concept to the transmission line propagation equations is discussed next. Consider, as in [3, 4], the line propagation function for each decoupled line mode A i = e - % 1 = e -7il le--Jarre
R2 R3
(21 )
where [Am] is the diagonal matrix with the natural component modes. For a frequency dependent [Q(w)] matrix, equation
[Aph] = [C1C2C3]
0 :][ RI
0
N2
0
0
R2 A3
(24) (25)
R3
thus giving [Aph] =- [C1R1]A1 + [C2R2]A2 + [C3R3]A3 = [M1JA1 + [M2]A2 + [M3]A3
(26)
In equation (26), [Mi] = [C/R/] = [3 × 3] idempotent coefficient matrix i (27) and the A i terms are as in equation (20). Equation (26) gives the phase-domain propagation function [Aph(W)] in terms of the natural propagation modes m i = e-Til, where the ~i s are the eigenvalues of [YZ] 1/2. Equation (26) then retains the natural eigenvalues of the problem while expressing the result in coupled phase coordinates. Since the idempotent coefficient matrices [Mi] = [C/Ri] are the products of a column of the eigenvectors matrix [Q] times a row of its inverse [Q]-l, the result is independent of the scaling factors used in the eigenvectors. The elements of the idempotent matrices as functions of frequency are, therefore, uniquely defined and, assuming one prevents columns swapping from one frequency point to the next [4, 16], the problems of the 'natural' frequency continuity of the eigenvector functions are eliminated. For the time domain implementation of the model, the idempotent matrices are synthesized by rational function
Phase-domain transmission line models." F. Castellanos et al.
approximations. Figure 8 shows an example of this synthesis for a single-circuit transmission line. Similar results have been obtained for extremely asymmetrical configurations of multicircuit lines. The table below summarizes the number of poles used for the example case of a single-circuit transmission line. Mode No. of poles
1
2
3
Yc
14
17
18
12
The idempotent functions being fitted are obtained with the help of the Newton-Raphson eigenvector tracing technique of [16]. The fitted idempotent matrices can be written in a very compact form using a partial fractions expansion in terms of common poles: n 1 [Mi] = [K~] + Z[K~] j=l S"~pj
(28)
Similarly, equation (2,0) can be written in the following form
247
0.1 f 0.05
f
w&id //
@
_g w&id
-0.05 0
0'.s
~ & i d 1'.5
2
2.5
Time ($ec) x 10-3 Figure 9. Voltage at the receiving end for conductor 5. Comparison between FDTP model (w), fd-line model (f) and id-line model (id)
Ai ---- tt=l s +p~J e-J~' Combining the two expressions, as indicated in equation (26), a synthesis of the phase propagation matrix in terms of rational functions and delays is obtained: k j=l
i
e--JWTI
/n+np / (.+rip3
1
] e-j~°"
(30)
) With this synthesis of the phase domain propagation function, the convolutions that appear when translating into time domain can. be calculated using recursive techniques [17]. Doing a similar synthesis of the line characteristic admittance matrix [Yo] = [YZ]-I/2[Y], the discrete-time domain equivalent circuit of the line becomes a constant admittance matrix with a history current source. At the present time the model is being implemented as production-level code. Details of this implementation as well as more extensive simulation results will be presented in future publications. In Figure 9 of this paper we show the simulation results obtained with the id-line model for the same case studied with the z-line model in Figures 6 and 7b. Comparing Fig~ares 9 and 7b we can see that both new models, z-line and id-line, give results which are very close to those of the reference solution with the FDTP program. In terms of computer time, the id-line model was about 6 times slower than the constant transformation matrix fd-line model in the EMTP. The same is true for the z-line model, the experience for a number of simulations with the id-line model is that the solution times tend to be about N times longer than for the constant matrix fd-line model, where N is the number of phases of the line.
IV. Conclusions The major stumbling block in modal domain modelling
for time domain simulations has been the frequency dependence of the transformation matrices that relate modal and phase coordinates. Two line models are presented in this paper that avoid the use of transformation matrices. The first model (z-line) separates the losses (resistance and internal inductance) from the propagation channel (external inductance and capacitance). Propagation is then simulated as ideal (at the speed of light in the medium) and equal for all modes. The delays and shaping effects of the losses are simulated by lumped coupled-impedance matrices. This model provides a full frequency dependent line model at a reasonable cost in terms of computer time. The main solution-time cost factor is related to the sectionalization of the line into a reasonable number of segments to simulate the actual distributed nature of the losses. In this regard, however, the model is particularly suited for the modelling of coronal and other nonlinear phenomena which can be represented as lumped in-between line sections. The second proposed model (id-line) adopts the 'opposite' philosophy to the first one. Instead of trying to idealize the line propagation characteristics, it tries to exactly maintain the individuality of the natural component modes of propagation. This is achieved by using the idempotents theory to express the coupled phase-domain propagation function as a linear combination (with matrix coefficients) of the independent modal-domain propagation modes. Both models, z-line and id-line, are numerically stable, and even though they were derived using very different conceptual and mathematical approaches, it is quite notable that their solution times and modelling accuracy are very similar. In their preliminary, un-optimized designs using an object-oriented approach with an ADA-95 compiler, they were about N times slower (N being the number of phases) than the constant transformation matrix fd-line model in the Microtran version of the EMTP. Microtran is the University of British Columbia's version of the EMTP and is one of the fastest EMTPs available. Microtran uses optimized production code compiled with a Fortran 77 compiler. The overall characteristics of the proposed new line
248
Phase-domain transmission line models: F. Castellanos et al.
models are, therefore, very promising in terms of accuracy, stability, and solution times.
transmission lines. Presented at PES Summer Meeting, Denver, 96 SM458-0-PWRD, July 1996. 9. Marti, J. R., Castellanos, F. and Santiago, N., Wide-band corona circuit model for transient simulations. IEEE Transactions on Power Systems, 1995, 10(2), 1003-1013.
V. References 1. Wedepohl, L. M., Application of matrix methods to the solution of travelling-wave phenomena in polyphase systems. Proceedings o f the IEEE, 1963, 110(12), 2200 2212. 2. Hedman, D. E., Propagation on overhead transmission lines. I: Theory of modal analysis. IEEE Transactions on Power, Apparatus and Systems, 1965, PAS-84(3), 200-205. 3. Marti, J. R., Accurate modelling of frequency-dependent transmission lines in electromagnetic transient simulations. IEEE Transactions on Power Apparatus and Systems, 1982, PAS-101(1), 147-157. 4. Marti, L., Simulation of transients in underground cables with frequency-dependent modal transformation matrices. IEEE Transactions on Power Delivery, 1988, 3(3), 1099 1110. 5. Angelidis, G. and Semlyen, A., Direct phase-domain calculation of transmission line transients using two-sided recursions. IEEE Transactions, 1995, PWRD-10(2), 941949. 6. Noda, T., Nagaoka, N. and Ametani, A., Phase domain modelling of frequency dependent transmission lines by means of an ARMA model. IEEE Transactions, 1996, PWRD-II(1), 401-411. 7. Gustavsen, B., Sletbak, J. and Henriksen, T., Calculation of electromagnetic transients in transmission cables and lines taking frequency dependent effects accurately into account. IEEE Transactions, 1995, PWRD-10(2), 1076 1084. 8. Nguyen, H. V., Dommel, H. W. and Marti, J. R., Direct phase-domain modelling of frequency dependent overhead
10. Huang, W.-G. and Semlyen, A., Computation of electromagnetic transients on three-phase transmission lines with corona and frequency dependent parameters. IEEE Transactions on Power Delivery, 1987, PWRD-2(3), 887-898. 11. Wedepohl, L. M., Frequency' Domain Analysis o f Wave Propagation in Multiconductor Transmission Systems. The University of British Columbia, Dept of Electrical Engineering, Vancouver, Canada, 1993. 12. Dommel, H. W., Digital computer solution of electromagnetic transients in single- and multiphase networks. IEEE Transactions, 1969, PAS-88(4), 388-399. 13. Dommel, H. W., E M T P Theory Book. Microtran Power System Analysis Corporation, Vancouver, Canada, 1992. 14. Marti, J. R., Garrett, B. W., Dommel, H. W. and Wedepohl, L. M., Transients simulation in power systems: frequency domain and time domain analysis. Canadian Electrical Association Transactions, Vol. 24, Part 4, Montreal, Quebec, 1985, p. 28. 15. Brandfio Faria, J. A., Overhead three-phase transmission lines--Non-diagonalizable situations. IEEE Transactions on Power Delivery, 1988, 3(4), 1348-1355. 16. Wedepohl, L. M., Nguyen, H. V. and Irwin, G. D., Frequency-dependent transformation matrices for untransposed transmission lines using Newton-Raphson method. IEEE Transactions on Power Systems, 1996, PWRS-11(3), 1538-1547. 17. Semlyen, A. and Dabuleanu, A., Fast and accurate switching transients calculation on transmission lines with ground return using recursive convolutions. IEEE Transactions on Power Apparatus and Systems, 1975, PAS-94, 561 571.
Electrical Power & Energy Systems, Vol. 19, No. 4, pp. 241-248, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved PII: S0142-0615(96)00039-7 o142-0615/97/$17.oo + 0.oo
Phase-domain multiphase transmission line models F Castellanos, J R M a r t i and F M a r c a n o Department of Electrical Engineering, The University of British Columbia, Vancouver, B.C., Canada V6T 1 Z4
models avoid the use of transformation matrices to relate the decoupled eigenmode propagation in the modal domain to the coupled-mode propagation in the phase domain. Since in their final form the equations are solved entirely in phase coordinates, these models can be regarded as phase-domain line models. They differ, however, from other recently proposed phase-domain models (e.g., [5-8]) in that the frequency dependent wave propagation functions are still first synthesized in the frequency domain, as in [3] and [4].
In general, multiphase t,,ansmission line and cable models for transient programs, such as the EMTP, base their solutions on transformations between the phase and modal domains. The main difficulty with these models is the synthesis of the frequency dependent transformation matrices. This paper presents two line models that circumvent this problem by writing the propagation functions directly in the phase domain and thus avoid the use of modal transformation matrices. The first model (developed in connection with our work on corona modelling) avoids the use of modal transformation matrices by separating the ideal-line travelling effect from the loss effects (resistance and internal inductance). The second proposed model is a full frequency dependent distributed parameter model based on idempotent decomposition. By using idempotents instead of eigenvectors a,: the basisfor modal decomposition, the problems associated with the indeterminacy of the frequency dependent eigenvector functions are eliminated. © 1997 Elsevier Science Ltd. All rights reserved
1.1 Lumped line model (z-line) The first proposed phase-domain line model (z-line, for 'z-lumped line') has been developed in connection with our work on a wide-band corona model [9]. This line model follows the basic ideas advanced in [10], except that it is formulated entirely in phase coordinates, thus eliminating the problems associated with modal transformation matrices. The premise here is that if the nonlinear corona model has to be lumped between relatively short line sections then advantage can be taken of the space discretization. This space discretization can be used to 'simplify' the modelling of the distortion part of the propagation function ([e-7(")x]). That is, the distortion due to the resistance and the internal inductance can be grouped into one lumped frequency dependent impedance matrix in phase coordinates [Z loss (0J)]while ideal propagation (at the speed of light in the medium) due to the external magnetic field [Lext] and capacitance [C] can be represented by an ideal-line segment. Since for ideal propagation all modes travel at the speed of light, the equations for this part of the model (ideal line) can be formulated directly in phase coordinates. This model is similar to the constantparameter line model in the EMTP (cp-line) where the resistance is lumped in the middle and at the end points of the line. As opposed to the cp-line model, however, the proposed z-line model can fully take into account the frequency dependence of the line parameters (in [Z l°ss(w)]) and exactly takes into account the asymmetry of any arbitrary line configuration without the need for a transformation matrix. The fd-line (frequencydependent) model in the EMTP works in modal coordinates and assumes that the transformation matrix relating mode and phase is constant and real and, therefore,
Keywords: transmission line modelling, EMTP simulation, phase-domain modelling, idempotent analysis
I. I n t r o d u c t i o n Overhead transmission lines and underground cables are the basic transmission links across a power system. Wave propagation and distortion along lines and cables affects the waveshapes reaching the system equipment and the associated stresses upon the equipment. Despite their importance in transient studies, the accurate modelling of transmission lines in time domain simulations is still not fully satisfactorily resolved. Good time-domain frequency dependent transmission line and cable models, based on Wedepohl and Hedman's theory [1, 2] of the modal decomposition of natural propagation modes (eigenmodes) are available in the EMTP program [3, 4]. The main problem with these models is the frequency dependence of the transformation matrices (matrices of eigenvectors) that connect the modal and phase domains. This paper introduces two new line models being developed at the University of British Columbia. These
241
Phase-domain transmission line models." F. Castellanos et al.
242 Corrections for losses and internal flux ;
7
Ideal line, only external flux "
;
where [ZY] or [¥Z] are full matrices that couple the propagation of voltage and current waves in all the phases. The series impedance matrix [Z] is a full matrix with elements of the form Zi j .~ Rii q-Ja3(Lij • ext + ALij )
[z g)]
(2)
where Rij is the resistance of the conductor plus the correction for the ground return effect, w is the frequency, Lext U is the inductance related to the external flux, and A L O. is the inductance related to the internal flux inside the conductor plus the correction for ground return. Matrix [Z] can thus be rewritten as
[Lext] [c]
[Z] = [Z l°ss] + j w [ L ext]
(3)
[Z] = [Z l°ss] -~-[Z ext]
(4)
Figure 1. Separation of basic effects in the z-line model it cannot correctly simulate strongly asymmetrical configurations. The only limitation of the proposed z-model is that the losses are assumed to be lumped and, therefore, a certain number of line sections are needed to simulate the actual distributed nature of these parameters. 1.2 Idempotent line model (id-line) The second proposed line model (id-line, for 'idempotentline') is a fully frequency dependent distributed-parameter line model. This model is conceptually similar to the current frequency dependent line models in the EMTP (fd-line and q-line, [3] and [4]), except that it overcomes the problems of these models with respect to the frequency dependence of the modal transformation matrices. In general, the main problem in synthesizing the elements of the transformation matrices as functions of frequency is that the eigenvectors that make up these matrices are only defined up to a complex constant factor in the case of distinct eigenvalues or are multiple-valued in the case of repeated eigenvalues. This means, for example, that different scalings ('normalizations') of the eigenvectors will result in different frequency functions for the elements of the transformation matrix. The question of which one is the 'right' normalization is very difficult to answer. The problem of multiple-choice eigenvector functions is aggravated in the case of repeated eigenvalues, which occurs in certain regions of the frequency spectrum. The eigenvectors for N repeated eigenvalues can lie anywhere in a given N-dimensional plane. The proposed id-line model solves these problems by avoiding the use of eigenvectors as base functions for modal decomposition. Instead, the phase-domain propagation function [e -Tphx] (matrix function) is expressed as a linear combination of the natural propagation modes e -7'x (scalar functions) with idempotent coefficient matrices. The idempotent matrices [11] result from the column-row products of the matrix of eigenvectors (modal transformation matrix) and its inverse, and are uniquely defined, independent of the particular scaling of the eigenvectors or of the multiplicity of the eigenvalues.
II. Proposed lumped line model (z-line) The modelling of transmission lines is based on the travelling wave equations, which can be expressed, for a given frequency ~ as, d2V
[ZY]V and d2I [YZ]I 2=
(1)
with Z!OSS tj = Ri j q_ju;/kLij
(5)
The shunt admittance matrix [Y] can be written as (assuming conductance G ~ 0) [Y] =jw[C],
with
[C] = [p]-i
(6)
[P] is the Maxwell coefficients matrix. The [Y] matrix does not require any corrections for ground return and contains only elements that depend on the capacitances (geometry) of the system. With equations (3)-(6), we can rewrite [YZ] as [YZ] = jw[C]([Z l°ss] + jw[Lext])
(7)
[VZ] = j w [ C ] [Z l°ss] - co2 [C] [L ext]
(8)
Equation (8) shows how the product [YZ] can be expressed as the combination of two main components: the first one related to the losses and ground effect corrections and the second one related to the ideal propagation. If we assume that the effect of the losses term is small compared with the effect of the ideal component, then we can lump the losses together and obtain the model shown in Figure l. This condition is met when the section length is small. In the general case of frequency dependent parameters the term [zl°ss(~)] is a function of both the frequency and the section length. I1.1 Modelling of the multiphase ideal line Once the impedance corrections ([Z loss (w)]) are extracted
from the complete model, the ideal transmission line can be easily modelled in the phase domain. Because the travelling time, r, is the same for all modes and the solution of the system of travelling wave equations is decoupled, Dommel's ideal line equation [12] for the time-domain single-phase case can be rewritten for the multiphase case, as illustrated in Figure 2. Using circuit theory, the terminal voltage vectors vk(t) and Vm(t) can be expressed as functions of the terminal current vectors ik(t ) and im(t), the characteristic impedance matrix [Zc], and the history source vectors ekh(t ) and emh(t). The history source vectors are calculated at each time step as functions of past values, through the expressions ekh(t) = Vm(t -- r) + [Zc]im(t -- r)
(9)
emh(t) : vk(t -- r) + [Zc]ik(t -- 7-)
(10)
Phase-domain transmission line models: F. Castellanos et al.
ik(t)
IZcl
kv~(t)~,
lZcl
ira(t)
ekll(t) ~ ~ + ) (~
m Vm(t)
emh(tt-~
Figure 2. Multiphase ideal transmission line model in the time domain
(where s = j ~ ) , which corresponds to a parallel R - L impedance in the continuous time domain. . Each element of [Zl°Ss(~)] is expressed as a sum of several of these blocks (total number of blocks = m), plus an additional resistance term, Riio, in the diagonal elements to satisfy the dc condition (Rii o -----Rii @ dc). With the subscriptfindicating a 'fitted' function:
m l°ssw ~ sKii(l) Diagonal elements (12) Z~ii ( ) = Rii o -'l- ~ s q- Pii(l)
with [Zc] = [y]--1([y] [zext]} 1/2
m ~loss, , = ~ sKij(1) Off-diagonal elements
(11)
In this case, the matrix [Z¢] represents the coupled system in the phase domain and is a full matrix. The implementation of this model in the EMTP is similar, with small modifications, to the standard coupled R - L model [13]. Like in the case of the single-phase ideal line model in the EMTP, the multiphase ideal line model is independent of the integration rule, and represents an exact solution of the line equations. 11.2 Modelling of the multiphase lumped loss section The elements of the matrix [zl°ss(w)] are frequency dependent functions, and they can be synthesized by rational function approximations, as in the frequency dependent line models in the EMTP [3, 4]. Mathematically, however, one has to be very careful when synthesizing coupled systems (as opposed to modal-domain decoupled systems). From network synthesis theory, there are a number of conditions regarding the relationship among poles and zeros of the elements of the matrix that need to be met to assure the stability of the resulting synthesis. In general, these conditions will not be met if each element of the matrix is synthesized independently of the other elements. A new synthesis procedure is used to achieve a coordinated synthesis considering all elements of the matrix together. This procedure preserves the stability of the full fitted matrix at all frequencies. The main concepts of the synthesis procedure used for the new z-line model are the following: . The basic fitting block is the function s K / ( s + p )
243
/% t~)
~'~' s + Pij0)
(13)
e The same number of blocks (m) is used for all the [zl°ss(w)] elements. e The Ks and p s are calculated in a coordinated way, which preserves the positive definitive condition of the [Zl°S~(w)] matrix with the fitted values. e A minimum least squares technique is used to optimize the fitting. Once the [zl°s~(w)] functions are represented in the form of equations (12) and (13) the system of voltagedrop equations is solved in the discrete time domain using the trapezoidal rule as the integration method. Figure 3 shows the fitting of some of the elements of [Z l°s~(~)] for a typical double-circuit transmission line. Nine blocks per function were used for these fits. 11.3 Short-section transmission line solution The phase domain solutions of the ideal line section and the [Zl°~ (w)] section can be combined to simulate a short section of a transmission line. For the short-section model, the part corresponding to [ZJ°~(~)] is divided into two sections. Each half section of [ZL°~(~)] is added at the ends of the ideal line section, as shown in Figure 4. A set of simulations was done to study the relation between the section length of the z-line model and the frequency of the study. The transmission line used for the testing was a typical flat-configuration three-phase 230 kV line. The base case is shown in Figure 5a and consists of one sinusoidal source attached to a node with
10
4
1 0 - -
original/ 120
0
=9
~10
1
[loo v
.~ 101 original
1(~ 0
m=9 -5
10o Frequency
(a)
(Hz)
10 s
0"l
1 0° 1 05 Frequency (Hz)
1
(b)
Figure 3. Typical elements of the lumped loss matrix and their fittings. (a) Diagonal element and (b) off-diagonal element
244
Phase-domain transmission line models. F. Castellanos et al.
[Zl°~))]/2
and [C]
[ZJ°~s)]/2
Figure 4. Proposed line-section for the z-line model different resistance terminations at the other ends. The source frequency was set equal to the frequency used for the calculation of the line parameters. The time-domain response of the z-line model was compared against the solution obtained with the constant-parameter model of the EMTP. By repeating this process for several frequencies an empirical relation between the length of the linesection and the frequency of the analysis was established. The results are shown in Figure 5b and they give the recommended maximum length of the z-line section that could be used for a study at a given dominant frequency.
11.4 Full-length transmission line solution and test cases
The solution for a complete transmission line is accomplished by connecting several of the z-line short-section models in cascade. The implementation of the full-length z-line model was done in a stand-alone program written in ADA-95 and several cases were tested. The results of these cases were compared against the frequency dependent (fd-line) model in the E M T P [3] and the F D T P (Frequency Domain Transients Program) of Wedepohl [11, 14]. Since the F D T P program solves the system entirely in the frequency domain (using an inverse Fourier transform to obtain the time-domain waveforms), its accuracy is not affected by issues of the frequency dependence of the line parameters. Figure 6 shows the general system and line configuration used for one of the test cases. Circuit 1 is energized at the beginning of the simulation (t = 0) with a three-phase sinusoidal voltage source with a peak value 1 p.u. The receiving end o f circuit 1 is connected to a three-phase Y-connected resistive load of 500 ~2/phase. Circuit 2 is connected to ground at the sending end through a small three-phase Y-connected resistive path of 10 ~/phase. The transmission line was represented using 40 sections
of 2.5 km each, for a bandwidth, according to Figure 5b of about 8 Khz (The graphic in Figure 5b was obtained allowing an average error of about 3 %). The time step A t of the simulation was chosen as 8.339 #s to exactly match the travelling time of the ideal line sections. For the L and C elements of the network (including the L s in [Zl°Ss(~)], a A t = 8.339 #s gives a bandwidth of about 12 K H z for a distortion of about 3% with the trapezoidal rule of The sectionalization integration ( f = ½fNyquist : ~ )1. requirements of the model (number of sections and size of At : 7-section) to achieve a certain bandwidth of accurate modelling are, therefore, not too different from those imposed by the distortion of the integration rule on the L and C components: 8 K H z for sectionalization versus 12 K H z for distortion in this case. Nine R - L blocks were used to fit the [Zl°SS(~v)] functions over an extended frequency range. A pole truncation algorithm was then applied to discard those poles close to and beyond the Nyquist frequency. This truncation procedure reduced the number of R L blocks used in the transient simulation to only four per function, with no sacrifice of the accuracy of the synthesis. The simulation results for this case are shown in Figure 7. The asymmetry in this case emerges from the line configuration and the untransposed characteristic of the system. The main goal of the simulation is to observe the induced voltages in parallel circuit 2. Both transient and steady state results are presented. The voltage at the receiving end of conductor 5 is shown in Figure 7 for simulations performed with the F D T P (considered as the reference solution), the frequency dependent fd-line model in the EMTP, and the proposed z-line model. The results with the constant transformation matrix fd-line model show a noticeable difference in magnitude and phase angle as compared to the full frequency dependent F D T P and z-line models. Figure 7b gives a magnification of the initial transient which emphasizes the differences. For the case presented the z-line model was about seven times slower in terms of computer time than the fd-line model. In general, we have found the model to be about N times slower than the fd-line, where N is the number of phases of the line (i.e., N = 3 for a single-circuit line, N = 6 for a double-circuit line, etc.). The current code, 2 10
a •
0
-
,
-
,
....
,
.....
0
d
0
I) I
0
AIO 0 0
¢
b
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CO
.~.
0 0 0 0
~
C
)
10~
f
,o;o
lo-1
o o
lg 101
................................. 102 103 104
105
106
Frequency(Hz)
(a).
(b).
Figure 5. Analysis of the z-line model frequency-length response. (a) Base case for analysis and,(b) z-line maximum section length vs maximum frequency of interest
Phase-domain transmission line models." F. Castellanos et al.
245
(Ymfit2 t=O.O
£
Circuit 1
alitl
I
1 p.u. ~
Unlransposed coupled double
"
93
I
fiat circuit
: \/ \/
length 100km
i, q
lOf~
..
"
?
J
Circuit 2
~au=~ 5~.0ASCP.(E~)
(a).
(an_~m_ hnms)
(b).
Figure 6. Case study. (a) One-line diagram of the study case and (b) line configuration
f
0.02
0.1
Jf
0.01
005
IlllWI1VV'
(
o_
J
-0.05~
-o.o,r,ll;i, -0.03~[~
0.01
0.02 0,03 Time(see)
0.04
t 0
(a).
0s
i
1'5
Time(sec)
i
2.
x 1 0 "s
(b).
Figure 7. Voltage at the receiving end of conductor 5. Comparison between FDTP model (w), fd-line model (f) and z-line model (z). (a) Receiving voltage and (b) initial transient however, is development code and, therefore, not yet optimized. III. I d e m p o t e n t
line model
(id-line)
The frequency dependent line and cable models of [3, 4] are based on solving the line travelling wave equations in the 'modal domain', u,;ing diagonalizing transformation matrices. That is, the original full-matrix propagation equations in phase coordinates,
dzVph dx 2
--
d2Iph [ZphYph] Yph and ~ = [YphZph] Iph
(14)
are transformed into diagonal-matrix equations in modal coordinates
d2Vm -- [ZmVm]Vm and d2Im = [YmZm]Im
(15)
by the transformations [P] = eigenvectors of [ZphYph] and
(16)
[Q] = eigenvectors of [YphZph]
(17)
The line model of [3] makes the simplification of assuming constant (frequency independent) eigenvector matrices [P] and [Q], while the cable model of [4] synthesizes the frequency-dependent elements of these matrices with rational functions. A fundamental problem of the eigenvector description (modal transformation matrices) is that the eigenvectors are not uniquely defined. In the case of distinct eigenvalues individual eigenvectors can be multiplied by an arbitrary complex-number scaling factor and the new [P] or [Q] matrices will still be valid diagonalizing matrices. In the case of N repeated eigenvalues the corresponding eigenvectors can be anywhere in an N-dimensional space. This freedom in the location of eigenvectors may lead to serious mathematical complications in formulations based on synthesis by continuous frequency-domain functions (for example, by rational-function approximations). The problems related to the indeterminacy of the eigenvector solution can be avoided if the line modelling problem is formulated not in terms ofeigenvector decomposition but in terms of idempotent decomposition. A full idempotent-based transmission line model is
246
Phase-domain transmission line models. F. Castellanos et al.
I ,
N
(21) leads to convolutions in the time domain. To avoid these numerically very expensive convolutions, reference [4] synthesizes the elements of [Q(w)] by rational functions. Difficulties encountered with this approach in the modelling of overhead lines seem to be related to the indeterminacy of the frequency functions that constitute [Q(w)], particularly in frequency regions with repeated eigenvalues. (An interesting theoretical analysis of problems related to repeated eigenvalues can be found in [15].) As indicated earlier, indeterminacy problems can be eliminated when the formulation is established in terms of idempotents instead of eigenvectors. The breaking up of equation (21) in terms of idempotents can be done as follows. (A three-phase line is used to explain the procedure, which applies equally to an N-phase line.) (1) Write [Q] in terms of its constituent columns:
=
Diagonal elements
0.8
0.6 ® "-0I g:
I[
=o.4 0.2
[Q] = [CIC2C3] 10 -2
10 0
10 2
10 4
10 6
10 a
(22)
where C i are the eigenvectors. (2) Write [Q]-I in terms of its constituent rows:
Frequency (Hz) Figure 8.
Fitting of the phase propagation matrix
[Q]-l =
elements
currently under development by our research group. Due to space limitations, however, only the fundamental principles of this model will be introduced in this paper. II1.1 I d e m p o t e n t - b a s e d p r o p a g a t i o n e q u a t i o n s The concepts and properties of the idempotents used here are taken from reference [11]. Basically an idempotent matrix [/,.] is a matrix that has the property [I2] = [Ii]
(19)
where ~-i is the time-delay (phase velocity) of component mode i. In the frequency dependent line and cable models of [3, 4], the propagation functions (distortion and travelling time) e -y'' are synthesized by a minimum-phase rational function pi(w) (distortion) plus a time delay (travelling time) Ai(oJ ) = pi(co)e -Jw~'i
(20)
Equation (20) defines (in a synthesis form suitable for modelling) each natural propagation mode of the line. In the conventional approach, the component modes are combined into a coupled-mode propagation matrix in the phase domain using the transformation matrix of eigenveetors [Q]:
[Aph ] = [Q] [Am][O] -1
(23)
• Even though it is not necessary in the derivation, it is interesting to mention here that when the eigenvectors of[P] and [Q] in equations (16) and (17) are normalized according to the Euclidean norm (vector length equal to one), then [Q]-I = [p]t and the row vectors in (23) correspond to the eigenvectors of [P]. (3) Write equation (21) in terms of the column and row partitions in (22) and (23)
[Aph ] = [Q] [Am][Q]-1
(lS)
and, of course, [Ii]" = [Ii] for any integer n. From this apparently innocuous property one can intuit that a matrix function response expressed as a power expansion based on idempotents will bring out the basic constituent responses of the system (eigenmodes). The application of this concept to the transmission line propagation equations is discussed next. Consider, as in [3, 4], the line propagation function for each decoupled line mode A i = e - % 1 = e -7il le--Jarre
R2 R3
(21 )
where [Am] is the diagonal matrix with the natural component modes. For a frequency dependent [Q(w)] matrix, equation
[Aph] = [C1C2C3]
0 :][ RI
0
N2
0
0
R2 A3
(24) (25)
R3
thus giving [Aph] =- [C1R1]A1 + [C2R2]A2 + [C3R3]A3 = [M1JA1 + [M2]A2 + [M3]A3
(26)
In equation (26), [Mi] = [C/R/] = [3 × 3] idempotent coefficient matrix i (27) and the A i terms are as in equation (20). Equation (26) gives the phase-domain propagation function [Aph(W)] in terms of the natural propagation modes m i = e-Til, where the ~i s are the eigenvalues of [YZ] 1/2. Equation (26) then retains the natural eigenvalues of the problem while expressing the result in coupled phase coordinates. Since the idempotent coefficient matrices [Mi] = [C/Ri] are the products of a column of the eigenvectors matrix [Q] times a row of its inverse [Q]-l, the result is independent of the scaling factors used in the eigenvectors. The elements of the idempotent matrices as functions of frequency are, therefore, uniquely defined and, assuming one prevents columns swapping from one frequency point to the next [4, 16], the problems of the 'natural' frequency continuity of the eigenvector functions are eliminated. For the time domain implementation of the model, the idempotent matrices are synthesized by rational function
Phase-domain transmission line models." F. Castellanos et al.
approximations. Figure 8 shows an example of this synthesis for a single-circuit transmission line. Similar results have been obtained for extremely asymmetrical configurations of multicircuit lines. The table below summarizes the number of poles used for the example case of a single-circuit transmission line. Mode No. of poles
1
2
3
Yc
14
17
18
12
The idempotent functions being fitted are obtained with the help of the Newton-Raphson eigenvector tracing technique of [16]. The fitted idempotent matrices can be written in a very compact form using a partial fractions expansion in terms of common poles: n 1 [Mi] = [K~] + Z[K~] j=l S"~pj
(28)
Similarly, equation (2,0) can be written in the following form
247
0.1 f 0.05
f
w&id //
@
_g w&id
-0.05 0
0'.s
~ & i d 1'.5
2
2.5
Time ($ec) x 10-3 Figure 9. Voltage at the receiving end for conductor 5. Comparison between FDTP model (w), fd-line model (f) and id-line model (id)
Ai ---- tt=l s +p~J e-J~' Combining the two expressions, as indicated in equation (26), a synthesis of the phase propagation matrix in terms of rational functions and delays is obtained: k j=l
i
e--JWTI
/n+np / (.+rip3
1
] e-j~°"
(30)
) With this synthesis of the phase domain propagation function, the convolutions that appear when translating into time domain can. be calculated using recursive techniques [17]. Doing a similar synthesis of the line characteristic admittance matrix [Yo] = [YZ]-I/2[Y], the discrete-time domain equivalent circuit of the line becomes a constant admittance matrix with a history current source. At the present time the model is being implemented as production-level code. Details of this implementation as well as more extensive simulation results will be presented in future publications. In Figure 9 of this paper we show the simulation results obtained with the id-line model for the same case studied with the z-line model in Figures 6 and 7b. Comparing Fig~ares 9 and 7b we can see that both new models, z-line and id-line, give results which are very close to those of the reference solution with the FDTP program. In terms of computer time, the id-line model was about 6 times slower than the constant transformation matrix fd-line model in the EMTP. The same is true for the z-line model, the experience for a number of simulations with the id-line model is that the solution times tend to be about N times longer than for the constant matrix fd-line model, where N is the number of phases of the line.
IV. Conclusions The major stumbling block in modal domain modelling
for time domain simulations has been the frequency dependence of the transformation matrices that relate modal and phase coordinates. Two line models are presented in this paper that avoid the use of transformation matrices. The first model (z-line) separates the losses (resistance and internal inductance) from the propagation channel (external inductance and capacitance). Propagation is then simulated as ideal (at the speed of light in the medium) and equal for all modes. The delays and shaping effects of the losses are simulated by lumped coupled-impedance matrices. This model provides a full frequency dependent line model at a reasonable cost in terms of computer time. The main solution-time cost factor is related to the sectionalization of the line into a reasonable number of segments to simulate the actual distributed nature of the losses. In this regard, however, the model is particularly suited for the modelling of coronal and other nonlinear phenomena which can be represented as lumped in-between line sections. The second proposed model (id-line) adopts the 'opposite' philosophy to the first one. Instead of trying to idealize the line propagation characteristics, it tries to exactly maintain the individuality of the natural component modes of propagation. This is achieved by using the idempotents theory to express the coupled phase-domain propagation function as a linear combination (with matrix coefficients) of the independent modal-domain propagation modes. Both models, z-line and id-line, are numerically stable, and even though they were derived using very different conceptual and mathematical approaches, it is quite notable that their solution times and modelling accuracy are very similar. In their preliminary, un-optimized designs using an object-oriented approach with an ADA-95 compiler, they were about N times slower (N being the number of phases) than the constant transformation matrix fd-line model in the Microtran version of the EMTP. Microtran is the University of British Columbia's version of the EMTP and is one of the fastest EMTPs available. Microtran uses optimized production code compiled with a Fortran 77 compiler. The overall characteristics of the proposed new line
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Phase-domain transmission line models: F. Castellanos et al.
models are, therefore, very promising in terms of accuracy, stability, and solution times.
transmission lines. Presented at PES Summer Meeting, Denver, 96 SM458-0-PWRD, July 1996. 9. Marti, J. R., Castellanos, F. and Santiago, N., Wide-band corona circuit model for transient simulations. IEEE Transactions on Power Systems, 1995, 10(2), 1003-1013.
V. References 1. Wedepohl, L. M., Application of matrix methods to the solution of travelling-wave phenomena in polyphase systems. Proceedings o f the IEEE, 1963, 110(12), 2200 2212. 2. Hedman, D. E., Propagation on overhead transmission lines. I: Theory of modal analysis. IEEE Transactions on Power, Apparatus and Systems, 1965, PAS-84(3), 200-205. 3. Marti, J. R., Accurate modelling of frequency-dependent transmission lines in electromagnetic transient simulations. IEEE Transactions on Power Apparatus and Systems, 1982, PAS-101(1), 147-157. 4. Marti, L., Simulation of transients in underground cables with frequency-dependent modal transformation matrices. IEEE Transactions on Power Delivery, 1988, 3(3), 1099 1110. 5. Angelidis, G. and Semlyen, A., Direct phase-domain calculation of transmission line transients using two-sided recursions. IEEE Transactions, 1995, PWRD-10(2), 941949. 6. Noda, T., Nagaoka, N. and Ametani, A., Phase domain modelling of frequency dependent transmission lines by means of an ARMA model. IEEE Transactions, 1996, PWRD-II(1), 401-411. 7. Gustavsen, B., Sletbak, J. and Henriksen, T., Calculation of electromagnetic transients in transmission cables and lines taking frequency dependent effects accurately into account. IEEE Transactions, 1995, PWRD-10(2), 1076 1084. 8. Nguyen, H. V., Dommel, H. W. and Marti, J. R., Direct phase-domain modelling of frequency dependent overhead
10. Huang, W.-G. and Semlyen, A., Computation of electromagnetic transients on three-phase transmission lines with corona and frequency dependent parameters. IEEE Transactions on Power Delivery, 1987, PWRD-2(3), 887-898. 11. Wedepohl, L. M., Frequency' Domain Analysis o f Wave Propagation in Multiconductor Transmission Systems. The University of British Columbia, Dept of Electrical Engineering, Vancouver, Canada, 1993. 12. Dommel, H. W., Digital computer solution of electromagnetic transients in single- and multiphase networks. IEEE Transactions, 1969, PAS-88(4), 388-399. 13. Dommel, H. W., E M T P Theory Book. Microtran Power System Analysis Corporation, Vancouver, Canada, 1992. 14. Marti, J. R., Garrett, B. W., Dommel, H. W. and Wedepohl, L. M., Transients simulation in power systems: frequency domain and time domain analysis. Canadian Electrical Association Transactions, Vol. 24, Part 4, Montreal, Quebec, 1985, p. 28. 15. Brandfio Faria, J. A., Overhead three-phase transmission lines--Non-diagonalizable situations. IEEE Transactions on Power Delivery, 1988, 3(4), 1348-1355. 16. Wedepohl, L. M., Nguyen, H. V. and Irwin, G. D., Frequency-dependent transformation matrices for untransposed transmission lines using Newton-Raphson method. IEEE Transactions on Power Systems, 1996, PWRS-11(3), 1538-1547. 17. Semlyen, A. and Dabuleanu, A., Fast and accurate switching transients calculation on transmission lines with ground return using recursive convolutions. IEEE Transactions on Power Apparatus and Systems, 1975, PAS-94, 561 571.