emtp simul(11)
TRANSCRIPT
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114 Power systems electromagnetic transients simulation
Input type (d):
y(z)
u(z)
=b/a(eat + 1/(at)(1 eat))z1 + b/a(1 1/(at)(1 eat))
(1 z1
eat
)
Table E.3 (Appendix E) summarises the resulting difference equation for the
family of exponential forms developed using root-matching techniques. The table
also contains the difference equations derived from trapezoidal integrator substitution.
The difference equations are then converted to the form:
(a0 + a1 z1)/(b0 + b1 z
1) or (a0 + a1z
1)/(1 + b1z1) if b0 is non-zero
Tables E.1 and E.2 give the coefficients of a rational function in z1 that repre-
sents each difference equation for the family of exponential forms, for admittanceand impedance respectively. It can be shown that the difference equation obtained
assuming type input (d) is identical to that obtained from the recursive convolution
technique developed by Semlyen and Dabuleanu [7].
5.5.1 Step response
A comparison of step responses is made here with reference to the simple switching
of a series RL branch, shown in Figure 5.2. Figure 5.15 shows the current magnitude
using the difference equations generated by Dommels method, root-matching forinput types (a), (b), (c) and (d) and the theoretical result for t = ( = 50s).
Figures 5.16 and 5.17 show the same comparison for t = 5 and t = 10, respec-
tively. Note that in the latter cases Dommels method exhibits numerical oscillation.
Root-matching type (a) gives the exact answer at each time point as its discretisation of
the input is exact. Root-matching type (b) gives the exact values but one time step late
as its discretisation of the input is a step occurring one time step later. Root-matching
type (c) is an average between the previous two root-matching techniques.
Current(amps)
Time (s)
Dommel Theoretical curve
0
20
40
60
80
100
0.00095 0.0012 0.00145
RM type A RM type B RM type C RM type D
Figure 5.15 Comparison of step response of switching test system fort =
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The root-matching method 115
Current(amps)
Time (s)
0
20
40
60
80
100
120
0.0006 0.0012 0.0018 0.0024 0.003
Dommel Theoretical curve
RM type A RM type B RM type C RM type D
Figure 5.16 Comparison of step response of switching test system fort = 5
C
urrent(amps)
Time (s)
0
20
40
60
80
100
120
0.0005 0.001 0.0015 0.002 0.0025 0.003
Dommel Theoretical curve
RM type A RM type B RM type C RM type D
Figure 5.17 Comparison of step response of switching test system fort = 10
Although from Figure 5.13 it would seem that root-matching type (d) should
provide the best approximation to an arbitrary waveform, this input resulted in sig-
nificant inaccuracies. The reason is that this discretisation is unable to model a pure
step, i.e. there will always be a slope, which is a function of t, as depicted in
Figure 5.18. However ift is sufficiently small then this method will provide a good
approximation to a step response.
Root-matching type (b) results in terms from the previous time step only, that is
only a current source but no parallel conductance. This can cause simulation problems
if a non-ideal switch model is used. If a switch is modelled by swapping between
high and low resistance states then even when it is OFF, a very small current flow
is calculated. This current is then multiplied by et/ and injected into the high
impedance switch and source, which results in a voltage appearing at the terminals. If
an ideal switch cannot be modelled, judicious selection of the switch parameters can
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116 Power systems electromagnetic transients simulation
t
U(t)
Time (ms)
Figure 5.18 Root-matching type (d) approximation to a step
Voltage(volts)
Time (s)
4
2
0
2
4
0.0046 0.0047 0.0048 0.0049 0.005
Dommel Theoretical curve
RM type A RM type B RM type C RM type D
Figure 5.19 Comparison with a.c. excitation (5 kHz) (t = )
remove the problem, however a better solution is to use a controlled voltage source
when applying the step in voltage.
5.5.2 Steady-state response
The second test system, shown in Figure 5.6, consists of an RL branch, excited by a
5 kHz current source. Figure 5.19 shows the voltage response using a 10s step length
for each of the difference equations. The theoretical answer is 1.86 sin(t), where
= 57.52. Root-matching types (a), (b) and (d) give good answers; however,
root-matching type (c) gives results indistinguishable from Dommels method.
It should be noted that as the excitation is a current source and root-matching
type (b) is also a pure current source, there are two current sources connected to one
node. Hence, in order to get answers for this system a parallel conductance must
be added to enable Kirchhoffs current law to be satisfied. The conductance value
must be large enough so as not to influence the solution significantly but not too
large otherwise instability will occur. However, from a stability viewpoint the poles
in the z-plane for the complete solution fall outside the unit circle when parallel
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The root-matching method 117
Voltage(volts)
0.0048 0.00485 0.0049 0.00495 0.005
Time (s)
8
4
0
4
8Dommel Theoretical curve
RM type A RM type B RM type C RM type D
Figure 5.20 Comparison with a.c. excitation (10 kHz) (t = )
resistance is increased. Using a voltage source rather than current source excitation
would eliminate the need for a parallel resistor in the root-matching type (b).
The same conclusions are found from a simulation using 10 kHz as the excitation
frequency and a step length of 10 s. The theoretical answer is 3.30 sin(t ),
where = 72.43. In this case root-matching types (a), (b) and type (d) give
good answers, and again, root-matching type (c) gives results indistinguishable from
Dommels method (this is shown in Figure 5.20).
5.5.3 Frequency response
The frequency response of each difference equation can be reconstructed from the
rational function by using the following equation:
Z(f) =
ni=0 ai e
jitni=0 bi e
jit(5.18)
which for root-matching, simplifies to:
1
Gequ
et/ejt
Gequ= ((1 cos(t)et/) + j sin(t))/Gequ (5.19)
The magnitude and phase components are:
|Z(f)| =
(1 cos(t)et/)2 + sin(t)2/Gequ
and
Z(f) = tan1 sin(t)
1 cos(t)et/The corresponding equation for an s-domain function is:
h(f) =
nk=0 ak (j)
k
1 +n
k=1 bk (j)k
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118 Power systems electromagnetic transients simulation
RM type a
RM type b
RM type c
RM type d
Dommel
Theoretical
1
1.5
2
2.5
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
20
40
60
80
100
120
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency (Hz)
Magnitude
Phase(degs)
Figure 5.21 Frequency response for various simulation methods
The results of calculations performed using MATLAB (the code is given in
Appendix F.4), are displayed in Figure 5.21. These results were verified by performing
injection tests into the appropriate difference equation using PSCAD/EMTDC sim-
ulation software. As expected, the root-matching methods (a) and (b) provide the
closest match to the theoretical magnitude response, while root-matching methods
(c) and (d) are similar to the trapezoidal rule. The phase response clearly shows the
phase advance and phase lag inherent in the various discretisations used in the vari-
ous root-matching methods. Root-matching method (a) shows the phase advance and
method (b) the phase lag expected. Root-matching methods (c) and (d) and trapezoidalintegration show a considerably better phase response.
The trapezoidal rule assumes a linear variation between time points. An expo-
nential form of the difference equation can also be derived assuming constant input
between solution points. Hence the exponential form of a circuit or transfer function
is not unique but depends on the assumed variation in input between time points.
5.6 Example
For the test system shown in Figure 5.2, if the switch is closed at t = 1.0 s the exactsolution is:
i(t) =VDC
R(1 e(t1.0)/)
for t 1.0 where = L/R.
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The root-matching method 119
For this example the parameters of the circuit are: R = 100 , L = 0.05 mH,
VDC = 100 V. If t = = 50s the difference equation obtained using the
trapezoidal rule is:
i(t + t) = 13
i(t + t) + 13
(v(t + t) + v(t))
For root-matching the difference equation is:
i(t + t) = i(t)e1 + v(t + t)(1 e1)
and the results are summarised in Table 5.3.
For t = 5 = 250 s the difference equations are:
i(t + t ) = 3
7i(t + t) + 5
7(v(t + t ) + v(t)) for the trapezoidal rule
i(t + t ) = i(t)e5 + v(t + t)(1 e5) for the root-matching method
and the corresponding results are summarised in Table 5.4.
Finally for t = 10 = 500s the difference equations are:
i(t + t ) = 23
i(t + t) + 56
(v(t + t) + v(t)) for the trapezoidal rule
i(t + t ) = i(t)e10 + v(t + t)(1 e10) for the root-matching method
and the results are summarised in Table 5.5.
Table 5.3 Response fort = = 50 s
Exact solution Trapezoidal rule Root-matching
1.0 0.0 0.0 0.0
1.0 + t 63.212056 33.333333 63.212056
1.0 + 2t 86.466472 77.777777 86.466472
1.0 + 3t 95.021293 92.2530864 95.021293
1.0 + 4t 98.168436 97.530864 98.168436
1.0 + 5t 99.326205 99.176955 99.326205
Table 5.4 Response fort = 5 = 250 s
Exact solution Trapezoidal rule Root-matching
1.0 0.0 0.0 0.0
1.0 + t 99.326205 71.428571 99.326205
1.0 + 2t 99.995460 112.244898 99.9954601.0 + 3t 99.999969 94.752187 99.999969
1.0 + 4t 100.000000 102.249063 100.000000
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120 Power systems electromagnetic transients simulation
Table 5.5 Response fort = 10 = 500 s
Exact solution Trapezoidal rule Root-matching
1.0 0.0 0.0 0.0
1.0 + t 99.995460 83.333333 99.995460
1.0 + 2t 100.000000 111.111111 100.000000
1.0 + 3t 100.000000 92.592593 100.000000
To demonstrate why root-matching is so good let us consider the exact response
at a discrete time tk , i.e.
i(tk ) =Vdc
R(1 e(tk 1.0)/) (5.20)
which, expressed as a function of a previous time point at tk t, becomes:
i(tk ) =Vdc
R(1 e(tk 1.0)/) =
Vdc
R(1 et/e(tk t1.0)/) (5.21)
Now the same must be true for the previous time point, hence from equation 5.20:
i(tk t) =Vdc
R(1 e(tk t1.0)/) (5.22)
Hence
e(tk t1.0)/ = 1 R
Vdci(tk t ) (5.23)
Substituting equation 5.23 in equation 5.21 gives:
i(tk ) =
Vdc
R
1 et/
1 R
Vdc i(tk t)
= et/i(tk t) +
Vdc
R
1 et/
(5.24)
which is exactly the difference equation for the root-matching method.
5.7 Summary
An alternative to the difference equation using the trapezoidal integration developed
in Chapter 4 for the solution of the differential equations has been described in this
chapter. It involves the exponential form of the difference equation and has been devel-
oped using the root-matching technique. The exponential form offers the following
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The root-matching method 121
advantages:
Eliminates truncation errors, and hence numerical oscillations, regardless of the
step length used.
Can be applied to both electrical networks and control blocks. Can be viewed as a Norton equivalent in exactly the same way as the difference
equation developed by the numerical integration substitution (NIS) method.
It is perfectly compatible with NIS and the matrix solution technique remains
unchanged.
Provides highly efficient and accurate time domain simulation.
The exponential form can be implemented for all series and parallel RL, RC , LC
and RLC combinations, but not arbitrary components and hence is not a replacement
for NIS but a supplement.
5.8 References
1 WATSON, N. R. and IRWIN, G. D.: Electromagnetic transient simulation of power
systems using root-matching techniques, Proceedings IEE, Part C, 1998, 145 (5),
pp. 4816
2 ANGELIDIS, G. and SEMLYEN, A.: Direct phase-domain calculation of trans-
mission line transients using two-sided recursions, IEEE Transactions on Power
Delivery, 1995, 10 (2), pp. 94173 WATSON, N. R. and IRWIN, G. D.: Accurate and stable electromagnetic transient
simulation using root-matching techniques, International Journal of Electrical
Power & Energy Systems, Elsevier Science Ltd, 1999, 21 (3), pp. 22534
4 CAMPOS-BARROS, J. G. and RANGEL, R. D.: Computer simulation of modern
power systems: the elimination of numerical oscillation caused by valve action,
Proceedings of 4th International Conference on AC and DC Power Transmission,
London, 1985, Vol. IEE Conf. Publ., 255, pp. 2549
5 WATSON, N. R. and IRWIN, G. D.: Comparison of root-matching techniques
for electromagnetic transient simulation, IEEE Transactions on Power Delivery,2000, 15 (2), pp. 62934
6 WATSON, N. R., IRWIN, G. D. and NAYAK, O.: Control modelling in electro-
magnetic transient simulations, Proceedings of International Conference on Power
System Transients (IPST99), June 1999, pp. 5448
7 SEMLYEN, A. and DABULEANU, A.: Fast and accurate switching transient
calculations on transmission lines with ground return using recursive convolutions,
IEEE Transactions on Power Apparatus and Systems, 1975, 94 (2), pp. 56171
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Chapter 6
Transmission lines and cables
6.1 Introduction
Approximate nominal PI section models are often used for short transmission lines
(of the order of 15 km), where the travel time is less than the solution time-step, but
such models are unsuitable for transmission distances. Instead, travelling wave theory
is used in the development of more realistic models.
A simple and elegant travelling wave model of the lossless transmission line has
already been described in Chapter 4 in the form of a dual Norton equivalent. Themodel is equally applicable to overhead lines and cables; the main differences arise
from the procedures used in the calculation of the electrical parameters from their
respective physical geometries. Carsons solution [1] forms the basis of the overhead
line parameter calculation, either as a numerical integration of Carsons equation,
the use of a series approximation or in the form of a complex depth of penetration.
Underground cable parameters, on the other hand, are calculated using Pollacks
equations [2], [3].
Multiconductor lines have been traditionally accommodated in the EMTP by a
transformation to natural modes to diagonalise the matrices involved. Original sta-bility problems were thought to be caused by inaccuracies in the modal domain
representation, and thus much of the effort went into the development of more accu-
rate fitting techniques. More recently, Gustavsen and Semlyen [4] have shown that,
although the phase domain is inherently stable, its associated modal domain may be
inherently unstable regardless of the fitting. This revelation has encouraged a return
to the direct modelling of lines in the phase domain.
Figure 6.1 displays a decision tree for the selection of the appropriate transmission
line model. The minimum limit for travel time is Length/c where the c is the speed
of light, and this can be compared to the time step to see if a PI section or travelling
wave model is appropriate. Various PI section models exist, however the nominal
(or coupled) PI, displayed in Figure 6.2, is the preferred option for transient solutions.
The exact equivalent PI is only adequate for steady-state solution where only one
frequency is considered.
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124 Power systems electromagnetic transients simulation
Is travelling time greater
than time step?
Is physical
geometry of line available
(i.e. conductor radius and
positions)?
PI section BergeronFrequency-
dependent
No
No Yes
Yes
Use travelling wave model
Use R, X& B informationFrequency-dependent
transmission line model
Start
Figure 6.1 Decision tree for transmission line model selection
R
ia
ib
ic
va va
vb
vc
vb
vc
R
R
Figure 6.2 Nominal PI section
6.2 Bergerons model
Bergerons model [5] is a simple, constant frequency method based on travelling wave
theory. It is basically the model described in Chapter 4. Here, the line is still treated
as lossless but its distributed series resistance is added in lump form. Although the
lumped resistances can be inserted throughout the line by dividing its total length into
several sections, it makes little difference to do so and the use of just two sections at
the ends is perfectly adequate. This lumped resistance model, shown in Figure 6.3,
gives reasonable answers provided that R/4 ZC , where ZC is the characteristic
(or surge) impedance. However, for high frequency studies (e.g. power line carrier)
this lumped resistance model may not be adequate.
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Transmission lines and cables 125
4R 4R2R
ZC CZ
ikm(t)
vk(t)
in (t)imk(t)
io (t)
vm (t)
Ik(t/2 )Io (t/2 )
Im (t /2 )In (t/2 )
ZC ZC
Figure 6.3 Equivalent two-port network for line with lumped losses
ikm (t)
Ik(t /2)
Im (t /2)
vk(t) vm (t)
imk(t)
Z0+R
4Z0+
R
4
Figure 6.4 Equivalent two-port network for half-line section
By assigning half of the mid-point resistance to each line section, a model of half
the line is depicted in Figure 6.4, where:
ikm(t ) =1
ZC + R/4vk (t) + Ik (t /2) (6.1)
and
Ik (t /2) =1
ZC + R/4vm(t /2)
ZC R/4
ZC + R/4
im(t /2) (6.2)
Finally, by cascading two half-line sections and eliminating the mid-point vari-
ables, as only the terminals are of interest, the model depicted in Figure 6.5 is obtained.
It has the same form as the previous models but the current source representing the
history terms is more complicated as it contains conditions from both ends on the line
at time (t /2). For example the expression for the current source at end k is:
Ik (t ) = ZC
(ZC + R/4)2(vm(t ) + (ZC R/4)imk(t ))
+R/4
(ZC + R/4)2(vk (t ) + (ZC R/4)ikm (t )) (6.3)
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126 Power systems electromagnetic transients simulation
ikm (t)
Ik (t )
Im (t )
vk(t) vm (t)
imk(t)
Z0+R
4Z0+
R
4
Figure 6.5 Bergeron transmission line model
In the EMTDC program the line model separates the propagation into low and high
frequency paths, so that the line can have a higher attenuation to higher frequencies.
This was an early attempt to provide frequency dependence, but newer models (in the
phase domain) are now preferred.
6.2.1 Multiconductor transmission lines
Equations 4.22 and 4.23 are also applicable to multiconductor lines by replacing the
scalar voltages and currents by vectors and using inductance and capacitance matrices.The wave propagation equations in the frequency domain are:
dVphase
dx
=
Zphase
[Iphase] (6.4)
dIphase
dx
=
Yphase
[Vphase] (6.5)
By differentiating a second time, one vector, either the voltage or current, may be
eliminated giving:
d2Vphase
dx2
=
Zphase
dIphasedx
=
Zphase
[Yphase][Vphase] (6.6)
d2Iphase
dx2
=
Yphase
dVphasedx
=
Yphase
Zphase
[Iphase] (6.7)
Traditionally the complication of having off-diagonal elements in the matrices of
equations 6.6 and 6.7 is overcome by transforming into natural modes. Eigenvalue
analysis is applied to produce diagonal matrices, thereby transforming from coupled
equations in the phase domain to decoupled equations in the modal domain. Each
equation in the modal domain is solved as for a single phase line by using modal
travelling time and modal surge impedance.
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Transmission lines and cables 127
The transformation matrices between phase and modal quantities are different for
voltage and current, i.e.
[Vphase] = [Tv][Vmode] (6.8)
[Iphase] = [Ti ][Imode] (6.9)
Substituting equation 6.8 in 6.6 gives:d2[Tv ]Vmode
dx2
=
Zphase
Yphase
[Tv][Vmode] (6.10)
Hence
d2Vmode
dx2
= [Tv]1
Zphase
Yphase
[Tv ][Vmode] = [][Vmode] (6.11)
To find the matrix [Tv ] that diagonalises [Zphase][Y
phase] its eigenvalues and eigen-
vectors must be found. However the eigenvectors are not unique as when multiplied
by a non-zero complex constant they are still valid eigenvectors, therefore some
normalisation is desirable to allow the output from different programs to be com-
pared. PSCAD/EMTDC uses the root squaring technique developed by Wedepohl for
eigenvalue analysis [6]. To enable us to generate frequency-dependent line models
the eigenvectors must be consistent from one frequency to the next, such that the
eigenvectors form a continuous function of frequency so that curve fitting can beapplied. A NewtonRaphson algorithm has been developed for this purpose [6].
Once the eigenvalue analysis has been completed then:
[Zmode] = [Tv ]1[Zphase][Ti ] (6.12)
[Ymode] = [Ti ]1[Yphase][Tv ] (6.13)
[Zsurge i ] =
Zmode(i,i)
Ymode(i,i)(6.14)
where [Zmode] and [Ymode] are diagonal matrices.
As the products [Zphase][Y
phase] and [Y
phase][Zphase] are different so are their
eigenvectors, even though their eigenvalues are identical. They are, however,
related, such that [Ti ] = ([Tv ]T)1 (assuming a normalised Euclidean norm, i.e.n
j=1 T2
ij = 1) and therefore only one of them needs to be calculated. Looking at
mode i, i.e. taking the ith equation from 6.11, gives:
d2Vmode idx2
= ii
Vmode i
(6.15)
and the general solution at point x in the line is:
Vmode i (x) = e ix VFmode i (k) + e
ix VBmode i (m) (6.16)