EMERSON EDUARDO RODRIGUES

1. 900
15
COMPUTER SIMULATION OF
SCALE-BRIDGING TRANSIENTS
IN POWER SYSTEMS
Kai Strunz and Feng Gao
Electromagnetic transients result from changes of the electric and magnetic energies that
are stored in lumped as well as distributed capacitances and inductances of the electric
power network. These transients cause temporary disturbances of voltage and current
waveforms. The frequency range of electromagnetic transients typically extends from
several hertz up to hundreds of kilohertz. Electromechanical transients also involve
oscillatory interactions due to the mechanical energy stored in the rotating shafts of
the generating units. These transients cause oscillations of the electric power over the lines
of the electric power networks at frequencies that typically range from fractions of a hertz
up to several hertz.
In a power system blackout, both types of transients do appear. In the 2003 North
America blackout, three 345-kV transmission lines operated by First Energy (FE)
were tripped due to the contact of trees, which led to electromagnetic transients. As
the transmission lines failed, "power flows shifted to other transmission paths, voltages on
the rest of FE's system degraded further" [1]. This involved the electromechanical
transients. The following cascading events were similar and scale-bridging transients
appeared throughout the blackout.
In the following sections, it is shown how diverse scale-bridging phenomena ranging
from electromagnetic to electromechanical transients are modeled efficiently. In
Section 15.1, the critical role of bridging instantaneous and phasor signals in this
context is explained. In Section 15.2, the construction of the network model is discussed.
In Section 15.3, the modeling of transformers, lines, and synchronous machinery is
elaborated upon in detail. In Section 15.4, the efficient simulation of a power system
blackout scenario, which covers diverse transients, is studied using a four-machine two­
area power system.
HandbookofElectrical Power SystemDynamics: Modeling. Stability. and Control. Edited by Mircea Eremia and
Mohammad Shahidehpour.
© 2013 by The Institute of Electrical and Electronics Engineers. Inc. Published 2013 by John Wiley & Sons. Inc.

2. BRIDGING OF INSTANTANEOUS AND PHASOR SIGNALS
15.1 BRIDGING OF INSTANTANEOUS AND PHASOR SIGNALS
When studying the effects of lightning strikes, switching or other phenomena that trigger
electromagnetic transients, it is of central interest to simulate the instantaneous values of the
distorted voltages and currents. The electromechanical transients are slower and effectively
lead to angle modulations of the generator terminal voltages since the machine rotors do not
tum at their steady-state angular frequencies. Here, it is most illustrative to simulate the
envelopes of AC voltages and currents as well as the average power transfers. In Figure 15.1,
the difference between tracking the instantaneous values of the natural waveform of an AC
voltage and the tracking of its envelope are illustrated. On the left, the AC voltage is zoomed
in for one cycle of duration Ilfc with the carrier frequencyIc of the voltage being equal to
50 Hz in European and 60 Hz in North Americanelectric power networks. On the right, five
cycles and the envelope are shown. From inspection, it is obvious that a much smaller time
step size isneededto accurately track the instantaneous valuesof thenaturalwaveformthan it
is fortracking the envelope. In simulatorsof thetype of the EMTP (electromagnetictransients
program) [2-4], the natural waveforms are represented. In simulators for electromechanical
transients, phasor techniques are used to represent the envelopes [5,6].
In the sequence of a power system blackout, both electromagnetic and electro­
mechanical transients can appear. For a scale-bridging simulation that can track both
electromagnetic and electromechanical transients, the application of analytic signals is
appropriate as they enable the representation of both natural and envelope waveforms.
An analytic signal is obtained by adding a quadrature component as an imaginary part to
the original real signal [7,8]. This quadrature component is obtained through the Hilbert
transform of the original real signal s(t) denoted as:
00
-;?I[s(t)] = � J s(r)
dr
IT t-r
-00
(15.1)
The analytic signal, marked by an underscore to indicate that it is complex, is then
obtained as follows:
�(t) = s(t)+ j -;?I[s(t)]
I-
.� �-++----.,It-----r
I­
...
"
U
Os 1
'lTc
Time
1
Tc
-f- -f- �r.o-
I I I I I I
I I I I I I
I I I I I I I I
I'r
i-- i--
--
Os 1 2 3 4 5
Tc Tc Tc Tc Tc
Time
(15.2)
Figure 15.1. Tracked AC voltages; solid light: natural waveform; solid bold : envelope waveform; x:
sampling points.
901

3. 902 COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
The effect of the creation of the analytic signal is particularly illustrative when the
Fourier spectrum of s(t) shows bandpass character, that is, when it is narrowly concentrated
about the carrier frequency fe. This is typical for voltages and currents that are subject to
electromechanical transients. While the Fourier spectrum of the real signal s(t) extends
to negative frequencies, it can be seen from Figure 15.2 that this is not the case for the
Fourier spectrum .1="[ �(t)l of the corresponding analytic signal �(t).
The analytic signal can be shifted by the frequency fs, which is hereafter referred to as
shift frequency, as follows:
(15.3)
Inserting the angular frequency Ws = 2nfs, (15.3) becomes
(15.4)
For the special case where the shift frequency is equal to the carrier frequency,
fs = fe or Ws = We, the complex envelope [7] is obtained:
(15.5)
Since le-j21ff,tl = 1, the magnitude is not changed through the shift operation. It
follows from (15.5) that the magnitude can readily be derived from the complex
envelope [7]:
I&[ �(t)ll =
1 �(t)1 (15.6)
The impact of the shifting on the Fourier spectrum is given in Figure 15.2. It can be
seen that the complex envelope is a low-pass signal whose maximum frequency is lower
than the one of the original real bandpass signal. In accordance with Shannon's sampling
theorem, a lower sampling rate can be chosen when tracking the complex envelope
rather than the original bandpass signal.
The method frequency-adaptive simulation of transients (FAST) [9,10] processes
analytic signals to enable a scale-bridging simulation. Compared to simulators of the
EMTP-type that process instantaneous signals or simulators for electromechanical tran­
sients that process phasor signals, FAST comprises the shift frequency as a simulation
parameter in addition to the time step size. If the shift frequency is set equal to the carrier
frequency, then the envelope is obtained and electromechanical transients are emulated
efficiently as it is for phasor signals. If the shift frequency is set to zero, then the
instantaneous values of the natural waveforms are obtained as it is the case in simulators
IF[s(t)JI IF[t:[,!:(t)lJI
set) +j1i[s(t)]
..
-Ie Ie -Ie Ie -Ie Ie I
Figure 15.2. Creation of the analytic signal and frequency shifting.

4. NETWORK MODELING
of the EMTP-type. The shiftable analytic signals can so bridge instantaneous and phasor
signals.
15.2 NETWORK MODELING
An electric network consists of branches, which are interconnected at nodes. Through the
application of nodal analysis techniques, the derivation of a network model is based on
straightforward rules. In a first step, the principle of modeling network branches
is introduced in Section IS.2.l. The construction of the nodal admittance matrix is
explained in Section 15.2.2. The key steps of the simulation procedure are summarized
in Section 15.2.3.
15.2.1 Companion Model for Network Branches
While the equations relating voltages and currents of resistive network branches are
governed by Ohm's law, the modeling of inductive and capacitive characteristics involves
differential equations. For the inductance in Figure IS.3a, the differential equation when
using analytic signals is
d lL(t) 2'L(t)
dt L
(15.7)
Expressing the current through (15.4), SUL(t)] = lL(t)e-jw,t, and insertion into
(15.7) yield:
d(S[ lL(t)]dw,t) 2'L(t)
dt L
(15.8)
which can be expanded as follows using the chain rule of differentiation:
d(S[ lL(t)])
= e-jw,1 (-'w i (t)+ 2'L(t))
dt ) S _L
L
(15.9)
Since simulations performed on digital computers require that all differential equa­
tions are transformed into difference equations, a numerical integration method needs to be
applied. Very popular in the simulation of electric networks is the trapezoidal method as
described in Section 10.3.3. On the left side of (15.9), the time differential dt is replaced by
the time step size r, which separates two instants t=kr and (k-l), at which a network
solution is established. The differential d(S[ lL(t)]) is replaced by S[ lLk]-S[ lL(k-1)]'
where k appears as the time step counter. On the right of (15.9), the average obtained at
VL(t)
h(t) +--
•••.'-'----'L�--
(a)
•
(b)
Figure 15.3. Single-phase inductance
and associated companion model.
903

5. 904 COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
instants hand (k - I)T is taken
T
= �e-jw,kT (-)·w i + -YLk)+
2
s_u L
+
1 -jW,(k-l)r ( . .
+ -YL(k-l))
2
e -)Ws ..!:.L(k-I) L
Back substitution of analytic signals for SUul and SUL(k-1)l yields:
i e-jw,kT _
i e-jw,(k-l)r
-Lk -L(k-l) �e-jw,kr (-)·w i + -YLk)+
with
T 2
s -Lk L
+
1 -jW,(k-l)r ( . . +
-YL(k-I))
2
e -)Ws ..!:.L(k-I) L
Multiplying both sides of (15.11) by eiw,kT and gathering like terms lead to
I - V C" I V
. T _iw r (2 -jwsT . T )
-Lk -
L(2+jWsT) - Lk+
2+jWsT -L(k-I)+
L(2+jWsT) - L(k-I)
This can be more clearly summarized as [9]:
.iLk
=
IL -YLk+!lu
eiw r (2 -jwsT . T )
!lu =
'
2+jwsT ..!:.L(k-l)+
L(2+jwsT)
-YL(k-l)
(15.10)
(15.11)
(15.12)
(15.13)
(15.14)
(15.15)
Equations (15.13), (15.14), and (15.15) model a single-phase inductor through a
companion model that consists of an admittance IL in parallel with a history current
source !lL as shown in Figure 15.3b. The term history current source stems from the fact
that it only contains information calculated at the preceding time step k - 1.
For the special unshifted case of fs =
0Hz, both the admittance and the history
current source are real: IL =
T/(2L), !lu =
.iL(k-l)+T/(2L) -YL(k-I). The companion
model then has exactly the same format as its counterpart developed for the EMTP [3] and
is suitable for simulating electromagnetic transients. For fs =
fe' the frequency-adaptive
companion model is suitable for simulating electromechanical transients. This can be
understood by applying the Z-transform to (15.12):
I (2)-
T V (2)+eiw,r (2 -jWsT -I I (2)+
T -I V (2))
-L -L(2+jwsT) -L 2+jwsTZ -L L(2+jwsT)Z -L
(15.16)

6. NETWORK MODELING
where 1L(Z) is the Z-transform of lLk, z-11L(Z) is the Z-transform of lL(k-I)' YL(Z) is
the Z-transform of .!lLk, and Cl YL(Z) is the Z-transform of .!lL(k-I)' Equation (IS.16)
can be rearranged as:
(IS.17)
Substitution of the exponential form of z = eiWcTin (1S.17) and setting Ws = We lead
to:
(IS.18)
Thus, for fs =fe' the admittance is XL = 1/(jweL) and is suitable for processing
phasors as it is desirable in the simulation of electromechanical transients. The frequency­
adaptive companion model is so suitable for the scale-bridging simulation of transients as a
function of the setting of the shift frequency.
Using analytic signals, the differential equation describing the behavior of a capaci­
tance is as follows:
d v
(t)
i (t) = C�
-c dt
(IS.19)
Expressing the voltage through (1S.14), S[.!ldt)] = .!ldt)e-jw,t, and insertion into
(IS.17) yield:
. dS[.!ldt)]eiw,1
.!.dt) = C
dt
(IS.20)
which can be expanded to:
d(S[.!ldt)])
= e-jw,1 (-l'W v
(t)+ ldt))
dt
s_c C
(IS.21)
Similar to the inductance model, the differential d(S[.!ldt)]) on the left of (IS.21) is
replaced by S[.!lCk] - S[.!lC(k-1)]' On the right of (IS.21) the average obtained at instants
kr and (k - 1)'[ is taken as follows:
S[.!lCk] - S[.!lC(k-l)] 1
e-jw,kr (-l'W v
+
lCk)+�e-jW,(k-l)T(-l'W v
+lC(k-l))
to:
T 2 S_Ck C 2
s- C(k-l) C
(IS.22)
Back substitution of analytic signals for S[.!lCk] and S[.!lC(k-I)] and rearranging lead
. _
C(2+jWsT) �iW,T( . (2 -jWsT)C )
.!.Ck- .!lCk+c -.!.C(k-l) - .!lC(k-l)
T T
(IS.23)
905

7. 906 COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
with
This can be written as:
iCk = Xc -YCk+ !iCk
Xc =
C(2+jWST)
T
�iWT(. (2 -jwsT)C )
!lCk = C" -J.C(k-I) -
T -YC(k-l)
(15.24)
(15.25)
(15.26)
Equations (15.24), (15.25), and (15.26) model a single-phase capacitance through a
companion model that consists of an admittance Xc in parallel with a history current
source !lc.
15.2.2 Direct Construction of Nodal Admittance Matrix
The nodal equation system for the network model is of the following form:
(15.27)
where Y is the nodal admittance matrix, v is the nodal voltage vector, andj is the nodal
current injection vector that includes the history current sources. If besides the ground
node there are N nodes in the network under study, then Y is of size N x N, and v andj are
of size N x 1. Using analytic signals, Y, v, andj become complex as indicated through
underlining:
(15.28)
In the following, it is introduced how to directly construct the network nodal
admittance matrix Y and nodal current injection vectorj through the stamping method
[11]. The time step counter k is dropped for the sake of clarity of explanation.
For a resistive branch with conductance G connected to nodes m and n, the admittance
matrix stamp contributed by this branch is as follows:
m n
0 0 0
ilY
m G -G
0 0 0
(15.29)
n
-G G
0 0 0
The admittance matrix stamp of (15.29) is an N x N matrix and contains only four
nonzero entries at positions (m, m), (m, n), (n, m), and (n, n). The sign is positive at the
diagonal positions and negative at the off-diagonal positions. If a branch is connected
from node m to the ground, then the admittance stamp will only have one nonzero

8. NETWORK MODELING
entry at (m, m):
�y m
m
(� GO.;.)
o 0 ···
(15.30)
The nodal admittance matrix Y is obtained by first conceptually removing all branches
from the network and thensuccessivelystamping the contributions of the companionmodels
into the emerging nodal admittance matrix in accordance with the network topology:
(15.31)
where b refers to all branches in the network. The construction of the network current
injection vectorjis based on the same concept. If a current source injects a current iinto node
n, then the corresponding current injection vector stamp is as follows:
o
�j= n (15.32)
o
The stamping process for constructing the network admittance matrix and the current
injection vector is illustrated for a simple circuit in Figure 15.4 . Since there are three nodes
in the circuit, that is, N= 3, all the admittance matrix stamps are 3 x 3matrix contributions.
.
...
...
G,
'G,
'G,
G,
....
.
...
......
.•......
2
Go
,.---.__-1
,
Gc i
.. .I
.••�v._••
•••••.
..-;./
........
I
G.
+
'Go
..
..
.
.......
...........
...
.
.
.
.. ..
.
.
.
.......
.
....
.. . ::::..-.::-:::.-.
.
...
.................
.
.
Gb 3 .... ..
....� ",
./
;
1"1+ ;./
! -
. •• -!-. . • ••• . .• • ••••• •--••••••••••••••••••••
j -
.•••..
!
)
'Go +
ic +
Figure 15.4. Example of stamping method.
j
907

9. 908 COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
The conductance Ga is connected to nodes 1 and 2 and therefore the corresponding
admittance stamp in accordance with (15.29) is
-Ga 0 )
Ga 0
o 0
Similarly, the admittance matrix stamp of Gb is as follows:
(15.33)
(15.34)
The conductance Gc is connected to node 2 and the ground, so with the application of
(15.30) its admittance matrix stamp becomes:
(15.35)
With all the admittance matrix stamps built, the network admittance matrix can then be
obtained by adding the stamps according to (15.31):
(15.36)
The current source connected to node 2inFigure 15.4provides excitation in that it injects
current ic into node 2. Therefore, its corresponding current injection vector stamp is
6.jc =(0 ic 0)T. The current is through the ideal voltage source Vs cannot be calculated
solely as a function of the source voltage element Vs but also depends on other nodal voltages.
It contributes a stamp 6.is =(0 0 is )T. The current injection vector is therefore:
(15.37)
In the following, the nodal voltage vector is partitioned into a set of voltages Vd that are
dependent and unknown and need to be calculated, and a set of voltages Ve that provide
excitation and are given through source terms. The partitioning leads to the following
modified version of equation (15.27):
(15.38)
Only the first row of (15.38) is needed for calculating the unknown nodal voltage
vector Vd. Therefore
(15.39)

10. MODELING OF POWER SYSTEM COMPONENTS
G.
G.+Gb -G. -Gb
-Ga G,,+Gc 0
-Gb 0 Gh
3
L.........................:=:......J
2 Ga
�}; c:::J�
1 I };�vs
':(�Gc
I
G�':(
-=- -=-
�"....."..=..." ...:=..",.....,.
Ga+Gh -Ga
-Go G.+Gc
VI
V2
=
o -GbV$
jc o
Figure 15.5. Example of the network equation reformulation.
or
(15.40)
Equation (15.40) is solved for Vd. The second term - Ydeve on the right-hand side of
(15.40)is a multiterminal Norton equivalent source that makes the excitation voltage sources
appear as current injections.
The vector.ie in the second row of (15.38) contains the excitation current injections
into the nodes at which the ideal voltage sources are incident, whereas ie contains the
unknown currents through the ideal voltage sources.
The reformulation of (15.27) leading to the partitioned counterpart (15.40) can be
illustrated by the example shown in Figure 15.5.
The voltages at nodes 1 and 2 are dependent while the voltage at node 3 is known. The
Norton equivalent source is here given by -GbVs.
15.3 MODELING OF POWER SYSTEM COMPONENTS
In the following, the scale-bridging modeling of multiphase inductors, transformers,
transmission lines, and synchronous machinery is discussed [10].
15.3.1 Multiphase Lumped Elements
In the continuous time domain, the magnetically coupled inductances shown in Figure 15.6
can be described in vector-matrix notation as follows:
where
diLl
-
dt
dh(t)
dh2
-
dt L-1=
dt
diLM
--
dt
dh(t) _ c1 ()
--- VL t
dt
L'M
f
(
L
"
L12
L21 L22 LzM
LMI LMM
(15.41)
(
'U
(t)
)
vdt)
VL(t)= .
VLM(t)
909

11. 910 COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
Figure 15.6. Current and volt­
age conventions for magneti­
cally coupled inductances.
whereM is the number of inductors, Lmm with m=1, 2,... ,M is the self inductance of the
mth inductor, and Lmn with m, n=1, 2,... ,M and m -I n is the mutual inductance between
mth inductor and nth inductor.
The frequency-adaptive companion model of the magnetically coupled inductances
can be developed in an analogous way as for the single inductance already described in
Section IS.1.2. Using analytic signals, (1S.41) becomes:
d i
(t)
�
=L-1 v (t)
dt -L (1S.42)
Substitution of the shifted analytic signals in analogy to (1S.9) results in the vector­
matrix format:
(1S.43)
Application of the trapezoidal rule to differentiate (IS.43) yields the companion model
of the magnetically coupled inductances [9]:
with
-dW'T(2 -jwsr i L-1 T V )
!LLk- 2 + . -L(k-I)
+
2 + . -L(k-I)
)WsT )WsT
(1S.44)
(1S.4S)
(1S.46)
While in (1S.14) the admittance YL is a scalar, it is a matrix in (IS.4S) for the
multiphase case. Similarly, while in (IS.IS) the history current source YJL is a scalar, it is a
vector in (1S.46).

12. MODELING OF POWER SYSTEM COMPONENTS
15.3.2 Transformer
Atransformermade of an ideal core with infinite permeability and a finite leakage inductance
is modeled by the equivalent circuit consisting of an ideal transformer with turns ratio a of
primary side to secondary side and a series inductance as depicted in Figure 15.7. The series
inductance that represents the leakage effects is shown on the primary side with the higher
voltage because the winding supporting the higher voltage is usually further away from
the transformer core. Alternatively, the leakage inductance may be split up to model the
respective leakage contributions on both sides.
In the discrete time domain, the leakage inductance can be represented by the
companion model developed in Section 15.1.2. This leads to the circuit model illustrated
by Figure 15.8.
For the currents shown in Figure 15.9, the following equations apply
.!:.Tlk=XL(J:Pk - aJ:Sk)+ !1Lk=XL(J:TIk - J:T3k - a(J:T2k - J:T4k))+ !1Lk
(15.47)
Primary Leakage Ideal Secondary
side inductance transformer side
= =
Figure 15.7. Conventions for single-phase trans­
former with leakage inductance.
Figure 15.8. Circuit model of single-phase transformer with leakage inductance.
=
In(I)
3---------'
t}Cn(t)
Excitation
inductance
II
LT4(t)
'----.. 4
t}CT4(t) Figure 15.9. Conventions for single-phase trans­
former with leakage and excitation inductance.
911

13. 912 COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
iT2k = -aiTlk (15.48)
(15.49)
(15.50)
The admittance XL and history source term !h are provided by (15.14) and (15.15).
Equations (15.47) to (15.50) can be reorganized in the vector-matrix format:
iTk = XT .!Tk+ !iTk (15.51)
where iTk = (iTlk, iT2k, iT3k, iT4k)T is the vector of transformer terminal currents and
.!Tk = ( �Tlk' �T2k' �T3k' �T4k)T is the vector of transformer terminal voltages. The
admittance stamp of the transformer model is therefore:
�h -aQL -QL aQL
-aQL a2 G aQL -a2 G
XT =
- L - L
(15.52)
-QL aQL QL -aQL
aQL _a2 G
- L -aQL a2 G
- L
The history source vector is !lTk = (!lLk, -a!lLk, -!lLk, a!lLk)T.
More detailed models of the transformer can be developed in accordance with the
simulation needs. Excitation effects, for example, can be emulated by connecting an
inductance in parallel with the above transformer model to account for the magnetization
of the transformer core. The circuit model is illustrated in Figure 15.9. Multiphase
transformers can be modeled by connecting primary and secondary sides of multiple
single-phase transformer models depending on the case under study in star- or delta­
configurations.
15.3.3 Transmission Line [10]
In order to enable the scale-bridging simulation of electromagnetic and electromechan­
ical transients on transmission lines, the line model is to represent both traveling wave
effects at time step sizes on the order of microseconds as well as slowly changing voltage
and current envelopes due to electromechanical transients at much larger time step sizes
as illustrated in Figure 15.1. In what follows, the scale-bridging line model is elaborated
upon.
15.3.3.1 Single-Phase Line Model. The propagation of voltages and currents on a
single-phase lossless line with length C, distributed inductance L', and capacitance C' as
shown in Figure 15.10 is given through the general solution by d'Alembert [12]:
(15.53)
in(t) = vn(t)jZo - ill(t - Twp) - VII(t - Twp)jZo (15.54)

14. MODELING OF POWER SYSTEM COMPONENTS
ill(t)=i(Okm,t) i(X,t)
I •• .
VII (t)1
=v(Okm,t)
=
x
in(t)=-i(R,t)
"2
V12(t)1
=v(R,t)
• I
X=R
Figure 15.10. Current and voltage conven­
tions for lossless single-phase line.
where Zo = JL'/c' is the characteristic impedance and Twp = RVL'c' is the propagation
time that the waves need to travel from one to the other end of the line. From (15.53),
(15.54) it can be seen that there is no topological coupling through a lumped element
between both ends of the line in the model because the voltages and currents at the opposite
end date from instant t - Twp rather than t. In digital simulation, this can only be modeled if
the wave propagation time exceeds the time step size, that is, T < Twp' If T 2: Twp, the line
can be modeled with a n-circuit giving a topological coupling between both ends.
A scale-bridging transmission model needs to be valid for any T, that is, for both
T < Twp and T 2: Twp' The switches are key to the model that is shown in Figure 15.11 [13].
For electromagnetic transients and T < Twp, the switches are open. For electromechanical
transients and T 2: Twp, the switches are closed to provide the topological lumped coupling
via a n-circuit cell, the parameters of which are developed in the following.
As all other models, the line model is formulated through a companion model with all
time-varying quantities represented through analytic signals:
11k = Xl .!'lk+ !1lk (15.55)
where 11k = Ullk' .i12k)T and .!'lk = (�llk' �12k)T are the terminal quantities identified in
Figure 15.11. The solution by d'Alembert in (15.53) and (15.54) is the starting point for
deriving Xl and !ilk' In order to calculate values of the time-varying quantities .ill' .i12,
�ll' and �12 at instant (t - Twp), the usage of linear interpolation is appropriate whenever
Twp is not an integer multiple of T. To enable such interpolation at any time step size, that is,
for T < Twp when the switches in Figure 15.11 are open and for T 2: Twp when these
switches are closed, a variable K is defined as follows:
K = ceil (T
;p
), K 2: 1 (15.56)
where ceil is a function that rounds a variable to the nearest integer greater than or equal to
itself. Then (K - 1)T < Twp :S KT where K > 1 corresponds to the case where T < Twp, and
Switches closedwhen /(=1
2p
Figure 15.11. Frequency-adaptive model of single-phase line model.
913

15. 914 COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
K = 1 corresponds to the case where r 2 Twp' Application of linear interpolation to shifted
analytic signals of S[jAt- Twp)] gives:
S[.L(t- Twp)] = (K -
T
;
p
)S[.L(t- (K - l)r)]+ (1 - K+
T
;
p
)S[.L(t- KT)]
(15.57)
Back substitution of the definition of shifted analytic signals in (15.4) and solving for
jAt- Twp) lead to:
(15.58)
with
(15.59)
(15.60)
The interpolation for S[.!l(t - Twp)] is performed in exactly the same way. Inserting
the results of these interpolations in the analytic signal versions of (15.53) and (15.54)
yields:
!.lk =
;0 (� �) .!lk+(�1 �
1
) (;0 .!l(k-I<+l)+;0 .!l(k-I<)+P !.l(k-I<+l)+(J !.l(k-I<))
(15.61)
For K > 1, the terms .!l(k-I<+l), .!l(k-I<)' !.l(k-I<+l)' !.l(k-I<) on the right-hand side of
(15.61) refer to values calculated in past time steps so that no topological coupling through
a lumped element is given between both ends of the line. Therefore, the history term !1lk as
well as the admittance stamp Xlk in (15.55) when K > 1 can be described by:
!1lk =
(�1 �
1
) (;0 .!l(k-I<+l)+;0 .!l(k-I<)+P !.l(k-I<+l)+(J !.l(k-I<)) if K > 1
(15.62)
y - y - � (1 0
) if K > 1
- I
- - to
-
20 0 1 (15.63)
For K = 1, both .!lk and !.lk appear on the right-hand side of (15.61), which implies a
topological coupling between both ends of the line. That is to say when K = 1, (15.61) is
. 1 (1 0
) ( 0
J.lk =
20 0 1 .!lk+
-1
-1
)(P (J . .
)
o 20 .!lk+
20 .!l(k-l)+P J.lk+(J J.l(k-l)
(15.64)

16. MODELING OF POWER SYSTEM COMPONENTS
which can be recast to:
(1 p) . 1
( 1
p 1 !:.lk=
Zo -p
-p) ( 0 -1 )(CJ . )
1 .!lk+ -1 0 ZO .!I(k-I)+CJ!:.I(k-l) (15.65)
Since
(1 p )-l 1 ( 1
p 1 =
1 - p2 -p
the left multiplication of (15.65) by (15.66) leads to:
. 1 1 ( 1
!:.lk=Z�
o p _p
-P )( 1 -P ).!lk
1 -p 1
(15.66)
1 ( 1
+�
p -p �
P )(:1
�1)(;0 .!l(k-l)+CJll(k-l)
)
(�
1 1 - p2
Zo -2p
1 - p2
I - p2
CJ p
-2p )
l+p2
- I - p2
(_1
"lk+--
1 - p2
-1 )(1 . )
p Zo .!l(k-l)+ !:.l(k-l)
(15.67)
Therefore, the history term !ilk as well as the admittance stamp %lk in (15.55) when
K=1 can be described by:
CJ ( p
!ilk=
1 _ p2 -1 �1 )(;0 .!l(k-l)+ ll(k-l)
) ifK=1 (15.68)
ifK=1
(15.69)
The matrix %IC in (15.69) provides the topological coupling forK=1, that is, for
T 2 Twp, as used when electromechanical transients are studied. It can also be described
through the n-circuit cell as shown in Figure 15.11.
915

17. 916
!ilk
COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
In summary, the two cases when K > 1 and K = 1 can be combined together:
-1)(p a . . )
o
Z
o !'I(k-K+I)+Z
o !'I(k-K)+P J.I(k-K+I)+a J.I(k-K)
a ( p
1 - p2 -1
-1 )(1 . )
p Z
o .!I(k-K)+ J.I(k-K)
.rID ifK > 1
.rID+ .rIC ifK = 1
ifK > 1
ifK = 1
(15.70)
(15.71)
15.3.3.2 Multiphase Line Model. In order to model a fully transposed m-phase
line, the m coupled phase quantities are transformed into m decoupled modal quantities
using eigenvalue analysis as, for example, discussed in [2,12,14]. In the special case of
the balanced three-phase line as shown in Figure 15.12, the decoupling can be achieved
using Oa{3 components [2,12,14] for both voltages and currents: !'l1abe = h !'110afl'
!'12abe = h !'120afl' illabe = h illOafl' and il2abe = h il20afl with the transform
matrix [12]:
1 v'2 0
1
�
1 1
v'2
TL = - (15.72)
v'3
1
1
v'2
-�
and v = ( v v V )T
V =( V V V )
T
V = ( v v V )T
-l1abe -lla' -lib' -lie , -llOafl -110, -lla' -l1fl ' -12abe -12a, -12b, -12e ,
V = ( V V v )
T
i = ( i i i )T
i = ( i i i )
T
-120afl -120' -12a' -12fl '-llabe -lla' -lib' -lie '-llOafl -110' -lla' -llfl '
il2abe = ( il2a, il2b, il2e)T
, and il20afl = (iI20, il2a, il2fl)
T
. The decoupling scheme is
shown in Figure 15.13.
bla
}C/lat
-
blb
t
-
illc
Dependent components
through mutual coupling
" "
,
, 1
t�2b
-
L/2c
t�2c
-
ina
Figure 15.12. Current and voltage conven­
tions for balanced three-phase line.

18. MODELING OF POWER SYSTEM COMPONENTS
Independent components
= = =
Figure 15.13. Decoupling of balanced three-phase line based on Qa{J components.
The three decoupled Oaf3 modes are dealt with in the same way as a single phase in the single­
phase model (15.55). In analogy to (15.55), a vector-matrix form is obtained as follows
( �llOafJk ) = X
/
OafJ
( .!llOafJk ) + ( !lllOafJk )
J:.120afJk .!120afJk !1120afJk
(15.73)
where XlafJO is formed by stamping the elements of three 2 x 2 matrices 1'/0, X1a, and XlfJ
into the corresponding positions:
1'/0(1, 1)
Xla(1,I)
XlafJO = X1fJ(I,1)
1'/0(2, 1)
Xla(2,I)
XlfJ(2,I)
1'/0(1, 2)
1'/0(2, 2)
Xla(1, 2)
X1a(2,2)
X1fJ(1,2)
XlfJ(2,2)
(15.74)
For example, the two ends of the ex component correspond to the second and fifth rows
in the current and voltage vectors; therefore, Xla is stamped into XlafJO at the four
intersections of the second and fifth rows and columns.
Transforming back to the original abc quantities with I'L
l gives
(I'L
l
l
J.
.
llabek ) = Y (I'L
l
.!llabek ) + (I'L
l
!lllabek )
7'- -IOafJ 7'-1 7'-1
� L J:.12abek � L .!12abek � L !l12abek
which can be recast as:
( J:.
,
:llabek ) = Y ( .!llabek ) + ( !lllabek )
-12abek
- labe
.!12abek !l12abek
(15.75)
(15.76)
where the 6 x 6 matrix Xlabe for representing the three-phase balanced line in the phase
domain is
Y _ (h
- labe- 0
o ) Y (I'L
l
h
-IOafJ 0 (15.77)
917

19. 918 COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
Similar procedures can be used for transmission lines with more than three phases.
When losses are represented by lumped multiphase resistances at the ends and the
middle of the lines, then the modal transform is also applied to the multiphase
resistances.
15.3.4 Synchronous Machine in dqO Domain
The modeling of synchronous machinery is essential to the study of electromechanical
transients. The Park transformation from abc to dqO variables is popular in synchronous
machine modeling because it allows for the representation of inductance terms independent
of the present rotor angle [15]. Adopting the Park transformation, the synchronous machine
modeling can be organized as shown in Figure 15.14 [16]. The three-phase machine
terminal voltage -.!'Mabc of the abc domain is mapped into the voltage -.!'MdqO of the dqO
domain by the Park transform. The machine model provides as output the three-phase
terminal current i.Mabc following a back transform from i.MdqO in the dqO domain to the abc
domain to ensure compatibility with the network model, which is also represented in the
abc domain.
The exciter model uses a reference voltage Vref and the real part of terminal voltage
-.!'Mabc to provide the field voltage Vf as input to the machine equations. The model of the
governor-turbine stage compares the rotor electrical angular speed Wr with the speed
reference Wref and provides the mechanical torque Cm as an input to the machine equations.
The modeling of those is described in Chapters 2 and 3.
The real part of i.MdqO is obtained from the electromagnetic and mechanical machine
equations which has as inputs the real part of -.!'MdqO' Vf, and Cm, and the imaginary part of
i.MdqO can be constructed as will be explained later in this section.
15.3.4.1 Electromagnetic and Mechanical Machine Equations. In Section
2.1.3.4, the electromagnetic and mechanical machine equations make use of the per unit
system are described. The electromagnetic equations (2.39'), (2.40'), (2.32')and (2.34')can
be written in matrix form and the time t is not per unit here:
Urer
Wref
,..----_----JMabc
Other
outputs
(15.78)
(15.79)
Figure 15.14. Block diagram organization of syn­
'----·iMabc chronous machine model in dqO domain.

20. MODELING OF POWER SYSTEM COMPONENTS
where
'I'M' = (Re[lrd'] 1/Ir 1/ID' Re[lrq,] 1/IQ' )T (15.80)
iM' = (Re[iMd'] ir iD, Re[..!:.Mq'] .
)T
lQ' (15.81)
VM' = (Re[-.YMd'] Ur 0 Re[-.YMq'] of (15.82)
R, 0 0 -wr,Lq, wr,LmQ,
0 -Rr 0 0 0
ZM' = 0 0 -RD' 0 0 (15.83)
wr,Ld, -wr,Lmd, -Wr,LmD' R, 0
0 0 0 0 -RQ'
The flux linkage equations (2.32') and (2.34') (see Section 2.1.2) can also be written in
matrix form:
1/IM' = LM,iM, (15.84)
where
-Ld' Lmd' LmD' 0 0
-Lmd' Lr LfD, 0 0
LM, = -LmD' LjD' LD, 0 0 (15.85)
0 0 0 -Lq' LmQ,
0 0 0 -LmQ' LQ,
and
Re[lro'] = -La,Re[io'] (15.86)
The electromechanical equations given in (2.12) (see Section 2.1.2) are given here in
per unit form:
dwr, 1
- = -
(C ' - C ' - D(w , - 1))
dt 2H m e r
do
dt
= WOWr'
(15.87)
(15.88)
The angle 0 (in electric radians) is the rotor position, at any instant t, with respect to a
synchronously rotating reference frame (see Figure 2.4). The air-gap torque is calculated
by:
Ce' = Re[lrd']Re[..!:.Mq'] - Re[Irq,]Re[..!:.Md'] (15.89)
919

21. 920 COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
15.3.4.2 Calculation of Real Part of Stator Current. By substitution of (15.84)
and (15.86) into (15.78) and (15.79), respectively, and converting the differential equations
into difference equations, the real part of !..MdqO at time step k can be obtained. To have a
noniterative solution, the values of VM' and Re[�o'1 are taken at time step k - 1. The value
of ZM' is also taken at time step k - 1 because ZM' is a function of the rotor electrical
angular speed Wr', which is a function of time. For iM, and Relio']' it is possible to take
their values at time step k because they can be moved to the left of the equations and
combined to the corresponding values at time step k. Therefore, the difference equations
for solving the real part of stator currents are as follows:
LM,iM'k - iM'(k-l) •
-- = ZM'(k-l)IM'k+ VM'(k-l)
Wo T
-Lo' Re[iO'kl - Re[iO'(k-1)l _ .
[ ]
-- -R,ReUo'kl + Re �O'(k-I)
Wo T
They can be reorganized as:
Therefore
. Lo'Re[iO'(k-1)l - TwoRe[�o'(k-1)l
ReUO'kl = -----'--
---'-
------'-
---'­
Lo' + TWoR,
(15.90)
(15.91)
(15.92)
(15.93)
(15.94)
(15.95)
15.3.4.3 Calculation of Imaginary Part of Stator Current. With the models
presented above, the real part of !..MdqO is obtained. In order to generate !..MdqO in
Figure 15.14, it is necessary to also construct the imaginary part of !..MdqO' For analytic
signals, the imaginary part is obtained by the Hilbert transform of the real part. To find an
equation for obtaining the imaginary part, it is helpful to consider the Park transformation:
where
i -p i
-Mabc- M -MdqO
cose -sine 1
cos (e -
2
;) -sin (e _
2
;) 1
cos (e+
2
;) -sin (e+
2
;)
(15.96)
(15.97)
The angle e is the angle by which the d axis leads the axis of phase a, as illustrated in
Figure 2.6.

22. MODELING OF POWER SYSTEM COMPONENTS
or
and
Phase a of the transformed signal becomes:
From the definition (1S.2), lMdqo is an analytic signal when
Substitution of (IS.99) in (1S.101) yields:
(IS.98)
(IS.99)
(IS.100)
(IS.101)
(IS.102)
In steady state, Re[ iMd], Re[ iMq] and Re[ iMO] are constant. In steady state (1S.102)
becomes:
Im[iMa] = Re[iMd]-;:?I[cos e] - Re[iMq]-;:?I[sin e] = Re[iMq]cos e + Re[iMd]sin e
(IS.103)
Subtracting (1S.103) from (1S.100) yields:
Equation (IS.104) should hold for any e, therefore:
(1S.lOS)
(IS.106)
(IS.107)
The same conditions also need to hold when phases band c are considered.
Equations (IS.lOS), (IS.106), and (IS.107) exactly hold for constructing the imagi­
nary part of lMdqo from the real part of lMdqo. In practical application, these equations are
also applicable when the system is subject to slow transients or fast transients. For slow
transients, Re[iMd], Re[iMq], and Re[iMO] change slowly, and (1S.103) provides an
acceptable approximation of (1S.102); for fast transients, natural waveforms are tracked so
that the imaginary part of an analytic signal is of little importance.
921

23. 922 COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
15.3.4.4 Calculation of Rotor Speed and Angle. With the obtained stator
currents, the flux linkage at time step k, 1/IM'k' can be calculated by (15.84), and the
air-gap torque at time step k, Ce'b can be calculated by (15.89). By converting differential
equations (15.87) and (15.88) into difference equations, the rotor electrical angular speed
Wr and rotor angle 0 at time step k can be obtained. Using the trapezoidal rule, (15.87) and
(15.88) become:
Wr'k - Wr'(k-l) (Cm'k - Ce'k - D(Wr'k - 1))+(Cm'(k-l) - Ce'(k-l) - D(Wr'(k-l) - 1))
r 4H
Equations (15.108) and (15.109) can be reorganized as:
(15.108)
(15.109)
4H - rD r
Wr'k = Wr'(k-l)+ (Cm'k+Cm'(k-l) - Ce'k - Ce'(k-l)+2D)
4H+rD 4H+rD
(15.110)
rwo
Ok = Ok-l+2 (Wr'k+Wr'(k-l)) (15.111)
On the right of (15.110), Cm'k is obtained from the output of the governor and turbine.
After Wr'k is obtained by (15.110), Ok is calculated by (15.111).
15.3.4.5 Integration with AC Network. It can be seen from Figure 15.14 that the
model accepts the three-phase terminal voltage that it shares with the network as input, and
injects the three-phase terminal current as output into the network. Therefore, from the
network side the synchronous machine model described above is in fact a voltage­
controlled current source. If a noniterative simulation process is adopted, then three-phase
terminal current injection at time step k, lMabek> is obtained based on terminal voltage
information available in time step k - 1. To attain a fast simulation process with good
numerical stability and accuracy, an adjustment current source lAabek and the adjustment
three-phase resistance GAabe are introduced at the machine terminal. This leads to the
interface as illustrated in the circuit model of Figure 15.15.
The per-phase resistance GA of GAabe is selected to reflect the characteristic impedance
[17] when the machine operates near open-circuit conditions:
iMAabck
G
_
r
A-L"+L"
d q
AC
network
(15.112)
Figure 15.15. Network interfacing of
synchronous machine model in dqO
domain.

24. APPLICATION SIMULATION OF BLACKOUT
where L� and L� are the subtransient inductances of the stator circuit. The adjustment
current source is as follows:
(15.113)
where OJrk is obtained at the same time as lAabck. The net current lMAabck flowing into the
network considering lMabck, lAabck and the current flowing through GAabc is given as
the sum of lMabck and an adjustment term:
(15.114)
The interface allows the characteristic impedance GA to be seen from the network for
sudden voltage changes.
From (15.112), it can be seen that GA increases as the time step size decreases. When
electromagnetic transients are studied at very small time step sizes, the adjustment term on
the right of (15.114) is therefore very small. When electromechanical transients are
studied, the adjustment term is small too, even though the time step size is much larger.
This is because the three-phase terminal voltage retains its sinusoidal shape for such low­
frequency transients and therefore .!Mabck_,eiW'kT:::::: .!Mabck. In steady state, the adjust­
ment term is zero because .!Mabck_leiw,kT = .!Mabck for any time step size.
15.3.4.6 Initialization. A simulation can start with zero initial conditions and can
then reach steady state fast. However, when the simulated network contains synchronous
machines, the following initialization method needs to be used so that the simulated network
can reach to steady state fast enough. The first step is to use the steady-state calculation from
power flow results to obtain machine quantities as described in Section 2.1.3.6. Then the
machines act as sinusoidal current sources until the entire network reaches steady state. After
that, the electromagnetic equations in the machine model are taken into account while the
electromechanical equations are not. This means the rotation speed of the machine is fixed.
When the transients fade away, the electromechanical equations can also be included and now
the machines are simulated by the full model described above.
15.4 APPLICATION SIMULATION OF BLACKOUT
The network depicted in Figure 15.16 is identical in structure to the two-area system
described in [15]. Two areas, each with two synchronous machines and loads, are
7
Area I
�nter
-
8
IIOkm
9
llOkm
Fault L C9
1 ILD9
Area 2
Figure 15.16. Four-machine two-area test system.
923

25. 924 COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
T A B l E 1 5.1 . Generator Parameters
R, = 0.0025 p.u.
XI' = 0.2 p.u.
D = O
Xd• = 1 .8 p.u.
Xq• = 1 .7 p.u.
Vb = 20 kY
X�. = 0.3 p.u.
X�. = 0.55 p.u.
Sb = 900 MYA
X�. = 0.25 p.u.
X�. = 0.25 p.u.
H = 6.5 (GI , G2),
T�O' = 8.0 s T�O' = 0.03 s
T�O' = 0.4 s T�o' = 0.05 s
H = 6. 175 (G3 , G4)
T A B l E 1 5.2. Transmission line Parameters
L: = 2.653e-6p.u'/km C; = 4.642e-6 p.u./km R: = 0.000 1 p.u./km Sb = 1 00 MYA
T A B l E 1 5.3. Transformer Parameters (per Phase)
L. = 0. 1 5 p.u.
Vb primary = 230/V3 = 1 32.8 kY, Vb secondary = 20 kY
a = (230/V3)/20 = 6.64
Sb = 900 MYA
connected through weak tie lines. In steady state, each generator provides a real power of
the order of 700 MW, while constant-impedance loads LD7 and LD9, respectively,
consume about 1000 and 1800MW. About Pinter = 400MW flows over the tie lines.
Reactive power also flows from area 1 to area 2. The scenario considered hereafter shows a
sequence of diverse phenomena that are common to blackouts [1]. It starts with a short
circuit, which can be caused by a line touching a tree. Such a short circuit triggers
electromagnetic transients. The missing line leads to an increase of the impedance between
the two areas weakening the tie further. This gives rise to electromechanical transients
involving inter-area power oscillations over the one remaining tie line. If no other actions to
stop the overload condition of the remaining tie line are undertaken in time, then a tripping
of this line to decompose the system into two islands may be necessary. If the formerly
importing area experiences a lack of power generation capability, then load shedding can
prevent further problems and stop the cascading events.
The inter-area tie lines were represented through the scale-bridging line model
introduced in Section 15.3.3. Other lines were represented by the 1t-models with lumped
parameters. The passive resistive and reactive loads were also modeled as lumped RLC
components. Each transformer was modeled by connecting three single-phase transformer
models as described in Section 15.3.2. The synchronous machines were represented by the
models presented in Section 15.3.4. Network parameters are shown in Tables 15.1, 15.2,
and 15.3 [15] where R: is the distributed resistance.
The results obtained are depicted in Figures 15.17, 15.18, and 15.19. From the voltage
of load LD9 in Figure 15.17, it can be seen that the simulation alternately tracks the
envelope or the natural waveform.
200 F- �-
-
-
-
__
-�r---""1
;;-
c
�� 0 · · ·
;;
'0
>
-200 � �
_
_
_
_
...JL--�L._
_-I Figure 15.17. Phase a voltage of load
5 LD9 : solid light: natural waveform; solid
bold : envelope.
o O.S 2 3
Ti me (s)
4

26. APPLICATION SIMULATION OF BLACKOUT
I I
1
2 13 14 15 16
� 380 .-�
�
�--
--
�
�
�
�
�
�
�--
--
,
�
-..; 378
""8
�
en 376
@
:l
� 374 L-
�
__
�
__
__
�
__
M-
�
__
-U-
�
__
__
---' Figure 15.18. Machine angular speed;
« 0 0.5 2 3 4 5 solid bold : G 1 ; dash dot bold : G2; solid
[
600
400
<;
� 200
0
Co
., 0
>
li
-200
-<
0
II 12
0.5
Time (s) light: G3; dash dot light: G4.
2 3 4 5 Figure 15.19. Interarea active power
Time (5)
The two-dimensional parameter control of the FAST method for the test case at different
times is illustrated in Figure 15.20, where each cross is an operating point representing the
simulation setting for a given time period, and the arrows with time points show the
transitions of the operating points. The x-axis stands for the time step size T, and the y-axis
stands for the shift frequency Is-
The simulation starts out in steady state at t = 0s. Therefore, the FAST method tracks
envelope waveforms at the beginning of the simulation. As shown in Figure 15.20, initially
the shift frequency is is equal to the carrier frequency of 60Hz and the time step size T
equals 2ms. As the permanent three-phase fault occurs at t1 = 0.4s at the center of one of
the lines interconnecting Bus 8 and Bus 9, the simulation parameters are reset tois = 0Hz
and T = 50 f.LS, and natural waveforms are now being tracked since electromagnetic
transients are expected. The simulation results in Figure 15.17 show a drop of the voltage
of load LD9, which therefore consumes less power. More energy is therefore absorbed by
the generation units' rotating shafts, which show speed-ups in Figure 15.18. At t = 0.5s, the
Is (Hz)
60
AC voltage disturbance,
AC steady state
X
0 *---+-----+----+---+---.
Figure 15.20. Obtained two-dimensional set-
Relay action, 2 r (ms) tings of shift frequency fs and time step size T in
fault study of transients in two-area system.
925

27. 926 COMPUTER SIMULATION OF SCALE-BRIDGING TRANSIENTS IN POWER SYSTEMS
breakers CBl and CB2 open to take out the faulted line. The tracking of instant natural
waveforms continues. By t2 = 0.57 s the electromagnetic transients due to the actions of the
breakers have sufficiently damped out to resume envelope tracking at is = 60 Hz and
" = 2 ms. However, the interconnection between both areas is now weakened due to the loss
of one of the tie lines. This triggers inter-area oscillations of the power Pinter as visible in
Figure 1 5. 1 9. The oscillation can also be noticed from Figure 1 5. 1 8. The angular speeds of
generators Gl and G2 within area 1 are in phase, and so are the angular speeds of
generators G3 and G4 within Area 2. As in inter-area oscillations, the generators in area 1
swing in anti-phase to those in Area 2. In FAST, these electromechanical transients are
emulated with envelope tracking at time step size " = 2 ms.
In the official report investigating the August 14, 2003 blackout in the USA and Canada
[ 1 ], it is said that "commonly used protective relays that measure low voltage and high current
cannot distinguish between the currents and voltages seen in a system cascade from those
caused by a fault. This leads to more and more lines and generators being tripped". If for the
same reason the remaining line in service between Bus 8 and Bus 9 is taken out because of its
overload, then an opening of CB3 at t3 = 2.6 s decouples area 1 and area 2 into two islands.
The zero power flow Pinter confirms this. After a brief period of electromagnetic transient and
tracking of natural waveforms following the opening of CB3, envelope tracking resumes at
t4 = 2.67 s at a larger time step as illustrated by Figure 1 5.20. The voltage envelope across
LD9 in Figure 1 5. 1 7 decreases steadily. This points to a lack of reactive power at the load LD9
following the islanding. At t5 = 3.60 s, a quarter of LD9 is shed to avoid further voltage drop.
This leads to the partial blackout in area 2. To simulate the shedding action accurately, the
natural waveforms are captured until t6 = 3.67 s. Then, envelope tracking resumes.
Over the entire simulated scenario, the simulation algorithm of FAST processes
analytic signals and can so adapt to diverse types of waveforms for an efficient simulation
of electromagnetic and electromechanical transients.
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