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Switching time bifurcation in a thyristor
controlled reactor
Article in IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications · April 1996
DOI: 10.1109/81.486445 · Source: IEEE Xplore
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-l: FUNDAMENTAL THEORY AND APPLICATIONS,VOL. 43, NO. 3, MARCH 1996
209
Switching Time Bifurcations in
a Thyristor Controlled Reactor
Sasan Jalali, Ian Dobson, Robert H. Lasseter, and Giri Venkataramanan
Abstract-Thyristor
controlled reactors are high power switching circuits used for static VAR control and the emerging technology of flexible ac transmission. The static VAR control circuit
considered in the paper is a nonlinear periodically operated
RLC circuit with a sinusoidal source and ideal thyristors with
equidistant firing pulses. This paper describes new instabilities in
the circuit in which thyristor turn off times jump or bifurcate as
a system parameter varies slowly. The new instabilities are called
switching time bifurcations and are fold bifurcations of zeros
of thyristor current. The bifurcation instabilities are explained
and verified by simulation and an experiment. Switching time
bifurcations are special to switching systems and, surprisingly,
are not conventional bifurcations. In particular, switching time
bifurcations cannot be predicted by observing the eigenvalues of
the system Jacobian. We justify these claims by deriving a simple
formula for the Jacobian of the Poincare map of the circuit and
presenting theoretical and numerical evidence that conventional
bifurcations do not occur.
Fig. I.
Single phase static VAR system.
Since the mid-1970’s, thyristor controlled reactors have
been used at the loaded ends of transmission lines to control
the reactive power supplied from a fixed shunt capacitor so that
voltage can be maintained when system loads or transmission
line configurations change [ 111, [20]. More recently, thyristor
I. INTRODUCTION
controlled reactors have been used as one of the economical
E study the stability of a high power switching circuit
alternatives for the emerging technology of flexible ac transwith a thyristor controlled reactor. This nonlinear cir- mission (FACTS) [2]. The expected benefits of flexible ac
vv
cuit can exhibit novel bifurcation instabilities and our main transmission include increased and controllable power flows
objective is to explain and verify the new instabilities using on transmission lines and’the enhancement of power system
simulation and experiment. A state space approach is used to stability.
derive a simple formula for the Jacobian of the Poincare map
Despitethe significance of high power circuits with thyristor
of the circuit and confirm the nature of the new instabilities.
controlled reactors, little nonlinear theory has been developed
A thyristor controlled reactor is a fixed inductor in series for their analysis. The standard approach is to replace the
with two oppositely poled thyristors as shown in the right thyristor controlled reactor by an average inductor model
most branch of Fig. 1. C.ontrolling the firing (switch on) of and then apply linear techniques to the resulting circuit [l 11,
the thyristors controls the thyristor conduction time and hence [20]. While this average inductor approximation is sometimes
the proportion of time for which the inductor is included effective for predicting steady state behavior, it fails to capture
in the circuit. One approximation which is often used to much of the circuit nonlinearity and it breaks down when large
understand the thyristor controlled reactor in a periodic steady harmonic distortions occur. Operating conditions with large
state is that it acts as a continuously variable inductor. The harmonic distortions are documented in [l], [12], [27], [13].
value of the inductance depends on the thyristor conduction While detailed time domain simulation is valuable in analyzing
time. The thyristor controlled reactor is often combined with nonlinear effects in thyristor controlled reactor circuits, there is
a fixed capacitor in parallel so that varying the thyristor
also a need to develop mathematical concepts and approaches
conduction time varies the effective impedance of the parallel so that the simulated or actual nonlinear phenomena may be
combination.
understood and predicted.
This paper describes new’ instabilities of a thyristor conManuscriptreceived June 5, 1993. This work was supported in part by
trolled
reactor circuit called switching time bifurcations in
the Electric Power Research Institute under Contracts RP4000-29, RP8010which thyristor switch off times jump or bifurcate as a
30, RP8050-03, RP8050-04, WO8050-03 and by the NSF Presidential Young
Investigator Grant ECS-9 I57 192. This paper was recommended by Associate
parameter is slowly varied. The thyristor controlled reactor
Editor M. Ilic.
circuit and its classical operation are described in Sections
S. Jalali is with Siemens Energy and Automation, Atlanta, GA 30202 USA.
II and III. The large harmonic distortions asfociated with
I. Dobson and R. H. Lasseter are with the Department of Electrical and
Computer Engineering, University of Wisconsin, Madison, WI 53706 USA.
switching time bifurcations are briefly summarized in Section
G. Venkataramanan is with the Department of Electrical Engineering,
IV.
Sections V and VI explain that switching time bifurcations
Montana State University. Bozeman, MT 59707 USA.
are fold bifurcations of the zeros of the thyristor current. SecPublisher Item Identifier S 1057.7 122(96)01665-O.
1057-7 I22/96$05.00 0 1996 IEEE
210
IEEETRANSACTlONSONClRCUlTS ANDSYSTEMS--I:FUNDAMENTALTHEORYANDAPPLICATIONS,VOL.43,NO.3,MARCH
tion VII presents simulations of a thyristor controlled reactor
circuit for static VAR control which shows switching time
bifurcations in detail: The circuit was developed by Bohmann
and Lasseter to study harmonic interactions and distortions
[l]. This simulation evidence for switching time bifurcations
is followed in Section VIII by experimental work .showing
switching time bifurcations on a single phase equivalent of
the static VAR compensator installed near Rimouski, Quebec
cw.
The occurrence of switching time bifurcations raises expectations that they should.be related to well known generically
occurring’ bifurcations. It is interesting that this is not the
case and that switching time bifurcations appear to be a
novel mechanism for instability. In particular, the occurrence
of the switching time bifurcations cannot be predicted from
eigenvalues of the system Jacobian. The remainder of the
paper is devoted to explaining how the special properties of
the thyristor controlled reactor circuit precludes conventional
bifurcations. Sections. IX, X, and XI compute the. Poincare
map [IO], [25] of the circuit and the Jacobian of the Poincare
map. The formula for this Jacobian is simplified considerably
by the special properties of the thyristor turn off and is the
same formula that would be obtained for fixed turn off times.
Section XII computes the eigenvalues of the Jacobian to show
that conventional bifurcations do not occur and explains how
the Jacobian simplification ensures that switching time bifurcations are not predicted by the eigenvalues of the Jacobian.
This paper differs from the initial conference paper [14] by
including experimental results and reworked theory.
We briefly review other approaches to bifurcations in
switching circuits. Switching circuits with high switching rates
and ideal switches are well approximated using averaging
methods [19], [17], [23] and the stability of the averaged
system can be investigated using bifurcation theory [21], [24].
The high power switching circuits addressed in this paper
have switching rates comparable to the 60 Hz frequency of the
voltage sources and averaging methods have not been shown
to be applicable. There are also simple low power switching
circuit models that exhibit bifurcations and chaos when the
nonlinear junction capacitance of a diode is modeled [4], [3].
In deriving the Jacobian of the Poincare map, we use
a state space analysis of switching circuits which overlaps
with contributions of other authors. The fundamental work
of Louis [18] computes PoincarC maps for switching circuits
including controls. The varying dimensions of the state vector
and switching conditions are discussed and formulas for the
propagation of first order deviations through switchings are
stated. The formulas show that Louis had used the Jacobian
simplification which is highlighted in Section X. Louis computes as an example the Jacobian of the Poincare map of an
acfdc convertor with a current regulator. Verghese et al. [26]
give a general approach to computing Poincare maps and their
Jacobians for switching circuits. Circuit controls and symmetries and the automation of the computations are discussed but
the Jacobian simplification and the varying dimension of the
state vector are not treated. The linearized dynamics of a series
resonant convertor are computed and studied. Grotzbach and
Lutz [9] computed Jacobians of Poincare maps of switched
Fig. 2.
1996
Classical operation of thyristor controlled reactor.
circuits including control actions. They develop Newton algorithms for computing steady state solutions and compute
eigenvalues for ac/dc convertors with controls. The extension
to nonlinear circuits and the derivation of averaged circuit
models are discussed. Dobson [6] derives the Jacobian of a
general switching circuit with RLC elements and ideal diodes
or thyristors with an emphasis on the Jacobian simplification.
The work in this paper has stimulated further work on the
thyristor controlled reactor circuit such as its novel transient
dynamics [22] and analyzing damping and resonance phenomena [5]. The effects of thyristor firing synchronization schemes
on the Jacobian are computed and presented in [ 161, [ 151.
II. SYSTEM DESCRIPTION
Fig. 1 shows a single phase static VAR compensator consisting of a thyristor controlled reactor and a parallel capacitor.
The controlled reactor is modeled as a series combination of an
inductor L, and resistance R,. The static VAR compensator is
connected to an infinite bus behind a power system impedance
of an inductance L, and a resistance R, in series.
The switching element of the thyristor controlled reactor
consists of two oppositely poled thyristors which conduct
on alternate half cycles of the supply frequency. A thyristor
conducts current only in the forward direction, can block
voltage in both directions, turns on when a firing pulse is
provided and turns off when the thyristor current, becomes
zero. (The phenomenon of thyristor misfire [22] is not addressed here.) The thyristors are assumed ideal so that detailed
nonlinearities in the turn on/off of the thyristors are neglected.
The dependence of the thyristor switch off times on the system
state causes circuit nonlinearity. The firing pulses are supplied
periodically and the system is controlled by varying the phase
4 (delay) of the firing pulses. The system is studied with the
phase as an open loop control parameter. In practice a closed
loop control would modify the firing phase.
III. CLASSICAL
ANALYSIS
The classical, idealized operation of a thyristor controlled
reactor is explained in Fig. 2. In this figure, the gray line
represents the thyristor controlled reactor voltage Vc(t) (cf.
Fig. 1) and the solid line represents the thyristor current.
If the thyristors are fired at the point where the voltage
Vc(t) is at a peak, full conduction results. The circuit then
operates as if the thyristors were shorted out, resulting in a
current which lags the voltage by 90’. If the firing is delayed
from the peak voltage, the current becomes discontinuous with
a reduced fundamental component of reactive current and a
reduced thyristor conduction time 0. As the phase angle 4
ranges between 90’ and 180” the thyristor conduction time 0
JALALI et al.: SWITCHING TIME BIFURCATIONS
(b)
(b)
I
Fig. 3.
time
Appearance of a new thyristor current zero. (a) 4 < 4*. (b) 4 =. d*
time
ranges between 180’ and 0”. The classical analysis assumes
that the voltage Vc(t) is a pure sinusoid.
IV. HARMONIC DISTORTION
The classical analysis is often applicable, but can, as demonstrated here and in [l] and [ 121, fail for certain circuit
parameters and operating conditions. Under these conditions,
both the voltage and the current waveforms become greatly
distorted with large harmonic components. This distortion is
associated with a resonance phenomenon in which the natural
frequencies of the circuit, from when the reactor is fully on
to when it is fully off, span an odd harmonic [l], [5]. This
harmonic distortion can lead to instabilities as switching times
suddenly change or bifurcate as follows. It is useful to recall
that fold bifurcations (also called saddle node bifurcations) can
either create or annihilate pairs of zeros of functions or vector
fields [lo], [25].
Fig. 4.
Disappearance of a thyristor current zero. (a) 4 <’ v.
(b) I$ = 4’.
Cc)4 > d-.
zero disappears. As the phase delay of the firing pulses slowly
increases, the fold in the dotted line rises until, passing through
the critical phase 4*, the current zero disappears and a later
zero of the thyristor current applies (see Fig. 4(b) and (c)).
The switching off time of the thyristor has suddenly increased
in a switching time bifurcation and stability has suddenly been
lost. Note how the zero of the actual thyristor current coalesces
with a zero of the virtual current indicated by the dotted line
and disappears in a fold bifurcation. As soon as the switching
time bifurcates and system stability is lost, a transient starts.
VII. SIMULATION
RESULTS
The switching time bifurcations are illustrated with the
single phase static VAR circuit of Fig. 1 described in [l]. On
a 1 MVA, 1 kV base, the source impedance has a per-unit
V. INSTABILITY WHEN A NEW
magnitude
of 7.35% with an angle of 89”, the reactor is a
THYRISTOR CURRENT ZERO APPEARS
series combination of 62.6% inductive reactance and a 3.13%
Fig. 3 describes one way in which the system can lose resistance. The capacitor bank has a capacitive reactance
stability. Suppose that harmonic distortion produces a fold or of 177%. Alternatively,, the per unit component values can
dip in the thyristor current as shown in Fig. 3(a). As the phase be specified as L,$ =0.195 mH, R, =0.9 mR, L, =1.66
delay 4 of the firing pulses slowly increases, the fold lowers mH, R, =31.3 mR, and C =1.5 mF. The firing pulses are
until, passing through the critical phase 4*, a new, earlier zero equidistant and the phase 4 is the relative phase of the firing
of the thyristor current is produced by a fold bifurcation of pulses with respect to the stiff ac source u(t) = sin wt shown
the thyristor current as shown in Fig. 3(b). (The new zero in Fig. 1. 0 denotes the conduction time of the thyristors.
of the thyristor current in Fig. 3(b) is in fact a double zero.)
In this example, the system impedance and the capacitor
The thyristor switching off time has suddenly decreased and bank have a natural frequency which is 4.9 times the fundathe stable operation of the system at the previous periodic mental frequency 60 Hz. If the controlled reactor is included
orbit has been lost. We call this qualitative change a switching in the circuit, the natural frequency shifts to 5.18 times the
time bifurcation. As soon as the switching time bifurcates, a fundamental frequency. Thus the natural frequencies of the
transient starts.
circuit from when the reactor is fully on to when it is fully
off span the fifth harmonic. This crossing of an odd harmonic
VI. INSTABILITY WHEN A THYRISTOR
indicates that system odd harmonics can be large and that
CURRENT ZERO DISAPPEARS
voltage and current wave forms can be significantly distorted
Fig. 4 explains another type of switching time bifurcation
Ul, [51.
Fig. 5 shows how the conduction time 0 of periodic orbits
in which the system loses stability as a thyristor current zero
varies as the phase 4 is varied. To simplify the calculations,
disappears. Fig. 4(a) shows a periodic solution for the thyristor
only periodic orbits which are half wave symmetric are comcurrent with the solid line. The gray line shows the thyristor
current that would occur if the thyristor did not switch off puted, using (16) and (17) from Section XI. As can be seen,
for negative current. This part of the current is referred to two separate sets of periodic solutions are computed. One set
starts at 0 = 180” and ends at 0 z 91”. The other starts at
here as “virtual”. The virtual current does not occur in circuit
operation but it is useful in understanding how the current 0 = 0” and ends at 0 M 60”. The classical model predicts
212
IEEE TRANSACTIONSON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS,VOL.43.NO.3,MARCH
h
ti
I..““.“..‘.‘.‘.“1
.
1996
Periodic orbit 1, 0 = 102 Deg.
t....“.““‘..‘.‘..‘....j
3
.
l . 1”
.
..
dl
5
r
45 l ~..*.~~*b****.~*~~...................~~
0..
.
20
Fig. 5.
60
100
.
.
.
.
140
Q (Degrees)
Conduction time d versus firing delay ~5.
only one set of solutions starting from (T = 0” and ending at
u = 180”.
In order to investigate how the system loses stability when
stable periodic orbits disappear, the ElectroMagnetic Transient
Program (EMTP) was used [7], [8]. Periodic orbits 1 through
6 in Fig. 5 were chosen to study in detail how harmonic distortion can cause periodic solutions to disappear in switching
time bifurcations.
The loss of a stable periodic solution at u near 91” is a
switching time bifurcation in which a new thyristor current
zero appears. The EMTP simulation in Fig. 6(a)-(c) shows
periodic solutions of the thyristor current at periodic orbits 1,
2, and 3 respectively. Note that as we move toward periodic
orbit 3, the harmonic distortion produces a fold in the thyristor
current. Fig. 6(c) shows that as the phase delay of the firing
pulses is slightly increased, the fold in the thyristor current
lowers and a new, earlier zero of the thyristor current is
produced and a transient starts.
The loss of a stable periodic solution at (T near 60” is a
switching time bifurcation in which a thyristor current zero
disappears. Fig. 7 shows how the periodic solutions behave
as the switching time bifurcation is approached. The plots in
Fig. ‘7 are on an expanded time scale so as to closely observe
the behavior of the thyristors as they turn off. The dotted lines
show the virtual current which would have occurred if the
thyristors did not turn off as the current decreased through
zero. As the phase is decreased, the periodic orbit progresses
through periodic orbits 4, 5, and 6 and the zeros of the virtual
and the actual current approach each other, coalesce, and
disappear. The stable periodic solution disappears when the
actual current zero disappears and the switch off time of the
thyristor suddenly increases. The initial portion of the transient
which occurs when the stable periodic solution disappears is
shown in Fig. 8. The system will eventually converge slowly
to another stable periodic solution.
Even though many systems enforce equidistant firing in
steady state operation, the firing pulses may or may not
I..
.,..
I :.
0.66
T;‘
?
9 0.5
‘;
2i
k 0.3
:,
0.67
Periodic
. .
I..
0.68
orbit
!a?1
0.69
2, 0 = 96 Deg,
a
Pi 0.1
E
-0.1
-0.3
-0.5
I . . ..I....I....I...
0.96
0.97
‘I’...1 (b)
0.98
0.99
Q
?
a
: 0.6
5
::
B 02.
Periodic
orbit
1
1
33
g
-0.2
I
-0.6
t
t
1
.
.
.
.
.
.
..I.........
1.275
*.*.......
1.3
I
.
.
.
.
.
. !"?i
1.325
time (seconds)
Fig. 6. A new thyristor current zero appears. (a) periodic orbit I. (b) Periodic
orbit 2. (c) Periodic orbit 3 up to 1.3 s. Q is increased by 2’ at 1.3 s.
be equidistant during transients. Therefore, the detail of the
transient depends heavily on the assumptions used in modeling
the control of the thyristor firing pulses. The intent of the
simulation results is to show the.existence of the transient as a
consequence of the switching time bifurcation rather than the
detail of the transient. However, the computations of steady
state periodic orbits as in Fig. 5 are valid for any firing control
scheme that enforces equidistant firings in steady state.
213
JALALI e, nl.: SWITCHING TIME BIFURCATIONS
230 kV
I
l~MVA
230124 kV
h
‘Is
‘yr^A
x
=ll%
t
431 PF
Fig. 9.
Rimouski static VAR compensator.
The actual and the-virtual
current zeros
. . . . , . . . . , . . . , , . . I . . .
0.133
0.134
0.135
0.136
-1.0
TABLE I
RIMOUSKI AND EXPERIMENTAL CIRCUIT COMPONENTS
Data
1%I. ( KL N )
Power
Fig. 7. Thyristor current on an expanded time scale for periodic orbits 4,
5, and 6.
Zbvse
ecrease
?
?
,a4
5
ar”
H
u
H
2 Degrees
-
A& (1.07 p.u.1
St,,(O.288 p.u.1
St (0.11 p.u.)
z&(0.016 p.u.)
Rimouski(34)
230 (132.8) kV
100 MVA
529 (1
566 Cl
152.3 (2
58.2 0
8.46 II
Experiment( 14)
115.0 v
301.4 VA
43.88 i2
46.95 R
12.64 61
4.83 61
0.70 cl
2
0
b
1 zo-
1..
.
I .
0.3
.
.
.
0.33
.
.
..I..
.
t
\L
1
. I
0.36
time (seconds)
Fig. 8. Two current zeros coalesce and disappear. Periodic orbit 6 up to 0.3
s. C$is decreased by 2’ at 0.3 s.
VIII.
100
EXPERIMENTAL
RESULTS
This section gives experimental results showing switching
time bifurcations in the single phase equivalent of the static
VAR compensator installed near Rimouski, Quebec [20, ch. 61.
This compensator is a three phase delta connected thyristor
controlled reactor and ungrounded wye-connected capacitor
banks interfaced to a 230 kV system through a step down
transformer as shown in Fig. 9.
This experiment used a 115 V, 60 Hz ac line which was
assumed to be a stiff and harmonic free source. The circuit
components were scaled as shown in Table I. The reactance
to resistance ratio was measured as ~20 for the inductors and
~70 for the capacitor. The control circuit used a zero voltage
detector synchronized directly across the ac line and a firing
pulse generator to build a train of equally spaced firing pulses.
120
140
160
# (Degrees)
Fig. 10. 6 versus 0.
The firing pulses are delayed by a time delay generator and
transmitted to the back-to-back thyristor modules through a
pair of fiber optics. The thyristors used were Westinghouse
model 707408 rated 1000 V and 200 A.
Fig. 10 shows how the conduction time 0 varies as the phase
d, is varied. The solid line is computed using (16) and (17)
from Section XI and the triangles show the experimental measurements. The solution at point A- is lost when the thyristor
firing is slightly increased and the system state converges to
a new steady state at point A+. This instability occurs when
the thyristor conduction time is. suddenly decreased due to
the appearance of a new earlier current zero. The steady-state
214
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS,VOL. 43, NO. 3, MARCH 1996
solution at point B- is lost when the thyristor firing phase
is slightly decreased upon which the system converges to a
new steady state at point B+. This instability occurs when
the thyristor turn off time is suddenly increased due to the
disappearance of the thyristor current zero.
In this example, the switching time bifurcations are associated with the two natural frequencies of the circuit (the
thyristor fully on and fully off) spanning the third harmonic.
For example, when the short circuit MVA of the system is
infinite’(X, = 0), the two natural frequencies do not span the
3rd harmonic and the bifurcation instabilities are absent.
IX.
POINCAR~
MAP
The PoincarC map of the static VAR system can be computed
by integrating the system equations and taking into account a
change in coordinates when the switchings occur. The thyristor
turn on time is controlled by the phase parameter 4 which
specifies the delay of the firing pulse. The thyristor turn off
occurs when the thyristor current decreases through zero. The
thyristors are modeled as short circuits when on and open
*circuits when off.
The system state vector z(t) specifies the thyristor current,
capacitor voltage and the source current:
IT(t)
K(t)
( 1s(t) )
x:(t) =
The system input u(t) is the source voltage function shown
in Fig. 1. u(t) is assumed to be periodic with period T.
During the conducting time of each of the thyristors, the
system dynamics are described by the following set of linear
differential equations
i=Ax+Bu
(1)
where
A=
-R,L,l
-C-l
0
(
L,l
0
-L;l
m$;;l)
andB=
(L;l).
During the off time of each thyristor, the circuit state is
constrained to -lie in the plane 1, = 0 of zero thyristor
current. In this mode, the system state vector y(t) specifies
the capacitor voltage and the source current:
y(t)=(K(t)
Is(t)
>
and the system dynamics are given by the linear system
G = PAPty
+ PBu
ON
(A,B)
0 1 0
where P is the projection matrix P =
0
0 1) .
(
Fig. 11 describes the system dynamics as the system state
evolves over a period T. A thyristor starts conducting at time
~$0. This mode as described by (1) ends when the thyristor
current goes through zero at time 40 + ~1. The nonconducting
mode as described by (2) follows the conducting mode and
continues until the next firing pulse is applied at time ~$1.This
(i;)
I
40 + Cl
40
41
1 (PfifpB)
41 +a
40 +T
i”~ ------+-c------Tze
J
Fig. 11. System dynamics from time &O to do + T
starts a similar on-off cycle which lasts until the next period
starts at time 40 f T.
The state at the switch on time do is denoted either by the
vector y/(40) or by the vector ~($0). These representations of
the state at the switch on time are related by
(3)
X(40) = ptY(do).
Equation (3) expresses the fact that the z representation of the’
state at a switch on is computed from the y representation by
adding a new first component which has value zero.
The state at the switch off time ~$0+ g1 is similarly denoted
either by ~(40 + ~1) or y(& + al) and these are related by
YY(40+ al) = Px(40
(4)
+ 01).
The matrix P in (4) may be thought of as projecting the vector
x onto the plane of zero thyristor current.
Given a time interval [sl, sz], it is convenient to write
f(., ~1,‘s~) for the map which advances the state at s1 to ihe
state at ~2. For example, a Poincark map which advances the
state by one period T starting at the switch on time 40 may
be written f(y(&), 40, $0 + T). (If the time s2 at the end of
the interval is a switching time, then there is ambiguity about
whether f(.,sl, ~2) evaluates to y(s2) or x(s2). We adopt
the convention that f(x(s1),sl,s2)
evaluates to x(s2) and
f(y(sl), sl, ~2) evaluates to y(sz). For example, the PoincarC
map f(~Y(do), 40~40 + T) evaluates to y(& + T).) If the
thyristor is on during all of the time interval [sl, sq], we write
f(z(si), sl, SP) as fon(x(si), ~1, sp) and if the thyristor is off
during [a, 921, we w&e f(y(sl),
~1, ~2) as .fo~(y(sl),
SI, SP).
fon(x(sl),~lr
32) or fod~(sl),sl,s2)
can be computed
by
integrating the corresponding linear system (1) or (2) over the
time interval [sl, ~‘21.
Now we construct in stages an expression for the Poincark
map f(~(do),
40,40 + T) in term of fan and .f,,ff and the
coordinate changes (3) and (4). At the start of the period at time
40, the state is expressed in the y coordinates as y/(40) or, using
in the 2 coordinates. The
(3), is expressed as x(40) = P”y(&)
state x(&,+~l)
is obtained by integrating the on linear system
(1) with initial state Pty(&,) from time 40 to 40 + 01:
x(40
(2)
OFF
1 (PAP’,PB)
+ Cl)
=fcn,(ptY(~o)>
=eAol
40740
(P’y@,,)
+ 01)
+ lo’
CA’Bu(7
+ (Oo)dr).
(5)
The switch off time s,ff = 40 4 o1 is dtitermined by an
equation including P which constrains the thyristor current
to be zero:
0 = (I - PtP)x(s&)
=
(6)
215
JALALI er al.: SWITCHING TIME BIFURCATIONS
The coordinate change (4) at the switch off and integration of
the off linear system (2) yields
Y(h)
= foff(P440
+ m)140
+ Olr41).
(7)
A half cycle map is given by combining (5) and (7):
foff(%n(ptY(40)r
do, 40 + m),40
. [x+2)
+
Df(x(tz),
The Poincare map may now be written by composing two
successive half cycle maps and then neglecting the gory details
of the time arguments:
STABILITY
(8)
(9)
OF PERIODIC SOLUTIONS
When the static VAR circuit is in steady state with a periodic
trajectory of period T, the Poincare map has a corresponding
fixed point. That is,
f(~(40), 40, do + T! = ~(401.
t2r t3)
eAborr-tz)
x(t2)
+ ePAP’(ts-sorr)pA(~
+
tl)y(tl)
+ function( son, tl, t2 ).
Differentiating with respect to y(tl) and keeping in mind that
the thyristor turn on time son is a fixed quantity (recall that
the thyristor firing is equidistant), we obtain
Df(y(tl),
tl, t2) = eA(t2--S,“)ptePAP’(s,,-tI).
(10)
.Let [t2, t3] be a fixed time interval including a thyristor
turn off at time s0a and no other switchings. The map
f(x(t2), t2, t3) advances the state x(t2) to the state y(tj) and
we want to compute the Jacobian Df(x(tZ),
t2, t3) which is
a 2 x 3 matrix.
f(z(tz),tz,t3)
= foff(Pf,,,(x(ta),tP,Soff
1 rsoff,t3)
eAct2 -) Bu(~)h
s
tz
_ ptp)
1
DS,E
(12)
Note that the two terms associated with sOff in the limits of
the two integrals of (11) cancel. The row vector DS,E is the
gradient of s,tf with respect to x(t2). The second term of (12)
may be written as
ePAPt(t3-sorf)PA(I
- PtP)x(s,R)Ds,tf
which vanishes according to the constraint equation (6) so that
we obtain the surprising and simple result
t2, t3) = ePAPt(ts-s,rf)peA(~urr-ttz).
(13)
Result (13) implies that the switch off time may be regarded
as constant when deriving the Jacobian.
The map advancing the state over the combined interval
[tl,
t3]
can be written as the composition
f(y(t1),t1,t3)
= f(f(y(t1),t1,tz),tz,t3)
and the chain rule yields
DfMh),
tl, t3) = Df(x(tz),
t2, kx)Df(dh),
tl; t2)
so that substitution from (10) and (13) gives
Df(y(h),tl,
t3) =
In particular, let tl = son = 40, sol = 40 + 01, t3 = 41 and
Tl = 41 - ~$0to obtain the Jacobian of the half cycle map:
Df(y(qbo),
= fon(ptf,ff(Y(tl),tl,s,,),Son,t2)
= eA(t2-s,,)ptePAPt(s,,,-
(11)
=
Soff
Df(x(tz),
The stability of the periodic orbit can be computed from
the Jacobian of the Poincare map evaluated at the fixed
point [lo], [25], [26]. In particular, the periodic orbit is
exponentially stable if the eigenvalues of the Jacobian lie
inside the unit circle. Since the thyristor turn off time s,,~
and the Poincare map are discontinuous at a switching time
bifurcation [22], [6], we assume when computing the Jacobian
in this section that the system is not exactly at a switching
time bifurcation.
To compute the Jacobian of the Poincare map, we first
compute the Jacobian of maps which advance the state from
the beginning to the end of a time interval containing one
switching. Let [tl, t2] be a fixed time interval including a
thyristor turn on at time s,, and no other switchings. The
map f(y(tl),
tl, t2) advances the state y(tl) to the state x(t2)
and we want to compute the Jacobian Df(y(ti),
ti, t2) which
is a 3 X 2 matrix.
f(Y(tl),tl;t2)
I
t3 ePAP”(t3--r)pBu(T)&.
s sow
ePAPt(t3--s,rr)peA(s,rf-tz)
[
X.
eA(t2-‘)Bu(r)d7
Differentiating with respect to x(h) and keeping in mind that
the thyristor turn off time .sOeis a function of x(ta), we obtain
+ ~lr41).
do, h), 41,40 + T)
f(~l(40), do,40 + T) 7 f(.f(~($o),
= f~ffPf”~P”f~ffPf~~~“(Y~~O)).
+ lIoff
$o, 41) = ePAPL(T1-gl)PeAulPt.
(14)
Applying the chain rule to (8) and using (14) for the Jacobians
of the half cycle maps yields the 2 x 2 Jacobian of the Poincare
map:
Df(y($o), 40,40 + T)
=e PAPt(T*-al)peAuzptePAP1(T~-ul)peAulpt.
(15)
One of the interesting and useful consequences of (15) is that
the stability of the periodic orbit only depends on the state and
the input via g1 and ~72.It is remarkable that (15) is also the
formula that would be obtained for fixed switching times 01
and 02; the varying switching times apparently introduce no
additional complexity in the formula, but the nonlinearity of
the circuit is clear since ~1 and 02 vary as a function of ~($0).
216
IEEE TRANSACTIONSON CIRCUITS AND SYSTEMS-L FUNDAMENTAL THEORY AND APPLICATIONS.VOL.43, NO. 3. MARCH 1996
XI. SIMPLIFICATION
FOR SYMMETRIC
PERIODIC
ORBITS
It is convenient to take advantage of symmetry when the
periodic orbits are half wave symmetric. Half wave symmetry
of a periodic orbit means that the thyristor firing pulses are
sent every half cycle and the orbit is half wave symmetric.
In particular, the two conduction times g1 = c2 = cr are
equal. Moreover, the system states at the half cycle are equal
in magnitude and opposite in sign to the system states at the
beginning of the cycle. That is, if y(&) is the system state at
the beginning of the period, then:
f’(Y(do), $0, do + T/2) = -Y/(40).
(16)
The conduction time g of the conducting thyristor is given by
the constraint equation
0 = (I - PP)x(&
+ LT).
(17)
The fixed points ~($0) corresponding to half wave symmetric
periodic orbits can be computed by solving (16) and (17)
simultaneously. In addition, the Poincark map Jacobian in (15)
simplifies to:
~f(y(40),qb0,~0
+ T) = (epAp’(T/2-n)reA”p’)z.(18)
XII. CONVENTIONALBIFURCATIONS
We review the instabilities expected from conventional
bifurcation theory. The thyristor controlled reactor circuit
becomes a discrete time nonlinear system when analyzed with
the Poincark map. Conventional bifurcation theory (e.g., [lo],
1251) describes several typical ways in which a discrete time
nonlinear system can become unstable as parameters vary.
The system is assumed to be initially operating in a stable
periodic fashion before the bifurcation occurs and the nature
of the bifurcation is determined by where a critical eigenvalue
of the Jacobian of the Poincare map crosses the unit circle.
(The stable periodic orbit disappears in a fold bifurcation if an
eigenvalue crosses the unit circle at 1, becomes modulated
with another frequency or becomes unstable in a Niemark
or secondary Hopf bifurcation if a complex conjugate pair
of eigenvalues crosses the unit circle and period doubles if
an eigenvalue crosses the unit circle at - 1.) Several authors
(e.g., [4], [3], 1211)have investigated instabilities such as Hopf
and period doubling bifurcations in averaged models of fast
switching power electronic circuits.
The Jacobian formula (18) can be used to rule out conventional bifurcations of half wave symmetric periodic orbits
in the thyristor controlled reactor. circuit. The Jacobian in
(18) is a complicated function of the initial state ~(40) and
the input u(t), .but it only depends on ~($0) and u(t) via
the switching time U. Thus to determine stability, it is only
necessary to test the stability of the Jacobian in (IS) as fl
varies over its range of 0 to 180”‘. Numerical computation of
the eigenvalues of the Jacobian in (18) as 0 varies over this
range as shown in Fig. 12 indicates that the absolute values
of the eigenvalues are less than 0.98 for both the simulation
and the experimental examples. This demonstrates that, for the
given component values, the circuits do not lose stability in a
conventional bifurcation. We conclude that the eigenvalues of
Fig, 12. Eigenvalues of Jacobian as c varies.
the Jacobian of the PoincarC map are strictly inside the unit
’ circle as a switching time bifurcation is approached; that is, the
eigenvalues give no warning of the switching time bifurcation.
The points in the eigenvalue locus of Fig. 12 at which an
eigenvalue approaches the unit circle can be predicted and
associated with resonance effects; see [5].
The switching time bifurcation can be detected as a zero
gradient of the thyristor current at the switch off time; what
may be surprising is that this zero gradient has no effect
on the Jacobian. As a thyristor current zero disappears in a
fold bifurcation, the gradient of the thyristor current evaluated
at the current zero tends to zero and one might expect this
to influence the Jacobian. In particular, the gradient of the
thyristor current tending to zero implies that the sensitivity
DSOff of the switch off time with respect to the initial state
becomes infinite. If the Jacobian contained a term including
Ds,tf (cJ: (12)), at least one eigenvalue of Df would leave the
unit circle and a conventional bifurcation would occur before
the switching time bifurcation was encountered. However, this
analysis is wrong because the Jacobian simplification shows
that the term in the Jacobian involving Dso~ vanishes (cf
the vanishing of the second term of (12)). Thus the Jacobian
simplification ensures that the fold bifurcation’ of the thyristor
current is not expressed in the eigenvalues of the Jacobian
and that the switching time bifurcation will occur. See [6] for
a more rigorous explanation. In the case of a switching time
bifurcation in which a new thyristor current zero appears, the
gradient at the thyristor current zero before the bifurcation does
not become zero as the bifurcation occurs and the Jacobian is
unaffected by the bifurcation.
XIII.
CONCLUSION
This paper studies instabilities in a thyristor controlled
reactor circuit used for reactive power control of power systems in which switching times change suddenly, or bifurcate
as a system parameter (phase of the thyristor firing) varies
slowly. The switching time bifurcations are explained and their
mechanisms are illustrated by simulation and experiment. In
JALALI er rd.: SWITCHING TIME BI~RCATIONS
211
[12] S. G. Jalali and R. H. Lasseter, “Harmonic interaction of power systems
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can cause a thyristor switch off time to disappear or a
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new thyristor switch off time to suddenly appear by fold
“Harmonic instabilities in advanced series compensators,” in
[ 131 -,
EPRI FACTS Conf., Boston, MA, May 1992, in Electric Power Research
bifurcations of the thyristor current. The consequence of the
Institute Report .TR-101784.
sudden change in switch off time is that stable periodic
[ 141 S. G. Jalali, I. Dobson, and R. H. Lasseter, “Instabilities due to
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switching time bifurcations are not explained by the analysis
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_ _ S. Pavliasevic, D. Maksimovic, “Using a discrete-time model for largesignal analysis of a current-programmed boost convertor,” in Power
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lation evidence of switching time bifurcations in a single phase [22] R. Rajaraman, I. Dobson, and S. Jalali, “Nonlinear dynamics and
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Sasan JaIali received the B.S., M.S., and Ph.D.
degrees in electrical engineering in 1988, 1990,
and 1993, respectively, from the University of
Wisconsin-Madison.
He joined Siemens Energy and Automation,
Atlanta, GA, in 1993. His interests include nonlinear
effects in utility power electronics and the analysis
and development of flexible ac transmission
systems.
Ian Dobson received the B.A. degree in mathematics from Cambridge, England in 1978 and the
Ph.D. degree in electrical engineering from Cornell
University, NY in 1989.
He worked for five years as a systems analyst
for the British firm EASAMS Ltd.; this included
simulation of switching circuits for plasma physics
power supplies on contract to Culham Laboratory,
UKAEA, England. He is now an associate professor
of electrical engineering with the University of
Wiscnsin-Madison. His current interests are applications of bifurcation theory and nonlinear dynamics, voltage collapse in electric
power systems and utility power electronics.
7.18
IEEE TRANSACTIONSON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS,VOL. 43, NO. 3. MARCH 1996
Robert H. Lasseter received the Ph.D. degree
in physics from the University of Pennsylvania,
Philadelphia, in 1971.
He was a Consultant Engineer with the General
Electric Company until 1980 when he joined the
University of Wisconsin-Madison, His main interest
is the application of power electronics to utility
systems including hardware, methods of analysis
and simulation.
Dr. Lasseter became a Fellow of the IEEE Power
Engineering Society in 1992.
Giri Venkataramanan received the B.E. degree
from the University of Madras, India, in 1986,
the M.S. degree from the California Institute of
Technology, Pasadena in 1987 and the Ph.D. degree
from the University of Wisconsin-Madison in 1992,
all in electrical engineering.
Currently he is an Assistant Professor of electrical
engineeering at Montana State University, Bozeman, His interests .include modeling, design and
control of power conversion systems and the introduction of pragmatism into engineering education.