1
Harmonic Modelling of Thyristor Bridges using a
Simplified Time Domain Method
P. W. Lehn, Senior Member IEEE, and G. Ebner
Abstract— The paper presents time domain methods for harmonic analysis of a 6-pulse thyristor bridge connected to an
ac network. Balanced firing of the converter and a balanced
interface transformer inductance are assumed. For the special
case when the ac network is linear, a closed form solution for the
harmonic injection of the converter is developed. For the more
general case of a nonlinear ac network, a modular converter
model is developed that relies on iterative methods. The converter
model module takes as input the ac voltage harmonics at the point
of common coupling and outputs the corresponding harmonic
current injection into the network. The models are first validated
against time domain simulation results. The results from the
developed models are then compared with those obtained from
approximate analytical techniques which assume zero dc ripple
current. For a system with typical parameter values, it is shown
that the zero ripple assumption may yield acceptable results for
some operating points, but highly erroneous results for others.
Index Terms— Thyristor bridge, HVDC, converter, harmonics,
steady state analysis
I. I NTRODUCTION
Power electronics converters contribute significantly to the
harmonic pollution of distribution and transmission networks.
In particular, converters that switch at line frequency, such
as thyristor bridges, inject large harmonic currents into the
network. The amplitude and phase of these harmonic current
injections may be significantly influenced by the presence of
resonance conditions on either the ac or dc side of the converter. Accurate prediction of the harmonic current injection
of a converter into the ac network must therefore take into
account dynamics of both the ac and dc side networks.
Harmonic interactions in thyristor bridges may be calculated
either in the frequency domain [1], [2] or in the time domain
[3], [4], [5]. The focus of this paper will be on the development
of simplified, computationally efficient time domain methods.
In order to solve for the steady state of a thyrsitor bridge
Grötzbach employed a state space formulation of the converter
[6]. For a simplified ac system, he analytically determined
initial conditions of the states that would lead to a steady
state solution. Based on this foundation he later determined the
influence of ac and dc side reactances on the ac side current
harmonics [3].
For the special case of a linear, balanced ac network Herold
and Weindl demonstrated the existence of a completely analytical solution for the steady state of the thyristor bridge [4].
Pivotal to this analytical solution was the need to model the
P.W. Lehn is with the Dept. of Electrical and Computer Engineering,
University of Toronto, Toronto, Canada (e-mail: lehn@ecf.utoronto.ca). G.
Ebner is with the Institute of Electrical Power Systems, University of
Erlangen-Nuremberg, Erlangen, Germany, (e-mail: ebner@eev.e-technik.unierlangen.de).
ac system all the way back to an ideal, harmonic free, infinite
bus. After finding the initial condition associated with the
steady state, a 1-period time domain simulation was employed,
followed by a Fourier Analysis, to determine converter current
harmonics.
Perkins [5] employed a more general iterative method,
similar to the one proposed by Dobson [7] for diode circuits,
that allows inclusion of arbitrary linear ac networks. Based
again on time domain concepts, iteration is employed to find
(i) the commutation angle of the converter and (ii) the initial
condition associated with the steady state. Based on the steady
state initial condition a 1-period time domain simulation could
be employed, followed by a Fourier Analysis, to determine
converter current harmonics.
All the above works employ ideal switching devices (zero
or infinite impedance) to avoid singularity and/or convergence
problems associated with high/low impedance switch models.
This leads to a converter representation with 2 state equations
during the commutation interval and only one state equation during the non-commutation interval (see commutation
and non-commutation circuit diagrams of Fig. 3). This time
varying dimension of the system significantly complicates the
analysis.
In this paper it is shown that an assumption of ideal
switching devices does not preclude the development of a
fixed dimension formulation of the thyristor bridge. It is then
demonstrated that this fixed dimension model of the converter
may be easily augmented to the state matrices of an arbitrary
linear time invariant, balanced ac network.
For analysis of ac networks that contain multiple converters,
a modular converter model is also developed. While solution
of the modular model requires Newton iteration, convergence
is extremely fast since: (i) dimension of the system Jacobian
is 1x1, (ii) an accurate initial estimate for the solution is
available, (iii) the equation being iterated is nearly linear in
the vicinity of the solution.
II. C ONVERTER WITH I DEAL S OURCE
Steady state analysis of a 6-pulse or 12-pulse converter is
most easily accomplished if:
• converter interface inductance and gating are balanced
• the ac network parameters are balanced
• the ac network is linear
• the infinite bus is balanced and free of harmonics.
Under these conditions a complete analytic model of the
unregulated converter may be developed. Key to this development is the observation that the state transition matrices of
the system depend only on the commutation interval length,
2
YD
YE
YF
YD
YE
YF
LD 5
/
LE
7
7
/G
5G
YGF
7
D
LD 5
/
LE
7
/G
7
5G
YGF
E
Fig. 3. The equivalent circuits of the converter: (a) during commutation and
(b) after commutation.
Fig. 1. Flow chart for finding the steady state solution of a thyristor bridge.
YD
YE
YF
LD
5
/
7
7
LE
/G
5G
LF
7
Fig. 2.
7
LGF
7
7
YGF
The simplified schematic diagram a thyristor bridge.
β [8]. Consequently, taking β as an input variable allows a
solution algorithm to proceed, as per Fig. 1, that outputs the
associated firing angle, α, and the associated initial condition
x(0) that leads to a steady state solution.
The following subsections detail the solution algorithm. For
simplicity, the ac and dc networks are first replaced by ideal
voltage sources, as shown in Fig. 2. It will later be shown
how this model may be extended to consider more general ac
network configurations.
A. State Transition Map
Under balanced operation, the 6-pulse converter displays 6th
period symmetry that may be exploited to simplify harmonic
analysis. Assuming continuous conduction, the two circuits
shown in Fig. 3 characterize the behavior of the converter.
The circuit of Fig. 3(a) holds during the commutation interval,
while the circuit of Fig. 3(b) holds for the remainder of the
6th period.
In general, the state transition map will depend on the
choice of state variables. In this work, a new state assignment
is selected to simplify analysis. States are selected as phase
currents ia and ib subject to the constraint that ia = 0 for the
entire interval after commutation.
The proposed selection of state is in contrast to the common
approach associating two state variables with the circuit of Fig.
3(a), and only one state variable with the circuit of Fig. 3(b).
By maintaining the same number of state variables during
commutation and non-commutation intervals, the proposed
approach avoids the need for ”projection” and ”injection”
matrices (as used in [5], [7]), or the need for partial matrix
inverses (as used in [4]), thereby simplifying implementation.
During commutation the state model of the system is given
by:
¸
¸
¸
·
·
·
d ia
ia
va
L1
= R1
+ B1
+ D1 vdc
(1)
ib
vb
dt ib
where
·
¸
· ¸
−2L − Ld −L − Ld
1
L1 =
D1 =
−L − Ld −2L − Ld
1
·
¸
·
¸
2R + Rd R + Rd
−2 −1
R1 =
B1 =
R + Rd 2R + Rd
−1 −2
After commutation phase a becomes open circuited and
ia = 0 for the remainder of the sixth period. To avoid
elimination of state variable ia we employ an “auxiliary
differential equation” of the form:
dia
= 0 t ² [β, π/3].
dt
Given that ia (β) = 0 by definition, this simple state model
yields the desired solution:
ia (t) = 0
t ² [β, π/3].
The state equations after commutation are therefore given
by:
·
¸
·
¸
·
¸
d ia
ia
va
L2
= R2
+ B2
+ D2 vdc , (2)
ib
vb
dt ib
3
where
·
L2 =
R2 =
1
0
·
0
−2L − Ld
0
0
0 2R + Rd
¸
·
B. Periodicity Constraint on State Trajectories
¸
0
D2 =
1
·
¸
0
0
B2 =
.
−1 −2
¸
Assuming the converter connection allows no zero sequence
currents to flow, the αβ frame equations of the system during
and after commutation are obtained by transformation of (1)
and (2):
·
·
·
¸
¸
¸
d iα
iα
vα
−1
−1
−1
= CL−1
R
C
+
CL
B
C
j
j
j
j
iβ
vβ
dt iβ
+ CL−1
j Dj vdc
(3)
where setting subscript j = 1 gives the equation during commutation and j = 2 gives the equations after the commutation.
The matrix C is a 2 × 2 variant of the Clarke Transform, as
given in the Appendix.
£
¤T
The ideal source voltage vector vα vβ
is represented
by a set of ideal oscillator equations, while the dc source
is represented by the equation dzdc /dt = 0 as per [9].
This allows formation of an equivalent autonomous system
equation:




x
x
d 
zac  = Aj  zac 
(4)
dt
zdc
zdc
with


−1
−1
CL−1
CL−1
CL−1
j Rj C
j Bj C
j Dj

Aj = 
0
Ωac
0
0
0
Ωdc
·
Ωac =
0
1
−1
0
¸
Ωdc = 0
and where the frequency of the ac source has been normalized.
Amplitude and phase information of the ac voltage
source, as well as amplitude information of the dc voltage
source is contained only in the initial conditions z(0) =
£
¤T
zac (0) zdc (0)
.
Over a sixth of a period the state transitions may be found
according to:




x(π/3)
x(0)
 zac (π/3)  = Φ  zac (0) 
(5)
zdc (π/3)
zdc (0)
where Φ is the state transition matrix of the system over a
sixth of a period given by:
Φ = eA2 (π/3−β) eA1 β .
(6)
It is important to note that the above state transition matrix
is only a function of the commutation angle β and does not
explicitly depend on the the firing angle of the converter.
For a time domain formulation, the conditions that must
be satisfied in the steady state are [10]: (i) period excitation,
(ii) periodicity of the state trajectories (iii) periodicity of the
switching events. Periodicity of the excitation is assumed
for any harmonic analysis and will not be discussed further.
Periodicity of state variables will be addressed in this section,
while periodicity of the switching times will be addressed in
the subsequent section.
Periodicity of the state trajectories requires that x(t+2π) =
x(t). For a balanced system this constraint may be converted
to an equivalent constraint on the state trajectories over a sixth
of a period [10]. Over a sixth of a period ac current and source
space vectors undergo a rotation of π/3 radians:
x(π/3) = Θac x(0)
where Θac is the π/3 rotation matrix:
·
¸
cos π/3 − sin π/3
Θac =
.
sin π/3 cos π/3
(7)
(8)
Dc quantities repeat ever sixth period, thus their state
rotation matrix is simply the unity matrix:
£ ¤
Θdc = 1 .
(9)
To apply the state periodicity constraint (7) to the state
trajectory equation (5), the state transition matrix (6) is first
evaluated for the specified commutation angle. It has the
form1 :

  p


x(π/3)
A
Np
Np
x(0)
ac
dc
 zac (π/3)  =  0 Ωp
0   zac (0)  .
ac
zdc (π/3)
0
0
Ωp
zdc (0)
dc
(10)
Applying the state periodicity constraint to (10) yields a
constraints on the initial conditions associated with the steady
state solution:
·
¸
£ p
zac (0)
p ¤
p −1
Nac Ndc
x(0) = (Θac − A )
. (11)
zdc (0)
Contrary to the analysis of VSC circuits [9], in analysis of
the thyristor bridge (11) provides only one of two constraints
necessary to solve for the steady state. A second constraint
must be imposed to ensure periodicity of the switching times.
C. Periodicity Constraint on Switching Times
Analysis leading up to (11) is based on the assumption that
the commutation interval length β is known a priori. In fact,
the commutation interval length is equal to β if and only if the
current ia in Fig. 3(a) reaches zero precisely at time t = β.
Mapping this constraint into the αβ-frame yields:
iα (β) = 0.
(12)
It is assumed that a unique firing angle α is associated with
each commutation interval length β. Thus we may adjust the
phase of the ac voltage vector until constraint (12) is satisfied.
1 It
p
may be easily proven that Ωp
ac = Θac and that Ωdc = Θdc .
4
Solution proceeds as follows. First x(β) is expressed in a from
similar to (10) by evaluating the state transition matrix eA1 β :

  β


A
Nβac Nβdc
x(β)
x(0)
 zac (β)  =  0 Ωβac
0   zac (0)  . (13)
zdc (β)
zdc (0)
0
0
Ωβdc
Introducing the constraint (11) and solving for x(β) yields:
·
¸
z (0)
x(β) = F ac
(14)
zdc (0)
and output of the network equations gives the ac input voltage
needed by the converter model of (3):
·
¸
vα
= yac .
(21)
vβ
This approach allows the influence of network parameters
on the operation of the converter to be determined under
balanced operation.
where
β
£
p −1
F = A (Θac −A )
Np
ac
Np
dc
¤ £ β
¤
+ Nac Nβdc . (15)
Both the dc and ac voltage information as well as the firing
angle information is contained in the initial condition vector
z(0). Assuming a dc voltage of Vdc , an ac voltage of V and
a firing angle of α, the associated initial condition vector is
given by (16):


·
¸
V cos(α + π/3)
zac (0)
z(0) =
=  V sin(α + π/3)  .
(16)
zdc (0)
Vdc
An expression for iα (β) is extracted from (14):
£
¤
iα (β) = F1,1 F1,2 F1,3 z(0).
(17)
Finally the constraint (12) is applied to determine the firing
angle α.
α = cos−1



q
−F1,3
2 + F2
F1,1
1,2

½
¾
Vdc 
F1,2
π
−1
+ tan
− .
V 
F1,1
3
(18)
In other words, a converter operated at the above calculated
firing angle will settle into a steady state with the stipulated
commutation interval length β. The initial conditions of the
states, x(0), associated with this solution may be determined
by applying initial condition (16) to (11). This yields a fully
analytic solution of the converter.
It is often necessary to study the harmonic interaction of
a converter with its filters and an existing AC network. In
this case, ac network equations must be added to the basic
converter equations. This may be done either by
• augmenting the abc-frame equations of of (1) and (2)
• augmenting the αβ-frame equations of of (3).
Generally, it is easier to employ the latter method, as it leads
to a more modular model of the system.
Network equations are therefore expressed in the form:
= Aac xac + Bac uac
(19)
= Cac xac
(20)
where input excitation uac comes from an ideal oscillator then
is used to represent the infinite bus within the ac network:
uac = zac ,
In the previous section, a set of linear network equations
were simply augmented to the converter model. A major
limitation of this classical approach is that no other switching
circuits may exist within the modeled ac network. Possible
interactions between neighboring converters cannot be studied.
To overcome this limitation, a fully modular harmonic
model of the converter can be developed subject to the
following reduced set of limitations:
•
•
converter interface inductance and gating are balanced
only characteristic harmonics exist in the ac network, i.e.
negative sequence 5, 11, 17, etc. and positive sequence
1, 7, 13, etc.
The model builds directly on the results of Section II.
Ac excitation is now provided not by merely a fundamental
frequency ac source, but by a set of harmonic sources all
summed together. The harmonic oscillator matrix Ωac is now
given by:
Ωac = diag(Ω1 , −5Ω1 , +7Ω1 , −11Ω1 , +13Ω1 , ...) (22)
where
·
Ω1 =
0
1
−1
0
¸
.
The phase angles of the bus voltage harmonics are defined
with respect to the phase angle of the fundamental:
vα + jvβ = V+1 6 0ejt + V−5 6 φ−5 e−j5t + V+7 6 φ+7 ej7t + ..
III. C ONVERTER WITH A RBITRARY AC N ETWORK
dxac
dt
yac
IV. C ONVERTER WITH AC S OURCE D ISTORTION
Assuming the fundamental of the bus voltage to have zero
phase, the necessary initial conditions for the oscillator states
are:


V+1 cos(0)
 V+1 sin(0) 


 V−5 cos(φ−5 ) 



zac (0) = 
(23)
 V−5 sin(φ−5 )  .
 V+7 cos(φ+7 ) 


 V+7 sin(φ+7 ) 
:
The solution algorithm time shifts the ac excitation voltage
to meet the commutation constraint iα (β) = 0. Introducing
harmonics on the bus voltage has one critical implication
on this solution algorithm. Since harmonics have their phase
angles defined relative to the fundamental, time shifting of
the fundamental results in an associated phase shift of all
5
Fig. 4.
Test system for validation of the analytical model.
harmonics. The required initial condition vector therefore has
the form:


V+1 cos(α + π/3)


V+1 sin(α + π/3)


 V−5 cos(φ−5 − 5(α + π/3)) 

·
¸ 
 V−5 sin(φ−5 − 5(α + π/3)) 
zac (0)

z(0) =
=
 V+7 cos(φ+7 + 7(α + π/3))  . (24)
zdc (0)


 V+7 sin(φ+7 + 7(α + π/3)) 




:
Vdc
Matrix F may be evaluated just as in (15), albeit with
β
larger matrix dimensions for Aβ , Np
ac and Nac . Assuming a
total of n ac side harmonics (including the fundamental) (17)
becomes:

T
F1,1
 F1,2 

 ·
¸


:
zac (0)

iα (β) = 0 = 
.
(25)
 F1,2n−1 
zdc (0)


 F1,2n 
F1,2n+1
All terms in the vector zac (0) have trigonometric dependance on firing angle α, hence a closed form solution to
constraint equation (25) does not exist. Instead the firing angle
is solved through iteration.
V. M ODEL VALIDATION
A test system, as depicted in Fig. 4, is employed to validate
the proposed model against time domain simulation results.
Parameters for the test system are given in Table I. A complete
analytical representation of the system is developed, as per
Sections II and III, and converter current harmonic injections
and bus voltage harmonics at the point of common coupling
are validated.
The operating point is specified by the selection of β =
18.00o . The resulting firing angle of the converter is solved
using the analytical model to be 45.98o . This firing angle is
used in the time domain simulation. Table II compares the α
and β values from simulation with those from the analytical
model. Table III compares the resulting voltage harmonics at
the point of common coupling (PCC) and the converter current
harmonics.
The results of Tables II and III clearly validate the accuracy
of the analytical model proposed in Sections II and III.
TABLE I
T EST SYSTEM PARAMETERS .
Quantity
Vs
Rs
Xs
Rl
Xl
XCl
XCf 1
Rf 1
Xf 1
XCf 2
Rf 2
Xf 2
Ac System
Value
10.00 kVln, peak
0.040 Ω
0.400 Ω
0.128 Ω
0.170 Ω
3858 Ω
14.0 Ω
0.0267 Ω
0.400 Ω
40.0 Ω
0.0186 Ω
0.280 Ω
Converter
Quantity
Vdc
Rd
Xd
R
X
System
Value
9.00 kV
0.200 Ω
3.600 Ω
0.100 Ω
1.400 Ω
TABLE II
VALIDATION OF A NALYTICAL M ODEL - A NALYTICALLY OBTAINED
FIRING AND COMMUTATION ANGLES COMPARED WITH SIMULATION
Quantity
β
α
Analysis
18.00o
45.98o
Simulation
18.12o
45.98o
TABLE III
VALIDATION OF THE ANALYTICAL MODEL - ANALYTICALLY OBTAINED
PCC VOLTAGE AND CURRENT HARMONICS COMPARED WITH SIMULATION
Harmonic
Number
h
+1
-5
+7
-11
+13
-17
+19
Analysis
h |
|Vpcc
(Vln, peak )
9223.9
282.28
124.90
53.01
24.72
15.52
20.55
Simulation
h |
|Vpcc
(Vln, peak )
9227.8
282.29
124.79
53.11
24.68
15.39
20.51
Analysis
|I h |
(Apeak )
1621.9
334.99
141.87
72.84
52.05
9.91
10.35
Simulation
|I h |
(Apeak )
1621.6
334.86
141.78
72.71
52.01
9.82
10.34
Next the modular converter model of Section IV is validated. The modular model takes as input voltage harmonics at
the PCC. For a given β value, it outputs the resulting converter
current harmonics. Again we assume β = 18.0o . Table IV
shows the PCC voltage harmonics applied to the converter
and the resulting converter current harmonics obtained from
simulation and from the modular converter model. In steady
state, the firing angle associated with these PCC harmonics
and the specified β value is found to be α = 44.63o .
Excellent agreement is again seen between the results from
time simulation and those obtained from the proposed method.
6
TABLE IV
VALIDATION OF THE MODULAR MODEL FOR AN ARBITRARY SET OF Vpcc
350
I5
HARMONICS
Input
h |
|Vpcc
(Vln, peak )
10000
4000
2000
0
0
0
0
Input
6 Vh
pcc
(degrees)
0
0
0
0
0
0
0
Output
|I h |
(Apeak )
Analysis
2763.6
612.18
207.42
124.70
85.74
22.04
20.00
Output
|I h |
(Apeak )
Simulation
2756.26
611.00
206.29
124.34
85.59
22.00
20.12
300
250
I5, I7 (Amps)
Harmonic
Number
h
+1
-5
+7
-11
+13
-17
+19
with dc ripple
no dc ripple
200
150
I7
100
In contrast to the previous model, no assumption on the
linearity of the ac network is made in the development of the
modular model. Thus the modular model may be interfaced to
an arbitrary network, provided the network is balanced.
VI. C OMPARISON WITH A PPROXIMATE M ODELS
Many classical texts analyze the thyristor bridge based on
an assumption of zero dc ripple current [11]. In other words,
they assume the dc smoothing reactor (Xd in Fig. 4) is
infinitely large. In this section, the accuracy of this zero ripple
assumption is investigated.
The system of Fig. 4 is first simulated with the nominal
smoothing reactance of the test system: Xd = 3.9872Ω. It is
then simulated with a near infinite smoothing reactance. Fig.
5 shows the resulting ac side 5th and 7th harmonic currents
as a function of the commutation angle β.
As may be seen from Fig. 5, the zero ripple assumption
yields fairly accurate results for some operating points (e.g.
for β > 30o ), however for other operating points (e.g. for
β < 30o ) large errors result. While the zero ripple assumption
sometimes yields sufficiently accurate results, there is no
obvious means of determining whether its results should be
trusted.
VII. C ONCLUSION
A time domain method for harmonic analysis of thyristor
bridges was presented. The complexity of the time domain
formulation was reduced through the introduction of an auxiliary differential equation. The auxiliary differential equation
makes the equations during commutation equal to the order
of the equations after commutation. This eliminates the need
for injection/projection matrices or the need for non-unique
partial matrix inverses, significantly simplifying the solution.
The proposed method leads to a fully analytic solution of the
converter equations provided the ac system is linear, balanced
and contains no other harmonic sources. A more general modular converter model is also developed. The modular model
may be interfaced to ac networks containing nonlinearities, and
other harmonic sources, however, the solution of the modular
model relies on iterative methods.
The proposed model is employed for a simple demonstrative
study investigating the accuracy of the common “zero dc
ripple” assumption. The study shows that significant errors
50
0
0
10
20
30
BETA (deg)
40
50
60
Fig. 5. Errors resulting from a zero dc current ripple assumption. 5th and
7th ac current harmonics with and without inclusion of dc ripple.
may occur when using simplified analytical models based on
a zero dc ripple assumption.
VIII. ACKNOWLEDGEMENTS
The authors would like to thank Prof. Dr. G. Herold of the
University of Erlangen-Nuremberg for hosting this research
project. Without his support this collaborative work would not
have been possible.
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