IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007
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A Harmonically Coupled Admittance Matrix
Model for AC/DC Converters
Yuanyuan Sun, Guibin Zhang, Wilsun Xu, Fellow, IEEE, and Julio G. Mayordomo, Member, IEEE
Abstract—This paper proposes a new method to model the
harmonic generating characteristics of ac/dc converters. The
model transforms the time domain nonlinear characteristic of the
converter into a frequency domain linear admittance matrix. The
matrix represents the coupling among the converter ac side harmonic voltages and currents accurately and it does not vary with
the harmonic conditions of the system. The proposed model opens
up a new way to conduct harmonic power flow studies. This paper
presents the theoretical foundation and analytical derivation of the
admittance model for both single-phase and three-phase bridge
rectifiers. Potential applications of the model are discussed.
Fig. 1. Single-phase bridge converter-schematic circuit for analysis.
Index Terms—Converter, harmonic analysis, harmonic model,
harmonics.
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I. INTRODUCTION
HE voltage and current waveforms in power systems
are often distorted by harmonics due to the presence of
harmonic-producing loads such as power electronic converters.
Over the past 20 plus years, various models have been proposed
to represent the harmonic sources and to support network-wide
harmonic analysis.
The predetermined harmonic current sources are the most
commonly used model for harmonic source representation [1].
The magnitudes and phases of the sources can be calculated
from the typical harmonic current spectrum of the harmonicproducing device. The model cannot provide adequate results
when the voltage distortions at the harmonic source buses are
high. In order to improve its accuracy, harmonic-voltage-dependent current source models have been proposed, which leads
to the well-known “iterative harmonic analysis” (IHA) method
[2]–[5]. To determine harmonic currents generated by a device,
the current source model is first determined from the device analytical equations or time domain simulation based on estimated
terminal harmonic voltages. The results are then used to compute bus harmonic voltages. The voltages are in turn used to
calculate more accurate harmonic current sources. This method
has gained industry acceptance. The IHA can be viewed as a
Gaussian iteration method to the harmonic power flow problem.
T
The Newton iteration method has also been proposed [6], [7]. In
order to calculate the Jacobian elements for the Newton method,
this approach generally requires knowing the analytical equations representing the harmonic generating characteristics of the
device. The resulting solution techniques can become very complicated.
In recent years, some researchers have shown that the relationship between the ac side harmonic voltages and currents
of a rectifier can be further clarified through a transformation
matrix [8] or through an admittance relationship [9]. The significance of these findings is that they reveal a new direction
to establish elegant harmonic models for power electronic devices analytically. Building on the work of [8] and [9], we have
managed to derive such a model for the bridge rectifier devices.
The result is that the time-domain nonlinear characteristic of
a converter is transformed into a frequency-domain linear admittance matrix with little approximation. Furthermore, it was
found that elements of the matrix are independent of the supply
system harmonic voltages, i.e., the matrix is constant as long as
the converter firing angle and commutation angle do not vary.
The model therefore opens up a new way to conduct harmonic
power flow studies.
Manuscript received January 29, 2007; revised April 23, 2007. This work was
supported by the China Scholarship Council Program under Grant [2006]3037,
the China Cheung Kong Scholars Program, and the Natural Sciences and Engineering Council of Canada. Paper no. TPWRS-00043-2007.
Y. Sun is with the school of Electrical Engineering, Shandong University,
Jinan, China (e-mail: sunyy@mail.sdu.edu.cn).
G. Zhang is with GP Technologies, Inc., Edmonton, AB T6G 2V4, Canada.
W. Xu is with the University of Alberta, Edmonton, AB T6G 2V4, Canada
(e-mail: wxu@ualberta.ca).
J. G. Mayordomo is with the Department of Electrical Engineering, Universidad Politécnica of Madrid, Madrid, Spain.
Digital Object Identifier 10.1109/TPWRS.2007.907514
II. MODEL DEVELOPMENT FOR SINGLE-PHASE
BRIDGE CONVERTER
Since the proposed model is entirely analytical, the derivation
of the model equations can be quite complicated. The model
for single-phase converter shown in Fig. 1 is developed first to
highlight the main steps.
A. Switching Function Modeling of the Rectifying Process
It is well known that the operating principle of an ideal converter without commutation process can be understood as the
process of modulating the ac side voltage with a switching function to obtain the dc link voltage [10]–[12], as follows:
0885-8950/$25.00 © 2007 IEEE
(1)
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007
is the rms value of th harmonic voltage and
where
is its phase angle, is the highest harmonic order available or
of interest.
Based on the modulation theory, the dc side voltage can be
with
in frequency dodetermined by multiplying
. For simplicity,
main. The result is a Fourier expression of
the result is shown in the following symbolic form:
(6)
where
is the dc component of the converter dc voltage,
is the rms value of th harmonic voltage and
is its phase
angle. Analytical expressions for these variables are available
but are omitted here to save space.
of the converter can be determined from
The dc current
the above voltage and the dc side impedance, in frequency domain, as follows:
Fig. 2. Modulation process of dc side voltage for single-phase converter.
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Fig. 2 shows this process where the dc voltage can be determined by multiplying the ac side voltage with the square wave. If the converter firing angle is
form switching function
, the switching function shifts right by degree, but the modulation process is still valid. As a result, a general expression,
assuming no commutation, for the voltage switching (or modulating) function is
(2)
(7)
where
is the th harmonic component of
. Note that
is zero.
(7) is based on assumption that the dc motor voltage
If
is not zero,
can also be derived but its form is slightly
more complex.
The ac current can be derived by the multiplication of Fourier
series of dc current and the current switching function as follows:
Equation (2) shows that thyristors 1 and 4 conduct between the
, whereas thyristors 2 and 3 are on
time interval of and
and
.
between
Similarly, the ac current can be determined by modulating the
as follows:
dc current with the current switching function
(8)
(3)
C. Analytical Expression for the Converter AC Current
. Since the analytical
For the ideal converter,
expression of the switching function is known, we can find its
frequency-domain expression analytically. The result is shown
in (4)
(4)
where
frequency of the system.
, and
is the fundamental angular
B. Analytical Determination of the Converter AC Current
generally contains harThe converter ac side voltage
monics. The general expression of this voltage can be written as
(5)
Equations (5)–(8) can be manipulated analytically. The final
result that relates the converter ac harmonic current to its terminal voltage has the following form:
SUN et al.: A HARMONICALLY COUPLED ADMITTANCE MATRIX MODEL FOR AC/DC CONVERTERS
3
,
The formulas to determine each of the elements in
and
are shown in (12)–(15) at the bottom of the page,
,
,
where
, and
D. Characteristics of the Admittance Matrices
(9)
A compact form for the above equation can be written as
(10)
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is the ac current phasor matrix,
is the ac voltage
where
is its conjugate matrix. When the dc
phasor matrix and
is taken into consideration, a new indepenvoltage source
dent vector matrix needs to be added to the above equation. The
result is
Despite its complexity in derivation, the final converter model
(11) can be constructed easily based on formulas for the matrix elements. Equation (11) is the harmonically coupled matrix model for the single-phase converter. It can be seen that
there is no approximation in the equations. The model is, therefore, an exact frequency-domain representation of the nonlinear
single-phase bridge converter. It effectively transforms a nonlinear element into harmonically-coupled admittance element.
It is interesting to note that this model has two distinct features
in comparison with common linear elements. The first one is
that it couples all harmonic voltages and current components.
Each harmonic voltage will contribute to the generation of harmonic currents at different frequencies. The second one is that
the current is not only a function of the supply voltage; it is also
a function of the voltage conjugate.
The most important characteristic of the admittance matrix
model, however, is its constant nature with respect to the supply
harmonic voltages: the elements of the matrix are independent
of the harmonic voltages applied to the device. This implies that
the model represents a linear component in the frequency domain. Because of the linearity, the matrices can be computed
(11)
(12)
(13)
(14)
(15)
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007
before the network harmonic condition is known. As will be discussed in Section VI, this feature leads to a new and elegant way
to conduct harmonic power flow analysis.
III. MODEL DEVELOPMENT FOR THREE-PHASE
BRIDGE CONVERTER
Fig. 3. Three-phase bridge converter-schematic circuit for analysis.
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The model deduction process for three-phase bridge converter is similar to that of the single-phase bridge converter. For
the component shown in Fig. 3 and assuming a balanced operating condition and no commutation, the switching functions
have the following form:
(16)
The dc voltage can thus be determined as
(17)
where
are the three-phase ac voltages. The equation for dc current is
similar to that of (7). Due to the balanced condition, we only
need to determine the phase-A ac current, which has the following form:
(18)
After extensive mathematical operations, the admittance matrix model for phase A is derived and it has the following form:
(19)
The compact form for (19) can also be written as
(20)
It can be seen that there is no triple order harmonics in the model.
If the dc voltage source is included, the model has the following
form:
SUN et al.: A HARMONICALLY COUPLED ADMITTANCE MATRIX MODEL FOR AC/DC CONVERTERS
The formulas to determine each of the elements in
are given in (22)–(25)
,
5
and
Fig. 4. Actual switching function for single-phase and three-phase converters.
(22)
when
conditions such as commutation. A complete model will be developed after the following analysis.
and
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A. Phase Angle of Fundamental Frequency Voltage
(23)
(24)
when
and
and
and
and
; when
; when
; when
(25)
and
when
where
,
,
,
, and
are integers.
It can be seen that the matrix model for three-phase converter
has the same characteristics as that for the single-phase converter. As a result, conclusions drawn for the single-phase converter model are also applicable to the three-phase converter.
IV. MODEL ENHANCEMENT
In this section, the proposed admittance matrix model will be
extended to take into account more realistic converter operating
If the formulas for the
matrix elements are examined in
, the
detail, one can find that the elements are functions of
phase angle of the fundamental frequency voltage. This pheessentially represents the shift
nomenon is expected since
degrees.
of the entire ac voltage and current waveforms by
From this perspective, the model is not strictly linear. However,
this complication can be overcome due to the following consideration: the fundamental frequency phase angles of network
bus voltages are mainly if not exclusively determined by the fundamental frequency power flow conditions of the system. It is
theoretically possible but practically rare that the harmonic currents produced by a converter are strong enough to change the
voltage phase angles at the fundamental frequency. As a result,
if we know the fundamental frequency load flow results,
angles for the converter buses become known quantities. The
matrices can be computed according to the formulas.
The problem then becomes how to compute the fundamental
frequency power flow when a network contains converter loads.
The simplest approach is to treat the converters as constant
power loads, which results in a conventional power flow solution process. A more sophisticated approach is discussed in
Section VI.
B. Modeling of the Thyristor Commutation Process
In reality, the change of the thyristor current from one steady
state value to another can not happen instantaneously due to the
existence of an inductance in the supply side of the converter.
The converter current is commutated from one phase to another.
Because of the commutation process, the switching functions
for the voltage and current modulation are no longer rectangular
waves and the function for voltage is different from that for the
current. Fig. 4 shows the actual switching functions needed to
take into account the commutation process accurately for the
single-phase and three-phase converter respectively, where in
the figure is the commutation angle for the thyristors.
It can be seen that the voltage switching function is close
to rectangular waveform. However, there are intervals of zero
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007
It is worth pointing out that the proposed current switching
function is an improvement over that used in [11]. Simulation
results have shown that the current waveforms will experience
large errors without this improvement.
The commutation reactance also affects the dc side current during the thyristor conducting period. The total circuit
impedance experienced by the current becomes
(28)
Fig. 5. Adopted switching function for single-phase and three-phase converters.
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values. The current switching function is no longer a rectangular
wave. The method proposed in this paper to deal with the complex switching functions is to approximate them with more sophisticated but manageable waveforms [10]–[12]. In this work,
we proposed to model the voltage switching function as a rectangular function shown in Fig. 5, which is similar with the
switching function used in [11] and [12].
As a result, the Fourier expression for the switching function
is the same as before, but the Fourier coefficients are changed
to the following:
is the commutation inductance. A multiplier of 2 is
where
used since the ac current passes through phase A commutation
inductance first, goes to the dc side, and returns through the
is the value
phase B (or C) commutation inductance.
that should be used to construct the matrix.
In summary, the commutation process has been included
through the modified Fourier coefficients of the switching
. The deducfunctions and augmented circuit impedance
tion process of the
matrix model becomes the same as that
matrix
presented earlier. As a result, the formulas for the
and their elements are the same. The only difference is that
the elements will be functions of the commutation angel and
inductance
as well.
According to [13], the commutation angles can be computed
approximately using the fundamental frequency parameters as
follows:
(29a)
(26a)
(26b)
(26c)
where
,
,
, and
is the Fourier coefficient
for single-phase and three-phase, respectively. If
, the
above equation becomes original switching functions (4) and
(16).
The switching function for the current modulation is more
complicated. The ac current during commutation is a function of
system impedance, firing angle, and the dc current. To simplify
the analysis and without introducing significant errors, a trapezoidal function is proposed to approximate the actual switching
function as illustrated in Fig. 5. As a result, the Fourier expressions of the switching functions in (4) and (16) also remain the
same. The only changes are the Fourier coefficients, which have
the following forms:
(27a)
(29b)
is the ac side inducwhere is the thyristor firing angle,
is the dc current mean value, and
is the ac side
tance,
phase voltage.
V. CASE STUDY RESULTS
The proposed model has been compared with the time domain simulation results. Waveform comparison is more desirable than harmonic spectrum comparison because the former
ensures that both the harmonic magnitudes and phase angles are
verified. The converter is supplied by harmonic voltage sources.
In the single-phase converter, only the odd order harmonics are
considered since the even order harmonics in the supply voltage
source are usually of much smaller amplitude. For the threephase converter, no triple order harmonic is presented in the
supply side. A lot of simulation work has been done and the following are some of the representative results. The system data
is shown in Table I.
A. No Commutation Process
(27b)
(27c)
Simulation results are shown in Figs. 6 and 7 for single-phase
converter. The waveforms of converter dc voltage, dc current
and ac current from the matrix method are compared with the
SUN et al.: A HARMONICALLY COUPLED ADMITTANCE MATRIX MODEL FOR AC/DC CONVERTERS
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TABLE I
PARAMETERS FOR BRIDGE CONVERTERS
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Fig. 8. Waveforms for three-phase converter switching function.
Fig. 6. Waveforms for V (t) S (t) and V (t) (single-phase converter).
Fig. 9. Waveforms for V (t) I (t) and I (t) (three-phase converter).
B. Impact of Commutation
Sample waveform comparison is shown in Figs. 10 and 11 for
the cases where there is a commutation overlap. As expected, the
waveforms do not match exactly since the matrix model uses
approximate switching functions. To further verify the model
accuracy at different commutation angles, an error index is
defined as follows:
Fig. 7. Waveforms for I (t) S (t) and I (t) (single-phase converter).
Matlab Simulink results. The figure shows that the results essentially overlap each other. This is expected since there is no
approximation for the matrix method in this case.
Figs. 8 and 9 show the results for the three-phase converter.
The three-phase switching functions for the ideal converter are
first illustrated in Fig. 8. Under ideal conditions, voltage and current switching function are the same for each phase. The converter dc voltage, dc current along with the ac current of phase
A is shown in Fig. 9. Again, the results have a good agreement.
(30)
where is the harmonic current phasor from Simulation and
is that from the matrix model. This index measures the relative
error between the results from the matrix method and those
from the time domain simulation. Such comparative results are
shown in Table II. It can be seen that the error increases with
the overlap angle but it is small from the practical application
perspective. If it is desirable to reduce the error further, more
sophisticated approximation to the switching functions could be
used.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007
is the converter
damental frequency. They are given data.
fundamental frequency line to line voltage obtained from the
power flow results. If commutation process is considered, angles and are determined by solving the following two nonlinear equations [14]:
(32a)
(32b)
Fig. 10. AC current waveform for single-phase converter ( = 30 ).
where
is the commutating reactance. and can be solved
from the above equations using, for example, the Newton aland
can then be obtained
gorithm. The dc magnitudes
from
(33a)
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(33b)
Accordingly, the matrix models for the converters can be established as follows:
Fig. 11. AC current (Phase A) waveform for three-phase converter ( = 30 ).
TABLE II
RELATIVE ERROR FOR DIFFERENT COMMUTATION ANGLES
(34)
is a known
where is the converter number in the network.
current vector that has included the impact of the fundamental
frequency voltage and the converter dc voltage source as follows:
VI. POTENTIAL APPLICATIONS OF THE
MATRIX MODEL
The proposed admittance model for converters has some significant applications. The most direct one is a new harmonic
power flow method that is noniterative. This method is explained
briefly here.
The method first computes the fundamental frequency power
flow by treating the converters as constant power loads. From
the power flow results, the converter firing angles and dc voltage
sources, assuming no commutation, can be determined using the
following equations [13]:
(35)
The unknowns in (34) are the converter ac side harmonic
voltage and current vectors. The rest of the network is linear and
can be modeled as a harmonically decoupled admittance matrix
as follows:
(36)
(31a)
(31b)
(31c)
where
and
is the power factor of the three-phase converter load
is the specified converter active power at the fun-
where is the harmonic order interested.
Equations (34) and (36) are both linear equations. They can be
solved together without iteration. The results are harmonic voltages at each bus that have included the impact of harmonic currents from all harmonic sources. This technique has the potential
SUN et al.: A HARMONICALLY COUPLED ADMITTANCE MATRIX MODEL FOR AC/DC CONVERTERS
[3] W. Xu, J. E. Drakos, Y. Mansour, and A. Chang, “A three-phase converter model for harmonic analysis of HVDC systems,” IEEE Trans.
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2005.
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to analyze systems with distributed harmonic sources. For such
cases, the impact of background harmonics in the form of diversity and attenuation effects must be considered [15].
The above noniterative solution method relies on some approximations. The main one is that the harmonic voltages will
not affect the converter firing angle (and overlap angle). If one
must model such an effect, an iterative solution for the firing
angle can be implemented. This iterative method has firing angles and overlap angles of the various converters as variables so
it leads to a new iterative harmonic power flow method. This is
the second potential application of the model.
The third application is that the model provides an excellent
platform to analyze the mutual couplings among the various harmonic components. The result will help us to understand the accuracy and characteristics of the harmonic current source model.
It is important to note that the current source model is the foundation for a number of harmonic-related subjects besides assisting harmonic power flow calculations. One example is the
harmonic source detection problem which is formed on the assumption that the harmonic sources can be approximated as harmonic current sources.
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VII. CONCLUSION
This paper has presented a new and elegant model for analyzing the harmonic characteristics of the ac/dc converters. The
model is in the form of harmonically-coupled admittance matrices that relate the ac side harmonic voltages and currents. Despite the complexity of derivation, the final matrix model can be
computed easily according to a set of formulas. The proposed
model has the following useful characteristics.
• The model is linear, which leads to the formulation of alternative harmonic power flow solution algorithms.
• The model is analytical. It will enable one to investigate the
harmonic coupling characteristics of the converters much
more thoroughly.
• The model is developed for a wide range of operating conditions. Unlike the linearized model, the proposed model
is operating point independent.
Our subsequent effort is to investigate the new harmonic
power flow algorithms described in Section VI, with the distributed harmonic sources as the main target of application.
Research is also underway to establish a similar model for the
thyristor-controlled reactors.
REFERENCES
[1] “Task force on harmonics modeling and simulation, “Modeling and
simulation of the propagation of harmonics in electric power networks
Part I: Concepts, models and simulation techniques”,” IEEE Trans.
Power Del., vol. 11, no. 1, pp. 452–465, Jan. 1996.
[2] H. W. Dommel, A. Yan, and W. Shi, “Harmonics from transformer
saturation,” IEEE Trans. Power Del., vol. PWRD-1, no. 2, pp. 209–215,
Apr. 1986.
Yuanyuan Sun received the B.Sc. degree in electrical engineering from Shandong University, Jinan, China, in 2003. She is currently pursuing the Ph.D degree in the School of Electrical Engineering, Shandong University.
Her main research interest is harmonics.
Guibin Zhang received the B.Sc. and M.Sc. degrees in electrical engineering
from Shandong University, Jinan, China, in 1995 and 1998, respectively, and
the Ph.D. degree from Zhejiang University, Hangzhou, China, in 2001.
He was a Postdoctoral Fellow with the University of Alberta, Edmonton, AB,
Canada, from 2003 to 2005. He is currently with GP Technology, Inc., Edmonton.
Wilsun Xu (M’90–SM’95–F’05) received the Ph.D. degree from the University
of British Columbia, Vancouver, BC, Canada, in 1989.
He was an Electrical Engineer with BC Hydro from 1990 to 1996. He is
presently a Professor with the University of Alberta, Edmonton, AB, Canada,
and an Adjunct Professor at Shandong University, Jinan, China. His main research interests are harmonics and power quality.
Julio G. Mayordomo (M’97) received the Dipl.-Eng. and Ph.D. degrees from
the Universidad Politécnica of Madrid (UPM), Madrid, Spain, in 1980 and 1986,
respectively.
Currently, he is a Professor at UPM. His main research interest is low frequency disturbances.