2004 35th Annual IEEE Power Electronics Specialists Conference
Aachen, Germany, 2004
Dynamic Modeling and Control of Three Phase Pulse Width Modulated
Power Converters Using Phasors
Giri Venkataramanan
Bingsen Wang
Department of Electrical and Computer Engineering
University of Wisconsin – Madison
1415 Engineering Drive
Madison, WI 53706
Email: bingsenwang@wisc.edu
Abstract—Although the application of power electronic
converters in ac power systems has been mostly limited to
unidirectional loads like motor drives, various evolving
applications such as power quality conditioners and distributed
generation systems feature complex dynamic interactions
affecting the operation of the ac power system. The focus of this
paper is to present systematic technologies for modeling
switching power converters in conjunction with their controls
to determine their dynamic properties and assess their
performance in an ac power network. The paper presents a
dynamic phasor-oriented modeling technique that is readily
compatible with classical power system analysis techniques. A
state space model that represents the dynamic properties of the
system in the magnitude-angle form is developed. The model
can be used for obtaining steady state small signal dynamic
properties at various operating conditions, and hence be used
for design of appropriate regulators. Application of the
technique is illustrated using a current source inverter example.
I.
INTRODUCTION
The application of three phase pulse width modulated
power converters is rapidly growing beyond adjustable
speed ac motor drives to include distributed generation
systems, power quality conditioners, etc. The design of
closed loop regulators and controllers for the power
converters in such applications have heavily drawn upon the
techniques used for ac motor control and dc-dc converters.
basis of recent advances in quasi-stationary phasor dynamic
modeling techniques developed for power system analysis
[7-10]. The specific focus of the paper is to present
systematic techniques for modeling three phase ac power
converters in conjunction with their controls to determine
their dynamic properties and assess their performance. An
example application of the technique for controlling a
current source inverter feeding a three-phase ac load is
presented in the paper.
II.
As an illustrative case for demonstrating the behavior of
power converter dynamics in an ac system, a simple example
consisting of a first order system at the ac port and a static
stiff dc source is being considered. Fig. 1 illustrates the
schematic of the power circuit of a three phase current
source inverter feeding a balanced R-C load.
A nominal application being considered here develops an
ac voltage regulator for the inverter system as illustrated in
Fig 2. The output voltage (Vac) is measured and compared
against a reference value (V*ac) to generate an error. The
error drives a voltage regulator Gv(s) that modifies the
modulation level of the inverter switches appropriately to
regulate the output voltage. In order to design the regulator
Gv(s), it is desirable to obtain the transfer function between
the modulation input and the output voltage.
The use of ac motor control techniques involves the
application of the synchronously rotating D-Q vector
coordinates to model the dynamic phenomena of the three
phase quantities under time varying excitation [1-3]. On the
other hand, the use of dc-dc converter control techniques
involves approximating the time varying excitation to be
‘slow’ enough such that they may be considered stationary,
while ensuring that the resulting controllers have wide
enough bandwidth to faithfully follow the time variations [46].
While both of these approaches provide reasonable
solutions to the control problem, they are not readily
compatible with the steady state modeling and control
techniques that are used to study ac power system dynamics,
namely phasors. The paper presents a phasor-oriented
modeling technique that is readily compatible with classical
power system analysis techniques. It is developed on the
0-7803-8399-0/04/$20.00 ©2004 IEEE.
CASE STUDY EXAMPLE
Idc
Vac
R
C
Fig. 1. Power circuit schematic of current source inverter feeding a current
source inverter with a balanced RC load
The most straightforward technique for approaching this
control problem is to develop a scalar transfer function
between the control input (modulation) and the output
voltage assuming both quantities to be stationary and use it
2822
2004 35th Annual IEEE Power Electronics Specialists Conference
to design a regulator with adequate controller bandwidth
wide enough to faithfully track large signal sinusoidal inputs
[4]. Such an approach also assumes that the three phase
quantities are decoupled from each other, i.e. varying the
modulation level in a particular phase affects the output
voltage only in that phase.
Aachen, Germany, 2004
In order to develop these phasor magnitude regulators, it
is necessary to develop the transfer function between the
magnitude of the modulation phasor input and the magnitude
of the ac voltage phasor output, which is discussed further in
the following section.
80
-
+
Gain (dB)
V*ac
60
40
Gv(s)
20
1
10
1
10
3
100
1 .10
100
1 .10
Idc
0
Phase (degree)
30
Current+ source
MVac
inverter
-
Vac
R
90
C
M Idc
60
120
3
Frequency (Hz)
Fig. 3. Bode plots of control to output transfer function of current source
inverter in the large signal instantaneous domain
Fig. 2. Block diagram of the proposed controller to regulate the output
voltage of the CSI
200
*
The frequency response of the control to output transfer
function (per phase) of the system is illustrated in Fig. 3. The
first order plant model may be represented as
a
50
(1),
0
a
v AC ( s )
1
= I dc R
m( s )
1+ s
100
V * & V (A)
G pL ( s ) =
Va
Va
150
ω pL
−50
−100
where ωpL = 1/RC. The first order plant may be controlled
using a PI regulator with an appropriately placed zero to
cancel the plant pole. The start-up and steady state time
domain response of the system with such a controller is
illustrated in Fig. 4. As may be observed from Fig. 4, the
response features a steady state error in magnitude and phase
angle. One of the classical approaches to overcome this
performance error is the design of controller in the
synchronously rotating reference frame [3].
Alternately, the control problem may be formulated in the
phasor domain, wherein the magnitude of the output voltage
phasor (VACa) is used as the feedback variable, compared
against a corresponding command value to generate a phasor
magnitude error. The controller Gv(s) acting upon the error
generates a reference modulation phasor magnitude (Ma).
The modulation phasor magnitude is then modulated by
three phase unit sinusoidal waveforms at the desired power
frequency ωe, to generate the switching signals for the
inverter. This approach closely follows the point of load
regulation of the magnitude of voltage, common in ac power
generation, transmission and distribution systems.
−150
−200
0
1
2
3
cycle of 60 Hz
4
5
Fig. 4. Computer simulation results of start-up response of the current
source inverter with feedback control implemented in the instantaneous
large signal domain
III.
PHASOR DYNAMIC MODELS FOR CIRCUIT
ELEMENTS
Let the voltage (vG) across a generic two terminal device
with a current (iG) flowing through it be defined as
vG = VGa cos VGθ
iG = I Ga cos I Gθ
(2),
following the sign convention shown in Fig. 5. In general,
the quantities VGa, VGq, IGa, IGq may be functions of time,
thus
accommodating
complex
waveforms,
while
corresponding to sinusoidal waveforms with arbitrary
amplitude and phase as a special case. The superscripts refer
2823
2004 35th Annual IEEE Power Electronics Specialists Conference
magnitude and phase angle respectively. When the
magnitudes are constant and phases are linearly increasing
with time, the waveforms are sinusoidal.
iL
iR
+
+
vL
-
vR
iC
-
iG
+
+
vC
vG
-
-
The instantaneous power absorbed by the generic device
may be determined to be the product of voltage and current
as
VGa I Ga
[cos(VGθ − I Gθ )(1 + cos 2 I Gθ ) − sin(VGθ − I Gθ ) sin 2 I Gθ ]
2
I aV a
= G G [cos( I Gθ − VGθ )(1 + cos 2VGθ ) − sin( I Gθ − VGθ ) sin 2VGθ ]
2
(3)
If the generic device is chosen to be a capacitor (C), the
energy stored in the capacitor may be expressed by
eC =
1
1
2
CvC = C VCa cosVCθ
2
2
[
]
2
(4).
The instantaneous power absorbed by the capacitor may be
obtained by determining the time derivative of (4) as,

deC
CVCa  dVCa
dV θ
= pC =
(1 + cos2VCθ ) − VCa C sin2VCθ  (5)

dt
2  dt
dt

Comparing (3) and (5) and substituting the generic
component’s subscript (G) with that of a capacitor (C), we
obtain,
dVCa 1 a
= I C cos(I Cθ − VCθ )
dt
C
dVCθ
1 a
I C sin(I Cθ − VCθ )
=
dt
CVCa
VRa = RI Ra ;
VRθ = RI θR
(8)
I CA = ωCVCA ;
I CΘ − VCΘ = π / 2
(9)
VLA = ωLI LA ;
VLΘ − I LΘ = π / 2
(10)
VRA = RI RA ;
VRΘ − I RΘ = 0
(11)
The uppercase superscripts refer is used to designate steady
state quantities. It may be noted that (9-11) are the familiar
classical steady state phasor solutions for these elements. In
essence, they represent the polar coordinate equivalent of the
more common D-Q model in the synchronously rotating
reference frame.
In general, while developing classical sinusoidal steady
state phasor solutions for ac circuits, the sinusoidal voltages
and currents are considered to be the projections of rotating
vectors of the constant magnitude and constant angular
velocity, and the steady state conditions expressed by (9-11)
are developed directly, without formulating the dynamic
phasor forms (6–8) Therefore, relationships (6-8) may be
considered to be extensions of the classical phasor solution
(9-11) to dynamic operating conditions, valid also when the
magnitude and phase functions of the voltage and current
have not yet reached their steady state. Furthermore, in the
case of balanced three phase systems, translations of the
solution by ±2π/3 in phase readily provide the transient
solution for all the three phases simultaneously.
IV.
(6)
The pair of equalities in (6) represents the magnitude and
phase dynamics of the capacitor voltage related to the
magnitude and phase of the capacitor current. They may be
construed as generalized phasor dynamic model for time
varying excitation of a capacitor. In a similar manner,
generalized phasor dynamic model for the inductor may be
derived as
dI La 1 a
= VL cos(VLθ − I θL )
dt
L
θ
dI L
1
= a VLa sin(VLθ − I θL )
dt
LI L
Similarly, an algebraic relationship between the voltage and
current for a resistor in the phasor form may be expressed as
For a steady state sinusoidal excitation, the solutions to
(6-7) may be readily deduced by setting the time derivative
amplitude terms to zero and those of phase by the angular
frequency of excitation ω. Thus, we may obtain
Fig. 5.
Sign convention of voltages and currents across various two
terminal devices.
pG =
Aachen, Germany, 2004
CURRENT SOURCE INVERTER MODEL
The ac dynamic models for the circuit elements in the
phasor domain can be coupled with switching power
converter models using averaged switching function models.
An ideal switch equivalent circuit schematic of the switching
circuit of the current source inverter is illustrated in Fig. 6.
The inverter consists of two single-pole-triple-throw (SP3T)
switches with a stiff dc current connected to the pole
terminals and three phase stiff voltages connected across the
throws. The switching functions that determine the converter
behavior may be defined as
1
H ij (t) = 
0
if t ij is closed
otherwise
(7)
for j = 1, 2 and i = 1, 2, 3
2824
(12).
2004 35th Annual IEEE Power Electronics Specialists Conference
Aachen, Germany, 2004
Idc
t11
Idc
t31
t21
+
MV ac
-
t22
t12
V ac
t32
R
M Idc
C
Fig. 6. Ideal switch equivalent circuit of the current source inverter
Furthermore, if the switch throws are operating at repetitive
switching frequency considerably higher than the pulse
width modulation frequency, the ith phase output current Ii(t)
can be expressed as Idc mi(t), where
t
mi (t ) =
1
[H i1 (τ ) − H i 2 (τ )]dτ , for i =1, 2, 3
T t −∫T
(13)
Through appropriate pulse width modulation strategy,
mi(t) are chosen such that the currents injected into the ac
network form a balanced three phase set. In this case, the
average output currents and the average dc voltage may be
expressed as


θ
 cos(ωt + M ) 

2π 
I o = I dc M a  cos(ωt + M θ −
)
3 

cos(ωt + M θ + 2π ) 
3 

3 a a
θ
Vdc = M Vac cos(M θ − I ac
)
2
Fig. 7 illustrates the equivalent circuit of the three phase
switching inverter using modulation-dependent voltage and
current sources to represent the power transfer mechanism
between the ac and dc ports. The dynamic dependence
between the ac voltages, dc current and the modulation
quantities in the phasor domain may be expressed as
follows:
(15)
θ
dVAC
1
θ
=
I DC M a sin(Mθ − VAC
) − ωe
a
dt
C VAC
[
d v

dt v
A
AC
Θ
AC
 1
−
  RC
=


ω
 − Ae
 V AC
G ps ( s) =
OVERALL SYSTEM MODEL
a
dVAC
Va 
1
θ
= I DC M a cos(Mθ − VAC
) − AC 
dt
C
R 
(16)
Once the steady state operating point has been
determined, a linearized small-signal state space model for
(15) can be determined at the steady state operating point
given by (16) as
(14),
where M and M are the modulation amplitude and
modulation angle, and Vaca and Vacq are the ac voltage
amplitude and phase angle, respectively.
V.
A
VAC
Θ
= I DC M A cos(M Θ − VAC
)
R
A
Θ
ω e CVAC
= I DC M A sin(M ΘAC − VAC
)

A
ω eV AC

v
1  v

−
RC 
A
AC
Θ
AC
A
 V AC


A
 M RC  A
 +  ω [m ]

 
 MA 


(17)
Using the small signal state space model (17), the small
signal transfer function from the input mA to the output vACA
may be readily determined as
q
a
Fig. 7. Phasor equivalent circuit model of the CSI feeding an RC load
]
where ωe is the ac excitation frequency. The steady state
relationships among quantities in (15) may be determined by
setting the time derivatives to be zero (with the uppercase
superscripts denoting steady state operating points)
A
v AC
( s)
= ( sI − A) −1 B
A
1
m ( s)
(18),
where the state space matrices are defined from (17) by
A
 V AC

 1
A 
−
V
ω
 A

e AC 
 RC
 M RC  .
A= ω
and
B
=

1
 ω 
− Ae −

 MA 
RC
V
 AC



The transfer function GpS(s) may be evaluated as
 sω pL

1 +
(ω pL + jω e )(ω pL − jω e ) (19)
G pS ( s) = G pL ( jω e )



s
s
1 + ω pL + jω e  1 + ω pL − jω e 



The discrepancies between the transfer function of
instantaneous large signal quantities in (1) and the transfer
function of the small signal phasor quantities in (19) may be
observed as follows:
a. The ‘dc gain’ of GpL(s) has been replaced by |GpL(jωe)|.
b. The real pole at ωpL has been separated into two conjugate
complex poles with imaginary parts given by jωe.
2825
2004 35th Annual IEEE Power Electronics Specialists Conference
Aachen, Germany, 2004
c. A real zero has been introduced, which is given by the
product of the newly introduced complex conjugate poles
divided by the original real pole.
It is further observed that if the characteristic dynamic
frequencies of the network ωpL are considerably higher than
the excitation frequency ωe, the transfer functions of the
instantaneous large signal model and the small signal phasor
model are indistinguishable. In other words, when the
imaginary part of the separated pair of complex conjugate
poles is much smaller than the real part, one of them
essentially appears to be ‘canceled’ by the newly introduced
real zero. However, as the characteristic dynamic
frequencies of the network are comparable or slower than
the ac excitation frequency, the instantaneous large signal
dynamic model and the small signal phasor dynamic model
are considerably different from each other.
Bode plots of the frequency response of the control to
output small signal transfer function, GpS(s) for a nominal
value R is shown in Fig. 8 (solid red curves). The large
signal transfer function of the system GpL(s) is also shown in
the figure (dashed blue curves) in order to highlight the
variation due to ac excitation.
In order to verify the rather preplexing small signal
dynamic phasor transfer function as predicted by the
analytical model described in (19), a laboratory scale
experimental system was built. The small signal transfer
function was measured using a frequency response analyzer
and the results from the operating condition corresponding
to R= 10 kW ; C= 1 mF; and ωe = 257.6 rad/s, are illustrated
in Fig. 9. The excellent agreement between the predicted
model and the experimental measurement is readily evident
from the figure.
80
Fig. 9. Experimental measurement of small signal control (current phasor
magnitude) to output (voltage phasor magnitude) transfer function
obtained using frequency response analyzer.
Furthermore, in order to examine the effect of the
perturbations in the dynamic behavior, the loop gain a closed
loop system with a simple PI regulator operating on phasor
quantities were developed. The loop gain plots ate nominal
(red solid curve), heavy (dashed blue) and light (dash-dot
brown) load conditions are shown in Fig. 10.
80
60
40
Gain (dB)
Gain (dB)
60
20
0
1
10
100
40
20
0
3
1 .10
20
0
1 .10
3
1 .10
10
100
1 .10
1
10
100
Frequency (Hz)
1 .10
4
0
30
30
60
Phase (degree)
Phase (degree)
3
1
90
120
150
60
90
120
150
1
10
100
3
1 .10
180
Frequency (Hz)
Small signal
Large signal
210
Fig. 8. Bode plot of small signal control (modulation phasor magnitude) to
output (voltage phasor magnitude) transfer function of the current source
inverter (for C=1 mF, R= 10 kW)
4
Nominal Load
Heavy load
Light load
Fig. 10. Bode plot of small signal loop gain transfer function of the current
source inverter with a PI controller at various loading conditions
2826
2004 35th Annual IEEE Power Electronics Specialists Conference
It may be observed from the loop gain plots, that the
closed loop system would be unstable at light load
conditions because of a negative value for phase margin.
The results from the time domain computer simulation of the
system are illustrated in Fig. 11. The top plots show the
output voltage waveforms, and the bottom plots show the
averaged ac current waveforms. The step change in load
from nominal value to light load is applied at the end of
three cycles. The instability in the output upon the step
change to a light load is clear from the figure, as predicted
by the small signal phasor dynamic transfer function.
150
va
vb
vc
100
Va,b,c (V)
50
0
Aachen, Germany, 2004
‘system identifier’ operating in real-time to determine the
load level using measured quantities.
On the other hand, an acceptable controller may be
designed based only on the knowledge of range of load
levels. If the equivalent load resistance at the heaviest and
the lightest anticipated load are known, the controller design
may be based on a nominal load level that may be arithmetic
or a geometric average of the upper and lower bounds on the
load, with adequate margins.
Fig. 12 illustrates the frequency response of loop gain of a
regulator designed to provide perfect cancellation at the
nominal load level (solid red curves). The variation of the
curves at heavy load (dashed blue curve) and light load (dotdashed brown curve) are also shown.
−50
40
−100
20
0
1
2
3
4
5
6
7
8
Gain (dB)
−150
8
i
a
ib
ic
6
Ia,b,c (A)
4
0
20
2
40
0
−2
3
1 .10
3
1 .10
1
10
100
1 .10
1
10
100
Frequency (Hz)
1 .10
4
90
−4
45
0
1
2
3
4
t (cycle of 60 Hz)
5
6
7
8
Phase (degree)
−6
Fig. 11. Time domain response of system with a simple PI controller
operating in the phasor domain
It has thus clear that the large signal transfer function of
the system cannot be used as-is for designing closed loop
regulators operating on phasor domain quantities,
particularly when the system has dynamic behavior that is
‘slower’ then the ac excitation frequency of the phasors.
Having definitively established the validity of the control
to output transfer function in the phasor domain through
laboratory experiments as well as computer simulations, a
suitable controller that accounts for the discrepancies in the
phasor domain transfer function may be developed.
A candidate control transfer function Gv(s) may be chosen
to be of the structure

G v ( s) =
1+
ω b 
s

ωˆ pL + jω e  1 +
s

ωˆ pL − jω e  (20)
s  sωˆ pL

1 +
(ωˆ pL + jω e )(ωˆ pL − jω e ) 
This structure provides pole-zero cancellation for the
plant, while featuring a integral loop gain transfer function,
when the plant poles and zeros of the small signal phasor
domain transfer functions are cancelled exactly by the
controller zeros and poles respectively. This approach
requires prior knowledge of the location of real pole of the
system ωpL, indicated by ‘^’ in the controller transfer
function.
Such a regulator will have to adapt to changing load
conditions, since ωpL depends on the load. It will consist of a
0
45
90
135
180
4
Nominal Load
Heavy load
Light load
Fig. 12. Bode plot of small signal loop gain transfer function of the current
source inverter with the proposed controller at various loading conditions
The operation of the using the proposed controller was
verified using computer simulations of the complete system
over the entire range of load and transient conditions.
Although the design of the regulator is based on the small
signal phasor transfer functions, the response of the system
to large signal disturbances was found to be satisfactory in
the simulations.
Fig. 13 provides selected time domain waveforms from
the simulations. Fig. 13 (a) shows the traces of three phase
load current waveforms and Fig. 13 (b) shows the traces of
three phase output voltage waveforms. The set of waveforms
on the top illustrates the system response during start up.
The set of waveforms in the middle illustrates the system
response during a load step change from nominal load to
light load. The set of waveforms on the bottom illustrates the
system response during a load step change from light load to
heavy load. All the traces indicate a fast transient response
of the output voltage, while maintaining adequate stability,
when compared to the case illustrated in Fig. 11.
2827
2004 35th Annual IEEE Power Electronics Specialists Conference
loads.
2
i
a
i
b
i
abc
I , , (A)
1
c
0
−1
−2
0
1
2
3
4
5
6
abc
I , , (A)
2
i
a
ib
i
1
c
0
−1
−2
0
1
2
3
4
5
6
abc
I , , (A)
20
ia
i
b
i
10
c
0
−10
−20
0
1
2
3
t (cycle of 60 Hz)
4
5
6
(a)
Va,b,c (V)
va
vb
vc
0
The behavior of the system dynamics in the small signal
phasor domain have been compared and contrasted with the
large signal instantaneous domain. The small signal phasor
domain transfer function has been analytically predicted and
subsequently verified using frequency response tests on a
laboratory prototype. The pitfalls of using the large signal
instantaneous domain transfer functions in the phasor
domain has been illustrating using frequency response
transfer functions at varying load conditions, and verified
using computer simulations.
A controller design approach based on pole-zero
cancellation at nominal operating conditions has been
proposed. Stable operation of the proposed controller has
verified using computer simulations over a range of
operating conditions.
The modeling approach can be conveniently used for
controller design and system stability analysis while
interfacing power converters with ac power networks.
200
100
Aachen, Germany, 2004
−100
−200
ACKNOWLEDGMENT
0
1
2
3
4
5
6
Va,b,c (V)
200
v
a
vb
vc
100
0
−100
−200
0
1
2
3
4
5
6
Va,b,c (V)
200
v
a
v
b
vc
100
0
The authors would like to acknowledge support from
Wisconsin Electric Machine and Power Electronics
Consortium (WEMPEC) at the University of WisconsinMadison. The authors also thank Paul Van Opens for
valuable assistance in developing the laboratory scale
hardware for the small signal model frequency response
tests.
REFERENCES
−100
−200
0
1
2
3
t (cycle of 60 Hz)
4
5
6
(b)
Fig. 13. Time domain response of system of the system with the proposed
regulator: (a) load current response; (b) load voltage response (from top to
bottom: startup, switching to light load, and switching to heavy load)
VI.
CONCLUSIONS
The application of dynamic phasor-oriented modeling
technique that is readily compatible with classical power
system analysis techniques in the design of controllers for
pulse width modulated power converters has been presented
in this paper. The technique represents a refinement of the
classical scalar approach to controller design for three phase
ac power converters. Furthermore, it is an extension of the
D-Q synchronously rotating reference frame controller
approach to the phasor modeling approach common in
practice of ac power systems. The analytical modeling
approach has been presented for a case study current source
converter, whose results have been verified using
experimental and computer simulations. The paper has
presented a detailed development for the dynamic phasor
model for the current source converters with an RC load.
The modeling approach can be systematically extended to
study voltage source inverters and to various other classes of
[1]
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