MODELLING GRID-CONNECTED VOLTAGE SOURCE INVERTER OPERATION
Erika Twining & Donald Grahame Holmes
Power Electronics Group
Department of Electrical and Computer Systems Engineering
Monash University, Clayton
Abstract
This paper presents the first stage of a research program that aims to explore interactions between
multiple power electronic converters connected to weak distribution networks. The paper describes
a simple averaging inverter model which allows converter systems to be rapidly and accurately
simulated. The model has been verified against both a switched inverter simulation and an
experimental system. It has also been used to tune a synchronous frame PI regulator to achieve an
improved response when operating into a distorted AC supply.
1. INTRODUCTION predictive current regulation (PCR) (ie. deadbeat
control) [6]. However, synchronous frame PI current
The electrification of rural and remote areas presents
regulation is still commonly used in many
significant challenges to Australian distribution
applications, as it is effective and relatively simple to
companies. Rural distribution networks are typically
implement. It was therefore deemed useful to
characterised by very low X/R ratios because of the
investigate its limitations, as an example of the use of
long distances involved, and consequently power
average modelling at a practical level.
quality issues such as poor voltage regulation, voltage
In Section 3, the averaging model, referred to here as
dips and harmonic distortion are common in these
an Average Switching Model (ASM), is developed and
networks. With growing demand and increased use of
shown to achieve accurate simulation results whilst
sensitive electronic equipment, the need to address
being significantly faster to execute than a full
these issues has become a priority. Recent
switched model. In Section 4, the effects of supply
developments in power electronic and digital control
distortion on the harmonic performance of the inverter
technologies have seen the design of a range of power
system are investigated through stability analysis
electronic based conditioning equipment, including
techniques and ASM simulations. It is shown that the
FACTS (Flexible AC Transmission Systems) devices
synchronous frame PI controller can be tuned to
(such as STATCOM s, UPFC s etc) and active
achieve improved harmonic response. The influence
interfaces for distributed generation systems (eg. PV,
of this result on AC filter design is explored. Finally,
wind etc.). However, despite their potential to improve
in Section 5, the accuracy of the ASM is verified
the power quality of weak grid environments [1-3],
against experimental results.
there remains a reluctance to incorporate power
electronic plant into distribution systems. This is in
2. SYSTEM DESCRIPTION
part due to unresolved issues relating to their
interaction with the existing distribution network [4].
The grid-connected VSI configuration modelled in
This paper presents the first stage of a research this paper is shown in Figure 1. For the purposes of
program which aims to explore interactions between this initial work, the DC side of the converter system
multiple power electronic converters connected to was connected to a resistive load and so the inverter
weak distribution networks. The paper describes a acts as an active rectifier. However, since the
simple averaging inverter model which allows grid- converter is bi-directional, the developed models can
connected converter systems to be rapidly and be applied to any type of inverter application without
accurately simulated without requiring the complexity loss of generality.
of full switched inverter models. In [5] a similar
2.1 Control Strategy
averaging model was shown to be a convenient tool
As noted in the introduction, a synchronous frame PI
for the evaluation of a system s dynamic performance.
current regulator was chosen to control the inverter.
The inverter system described in this paper is a three-
Synchronous frame controllers operate by
phase grid connected Voltage Source Inverter (VSI)
transforming the three-phase AC currents ia,ib and ic
configuration commonly used in STATCOM devices
in the stationary frame, into the DC components
and distributed generation interfaces. A synchronous
id and iq in the synchronously rotating frame. This
frame PI current regulator was chosen to control the
inverter. There has been some debate in literature
allows the steady-state error that is normally
regarding the performance of this control strategy in
associated with the application of PI control to AC
relation to other strategies such as hysteresis and
quantities [7] to be eliminated, and also provides
analog and digital control functions available in
Simulink, the PSB contains built-in models for power
systems components, such as transmission lines, and
power electronics devices such as inverters. It is
therefore possible to accurately simulate VSI systems
such as the one described above. However, each of
the non-linear switching devices is modelled
explicitly. Therefore very small time steps, and
consequently long simulation times, are required to
accurately represent the VSI operation. Tests have
shown that even with appropriate starting conditions,
Figure 1: Grid-connected VSI times in the order of several minutes are required to
simulate the inverter operation over one or two
independent control of real and reactive power flow.
fundamental cycles. It is clear that such a
The synchronous transformation is:
computationally intensive model would not be feasible
for distribution system applications involving multiple
îÅ‚ 2Ä„ 2Ä„ Å‚Å‚
öÅ‚ öÅ‚
(¸ )
ìÅ‚ ìÅ‚
ïÅ‚cos cosëÅ‚¸ - 3 ÷Å‚ cosëÅ‚¸ + 3 ÷łśłîÅ‚iaÅ‚Å‚
inverters. It was therefore necessary to find an
id
îÅ‚ Å‚Å‚ 2
ïÅ‚ibśł
íÅ‚ Å‚Å‚ íÅ‚ łłśł
(1)
ïÅ‚
=
ïÅ‚iqśł
alternative which would accurately represent the
ïÅ‚
3 öÅ‚ öÅ‚
ïÅ‚sin sinëÅ‚¸ - 2Ä„ ÷Å‚ sinëÅ‚¸ + 2Ä„ ÷łśłïÅ‚ śł
ðÅ‚ ûÅ‚
(¸ )
ìÅ‚ ìÅ‚ śł
ic
dynamic interaction between inverters and distribution
ïÅ‚
3 3
íÅ‚ Å‚Å‚ íÅ‚ łłśłðÅ‚ ûÅ‚
ðÅ‚ ûÅ‚
systems without the high level switching detail.
Once in the synchronous frame, the quantities
3.2 Average Switching Model
id and iq are regulated using two conventional PI
In most cases, it is reasonable to assume that the VSI
feedback control loops one for each current.
switching frequency is significantly higher than the
A third PI controller is used to maintain the DC link
power system frequency and will have negligible
voltage at a specified value. This controller acts as an
impact on the inverter control loop dynamics.
outer control loop, providing part of the real current
Therefore, the inverter switches can be replaced by a
demand to the inner current control loop as shown in
function representing their averaged value [5].
Figure 2. (Note that in a complete system, the
Providing the VSI does not saturate, the output of the
remainder of the real and reactive current references
control loops then command the average value of the
would be generated by higher level control loops.
VSI output voltage phasor, u1, and the operation of the
However, the operation of such higher level control
entire inverter and its output filter system can be
loops is beyond the scope of this paper). The operation
modelled using a continuous state space model.
of the DC voltage control loop is decoupled from the
The following (conventional) state-space model
current regulator by giving it a significantly longer
represents the AC filter in the synchronous dq frame:
time constant.
X = AX + BU Y = CX (2)
3. SYSTEM MODELLING
where:
T
3.1 Switched Model X = [i1d i1q i2d i2q ucd ucq]
A complete switched model of the inverter system has
T T
U = [u1d u1q u2d u2q] Y = [i2d i2q]
been developed using the Power Systems Blockset
(PSB) available in the Matlab Simulink package. This
R1
îÅ‚ Å‚Å‚
1
package uses numerical integration to solve
ïÅ‚- L1 É 0 0 - L1 0 śł
differential equations. In addition to the range of ïÅ‚
R1 0 0 0 - 1 śł
- É -
ïÅ‚
L1 L1 śł
ïÅ‚
Vdc
R1 É - 1 0 śł
ïÅ‚ śł
0 0 -
Vdc
L1 L2 śł
ïÅ‚
A =
error
Id*
PI
R1 0 - 1
Vdc* ïÅ‚ śł
Controller 0 0 -É -
ïÅ‚ L1 L2śł
ïÅ‚ śł
Id 1 1
0 - 0 0 É
PI
Demanded
ïÅ‚ C C śł
Controller f f
Voltage
ïÅ‚ śł
PWM
1 1
Measured
abc dq dq abc 0 0 - - É 0
ïÅ‚ śł
Modulator
Currents
C C
f f
ðÅ‚ ûÅ‚
PI
Controller
Iq
Iq*
Figure 2: Synchronous Frame Control Strategy
As mentioned above, the inverter system is non-linear
îÅ‚ 1 Å‚Å‚
L1 0 0 0 śł
ïÅ‚
and cannot be solved analytically. Therefore
ïÅ‚ 1 śł
0
Equations (2) to (6) have been used to create a closed
L1 0 0 śł
ïÅ‚
loop model of the system in Simulink. As the high
ïÅ‚ śł
1
0 0 -
B = L2 0
ïÅ‚ śł
frequency switching operations are not included in the
ïÅ‚ śł
1
0 0 0 -
ASM the computation requirements are significantly
ïÅ‚ L2 śł
less than those of the switched model, resulting in a
ïÅ‚ śł
0 0 0 0
ïÅ‚ śł
greatly reduced simulation time.
0 0 0 0
ïÅ‚ śł
ðÅ‚ ûÅ‚
3.3 Comparison of Simulation Models
0 0 1 0 0 0
îÅ‚ Å‚Å‚
C =
ïÅ‚0 0 0 1 0 0śł
The system parameters used in these simulation
ðÅ‚ ûÅ‚
studies are given in Table 1. Figures 3 and 4 show the
where R1 , R2 = resistance values associated with
phase currents obtained from the switched model and
L1 , L2
the averaged model respectively for a step change in
*
The DC voltage is defined by Equation 3.
demanded reactive current, iq . It can be seen that the
(u1di1d + u1qi1q)
switched model phase current contains high frequency
dVdc il
= - (3)
components due to the switching operation. However,
dt CdcVdc Cdc
the average value of this current is in close agreement
The overall inverter system may be represented by the
with the results from the ASM. Furthermore, the time
state-space equation:
taken for the ASM to simulate the system operation
was approximately 5 seconds compared to 6 minutes
X = f ( X ,U ) (4)
for the switched model. This is a significant
where:
improvement of nearly two orders of magnitude.
T
From these results, it may be concluded that the ASM
X = [i1d i1q i2d i2q ucd ucq Vdc il ]
is a suitable tool for studying the application of
T
U = [u1d u1q u2d u2q] multiple power electronic converters connected to a
power distribution network.
The inverter system defined by Equation 4 is non-
PWM Converter
linear because f ( X ,U ) is a non-linear function. This
Rating 10kVA
is because the differential equation defining the state-
Switching Frequency 5kHz
variableVdc includes an inverse relationship (ie.
AC Supply Voltage, u2 415V(l-l)
1 / V ) and two terms which involve a product
dc
AC Filter
between a state-variable and a system input (ie. u1d i1d
Inverter inductance, L1 6.5mH (0.12 p.u.)
and u1qi1q ) (ref. Equation 3).
Supply Inductance, L2 1mH (0.02 p.u.)
Shunt Capacitance, Cf
15µF (615 p.u.)
To obtain the closed loop response, the inverter
DC Link
outputs (filter inputs), u1d and u1q, are taken from the
DC Voltage, Vdc 700V
outputs of the inner loop PI controllers, as:
DC Capacitance, Cdc
2200µF
Ki
îÅ‚ Å‚Å‚
*
îÅ‚i1d
u1d Table 1: VSI System parameters
îÅ‚ Å‚Å‚ ïÅ‚K p + s 0 śł - i1d Å‚Å‚
= (5)
ïÅ‚u śł ïÅ‚
*
Ki śłïÅ‚ 1q - i1q śł
1q
ïÅ‚ śł
ðÅ‚ ûÅ‚ ïÅ‚ śłðÅ‚i ûÅ‚
0 K +
p
4. PERFORMANCE OF INVERTER UNDER
ïÅ‚ śł
ðÅ‚ s ûÅ‚
DISTORTED SUPPLY CONDITIONS
* *
where i1d and i1q are the reference currents. K and
p
Initially, the gains of the PI controllers described
Ki are the proportional and integral gain constants above were tuned for fundamental response using the
full switched simulation model and assuming a
respectively. These gain constants are set by tuning
sinusoidal supply voltage. However, experimental
the controller for optimal response.
investigations carried out for this work indicated that
The DC voltage is maintained at a constant value
small levels of supply voltage distortion can result in
using a PI controller which provides the real current
significant harmonic current distortion from the VSI
reference of:
with these tuning conditions. Stability analyses and
ëÅ‚ ASM simulations have been used to develop a
Ki' öÅ‚ *
* '
ìÅ‚ ÷Å‚
i1d = K + (Vdc - Vdc) (6)
p
theoretical and practical understanding of the system
ìÅ‚ ÷Å‚
s
íÅ‚ Å‚Å‚
robustness and its response to such low order
*
where Vdc is the DC voltage target. harmonic distortion. The results are presented in the
following sections.
10
1 0
10
8
8
8
6
6
6
4
4
4
2
2
2
0
Ia(A)
00
-2
-22
-
-4
-44
-
-6
-6
-6
-8
-8
-8
-10
0 0.05 0.1 0.15
-10
-10
0 0.05 0.1 0.15
0 0 .0 5 0.1 0 .15
Time (sec)
Time (sec)
Figure 3: Switched model simulation results Figure 6 ASM simulation results phase current
Figure 4: ASM simulation results phase current
phase current
Under these conditions the open loop transfer function
4.1 Stability Analysis
of the system is linear and is defined by:
In order to apply classical stability analysis
H (s) = Vdc *GAC (s)* H (s) (9)
ol f
techniques, the non-linear system described in Section
3.4.2 has to be linearised around a given operating
The frequency response of the open loop system is
point. This may be achieved with small-signal
shown in Figure 5. As expected, there is a large gain
analysis. However, by making some reasonable
at the fundamental frequency caused primarily by the
assumptions about the system operation, the analysis
integral term of the PI controller. This gain eliminates
is greatly simplified as shown below.
steady-state error at the system frequency. There is
Assuming a balanced system and ignoring cross-
also a resonance point introduced by the AC filter. It
coupling terms, the synchronous frame PI controller
should be noted that the digital sampling introduces
transformed into the stationary frame can be
additional phase delay which is not included here.
approximated by single-phase resonant controllers
The harmonic performance of the system relates to the
which are described by the following transfer function
bandwidth of the controller ie. the higher the
[7]:
bandwidth the lower the current distortion. The
2Ki s bandwidth is determined by the magnitude crossover
GAC (s) = K + (7)
p
2 2 point (gain = 0dB) on the bode plot (ref. Figure 5). It
s +É
0
can be seen that increasing the proportional gain
increases the bandwidth of a PI controller. The limit
where É0 is the AC angular frequency, 100Ä„ rad/s.
on stability is the phase at the crossover point. Clearly
This approximation is justified by the fact that it is the
there is a tradeoff between stability and level of
proportional gain which dominates the response of the
current distortion. Finding an acceptable compromise
controller at the frequencies of interest (ie. harmonic
between harmonic performance and transient stability
frequencies) whereas the resonant and cross-coupling
requires both simulation and experimental
terms only impact the system response at near the
investigation to suit a particular case.
system frequency [7].
300
For a balanced system the transfer function of the AC
filter for each phase is given by:
200
i1 C f L2s2 + C f R2s + 1 100
H (s) = = (8)
f
u1 as3 + bs2 + cs + d
0
where:
-100
a = C L1L2 100
f
b = C L1R2 + C L2 R1
0
f f
To
:
Y(
c = L1 + L2 + C R1R2 d = R1 + R2
f 1) -100
Then the DC voltage is assumed to be constant. This
-200
is a reasonable assumption if the DC capacitance is
10-2 10-1 100 101 102 103 104 105
large or if DC compensation is included in the control
Frequency (rad/sec)
algorithm.
Figure 5: Open loop frequency response of
simplified VSI system.
Phase Current (A)
Phase Current (A)
M
Phase (deg)
agnitude (dB)
Traditionally, more complex current regulation
25
schemes, such as hysteresis and predictive current
regulation, have been employed in applications where
20 Kp 5*Kp
supply distortion is an issue. However, the
observations detailed above suggest that an additional
15
compensation controller could be used to introduce a
phase lead at the crossover point, which would in
10
principle allow an increased bandwidth. The
advantage of such a controller would be its simplicity
5
and ease of implementation. This concept will be the
subject of future investigations.
0
0 0.05 0.1 0.15 0.2 0.25
4.2 Simulation Results
L1 (p.u.)
The ASM was used to further investigate the
Figure 7: Effect of filter inductance, L1, on
performance of the three-phase system under distorted
current distortion.
supply conditions. Two values of proportional gain
were considered, Kp and 5Kp, where Kp is the value
frequency and thereby introduce an undesirable
tuned for fundamental response. In both cases, 2% of
harmonic resonance condition.
5th harmonic distortion was added to the supply
In the previous sections it was shown that supply
voltage and the system parameters were taken as those
voltage distortion can cause significant levels of phase
given in Table 1. The resulting phase currents are
current distortion. While it is possible to reduce
shown in Figure 6. It can be seen that the phase
current distortion by increasing the size of the filter
current distortion (9.2%) for the original value of
inductance, this also increases the system cost. It is
proportional gain is significantly greater than that with
therefore of interest to know the minimum inductance
the increased proportional gain (5.5%). It should be
required to achieve an acceptable low order harmonic
noted that the 5th harmonic was dominant and the
performance.
percentage distortion decreased with increasing load.
Using the ASM, the inverter inductance, L1, was
These results confirm that it is possible to achieve an
varied between 0.05 p.u. and 0.20 p.u. for two values
improved current regulation response under distorted
of proportional gain, Kp and 5Kp. 2% of 5th harmonic
supply conditions by simply increasing the
distortion was again added to the supply. The results
proportional gain of the PI controllers.
are summarised in Figure 7, where it can be seen that
4.3 Filter Design
for the original value of proportional gain, the
The primary function of the AC filter is to filter out harmonic distortion in the supply current is sensitive
the high frequency components caused by the inverter to the value of inductance in the AC filter. The current
switching operation. However, the filter also affects distortion recorded for the higher value of
the low order harmonic performance of the system. proportional gain was less sensitive to filter
The design of AC filters for grid-connected inverter inductance and was significantly lower across the
systems is not well covered in literature. range considered. However, in this case the system
was unstable for inductance values below 0.07 p.u.
In Section 4.2 it was observed that the AC filter
introduces a point of resonance above the system
frequency. The AC filter should be designed such that
5. EXPERIMENTAL VERIFICATION
this resonance point does not occur at a harmonic
The results presented in the previous sections have
10
been confirmed experimentally. The experimental
5
system was based on a DSP inverter control card and
the system parameters were those specified in Table 1.
0
Kp
The PI constants were matched to the simulation
-5
ITHD=9.2%
studies.
-10
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Tests showed that there was a low level
Time (sec)
10 (approximately 1.5%) of harmonic distortion in the
supply, with the 5th and 7th harmonics dominating.
5
Using the original value of proportional gain, Kp, the
0
supply current distortion measured at approximately
5Kp
-5
ITHD=5.5%
8%. When the proportional gain was increased to 5Kp,
-10
the current distortion decreased to below 4% as
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Time (sec)
expected. In both cases, the 5th and 7th harmonics were
Figure 6: ASM simulation results with 5% of 5th dominant.
harmonic distortion added to supply.
Supply Current THD (%)
Phase Current (A)
Phase Current (A)
20
These results confirm the value of the averaging
15
technique in studying the operation of multiple power
electronic converters connected to power distribution
10
systems.
5
Using the developed model, it has been shown that
0
synchronous frame PI current regulators can be tuned
-5
to achieve an improved response when operating into
a distorted supply. If the regulator is tuned for a
-10
distorted supply rather than a sinusoidal supply, an
-15
acceptable harmonic performance can be achieved
-20
with a lower value of inductance in the AC filter thus
0.02 0.04 0.06 0.08 0.1
reducing system costs.
Time (sec)
Figure 8: Experimental results for step
7. REFERENCES
load change
[1] H. C. van Nierkerk and I. W. Hosajer, "The Use of
15
Series Injection to Eliminate Voltage Distortion in
10
Low and Medium Voltage Network", Proceedings
of the Industrial and Commercial Power Systems
5
Technical Conference, pp. 1-6, 2000.
0
[2] Z. Saad-Saoud, M. L. Lisboa, J. B. Ekanayake, N.
Jenkins, and G. Strbac, "Application of
-5
STATCOMs to wind farms," IEE Proceedings.
-10
Generation, Transmission & Distribution, vol. 145,
pp. 511-518, 1998.
-15
0 0.02 0.04 0.06 0.08 0.1
Time (sec)
[3] S. Thiel, C.-H. Mostert, and J. H. R. Enslin,
"Universal power electronic solution to low-cost
Figure 9: ASM simulation results for step
rural electrification", Proceedings of the 4th IEEE
load change
AFRICON Conference, pp. 335-340, 1996.
[4] G. Ledwich and H. Sharma, "Connection of
Figure 8 shows the experimental results obtained for a
Inverters to a Weak Grid", Proceedings of the
step change in load using the proportional gain of
IEEE Power Electronics Specialists Conference,
5Kp. The ASM was used to simulate this transient
2000, vol. 2, pp. 1018-1022, 2000.
response. The results, shown Figure 9, are in close
agreement with the experimental results. The
[5] M. B. Lindgren, "Analysis and simulation of
harmonic current distortion recorded for the ASM
digitally-controlled grid-connected PWM-
simulation was also similar to the measured result.
converters using the space-vector average
The minor differences between the experimental and
approximation", Proceedings of the 5th IEEE
simulation results can be explained by phase
Workshop on Computers in Power Electronics, pp.
imbalance and the inaccuracies associated with
85-89, 1996.
measurement of model parameters such as supply
[6] N. Abdel-Rahim and J. E. Quaicoe, "Modeling and
impedance and supply distortion.
analysis of a feedback control strategy for three-
phase voltage-source utility interface systems",
6. CONCLUSION
Proceedings of the 29th IAS Annual Meeting, Part
2 (of 3), pp. 895-902, 1994.
This paper has described an averaging inverter model
which allows converter systems to be rapidly and
[7] D. N. Zmood, D. G. Holmes, and G. Bode,
accurately simulated. The accuracy of the averaged
"Frequency domain analysis of three phase linear
model for a grid-connected converter system has been
current regulators", Proceedings of the IEEE IAS
verified against both a switched inverter model and an
Annual Meeting. V 2 pp. 818-825, 1999.
experimental system. Furthermore, it has been shown
that the averaged model is close to two orders of
magnitude faster than the equivalent switched model.
Phase Current (A)
Phase Current (A)
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