MODELLING GRID CONNECTED VOLTAGE SOURCE INVERTER OPERATION

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.

INTRODUCTION

The electrification of rural and remote areas presents
significant challenges to Australian distribution
companies. Rural distribution networks are typically
characterised by very low X/R ratios because of the
long distances involved, and consequently power
quality issues such as poor voltage regulation, voltage
dips and harmonic distortion are common in these
networks. With growing demand and increased use of
sensitive electronic equipment, the need to address
these issues has become a priority. Recent
developments in power electronic and digital control
technologies have seen the design of a range of power
electronic based conditioning equipment, including
FACTS (Flexible AC Transmission Systems) devices
(such as STATCOM’s, UPFC’s etc) and active
interfaces for distributed generation systems (eg. PV,
wind etc.). However, despite their potential to improve
the power quality of weak grid environments [1-3],
there remains a reluctance to incorporate power
electronic plant into distribution systems. This is in
part due to unresolved issues relating to their
interaction with the existing distribution network [4].

This paper presents the first stage of a research
program which aims to explore interactions between
multiple power electronic converters connected to
weak distribution networks. The paper describes a
simple averaging inverter model which allows grid-
connected converter systems to be rapidly and
accurately simulated without requiring the complexity
of full switched inverter models. In [5] a similar
averaging model was shown to be a convenient tool
for the evaluation of a system’s dynamic performance.

The inverter system described in this paper is a three-
phase grid connected Voltage Source Inverter (VSI)
configuration commonly used in STATCOM devices
and distributed generation interfaces. A synchronous
frame PI current regulator was chosen to control the
inverter. There has been some debate in literature
regarding the performance of this control strategy in
relation to other strategies such as hysteresis and

predictive current regulation (PCR) (ie. deadbeat
control) [6]. However, synchronous frame PI current
regulation is still commonly used in many
applications, as it is effective and relatively simple to
implement. It was therefore deemed useful to
investigate its limitations, as an example of the use of
average modelling at a practical level.

In Section 3, the averaging model, referred to here as
an Average Switching Model (ASM), is developed and
shown to achieve accurate simulation results whilst
being significantly faster to execute than a full
switched model. In Section 4, the effects of supply
distortion on the harmonic performance of the inverter
system are investigated through stability analysis
techniques and ASM simulations. It is shown that the
synchronous frame PI controller can be tuned to
achieve improved harmonic response. The influence
of this result on AC filter design is explored. Finally,
in Section 5, the accuracy of the ASM is verified
against experimental results.

SYSTEM DESCRIPTION

The grid-connected VSI configuration modelled in
this paper is shown in Figure 1. For the purposes of
this initial work, the DC side of the converter system
was connected to a resistive load and so the inverter
acts as an active rectifier. However, since the
converter is bi-directional, the developed models can
be applied to any type of inverter application without
loss of generality.

2.1

Control Strategy

As noted in the introduction, a synchronous frame PI
current regulator was chosen to control the inverter.
Synchronous frame controllers operate by
transforming the three-phase AC currents

and

in the stationary frame, into the DC components

i and

in the synchronously rotating frame. This

allows the steady-state error that is normally
associated with the application of PI control to AC
quantities [7] to be eliminated, and also provides

independent control of real and reactive power flow.
The synchronous transformation is:

( )



































 +











 −











 +











 −













sin

cos

(1)

Once in the synchronous frame, the quantities

i and

are regulated using two conventional PI

feedback control loops – one for each current.

A third PI controller is used to maintain the DC link
voltage at a specified value. This controller acts as an
outer control loop, providing part of the real current
demand to the inner current control loop as shown in
Figure 2. (Note that in a complete system, the
remainder of the real and reactive current references
would be generated by higher level control loops.
However, the operation of such higher level control
loops is beyond the scope of this paper). The operation
of the DC voltage control loop is decoupled from the
current regulator by giving it a significantly longer
time constant.

SYSTEM MODELLING

3.1

Switched Model

A complete switched model of the inverter system has
been developed using the Power Systems Blockset
(PSB) available in the Matlab Simulink package. This
package uses numerical integration to solve
differential equations. In addition to the range of

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,
times in the order of several minutes are required to
simulate the inverter operation over one or two
fundamental cycles. It is clear that such a
computationally intensive model would not be feasible
for distribution system applications involving multiple
inverters. It was therefore necessary to find an
alternative which would accurately represent the
dynamic interaction between inverters and distribution
systems without the high level switching detail.

3.2

Average Switching Model

In most cases, it is reasonable to assume that the VSI
switching frequency is significantly higher than the
power system frequency and will have negligible
impact on the inverter control loop dynamics.
Therefore, the inverter switches can be replaced by a
function representing their averaged value [5].
Providing the VSI does not saturate, the output of the
control loops then command the average value of the
VSI output voltage phasor, u

, and the operation of the

entire inverter and its output filter system can be
modelled using a continuous state space model.

The following (conventional) state-space model
represents the AC filter in the synchronous dq frame:

(2)

where:

[

]

[

]

[

]













−

Figure 1: Grid-connected VSI

PI
Controller

abc

Measured
Currents

PI
Controller

abc

PWM

Modulator

Vdc*

Vdc

Vdc
error

Id*

Iq*

Demanded
Voltage

Figure 2: Synchronous Frame Control Strategy













−













where

R ,

R = resistance values associated with

L ,

The DC voltage is defined by Equation 3.

(

)

−

(3)

The overall inverter system may be represented by the
state-space equation:

)

(

(4)

where:

[

]

[

]

The inverter system defined by Equation 4 is non-
linear because

)

(

is a non-linear function. This

is because the differential equation defining the state-
variable

includes an inverse relationship (ie.

) and two terms which involve a product

between a state-variable and a system input (ie.

and

) (ref. Equation 3).

To obtain the closed loop response, the inverter
outputs (filter inputs), u

and u

, are taken from the

outputs of the inner loop PI controllers, as:













−

























(5)

where

and

are the reference currents.

K and

K are the proportional and integral gain constants

respectively. These gain constants are set by tuning
the controller for optimal response.

The DC voltage is maintained at a constant value
using a PI controller which provides the real current
reference of:

(

)

−















(6)

where

is the DC voltage target.

As mentioned above, the inverter system is non-linear
and cannot be solved analytically. Therefore
Equations (2) to (6) have been used to create a closed
loop model of the system in Simulink. As the high
frequency switching operations are not included in the
ASM the computation requirements are significantly
less than those of the switched model, resulting in a
greatly reduced simulation time.

3.3

Comparison of Simulation Models

The system parameters used in these simulation
studies are given in Table 1. Figures 3 and 4 show the
phase currents obtained from the switched model and
the averaged model respectively for a step change in

demanded reactive current,

i . It can be seen that the

switched model phase current contains high frequency
components due to the switching operation. However,
the average value of this current is in close agreement
with the results from the ASM. Furthermore, the time
taken for the ASM to simulate the system operation
was approximately 5 seconds compared to 6 minutes
for the switched model. This is a significant
improvement of nearly two orders of magnitude.
From these results, it may be concluded that the ASM
is a suitable tool for studying the application of
multiple power electronic converters connected to a
power distribution network.

PWM Converter
Rating
Switching Frequency

10kVA
5kHz

AC Supply Voltage, u

415V(l-l)

AC Filter
Inverter inductance, L

Supply Inductance, L

Shunt Capacitance, C

6.5mH (0.12 p.u.)
1mH (0.02 p.u.)
15

µF (615 p.u.)

DC Link
DC Voltage, V

DC Capacitance, C

700V
2200

µF

Table 1: VSI System parameters

PERFORMANCE OF INVERTER UNDER
DISTORTED SUPPLY CONDITIONS

Initially, the gains of the PI controllers described
above were tuned for fundamental response using the
full switched simulation model and assuming a
sinusoidal supply voltage. However, experimental
investigations carried out for this work indicated that
small levels of supply voltage distortion can result in
significant harmonic current distortion from the VSI
with these tuning conditions. Stability analyses and
ASM simulations have been used to develop a
theoretical and practical understanding of the system
robustness and its response to such low order
harmonic distortion. The results are presented in the
following sections.

4.1

Stability Analysis

In order to apply classical stability analysis
techniques, the non-linear system described in Section
3.4.2 has to be linearised around a given operating
point. This may be achieved with small-signal
analysis. However, by making some reasonable
assumptions about the system operation, the analysis
is greatly simplified as shown below.

Assuming a balanced system and ignoring cross-
coupling terms, the synchronous frame PI controller
transformed into the stationary frame can be
approximated by single-phase resonant controllers
which are described by the following transfer function
[7]:

)

(

(7)

where

is the AC angular frequency, 100

π rad/s.

This approximation is justified by the fact that it is the
proportional gain which dominates the response of the
controller at the frequencies of interest (ie. harmonic
frequencies) whereas the resonant and cross-coupling
terms only impact the system response at near the
system frequency [7].

For a balanced system the transfer function of the AC
filter for each phase is given by:

)

(

(8)

where:

Then the DC voltage is assumed to be constant. This
is a reasonable assumption if the DC capacitance is
large or if DC compensation is included in the control
algorithm.

Under these conditions the open loop transfer function
of the system is linear and is defined by:

)

(

)

(

)

(

(9)

The frequency response of the open loop system is
shown in Figure 5. As expected, there is a large gain
at the fundamental frequency caused primarily by the
integral term of the PI controller. This gain eliminates
steady-state error at the system frequency. There is
also a resonance point introduced by the AC filter. It
should be noted that the digital sampling introduces
additional phase delay which is not included here.

The harmonic performance of the system relates to the
bandwidth of the controller ie. the higher the
bandwidth the lower the current distortion. The
bandwidth is determined by the magnitude crossover
point (gain = 0dB) on the bode plot (ref. Figure 5). It
can be seen that increasing the proportional gain
increases the bandwidth of a PI controller. The limit
on stability is the phase at the crossover point. Clearly
there is a tradeoff between stability and level of
current distortion. Finding an acceptable compromise
between harmonic performance and transient stability
requires both simulation and experimental
investigation to suit a particular case.

0 .0 5

0 .1

0 .1 5

-1 0

-8

-6

-4

-2

1 0

Ia (A )

Figure 6 ASM simulation results – phase current

0.05

0.1

0.15

-10

-8

-6

-4

-2

Time (sec)

e Cu

)

Figure 3: Switched model simulation results –

phase current

0.05

0.1

0.15

-10

-8

-6

-4

-2

Time (sec)

has

e Current

)

Figure 4: ASM simulation results – phase current

Frequency (rad/sec)

has

e (

deg)

agni

tude (

)

-100

100

200

300

-2

-1

-200

-100

100

To
:
Y(
1)

Figure 5: Open loop frequency response of

simplified VSI system.

Traditionally, more complex current regulation
schemes, such as hysteresis and predictive current
regulation, have been employed in applications where
supply distortion is an issue. However, the
observations detailed above suggest that an additional
compensation controller could be used to introduce a
phase lead at the crossover point, which would in
principle allow an increased bandwidth. The
advantage of such a controller would be its simplicity
and ease of implementation. This concept will be the
subject of future investigations.

4.2

Simulation Results

The ASM was used to further investigate the
performance of the three-phase system under distorted
supply conditions. Two values of proportional gain
were considered, K

and 5K

, where K

is the value

tuned for fundamental response. In both cases, 2% of
5

harmonic distortion was added to the supply

voltage and the system parameters were taken as those
given in Table 1. The resulting phase currents are
shown in Figure 6. It can be seen that the phase
current distortion (9.2%) for the original value of
proportional gain is significantly greater than that with
the increased proportional gain (5.5%). It should be
noted that the 5

harmonic was dominant and the

percentage distortion decreased with increasing load.
These results confirm that it is possible to achieve an
improved current regulation response under distorted
supply conditions by simply increasing the
proportional gain of the PI controllers.

4.3

Filter Design

The primary function of the AC filter is to filter out
the high frequency components caused by the inverter
switching operation. However, the filter also affects
the low order harmonic performance of the system.
The design of AC filters for grid-connected inverter
systems is not well covered in literature.

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
this resonance point does not occur at a harmonic

frequency and thereby introduce an undesirable
harmonic resonance condition.

In the previous sections it was shown that supply
voltage distortion can cause significant levels of phase
current distortion. While it is possible to reduce
current distortion by increasing the size of the filter
inductance, this also increases the system cost. It is
therefore of interest to know the minimum inductance
required to achieve an acceptable low order harmonic
performance.

Using the ASM, the inverter inductance, L

, was

varied between 0.05 p.u. and 0.20 p.u. for two values
of proportional gain, Kp and 5Kp. 2% of 5

harmonic

distortion was again added to the supply. The results
are summarised in Figure 7, where it can be seen that
for the original value of proportional gain, the
harmonic distortion in the supply current is sensitive
to the value of inductance in the AC filter. The current
distortion recorded for the higher value of
proportional gain was less sensitive to filter
inductance and was significantly lower across the
range considered. However, in this case the system
was unstable for inductance values below 0.07 p.u.

EXPERIMENTAL VERIFICATION

The results presented in the previous sections have
been confirmed experimentally. The experimental
system was based on a DSP inverter control card and
the system parameters were those specified in Table 1.
The PI constants were matched to the simulation
studies.

Tests showed that there was a low level
(approximately 1.5%) of harmonic distortion in the
supply, with the 5

and 7

harmonics dominating.

Using the original value of proportional gain, K

, the

supply current distortion measured at approximately
8%. When the proportional gain was increased to 5Kp,
the current distortion decreased to below 4% as
expected. In both cases, the 5

and 7

harmonics were

dominant.

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

-1 0

-5

THD

=9.2%

Tim e (sec)

has

e C

)

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

-1 0

-5

THD

=5.5%

Tim e (sec)

has

e C

)

Figure 6: ASM simulation results with 5% of 5

harmonic distortion added to supply.

0.05

0.1

0.15

0.2

0.25

L1 (p.u.)

Supply Current THD (%)

5*Kp

Figure 7: Effect of filter inductance, L

, on

current distortion.

Figure 8 shows the experimental results obtained for a
step change in load using the proportional gain of
5Kp. The ASM was used to simulate this transient
response. The results, shown Figure 9, are in close
agreement with the experimental results. The
harmonic current distortion recorded for the ASM
simulation was also similar to the measured result.
The minor differences between the experimental and
simulation results can be explained by phase
imbalance and the inaccuracies associated with
measurement of model parameters such as supply
impedance and supply distortion.

CONCLUSION

This paper has described an averaging inverter model
which allows converter systems to be rapidly and
accurately simulated. The accuracy of the averaged
model for a grid-connected converter system has been
verified against both a switched inverter model and an
experimental system. Furthermore, it has been shown
that the averaged model is close to two orders of
magnitude faster than the equivalent switched model.

These results confirm the value of the averaging
technique in studying the operation of multiple power
electronic converters connected to power distribution
systems.

Using the developed model, it has been shown that
synchronous frame PI current regulators can be tuned
to achieve an improved response when operating into
a distorted supply. If the regulator is tuned for a
distorted supply rather than a sinusoidal supply, an
acceptable harmonic performance can be achieved
with a lower value of inductance in the AC filter thus
reducing system costs.

REFERENCES

[1] H. C. van Nierkerk and I. W. Hosajer, "The Use of

Series Injection to Eliminate Voltage Distortion in
Low and Medium Voltage Network", Proceedings
of the Industrial and Commercial Power Systems
Technical Conference, pp. 1-6, 2000.

[2] Z. Saad-Saoud, M. L. Lisboa, J. B. Ekanayake, N.

Jenkins, and G. Strbac, "Application of
STATCOMs to wind farms," IEE Proceedings.
Generation, Transmission & Distribution, vol. 145,
pp. 511-518, 1998.

[3] S. Thiel, C.-H. Mostert, and J. H. R. Enslin,

"Universal power electronic solution to low-cost
rural electrification", Proceedings of the 4th IEEE
AFRICON Conference, pp. 335-340, 1996.

[4] G. Ledwich and H. Sharma, "Connection of

Inverters to a Weak Grid", Proceedings of the
IEEE Power Electronics Specialists Conference,
2000, vol. 2, pp. 1018-1022, 2000.

[5] M. B. Lindgren, "Analysis and simulation of

digitally-controlled grid-connected PWM-
converters using the space-vector average
approximation", Proceedings of the 5th IEEE
Workshop on Computers in Power Electronics, pp.
85-89, 1996.

[6] N. Abdel-Rahim and J. E. Quaicoe, "Modeling and

analysis of a feedback control strategy for three-
phase voltage-source utility interface systems",
Proceedings of the 29th IAS Annual Meeting, Part
2 (of 3), pp. 895-902, 1994.

[7] D. N. Zmood, D. G. Holmes, and G. Bode,

"Frequency domain analysis of three phase linear
current regulators", Proceedings of the IEEE IAS
Annual Meeting. V 2 pp. 818-825, 1999.

-20

-15

rre

t (A)

-10

-5

0.1

0.06

0.08

0.02

0.04

Time (sec)

Figure 8: Experimental results for step

load change

0.02

0.04

0.06

0.08

0.1

-15

-10

-5

Time (sec)

rre

)

Figure 9: ASM simulation results for step

load change