MODELLING GRID-CONNECTED VOLTAGE ...

MODELLING GRID-CONNECTED VOLTAGE SOURCE INVERTER OPERATION, elektronika, elektronika, INWERTERY

[ Pobierz całość w formacie PDF ]
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
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.
2.
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
,
in the stationary frame, into the DC components
q
i
i
and
i
a
b
c
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
i
d
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.
Figure 1: Grid-connected VSI
independent control of real and reactive power flow.
The synchronous transformation is:
2
π
2
π
()
i
cos
θ
cos
θ
−
cos
θ
+
a
i
2
3
3
d
(1)
=
i
b
i
3
2
π
2
π
()
q
sin
θ
sin
θ
−
sin
θ
+
i
c
3
3
Once in the synchronous frame, the quantities
q
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
1
, 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:
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.
i
d
(2)
X
=
AX
+
BU
Y
=
CX
3.
SYSTEM MODELLING
where:
[
]
T
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
X
=
i
i
i
i
u
u
1
1
2
2
d
q
d
q
cd
cq
[
]
[
]
T
T
U
=
u
u
u
u
Y
=
i
i
1
1
2
2
2
2
d
q
d
q
d
q
R
1
1
−
ω
0
0
−
0
L
L
1
1
R
1
1
−
ω
−
0
0
0
−
L
L
1
1
R
V
dc
1
0
0
−
1
ω
−
0
L
L
V
dc
err
or
A
=
1
2
I
d
*
R
PI
Controller
1
1
V
dc
*
0
0
−
ω
−
0
−
L
L
1
2
1
1
0
−
0
0
ω
I
d
PI
Controller
C
C
Demanded
Voltage
f
f
1
1
PWM
Modulator
0
0
−
−
ω
0
Measured
Currents
abc
dq
dq
abc
C
C
f
f
PI
Controller
I
q
I
q
*
Figure 2: Synchronous Frame Control Strategy
 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.
1
0
0
0
L
1
1
0
0
0
L
1
1
0
0
−
0
B
=
L
2
1
0
0
0
−
L
2
0
0
0
0
0
0
0
0
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,
*
0
0
1
0
0
0
C
=
0
0
0
1
0
0
where
R
,
R
= resistance values associated with
L
The DC voltage is defined by Equation 3.
(
L
,
2
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.
)
u
i
+
u
i
dV
i
1
d
1
d
1
q
1
q
dc
l
=
−
(3)
dt
C
V
C
dc
dc
dc
The overall inverter system may be represented by the
state-space equation:
)
X
=
f
(
X
,
U
(4)
where:
[
]
T
X
=
i
i
i
i
u
u
V
i
1
d
1
q
2
d
2
q
cd
cq
dc
l
[
]
T
U
=
u
u
u
u
1
d
1
q
2
d
2
q
The inverter system defined by Equation 4 is non-
linear because
PWM Converter
Rating
Switching Frequency
Xf
is a non-linear function. This
is because the differential equation defining the state-
variable
(
,
U
)
10kVA
5kHz
includes an inverse relationship (ie.
V
dc
AC Supply Voltage
, u
2
415V(l-l)
/1
) and two terms which involve a product
between a state-variable and a system input (ie.
V
dc
AC Filter
Inverter inductance, L
1
Supply Inductance, L
2
Shunt Capacitance, C
f
u
d
i
6.5mH (0.12 p.u.)
1mH (0.02 p.u.)
15
1
1
d
and
) (ref. Equation 3).
u
q
i
1
1
q
µ
F (615 p.u.)
To obtain the closed loop response, the inverter
outputs (filter inputs),
u
1d
and
u
1q
, are taken from the
outputs of the inner loop PI controllers, as:
DC Link
DC Voltage, V
dc
DC Capacitance, C
dc
700V
2200
µ
F
K
i
K
+
0
*
1
u
Table 1: VSI System parameters
i
−
i
p
1
s
d
d
1
d
=
(5)
K
*
1
u
i
−
i
1
i
q
0
K
+
q
1
q
p
4.
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.
s
1
d
1
q
where
i
and
i
are the reference currents.
K
and
p
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:
'
(
)
K
*
1
'
*
i
i
=
K
+
V
−
V
(6)
d
p
dc
dc
s
dc
where
is the DC voltage target.
V
 10
1 0
10
8
8
8
6
6
6
4
4
4
2
2
2
0
Ia(A)
0
0
-2
-2
-2
-4
-4
-4
-6
-6
-6
-8
-8
-8
-10
0
0.05
0.1
0.15
-10
-10
Time (sec)
0
0
0.05
0 .05
0.1
0.1
0.15
0 .15
Time (sec)
Figure 3: Switched model simulation results –
phase current
Figure 6 ASM simulation results – phase current
Figure 4: ASM simulation results – phase current
Under these conditions the open loop transfer function
of the system is linear and is defined by:
)
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]:
H
(
s
)
=
V
*
G
(
s
)
*
H
(
s
(9)
ol
dc
AC
f
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
2
K
s
i
G
(
s
)
=
K
+
(7)
AC
p
2
2
0
s
+
where
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:
is the AC angular frequency, 100
π
ω
0
both
simulation
and
experimental
investigation to suit a particular case.
300
200
2
C
L
s
+
C
R
s
+
1
i
f
2
f
2
100
1
H
(
s
)
=
=
(8)
f
3
2
u
as
+
bs
+
cs
+
d
1
0
where:
-100
a
=
C
L
1
L
100
f
2
b
=
C
L
R
+
C
L
R
f
1
2
f
2
1
0
To
:
Y(
1)
c
=
L
+
L
+
C
R
R
d
=
R
+
R
1
2
1
2
f
1
2
-100
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.
-200
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
Frequency (rad/sec)
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.
25
Kp
5*Kp
20
15
10
5
0
0
0.05
0.1
0.15
0.2
0.25
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
p
and 5K
p
, where K
p
is the value
tuned for fundamental response. In both cases, 2% of
5
th
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
th
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.
L1 (p.u.)
Figure 7: Effect of filter inductance, L
1
, on
current distortion.
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
1
, was
varied between 0.05 p.u. and 0.20 p.u. for two values
of proportional gain, Kp and 5Kp. 2% of 5
th
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.
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
5.
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
th
and 7
th
harmonics dominating.
Using the original value of proportional gain, K
p
, 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
th
and 7
th
harmonics were
dominant.
10
5
0
K
p
I
THD
=9.2%
-5
-10
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (sec)
10
5
0
5K
p
I
THD
=5.5%
-5
-10
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (sec)
Figure 6: ASM simulation results with 5% of 5
th
harmonic distortion added to supply.
[ Pobierz całość w formacie PDF ]