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Journal of Energy and Power Engineering 8 (2014) 749-756
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Finite Set-Model Predictive Current Control of
Three-Phase Voltage Source Inverter for RES
(Renewable Energy Systems) Applications
Ali Almaktoof, Atanda Raji and Tariq Kahn
Electrical Engineering Department, Cape Peninsula University of Technology, Bellville 7535, Republic of South Africa
Received: November 26, 2013 / Accepted: January 10, 2014 / Published: April 30, 2014.
Abstract: This paper focuses on a combination of three-phase VSI (voltage source inverter) with a predictive current control to provide
an optimized system for three-phase inverters that control the load current. A FS-MPC (finite set-model predictive control) strategy for
a three-phase VSI for RES (renewable energy systems) applications is implemented. The renewable energy systems model is used in
this paper to investigate the system performance when power is supplied to resistive-inductive load. With three different cases, the
evaluation of the system is done. Firstly, the robustness of control strategy under variable DC-Link is done in terms of the THD (total
harmonic distortion). Secondly, with one prediction step, the system performance is tested using different sampling time, and lastly, the
dynamic response of the system with step change in the amplitude of the reference is investigated. The simulations and result analyses
are carried out using Matlab/Simulink to test the effectiveness and robustness of FS-MPC for two-level VSI with AC filter for
resistive-inductive load supplied by a renewable energy system.
Key words: Finite set-model predictive control, three-phase voltage source inverter, renewable energy system application, AC filter,
Matlab/Simulink.
1. Introduction
The use of RES has become very important in recent
years, due to environmental concern and the increasing
demand of energy. Nowadays, PV and wind power can
be used with maximum efficiency by using appropriate
power converter. The increase in the power of these
resources leads to the need of research on new control
strategies and good topology of high power converters
to combine performance and efficiency demanded.
From the new control strategies for power converters
developed in the last years, the use of MPC (model
predictive control) and FS-MPC (finite set-model
predictive control) in particular is a very interesting
alternative due to its good dynamic behavior and
Corresponding author: Ali Almaktoof, M.Sc., research
fields: control of power converters, model predictive control,
multilevel converters and renewable energy systems
applications. E-mail: [email protected].
flexibility. Considering the trend of increasing the
power rating of the RES resources and the use of VSI
(voltage source inverter) converter is an interesting
alternative because they can work in the medium
voltage range and provide higher quality voltages and
currents. Voltage source converters have been
extensively studied in the last decades in most
industrial sectors for many applications. By
considering the increasing energy demands and power
quality and efficiency, a control and power conversion
using power electronics has become an important topic
today.
Predictive control for power electronics has been
presented since 1980’s [1]. Several advantages make
the predictive control suitable for the control of power
converters, for instance, easy to understand and
implement and can be applied to different kinds of
voltage source converters. It requires a high amount of
750
Finite Set-Model Predictive Current Control Of Three-Phase Voltage Source Inverter
for RES (Renewable Energy Systems) Applications
calculations, compared to classic control methods, but
the fast microprocessors available today have made it
possible to implement predictive control for VSI
converters and predictive control takes advantages,
when compared to traditional PWM (pulse width
modulation) methods [2, 3].
MPC for power converters and drives is to take
advantage of the inherent discrete nature of power
converters. Since power converters have a finite
number of switching states, the MPC optimization
problem can be simplified and reduced to the
prediction of the system behaviour only for those
possible switching states. Then, each prediction is used
to evaluate a cost function (also known as quality or
decision function), and consequently, the state with
minimum cost is selected and generated. This control
method is known as an FS-MPC, since the possible
control actions (switching states) are finite and it is
proposed in this paper. It has been successfully applied
to a wide range of power converter and drive
applications [2], in a three-phase inverter [4-7] and a
matrix converter [8] and flux and torque control of an
induction machine [9]. An example of different
variables controlled using a single cost function is
presented in Ref. [10], where the current is controlled
while, at the same time, minimizing the switching
frequency and balancing the DC-link voltages in an
inverter. In all these works, the switching states are
changed at equidistant time instants. Ref. [11] showed
that the problem of high number of calculations can be
simplified by considering the application of the same
voltage vector for several sampling periods. However,
there is no improvement for more than two prediction
steps and it was observed that the THD (total harmonic
distortion) is increased when a three-step prediction is
considered,
because
the
assumptions
and
approximations are considered for the model, which
are not valid for such long prediction horizons.
In general contribution, this paper presents the
combination of the power rate and power quality of
VSI converters together with the performance and
flexibility of MPC for their application. In this paper,
an FS-MPC for three-phase two-level VSI with
resistive-inductive load supplied by a renewable
energy system (wind and PV) is presented. The
powerful and robustness of the proposed control
method are evaluated through simulations results.
Section 2 presents the power converter model that is
used in this paper. Firstly, the time-continuous model is
presented and then discretized. The control scheme is
developed in Section 3 that is used to control the power
converter model in Section 2. Matlab/Simulink
modeling and simulation work is presented in the
penultimate section. Conclusions are presented in the
last section of the paper.
Fig. 1 shows a VSI with AC filter for
resistive-inductive load supplied by a PV and wind
turbine. In this paper, an FS-MPC for three-phase VSI
with resistive-inductive load is presented. The
powerful and robustness of the proposed control
method are evaluated through simulations results.
2. 2 Converter Model
2.1 Voltage Source Inverter Layout
A two-level VSI three-phase power converter is the
least complicated multiple levels VSI, because it
presents only two voltage levels. The topology of the
inverter considered in this paper is depicted in Fig. 2.
A two-level converter consists of six switches in each
leg instead of two. The circuit is operated by switching
S1, S2, S3, S4, S5 and S6. The inverter uses two pairs of
complementary controlled switches in each inverter
phase or leg, (S1, S2), (S3, S4) and (S5, S6) as shown in
Fig. 2. The two switches in each inverter phase or leg
operate in a complementary mode in order to avoid
short circuiting the DC source.
The state of the switches is determined according to:
Phase/leg a
1
0
if S ON and S OFF
if S OFF and S ON
(1)
Phase/leg b
1
0
if S ON and S OFF
if S OFF and S ON
(2)
Phase/leg c
1 if S ON and S OFF
0 if S OFF and S ON
(3)
Finite Set-Model Predictive Current Control Of Three-Phase Voltage Source Inverter
for RES (Renewable Energy Systems) Applications
751
Fig. 1 VSI with AC filter for resistive-inductive load for
RES applications.
Fig. 3 Voltage vectors generated by 2-level VSI.
L fL = L f + L L
Fig. 2 Two-level voltage source inverter circuit topology.
This leads to eight different switching possibilities
voltage vectors because V0 and V7 produce the same
voltage vector (V0 = V7). That means a three-phase
two-level voltage source converter can deliver only 7
different voltage vectors, although there are 8 different
switching combinations, as can be seen in Fig. 3. The
switching states define the value of the output voltages
are determined according to:
(4)
vaN = SaVDC
vbN = SbVDC
(5)
vcN = ScVDC
(6)
where, VDC is the DC source voltage and can be
expressed the output voltage vector in vectorial form by:
(7)
where,
2 /3 where, α
e
/
.
2.2 Load Model
The first step that the differential equation of the
load current is applied to obtain the continuous-time
state-space equations of the load for each phase as in
Eq. (8), before that L filter is added to resistive-inductive
load to be LfL = Lf + LL as shown in Fig. 4.
RL
LL
RL
Lf
LL
RL
n
VDC (t) = RLi(t) + LfL
the voltages VDC and -VDC can be switched. It can be
generated by the inverter result in only seven different
LL
Lf
Fig. 4 Representation of the AC filter with inverter load.
and consequently, eight voltage vectors are obtained,
noticed in Fig. 3 that the voltage vectors which are
Lf
(8)
Using Clarke transformation, the equations can be
expressed in the stationary α-β frame.
Clarke transformation is defined as following:
α = 2/3(a – 0.5b – 0.5c)
(9)
β = 2/3(0.5√3b – 0.5√3c)
(10)
Applying Clarke transformation on the load current in
Eq. (8), the equations can be expressed in the stationary
α-β frame as in Eq. (11). Then, the continuous-time
state-space equation of the load result to:
0
0
(11)
0
0
2.3 Discrete-Time Load Model
A discrete-time equation for the future load current
is obtained by using Euler-forward equation as in Eq.
(12) in order to obtain discrete-time system
representation. The derivation of the state variables is
approximated as follows:
(12)
where, Ts is the sampling time and k is the current
sampling cycle. The state variable is denoted with x.
Then, the discrete-time load model can be calculated
to:
752
Finite Set-Model Predictive Current Control Of Three-Phase Voltage Source Inverter
for RES (Renewable Energy Systems) Applications
1
1
1
0
0
1
0
(13)
0
Eq. (13) is used to predict the load current for each
switching possibility. The cost function g is evaluated
for each of the seven possible voltage vectors
generated by this inverter to calculate the future value
of the load current. The voltage vector that minimizes
the cost function is selected and applied during the
next sampling instant.
3. Control Algorithm
3.1 Finite Set-Model Predictive Control
An FS-MPC does not need use a modulator.
FS-MPC has been used as a current controller [12, 13]
for two- [2-4], three- [14] and four- [15, 16], level
inverters. Fig. 5 shows VSI converter under FS-MPC,
where: iref represents the reference current for the
predictive current control, i(k) is the m measurements
taken at time k and i(k+1) are the predicted values of the
m states for n possible switching states at time k+1. The
error between the reference and predicted values is
obtained to minimize the cost function and the
switching state that minimizes the cost function is
chosen. The chosen state’s switching signals, S, are
then applied to the converter.
In order to reduce the computational effort which
gives rise to multiple possibilities (eight different
switching possibilities for one prediction step), the best
one of the seven voltage vectors is determined; after
that the best switching state which delivers this voltage
vector is determined. In order to determine the best one
of the seven voltage vectors that should be applied in
the next sampling cycle k + 1, the 1-norm cost function
has to be minimized, for the current control of a 2-level
VSI, a simple cost function can be defined in absolute
value term with one prediction step and n prediction
step respectively, as:
Fig. 5 FS-MPC block diagram.
1
1
1
1
(14)
∑
(15)
In order to simplify the calculations, it can be
assumed that the current reference values are constant
in Eq. (15) with the prediction horizon for small
sampling time TS as in Eq. (16). This approximation is
considered in this paper.
(16)
For more accurate approximation, the reference
1 for α-β axis required by Eq.
current values
(14) have to be predicted from the present and previous
values of the current reference using a second-order
extrapolation given by:
1
3
3
1
2 (17)
Eq. (17) has been obtained from the quadratic
Lagrange extrapolation formula [3, 17].
In general, the control algorithm as shown in Fig. 5
can be summarized to the following steps [18]:
(1) Measure the load currents;
(2) Predict the load currents for the next sampling
instant for all the possible switching states;
(3) Evaluate the cost function for each prediction;
(4) Optimal switching state is selected which
minimizes the cost function;
(5) Apply the new switching state.
In the current control case, the cost function is
minimized which obtained from the error between the
reference current and predicted current for produce the
switching state and applied.
3.2 Prediction Step
For prediction step i(k + 1), the controller calculates
the measured currents at k + 1 for all possible switch
states at k in order to bring it closer to the reference
Finite Set-Model Predictive Current Control Of Three-Phase Voltage Source Inverter
for RES (Renewable Energy Systems) Applications
value. In the next step, i(k + 2), the controller thus
calculates the measured currents at k + 2 for all
possible switch states at k + 1. The output current with
RL and LfL can be expressed as:
2
2
1
1
0
0
1
1
0
753
sinusoidal reference current was applied to the system
and the amplitude of the reference current was set to 4
A and frequency of 50 Hz per phase. A
resistive-inductive load is connected to the output of
the VSI. Table 1 shows the parameters used for the
simulations.
4.1 Control Strategy Robustness under Variable
DC-Link
1
When an inverter is controlled directly it gives rise to
multiple possibilities. In this case, the inverter leads to
8 different switching possibilities and consequently,
eight voltage vectors are obtained for one prediction
step, with long prediction horizon the calculation effort
rises, for two prediction steps 64 possibilities.
As mentioned in Section 2.1, the voltage vectors
which are generated by the inverter result in only seven
different voltage vectors because V0 and V7 produce
the same voltage vector, that means a three-phase
two-level voltage source converter can deliver only
secen different voltage vectors, for two prediction steps
with 49 possibilities.
The first simulation shows the robustness of the
proposed control method under variable DC-link
voltages, when the DC-link voltage was changed from
90 V to 150 V for Ts = 25 µs. Fig. 6 shows the output
currents for different values of DC-link voltage. It can
be observed from Fig. 6 that the proposed control
algorithm has the ability to track sinusoidal reference
currents although the DC-link voltage was changed
around the desired voltage. In Fig. 6a, DC-link was set
to 90 V and the THD was 2.5%, whereas in Fig. 6b,
DC-link was set to 110 V and THD was decreased to
0.83% and in Figs. 6c and 6d, the DC-links were set to
130 V and 150 V and the THDs were decreased to 0.96%
and 1.11%, respectively. It can be noticed that lower
than 110 V the THD are increased a lot whereas higher
than 110 V the THD and amplitude of the output
current are increased marginally, because of the AC
filter was designed for ±110 V. Table 2 shows the THD
and fundamental output current for variable DC-link
voltages.
It can be observed from this simulation that the
predictive control method has has the ability to to track
sinusoidal reference currents and shows excellent
tracking behavior with all DC-link voltage values.
4. Simulation Results
Table 1
(18)
0
1
With long prediction step/horizon the load current
result to:
1
1
0
0
1
1
0
1
(19)
0
1
FS-MPC strategy for three-phase two-level VSI has
been simulated with Matlab/Simulink, in order to
evaluate the performance of the proposed control
algorithm and check the performance and robustness of
the proposed predictive control method for a VSI with
AC filter supplied by a PV and wind turbine. A
Parameters used for the simulations.
Parameter
Value
Load resistor, R
10 Ω
Load inductor, L
10 mH
AC filter, L
10 mH
DC link voltage, VDC
100 V
Reference amplitude current, iref
4A
Sampling time, Ts
25 μs , 75 µs
Finite Set-Model Predictive Current Control Of Three-Phase Voltage Source Inverter
for RES (Renewable Energy Systems) Applications
754
Table 2 THD and output current for variable DC-link
voltages.
THD (%)
2.50
0.83
0.96
1.11
Fundamental (50 Hz)
3.977
3.998
4.000
4.002
4.5
4
3.5
3
2.5
4
i L & i re f ( A )
DC-Link value
90
110
130
150
6
2
0.041
0.043
0.044
-2
-4
-6
0.02
6
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
0.045
0.05
0.055
0.06
Time (s)
4
iL & iref (A)
0.042
0
(a)
2
100
0
-2
50
-6
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
Time (s), THD = 2.5%
(a)
6
i L & i ref (A)
4
-50
0
-100
0.02
-2
-6
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
Time (s), THD = 0.83%
(b)
4
2
0
-2
-4
-6
0.02
0.025
0.03
0.035
0.04
Time (s)
6
i L & iref (A)
0
2
-4
0.025
0.03
0.035
6
0.04
Time (s), THD = 0.96%
(c)
0.045
0.05
0.055
0.06
4
iL & i ref (A)
v  (V )
-4
(b)
Fig. 7 (a) The load current and voltage for a sampling time
Ts = 75 µs; (b) the load current and voltage for a sampling
time Ts = 25 µs.
with Fig. 7b for sampling time Ts = 75 µs, the ripple
has been reduced when sampling time is reduced for Ts
= 25 µs. However, by reducing the sampling time, the
switching frequency is increased as can be seen by
comparing the load voltage vα in Figs. 7a and 7b.
2
4.3 With Step Change in the Amplitude of the Reference
0
-2
The second simulation shows the control result for
-4
-6
0.02
0.025
0.03
0.035
0.04
Time (s), THD = 1.11%
(d)
0.045
0.05
0.055
0.06
Fig. 6 Output currents for different values of DC-link
voltage.
4.2 With One of Prediction Step for Sampling Time Ts
= 25 µs and Ts = 75 µs
The robustness of the proposed control method was
tested with one prediction step for sampling time Ts =
25 µs and Ts = 75 µs. Current and voltage in the load
are shown in Fig. 7. It can be seen that the control
algorithm shows excellent tracking behavior and
noticed that how the output currents were tracking their
references for both sampling times. In Fig. 7a, it can be
observed that there is a noticeable ripple compared
sinusoidal reference values, when a step change in the
amplitude of the reference iα was changed from 4 A to 2
A at 0.042 s and the amplitude of the reference iβ was
kept at 4 A, as shown in Fig. 8. It can be seen in Fig. 8,
how the load current iα reaches to its reference with fast
dynamic response while iβ is not affected by the step
change.
The result for step changes in the amplitude of the
references iα and iβ were changed from 4 A to 2 A and 4
A to 3 A at 0.042 s and 0.048 s, respectively, are shown
in Fig. 9a. It can be observed in Fig. 9b for the same test
that during the step change of the currents iα and iβ the
load voltage vα is kept at its maximum value until the
reference current iα is achieved.
Finite Set-Model Predictive Current Control Of Three-Phase Voltage Source Inverter
for RES (Renewable Energy Systems) Applications
10
4
2
(A )
5
L
& i , i
L
0.04 0.041 0.042 0.043 0.044 0.045 0.046
re f
i , i
re f
0
-5
-10
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
Fig. 8 Load current with Sinusoidal reference steps for a
sampling time Ts = 25 µs.
6
2
L
L
& i  , i  (A )
4
re f
i , i
ref
0
-2
-4
-6
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.06
0.07
0.08
Time (s)
(a)
100
v  (V )
50
0
-50
-100
0.02
0.03
0.04
0.05
Time (s)
(b)
Fig. 9 Sinusoidal reference steps: (a) load current with two
step changes; (b) load voltage vα changes during the
reference steps.
This simulation clearly demonstrates the ability of
the proposed control algorithm to track sinusoidal
reference currents. It can be observed from this
simulation that the predictive control method has fast
dynamic response with inherent decoupling between iα
and iβ. It is again, the algorithm that shows excellent
tracking behavior.
5. Conclusions
In this paper, an FS-MPC strategy for a three-phase
VSI for RES applications has been presented and
implemented. The RES model has been used in this
paper to investigate the system performance when
power is supplied to resistive-inductive load. The
proposed control does not need to use modulator. The
computational effort which gives rise to multiple
755
possibilities with long prediction horizon has been
reduced analytically. The control algorithm has been
evaluated with three different cases through
simulation results. First of all, the robustness of
control strategy under variable DC-Link has been
done in terms of the THD, the simulation results show
that the predictive control method has the ability to
track sinusoidal reference currents and shows
excellent tracking behavior with all DC-link voltage
values. Secondly, the robustness of the proposed
control method with one prediction step using
different sampling time has been assessed; the
assessment has been done by checking the load
current and voltage between the reference and actual
load. It has been noticed that the control algorithm
provides very good current tracking behaviour. Lastly,
with step change in the amplitude of the reference, the
simulation results show that the predictive control
method has fast dynamic response with inherent
decoupling between iα and iβ and also shows the load
voltage vα is kept at its maximum value during the
step change until the reference current iα is achieved.
Simulation results show that FS-MPC strategy for
two-level VSI with AC filter for resistive-inductive
load supplied by a RES gives very good performance
under these conditions.
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