Modular Approach to Implement Model Predictive Control on Three Phase Voltage Source Inverter

International Journal of Power Electronics and Drive System (IJPEDS)
Vol. 9, No. 2, June 2018, pp. 897~904
ISSN: 2088-8694, DOI: 10.11591/ijpeds.v9.i2.pp897-904  897
Journal homepage: http://iaescore.com/journals/index.php/IJPEDS
Modular Approach to Implement Model Predictive Control on
Three Phase Voltage Source Inverter
Muhammad Abbas Abbasi1
, Abdul Rashid Bin Husain2
, Hasan Alqaraghuli3
1,2,3
Department of Control and Mechatronics Engineering, Universiti Teknologi Malaysia (UTM), Malaysia
1
The Islamia University of Bahawalpur, Pakistan
Article Info ABSTRACT
Article history:
Received Dec 21, 2017
Revised Jan 22, 2018
Accepted Feb 7, 2018
A modular abstraction is presented to implement model predictive control
(MPC) on a three phase two level voltage source inverter to control its output
current. Traditional ways of coded implementation do not provide insights
into the complex nature of MPC; hence a more intuitive, logical and flexible
approach for hardware implementation is conceptualized in the form of
signal flow graphs (SFGs) for estimation, prediction and optimization.
Simulation results show good performance of the approach and easier code
generation for real time implementation. RL load is assumed for the inverter
and the importance of choosing load inductance and sampling time ratio is
emphasized for better control performance.
Keyword:
Current control
Model predictive control
Modular MPC
Three phase inverter
Copyright © 2018 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Abdul Rashid Bin Husain,
Department of Control and Mechatronics Engineering,
Universiti Teknologi Malaysia (UTM)
81310 Skudai, Johor, Malaysia
Email: abrashid@utm.my
1. INTRODUCTION
Three phase two level voltage source inverters are widely used in power electronics applications
especially in drive control [1], [2]. Controlling output current of such inverter topologies is also widely
researched topic. Model predictive control (MPC) is emerging as the most promising digital control method
in the field of power electronics and drive applications . It has extensively been studied for induction and
synchronous motor drives, front end rectifiers, multilevel inverters, and DC-DC converters [3], [4]. MPC
demand higher computational costs and its applications will rapidly increase in the field of power electronics
with the advent of high speed microprocessors and digital signal processors. MPC provide numerous benefits
over traditional control methods such as easier handling of system constraints, parameter variations and
easier incorporation of control tasks in the optimization cost function [5]–[7]. Most of the control strategies
treat converters to be linearized systems ignoring their highly nonlinear nature. MPC takes this factor into
account.
A model based predictive control for VSI is presented and discussed in detail in [1]. A space vector
(SV) representation is used to model and analyze the system. An RL load and a simple back EMF model is
used to investigate the performance of the controller. Controller consists of three stages of implementation
i.e. estimation of back EMF, prediction of future currents, and optimization of a cost function involving
current error. The estimation and prediction is based on Euler’s forward approximation formula and
prediction horizon of 1 is assumed.
Most of the controller applications are implemented in the form of C codes, making the working and
troubleshooting of the controller obscure and difficult. Instead of directly implementing the control scheme in
the form of codes, a signal flow graph representation and automatic code generation of the controller is

 ISSN: 2088-8694
Int J Pow Elec & Dri Syst, Vol. 9, No. 2, June 2018 : 897 – 904
898
suggested here for easier modifications, understanding and implementation. A module based SFG will make
it also possible to adopt model referencing for incorporating prediction horizon greater than 1 by simply
repeating the appropriate modules. Tustin discretization rule is adopted for better results and a prediction
horizon equal to 2 is used. Moreover the role of a special resistance L/Ts is elaborated with simulation results
as a gain factor.
In the first section, mathematical model of inverter and load is presented. In section II, detailed
discussion on the proposed MPC approach is given, followed by simulation results and conclusion in section
III and IV respectively.
2. INVERTER AND LOAD MODEL
The inverter and load model is adopted from [8]. The circuit diagram of the system is shown in
Figure 1.
a
S b
S c
S
a
S b
S c
S
a
i
b
i
c
i
dc
V
a
b
c
N
e_back
L
R
n
Figure 1. Three phase voltage source inverter with RL load
A dc source feeds the inverter with a constant supply voltage dc
V . Switches of the inverter are
operated in bipolar fashion i.e. no two switches in the same leg are turned ON or OFF simultaneously.
Therefore the lower switches are represented with a complimentary symbol. Inverter is driving a three phase
RL load with a sinusoidal back EMF _
e back . A space vector representation will be assumed throughout
the discussion. Space vector for three phase quantities ,
x y and z is defined as:
 
2 2
3 3
2
, ,
3
j j
sv x y z x ye ze
 
  
 
 
 
(1)
The inverter voltage space vector can be defined as:
 
2 2
3 3
2
, ,
3
j j
aN bN cN aN bN cN
sv v v v v v e v e
 
  
 
 
 
(2)
Where
aN a dc
bN b dc
cN c dc
v S V
v S V
v S V



(3)
Based on (2) and (3) a table (Table 1) of inverter voltage vectors can be generated for each possible
switching state. Note that 0 7
0
v v
  lie on the origin. Voltage vectors are shown in Figure 2.
The load equation in the form of space vectors can be written as [9]:
_
di
v iR L e back
dt
   (4)
Following vectors are assumed in (4):

Int J Pow Elec & Dri Syst ISSN: 2088-8694 
Modular Approach to Implement Model Predictive Control on Three… (Muhammad Abbas Abbasi)
899
 
 
 
, ,
, ,
_ _ , _ , _
aN bN cN
a b c
a b c
v sv v v v
i sv i i i
e back sv e back e back e back



(5)
3. MODULAR MPC CONTROLLER
The block diagram of the traditional way to implement MPC for current control is shown in Figure
3.
Current References
MPC Inverter Model Load Model
Current Measurements
Figure 2. Traditional implementation of MPC
Table 1. Inverter Switching States and Resulting Voltage Vectors
V1
V2
V6
V5
V4
V3
V1
V2
V6
V5
V4
V3
Figure 3. Voltage vectors Figure 3. Voltage Vector addition for prediction
horizon N=2

 ISSN: 2088-8694
900
For predicting currents, we can discretize (4) using Euler’s forward rule
( ) ( 1) ( )
s
dy t y k y k
dt T
 
 to
get the following approximation [10]:
         
 
1| 1 _
o
o
R
i k k i k R v k e back k
R
     (6)
Where  
1|
i k k
 is the predicted value of current, one sampling step into the future 1
k  and the
prediction is made at time instant k . o
s
L
R
T
 is a special ratio and can be changed by changing the sampling
time s
T [11]. Note that  
v k can take 7 values depending upon the switching combination applied to the
inverter. We will assume all the possible switching states to predict 7 values of the current and choose the
best switching state which produces minimum current error.
For better performance, we will assume prediction horizon equal to 2. Therefore, at time instant
2
k  the current can be predicted to be:
         
 
2 | 1 1| 1 _ 1
o
o
R
i k k i k k R v k e back k
R
        (7)
Since back EMF is slow varying, therefore we can safely assume    
_ 1 _
e back k e back k
  .
Putting (6) into (7) with  
1
1
o
R
c
R
  and 2 o
c R
 and simplifying:
       
     
1 2 1 1 2 2
2 | 1 _
i k k c i k c c v k v k c c c e back k
       (8)
The underlined part in (8) is significant since it represents the voltage vector addition to predict
current two instants into the future. Each vector from 0
v to 6
v is added to the scaled version of entire vector
space. Ignoring 0
v addition, we can anticipate 36 sums. This situation is shown for 1
v addition in fig. 4. Note
that c1 is less than 1, therefore the added vectors are smaller in lengths. In the same way we can get all the
sums and evaluate (8) for different vectors. Number of sums can further be reduced by considering the fact
that o
R is larger than R which will give 1
1
c  .
For better estimation of back EMF, we will use Tustin’s Bilinear Transformation [12] i.e.
1
1
2 1
1
s
z
s
T z





 
 
 
. Applying Laplace transform on (4) will result in:
_
V RI sLI E back
   (9)
After discretization:
     
1 1 1
_ 1 2 2 _
o o
E back z V R R I R R z I z E back
  
       (10)
1
z

represents unit delay and the system is assumed relaxed. Based on equation (6) and equation (10)
the implementation block diagrams are shown in Figure 5 and Figure 6. These block diagrams can be
separately implemented for real and imaginary parts of the system or multiplexed as a vector.

901
-(R+2Ro)
Z-1
-(R-2Ro) +
+
Z-1
+
Z-1
i(k)
vo(k)
e_back(k)
1-RTs /L +
+
-1
i(k)
v(k)
i(k+1|k)
e_back(k)
Ts /L
1
7
1
7
Figure 4. Back EMF estimation module Figure 5. Current Prediction
We can repeat the prediction module twice to get the predictions  
2 |
i k k
 and the process can be
extended to further predictions easily. However, the number of computations will increase
2
N times. There
are various techniques available to reduce these calculations.
After prediction and estimation, we have to choose the best switching state to be applied at the next
sampling interval based on the following cost function.
       
2
, ,
1
| |
ref ref
N
g i k N i k N k i k N i k N k
   

       
 (11)
Here, the currents are referred to the stationary  frame. Assuming that we choose 1
N  , from
fig. 6, we see that there are 7 predictions for current at time instant 1
k  , hence 7 values of cost function in
(11) can be obtained by the implementation shown in Figure 7. The diagram uses basic building blocks
available in most simulation software for code generation. After the calculation of 7 error values between
future reference current and 7 predicted currents, their absolute value is computed. Next block finds the
minimum absolute error ( _
g opt ). To find the voltage vector which produces this minimum error, we
compare _
g opt with all the other values. Only one comparison produces 1 which is multiplied with the
index values to determine optimal state _
state opt to be applied at next sampling interval. For prediction
horizon greater than 1, this module can easily be replicated as required. The overall implementation block
diagram is shown in Figure 8.
-
i(k+1|k)
iref(k+1)
||u||
7
7
min
==
×
0
×
1
×
2
×
6
.
.
.
+
1
1
7
State_opt
g_opt
Load Inverter
Back EMF
Estimation
Current
Prediction
Optimization
v0, v1,…v6 iref(k)
vo(k)
i(k)
i(k)
e_back(k) i(k+N|k)
g_opt
state_opt
Figure 6. Selection of optimal switching states Figure 7. Implementation Block Diagram

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4. SIMULATION RESULTS
The proposed modular approach is implemented in Matlab/Simulink for 10 , 8
L mH R
  . Peak
amplitude of back EMF is assumed to be 120V and frequency of 50 Hz while DC source of 450V is
considered. Figure 9 shows the simulation results of output currents for a sinusoidal reference of amplitude
12A and frequency 50 Hz, for two different values of sampling time i.e. 100 s
 and 20 s
 . Since decreasing
the value of sampling time will require a higher switching frequency, it will significantly increase the quality
of the output currents at the same time giving a boost to the computational demands. Practically, it becomes
computationally heavier to use smaller values of sampling time. Therefore a compromise is made while
choosing the sampling time for less distorted current waveforms.
Figure 8. Effect of sampling time on output current Figure 9. Effect of L/Ts ratio on output currents
Figure 10. Optimal states and cost function for different values of sampling time
Figure 10, on the other hand shows the effect of the ratio L/Ts on the current waveforms. Only
single phase current is showed to explain the effect. If the ratio is kept higher i.e. s
T smaller, current
waveforms are less distorted. If we decrease the ratio by increasing s
T , the current THD increases and
distortions occur. If this ratio becomes equal to R, current predictions are purely made on the basis of
estimated back EMF and voltage vectors. For better results, this ratio should be kept around 100 to 200.
Finally Figure 11 gives the plots of the optimal costs and optimal states calculated by the MPC
controller for a limited frame of time. There is no significant effect of sampling time on the optimal states
0 0.01 0.02 0.03 0.04 0.05 0.06
-10
-5
0
5
10
Time (sec)
Current
(A)
Three phase cureents at Ts=100 us
ia
ib
ic
0 0.01 0.02 0.03 0.04 0.05 0.06
-10
-5
0
5
10
Three phase cureents at Ts=20 us
Time (sec)
Current
(A)
0 0.01 0.02 0.03 0.04 0.05 0.06
-20
-10
0
10
20
Time (sec)
Current
(A)
Effect of L/Ts ratio on Output Currents: L/Ts = 200
Phase a Current
0 0.01 0.02 0.03 0.04 0.05 0.06
-20
-10
0
10
20
L/Ts = 100
Time (sec)
Current
(A)
0 0.01 0.02 0.03 0.04 0.05 0.06
-20
-10
0
10
20
L/Ts = 20
Time (sec)
Current
(A)
0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08
0
1
2
3
4
5
6
7
Time (sec)
Switching
Vector
Optimal Switching Vectors
Ts=100 us
Ts=20 us
0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08
0
0.5
1
1.5
2
2.5
3
Optimal costs
Time (sec)
Cost

903
chosen for the inverter, however optimal costs are significantly reduced when smaller sampling times are
used.
5. CONCLUSION
A modular way to interpret and implement model predictive control (MPC) for a three phase two
level inverter is investigated. The basic stages involved in MPC implementation are broken down to simple
modules based on SFGs and fundamental building block found in all digital control implementation and
simulation software. The approach provides more intuition, flexibility and ease of code generation and
modification of the MPC hardware implementation. This approach can be easily extended to incorporate
higher prediction horizons and complex discretization methods for estimation. Future work may involve the
extension of the approach to other applications such as drive control, increasing the prediction horizon by
using vector representation and reduction of effective switching states through state transition diagrams.
ACKNOWLEDGEMENTS
Authors would like to acknowledge Universiti Teknologi Malaysia (UTM), The Islamia University
of Bahawalpur (IUB) Pakistan and Higher Education Commission (HEC) of Pakistan for providing financial
support to conduct this research.
REFERENCES
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[2] P. Cortés, M. P. Kazmierkowski, R. M. Kennel, S. Member, D. E. Quevedo, and J. Rodríguez, “Predictive Control
in Power Electronics and Drives,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4312–4324, 2008.
[3] A. Iqbal, S. Moinoddin, and K. Rahman, “Finite State Predictive Current and Common Mode Voltage Control of a
Seven-phase Voltage Source Inverter,” Int. J. Power Electron. Drive Syst., vol. 6, no. 3, 2015.
[4] G. Mirzaeva, G. C. Goodwin, B. P. McGrath, C. Teixeira, and M. E. Rivera, “A Generalized MPC Framework for
the Design and Comparison of VSI Current Controllers,” IEEE Trans. Ind. Electron., vol. 63, no. 9, pp. 5816–5826,
2016.
[5] K. Shen and J. Zhang, “Modeling error compensation in FCS-MPC of a three-phase inverter,” in PEDES 2012 -
IEEE International Conference on Power Electronics, Drives and Energy Systems, 2012.
[6] H. H. Goh, A. Aida, S. S. Lee, S. Y. Sim, and K. C. Goh, “Predictive Direct Power Control ( PDPC ) of Grid-
connected Dual-active Bridge Multilevel Inverter ( DABMI ),” Int. J. Power Electron. Drive Syst., vol. 8, no. 4, pp.
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2017.

 ISSN: 2088-8694
904
BIOGRAPHIES OF AUTHORS
Muhammad Abbas Abbasi received his B.Sc. degree in Electronics Engineering from
The Islamia University of Bahawalpur, Pakistan in 2004 and M.S. degree in
Electrical Engineering from University of Engineering & Technology (UET), Taxila,
Pakistan in 2012. Currently he is pursuing his PhD degree in Universiti Teknologi
Malaysia (UTM). He is a full time faculty member at The Islamia University of
Bahawalpur, Pakistan and has also worked in different control industries in Pakistan.
He has taught courses in Control Engineering, Digital Signal Processing, FPGA
based system Design and Electronics Systems. His research interests include Model
Predictive Control of Power Converters and Drives, Predictive Direct Torque
Control, FPGA based Control, and Signal Processing.
A. R. Husain received the B.Sc. degree in electrical and computer engineering from
The Ohio State University, Columbus, Ohio, U.S.A., in 1997, M.Sc. degree in
Mechatronics from University of Newcastle Upon Tyne, U.K., in 2003, and Ph.D. in
Electrical Engineering (Control) from Universiti Teknologi Malaysia (UTM) in 2009.
Before joining UTM, he worked as an engineer in semiconductor industry for several
years specializing in precision molding and IC trimming process. He has taught
courses in introduction to electrical engineering, microcontroller based system,
modeling and control, electric motor and drives, and real-time control system. His
research interests include control of dynamic system, real-time control system, and
mechatronic system design.
Hasan Alqaraghuli received the B.S. degree from University of Technology,
Baghdad, Iraq, in 2014 and the MEng degree from Universiti Teknologi Malaysia,
Johor Bahru, Malaysia, in 2016. He is currently working towards the Ph.D. degree in
Universiti Teknologi Malaysia, Johor Bahru, Malaysia. His research interests include
model predictive control, motor drives, sensorless control of AC motors and electrical
vehicle.

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