An efficient predictive current controller with adaptive parameter estimation in 3-ϕ inverter

International Journal of Power Electronics and Drive System (IJPEDS)
Vol. 12, No. 2, Jun 2021, pp. 858∼869
ISSN: 2088-8694, DOI: 10.11591/ijpeds.v12.i2.pp858-869 r 858
An efficient predictive current controller with adaptive
parameter estimation in 3-Φ inverter
Haddar Mabrouk, Allaoua Boumediene
Smart Grids & Renewable Energy (SGRE) Laboratory, University of Tahri Mohammed, Bechar, Algeria
Article Info
Article history:
Received Dec 1, 2020
Revised Feb 14, 2021
Accepted Mar 19, 2021
Keywords:
Filter
Grid-connected
Inverter
MRAS observer
Predictive control
ABSTRACT
In this paper, a detail design and description of a predictive current control scheme are
adopted for three-phase grid-connected two-level inverter and its application in wind
energy conversion systems. Despite its advantages, the predictive current controller is
very sensitive to parameter variations which could eventually affected on system sta-
bility. To solve this problem, an estimation technique proposed to identify the value
of harmonic filter parameter based on Model reference adaptive system (MRAS). Lya-
punov stability theory is selected to guarantee a robust adaptation and stable response
over large system parameter variation. The simulation results shows the efficiency of
the proposed techniques to improve the current tracking performance.
This is an open access article under the CC BY-SA license.
Corresponding Author:
Haddar Mabrouk
Department of Electrical Engineering
SGRE Laboratory, University of Tahri Mohammed, Bechar, Algeria
B.P 417 route kenadsa 08000, Bechar, Algeria
Email: haddar.mabrouk@univ-bechar.dz
1. INTRODUCTION
In recent years, the grid-connected inverters have made a giant strides in the various applications of
industries and renewable energies , that embody by introduce a new concepts of advanced control in detriment
to other conventional controls. The grid-connected inverters is widely used in wind energy systems. The
simplified structure of the grid-connected inverters is illustrated in Figure 1. Where, it’s possible to replace
the maximum amount of power that can be withdrawn from the wind turbine, generator, and rectifier flowed
through the inverter by adequate variable DC-current source without causing any damage in system caracteristic
[1]-[6].
(a) (b)
Figure 1. Grid-connected inverter in a wind energy concersion system,
(a) General structure, (b) Simplified structure
Journal homepage: http://ijpeds.iaescore.com

Int J Pow Elec & Dri Syst ISSN: 2088-8694 r 859
On the other side, The progress of the micro processor unit allowed the possibility to improve the
controllers design with very short sampling time and good performance. The model predictive control ( MPC)
is one of the most important milestones of development, as it recently occupied a large part of studies and
the researchers’ effort as promise control methods because of its: (1 simple theory concept, (2 easy to im-
plement (3 flexible and promising for digital control, (4 fast dynamic response, [2] ,[3]. However, despite of
its advantages, the MPC is sensitive to the system parameter variations during the operating conditions. This
results of perturbations influences on the overall stability of the system, which makes MPC need to adaptation
or estimation algorithm [7]. The MPC applied for power converters and electrical drives include two main
categories: Continuous Control Set MPC with a modulation stage and Finite Control Set MPC (FCS-MPC)
without a modulation stage [8], [9]. In FCS-MPC, the control objectives in cost function may be either main or
additional [7]-[11], the first one using physics quantities such as current, voltage, power, torque, etc. It evalu-
ates error tracking between any predicted variable value and its reference. While the second one is considering
as sub-objectives control for example switching frequency reduce. The additional control objective multiplying
by suitable values of weighting factors witch’s reflecting its importance in optimization criteria. The algorithm
of FCS-MPC minimizes the cost function and applies the generated switching signal at end each the sampling
time directly on power converter [11], [12].
The robustness of FSC-MPC against parameter variations still major concern in many studies [12].
Addressing the issue, a many control strategies have been proposed to estimate the parameters variable in power
converter applications. In [9], an analytical approach is proposed to exam the influence of model parametric
uncertainties on the prediction of FSC-MPC error for current control in three phase two-level inverter. In [13],
the authors proposed adaptive observer to accurately estimate two variable: speed and flux, this observer based
encoderless FCS-PTC (predictive torque control) helps keeping the stability for induction machine. The Model
Reference Adaptive System (MRAS) observer is used in [14], where the authors estimate line inductance to
improve robustness for sensorless predictive control of Active front End (AFE) rectifiers. In [15], a control
scheme based on a computationally finite-set model predictive power control (FS-MPPC) for grid-connected
photovoltaic systems is proposed, the stability of (FS-MPPC) against inductance variation is verified by means
of novel online finite-set model inductance estimation technique. In [16], an extended Kalman filter (EKF) is
proposed to estimate the system parameter ( filter grid impedance) for AFE.
However, the MRAS technique has been approved to be a powerful tool for parameter estimation, the
MRAS is widely used in power electronics application and motor drives system to mainly estimate: (1 the filter
parameter [16], (2) the machine state variables (speed, flux,...) for sensorless drive systems [18], [19],...etc,
but there are limited researches based on MRAS for grid-connected inverter [16]. The MRAS consists of two
models, the reference one and the adaptive one. The difference between the outputs of these two models is then
used in an adaptation mechanism such as lyapunov theory to adjust the parameters in the adaptive model until
the response of the main two models become consistent and the tracking error converges to zero [19], [20].
On this light, the performance of predictive current controller for the On this light, the performance
of predictive current controller for the three-phase grid-connected two-level inverter entirely depends on its
mathematical model as well as the accuracy of the parameters that may assumed in theory constant and known
but practically not so. In this paper, the filter parameters may easily affected due to the heat risen, magnetic
saturation and system lifetime, etc. This paper presents a structure that combines two types of controller; the
first one is a FCS-MPC witch generate a switching states for the grid-connected inverter but it is influenced
by parameters variation, this could yield negative results as inaccurate prediction of the future behavior due
to an incorrect system model and so the selection of the incorrect switching states. The second is a classic
which relies on the MRAS observer to cope with the aforementioned drawback through a parameter estimation
method based on Lyapunov stability theory. Therefore, it is possible to avoid the deterioration of the control
performance. There are a few researches report on the application the MRAS observer to grid-connected
inverter [17]. So, this novel structure brings great advantages such as simple control scheme, No inner control
loop, and No modulator. Additionally, although the proposed estimation technique is classic, it can make the
system operate robustly and independently from mathematical model.
This paper is arranged as follows: Section (2 discusses the system structure and modeling. Section (3
and section (4, present the theory behind the voltage oriented vector and the predictive current control strategy,
respectively. The identification of the filter parameter based on MRAS observer are presented is given in
Section (5. Section (6 contains The simulation results and analysis, and finally conclusions are discussed in
Section (7.
An efficient predictive current controller with adaptive parameter estimation... (Haddar Mabrouk)

860 r ISSN: 2088-8694
2. SYSTEM STRUCTURE AND MODELING
The conventional power circuit of a three-phase grid-connected two-level inverter applied to wind
energy is modeled as shown in Figure 2. Where, a harmonic filter (rg,Lg) takes place between the grid and the
inverter. The models of the three-phase inverter and grid current dynamic are briefly described in the following
subsections..
2.1. Inverter model
The two-level inverter is composed of six bidirectional switches. Since there are three phases with two
operating modes in each phase, the inverter is able to generate only 23
= 8 states as different possible output
voltages. The operating modes of the inverter are summarized as follows [2]:
Sx + S̄x = 1 for x ∈ {a, b, c} (1)
Figure 2. Topology of grid-connected two-level inverter in wind energy conversion systems
By means of the space vector tool, The output voltage generated by the inverter can expressed in terms
of the different switching states by

vαi
vβi

= Vdc
2
3



1 −
1
2
−
1
2
0
√
3
2
−
√
3
2





Sa
Sb
Sc

 (2)
Where Vdc is DC link voltage, vαi and vβi are α and β component of the inverter output voltage
vectors. Figure 3 shows the eight output inverter voltage vectors in space vectors corresponding the all possible
switching states which including six different active voltage vectors (V1 to V6) and two other zero vectors (V0
and V7).
Figure 3. Space vectors generated by the inverter
The inverter voltage component expressed in the dq-frame rotating at angle θg can be related to the
αβ component by

vdi
vqi

=

cos θg sin θg
−sin θg cos θg

vαi
vβi

(3)
Where vdi and vqi are dq-axis components of the inverter output voltage.
Int J Pow Elec Dri Syst, Vol. 12, No. 2, Jun 2021 : 858 – 869

Int J Pow Elec Dri Syst ISSN: 2088-8694 r 861
2.2. Grid current dynamic model
The continuous-time grid current dynamic equation of three-phase grid-tied inverter can be written in
the dq-frame as [1]-[4]
didg
dt
= −
rg
Lg
idg + ωgiqg +
1
Lg
(vdi − vdg)
diqg
dt
= −ωgidg −
rg
Lg
iqg +
1
Lg
(vqi − vqg)
(4)
Where vdg and vqg are dq-axis components of the grid voltage and ωg is the angular fraquency of the
grid, rg filter resistance and Lg is filter inductance.
The grid active and reactive powers are expressed by [19]:
Pg =
3
2
{vdgidg + vqgiqg}
Qg =
3
2
{vqgidg − vdgiqg}
(5)
3. VOLTAGE ORIENTED VECTOR (VOC)
In the literature, the phase-locked loop (PLL) is needed not only to find the grid voltage angle θg for
the grid synchronization, but also to realize the voltage oriented control (VOC). Where, the dq-axis components
of the grid voltage is oriented to be align only with its d-axis component and eliminate the second component.
That means, the active power equation and reactive power equation are simplified to [20]
Pg =
3
2
vdgidg
Qg = −
3
2
vdgiqg
(6)
As described previously, It is noted that the d-axis current reference is i∗
dg related to the grid active
power. It can be adjusted through PI controller which maintains the measured DC-link voltage vdc at its given
reference value v∗
dc , we can write
i∗
dg = (kp +
ki
s
)(v∗
dc − vdc) (7)
The q-axis current reference i∗
qg can be calculated from reference reactive power of the grid Q∗
g as
i∗
qg(k) = −
Q∗
g
1.5v∗
dg
(8)
4. PREDICTIVE CURRENT CONTROL (PCC)
According to the 8 switching possible states for two-level inverter, the predictive current controller
exploit the discrete-time model with one-step for grid-connected inverter to predict at the next instant the
future behavior of the controlled grid current. It consists of three major subsystems: extrapolation of reference
currents, predictive model, and cost function minimization. At each sampling time, cost function values are
calculated for all of possible commutation states, based on predefined criterion. Then, the smallest value of the
cost function will be used to determine the optimal switching state applied to the inverter at the next period [2].
Approximating the grid current derivatives by [1]-[3]
dig
dt
≈
ig(k + 1) − ig(k)
Ts
(9)

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The discrete-time model on dq-frame of grid currents at (k + 1) state is expressed as follows:

idg(k + 1)
iqq(k + 1)

=

1 − rgTs/Lg 0
0 1 − rgTs/Lg

idg(k)
iqg(k)

+

Lg 0
0 Lg

vdi(k) − vdg(k)
vqi(k) − vqg(k)

(10)
However, the future reference of the grid current i∗
g,dq(k + 1) can be estimated by means of current
and previous value of the reference current with the help of second-order Lagrange extrapolation as follows
[2], [3].

i∗
dg(k + 1)
i∗
qq(k + 1)

= 2

idg(k)
iqg(k)

−

i∗
dg(k − 1)
i∗
qq(k − 1)

(11)
In this paper, the first target of the predictive current controller is to achieve the smallest current
error between the extrapolated reference currents and the predicted current. The second target is to reduce
significantly the switching frequency by adjusting the commutation number between two successive sampling
instants. In situations where the switching loses are important. In brief, These all objectives can be expressed
in the form of a cost function g to be minimized. The cost function summarizes the desired behavior of the
inverter can be obtained using the error squared such as :
g = i∗
dg(k + 1) − idg(k + 1)
2
+ i∗
qg(k + 1) − iqg(k + 1)
2
+ λsw (S(k) − S(k − 1))
2
(12)
Where λsw is weighting factors for switching frequency reduction, S(k) and S(k − 1) are the present
switching state and past applied switching state respectively. Figure 4 shows the proposed predictive current
controller for the grid-connected two-level inverter. In this scheme, getting the angle θg from the measured grid
voltage is done via PLL method to make vdg equal to Vg and also to transforme the variables measured ( vg
and ig) from abc-frame to dq -frame. Once reference grid currents i∗
g,dq(k) is calculated, the PCC procedure
is established to select the optimal switching signal applied to the inverter. Moreover, the scheme involves the
MRAS observer to improve the robusrness of PCC by identifying the filter parameter.
Figure 4. Proposed predictive current controller scheme for grid-connected two-level inverter.
5. IDENTIFICATION OF THE HARMONIC FILTER PARAMETERS BASED ON MRAS
In the real time, Both values of harmonic filter parameters (inductance and resistance) linked between
the inverter and the grid are changed under the operating conditions that reduce system efficiency in term of
current quality. So that, these parameters are estimated based on the model reference adaptive system (MRAS).

Figure 5 presents the basic structure of MRAS, wich consists of three parts: a reference model ,
adaptive model, and an adaptive controller.
Figure 5. Basic structure of MRAS
The equation of state for reference model:
(4) ⇔
dig
dt
= Aig + Bu (13)
Where ig =

idg iqg
T
, A =

α ωg
−ωg α

, α = −
rg
Lg
, B = −
1
Lg
, u =

ud uq
T
, ud = vdi −vdg,
uq = vqi − vqg
The adaptive model is given as follows:
dîg
dt
= Aîg + Bu (14)
Where îg =

îdg îqg
T
, Â =

α̂ ωg
−ωg α̂

, α̂ = −
r̂g
L̂g
, B̂ = −
1
L̂g
, u =

ud uq
T
ig and îg are the output current of reference model and adaptive model respectively. The measured grid
current is tuning as the reference model witch is strongly reliant on the rated filter parameters (rg and Lg), and
the estimated grid current is tuning as the adaptive model witch uses a value of estimated filter parameters (r̂g
and L̂g), and then, the output current error of two models is used by adaptive controller to adjust the parameter
in adaptive controller until the identification becomes asymptotically stable and the current error approaches
zero as small as possible.
The current error is defined as e = ig − îg = [ ed eq ]T
, the error state equation can be written:
ė = Ae + (A − Â)îg + (B − B̂)r (15)
Letting aI = (A − Â), b = (B − B̂) =
1
Lg
−
1
L̂g
,Z = [ ˆ
ig u ]T
and ∆T
= [ a b ]T
The (15) can be written as
ė = Ae + ∆T
Z (16)
Considering a Lyapunov function candidate as follow:
V (e, t) =
1
2
(eT
Pe + ∆T
Z∆) (17)
Where P = I2×2 and A =

λ1 0
0 λ2
−1
λ1 and λ2 are arithmetic number.
Starting from Lyapunov stability concept, the Lyapunov function must respond the following three
necessary conditions to ensure that the identification is asymptotically stable [22]-[25]:
1) V (e, t) 0,

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2) V̇ (e, t) 0,
3) V (e, t) −→
8
as |e| −→
8
.
It is evident that the first and third conditions are met. The second condition can be discussed as
follows:
V̇ (e, t) =
1
2
(ėT
Pe + eT
Pė + ˙
∆T
Z∆ + ∆T
Z ˙
∆)
=
1
2
(eT
(PA + PAT
)e + a(îded + îqeq +
ȧ
λ1
)
+ b(uded + uqeq +
ḃ
λ2
)
(18)
For PA + PAT
= 2αI 0 it is clear that eT
(PA + PAT
) is negative definite.
The term V̇ (e, t) must be negative definite, this yields
ȧ
λ1
+ îdg(idg − îdg) + îqg(iqg − îqg) = 0 (19)
ḃ
λ2
+ ud(idg − îdg) + uq(iqg − îqg) = 0 (20)
The adaptive law can be easily expressed as following:
r̂g
L̂g
=
rg
Lg
− λ1
Z t
0
n
îdg(idg − îdg) + îqg(iqg − îqg)
o
dt (21)
1
L̂g
=
1
Lg
+ λ2
Z t
0
n
ud(idg − îdg) + uq(iqg − îqg)
o
dt (22)
6. SIMULATION RESULTS
To evaluate the efficiency of the considered predictive current control model with MRAS oberver for
grid connected two-level inverter under various values of parameter filter, the whole simulation studies are
implemented by means of MATLAB/simulink tools. The main parameters are indicated in Table 1.
Table 1. Simulation parameters
Parameter Nomenclature Value
Rated Line-to-Line Voltage (rms) [V] Vg 690
Rated current (rms) [A] Ig 627.6
Rated active power [kw] Pg 750
DC-link voltage [V] Vdc 1220
DC-link capacitor [µF] Cdc 16714
Filter resistance [mΩ] rg 95.25
Filter inductance [mH] Lg 0.3368
Sampling time [µs] fs 20
Frequency of the grid [Hz] fg 50
The results presented in three scenario:
• Ideal Case operation: In this case , the filter parameter are setting on rated values
• Non-ideal Case operation without MRAS: In this case , the parameter filter are changed to show the
influence of parameter variation on whole performance of system.
• Non-ideal Case operation with MRAS: In this case , the MRAS observer is used to estimate the filter
parameter and to enhance the performance of system

6.1. Ideal case operation
The performance of the proposed predictive current controller scheme for three-phase grid-connected
inverters considering the rated value of filter parameter is tested by applying a DC-link voltage reference V ∗
dc =
1220 V, and a zero reference power reactive Q∗
g = 0 to unitary the power factor. Figure 6 (a) shows the
weighting factor impact λsw on the grid current quality via Total Harmonic Distortion (THD) and the average
switching frequency fsw. It’s clearly noted when the weighting factor is increasing, the of the grid current is
changed progressively from 1.94 to 3.57 %, while the average switching frequency is reduced considerably
from 8.86 to 3.56 [KHz]. For this reason, the weighting factor can be adjusting at 1700 for operating the
inverter at fsw = 3.703 [KHz] and for having a suitable value of THD at 3.30 %.
(a) (b)
(c) (d)
Figure 6. Simulation results for case ideal operation,
(a) THD % and fsw versus λsw variation, (b) DC-link voltage [V],
(c) dq-axis grid current [A] (peak), (d) waveforms of phase-a grid current iag [A] (peak)
.
Figure 6 (b) shows the obtained simulation result where the DC-link voltage is controlled to maintain
at its reference value 1200 V and it is stable during steady state operation. This simulated response is obtained
with a PI controller having a kp = 6.90 and a ki = 1.952 103
. The dq-axis grid current components either the
reference i∗
g,dq or the measured ig,dq are tracking each other perfectly as shown in Figure 6 (c). Consequently,
The amplitude of the reference d-axis grid current component i∗
dg created from the voltage PI controller is
equal to 887.5 A (peak). In addition, the grid-connected inverter is controlled to supply zero reactive power
corresponding the zero reference q-axis grid current. Figure 6 (d) illustrates the waveform of a-phase grid
current, where the peak value of iag is equal to idg.
6.2. Non-ideal case operation without MRAS
It’s the case where the variation in filter parameter has been considered as follows:
• Filter resistance and inductance are 50 % lower than its nominal value
• Filter resistance and inductance is also 100 % of its nominal value
• Filter resistance and inductance are 150 % higher than its nominal value
These values are applying at the following moments 0.1 and 0.2 (s) respectively. Figure 7 shows the
effect of parameters variation of both rg and Lg on the overall performance of the controller as well as its
stability. Under these conditions, it is clear to note the tracking errors between the measured values of idg, iqg,
and Vdc and their references, which leads to unbalanced value of THD especially when both the resistance and
the inductance are higher than the nominal values as presented in Figure 8.

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(a) (b)
Figure 7. Simulation results for case non-ideal operation without MRAS,
(a) DC-link voltage [V], (b) dq-axis grid current [A] (peak).
(a) (b) (c)
Figure 8. Simulation results explains the influence of filter parameters variation on the performance of PCC
rg and Lg are: (a) 50 % lower than its nominal value , (b) aquals to nominal value,
(c) 150 % higher than its nominal value.
6.3. Non-ideal case operation with MRAS
Figure 9 shows the dynamics performance of PCC scheme when the MRAS observer based on lya-
punov function is enable under parameters variation. The adaptive gains are tuned using a tuning method
(trial and error) to give suitable performance. After some tuning, the adaptive gains are setting as following:
λ1 = 0.625 and λ2 = 6.795. the simulation results consists of two parts: in the first one, the filter parameters
are increasing and switching from 50 % of its nominal value to nominal value, and in the second one the filter
parameters are decreasing and switching from 150 % of its nominal value to nominal value.

(a) (b)
Figure 9. Simulation results for RL filter parameters estimation based on MRAS on server
(a) starts from 50 % of its nominal value to nominal value; (b) starts from 150 % of its nominal value to
nominal value
The switching is applied at the moment 0.15 (s). In the both simulation, the MRAS observer track and
make the estimated values converges to parameter variation. Moreover, the estimated current îg and the mea-
sured current ig tracks each other. Despite large error at the same time where the switching is applied. the error
current e converges to zero. It is worth noting that the estimate value of the filter resistance is more accurate
than the estimated value of the filter inductance with steady-state estimation error in range of 1.763 10−4
and
0.1 % respectively.
To show the robustness of PCC controller with a MRAS observer for a three-phase grid-connected
inverters, a comparison of performance can be done through two cases: The first is in agreement with the second
scenario, where the filter parameters are changed in the absence of a MRAS observer, while PCC controller is
operating on the normal values. As for the second case, which corresponds to the third scenario, the change
of the filter parameters are in the presence of the MRAS observer, which estimates the filter parameters so that
the PCC controller uses them. The comparison was done by measuring THD and fsw , as shown in Figure 10.
Although the filter parameters are changed, the MRAS observer helps the PCC controller to select a switching
states that corresponds to the minimal values of fsw with slight difference in THD.
(a) (b)
Figure 10. Comparison of system performance by measuring;
(a) THD %; (b) fsw [KHz].

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7. CONCLUSION
Sensitiveness to system parameter variations and its influences on three-phase grid-connected inverters
for wind energy conversion systems has been treating in this paper. A MRAS estimator based on lyapunov
function is developed to identify the RL filter parameter. the performance is verified via computer simulation,
and various scenario has been studied. the simulation results show that the proposed MRAS estimator gives
a good performance and the stability is guaranteed against parameter variations. In the future, a practical
implementation of predictive current control model with MRAS observer for grid connected two-level inverter
is planned. Moreover, a comparison between the adopted MRAS observer and others techniques such as Neural
Network (NN) and Extended Kalman Filter (EKF) will be carried out.
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BIOGRAPHIES OF AUTHORS
Haddar Mabrouk is received his Master’s degree in ”Industrial Electronics” and Bachelor’s Degree
in ”Electronics, Electrotechnics, Automatism” (EEA) from Ferhat Abbes University, Setif, Algeria
in 2012 and 2009, respectively. Actually, He is Ph.D student at Tahri Mohammed University, Bechar,
Algeria. His research subject is about ”modern control techniques applied on the power converters
dedicated to wind energy systems”, specially focused on predictive control of power converters.
His scientific interest include power converters, Advanced control and renewable energy. He is af-
filiated as student member with the Smart Grids Renewable Energies (SGRE) Laboratory, bechar,
Algeria.
Allaoua Boumediene was born in Bechar (Algeria). He received the Ph.D degree in electrical en-
gineering from the University of Tahri Mohamed Bechar. Currently, He is a Professor of Electri-
cal Engineering at Faculty of Technology, Bechar, Algeria. His Research Interests Include Power
Electronics, Smart Grid, Renewable Energies and Intelligent Controls for All Electric Hybrid Power
Vehicles. He is the team member of intelligent energy management in the on-board systems in the
smart grid Renewable Energies (SGRE) laboratory, bechar, Algeria.

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