Robust Integral Linear Quadratic Control for Improving

PV System Based on Four Leg Interleaved Boost Converter

MOHAMED CHERIF DAIA EDDINE OUSSAMA1, CHEBABHI ALI2, KESSAL ABDELHALIM1

1LPMRN Laboratory, University of Bordj Bou Arreridj, ALGERIA.

2GE Laboratory, Faculty of Technology, University of M’sila, ALGERIA.

Abstract- In this paper, a new robust integral linear quadratic controller (ILQC) is proposed for Four Leg interleaved boost

converters (FLIBCs) uses in the photovoltaic systems. Compared to classical boost converters (CBC), IBCs are used in

the high power and voltage application. Therefore, the IBC can convert a high-current low-voltage input to a low-current

high-voltage output and presents higher efficiency, lower current ripple, and better reliability. In order to enhance the

photovoltaic system robust performances as reliability and efficiency of the converter, the proposed robust ILQC is

calculating with the consideration of equal current sharing. Results of the proposed technical are compared with those of a

classical boost converter (CBC) and FLIBC based on PI control. Performances of FLIBC based on proposed ILQC are

tested in several simulations using Sim Power Systems and S-Function of MATLAB/SIMULINK. It is observed that the

ILQC based FLIBC is maximizes the conversion efficiency of photovoltaic systems, improving the response time, reduce

the overshoot of the waveforms, and decrease the current ripple. Compared to classical PI control, the proposed robust

ILQC can increase the efficiency of conversion under different irradiance levels.

Keywords: Photovoltaic system; Four Leg Interleaved Boost Converter (FLIBC); Integral Linear Quadratic Controller

(ILQC); Power Quality; Steady-State Error.

Received: May 29, 2021. Revised: February 12, 2022. Accepted: March 14, 2022. Published: April 21, 2022.

1. Introduction

In recent years, renewable energies have become a

major research topic due to the high prices of traditional

energies, and the emergence myriad environmental

problems such as pollution and global warming resulting

from these traditional energies. In the last few years,

Photovoltaic system (PV) is become increasingly

important as a green energy resource that is among the

most widely used as a promising technology to replace

the traditional energies[1]–[3]. DC/DC converter is one

of the important parts that used in photovoltaic systems

to control the delivered power/voltage and to boost the

Photovoltaic output voltage into higher voltage level [4].

Several DC–DC converters topologies were proposed

and used in the vast literature related to Photovoltaic

systems, one of the important power converters that used

is the DC/DC boost converter, but it still not able to give

the demanded of load power if the load voltage level is

higher than the input voltage level [5]. To overcome the

conventional boost converter problems, an Interleaved

Boost Converter (IBC) has been proposed in [6]–[10].

IBC consists of parallel CBC connected to the same

source and the same output. The feature of that topology

is sharing the input current among the phases, reducing

the input and output current ripple and the output voltage

ripple[11]. The interleaved DC–DC boost converter is

extensively utilized to boost the voltage into high

voltage ratio, due to its advantages compared to DC/DC

boost converter as a low current-ripple, high efficiency,

better reliability and in particular, the IBC can convert a

high-current low-voltage input to a low-current high-

voltage output [9], [10], [12]. Technical challenges of

the IBC driving researchers to elaborate control

strategies for IBCs based PV systems to improve their

performances and to ensure maximum power point

tracking of a photovoltaic system. In[12], an high

voltage gain IBC with MPPT based on radial basis

function network is compared with conventional MPPT

based on P&O and fuzzy logic at different irradiation

levels. In[13], output voltage control are proposed based

on PI control for four phase IBC. Furthermore, Yin et

Tun demonstrated the good performances gives in the

application of linear PI control for average input control

of two-phase IBC [14]. Nevertheless, during the

parameter variations and the coupled control channels of

IBC, the manner of control was presented aren’t suitable

for the PV systems and may lead to a lack of robustness

to operating conditions. The PV system based IBC show

highly nonlinear behaviour making linear controllers not

effective. Researcher shows that several intelligent and

advanced nonlinear controllers have been proposed and

widely used for IBCs [15]–[17] to decouple the control

channels, improving the dynamics of linear PI regulators

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DOI: 10.37394/23202.2022.21.4

Mohamed Cherif Daia Eddine Oussama,

Chebabhi Ali, Kessal Abdelhalim

E-ISSN: 2224-2678

Volume 21, 2022

to have increase the robustness, the stabilization, gives

good regulation on the dc voltage and the currents by the

elimination of input and output current ripple and the

output voltage ripple. The advanced nonlinear control

takes an important part in the PV based on IBC. To

ensure maximum power point tracking of a PV system

based on IBC, to regulate the output voltage, reduce the

inductor current ripple, and also to ensure the sharing of

total current carried between the different converter

phases, El Fadilet al.[18]proposed a nonlinear adaptive

sliding mode controller of a three-phase IBC to ensure

asymptotical stability. However, the adaptive law is

limited to the external parameter variations. Thounthong

et al. proposed a control law based on the differential

flatness for IBC which given a solution to attain the

maximum power point tracking without using a

complicated algorithm [19]. In [20]a sliding mode

controller was introduced to enhance the performance of

the IBC to achieve the robustness and stability and

taking into consideration the nonlinearity of the PV

system based on IBC, this control is examined with

classical PI controller to prove the high performance of

the presented control. In [21] a robust control has been

applied to an IBC using a hybrid strategy. Mohammad

Rasool Mojalli zadeh et al. proposed a switched linear

control to improve the performance of the PV system

based on IBC[22]. On the other hand, a simple linear

quadratic controller (LQC) proposed in [23] compared

with classical PI regulator in terms of robustness,

references tracking under external parameter and loads

variations, this technique offers a high good performance

and is insensitive to external parameter and loads

variations. Habib et all. [24]compared between the LQC

based GA technique and the PI controller under

undulation of current and load, as well as voltage

variations. The LQC is a robust control technique that

gives optimal control for linear systems with a given

weighting matrices Q and R proposed in [23], [24].

Therefore, the dynamic performance of LQC uses in the

PV reference maximum power point trackingcan

deteriorate with some steady-state error introduced due

to PV is subjected to vary with time [25].

To reduce this steady-state error and to increase the

performance of a LQC, several other researchers are

proposed a small modification of LQC by the

introduction of integral action at the recently LQC. In

[24], an LQR controller based on Genetic algorithms

(GA) for two phases interleaved boost converter of fuel

cell voltage regulation is proposed. In [25], a hybrid

integral LQC (ILQC) is proposed for two phases

interleaved boost converter based microgrids under

power quality events which compared the performance

between the ILQC technique and the classical LQC. In

this research work, a robust ILQC technique of the

FLIBC based PV system is developed and proposed. The

proposed technique is based on the integral action for

reduce the steady-state error and to increase the

performance in the two control loops of MPPT, which

used to the PV voltage control loop for maintain a

constant DC voltage at the desired value and to generate

the reference current of the current control loop which

permitting a good extraction and permanent of the

maximum power from the PV system. The outputs of

current control loop are the duty cycles of FLIBC which

shifted by (360/4) degree from each other.The

performance of the proposed controller is proven by

comparing its response with CBC and FLIBC based PI

controller through simulation tests using

Matlab/Simulink based Sim Power Systems and S-

Function, in order to evaluate the success, performance,

robustness, effectiveness, and the ability of this technical

to respond with minimal steady-state errors, lower

voltage and current ripples under any external

disturbance and parameter variations.

2. Mathematical Modeling Of FLIBC

The interleaved DC–DC boost converter is extensively

utilized in PV sources to boost the voltage into high

voltage ratio and to maximize the efficiency of

conversion as shown in Fig. 1, due to its advantages as a

low current-ripple, high efficiency, better reliability and

in particular, the FLIBC can convert a high-current low-

voltage input to a low-current high-voltage output. The

schematic of FLIBC is consist four boost converters

connected in parallel to the same PV system and output

filtering capacitor. The switches have the same

switching frequency and 90-degree phase shift. The

inductor resistance is neglected. The resistor R is the

load.

Fig. 1 Four legs interleaved boost converter topology.

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Mohamed Cherif Daia Eddine Oussama,

Chebabhi Ali, Kessal Abdelhalim

E-ISSN: 2224-2678

Volume 21, 2022

Further, the differential equations that describe the

appropriate dynamic model of the ICB topology are

required to design the control. By evaluating the

derivative of the four inductor currents and output

capacitor voltage corresponding to the state of circuit

when the switch Sj is ON, give the following dynamic

equations:

PV PV in

L V ; j 1, 2,3, 4 (1)

C I I (2)













When the switch Sj is OFF, the dynamic equations are

given by:

PV o

PV PV in

L V V ; j 1, 2,3, 4 (3)

C I I (4)

  











Where ij is the inductor current, Vpv, and Ipv are PV

system voltage and current respectively, Iin the FLIBC

input current. L=L1=L2=L3=L4 input inductor and C1

input capacitor.

By using the switch state Sjϵ {0,1}, the differential

equation describes the FLIBC dynamic performances are

presented in (5) and (6):

PV j o

PV PV in

L V (1 S )V ; j 1, 2,3, 4 (5)

C I I (6)

   











The average model of PV system is used to get the state

space form. By replacing the switch state Sj by its

average value dj during a sampling period (<Sj>= dj).The

differential equation describes the FLIBC dynamic

performances are given by:

PV j o

PV PV in

L V (1 d )V ; j 1, 2,3, 4 (7)

C I I (8)

   











3. Control Approach

Fig. 2 shows the control approach. It comprises two

parts, first part is the MPPT algorithm, and the other part

is a dual loop control (two cascade PV current and

voltage loops). The MPPT algorithm provides the PV

system voltage reference to reach the maximum power

point (MPP).The output of the voltage control loop act

as a reference value of current control loop to ensure the

equal sharing of the current between the phases of

FLIBC. The State feedback control strategy has been

applied to allocate the poles of the closed-loop system.

ILQC control allows calculating the state feedback gain

by minimizing the performance index (PI) J. The

optimization of PI is done by selecting two matrices Q

and R, the weighting matrices for the state variable and

the input variable, respectively. To design the ILQC

controller, a state-space plant is required.

Fig. 2 the control scheme of FLIBC based on ILQC

technique.

Consider a Linear time-invariant system (LTI) given by

its general form of state-space model:

x(t) A x(t) B u(t) (9)

y(t) C x(t) D u(t)









Where x(t) is the state vector, u(t) is a control vector; A,

B, C, and D are the state matrix, control matrix, output

matrix, and feed-forward matrix, respectively. For the

infinite horizon LQC problem, the time-invariant

quadratic PI supposes the form:

J (x (t) Q x(t) u (t) R u(t)) dt (10)







Where Q is symmetric, positive semi definite matrix and

R is symmetric, positive definite matrix.

In order to drive the PV system to their MPP and

maximize the efficiency of conversion, the control that

optimizes the PI is given by:

u(t) Kx(t) (11)

And K presented as follow:

K R B P (12)





Where P is the solution of algebraic Riccati Equation

(ARE), provided by the following equation:

T 1 T

A P PA PBR B P Q 0 (13)



   

3.1. Proposed ILQC-MPPT Voltage and Current

Controller Loops

In order to extract the optimal and permanent maximum

power from the PV system, an ILQC is developed for

the two PV current and voltage loops to track the PV

voltage to MPP voltage, and keep it constant at the

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Mohamed Cherif Daia Eddine Oussama,

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E-ISSN: 2224-2678

Volume 21, 2022

desired value by the adjusting of FLIBC duty cycles.

The first ILQC loop is proposed to insure the PV voltage

regulation and generate the inductance reference current

for the second proposed current ILQC loop which

insures the PV current regulation to generate the FLIBC

duty cycles with lower ripple in the voltage and current,

and with minimal steady-state errors. ILQC-MPPT law

depends on the PV voltage error; it represents the

movement of the MPP operating point on the PV

characteristics.

a) ILQC Voltage Control Loop

Let us consider the state input and output vector as

follows:

PV PV in PV

x [V ] u [I I ] y V (14)   

From (8), (9), and (14) the outer loop system matrices

are as follows:

A 0 B C 1 D 0 (15)

   

To eliminate the steady-state error, an integral action is

suggested.The new state space is given by the following

presentation:

 

A 0 B 0

X u r

C 0 0 1 (16)

y C 0 x

   

     

   

   

     



     

   







So, the new matrices become as follows:

 

00 C

A B C 1 0 D 0 (17)

10 0



 

   

 



 



Where:

 

K K K (18)

b) ILQC Current Control Loop

To design the current controller, a state space is required

where the state vector, input, and output vector

considered as:

j PV j o j

x [i ] u [V (1 d )V ] y i (19)    

From (7), (9) and (19) the inner loop matrices are

defined as:

A 0 B C 1 D 0 (20)

   

Scale the reference with gain N will scale the output to

the desired level.

N (C(BK A) B) (21)





The duty cycle dj that will be delivered to the PWM

block derived from the control vector of the inner loop

where:

j PV

d 1 (22)





Fig. 3 shows a block diagram of proposed ILQC-MPPT

voltage and current controller loops.

Fig. 3. Block diagram of proposed ILQC-MPPT

voltage and current controller loops.

4. Results and Discussion

In order to validate the proposed ILQC technique, a

four phases interleaved DC-DC boost converter based

PV system simulation model has been developed using

Sim Power System and S-Function of

MATLAB/Simulink. Block diagram of the control

schemes are shown in Fig. 2. The system has been

simulated under varying irradiation and the temperature

has been maintained constant (25º C) as shown in Fig. 4.

The subjects of these simulations are the study of

following aspects: (a) The PV system output current, the

converters input current, and the improvement of PV

system output power quality for FLIBC rating in

comparison with CBC controlled by conventional PI due

to four phases IBC controlled by conventional PI and

proposed ILQC. (b) The effects of proposed ILQC for

the response time and overshoot, the current ripple, the

error between the PV power and their MPP reference,

compensation of four phases interleaved DC–DC boost

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converter currents, PV system output current and

converter input current under changing irradiation. The

system and controllers simulation parameters are

presented in Table 1 and Table 2 respectively.

Performance comparison of the all converter topology

and their controllers (CBC based PI controller, FLIBC

based PI controller, and FLIBC based ILQC) is given in

the figures (Figs. 4–11).

Table 1. the Simulink model parameter values

PV system

Vco

21.83 V

Vmpp

17.27 V

Isc

5.33 A

Impp

4.93 A

Pmax

4x85.15 W

FLIBC

CPV

63uF

3.2 uF

L1=L2=L3=L4

8 mH

320

50 KHz

Table 2. The PI and ILQC parameter values

ILQC

Voltage control

loop

Kp= 0.2

Ki= 161.28

0.01 0

0 2800







R= 0.1

Current control

loop

Kp=50.27

Ki= 78977

Q= 342

R= 0.0171

Fig. 4 the solar irradiation used in the simulation.

Fig. 5. PV system output current for all converters topology and control (CBC based PI controller, FLIBC based PI

controller, and FLIBC based ILQC).

Fig. 5 illustrated the PV system output current behaviors

under irradiation varying from 600 to 1000 W/Km2 and

IpvMPP reference current varying from 3 to 5 A and

inversely in the all converter topologies and controllers.

The comparison of these behaviors show that the PV

system output current track perfectly the IpvMPP reference

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current with zero steady state error in the all converter

topologies and controllers, it can be observed that the

overshoot and ripple in the all points (A,B,C and D) and

periods (A,B,C and D) (zoom of all points) are greatly

reduced with very small response time in the case of

ILQC based FLIBC compared to others converter

topologies and controllers, as shown in the four zoom of

Fig. 7. It is also clearly observed in all points and

periods that the ILQC reject the all perturbation at the

variation of irradiance. The comparative study of

overshoot, ripple and response time based on the

simulation results of the all converter topology and their

controllers has been achieved and presented in Table III.

The input current of all converters and four inductors

currents are shown in Fig. 6, 7 and 8 respectively. The

comparison of the input current of all converters under

varying irradiance in terms of ripple, steady state error,

overshoot, and response time is shown in Figs. (6 and 7),

it is observed that ILQC based FLIBC has enhanced its

performance than the others converter topologies and

controllers as is lesser rise time, very better response

time, zero overshoot and steady state, and more

robustness under all perturbation at the variation of

irradiance. This comparison is detailed in Table III.

Similarly, Fig. 8 shows the four inductors currents

behaviors for the FLIBC based on ILQC and PI,

respectively. It is observed that the four inductors

currents are equal and very low ripple in both ILQC

based FLIBC and PI based FLIBC, and each inductor

current equal to one-fourth of the FLIBC input current in

all points and periods under all varying irradiance which

confirmed that the FLIBC is capable to ensure the equal

current sharing between four inductors.

The behaviors of PV system output power in the all

converter topologies and controllers are shown in Fig. 9.

Based on these behaviors,it is observed thatthe

disturbances of the irradiation changes are rejected in the

all converter topologies and controllers, and the

behaviors increases the power conversion efficiency of

the FLIBC topology compared to the CBC topology as

shown in the four zoom of Fig. 9. It is also clearly

observed in all points and periods that the ILQC based

FLIBC converges to MPP with very small response time

and zero overshoot and zero steady state error compared

to PI based CBC and PI based FLIBC under the all

perturbation at the variation of irradiance, which

confirms the effectiveness and the good dynamic

performances of the PV system based on FLIBC

controlled by ILQC in terms of power and current

quality.

Fig. 6. Input current ripple for all converter topology and controllers.

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Mohamed Cherif Daia Eddine Oussama,

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Fig. 7. Input current response time and overshoot for all converter topology and controllers.

Fig. 8. Four legs interleaved boost converter inductors currents.

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Fig. 9. PV system output power for all converter topology and controllers.

Table 3 show a comparison between the ripple value in

the input current, output current, inductors current and

output voltage in each converter scheme. The

comparison show that the ripple value is reduced with

the FLIBC topology and the ILQC control show

superiority compared to the PI controller in this point of

view.

Table 3. Ripple of the input current, output current, and

the output voltage in each case.

CBC with PI

FLIBC with

ILQC

Iin(A)

0.1760

0.0703

0.0682

Io(A)

0.0022

0.0004

0.0003

IL(A)

0.1611

0.1594

Vo(V)

1.3740

0.1110

0.1050

5. Conclusion

In this paper an interleaved DC-DC boost converter

connected to a PV system based ILQC is proposed. The

proposed scheme allows controlling the PV system

voltage and assuring extracting the maximum power and

the equal sharing of input current between each phase of

FLIBC.

The Results of ILQC are compared with the results of

CBC and FLIBC based PI which shown that the

proposed ILQC is more satisfactory and improves the

performance of the system. Therefore, the FLIBC based

ILQC is suitable to use it for enhance the conversion

efficiency in the photovoltaic applications.

In this research work, a FLIBCbased on ILQC

techniquehas beendeveloped and proposed for the PV

system application. The proposed technique is based on

the integral action for reduce the steady-state error and to

increase the performance in the two control loops, which

used to the PV voltage control loop for maintain a

constant DC voltage at the desired value and to generate

the reference current of the current control loop which

permitting a good extraction and permanent of the

maximum power from the PV system. FLIBC is

proposed to reduce allcurrent ripples, sharing the FLIBC

input current in equal between the four leg inductors,

and to reduce the power switches problems, thus

enhancing the efficiency of the FLIBC.To validate the

performance and the effectiveness of the proposed

FLIBCbased on ILQC it has been compared with CBC

based on PI and FLIBCbased on PI through simulation

tests using Matlab/Simulink based Sim Power Systems

and S-Function under varying irradiance. The simulation

comparative standard performance and robustness results

for all converter topologies and controllers

demonstratethat the proposed FLIBC based on ILQC

performed better than the CBC based on PI and

FLIBCbased on PI.

The proposed FLIBC based on ILQC can be used in PV

system for several power applications such as renewable

energy sources, electric vehicles, motor drives, battery

chargers, and power quality enhancement in grids, which

gives good dynamic performance as response time and

lower current ripple, as well as in PV system output

power and voltage. The advantageous of the uses of

FLIBC for maximum power point extraction from PV

system are the good dynamic response time and very

lower ripple.

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DOI: 10.37394/23202.2022.21.4

Mohamed Cherif Daia Eddine Oussama,

Chebabhi Ali, Kessal Abdelhalim

E-ISSN: 2224-2678

Volume 21, 2022

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WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2022.21.4

Mohamed Cherif Daia Eddine Oussama,

Chebabhi Ali, Kessal Abdelhalim

E-ISSN: 2224-2678

Volume 21, 2022