Nonlinear control strategy of single-phase unified power flow controller

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 11, No. 4, August 2021, pp. 2864~2875
ISSN: 2088-8708, DOI: 10.11591/ijece.v11i4.pp2864-2875  2864
Journal homepage: http://ijece.iaescore.com
Nonlinear control strategy of single-phase unified power flow
controller
Younes Abouelmahjoub1
, Mohamed Moutchou2
1
Department of Industrial Science and Technology, LABSIPE Lab, National School of Applied Sciences,
Chouaib Doukkali University in EL Jadida, Morocco
2
Department of Electrical Engineering, National Higher School of Arts and Grafts,
Hassan II University in Casablanca, Morocco
Article Info ABSTRACT
Article history:
Received Aug 13, 2020
Revised Dec 29, 2020
Accepted Jan 19, 2021
In this work we propose a nonlinear control strategy of single-phase unified
power flow controller (UPFC), using in order to enhance energy quality
parameters of a perturbed single-phase power grid supplying nonlinear loads.
The control objectives are: i) The current harmonics and the reactive power
compensation, that ensure a satisfactory power factor correction (PFC) at the
point of common coupling (PCC); ii) compensation of the voltage
perturbations (harmonics and sags of voltage) in order to ensure the desired
level, of load voltage, without distortion; iii) DC bus voltage regulation. The
considered control problem entails several difficulties including the high
system dimension and the strong system nonlinearity. The problem is dealt
with by designing a nonlinear controller with structure including three
control loops. The inner-loop regulator is designed using the Lyapunov
technique to compensate the current harmonics and reactive power. The
intermediary-loop regulator is designed using the Backstepping technique to
compensate the voltage perturbations. The outer-loop regulator is designed
using a linear PI to regulate the DC bus voltage. The control stability is
proved theoretically and through simulations, these latter show the
effectiveness and strong robustness of the proposed control, and prove that
the above-mentioned objectives are achieved.
Keywords:
Backstepping technique
Harmonics compensation
Lyapunov design
Nonlinear control
Power factor correction
UPFC system
This is an open access article under the CC BY-SA license.
Corresponding Author:
Younes Abouelmahjoub
Department of Industrial Science and Technology
National School of Applied Sciences, Chouaib Doukkali University
Azemmour road, National No. 1, El Haouzia BP: 1166 El Jadida 24002, Morocco
Email: abouelmahjoub.y@ucd.ac.ma
1. INTRODUCTION
Nowadays, the increased use of computer equipments and power electronics-based devices on
electrical grids contributes to the degradation of the electrical energy quality. In fact, the power electronics
dedicated to electrical engineering as well as the electronics of computer equipments essentially contribute to
the proliferation of harmonic disturbances. These devices, called nonlinear or distorting loads, generate
harmonic currents which cause the distortion of the voltage waveform at the PCC (due to load current in the
grid impedance) [1]. In addition, the presence of these harmonic disturbances in electrical installations
becomes a real 'headache' for producers and users of electricity in the industrial, tertiary and domestic sector.
Now, the concerns of distributors and consumers of electricity focus on improving the power factor.
The harmonic pollution that affects the electricity supply grid has led electricity producers and distributors to

Int J Elec & Comp Eng ISSN: 2088-8708 
Nonlinear control strategy of single-phase unified power flow controller (Younes Abouelmahjoub)
2865
take this new constraint seriously into account with a view to finding solutions for a better of the electrical
energy quality.
In recent times, active power filters (APFs) have proved to be the best modern solution to cope with
electrical power quality issues. Various APFs configurations exist namely: shunt APF [2-4], series APF [5]
and UPFC system [6], which combines both parallel and series structures of APF. The UPFC system injects
at PCC, appropriate current and voltage signals of harmonics compensation, which ensure the cancellation of
all disturbing harmonics, in the electrical distribution networks, upstream of the PCC [7-15].
In this study, we are interested in single-phase UPFC system connected between the perturbed
power grids and the nonlinear loads. The problem of controlling single-phase UPFC system has arouse a
great of interest over the last years. In this respect, several control strategies have been proposed. The work
[16] presented a microcontroller program used to control the single-phase UPFC system. The obtained
experimental result showed a good agreement with the simulation result. The paper [17] proposed a real-time
control system for a single-phase UPFC, both obtained simulation and experiment results have proved the
effectiveness of the proposed control system. In [18], an attempt was made to separate the series part and the
shunt part of UPFC, this gives the possibility to install series and shunt parts of UPFC at different required
locations. Finally the responses of two modified DC link UPFC are compared. In [6], an implementation for
controller to control the single-phase UPFC using the DSP-TMS320C31 is developed. In [19], a new hybrid
technique which combines the radial basis function (RBF) neural network with the sliding mode technique is
proposed to design a UPFC system for power ﬂow control of an electric power transmission system. In [20],
a proportional resonant (PR) controller is used to generate switching signals for each leg of the input full-
bridge converter; by synthesizing an appropriate injection voltage, the half-bridge inverter controls the power
flow of the transmission line; in addition to simulation of system by MATLAB/Simulink, an experiment
realization on the UPFC was carried out. In all the previous studies, the objective was controlling the power
flow in the transmission systems between the power grids and the consumer loads in order to improve the
quality of electrical energy; but note that the performances of the controllers which were proposed in
[6, 16-21] were not quantified by the indication of the current THD and the voltage THD. Furthermore, in
most previous works, the grid internal impedance was supposed to be zero [6, 16, 18, 19]. In real life, this
impedance is nonzero and must be considered in single-phase power grid model.
In the present paper, the problem of controlling single-phase UPFC system, associated with
perturbed single-phase power grids supplying nonlinear loads, is addressed considering reduced-part
topology. A novel nonlinear controller is designed and formally shown to meet the PFC requirement, the
compensation of the voltage perturbations, and the regulation of DC bus voltage. A major feature of the new
control design is that none of the limitation of previous controllers is present i.e. the grid internal impedance
is not neglected. The structure of proposed nonlinear controller contains three control loops. The inner-loop
is designed, using the Lyapunov technique, to ensure a perfect PFC. The intermediary-loop is designed, using
the Backstepping technique, to compensate the voltage perturbations. The outer-loop is established in order
to regulate the DC bus voltage, through filtered PI regulator. The closed-loop control system analysis is
presented in this paper to proof, that all control objectives are actually achieved by the proposed controller.
Several simulation results show that additional robustness features are reached. Compared to the controllers
previously cited, the new nonlinear controller enjoys several features including:
 The present controller is designed for the single-phase UPFC system with reduced topology, this latter
features less switches and a smaller number of gate drivers, compared to the Two-Leg Full-Bridge
topology used in [6, 18, 19, 20]. As a result, the present nonlinear controller is simpler to implement
because it involves less control signals to generate and apply.
 The control design proposed, in works [6, 16, 18, 19], relies upon very restrictive assumption e.g. the
internal impedance of the single-phase power grid was supposed to be zero [6, 16, 18, 19] which entails
an approximate model used in control design (because the model dimension is smaller than that of the
true system). The present study does not rely on this above assumption. Therefore, the system model used
in the control design is of higher dimension leading to a higher performances controller.
 The performances of the proposed controller were quantified by calculation of the THD values of current
and voltage. Such a study based in terms of THD current and voltage was missing in the all previous
works [6, 16-21].
 By using a rigorous theoretical analysis, the control objectives (i.e. PFC, compensation of voltage
perturbations and DC voltage regulation) are actually achieved. Such a formal analysis was missing in the
all previous works [6, 16-21].
This paper organized as follows: Section 2 is devoted to the description and modeling of the single-
phase UPFC system. The design of the nonlinear controller is treated in section 3. The closed-loop control
system analysis is presented in section 4. In section 5, the controller performances are illustrated by several
numerical simulations.

 ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 11, No. 4, August 2021 : 2864 - 2875
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2. PROBLEM FORMULATION
2.1. Single-phase UPFC topology
The proposed single-phase UPFC system is shown in Figure 1. It contains two inverters back-to-
back, connected to the DC bus side including two identical energy storage capacitors 𝐶𝑑𝑐. The IGBT-diode
based inverters operate in accordance with PWM [22, 23]. From the AC side, the single-phase UPFC system
is connected on the one hand in parallel with the perturbed power supply grid through filtering inductor
(𝐿𝑝, 𝑟𝑝), on the other hand in series with a nonlinear load via filtering inductor (𝐿𝑠, 𝑟𝑠), capacitor 𝐶𝑠 and
current transformer. The perturbed power grid is modeled by a disturbed voltage source 𝑣𝑔 in series with an
internal impedance formed by a resistor 𝑟
𝑔 and an inductor 𝐿𝑔.
Figure 1. Single-phase UPFC system
The switching functions and of the single-phase UPFC system are defined as:
;
The load current , in steady-state, is a periodic signal that can be expressed as (1).
(1)
where is the current harmonic amplitude of order , is the « h » harmonic phase at the origin.
2.2. Single-phase UPFC modeling
The instantaneous model of the single-phase UPFC system is given by:
(1 )( )
g
g g g g s L
di
L r i v v v
dt
     (2a)
0
(1 )( )
2 2
fp p d
p p fp s L
di v v
L r i v v
dt

      (2b)
2
(1 )( )
s
s s fs s L
dv
C m i m i
dt
  (2c)
p
 s

1 3
p
1 3
1 if S is ON and S is OFF
1 if S is OFF and S is ON



 


2 4
s
2 4
1 if S is ON and S is OFF
1 if S is OFF and S is ON



 


 
L
i t
1
( ) ( )
L h g h
h
i t Ι sin h t
 


 

h
I h h


2867
0
(1 )( )
2 2
fs s d s
s s fs
s
di v v v
L r i
dt m

     (2d)
0
(1 )( )
dc p fp s fs
dv
C i i
dt
 
   (2e)
(1 )( )
d
dc fp fs
dv
C i i
dt
   (2f)
where 𝑚𝑠 is the transformation ratio of current transformer,𝑣0 = 𝑣1 + 𝑣2 and 𝑣0 = 𝑣1 − 𝑣2.
The model (2a-f) is useful for building up an accurate simulator of the UPFC system. However, it
cannot be based upon in the control design as it involves binary control inputs, namely 𝜇𝑝 and 𝜇𝑠. This type
of difficulty is generally overcome by resorting to average models where instantaneous signals are replaced
by their average shapes. The signals are averaged over cutting intervals [22, 24]. The average model of the
single-phase UPFC system is expressed as (3a), (3b), (3c), (3d), (3e), (3f):
1 1 3
(1 )( )
g g g L
x L r x v x v
     (3a)
5 6
2 2 3
(1 )( )
2 2
p
p p L
u x x
x L r x x v
      (3b)
2
3 4
(1 )( )
s s s L
x C m x m i
  (3c)
5 6 3
4 4
(1 )( )
2 2
s
s s
s
u x x x
x L r x
m
     (3d)
5 2 4
(1 )( )
dc p s
x C u x u x
   (3e)
6 2 4
(1 )( )
dc
x C x x
   (3f)
where: 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6, 𝑢𝑝, 𝑢𝑠 are respectively the averaged variables , , , , , , , .
3. CONTROLLER DESIGN
The proposed nonlinear controller for the system (3a-f) represented by Figure 2 will be developed in
three major steps, respectively devoted to: i) the inner loop design, ii) the intermediary loop design, iii) and
the outer loop design. The first step is to design an inner-loop control, using the Lyapunov technique, to
ensure a perfect PFC. The second step is to design an intermediary-loop control, using the Backstepping
technique, to ensure the compensation of voltage perturbations in the power grid. In the third step, an outer-
loop control, involving filtered PI regulator, is built-up to achieve DC bus voltage regulation.
3.1. Current inner-loop design
According to the PFC requirement, the current 𝑥1 provided by the single-phase power supply grid
must be a sinusoidal signal in phase with the fundamental of grid voltage namely 𝑣𝑔1. To this end, the current
𝑥2 injected by the single-phase UPFC system should follow as closely as possible its reference 𝑥2
∗
as (4).
(4)
where 𝑣𝑔1 = 𝐸1sin(𝜔𝑡), and 𝛽 is any positive constant. As a matter of fact, the latter is allowed to be time-
varying but it must converge to a constant value. That is, 𝛽 stands as an additional control input. To achieve
the PFC objective, we introduce the tracking error on the filter current .
(5)
g
i fp
i s
v fs
i 0
v d
v p
µ s
µ
*
2 1
L g
x i v

 
2
x
1 2 2
*
z x x
 

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Using (3b), the time-derivative of (5) yields the following dynamics of the error 𝑧1.
1 2 5 6 3 1 1
(( ( 2) ( 2) ) ) ( )
p p L p L g g
z r x u x x x v L di dt v v
 
         (6)
The control variable, noted 𝜇𝑝, appears in (6) after a single derivation of the error 𝑧1. This control
variable must now be determined in order to make 𝑧1-system globally asymptotically stable. To this end, we
introduce the following Lyapunov function candidate:
2
1 1 2
V z
 (7)
Its dynamic is given by:
(8)
To ensure 𝑧1-system global asymptotical stablity, it is sufficient to choose the control law 𝜇𝑝 so that
𝑉̇1 = −𝑐1𝑧1, and then we obtain:
(9)
where 𝑐1 is any positive parameter. Comparing (9) and (6) yields the following control law:
2 3 6 1 1 1 1 5
(2 2 2 2 2 2 2 )
L
p p L p p p g p g
di
u r x x x v L c z L L v L v x
dt
 
        (10)
As this control law 𝜇𝑝 involves the dynamics of the signal 𝛽, it follows from (6) that the signal 𝛽
and its first time-derivative must be available.
3.2. Voltage intermediary-loop design
In order to compensate the voltage disturbances at the PCC, the load voltage 𝑣𝐿 must be a sinusoidal
signal at the terminals of sensitive load:
(11)
According to the voltage disturbances compensation requirement, the series voltage 𝑥3 injected by
the single-phase UPFC system should follow as closely as possible its reference signal 𝑥3
∗
as (12).
(12)
where 𝑣𝑝𝑐𝑐 is the voltage at the point of common coupling.
The proposed regulator must force the voltage 𝑥3 to track its reference signal 𝑥3
∗
. The synthesis used
is known as the Backstepping technique [25] and is carried out in two steps.
 Step 1: Stabilization of tracking error 𝑧2.
(13)
The time-derivative of (13) yields the following dynamics of the error 𝑧2:
 
2 *
2 4 3
( ) ( )
s s s L s
z m x C m i C x
   (14)
We use the following Lyapunov candidate function:
2
2 2 2
V z
 (15)
1 1 1
V z z

1 1 1
z c z
 
*
max sin( )
L L L
v v V t

 
* *
3 pcc L
x v v
 
2 3 3
*
z x x
 

2869
Its dynamic is given by:
 
2 *
2 4 3 2
( ) ( )
s s s L s
V m x C m i C x z
   (16)
Consider that (𝑚𝑠𝑥4/𝐶𝑠) is the effective control, 𝜎 is the stabilizing function, it is sufficient to take:
2 *
2 2 3
( )
s L s
c z m i C x
     (17)
where 𝑐2 is a positive parameter.
To study the stability of the above control, we define the following corresponding error 𝑧3:
3 4
( )
s s
z m x C 
  (18)
Then, we get the equation of 𝑧2 dynamics, and that of 𝑉2 derivative:
(19)
(20)
 Step 2: Stabilization of the subsystem (𝑧2, 𝑧3)
In this step 2, we present the design of the controller that makes the errors (𝑧2, 𝑧3) to tend to zero. The
𝑧3 dynamics is given in (21):
 
 
3 4 5 6 3
( 2) ( 2) ( )
s s s s s s
z m r x u x x x m C L 
      (21)
The control variable, noted 𝑢𝑠, appears for the first time in (21). Let us consider the following
Lyapunov function.
2
3 2 3
( 2)
V V z
  (22)
Using (20), the time-derivative of 𝑉3 is given by:
(23)
To ensure that the (𝑧2, 𝑧3)-system to be globally asymptotically stable, it is sufficient to choose the
control law so that which, due to (23), amounts to ensuring that:
(24)
where 𝑐3 is a positive parameter.
From the (21) and (24), we deduce the following backstepping control law 𝑢𝑠:
 
4 6 3 2 3 3 5
2 (2 ) 2 ( )
s s s s s s
u r x x x m C L z c z m x

       (25)
Remark: The control laws (10) and (25) involve a division by the DC bus voltage 𝑥5, there is no risk
of singularity (division by zero) in steady state because, in practice, the DC bus voltage remains all the time
positive. Otherwise, the two power converters of the UPFC system cannot work.
3.3. Voltage outer-loop design
According to the DC bus voltage regulation requirement, the outer-loop regulator generates an
adjustment law for the signal 𝛽 so that the DC bus voltage 𝑥5 is regulated to its reference value 𝑥5
∗
. To this
end, the relation between 𝛽 and the voltage 𝑥5 is established see in Figure 2.
2 2 2 3
z c z z
  
2
2 2 2 2 3
V c z z z
  
 
2
3 2 2 3 2 3
V c z z z z
   
s
u 2 2
3 2 2 3 3
V c z c z
  
3 2 3 3
z z c z
  

 ISSN: 2088-8708
2870
Figure 2. Synoptic scheme of the multi-loop nonlinear control of UPFC system
 The DC bus voltage 𝑥5 varies, in response to the signal 𝛽, according (26a):
5
x  1 1 2 3 2 1 2 3 5
( , , , , ) ( , , , , , ) 2
K f z z z f z z z t x
   
 
 
  (26a)
 Then, the squared-voltage 𝑦𝑑 = 𝑥5
2
varies, in response to the signal 𝛽, according to the following time-
varying linear (26b):
1 1 2 3 2 1 2 3
( , , , , ) ( , , , , , )
d
y K f z z z f z z z t
   
   (26b)
with 2
max 1 1 1 1 1
1
(4 ( ) 4 ( ) 2 )
L p h dc
h
K V I cos E I cos r I C
 


   
  2
1 1 2 3 1 1 1 1 1 1 1 1
2 2 2 2
1 1 1 1 2 1 1
, , , , (4 ( ) ( ) 2 ( ( ) sin( )) 2 ) )
(2( ) ) 2(( ) ) (4 ( ) )
g p g g dc
g p dc g p dc p p dc
f z z z E I r r cos E I L cos E C
r r E C L L E C z r z z L c z C
        
 
    
       
The expression of 2 1 2 3
( , , , , , )
f z z z t
  is complex and so is no cited in this paper.
The signal 𝛽 stands as a control input of the system (26b). The problem is to design for 𝛽 a tuning
law so that the squared voltage 𝑦𝑑 = 𝑥5
2
tracks its reference 𝑦𝑑
∗
= 𝑥5
∗2
. At this stage, the signal 𝛽 and its first
time-derivative must be available, the following filtered PI regulator is adopted:
6 4 4 5 5 6
( ) ( )
c c z c z c s
    (27a)
with and , where denotes the Laplace variable, and the parameters
4 5 6
( , , )
c c c , of the outer-loop regulator, are any positive real constants. The choice of these parameters will
be clearly mentioned in the following analysis so that the control objectives are achieved. For now, let us
note that (27a) implies that can be computed using (27b):
6 4 4 5 5
( )
c c z c z
 
   (27b)
*
4 d d
z y y
  5 4
0
t
z z d
  s


2871
4. CONTROL SYSTEM ANALYSIS
The following notations are needed to formulate the results:
2 2
2 6 1 1 1 1 1 1 1 1 1 1
((4 ( ) ( ) 2 ( ( ) sin( )) (2( ) 2 ) ))
p g g g p g o dc
k c k E I r r cos E I L cos r r E E C
    
       
, 1 g
 
 , 2
1 1
2 ( )
o g p dc
k E L L C

   ,
2
0 1 2 3 5 6 1 1 2 5 6 1 1 2 3 5 6 1 1 5 6 2
a c c c c c k c c c c k c c c c c k c c c k
     ,
2
1 1 2 3 5 6 1 4 5 6 1 2 5 6 2 3 1 2 3 1 2 3 4 5 6 1
2 3 5 6 1 2 3 5 6 1 2 3 5 6 5 6 1 4 6 2
( ( ) ( ( ) ) )
( ( ) )
a c c c c c c c c c c c c c c c c c c c c c c c c k
c c c c c c c c c c c c c c c c c c c k
        
     
,
2
2 4 6 5 6 1 2 3 5 6 1 2 3 1 4 2 3 4 5 6 1 4 2 3 6 5 1
2
2 3 5 6 1 5 6 2 3 5 6 1 2 3 5 6 2 1 2 3 6 1 6 4 5 6
( ( ) ( ( )(1 ) ) ( ))
( ( ) ( ) )
a c c c c c c c c c c c c c c c c c c c c c c c c c k
c c c c c c c c c c c c c c c c k c c c c c c c c c
          
        
,
2 2
3 5 6 2 5 6 1 5 6 2 3 5 6 2 3 2 5 6 2 3 1 2 3 4 5 6 2 3 6
2
1 4 5 6 1 5 6 1 1 5 6 2 1 2 3 4 6 1 1 2 3 6 1 2 3 6 4 5 6 1 1
( ) ( ) ( )
( ) ( )
a c c k c c k c c c c c c c c k c c c c k c c c c c c c c
c c c c c c c k c c c k c c c c c k c c c c c c c c c c c k c
         
          
,
2
4 5 6 2 5 6 1 4 5 6 4 5 6 2 3 6 2 3 2 3 1 4 6 1 2 3
( ) ( ) ( ) ( ) ( ) 1
a c c k c c k c c c c c c c c c c c c c c c c c c c
           
5 1 2 3 6 4 1
(1 )
a c c c c c k
     .
Theorem 1. We consider the single-phase UPFC system shown in Figure 1, represented by its
average model (3a-f), in closed-loop with the nonlinear controller including the following components:
 The inner-loop regulator (10), where is any positive parameter.
 The intermediary-loop regulator (25), where and are any positive parameters.
 The outer-loop regulator (27a-b), where 4 5 6
( , , )
c c c are any positive parameters, the choice of which will
be clearly done later.
Then, the global closed-loop control system has the following properties:
 The tracking error 𝑧2 = 𝑥2 − 𝑥2
∗
vanishes exponentially fast with 𝑥2
∗
= 𝑖𝐿 − 𝛽𝑣𝑔1.
 The tracking errors 𝑧2 = 𝑥3 − 𝑥3
∗
and 3 4
( )
s s
z m x C 
  vanish exponentially fast with .
 The augmented state vector 𝑍(𝑡) = (𝑧1 𝑧2 𝑧3 𝑧4 𝑧5 𝑧6)𝑇
undergoes state (28a):
𝑍̇ = 𝑓(𝑡, 𝑍) (28a)
Where
  1 1 2 2 3 2 3 3 1 2 4 6 4 4 5 5 6
, ( ) ( ) ( ) ( ( ) ( , )) ( )
T
f t Z c z c z z z c z K f Z f Z t z c c z c z z
          
 
  (28b)
 The control parameters (𝑐1, 𝑐2, 𝑐3, 𝑐4, 𝑐5, 𝑐6) are chosen so that the following inequalities hold:
1 5 4 3 2 5 2 1 5 1 1 3 2 5 3 1 3 5 2 5 1 2 0
1 1 0 5
2 2
3 1 5 2 2 1 1 5 3 1 5 2 2 1 1 5
1 3 2 5 0
, , 0, 0, 0, 0, 0
( )
(( ) ) ( ) 0, ((( ) ) ( )) 0
( )
b a a a b a a a a b b a b a a b a a b a b b a
b a a a
a b a b b b a a a b a b b b a a
b a b a a
           

 
        
 
 
 
(29)
Then, there exist positive constants 𝜀∗
and 𝜂∗
such that for all 0 < 𝜀 < 𝜀∗
, the system (28a-b) has a
unique exponentially stable (2 )
g
  -periodic solution ( , )
Z t  with the property * *
0
( , )
Z t Z
  
 
(where  
*
0 0 5 0
0 0 0 0
T
Z c
 
 ). With
2
4 4 3
0
3
4
2
k k k K
k

    


, 2
3 1
2( )
p g dc
k r r E C
   ,
2
4 1 1 1 1 1 1 1 1
(4 ( ) ( ) 2 ( ( ) sin( )) 2 )
g p g g dc
k E I r r cos E I L cos E C
   
    
5. NUMERICAL SIMULATION
The control system described in Figure 2 is simulated using the MATLAB/SIMPOWER toolbox
(V.R2013a). The controlled system is a single-phase UPFC which is connected between the disturbed power
supply grid and the nonlinear load based on a bridge rectifier, the latter supplying a load composed of resistor
in series with inductor 𝐿. The inductor 𝐿0 protects the bridge rectifier against abrupt voltage changes. The
numerical values, of single-phase UPFC system parameters, are placed in Table 1.
6
z 

1
c
2
c 3
c
* *
3 pcc L
x v v
 
R

 ISSN: 2088-8708
2872
Table 1. Single-phase UPFC system characteristics
Parameters of Symbol Values
Single-phase power grid , , 220√2𝑉/50 𝐻𝑧, ,
Single-phase UPFC system p
r , p
L , , , s
C , dc
C , 3 𝑚𝐻 , , 3 𝑚𝐻, 500 𝜇𝐹, 9000 𝜇𝐹
Nonlinear load (Bridge rectifier
and RL circuit)
, , , 20  , 500 𝑚𝐻
5.1. Control performances in presence of harmonics at power grid voltage
The simulation vises at illustrating the controller behavior in response to step changes of the DC bus
voltage reference 𝑣0
∗
. Taking into account the system parameters values of Table 1, the simulation profile is
described by Figure 3 which shows that the DC bus voltage reference switches from 800 V to 900 V at time
(0.25 s) and return up to its nominal value at time (0.6 s) while the load is kept constant (𝑅=20 Ω,
𝐿=500 mH). In this case, the circuit is supplying by a polluted power grid which its voltage contains the
harmonics. The resulting controller performances are illustrated by Figures 3-10. The DC bus voltage 𝑣0
converges to its reference value with a good accuracy see in Figure 3. Furthermore, it is observed that the
voltage ripple oscillates at the frequency of 2𝜔𝑔, but its amplitude is quite small see in Figure 3 compared to
the average value of the signal 𝑣0. Figure 4 shows the waveform of the load current 𝑖𝐿. It is seen that the
latter presents a serious harmonic distortion. Figure 5 reveals a serious distortion in the voltage 𝑣𝑔 of the
power supply grid. Its calculated harmonic distortion rate is around 𝑇𝐻𝐷𝑣𝑔(%) = 24.58%, which
recommends performing compensation. To better appreciate the controller performances, Figure 6 clearly
shows that the grid current 𝑖𝑔 remains sinusoidal all the time and in phase with the fundamental of the grid
voltage namely 𝑣𝑔1. This confirms that a satisfactory PFC is well ensured. Also, Figure 7 shows that the
voltage 𝑣𝐿, at the terminals of the nonlinear load after compensation, becomes sinusoidal. This confirms that
the voltage perturbations compensation is well ensured. Figure 8 shows the load current spectrogram, where
the THD value of this current equal to 32.04%. Figure 9 shows that the THD value of grid current after
compensation is very low (3.82%). This latter value is below the limits of the IEEE-519 standard. Figure 10
shows the grid voltage spectrogram, where its THD value equal to 24.58%. Figure 11 shows that the THD
value of load voltage after compensation is very low (0.68%). This latter value is below the limits of the
IEEE-519 standard.
Figure 3. DC bus voltage and its reference Figure 4. Load current 𝑖𝐿
Figure 5. Distortion in the grid voltage 𝑣𝑔 Figure 6. Current and fundamental voltage of grid
1
E g
r g
L 50m 6
0.5 10 H


s
r s
L 80m 80m
o
L R L 5mH
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
750
800
850
900
950
Time(s)
vo*
vo
0.1 0.12 0.14 0.16 0.18 0.2
-10
-5
0
5
10
Time(s)
0 0.02 0.04 0.06
-400
-300
-200
-100
0
100
200
300
400
Time(s)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-200
-150
-100
-50
0
50
100
150
200
Time(s)
0.25x vg1
ig

2873
Figure 7. Load voltage after compensation Figure 8. FFT analysis of load current
Figure 9. FFT analysis of grid current after
compensation
Figure 10. FFT analysis of grid voltage
Figure 11. FFT analysis of load voltage after compensation
5.2. Control performances in presence of a sags at power grid voltage
Figure 12 to Figure 16 illustrating the behavior of the proposed controller in the presence of the
voltage sags in the power grid. Indeed, Figure 12 reveals a sag in the signal of the grid voltage 𝑣𝑔, the depth
of which is 90% of the nominal voltage and its duration is 100 ms, which exceeds the values fixed by the
standard EN 50160. Figure 13 clearly shows that the grid current 𝑖𝑔 remains sinusoidal all the time and in
phase with the grid voltage 𝑣𝑔. This confirms that a satisfactory PFC is well ensured. As for Figure 14, it
shows that the voltage 𝑣𝐿 at the terminals of the nonlinear load after compensation is a sinusoidal signal, in
this case the voltage sag decreases sharply and its depth becomes 7% which is less than the value of 10%
imposed by standard EN 50160. The harmonics spectrum of the load current is plotted in Figure 15, it is
noted that the THD value of this current equal to 32,04%. It is observed in Figure 16 that the THD value of
grid current equal to 3.82%. This latter value is below the limits according to IEEE-519 standard.
0.1 0.12 0.14 0.16 0.18 0.2
-400
-300
-200
-100
0
100
200
300
400
Time(s)

 ISSN: 2088-8708
2874
Figure 12. A sag in the signal of grid voltage vg Figure 13. current and voltage of grid
Figure 14. Load voltage after compensation Figure 15. FFT analysis of load current
Figure 16. FFT analysis of grid current after compensation
6. CONCLUSION
The problem of controlling the single-phase UPFC system of Figure 1 is addressed in presence of
disturbed single-phase power grid supplying nonlinear loads. The complexity of the control problem resides
in the high dimension and the nonlinearity of the system dynamics. The proposed nonlinear controller,
including the inner regulator (10), the intermediary regulator (25) and the outer regulator (27a-b), is
developed by using various tools of control e.g. Lyapunov technique, Backstepping technique. It is shown
that all control objectives are achieved, including PFC requirement, voltage perturbation compensation, and
DC bus voltage regulation. The simulation results illustrate and prove the high performances of the proposed
controller and its strongest robustness. The extension of the present study to the case of three-phase UPFC
system is underway, and also an adaptive nonlinear control strategy taking into account the uncertainty on the
grid impedance applied to single-phase UPFC system is underway.
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0.2 0.25 0.3 0.35 0.4 0.45 0,5
-400
-300
-200
-100
0
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Time(s)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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2875
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