Design of a new backstepping controller for control of microgrid sources inverter

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 12, No. 4, August 2022, pp. 4469~4482
ISSN: 2088-8708, DOI: 10.11591/ijece.v12i4.pp4469-4482  4469
Journal homepage: http://ijece.iaescore.com
Design of a new backstepping controller for control of
microgrid sources inverter
Mahmoud Zadehbagheri1
, Mohammad Javad Kiani1
, Tole Sutikno2,3
, Rasoul Arvin Moghadam4
1
Department of Electrical Engineering, Islamic Azad University, Yasooj Branch, Yasooj, Iran
2
Department of Electrical Engineering, Universitas Ahmad Dahlan, Yogyakarta, Indonesia
3
Embedded System and Power Electronics Research Group, Yogyakarta, Indonesia
4
Department of Biomedical Engineering, Yasouj University of Medical Sciences, Yasooj, Iran
Article Info ABSTRACT
Article history:
Received Dec 7, 2021
Revised Mar 30, 2022
Accepted Apr 18, 2022
Emergency power supply is becoming an important capability for many
home or industrial electronic and computer devices. Therefore, the
performance of the designed uninterruptible power supplies (UPS) inverters
has low distortion at the output voltage. Initially, such inverters were
controlled by proportional integral (PI) control classic rules. This method is
difficult to understand the limitations of stability and to apply transient
response to strong external disturbances. In this paper, an inverter is
simulated and offered for single-phase and three-phase voltage controlled by
a non-linear controller. For this purpose, a comparison has been made
between the controller performance and the PI controller. In the first step,
there is a backstepping regulator that uses the stability tool next to the
Lyapunov function. And the other regulator operates according to the PI
method. The performance of these two regulators is simulated during a
change in reference or a load change in MATLAB. Also, a method of
feedback voltage control based on the Lyapunov theory for controlling of the
distributed generation (DG) unit independent inverter is presented. The
proposed controller is not only simple, but also against the sudden changes
in load and the unspecified system is resistant.
Keywords:
Backstepping controller
Distributed generation inverter
Lyapunov stability
Microgrid
Nonlinear loads
This is an open access article under the CC BY-SA license.
Corresponding Authors:
Mahmoud Zadehbagheri, Mohammad Javad Kiani
Department of Electrical Engineering, Islamic Azad University
Yasooj Branch, Yasooj, Iran
Email: m.zadehbagheri@iauyasooj.ac.ir, kianiph@iauyasooj.ac.ir
1. INTRODUCTION
Distributed generation sources have enormous advantages so that it is expected to play an important
role in the operation and planning of power systems in the near future. The status is changing, economic
stimuli, technology growth, and environmental issues are shifting the configuration and current view of the
power grid from generation to distribution [1]. Every microgrid is consisted number of new energy sources
that require a voltage source or inverter to manage the voltage and power of each source to use the power
output of these resources and to connect to the network. Converters such as a voltage source inverter (VSI)
are commonly used as power interfaces in many applications including uninterruptible power supplies (UPS)
and renewable energy systems (such as solar cells and wind turbines) [2]. Also, the good performance of the
inverter independently and connected in the network, requires accurate voltage and frequency control to
maintain the function and control the output power. The proper control of the voltage and frequency for the
operation of the inverter in systems such as emergency power systems (EPS) and microgrid systems are
required [3].

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Different control schemes for the control of inverters in the grid are provided in the relevant
literature. In the study [4], a proportional derivative (PD) controller with a proportional integral (PI)
controller structure for the photovoltaic (PV) system is presented. The controller detects a voltage reference
that shows with the specified error and the algorithm for maximum detection of the P, Q point of power. The
main problem with the maximum power tracking algorithm is that in the steady state, the operating point
varies around the maximum power point. Chapuis et al. [5] emphasizes the control of a single-phase direct
current-alternating current (DC-AC) compensator used to provide uninterrupted power supply. The control
goal in this field is to produce a sine voltage at the output of the system with amplitude and frequency fixed
by the reference signal. Abdel-Rahim and Elshafei [5] have proposed the application of hierarchical fuzzy
logic controllers for UPS applications. The proposed control scheme consists of two fuzzy controllers
implemented in a nesting mode to create two control loops. Sustainability has a fundamental role in the
theory and engineering of systems.
There are various sustainability issues that are being studied in the study of dynamical systems.
Usually, the stability of points of equilibrium is checked by Lyapunov [6]. In this paper, we intend to control
the output voltage of an inverter of distributed generation source in island condition by using a backstepping
controller under ohmic, inductive and nonlinear load conditions, and the stability of the system is studied
according to Lyapanov theory. Finally, it is compared with classic controllers. Inverter output voltage control
is a distributed generation (DG) unit under different load conditions such as ohmic load, ohmic-inductive
load, nonlinear load and no-load. The backstepping controller has a steady state error close to zero in
following the reference signal. Backstepping controller is resistant to unmodulated dynamics and
disturbances and noise.
2. METHOD
2.1. The structure of a single-phase inverter and mathematical modeling
The pulse width modulation (PWM) single-phase inverter shown in Figure 1. It consists of two arms
with bi-directional couplings insulated gate bipolar transistors (IGBTs) or metal-oxide semiconductor filed
effect transistor (MOSFET) with reverse parallel diodes that operate complementarily. The control signal μ is
generated by the PWM generator and has a value between {1, -1} and in summary the binary commands μ1
and μ2 are of the two switching parts [7], [8].
μ = μ1 – μ2
𝜇 = {
: 𝜇1 = 1 𝑎𝑛𝑑 𝜇2 = 0 → (𝑘1, 𝑘2
′ )𝑂𝑁 𝑎𝑛𝑑 (𝑙2, 𝑘1
′ )𝑂𝐹𝐹
−1: 𝜇1 = 0𝑎𝑛𝑑 𝜇2 = 1 → (𝑙2. 𝑙1
′ )𝑂𝑁 𝑎𝑛𝑑 (𝑘1, 𝑘2
′ )𝑂𝐹𝐹
(1)
Figure 1. Design of a single-phase inverter with an LC filter [7]
For mathematical modeling of the inverter with a LC filter, uses mesh law and the mathematical
equations node:
𝐿
𝑑𝑖𝐿
𝑑𝑡
= 𝑣𝐴𝐵 − 𝑉
𝑠 (2)
𝑐
𝑎𝑣𝑠
𝑑𝑡
= 𝑖𝐿 − 𝑖𝑠 (3)

Int J Elec & Comp Eng ISSN: 2088-8708 
Design of a new backstepping controller for control of microgrid sources inverter (Mahmoud Zadehbagheri)
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The output voltage of the inverter 𝑣𝐴𝐵, depending on the switch state (μ control signal), can have two values:
𝑣𝐴𝐵 = {
𝐸 𝑤ℎ𝑒𝑛 𝑜𝑛𝑙𝑦 (𝑘1, 𝑘2
′ )𝑎𝑟𝑒 𝑂𝑁 𝑖𝑒 𝜇 = 1
−𝐸 𝑤ℎ𝑒𝑛 𝑜𝑛𝑙𝑦 (𝑘2, 𝑘1
′ )𝑎𝑟𝑒 𝑂𝑁 𝑖𝑒 𝜇 = −1
(4)
Therefore, one can conclude:
𝑣𝐴𝐵 = 𝜇𝐸 (5)
This model is a system with variable structure: 𝑣𝐴𝐵 is not a continuous variable and can be two
discrete values of E and -E, so it is inappropriate to design a continuous control rule. To overcome this
problem, we often have to use the average model (widely used for modeling static converters) [8], [9] the
switching period in this case is assumed to be much smaller than the dynamic system. In our study, this is
very acceptable. We have:
𝐶𝑥̇1 = 𝑥2 − 𝑖𝑠 (6)
𝐿𝑥̇2 = 𝑢𝐸 − 𝑥1 (7)
Here, 𝒙𝟏 and 𝒙2 represent the average value of sampling period of the Vs output voltage across the capacitor
C and current iL in the inductor L. The control variable 𝑢 𝜖 [−1,1] has between values -1 and 1 and represents
the Average value of the control signal μ which is formed by rectangular pulse width modulation [10], [11].
It can be shown that a three-phase four-leg inverter (TPFLI) is equivalent to three independent single-phase
inverters. It makes sense to design a voltage and current controller for single-phase parallel inverters instead
of a direct controller design four-leg inverters. Then apply the same controllers to the four-leg parallel system
using the T_(4_qdo) conversion matrices and vice versa. The noteworthy point here is that all the controllers
are designed and implemented in the stationary device and the conversion is used only for modeling a
four-leg inverter to three single-phase inverters. Figure 2 shows the proposed control structure for a four-leg
inverter. In this figure, the two blocks of conversion from the abc to qdo and vice versa are the same
transformations 𝑇4_𝑞𝑑𝑜 𝑎𝑛𝑑 𝑇4_𝑞𝑑𝑜
−1
[12].
Figure 2. Block diagram of control strategy of a four-leg inverter [12]
Here, the voltage loop reference of a harmonic-free sinusoidal signal is considered. In the control
block, there are three independent controllers of parallel single-phase inverters whose design method we have
described, which are used to apply to the content of d, q and o voltage and current signals. Each of these
sections has two voltage controllers and a current divider. The three outputs of the axes d, q and o are
returned to the abc device by the reverse conversion 𝑇4_𝑞𝑑𝑜
−1
. The output of the controllers in the abc device,
as a reference, enters the space vector pulse width modulation (SVPWM) modulation block, which cannot be
addressed in this section [13].

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2.2. Design of backstepping controller
The proposed controller allows the converter to provide a fully sine voltage with constant amplitude
and frequency independent of load. The output voltage should follow the reference signal [14], [15].
𝑥1
∗(𝑡) = 𝑉√2 𝑠𝑖𝑛(𝜔𝑡) (8)
In which, 𝑣 = 230, 𝑓 = 50 𝐻𝑧(𝜔 = 2𝜋𝑓), the value of the root mean square (RMS) voltage and the
reference sine wave frequency, respectively. In control theory, backstepping is a technique developed in 1990
to design the stability control of dynamic systems. These systems are made of irreducible subsystems that can
be stabilized using other methods [16]. Due to this recursive structure, the designer can start the design
process at the known-stable system and "back out" new controllers that progressively stabilize each outer
subsystem. The process terminates when the final external control is reached [17]. Hence this process is
known as backstepping. We consider 𝑍1as tracking error and we define it:
𝑍1 = 𝐶(𝑥1 − 𝑥𝑖) (9)
The dynamics:
𝑧̇1 = 𝐶(𝑥̇1 − 𝑥̇1
∗
) (10)
𝑍̇2 = 𝑥2 − 𝑖𝑠 − 𝐶𝑥̇1
∗
(11)
The proposed Lyapunov function is defined as (12) [18].
𝑣1 =
1
2
𝑧1
2
(12)
Derived from time:
𝑉̇1 = 𝑍̇1𝑍̇1 (13)
By choosing:
𝑍̇1 = −𝑘1𝑧1 → 𝑧1(𝜏) = 𝑧1(0)𝑒−𝑘1𝜏
(14)
Here, 𝑘1is a positive constant. This leads to a candidate function of Lyapunov, which has a completely
negative dynamic. So, we get:
𝑉̇1 = −𝑘1𝑧1
2
(15)
As a result, the asymptotic stability is achieved and 𝑍1ultimately aspires to zero [19], [20]. In the system
𝑋2(7), it is like a virtual control input. So 𝑍1can be set to zero if: 𝑥2 = 𝑥2
∗
𝑥2
∗
= −𝑘1𝑧1 + 𝑖𝑠 + 𝐶𝑥̇1
∗
(16)
Here, 𝑋2 is called constant stability. A new variable error is defined between the virtual value and the desired
value [21], [22].
𝑧2 = 𝑥2 + 𝑥2
∗
(17)
This can be deduced from (10), (16), and (17).
𝑍̇1 = −𝑘1𝑧1 + 𝑧2 (18)
The Z2 time derivative is calculated [23]:
𝑥̇2
∗
= 𝑥̇2 − 𝑥̇2
∗
(19)

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𝑍̇2 =
1
𝐿
(𝑢𝐸 − 𝑥1) − 𝑥̇2
∗
(20)
The actual control system should be considered. The system stability problem presented in (18) and (21) can
be understood by the following Lyapunov function [24]–[26]:
𝑣2 =
1
2
𝑧1
2
+
1
2
𝑧2
2
(21)
𝑉̇2 = 𝑍̇1𝑍1 + 𝑍̇2𝑍2 (22)
𝑣̇2 = −𝑘1𝑧1
2
+ 𝑧2(𝑧1 + 𝑍̇2) (23)
Applying the (24):
(𝑧1 + 𝑍̇2) = −𝑘2𝑧2 (24)
we get:
𝑉̇2 = −𝑘1𝑧1
2
− 𝑘2𝑧2
2
(25)
Therefore, (20) and (24) lead to the development of a Backstepping controller [27].
𝑢 = −
𝐿
𝐸
(𝑧1 + 𝑘2𝑧2 −
𝑥1
𝐿
− 𝑥̇2
∗
) (26)
The control rule is chosen so that V2<0 and allows the (𝑍1 , 𝑍2)system to be asymptotically stable [28], [29].
3. RESULTS AND DISCUSSION
In this section, the system and the simulation results are expressed under different conditions such as
ohmic load, inductive load and non-linear. Simulation is based on MATLAB software. Simulation to
determine the response of supposed controllers relative to the transient state and the steady state under four
the following is conducted: i) symmetrical resistive load (transient behavior from 0% to 100%),
ii) symmetrical resistive load (transient behavior from 100% to 0%), iii) inductive ohmic load, and
iv) nonlinear load. The simulated inverter parameters are shown in Table 1, where fs is the switching
frequency of the system.
Table 1. Parameters of simulated single-phase inverter
Parameter Value
VDC 100 V
Vout 50 peaks
fS 15000
F 50
R (full load) 200
Single-phase nonlinear load is designed according to IEC62040-3 standard, shown in Figure 3. Also,
the reference signal with amplitude 50 and frequency 50 Hz is shown in Figure 4. The parameters of
single-phase non-linear load are shown in Table 2. Figure 5 shows the schematic diagram of the controller.
As shown in Figure 5, the capacitor current (ic), the load output voltage (vo) and the output current (io) are
considered as inputs of the control system. The crest factor (CF) and power factor (PF) of nonlinear load are
2.75 and +0.7, respectively [30].
Table 2. Parameters of single-phase non-linear load [31]
Parameter Value
Rs 2 Ω
RNL 18 Ω
CL 8200 μF

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Figure 3. Single-phase nonlinear load according to IEC62040-3 [32]
Figure 4. Waveform of the reference signal
Figure 5. General block diagram of the assumed control system
Simulation results of assumed controller, includes load output voltage (V), output current (A), which
is considered as the input of the control system, the error between the output voltage and the reference signal
and harmonic analysis of steady stat are shown in Figure 6(a) to (d). The simulation results to determine the
response of the assumed controllers to the transient state under symmetric resistive load, inductive ohmic
load and nonlinear load are shown in Figure 7(a) to (d). As can be seen from the figure, the transient response
of the system can be observed and investigated in two situations. The state where all these loads are
disconnected from the system (a), and the state where the resistance load, the inductive ohm load and the
nonlinear load are connected to a system (b, c, d).

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(a)
(b)
(c)
(d)
Figure 6. Simulation results of controller: (a) output voltage (v), (b) output current (A), (c) error between
output voltage and reference signal, and (d) harmonic analysis of steady stat

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(a)
(b)
(c)
(d)
Figure 7. Simulation results of the assumed controller: (a) transient response (all load is disconnected),
(b) transient response of a nonlinear load, (c) transient response to the inductive ohmic load, and
(d) transient response to resistance load

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Figure 8 illustrates the schematic of a three-phase DC/AC inverter with an LC filter. This schematic
includes a three-phase inverter, an LC output filter connected to local loads. The LC filter function eliminates
the harmonic components of the inverter output voltage due to the high frequency switching operation. So,
this filter is an essential part of circuit and non-removable. The system model as shown in Figure 8, can be
reduced to the unit's single-phase equivalent circuit according to Figure 5. So, based on the previous section,
for each phase, the dynamically model is extracted, and a controller is designed. The proposed controller is
investigated in a three-phase system under conditions of symmetric ohmic load, asymmetric ohmic load,
inductive ohmic load, and nonlinear load to control the voltage of a DG unit in island mode.
Figure 8. Schematic of a three-phase DC/AC inverter with LC filter
Simulation to specify the controller response assumed to be transient and steady state under the
following four conditions: i) symmetrical resistance load, ii) asymmetrical resistive load, iii) inductive ohmic
load, and iv) nonlinear load. The parameters of each phase of the inverter simulated are given in Table 1.
Three phase nonlinear load is designed according to IEC62040-3, as shown in Figure 9.
Figure 9. Three phase nonlinear load according to IEC62040-3 [34]
Also, a three-phase reference signal with amplitude 50 and a frequency of 50 Hz is shown in
Figure 10(a), hat each phase has a phase difference of 120 degrees with two other phases. The parameters of
three-phase non-linear load are shown in Table 3 [33], [34]. Simulation results of controller, includes load
output voltage (three-phase), load output current (three-phase), voltage during resistive load, voltage under
no-load, voltage resistive load during asymmetric resistive load, and voltage during nonlinear load are shown
in Figure 10(b) to (g). In this section, the circuit of Figure 11 is simulated in MATLAB software. Table 4
shows the numerical results of the steady state analysis in the three-phase state for one of the phases (phase a)
in the no load, symmetric resistance load, asymmetric resistance load, and nonlinear load.
Table 3. Parameters of three-phase non-linear load [34]
Parameter Value
Rdc 500 Ω
Cdc 3300 μF

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(a)
(b)
(c)
(d) (e)
(f) (g)
Figure 10. Simulation results of controller: (a) three-phase reference signal, (b) load voltage, (c) load current
(d) voltage during resistive load, (e) voltage under no-load, (f) voltage resistive load during asymmetric
resistive load, and (g) voltage during nonlinear load
Phase (b)
Phase (c)
Phase (a)
Phase (a) Phase (b) Phase (c)

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Figure 11. Block diagram of the three-phase voltage control system
Table 4. Results of steady state analysis of phase (a)
Load type Output
voltage peak
Effective output
voltage
THD
(%)
2th
harmonic
3th
harmonic
4th
harmonic
5th
harmonic Robustness
No load 47.82 33.82 0.49 0.22 0.09 0.25 0.17
Robust to
parameters
Symmetric
resistance load
47.74 33.76 0.55 0.15 0.21 0.28 0.10
Asymmetric
resistance load
47.69 33.72 0.76 0.25 0.35 0.41 0.16
Nonlinear load 47.28 33.43 3.02 1.37 1.58 1.34 0.96
In this section, we use a classical controller to compare the results of the proposed controller in this
paper. PI controllers are created by combining two proportional and integral control operations. This
controller is a closed loop control algorithm and method using the concept of feedback that is used in many
industrial processes to control. The simulation results are expressed with a PI controller for a single-phase
system. The simulation results show that the PI controller is sensitive to load variations and its steady state
error varies under different conditions [35]. Therefore, the proposed controller is a robust controller that is
not only simple but also resistant to sudden load changes. And system uncertainties are also resistant.
Figure 12 (in appendix) shows the voltage control based on the PI controller. According to the figure of the
output voltage of the controller, error between output voltage and reference signal (under all load conditions),
and harmonic analysis of the steady stat are shown.
4. CONCLUSION
In this paper, a method of feedback voltage control based on the Lyapunov theory for controlling of
the DG unit independent Inverter is presented. The proposed controller is not only simple, but also against
the sudden changes in load and the unspecified system is resistant. In addition, the stability of the control
system of the closed loop is also proven by the method of the Lyapunov. To provide validation of the
proposed control method, simulations are provided through the MATLAB software. Finally, the simulation
results have shown that the proposed control method provides convincing voltage regulation, such as fast
dynamics, lower steady state error, and low total harmonic distortion (THD) for various loads, including
linear and nonlinear loads. It is also shown by changing the amplitude of the reference signal that follows the
reference signal controller and causes the steady state error between the output voltage signal and the
reference signal of the voltage to zero.
Some suggestions for continuing research in this area can be expressed as follows: Investigation and
presentation of a robust controller to improve of voltage and frequency stability in a microgrid in island
mode. Investigation and presentation of a feedback controller based on Lyapunov theory to improve the
damping of low frequency fluctuations in a power system. Investigation and presentation of a feedback
controller based on Lyapunov theory to control power directly in the microgrid. Provide a feedback controller
based on Lyapunov theory to control the current of a grid-connected inverter.

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APPENDIX
(a)
(b)
(c)
Figure 12. Voltage control based on the PI controller (a) output voltage, (b) error between output voltage and
reference signal, and (c) harmonic analysis of the steady stat
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BIOGRAPHIES OF AUTHORS
Mahmoud Zadehbagheri was born in Yasouj, Iran in October 1979. In 2003 he
received his B.S. in Electrical Engineering from Kashan University and in 2008 he received his
M.S. in Electrical Engineering from the Islamic Azad University, Najafabad Branch. He
received the PhD degree in Electrical Engineering from Hakim Sabzvari University and
Universiti Teknologi Malaysia (UTM), Johor, Skudai, Malaysia in 2017. He is with the faculty
of the Electrical Engineering Department, Islamic Azad University of Yasouj. His research
interests include the fields of power electronics, electrical machines and drives, FACTS
devices and power quality. He can be contacted at email: m.zadehbagheri@iauyasooj.ac.ir.
Mohammad Javad Kiani was born in Yasouj, Iran. He received his B.S. and
M.S. degrees in Electrical Engineering from the K. N. Toosi University of Technology,
Tehran, Iran, in 2003 and 2008, respectively. He received his Ph.D. degree in Electrical
Engineering from the Universiti Teknologi Malaysia (UTM), Johor, Skudai, Malaysia, in 2017.
He is presently a faculty member in the Department of Electrical Engineering, Islamic Azad
University, Yasouj, Iran. His current research interests include nanotechnology and
application, power electronic, power systems, controller, and intelligence network. He can be
contacted at email: kianiph@iauyasooj.ac.ir.
Tole Sutikno is a Lecturer in Electrical Engineering Department at the
Universitad Ahmad Dahlan (UAD), Yogyakarta, Indonesia. He received his B.Eng., M.Eng.
and Ph.D. degrees in Electrical Engineering from Universitas Diponegoro, Universitas Gadjah
Mada and Universiti Teknologi Malaysia, in 1999, 2004 and 2016, respectively. He has been
an Associate Professor in UAD, Yogyakarta, Indonesia since 2008. He is currently an Editor-
in-Chief of the TELKOMNIKA since 2005, and the Head of the Embedded Systems and
Power Electronics Research Group since 2016. His research interests include the field of
industrial applications, industrial electronics, industrial informatics, power electronics, motor
drives, renewable energy, FPGA applications, embedded system, image processing, artificial
intelligence, intelligent control, digital design and digital library. He can be reached via email:
tole@ee.uad.ac.id.
Rasoul Arvin Moghadam was born in Yasouj, Iran in February 1988. In 2003, he
received his B.S. in Electrical Engineering from Islamic Azad University, Yasouj Branch and
in 2008 he received his M.S. in Electrical Engineering from the Islamic Azad University,
Yasouj Branch. He is currently studying for his doctorate at the Islamic Azad University of
Yasouj Branch. His research interests include context FACTS, power electronic and power
quality in power system. He can be contacted at email: rasoolarvin@gmail.com.

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