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Engineering, Technology & Applied Science Research Vol. 12, No. 4, 2022, 8960-8966 8960 
 

www.etasr.com Nebili et al.: Decoupling Control Applied to the Smart Grid Power Dispatching Problem 

 

Decoupling Control Applied to the Smart Grid Power 

Dispatching Problem 
 

Salim Nebili 

Department of Physics, Laboratory of 

Electrical and Energetic Systems, Faculty 

of Sciences of Tunis, University of Tunis 

El Manar, Tunis, Tunisia 

salim.nebili@tunisietelecom.tn 

Ibrahim Benabdallah 

Department of Physics, Laboratory of 

Electrical and Energetic Systems, Faculty 

of Sciences of Tunis, University of Tunis 

El Manar, Tunis, Tunisia 

ibrahim.benabdallah@gmail.com 

Adnen Cherif 

Department of Physics, Laboratory of 

Electrical and Energetic Systems, Faculty 

of Sciences of Tunis, University of Tunis 

El Manar, Tunis, Tunisia 

adnen2fr@yahoo.fr 
 

Received: 15 May 2022 | Revised: 9 June 2022 | Accepted: 13 June 2022 

 

Abstract-This paper presents controller designs of the decoupling 

issue dedicated to renewable decentralized generators or new 

generation grids known as smart grids. The control methods 

were based on a three-phase voltage source grid representation 

that integrated wind and photovoltaic generators. The utilized 

decoupling strategies (Boksenbom and Hood, Zilkind and 

Luyben, and inverted decoupling) allowed the optimization of 

energy transfers and costs through the routing grid distribution 

path. Simulations were conducted in Matlab/Simulink. The 

trustworthiness of the multivariable control system design in 

addition to the tuning method was established using three test 

batches, totally decoupled and stable from input dependencies. 

Keywords-decoupling; smart grid; demand-response; power 

dispatching; controller design 

I. INTRODUCTION  

Methods presenting only one output controlled by a single 
deployed variable are classified as Single-Input Single-Output 
(SISO) systems. Several processes do not imitate such a simple 
control shape. Each unit operation needs to control no less than 
two variables, and consequently, there are frequently at least 
two control loops to deal with. Systems that involve more than 
one control loop are known as Multi-Input Multi-Output 
(MIMO) or multivariable systems. The electrical smart grid is 
essentially a massive MIMO system. Due to the interactions of 
the flow power offer and demand, it is not easy to design a 
robust controller for each path autonomously to lead an input x 
to an output y without affecting any other output. Many studies 
have focused on several loop control structures to diminish 
these interactions.  

Decoupling methods are mostly used in industrial processes 
but are still applicable for many electrical system parameter-
decoupling purposes. A simplified, ideal, and inverted 
decoupling comparative study was presented in [1], which was 
the base for many classic studies. Three decoupling methods 
were considered to deal with stability, robustness, and 
implementation aspects, while the complexity of the decoupler 
elements depended on the overall system size. An improvement 
of the two inverted decoupling methods was presented in [2] 
for the case of a stable linear multivariable class of processes, 

through no minimum-phase zeros and multiple time delays. As 
some multivariable processes were still unable to be decoupled 
due to stability issues, alternative decoupling with a unity 
feedback structure was suggested. The PI/PID was designed 
using internal model control to decouple the processes. In the 
case of multiplicative input uncertainty, the low bounds of the 
control parameters were intended to guarantee the robust 
stability of the system. As the disturbances influence the way 
of the control synthesis of a MIMO system based on ideal 
decouplers and a PID regulator, single controller performance 
and robustness from the closed-loop could be prolonged to all 
loop systems [3]. An improved decoupling approach allowed 
more flexibility in the choice of transfer functions of the 
decoupled apparent process for general n×n processes, where 
the decoupler elements’ complexity appeared independent of 
the system's size [4]. A decoupling internal model controller 
design method was proposed in [5] for industrial MIMO time-
delay systems, based on a controlled model adjoint matrix. A 
decoupling compensator and an internal model controller, 
Butterworth low-pass filter, were designed for decoupling the 
generalized controlled object. The last filter aimed to improve 
the disturbance rejection aptitude. In [6], a method of finding 
the forced oscillation source was proposed. The relation 
between the observable outputs and the mechanical power 
problems in a unit was decoupled by sensibly designing a 
Luenberger observer using an aging structure task. A MIMO 
control system with multiple time delays and a model 
predictive control was presented in [7], describing a method to 
design MIMO filtered Smith predictors for n×n processes 
through multiple time delays based on the decentralized direct 
decoupling structure. This allowed simple tuning of the 
primary controller, making simpler the problem toward 
multiple single loops, if realizable, and it could also be used to 
eliminate the interactions among process variables.  

The decoupling method is also applicable to electrical grid, 
microgrid, and smart grid annexed domains facing new 
intermittent and highly variable system challenges and 
expectations [8-17]. Modeling approaches were proposed in [8] 
using analytical tools for control design purposes of step-up 
grid-connected PV systems, such as voltage source 

Corresponding author: Salim Nebili 



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www.etasr.com Nebili et al.: Decoupling Control Applied to the Smart Grid Power Dispatching Problem 

 

representation of the bulk capacitor interacting with the grid-
connected inverter and considering the inverter of a double-
stage PV system as a Norton equivalent. Furthermore, both 
ideal and real DC/DC converter models for the PV module 
were presented through four mathematical models, stated in 
state-space representation. A decoupled inverter connected to a 
grid was exposed in [9], using dynamic online grid impedance 
measurements in a micro grid application. The controller was 
implemented in a synchronous reference frame and PI 
controllers were used, while mutual coupling was introduced 
between the d and q control loops. The transformation into a 
synchronous reference was accurately decoupled using the 
dynamically measured grid impedance and feedforward 
control. Decoupling permits autonomous control of active and 
reactive powers following changes, and online real impedance 
measurement was proposed to make the decoupling more 
precise. In [10], a multivariable-droop synchronous current 
converter decoupling control strategy was proposed for the 
current management of weak grid/micro grid applications, 
offering a traditional forward design method that employs the 
loop determining technique. The controller mixes current 
regulation, management, and limitations that apply to weak and 
micro grids. The multivariable synchronous current converter 
allowed larger decoupling of d- and q-axis currents in weak 
and/or resistive grids, anywhere these control variables are 
highly coupled due to the converter’s large load-angle 
operation. 

In [11], the evolution allocation for the decoupling of the 
electrical network was studied using pure channel component 
transform, addressing issues such as robustness. Difficulties 
were efficiently overcome using singular values instead of 
proper decomposition. An algorithm was developed to show 
how the adopted transformation can be used for voltage 
stability analysis. In [12], a distributed generator strategy 
control was investigated to assist a conventional local grid by 
decoupling phase sequences, frequency domain unbalances, 
and harmonic compensation for grid-connected inverters 
through unbalanced loads. This strategy was considered 
sequential-asymmetric for unbalanced and harmonic voltage 
correction. Moreover, a Norton equivalent model in frequency-
domain was used. This study showed that following a 
frequency-domain decoupled method, the fundamental 
positive, harmonic symmetrical, fundamental zero, and 
negative sequence components can be regulated independently. 

In [13], a virtual synchronous generator control method was 
proposed for extensive renewable energies connected to a grid. 
A power decoupling control method based on linear active 
disturbance rejection control was developed to eliminate output 
power deviation. Using the coupling term estimates and 
disturbances from an extended state observer, the aforesaid 
matter triggered by power coupling can be lessened. 
Furthermore, the stability of the controller was analyzed. In 
[14], essential limitations were defined for the islanded mode 
of operation of micro grids which cannot be absorbed by 
conventional FP methods. Therefore, updates were designed 
for conservative approaches by including network frequency as 
a variable and avoiding the network soft bus. These particular 
aspects of the Newton-Raphson method modified the designs 
of the Jacobean matrix. Extended power factor equations were 

decoupled by reformulating the Jacobean matrix based on the 
fact that lines are mostly resistive. Furthermore, two 
convergence improvement methods were also incorporated. 
Dual-buck inverters were examined in [15], reassessing 
volume, system weight, and power density of dual-buck single-
phase grid-tied PV inverters. A new active power decoupling 
buffer and grid-tied photovoltaic inverter integration with 
single-inductor dual buck topology were proposed, using a 
single-loop direct input current ripple control method. Several 
concerns arise in the case of pulse width modulation using a 
high-voltage current source converter based on fully controlled 
devices, such as high switching losses, large DC voltage 
fluctuations, and thin power operation range [16]. To resolve 
these difficulties, a current source converter-based HVDC was 
proposed using fundamental frequency modulation presenting 
the topology, an independent control strategy, and a 
mathematical model. A synchronized control strategy was 
proposed to decouple active and reactive powers with an outer 
and inner current controller loop to lead to lower DC voltage 
fluctuations, lower switching losses, and a broader power 
operation range. 

This study investigated a Two-Input Two-Output (TITO) 
grid system, as a MIMO system could be devised into several 
TITO subsystems. The main goal was to employ decoupling 
methods for grid part energy aggregation, aiming to decouple 
demand response paths based on statue and output feedback 
control methods.  

II. CONTROLLER DESIGN AND METHODS 

The control strategy of the phase grid-connected PV, wind, 
and storage system inverter is presented in Figure 1. The 
elements were taken from [17-24]. 

 

 
Fig. 1.  Overall grid-connected system design. 

The system can be described in the state representation by: 

��. � �� � ��	 � 
� � ��    (1) 
where A is the characteristic matrix, B is the order matrix, C is 
the output matrix, and D is the coupling matrix. A control law 
was defined by the output feedback as: � � 
 � �. �    (2) 



Engineering, Technology & Applied Science Research Vol. 12, No. 4, 2022, 8960-8966 8962 
 

www.etasr.com Nebili et al.: Decoupling Control Applied to the Smart Grid Power Dispatching Problem 

 

where V is the new entries vector of the dimension (m.1), K is 
the matrix of dimension (m.l), m is the number of outputs, l is 
the number of entries, y is the output vector, and e is the input 
vector. K is calculated by solving: |�� � ��
| � 0    (3) 

The governing equations of the network filter defined in 
Figure 1 are: 

⎩⎪⎨
⎪⎧������� � ���� � 
� � ��� �� �� � �!�! � 
! � �!�"��"�� � �#�# � 
# � �#

    (4) 

The fundamental principles of simplification of the 
equations of states, such as the Park model, can be used to 
either substitute or eliminate the effect of one or several 
equations, to implement and succeed in the decoupling 
methods by delimitating the effects of inputs on outputs. The 
relative Park equation model from the overall grid-connected 
system design is given by: 

$�%��%�� � �� �� � 
� � ���&��&�� � �' �' � 
' � �'     (5) 
Input and output voltages are linked by values belonging to 

ℝ: 

�
� � �� �� � �' �'
' � (� �� � (' �'     (6) 
Using (5):  

$�%��%�� � ��� �� � 
� � ���&��&�� � ��' �' � 
' � �'     (7) 
Substituting (6) in (7) gives: 

�%��%�� � ��� �� � )�� �� � �' �' * � ���&��&�� � ��' �' � )(� �� � (' �' * � �'     (8) 
� � +,-%�% 00 ,-&�& . , � � /

)�,0%*�% ,0&�%,1%�& 2�,1&3�&
4 


 � 51 00 17 , � � 50 00 07 
(9) 

The transfer function matrix G(s) can be easily obtained 
from the state representation using: 8)9* � 
)9� � �*,�� � �    (10) 

In the general case, the matrix D is null and the calculation 
of G(p) is simplified as: 8)9* � 
)9� � �*,��    (11) 

8)9* � :8�� 8�!8!� 8!!;    (12) 

The used units are L1 = L2 =16.58.10
-3

H, and r1 = r2 = 
0.98Ω. The transfer matrix is obtained from the A, B, and C 
matrices: 

8)9* � + <.<=>�<<?@<.=> <.<=>�<<?@<.=>,<.<!A�<<?@�.=! <.<B!�<<?@�.=!.    (13) 
III. SIMULATION RESULTS  

The grid inputs are two photovoltaic conversion chain 
outputs by two different steps:  

A. Without Decoupling 

After obtaining the coupled system block diagram and the 
transfer matrix G(p) elements, the wiring was designed in 
Matlab/Simulink to visualize the evolution of its outputs as a 
function of time. The simultaneous excitation of e1 and e2 
inputs is shown in Figure 2. 

��� � C�8�� � C!8�!�! � C�8!� � C!8!!     (14) � � 8C    (15) 
with: 

� � D��, �! FG , C � DC�, C!FG  , 8 � :8�� 8�!8!� 8!!;  
 

 
Fig. 2.  Coupled system block Simulink diagram. 

 

 

Fig. 3.  Grid inputs. 

At t0=0s, a unit step is performed at each system input. 
Then the amplitude of e1 is reduced to 0.5 at t=1500s without 
modifying the second input. After that, at t=2500s, the second 
input e2 decreases to 0.5, without modifying the first input e1. 
Figure 4 shows the evolution of each output y1 and y2. 

0 500 1000 1500 2000 2500 3000 3500 4000
0

0.2

0.4

0.6

0.8

1

1.2

Time (seconds)

e
1

0 500 1000 1500 2000 2500 3000 3500 4000
0

0.2

0.4

0.6

0.8

1

1.2

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Fig. 4.  Grid outputs without decoupling 

The evolution of outputs of the coupled system shows that 
the change in the first input could vary and affect the second 
output. The following subsection introduces some decoupling 
methods to obtain an overall system made of TITO subsystems 
varying in parallel to Figure 5. In particular, the input e1 should 
only affect the first output y1, and the input e2 should only 
affect y2. The chosen methods were Boksenbom and Hood [2, 
10], Zalkind and Luyben [2], and a method based on inverted 
decoupling [2, 4]. 

 

 

Fig. 5.  The TITO overall system. 

B. Decoupling Using Boksenbom and Hood Method 

Using (12), the decoupling matrix 8H∗ can be given by:  
8H∗ � :8H,��∗ 8H,�!∗8H,!�∗ 8H,!!∗ ;....(16) 

Reference signals are shown by w=[w1, w2]:  8C � � , C � 8H∗DK � �F8H∗DK � �F � �    (17) � � D� � 88H∗F,�88H∗K    (18) � � D� � 88H∗F,�88H∗ � LMNODP�, P!F    (19) 
X is a diagonal matrix: 

88H∗ � :8�� 8�!8!� 8!!; :8H,��∗ 8H,�!∗8H,!�∗ 8H,!!∗ ; :Q� 00 Q!;    (20) 

⎩⎪⎨
⎪⎧Q� � 8��8H,��∗ � 8�!8H,!�∗0 � 8��8H,�!∗ � 8�!8H,!!∗0 � 8!�8H,��∗ � 8!!8H,!�∗Q! � 8!�8H,�!∗ � 8!!8H,!!∗ ⎭⎪⎬

⎪⎫
    (21) 

8H,�!∗ � ,U� UV,  ∗U��  ,   8H,!�∗ � ,U �UV,��∗U      (22) 
As the decoupled matrix functions do not influence the 

evaluation of the diagonal matrix decoupling, according to this 
method principle, the PI/PID regulators must be chosen 
arbitrarily. The stability condition of the system has to be 
respected, as follows: 

8H,��∗ � 8H,!!∗ � ?@<.�?     (23) 
The Simulink transfer functions block wiring which 

characterizes this method are shown in Figure 6. 

 

 
Fig. 6.  Boksenbom and Hood block Simulink diagram. 

The same conditions were applied in this case. At t0=0s, a 
unit step was performed for e1 and e2. Then, e1 was lowered to 
0.5 at t=1500s without changing e2. Next, as t=2500s, e2 
declined to 0.5, without changing e1. The evolution of y1 and 
y2 outputs is shown in Figure 7. 

 

 

 
Fig. 7.  Boksenbom and Hood grid outputs. 

The proposed control system is perceived as having good 
decoupling. Due to error approximation, the decoupling system 
interactions are thoughtful. Thus, a PI/PID must be 
implemented. 

C. Decoupling Using Zlikind and Luyben Method 

Using (11): � � 8C∗, C∗ � 8H∗C ,   C � 8H DK � �F    (24) 

0 500 1000 1500 2000 2500 3000 3500 4000
0

0.05

0.1

0.15

0.2

0.25

Time (seconds)

y
1

0 500 1000150020002500300035004000
0

0.005

0.01

0.015

0.02

0.025

0.03

Time (seconds)

y
2

Without decoupling

0 500 1000 1500 2000 2500 3000 3500 4000
0

0.2

0.4

0.6

0.8

1

1.2

1.4

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y
1

0 500 1000 1500 2000 2500 3000 3500 4000
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y
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are chosen as follows:  � � 88H∗C � 88H∗8H DK � �F    (25) � � 88H∗ � diagDP�, P!F 8H∗8 ,�� , 8 ,� � adj)8*/ det) 8* (26) 
where adj(G) is the adjoint of the matrix of G, and det(G) is the 
determinant of the matrix G given by: det) 8* � 8��8!! � 8�!8!�    (27) 

adj)8* � : 8!! �8�!�8!� 8�� ;    (28) 
8H∗ � 8 ,�� :8!!P� 8�!P!8!�P� 8��P!; ��_�)U*    (29) 

The chosen form is a unit diagonal given by: 8H,��∗ � 8H,!!∗ � 1    (30) 
therefore: 

8H,�!∗ � � U� U��  ,   8H,!�∗ � � U �U      (31) 
Gc1 and Gc2  must be PI/PID regulators chosen as follows: 8H� � 8H! � ?@<.�?     (32) 
The Simulink transfer functions block wiring of this 

method is shown in Figure 8. Figure 9 shows the evolution of 
the outputs y1 and y2 applying the same conditions as in the 
other methods. 

 

 
Fig. 8.  Zlikind and Luyben Simulink block diagram. 

 

 

Fig. 9.  Zlikind and Luyben grid outputs. 

D. Decoupling Using Inverted Decoupling Method 

Gc11 and Gc22 are placed in the forward direction which can 
be PI or PID controller. Using (11) and (12), it is necessary 
that: 

��! � � U� U��  , �!� � � U �U      (33) 
The Simulink transfer wiring function wedges describing 

this method are shown in Figure 10. 

 

 

Fig. 10.  Inverted decoupling block Simulink diagram. 

 

 

Fig. 11.  Inverted decoupling grid outputs. 

The inverted decoupling processes shown in Figure 11 does 
not contain bulk sums of transfer functions. Consequently, the 
diagonal controllers’ fine-tuning becomes more 
accommodating in higher multivariable processes with solid 
cross-couplings, even if the system elements have simple 
changing aspects.  

IV. RESULT COMPARISON AND DISCUSSION 

The three methods obtained perfect decoupling, but the 
variation in the calculation of the terms that characterize each 
method and the simplicity of their use should be noted. In the 
Boksenbom and Hood method, the functions and the PI 
regulator were chosen in the decoupling block, but in the 
Zalkind and Luyben method they are taken equal to 1, which is 
simpler, and they do not appear based on a reduced decoupling 
block. The Zalkind and Luyben method requires the 
computation of the inverse G(p), thus the determinant of G 
must be non-zero, which is not necessary for the two other 

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y
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methods. The addition of a compensation block consisting of a 
PI controller in the direct chain for the Zalkind and Luyben 
decoupling method has the same effect as its addition in the 
decoupling block by the method based on the reduced 
decoupling block, so there is only a variation in the structure. 
Its absence in the Boksenbom and Hood method makes it lose 
flexibility in the implementation of the non-interactive control 
grouping, especially regarding the stability of the system. 
Using these criteria, the method based on a reduced decoupling 
block is the simplest because it requires fewer computations 
and has more flexibility via the PI controller. 

A multivariable control system design employing a fine-
tuning filter based on an inverted decoupler, using statue or 
output feedback control methods, was presented. The high 
closed-loop system enactment was attained for predefined 
robustness by eliminating unnecessary control exertion and 
introducing adequate filtration of the controlled variables. This 
was verified by the results which treat an extremely 
intermittent PV source and chain outputs. The proposed 
demand response path methods improved inverted decoupling. 
The disturbance response was minimized within a narrow 
range for the intermitting nature of renewable energy processes 
known for their instability. 

V. CONCLUSION 

The cogency of the proposed statue feedback multivariable 
control system design and tuning was confirmed using three 
methods. High closed-loop system performance was attained 
by avoiding unnecessary control exertion using the suitable 
controlled variables filtration. The three methods, Boksenbom 
and Hood, Zilkind and Luyben, and inverted decoupling, 
eliminated the opposing effects caused by non-minimum phase 
zeros and achieved dynamic decoupling.  

The simulation results showed that the three proposed 
methods could establish a decent dynamic decoupling, solid 
robustness, and immunity against trouble. Results proved that 
they succeeded very effectively to limit the interactions of the 
inputs to the relative outputs, which opens the road to the 
generalization of these decoupling methods for the entire end-
to-end chain of different energy sources to improve and 
optimize energy dispatching. Inverted decoupling gives more 
effective and precise results compared to Boksenbom and 
Hood and Zilkin and Luyben, given the additional regulation 
blocks Gc11 and Gc22, which leads to maintaining the input 
magnitudes. Future work should assess more than two inputs 
and study a big-scale system matrix, rearranging it to a 2 by 2 
matrix. 

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AUTHORS PROFILE 

 

Salim Nebili was born in Gafsa, Tunisia, in 1973. He 
received the Engineering Degree from the National 
Engineering School of Gabes (ENIG) in 1998 and a 
Master's degree in 2002 from the National Engineering 
School of Tunis (ENIT). He is a member of the Analysis 
and Treatment of Electrical and Energetic Signals and 
Systems Laboratory. His research interests include 
electrical engineering, renewable energies, and smart 
grids. 

 

Ibrahim Benabdallah was born in Tunis, Tunisia, in 
1984. He received a Master's and Doctor’s degree in 
Electronics from the Faculty of Sciences of Tunis (FST). 
He is a member of the Analysis and Treatment of 
Electrical and Energetic Signals and Systems Laboratory. 
His research interests include electrical engineering, 
renewable energies, and smart grids. 

 

 

Adnen Cherif received Engineer, Master, and PhD 
degrees from the National Engineering School (ENIT) of 
Tunis, Tunisia. Since 1997, he is a senior professor and 
responsible for the Analysis and Treatment of Electrical 
and Energetic Signals and Systems (ATEESS) Laboratory 
in the Faculty of Science of Tunis, Tunisia. His fields of 
interest concern digital signal processing, speech 

processing, energetic systems, renewable energies, and smart grids.