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RESEARCH ARTICLE
Robust energy-based nonlinear observer and voltage control
for grid-connected permanent magnet synchronous
generator in the tidal energy conversion system
Youcef Belkhier
1
| Abdelyazid Achour
1
| Farid Hamoudi
2
| Nasim Ullah
3
|
Boubekeur Mendil
1
1
Laboratoire de Technologie Industrielle
et de l'Information (LTII), Faculté de
Technologie, Université de Bejaia, Bejaia,
Algeria
2
Laboratoire de Maitrise des Energies
Renouvelables (LMER), Faculté de
Technologie, Université de Bejaia, Bejaia,
Algeria
3
Department of Electrical Engineering,
College of Engineering, Taif University
KSA, Taif, Saudi Arabia
Correspondence
Youcef Belkhier, Laboratoire de
Technologie Industrielle et de
l'Information (LTII), Faculté de
Technologie, Université de Bejaia, Targa
Ouzemour, Bejaia 06000, Algeria.
Email: belkhieryoucef@outlook.fr
Funding information
Taif University, Grant/Award Number:
TURSP-2020/144
Summary
The conversion of the tidal power captured by the marine current turbine to
electrical power depends highly on the applied control strategy. However, in
the presence of nonlinearities, parametric uncertainties, and external distur-
bance, the controller design process is challenging task. This paper proposes
an energy-based control approach for a grid-connected permanent magnet syn-
chronous generator (PMSG)-based marine current energy conversion system.
This method combines a new passivity-based voltage control (PBVC) with a
nonlinear observer. The PBVC designs the desired dynamic of the system,
while the nonlinear observer is used to reconstruct the measured signals. The
main contributions and motivation of this work include the extraction of maxi-
mum power from the tidal current, integrating it to the grid, and forcing the
closed-loop system's energy to passive state. The mentioned objectives are
achieved by reshaping system's energy and introducing a damping control
term that compensates the nonlinear phenomena in a damped way rather by
the cancellation and it also establishes a duality concept between the observer
and the PBVC. Two steps are required to design the proposed controller: In the
first step, the reference current is formulated based on the reference torque
using a proportional-integral-derivative controller. As a second step, the over-
all control law is computed by the passivity-based combined nonlinear
observer and voltage controllers. To validate the performance of the system
under the proposed control, a comparison with the second-order sliding mode
control and the conventional proportional-integral control methods is pres-
ented. The proposed method is tested in MATLAB/Simulink environment
under different operating conditions and from the presented results it is evi-
dent that the proposed controller showed robustness against parameter
changes and ensured fast convergence of the states.
KEYWORDS
nonlinear control, nonlinear observer, passivity-based control, robust control, tidal renewable
energy
Received: 16 November 2020 Revised: 20 February 2021 Accepted: 1 March 2021
DOI: 10.1002/er.6650
Int J Energy Res. 2021;1–19. wileyonlinelibrary.com/journal/er © 2021 John Wiley & Sons Ltd. 1
1|INTRODUCTION
In recent times, tidal power is considered as the most
promising renewable power generation source due to its
inherent advantages such as predictability and negligible
environmental effects.
1-3
The majority of marine current
conversion technologies are based on permanent magnet
synchronous generators (PMSG) due to its numerous
advantages such as high-power density, low cost, and
favorable electricity production. However, the maximum
power that can be captured by the marine current turbine
is nonlinear and it depends on several factors. Also, the
conversion of the tidal power extracted by the turbine to
electrical power through the PMSG greatly relies on the
applied control strategies and due to the nonlinear prop-
erties of the PMSG, the controller design process is often
very complicated. Furthermore, reactive power and DC-
link over-voltage supports are the necessary conditions to
connect the conversion system to the grid.
4
To addresses
the aforementioned issues, during the last decades' exten-
sive control theories have been investigated. A detailed
review of the control strategies developed for PMSG-
based tidal energy conversion systems is presented in Ref-
erence 5. For maximum power harnessing, a sliding
mode control (SMC) strategy has been proposed in Refer-
ence 6. However, several factors such as unpredictable
changes in the parameters and sudden variations in the
marine current velocity have not been taken into consid-
eration. In Reference 7, a novel active disturbance
rejection control (ADRC) method is reported for the
PMSG-based marine energy conversion system discussed
in Reference 6. The ADRC strategy treats the parameter
uncertainties or changes as an element to be rejected
which can be canceled during the control design. Com-
pared to SMC and the classical proportional-integral
(PI) control methods, the ADRC method
7
shows clear
improvements in the performance of the power conver-
sion system. In Reference 8, a jaya-based sliding mode
approach is reported to enhance the performances of a
tidal conversion system. The authors proposed an associ-
ation of the tidal system with a superconducting mag-
netic energy system, for which, the jaya-based controller
is applied. However, this association improves the costs
and maintenances time of the conversion system. In Ref-
erence 9, a fuzzy SMC that adaptively extracts the maxi-
mum tidal power under swell effects is developed.
However, the PMSG parameter changes and uncer-
tainties have not been incorporated in the controller
design. A magnetic equivalent circuit method-based
second-order sliding mode is proposed in Reference 10,
however, external disturbances and parameter changes
have not been considered. To extract the maximum
power from the tidal current under large parametric
uncertainties and nonlinearities, a nonlinear observer-
based second-order SMC combined with a predictive con-
trol was developed in Reference 11. A linear-quadratic
controller is proposed in Reference 12 using a real profile
of the tidal current speed. In Reference 13, a perturb and
observe algorithm is proposed to track the maximum
tidal power. Tilt-based fuzzy cascaded control combined
with a new Q-network algorithm has been investigated
by Reference 14.
In the same context, the present paper proposes a novel
passivity-based voltage control (PBVC), combined with a
nonlinear observer that maintains the PMSG operating
conditions at the optimal torque. The proposed method
enforces the system to the passive state by introducing a
damping term and reshaping its energy to regulate the
physical variables of the system to their desired values and
also forces the PMSG to track velocity. The system is
decomposed into two interconnected subsystems with neg-
ativefeedback.ThePBVCisselectedtocontroltheelectri-
cal subsystem, while the mechanical dynamics are treated
as a “passive disturbance.”Unlike the aforementioned
nonlinear controls, which are usually neglecting the PMSG
mechanical dynamics,
15
the PBC introduced in Reference
16 has been adopted as a solution for a PMSG in Reference
17. In Reference 18, PBC combined with a fuzzy function
and integral sliding-mode is reported. In Reference 19, an
interconnected and damping assignment PBC strategy is
investigated. In Reference 20, an adaptive PBC is devel-
oped. The authors in Reference 21, investigated a PBC
combined with a linear feedback strategy.
The main contributions and motivation of this work
include the extraction of maximum power from the tidal
current, integrating it to the grid, and makes the closed-
loop system passive. The other aims of the study consist
to maintain the generated reactive power and DC-link
voltage at their reference values, despite the disturbances
related to the PMSG nonlinearities. Special attention is
given to the machine-side converter of the PMSG, as it's
the bridge between the tidal turbine and the grid. Fur-
thermore, the robustness against parameter changes has
been taken into considerations. The contribution and
originality of the present paper are summarized below:
•A new passive nonlinear observer is combined with a
PBVC controller for optimal performance of a PMSG
based tidal current conversion system and a
proportional-integral-derivative (PID) is introduced to
enhance the robustness of the proposed approach
against various uncertainties of the PMSG.
•The essential characteristic of this strategy is its duality
between the controller strategy and the observer.
•An important feature of the proposed controller is the
compensation of nonlinear terms by imposing a
2BELKHIER ET AL.
desired damped transient, rather than by exact
zeroing.
•The global stability of the system and the exponential
convergence of the current tracking error have been
analytically proven. The controller ensures fast conver-
gence, guarantees stability by taking under consider-
ation the nonlinearities of the plant, and approximates
the unstructured dynamics of the PMSG.
The rest of the paper is organized as follows: System
description is established in Section 2. Section 3 deals
with the formulation of PBVC control strategy. In Sec-
tion 4, the observer-based PBVC controller is presented.
In Section 5, the grid-side converter (GSC) controller is
discussed. Section 6, presents the numerical validation of
the presented control strategy. Finally, the main conclu-
sions are presented in Section 7.
2|MATHEMATICAL MODEL
The produced power via the generator is controlled by
the proposed nonlinear observer-PBVC controller applied
to the machine-side converter (Figure 1).
2.1 |Marine current power
The tidal power captured via the turbine and its related
output torque T
m
, are expressed as follows
9
:
Pm=1
2ρCpβ,λðÞAv3
tð1Þ
Tm=Pm
ωm
ð2Þ
Cpβ,λðÞ=0:5116
λi
−0:4β−5
e−21
λi
ð3Þ
λ−1
i=λ+0:08βðÞ
−1−0:035 1 + β3
−1ð4Þ
λ=ωmR
vt
ð5Þ
where βdenotes the pitch angle, v
t
denotes the tidal
speed, ρrepresents water density, λdenotes tip-speed
ratio, C
p
represents the power coefficient, Arepresents
the swept area of the blades, Rdenotes the radius of the
blades, and ω
m
denotes the rotor speed.
2.2 |PMSG αβ-reference frame model
The proposed control design needs the αβ-model of the
PMSG expressed below
4,22
:
Lαβ
diαβ
dt +ψαβ θe
ðÞpωm=vαβ −Rαβiαβ ð6Þ
Jdωm
dt =Tm−Teiαβ,θe
−ffvωmð7Þ
Teiαβ,θe
=ψT
αβ θe
ðÞiαβ ð8Þ
where T
e
denotes the electromagnetic torque,
ψαβ θe
ðÞ=ψf
−sin θe
ðÞ
cos θe
ðÞ
is the flux linkages vector,
iαβ =iα
iβ
represents the stator current vector,
Lαβ =Lα0
0Lβ
denotes the stator induction matrix, f
fv
is
the viscous coefficient, Rαβ =RS0
0Rs
represents the sta-
tor resistance matrix, vαβ =vα
vβ
denotes the voltage sta-
tor vector, θ
e
represents the electrical angular, and ω
m
denotes the PMSG speed.
FIGURE 1 Tidal conversion
system under Simulink [Colour
figure can be viewed at
wileyonlinelibrary.com]
BELKHIER ET AL.3
3|DESIGN OF THE PBVC
STRATEGY
The design of observer-based PBVC control requires a
number of steps: First, it is necessary to calculate an
Euler-Lagrange model that establishes appropriate input
and output passive vectors relationship. Second, the sys-
tem is decomposed into two interconnected subsystems
with negative feedback. Finally, the last step identifies
the non-dissipative terms in the system model. The con-
troller design process is depicted in Figure 2. Main idea
for introducing the PBVC control is to make the dynam-
ics of the closed-loop system as passive and this is
achieved by introducing a damping term and reshaping
system's energy.
3.1 |Passive feedback decomposition of
the PMSG αβ-model
The electrical dynamics (6) of the PMSG are re-arranged
as follows:
Xe:Ve=vαβ
−ωm
!Ye=iαβ
Tm
ð9Þ
The mechanical dynamics (7) can be re-expressed as
follows:
Xm:Vm=−Te+Tm
ðÞ!Ye=−ωm=−Te+Tm
ðÞ
Js +ffv
ð10Þ
Based on the above decomposition, one can formulate
the following lemma given below:
Lemma 1. According to the aforementioned conditions,
the dynamics of PMSG in the αβ-frame can be
decomposed into two passive feedback inter-
connected subsystems, that is, electrical subsystem
P
e
and mechanical subsystem P
m
.
Proof. See Appendix A.
3.2 |Passivity property of PMSG
Lemma 2. The dynamics of PMSG in αβ-mode
described by (6) to (8) are passive, if X=iT
αβ,ωm
hi
T
FIGURE 2 Passivity
concept design [Colour figure
can be viewed at
wileyonlinelibrary.com]
4BELKHIER ET AL.
and Y=vT
αβ,Te
hi
Tare defined respectively as the
inputs and outputs of the PMSG.
Proof. See Appendix B.
3.3 |Controller law computation
From the system of equations presented in (6) to (8) the
following desired dynamics are formulated:
v
αβ =Lαβ
di
αβ
dt +ψαβ θe
ðÞpωm+Rαβi
αβ ð11Þ
Tm=Jdω
m
dt −T
ei
αβ,θe
−ffvω
mð12Þ
where T
edenotes the desired electromagnetic torque, ω
m
represents the speed of the turbine, v
αβ denotes the refer-
ence voltage, and i
αβ denotes the reference current.
Therefore, to ensure convergence of the dynamics to
zero, that is, the error between the measured and the ref-
erence dynamics, it is required to formulate v
αβ
. Equa-
tions (11) and (12) are reduced to the following
expressions:
vαβ −v
αβ =Lαβ
dεi
dt +Rαβ i
αβ −iαβ
ð13Þ
Jdω
m
dt −T
ei
αβ,θe
−ffv ω
m−ωm
=0 ð14Þ
The desired energy function V
fεi
ðÞis expressed as
follows:
V
fεi
ðÞ=1
2εT
iLαβεi
ð15Þ
where εi=i
αβ −iαβ
represents the current tracking
error. The time derivative of V
fεi
ðÞalong the trajectory
(13), yields the following expression:
_
V
fεi
ðÞ=−εT
iRαβεi+vαβ −v
αβ
ð16Þ
The desired control law is formulated as below:
vαβ =v
αβ −Biεið17Þ
where the term B
i
=b
i
I
2
represents a positive definite
two-by-two matrix and I
2
is an identity matrix
Remark 1. With a suitable gain b
i
, the positive definite
matrix B
i
addresses the issue of the parameter
uncertainties of the system and it also improves the
tracking error convergence.
The proof of the convergence is given in Appendix C.
The PMSG operates at an optimal speed if the desired
current is taken as follows
15
:
i
αβ =2T
e
3pψf
−sin θe
ðÞ
cos θe
ðÞ
ð18Þ
From quation (14), the desired torque is expressed by
the following relation:
T
e=Jdω
m
dt −ffv ω
m−ωm
ð19Þ
where εm=ω
m−ωm
represents the speed error. The
objective is to minimize the speed error between the
PMSG and the marine current turbine. From Equa-
tion (19), the desired torque T
ehas two drawbacks: its
dependence on parameters (J,f
fv
) and it is an open loop.
23
Thus, a PID controller is selected to guarantee conver-
gence of ε
m
, eliminates the static error, and ensures
robustness. T
eis calculated as follows:
T
e=Jdω
m
dτ−kpεm−kiðt
0
εmdτ−kd
dεm
dτð20Þ
where k
p
>0, k
i
>0, and k
d
>0.
Remark 2. The proposed method shown in Figure 2
ensures the convergence of speed and current track-
ing errors, where the PID controller of Equation (20)
formulates the reference current given in Equa-
tion (18). It ensures the convergence of ε
m
using the
damping term “B
i
ε
i
”expressed in the voltage
expression of v
αβ
(Equation (17)). The condition
expressed in Equation (17) ensures a negative time
derivative of the energy term V
fεi
ðÞwhich guaran-
tees the stability of the system.
4|PROPOSED CONTROLLER
The control problem consists of designing a nonlinear
observer that asymptotically reconstructs the states (the
currents and the rotor speed signals) from the measure-
ments. To observer is designed based on the passivity
approach. Duality is established between the nonlinear
observer and the PBVC controller.
BELKHIER ET AL.5
4.1 |Observer design based on PBVC
approach
From the PMSG model presented in Equations (6) to (8)
and the voltage controller expression of Equation (17),
the proposed nonlinear observer is formulated as follows.
Lαβ
d
^
iαβ
dt +ψαβ θe
ðÞp^
ωm=vαβ −Rαβ^
iαβ −Г^
εi
Jd^
ωm
dt =Tm−Te^
iαβ,θe
−ffv ^
ωm−ρm^
εm
Te^
iαβ,θe
=ψT
αβ θe
ðÞ
^
iαβ
8
>
>
>
>
>
<
>
>
>
>
>
:
ð21Þ
where ^
iαβ is the estimated currents, ^
ωmis estimated rotor
speed, ^
εm=^
ωm−ωm
ðÞrepresents the estimated speed error,
and Г=ρ
i
I
2
>0, ρ
m
> 0 are the observer gains. Finally,
^
εi=^
iαβ −iαβ
is the estimated current error. Equations
(6) to (8) and Equation (21) are rewritten as follows:
D_
X+A θe
ðÞ
X+WX=Nvαβ +ζ
D_
^
X+A θe
ðÞ
^
X+W^
X=Nvαβ +ζ−Kℇ
(ð22Þ
where W = diag{R
αβ
,f
fv
}, N = diag{I
2
, 0}, ζ=
0
0
−Tm
2
6
43
7
5,
^
X= ^
iT
αβ,^
ωm
hi
Trepresents the observer states,
ℇ=^
εT
i,^
εm
T, K = diag{Г,ρ
m
}, A θe
ðÞ=−ATθe
ðÞ=
0ψαβ θe
ðÞp
−ψT
αβ θe
ðÞp0
"#
, and D = diag{L
αβ
,J}. The
observer model is re-formulated as follows:
D_
ℇ+A θe
ðÞℇ+W+KðÞℇ=0
3×1ð23Þ
The estimated error ℇis asymptotically stabile and its
convergence proof is provided in Appendix D.
4.2 |Combined nonlinear observer-
PBVC strategy
In this section, the combination of the PBVC control and
the observer is the re formulated. Thus, from Equa-
tion (21), the dynamics of the proposed observer is
deduced as follows:
Lαβ
d^
iαβ
dt +ψαβ θe
ðÞp^
ωm+Rαβ^
iαβ =vαβ −Г^
iαβ −i
αβ
Jd^
ωm
dt =Tm−Te^
iαβ,θe
−ffv ^
ωm−bm^
ωm−ω
m
8
>
>
<
>
>
:
ð24Þ
where b
m
> 0. Therefore, the desired torque T
eof Equa-
tion (20) is expressed as follows.
T
e=Jdω
m
dτ−kpεω−kiðt
0
εωdτ−kd
dεω
dτð25Þ
where εω=^
ωm−ω
m
. Then, the voltage controller of the
PMSG becomes as follows:
vαβ =Lαβ
di
αβ
dt +Rαβi
αβ +ψαβ θe
ðÞp^
ωm−Bi^
iαβ −i
αβ
ð26Þ
4.3 |Global stability property
Considering the systems of equations presented in (6) to
(8), (21) and (24) to (26), the following global stability
property is deduced:
D_
ε+G θe
ðÞε+F θe
ðÞℇ=ζð27Þ
where G θe
ðÞ=
Rαβ +Bi
02×1
−ψT
αβ θe
ðÞ ffv +bm
2
43
5,
Fθe
ðÞ=
Гψαβ θe
ðÞ
01×2ffv +ρm
2
43
5, and ε=εT
i,εm
T. Thus, the
following proposition is formulated to demonstrate the
global stability of the closed-loop observer-PBVC control
system:
Proposition 1. The closed-loop system described by
Equation (27) is globally stable only if the following
conditions are satisfied:
ρm>ρe+Rs
bi>ρe
4+R2
s
bm>ρm
4+f2
fv
8
>
>
>
<
>
>
>
:
ð28Þ
Proof. To prove the convergence of the error term
e
o
=[ε
T
,ℇ
T
]
T
, the energy function is defined as
follows:
Vobs ε,ℇðÞ=1
2εTDε+1
2ℇTDℇð29Þ
Tracking the time derivative of V
obs
(ε,ℇ)alongthe
trajectories of (24) and (27), yields the following
expression:
6BELKHIER ET AL.
Vobs ε,ℇðÞ=−1
2εTGθe
ðÞε−1
2εTFθe
ðÞℇ−1
2ℇTW+KðÞℇð30Þ
By re-arranging Equation (30), one obtains the follow-
ing form:
Vobs ε,ℇðÞ=−eT
oZθe
ðÞeoð31Þ
where Z θe
ðÞ=
Gθe
ðÞ 1
2Fθe
ðÞ
1
2FTθe
ðÞ W+KðÞ
2
6
43
7
5, and from Equa-
tion (31), one can deduce that, if the matrix Z(θ
e
) is posi-
tive definite, then, the system is asymptotically globally
stable. This condition is verified only if the following
inequality is satisfied:
Gθe
ðÞW+KðÞ−1
4Fθe
ðÞFTθe
ðÞ=Z11 Z12
Z21 Z22
>0 ð32Þ
where
Z11 =Rαβ Г+Bi
ðÞ−1
4Г2+ψαβ θe
ðÞψT
αβ θe
ðÞ
+ГRαβ +Г2
−1
4Г2+ψαβ θe
ðÞψT
αβ θe
ðÞ
Z12 =ρm
4ψαβ θe
ðÞ
Z21 =ψT
αβ θe
ðÞ
ρm
4−Г+Rαβ
Z22 =ffv bm+ρm
ðÞ−ρ2
m
4
One can see that the inequality (32) is satisfied only if the
conditions in (28) are satisfied. Thus, Z(θ
e
) is the positive
definite matrix.
5|PI CONTROLLER FOR THE GSC
CONVERTER
Figure 3 depicts the conventional PI control strategy for
the GSC converter. The mathematical model of the GSC
converter is formulated below
24-26
:
Vd
Vq
=Rf
idf
iqf
+
Lf
didf
dt −ωLfiqf
Lf
diqf
dt +ωLfidf
2
6
6
43
7
7
5+Vgd
Vgq
ð33Þ
where V
gd
and V
gq
are the grid voltages, i
df
and i
qf
are
the grid currents, V
d
and V
q
denotes the inverter voltages,
ωdenotes the network angular frequency, R
f
represents
the filter resistance, and L
f
is the filter inductance.
The dynamics of the DC link voltage is expressed as
follows
26
:
CdVdc
dt =3
2
vgd
Vdc
idf +idc ð34Þ
where Cdenotes the capacitance of the DC-link, i
dc
repre-
sents the GSC current, and V
dc
denotes the voltage of the
DC-link. Expression for PI current loop is given as follows:
VPI
gd =kd
gp iref
df −idf
−kd
gi Ðt
0
iref
df −idf
dτ
VPI
gq =kq
gp iref
qf −iqf
−kq
gi Ðt
0
iref
qf −iqf
dτ
8
>
>
>
<
>
>
>
:
ð35Þ
where kd
gp >0, kd
gi >0, kq
gp >0, kq
gi > 0. The q-axis current
iref
qf is expressed by the following relation:
iref
qf =kdcp Vdc_ref −Vdc
ðÞ−kdci ðt
0
Vdc_ref −Vdc
ðÞdτð36Þ
where k
dcp
> 0 and k
dci
>0.
Finally, the reactive and active powers are given by
the following expressions:
Pg=3
2Vgdidf
Qg=3
2Vgdiqf
8
>
<
>
:
ð37Þ
6|NUMERICAL VALIDATION
To evaluate the performance of the proposed passivity-
based nonlinear observer and voltage controller, exten-
sive simulations have been performed in MATLAB/
Simulink environment. The performance of the proposed
control is tested with a 1.5 MW tidal turbine-driven
PMSG energy system. The reference command for V
dc
and Q
g
are chosen as 1150 V and zero, respectively. Sys-
tem parameters are listed in Table 1. The initial condi-
tions used in simulation are given as follows: b
i
= 250,
ρ
m
= 600, b
m
= 150 and ρ
e
= 700, [ω
m
(0), i
dq
(0)] = [0,
0, 0], V
dc
(0) = 0 and i
dqf
(0) = [0, 0]. From the imposed
pole location, the PBVC-PID controller gains are selected
as: k
p
=5,k
i
= 100, and k
d
= 0.5. The DC-link PI
BELKHIER ET AL.7
controller gains are selected as: k
dcp
= 5 and k
dci
= 500.
The GSC current PI controller gains are kd
gp =kq
gp = 9 and
kd
gi =kq
gi = 200. The performance of the proposed control-
ler is compared with classical (PI) and the second-order
SMC methods.
7
The controllers are tested in the follow-
ing three scenarios; in the first test system parameters are
assumed to be known and fixed. The second test is per-
formed to validate the robustness of the proposed con-
troller under disturbances and parameter uncertainties.
Three sub-tests are carried out in this part. First, +50%
and +100% stator resistance R
s
variations are provoked.
Second, +50% and +100% change of the total inertia Jis
imposed. Finally, simultaneous variations of +100% in
R
s
and +100% in Jare imposed. In the third test, a com-
parison of the robustness performance of the proposed
strategy, SMC, and the PI against parameter changes is
presented.
6.1 |Controllers performance with fixed
system parameters
Figure 4 shows the speed profile of the marine current.
Figure 5 shows the comparison of the electromagnetic
FIGURE 3 Grid-side converter control strategy [Colour figure can be viewed at wileyonlinelibrary.com]
8BELKHIER ET AL.
torque T
e
(i
αβ
,θ
e
) under all three variants of the control
schemes. The presented results show that RPB-NOVC
allows a peak torque of 0.025 kNm. In comparison to the
proposed control scheme, the second-order SMC and PI
FIGURE 4 Tidal speed
FIGURE 5 Electromagnetic torque [Colour figure can be
viewed at wileyonlinelibrary.com]
FIGURE 6 DC-link voltage [Colour figure can be viewed at
wileyonlinelibrary.com]
TABLE 1 System parameters
PMSG parameter Value
Water density (ρ) 1024 kg/m
2
Stator resistance (R
s
) 0.006 Ω
Tidal turbine radius (R)10m
Stator inductance (L
dq
) 0.3 mH
Pole pairs number (p)48
Flux linkage (ψ
f
) 1.48 Wb
Total inertia (J) 35 000 kg m
2
DC-link capacitor (C) 2.9 F
Grid voltage (V
g
) 574 V
DC-link voltage (V
dc
) 1150 V
Grid-filter resistance (R
f
) 0.3 pu
Grid-filter inductance (L
f
) 0.3 pu
FIGURE 7 Zoom on reactive power [Colour figure can be
viewed at wileyonlinelibrary.com]
TABLE 2 Initial parameter results of the control strategies
Control RPB-NOVC SMC PI
Variation R
s
and JR
s
and JR
s
and J
T
em
0.021 0.21 0.19
ϵ(V
dc
) ±0.07 ± 0.2 ±0.42
ϵ(Q
g
) ±0.00006 ±0.00012 ±0.00015
P
g
0.021 0.019 0.017
FIGURE 8 Active power [Colour figure can be viewed at
wileyonlinelibrary.com]
FIGURE 9 Electromagnetic torque with change of +50% and
+100% of R
s
[Colour figure can be viewed at
wileyonlinelibrary.com]
BELKHIER ET AL.9
controllers offer a sluggish transient response and the
generated peak torques are 0.018 and 0.021 kNm, respec-
tively. It is also observed that with the proposed control
scheme and during the steady-state, the electromagnetic
torque shows a constant tendency without any fluctua-
tion. Figure 6, shows the tracking response of the DC-
link voltage. From the given results, transient peak over
and undershoots of +0.2, +0.42 V and −0.2, −0.42 V are
observed with SMC and PI controllers respectively, while
with the proposed control scheme, the recorded over-
shoot and undershoot in the DC link voltage are +0.07
and −0.07 V, respectively. Moreover, the proposed con-
trol ensures oscillation-free voltage tracking response.
Figure 7 shows the reactive power tracking comparison
under the proposed, SMC and PI control schemes, respec-
tively. From the presented results, a peak error of 1.5e−4
is observed with PI, 1.2e−4 with SMC and 0.6e−4is
observed with the proposed control method. Although
the reactive power tracking errors with all the three vari-
ants of control schemes are well-bounded. However, as
shown in Table 2, the proposed controller shows better
convergence criterion and lowest steady-state error. Simi-
larly, Figure 8 shows the active power tracking
response of the system under all three variants of the
control schemes. From the presented results, it is evi-
dent that in the case of the proposed control scheme
the average active power integrated into grid is higher
than the SMC and PI controllers. In summary, under
fixed system parameters, the proposed control scheme
showed better control performance as compared to the
other two variants of the controllers namely PI and
SMC methods.
6.2 |Proposed controller performance
under parameter changes
Case 1. Stator resistance R
s
variations
To test the performance of the system under proposed
control, variations of 1.5R
s
and 2R
s
are introduced in
the system model. Figure 9 shows the simulation results
of electromagnetic torque and it is observed that a vari-
ation of +50% and +100% in R
s
increases the electro-
magnetic torque by 9.52%, however, this variation does
not influence the steady-state dynamics of the electro-
magnetic torque. Similarly tracking response of DC-link
voltage with different values of R
s
is shown in
Figure 10. From the presented results it is observed that
the proposed controller exhibits robustness property to
such variations in R
s
and the transient tracking voltage
response in all the three cases nearly similar. The quan-
titative results recorded from Figure 10 are tabulated in
Table 3. The measured error in DC link voltage (ϵ(V
dc
))
is approximately ±0.004% which is very close and better
to the recorded error of ±0.006% (see Table 2).
Figure 11 confirms the robustness of the proposed con-
trol scheme for tracking and regulation of the reactive
FIGURE 10 DC-link voltage with change of +50% and +100%
of R
s
[Colour figure can be viewed at wileyonlinelibrary.com]
TABLE 3 Robustness comparison of the control strategies
Control RPB-NOVC SMC PI
Variation 1.5R
s
1.5 J
1.5R
s
and 1.5 J1.5R
s
1.5 J
1.5R
s
and 1.5 J1.5R
s
1.5 J
1.5R
s
and 1.5 J
T
em
+0.002 +0.0005 +0.002 −0.005 −0.006 −0.002 −0.003 −0.005 −0.004
ϵ(V
dc
) ±0.05 ±0.07 ±0.05 ±0.7 ±0.27 ±0.4 ±0.8 ±0.3 ±0.42
ϵ(Q
g
) ±0.00005 ±0.00006 ±0.00005 ±0.00012 ±0.00014 ±0.00012 ±0.0014 ±0.00013 ±0.00015
P
g
+0.004 +0.001 +0.004 −0.007 −0.002 −0.006 −0.005 −0.002 −0.007
FIGURE 11 Reactive power with change of +50% and +100%
of R
s
[Colour figure can be viewed at wileyonlinelibrary.com]
10 BELKHIER ET AL.
power to its reference value under the variation of R
s
.
From the presented results and the quantitative obser-
vations tabulated in Table 3, it is observed that the
recorded error ϵ(Q
g
)is±5e−5approximatelyanditis
very close to the measured error of ±6e−5underfixed
parameters. The active power tracking response with
variations R
s
is plotted in Figure 12, and it is observed
that the power integrated to grid increased by 19.04%
FIGURE 12 Active power with change of +50% and +100% of
R
s
[Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 13 Electromagnetic torque with change of +50% and
+100% of J[Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 14 DC-link voltage with change of +50% and +100%
of J[Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 15 Reactive power with change of +50% and +100%
of J[Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 17 Electromagnetic torque with change of +100% of
R
s
and +100% of J[Colour figure can be viewed at
wileyonlinelibrary.com]
FIGURE 18 DC-link voltage with change of +100% of R
s
and
+100% of J[Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 19 Reactive power with change of +100% of R
s
and
+100% of J[Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 16 Active power with change of +50% and +100% of
J[Colour figure can be viewed at wileyonlinelibrary.com]
BELKHIER ET AL.11
under these changes in R
s
(see Table 3). This is due to
the increased electromagnetic torque shown in
Figure 9, since the active power is proportional to the
torque as illustrated by Equation (2). The system
dynamic is not influenced by R
s
due to the fact that the
resistance variations are compensated by the imposed
damping gain b
i
in the PBVC design (see Equation 17).
The control robustness is ensured when that damping
factor is chosen larger than R
s
. Also, the significant
improvement of the performances is due to the parame-
ter “r
3
”in the nonlinear observer as explained in
“Remark 3”(Appendix D). From this test, it is con-
cluded that the proposed controller is robust to the vari-
ations in the stator resistance. Moreover, a significant
improvement of the performances presented in Sec-
tion 6.1 is observed.
FIGURE 20 Active power with change of +100% of R
s
and
+100% of J[Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 21 Electromagnetic
torque response comparison. A, +50% of
R
s
. B, +50% of J. C, 50% of R
s
and +50%
of J[Colour figure can be viewed at
wileyonlinelibrary.com]
12 BELKHIER ET AL.
Case 2. Variations in the total inertia J
To confirm the superiority of the proposed control
method, variations of +50% and +100% are introduced
in the total inertia J. With the above variations in J,the
electromagnetic torque is compared and shown in
Figure 13. From the presented results and with differ-
ent values of J, the electromagnetic torque T
em
remains
constant in the steady state. The measured steady-state
electromagnetic torque T
em
is approxiamrlty 0.021 kNm
and once can observe the variations in Jdo not affect
the robustnes of the proposed control scheme. More-
over, the recorded measurements are tabulated in and
Table 3, and it observed that with +50% and +100% var-
iation in Jdoes not influence the peak magnitude of
T
em
which was recorded for the fixed parameter case.
Similar observations are recorded for V
dc
and Q
g
and
theresultsareshowninFigures14and15respectively.
The robustness of the controller is ensured due to the
integrated nonlinear observer and the controller man-
aged to maintain the same response obtained with fixed
parameters test case 1 (see Tables 2 and 3). Moreover,
from Figure 16, it can be noticed that the extracted
power P
g
is slightly increased as compared to the previ-
ous case shown in Figure 8, which means that no influ-
ence of this perturbation is observed on the
performance of the system. Finally, it is also concluded
that the proposed control scheme is also robust to the
variation in J. The robustness is ensured against the
FIGURE 22 DC-link voltage
response comparison. A, +50% of R
s
.B,
+50% of J. C, +50% of R
s
and +50% of J
[Colour figure can be viewed at
wileyonlinelibrary.com]
BELKHIER ET AL.13
variation in the inertia moment Jby the new PID con-
troller computed by the PBVC-Observer given by
Equation (25).
Case 3. simultaneous change R
s
and J.
To test the performance of the system under proposed
control, simultaneous variations of +100% in R
s
and
+100% in Jare imposed on the simulation model of the
system. With the application of subject variations in the
parameters the electromagnetic torque is the same as in
case 1 (0.025 kNm) as shown in Figure 17 and the mea-
sured steady-state torque is tabulated in Table 3. The
recorded results of electromagnetic torque confirm the
robustness of the proposed control scheme under the
simultaneous variations in the parameters. The tracking
responses of V
dc
and Q
g
are shown in Figures 18 and 19
respectively. The recorded voltage error ϵ(V
dc
) and reac-
tive power error ϵ(Q
g
) are approximately the same as in
case 1, that is, ±0.05 and ±5e−5, respectively. The mea-
sured values are also tabulated in shown in Table 3.
While comparing the active power integration to the grid
(Figure 20), with the proposed controller the system gen-
erates a higher active power (19.04% more than test
case 2) which is the same response as in case 1. With
simultaneous variations of Jand R
s
, the proposed control-
ler showed robustness due to the dissipation control part.
This has validated the theoretical results of Section 4.
FIGURE 23 Reactive power
comparison. A, +50% of R
s
. B, +50% of
J. C, +50% of R
s
and +50% of J[Colour
figure can be viewed at
wileyonlinelibrary.com]
14 BELKHIER ET AL.
6.3 |Controllers performance
comparison with parameter variations
In this section, the performance of the system is tested
with all three variants of the control schemes that include
the proposed, SMC, and PI schemes. The parameter R
s
is
varied by +50% and +100% of the variation is imposed
on parameter J. The tracking responses of the electro-
magnetic torque with all three variants of the control
schemes and with 50% variation in R
s
, 50% variation in J
and simultaneous variation in both R
s
and Jare shown in
Figure 21A-C respectively. From Figure 21A, a fixed elec-
tromagnetic steady-state torque of 0.021 kNm is recorded
with the proposed controller, while in the case of both
SMC and PI controllers, the measured steady-state elec-
tromagnetic torque is approximately 0.015 kNm. Similar
behavior is also observed in the results presented in
Figure 21B, where a steady-state torque of 0.021 kNm is
recorded with the proposed control scheme, while with
PI and SMC controllers, the measured torque is around
0.014 kNm. In the third case, the measured steady-state
electromagnetic toque us around 0.021 kNm with the
proposed controller, while with PI and SMC controllers,
the measured values are 0.015 and 0.018 kNm, respec-
tively. The measured torque in the above three cases is
also tabulated in Table 3.
The tracking responses of the DC link voltage with all
three variants of the control schemes and with 50% varia-
tion in R
s
, 50% variation in Jand simultaneous variation
in both R
s
and Jare shown in Figure 22A-C respectively.
From Figure 22A-C, the transient overshoot of 1.8 V and
undershoot of −1.9 V is observed with the proposed con-
trol scheme under 50% variation in R
s
, 50% variation in J
and simultaneous variation of both the parameters.
While in case of PI and SMC controllers, both the
recorded overshoots and undershoots are larger as
FIGURE 24 Active power
comparison. A, +50% of R
s
. B, +50% of
J. C, +50% of R
s
and +50% of J[Colour
figure can be viewed at
wileyonlinelibrary.com]
BELKHIER ET AL.15
compared to the proposed controller. Another important
aspect of the proposed controller is the improved
damping of the transient oscillations in the DC link volt-
age. It is recorded from the presented results, that the
transient oscillations in the DC link voltage are settled at
around 0.8 second at all test conditions while with PI and
SMC controllers it takes a longer time to settle down. The
corresponding errors in DC link voltage are tabulated in
Table 3.
The reactive power tracking and regulation responses
with all three variants of the control schemes and with
50% variation in R
s
, 50% variation in Jand simultaneous
variation in both R
s
and Jare shown in Figure 23A-C
respectively. From Figure 23A-C, it is evident that under
all three variants of control schemes, the reactive power
tracking response is bounded. However, in case of the
proposed control scheme, the convergence of reactive
power to its reference power is faster as compared to the
other two control variants. The quantitative data mea-
sured from the presented results are tabulated in Table 3.
The active power tracking responses with all three vari-
ants of the control schemes and with 50% variation in R
s
,
50% variation in Jand simultaneous variation in both R
s
and Jare shown in Figure 24A-C respectively. From the
presented results, the measured steady-state active power
is around 0.025, 0.021, and 0.025 MW with 50% variation
in R
s
, 50% variation in Jand simultaneous variation in
both R
s
and J, respectively. While in case of PI and SMC
controllers reduced active power is observed in all the
three cases. Figure 25 shows that the action of the pro-
posed control scheme, a perfect sinusoidal voltage is
absorbed by grid.
7|CONCLUSION
In this paper, a new robust control method is presented
to improve the dynamic performance of the PMSG driven
marine current power conversion system. The proposed
passivity-based combined nonlinear observer-voltage
controller is adopted to extract the maximum power from
the tidal current. The proposed controller is tested under
three test conditions that include the following: (a) Fixed
system parameters, (b) +50% and +100% variation in
rotor resistance R
s
and moment of inertia J,
(c) Simultaneous variation in both rotor resistance R
s
and
moment of inertia J, the results obtained under these test
conditions are also compared with classical PI control
and SMC method. From the presented results it is con-
cluded that the proposed controller showed an improved
mixing of power under all test conditions to the grid
which is higher by 19% as compared to the classical PI
and SMC controllers.
Moreover, the proposed controller under all test con-
ditions showed a faster transient response as compared to
the classical PI and SMC controllers. This is evident from
the presented results, that with the proposed control
method the transient oscillations in the DC link voltage
are settled at around 0.8second at all test conditions
while with PI and SMC controllers it takes longer time to
settle down.
Similarly, under all test conditions, the average steady
state electromagnetic torque is noted around 0.021 kNm
while the PI and SMC controllers the measured values of
torque is approximately 0.014 and 0.018 kNm,
respectively.
Future works will be focused on:
•Experimental validation of the proposed RPB-NOVC.
•The adaptation of the PID fixed gains by introducing a
fuzzy logic controller to reduce their sensitivity to the
parameter changes.
ACKNOWLEDGEMENT
The authors acknowledge the funding of Researchers
Supporting Project number (TURSP-2020/144), Taif Uni-
versity, Taif, Saudi Arabia.
DATA AVAILABILITY STATEMENT
The authors confirm that there is no data to share within
this paper.
ORCID
Youcef Belkhier https://orcid.org/0000-0002-7520-2286
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How to cite this article: Belkhier Y, Achour A,
Hamoudi F, Ullah N, Mendil B. Robust energy-
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er.6650
BELKHIER ET AL.17
APPENDIX A: PROOF OF LEMMA 1
First, from P
e
, the total energy H
e
is expressed as follows:
He=1
2iT
αβLαβ iαβ +ψT
αβiαβ ðA1Þ
The time derivative of H
e
along (6), yields the follow-
ing expression:
_
He=−iT
αβRαβ iαβ +YT
eVe+d
dt ψT
αβiαβ
ðA2Þ
Integration on both sides of (A2) along [0 T
e
], gives
the following relation:
HeTe
ðÞ
−He0
ðÞ
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
Stored Energy
=ðTe
0
iT
αβRαβ iαβ dτ
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
Dissipated Energy
+ðTe
0
YT
αβVedτ+ψT
αβiαβ
hi
Te
0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Supplied Energy
ðA3Þ
where H
e
(T
e
)≥0 and H
e
(0) represents the initial stored
energy. By integrating Equation A3), a dissipation
inequality is deduced, which is given as follows:
ðTe
0
YT
eVedτ≥λmin Rαβ
ðTe
0
iαβ
2dτ−He0ðÞ+ψT
αβiαβ
hi
Te
0
ðA4Þ
where k.krepresents the vector norm of the standard
Euclidian.
From Equation (A4), it is deduced that P
e
is passive.
Then, the transfer function F
m
(s) is expressed by the fol-
lowing expression:
FmsðÞ=YmsðÞ
VmsðÞ=1
Js+ffv
ðA5Þ
It can be deduced that P
m
is passive, since F
m
(s)is
strictly positive. Thus, the PMSG model is decomposable
into two passive subsystems.
APPENDIX B: PROOF OF LEMMA 2
Let us define H
m
(Hamiltonian) of the PMSG, given as follow:
Hm_
iαβ,ωm
=1
2
_
iT
αβLαβ _
iαβ +ψT
αβ
_
iαβ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Electrical
Energy
+1
2Jω2
m
|fflffl{zfflffl}
Mecanical
Energy
ðA6Þ
The time derivative of the H
m
along the
Equations (6) to (8) yields the following expression:
dHm_
iαβ,ωm
dt =−
d_
iT
αβR_
idq
dt +yTν+d
dt ψT
αβ
_
iαβ
ðA7Þ
where R= diag{R
αβ
,f
fv
} is the symmetrical defines posi-
tive matrix. Integrating both sides of Equation (A7) in
[0 T
m
] yields:
HmTm
ðÞ−Hm0ðÞ
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
Stored
Energy
=−ðTm
0
_
iT
αβR_
iαβ dτ
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
Dissipated
Energy
+ðTm
0
yTνdτ+ψT
αβ
_
iαβ
hi
Tm
0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Supplied
Energy
ðA8Þ
where H
m
(T
m
)≥0 and H
m
(0) is the initial energy stored.
By integrating Equation (A8), a dissipation inequality is
deduced, given as follows:
ðTm
0
yTνdτ≥λmin R
fg
ðTm
0
_
iαβ
2dτ−Hm0ðÞ+ψT
αβ
_
iαβ
hi
Tm
0
ðA9Þ
Thus, the relationship M is passive, and the PMSG is
also passive.
APPENDIXC:PROOFOFTHE
EXPONENTIAL STABILITY FOR CURRENTS
Considering Equation (16), which by the Rayleigh quo-
tient and the matrix L
αβ
positivity, guarantees the follow-
ing inequality:
0≤λmin Lαβ
εi
kk
2≤V
fεi
ðÞ≤λmax Lαβ
εi
kk
2ðA10Þ
where λ
max
{L
αβ
}andλ
min
{L
αβ
} are the matrices and
L
αβ
represents the maximum and the minimum
eigenvalues.
The time derivative of (16) along (17) and (18), which
by the Rayleigh quotient and the dissipation term
R
αβ
+B
i
positivity, guarantees the following inequality:
V
fεi
ðÞ=−εT
iRαβ +Bi
εi≤−λmin Rαβ +Bi
εi
kk
2,8t≥0ðA11Þ
where λ
min
{R
αβ
+B
i
} > 0 and it represents the matrix
R
αβ
+B
i
minimum eigenvalue.
From (A10) and (A11), we deduce the following
inequality:
18 BELKHIER ET AL.
_
V
fεi
ðÞ=−r1V
fεi
ðÞ ðA12Þ
where r1=λmin Rαβ +Bi
fg
λmax Lαβ
fg>0.
By integrating (A12), we obtain the following
inequality:
_
V
fεi
ðÞ≤V
f0ðÞe−r1tðA13Þ
From (A10) and (A13), we get:
εi
kk
≤r2εi
kk
e−r1tðA14Þ
where r2=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λmin Lαβ
fg
λmix Lαβ
fg
r>0.
Therefore, with the rate of convergence r
1
the error ε
i
is exponentially decreasing.
APPENDIX D: PROOF OF THE ASYMPTOTIC
STABILITY AND EXPONENTIAL
CONVERGENCE FOR THE ESTIMATED
ERROR ε
Let us define the following desired Lyapunov function
given below:
Vob ℇðÞ=1
2ℇTDℇðA15Þ
The time derivative along (23) of V
ob
(ℇ), yields the fol-
lowing relation:
Vob ℇ
ðÞ
=−1
2ℇTW+K
ðÞ
ℇðA16Þ
The observer estimated error ℇ= 0 is asymptotically
stable since K = K
T
>0.
Following the same steps as in Appendix C, one
obtains the following expression:
ℇ
kk
≤r4ℇ
kk
e−r3tðA17Þ
where r3=λmin W+K
fg
λmax Lαβ
fg
> 0, and r4=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λmin Lαβ
fg
λmix Lαβ
fg
r>0.
Thus, it can be deduced that the speed tracking errors
and the estimated current of the observer are asymptoti-
cally stable.
Remark 3. The damping gain “b
i
”permits a good conver-
gence rate of “r
1
”and “r
3
,”which should be posi-
tive. To avoid divergence of v
αβ
, it is aimed to use a
high but limited value for this gain. This limitation
can be realized by simulation tests. The nonlinear
observer allows us to show an optimal choice with
b
i
= 250.
BELKHIER ET AL.19
