micromachines
Article
Characteristic Study of a Typical Satellite Solar Panel under
Mechanical Vibrations
Xin Shen 1 , Yipeng Wu 1, * , Quan Yuan 1 , Junfeng He 1 , Chunhua Zhou 1,2, * and Junfeng Shen 2
1
2
*
State Key Laboratory of Mechanics and Control for Aerospace Structures, Nanjing University of Aeronautics
and Astronautics, Nanjing 210016, China; shenxin0413@nuaa.edu.cn (X.S.)
Shanghai Institute of Satellite Engineering, Shanghai 201109, China
Correspondence: yipeng.wu@nuaa.edu.cn (Y.W.); zch777@126.com (C.Z.)
Abstract: As the most common energy source of spacecraft, photovoltaic (PV) power generation has
become one of the hottest research fields. During the on-orbit operation of spacecraft, the influence
of various uncertain factors and the unbalanced inertial force will make the solar PV wing vibrate
and degrade its performance. In this study, we investigated the influence of mechanical vibration
on the output characteristics of PV array systems. Specifically, we focused on a three-segment solar
panel commonly found on satellites, analyzing both its dynamic response and electrical output
characteristics under mechanical vibration using numerical simulation software. The correctness of
the simulation model was partly confirmed by experiments. The results showed that the maximum
output power of the selected solar panel was reduced by 5.53% and its fill factor exhibited a decline
from the original value of 0.8031 to 0.7587, provided that the external load applied on the panel
increased to 10 N/m2 , i.e., the vibration frequency and the maximal deflection angle were 0.3754 Hz
and 74.9871◦ , respectively. These findings highlight a significant decrease in the overall energy
conversion efficiency of the solar panel when operating under vibration conditions.
Keywords: photovoltaic modules; solar panel; mechanical vibration; power generation
1. Introduction
Citation: Shen, X.; Wu, Y.; Yuan, Q.;
He, J.; Zhou, C.; Shen, J. Characteristic
Study of a Typical Satellite Solar Panel
under Mechanical Vibrations.
Micromachines 2024, 15, 996. https://
doi.org/10.3390/mi15080996
Academic Editor: Igor Medintz
Received: 12 July 2024
Revised: 29 July 2024
Accepted: 29 July 2024
Published: 31 July 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
As one of the most significant parts of renewable energy, solar energy has become a
hotspot for research worldwide in recent years due to its sustainability and universality.
Photovoltaic (PV) systems, which convert the light energy of the sun into electrical energy,
represent the most common application of solar power [1]. For example, large-scale
solar power plants and rooftop solar panels are now a common sight. And solar powergeneration systems are increasingly being mounted on various types of mobile carriers,
including cars, satellites, ships, and drones. Additionally, the new concept of floating PV
systems on water surfaces is gaining visibility among the public.
Normally, researchers think that the efficiency of photovoltaic power generation is
mainly affected by solar radiation and temperature [2–4]. For this reason, Cuce et al. [5]
experimentally demonstrated that the photogenerated output current, short-circuit current,
and other parameters of solar arrays are directly proportional to the light intensity, and
the equivalent light intensity is related to the equivalent area of the array perpendicular to
the direction of the light. Localized shadows often appear in PV arrays due to interference
from the projected coverings of houses, plants, localized dark clouds, surface dust, etc.
In this case, the PV system has multiple local maximum power points (MPPs), i.e., its
power–voltage (P-V) curve exhibits a multi-peak characteristic [6–8]. Abdulmawjood
et al. [9] analyzed the impact of different shading patterns on the P-V characteristics
through performing a set of simulations with different array configurations. It was found
through experiments that the power dropped significantly when the shading increased.
Under localized shadows, PV arrays also produce hotspot effects, resulting in multi-peakoutput P-V curves, at which time the conventional maximum power point tracking (MPPT)
Micromachines 2024, 15, 996. https://doi.org/10.3390/mi15080996
https://www.mdpi.com/journal/micromachines
Micromachines 2024, 15, 996
2 of 15
is prone to local extreme points and failure [10]. In addition, as the main factor affecting
the performance of photovoltaic power generation, solar radiation’s own randomness and
cyclical changes will make the power of photovoltaic power generation intermittent and
fluctuating [11,12].
In consideration of the application environment of PV systems, it is easy to find that
mechanical vibration always exists and may also have an influence. However, there has not
been a lot of public interest in this. Although not particularly emphasized in engineering,
some scholars have conducted research in response to such questions. Domestic largescale photovoltaic power plants are mainly distributed in the northwest and other lightrich regions of China, where the windy climate also brings a negative impact on the
efficient operation of photovoltaic power-generation systems. With the help of experiments,
scholars have demonstrated that wind-induced vibration leads to oscillations of output
current [13,14], and they have inferred that, in the low-frequency range, vibration-induced
current transients and oscillations in the output of PV modules are the main sources of
distortion. Vibrations from the carrier motion itself or from the carrier engine operation
can also cause the oscillation of the PV module mounted on the carrier surface. Studies by
Zhang [15] investigated PV components on a train, proving that the carrier vibration will
lead to changes in the output characteristics, which will in turn lead to the failure of the
original MPPT control. Vidović et al. [16] focused on ocean floating solar power-generation
devices, and combined a static solar radiation energy calculation model with the dynamic
swing of the float angle to determine that the radiation energy received by the float and
photovoltaic cells under the influence of waves was 87.5% of the solar radiation energy
received by photovoltaic cells fixedly installed at the optimal receiving angle under the
same conditions.
Although mechanical vibration energy can be collected and converted into electrical
energy [17,18], the power-generation reduction in PV systems does exist and strongly
complicates the MPPT algorithm under vibration conditions. Moreover, as the sole power
source for most satellites, solar arrays are usually large outreach structures, and their size
is increasing due to their growing number of tasks and power requirements [19]. During
on-orbit operation, the influence of various uncertain factors in the space environment
and the unbalanced inertial force can easily cause solar arrays to vibrate and degrade their
performance [20]. Therefore, to gain a deeper understanding of the impact of mechanical vibration on the output characteristics of PV systems and to meet the engineering application
needs, this study takes a classical three-segment satellite solar panel as the research object
and mainly calculates its dynamic response and electrical performance under mechanical
vibration, and the effects are systematically discussed. This study provides new insights
for future research on structural vibration suppression and the MPPT algorithm in satellite
solar panels, and the research process is also applicable to the analysis of ground-based PV
systems subject to vibration, especially in times of dramatic climate change.
This paper is structured as follows: Section 2 outlines the structural dynamics and
photovoltaic power modeling of a typical satellite solar panel. Section 3 details the characterization of the solar panel under specific conditions. Finally, the conclusions are presented
in Section 4.
2. Structural Dynamics and Photovoltaic Power Model of a Typical Solar Panel
2.1. Structural Dynamics Model of a Typical Solar Panel
A satellite solar panel is a large-span, highly flexible, external extension structural
system, but the individual rectangular panels that constitute its basic units can be well
described using a linear model [21]. Therefore, in this paper, the three-segment solar panel
is represented by a typical model of three elastic rectangular thin plates connected with
micro-torsion springs. As shown in Figure 1, Plate-1 was fixedly attached to the spacecraft
body and connected to Plate-2 by two torsion springs k1 and k2 , and Plate-2 was connected
to Plate-3 by torsion springs k3 and k4 . In the figure, a and b represent the side lengths of a
single rectangular plate in the x and y directions, respectively, while y1 and y2 represent the
Micromachines 2024, 15, 996
micro-torsion springs. As shown in Figure 1, Plate-1 was fixedly attached to the spacecraft
body and connected to Plate-2 by two torsion springs k1 and k2, and Plate-2 was connected
to Plate-3 by torsion springs k3 and k4. In the figure, a and b represent the side lengths of a
single rectangular plate in the x and y directions, respectively, while y1 and y2 represent
3 of 15
the connection positions of torsion springs k1, k3 and k2, k4 in the y direction, respectively.
The follow-up studies were based on the above model with the following assumptions
[21,22]:
connection positions of torsion springs k1 , k3 and k2 , k4 in the y direction, respectively. The
studies
based on
the above
model with
theconnected
following by
assumptions
[21,22]:
1.follow-up
The solar
arraywere
consisted
of three
rectangular
plates
torsion springs,
and
the
substructures,
such
as
rectangular
plates
and
springs,
can
be
described
byand
a
1.
The solar array consisted of three rectangular plates connected by torsion springs,
linear
model.
This
means
that
the
deformation
of
the
plates
and
the
torsion
springs
the substructures, such as rectangular plates and springs, can be described by a linear
was
within
the means
linear elasticity
range, and the
nonlinearity
of
model.
This
that the deformation
ofgeometric
the platesand
andmaterial
the torsion
springs was
the
structure’s
deformation
notand
considered.
within
the linear
elasticity was
range,
the geometric and material nonlinearity of the
2. The
rigid-body
motion of was
the spacecraft
flight was fixed, and only the elastic vibrastructure’s
deformation
not considered.
the panel is
discussed.
first plate
waswas
fixed
to the
body, vibration
and the
Theofrigid-body
motion
of theThe
spacecraft
flight
fixed,
andspacecraft
only the elastic
2. tion
reference
coordinate
system
is
shown
in
Figure
1.
of the panel is discussed. The first plate was fixed to the spacecraft body, and the
3. The
two plates
were connected
two torsion
springs,
which had rotational freedom
reference
coordinate
system isby
shown
in Figure
1.
the y-axis.
The torsion
and joints
were
small,
and
their structures
3. only
Thearound
two plates
were connected
bysprings
two torsion
springs,
which
had
rotational
freedom
ensured
the effective
transmission
ofsprings
torque, and
shear
force,
andsmall,
axial and
force.
only around
the y-axis.
The torsion
joints
were
their structures
ensured
effective
transmission
of torque,
force, and
axial force.
4. The
panel the
formed
a stable
system when
it wasshear
completely
unfolded,
meaning that
4. the
The
panel formed
a stable
system
when ittorsion
was completely
unfolded,
meaning that the
rectangular
plates
and the
connecting
springs were
well fixed.
rectangular
and the
connecting
springs
wereneglected.
well fixed.The axial and
5. The
geometryplates
and mass
of the
torsion torsion
spring joints
were
5. shear
The deformation
geometry and
of the springs
torsion was
spring
were neglected.
The axial
and
ofmass
the torsion
notjoints
considered;
only the change
in the
shear angle
deformation
of the torsion springs was not considered; only the change in the
torsion
was considered.
torsion
angle was
considered.
6. The
longitudinal
stiffness
of the solar panel was very large, and the waving vibration
6. was
Thenegligible.
longitudinal stiffness of the solar panel was very large, and the waving vibration
was negligible.
y
y2
k2
k4
y1
k1
k3
b
x
a
Figure 1. The schematic solar panel structure with three sub-plates.
Figure 1. The schematic solar panel structure with three sub-plates.
2.1.1. Deflection Model of the Solar Panel
2.1.1. Deflection Model of the Solar Panel
The key to determining the dynamic response of the structure is to determine the form
The key to
determining
thevalue
dynamic
thereferring
structuretoisFigure
to determine
theto
of deflection
and
the deflection
of theresponse
structure.ofStill
1, it is easy
form
of deflection
andfixed
the deflection
value
structure.
Still
referringto
totorsion
Figure springs
1, it is
see that
Plate-1 was
on one side,
freeof
onthe
both
sides, and
connected
easy
to see
thattwo
Plate-1
fixed onwere
one side,
free the
on both
and connected
on one
side;
sideswas
of Plate-2
free and
othersides,
two were
connectedtototorsion
torsion
springs
on
one
side;
two
sides
of
Plate-2
were
free
and
the
other
two
were
connected
to
springs; and three sides of Plate-3 were free and the other was connected
to torsion
springs.
torsion
springs;
and
three
sides
of
Plate-3
were
free
and
the
other
was
connected
to
torsion
The three plates represent all three possible cases with different boundary conditions in
springs.
The three
platesstructures.
represent all three possible cases with different boundary condimulti-plate
connected
tions in
structures.
Asmulti-plate
shown in connected
Figure 2, the
boundary conditions for the solidly supported edges of
As shown
in Figure 2, the
boundary
conditions
thewsolidly
supported
of
Plate-1
were displacement
boundary
conditions.
w1for
(x,y),
w3 (x,y) edges
represent
2 (x,y), and
Plate-1
were displacement
conditions.
w1(x,y),
w2(x,y), and w3(x,y)
represent
the
the deflections
at the endsboundary
of the three
thin plates,
and approximately
satisfy
the bending
deflections
at the ends
of the three
plates, andedges.
approximately
satisfy
bending
momoment boundary
conditions
at thin
the connected
According
to thethe
different
boundment
boundary conditions
at the
connected
edges.the
According
differentof
boundary
ary conditions,
the deflection
functions
adopted
form of to
thethe
separation
variables
conditions,
deflection
functions
adopted
the form
of the
separation
variables
wi (x,y) = Athe
deflection
function
for each
plate
is listed of
below
[21]. wi(x,y)
i (y)B
i (x), and the
Plate-1:
= Ai(y)B
i(x), and the deflection function for each plate is listed below [21].
Plate-1:
n
n
b2
iπy1
iπy
b2
iπy2
iπy
w1 ( x, y) = (c1 + c2 ∑ 2 2 cos
cos
+ c3 ∑ 2 2 cos
cos
b
b
b
b
i
π
i
π
i =1
i =1
(1)
πx
2
2
+c4 y + c5 y) × (1 − cos
+ c6 x )
2a

i 1 i
b
2
b
 c4 y  c5 y）（1  cos
Micromachines 2024, 15, 996
x
2a
i 1 i

n
2
b
b
(1)
2
 c6 x ）
Plate-2:
4 of 15
 c7  c8
w（
2 x, y）（
n
b
i y
b
i y2
i y
 i  cos b cos b  c  i  cos b cos b
i 1
Plate-2:
i y1
2
2
9
2
i 1
2
2
(2)
2
3


（x  a）
 c10 y  cn11 y）
（x  a）iπy
（x  iπy
c13
a）
n bc214
b2 1  c12
iπy
2
1
2
iπy
cos
cos
+ c9 ∑ 2 2 cos
cos
2 π2
b
b
b
i
i
π
i =1
i =1
b
+c10 y2 + c11 y) × 1 + c12 ( x − a) + c13 ( x − a)2 + c14 ( x − a)3
( c7 + c8 ∑
w2 ( x, y) =
Plate-3:
Plate-3:
 c15  c16
w（
3 x, y）（
n
b2
i 
i 1
w3 ( x, y) =
n
2
cos
2
b2
n
i y1
i y2
i y
b2
i y
cos
 c17
cos
cos
2 2
b
b
b
b
i 1 i 

iπy
iπy
n
b2
1
2c
2
(cc15 +
+ cx17 2
∑a）
 cos （bx cos
 2 cos
 c∑
2a）b c21
（
18 y 16
19 y）
i2
π 21  c20
i2 π

iπy2
iπy
cos
b
b
(2)
(3)
(3)
i =1
i =1
+c18 y2 + c19 y) × 1 + c20 ( x − 2a) + c21 ( x − 2a)2
where cm (m = 1, 2, …, 21) in the functions are coefficients to be solved, and n is the truncation
term.
where
cm (m = 1, 2, . . ., 21) in the functions are coefficients to be solved, and n is the
truncation term.
y
z
b
x
O
a
Figure
2. The
thinthin
plate
andand
the the
corresponding
coordinates.
Figure
2. The
plate
corresponding
coordinates.
Based
energy
equation
of the
system
constraint
torsion
spring
Based
on on
thethe
energy
equation
of the
system
andand
thethe
constraint
thatthat
thethe
torsion
spring
only
rotational
degree
of freedom,
the coefficient
cmselected
in the selected
deflection
hashas
only
oneone
rotational
degree
of freedom,
the coefficient
cm in the
deflection
funcwas calculated
using
Ritz method.
It was assumed
the
solar
cellwas
wing
tionfunction
was calculated
using the
Ritzthe
method.
It was assumed
that thethat
solar
cell
wing
was
combined
with
three
square
plates
of
the
same
material,
which
was
isotropic
combined with three square plates of the same material, which was isotropic and ortho-and
orthotropic.
three
plates
were subjected
to a uniform
load
q. The connection
tropic.
The threeThe
plates
were
subjected
to a uniform
load q. The
connection
positionspositions
were
were
as yand
and ythe
0.75b,
the side
a square
= bdensity
= 1 m, the
1 = 0.25b
2 = side
taken
astaken
y1 = 0.25b
y2 = 0.75b,
lengths
of lengths
a squareofplate
as a =plate
b = 1as
m,athe
ρ =3,1700
kg/m3 , the
modulus as
of E
elasticity
as11EPa,
= 1.5
1011 Pa,ofthe
modulus
as ρdensity
= 1700 as
kg/m
the modulus
of elasticity
= 1.5 × 10
the ×
modulus
stiffness
as of
stiffness
as
D
=
0.5D
and
D
=
1.215D
,
and
Poisson’s
ratio
as
µ
=
2µ
=
0.3.
The
four
2
1
3
1
1
2
D2 = 0.5D1 and D3 = 1.215D1, and Poisson’s ratio as µ1 = 2µ2 = 0.3. The four micro-torsion
micro-torsion
springs
had
the
same
coefficient
of
elasticity,
i.e.,
k
=
k
=
k
=
k
=
300D
2 the3 truncation
4
1,
springs had the same coefficient of elasticity, i.e., k1 = k2 = k3 = k4 = 300D11, and
and
the
truncation
term
n
was
chosen
to
be
15.
Then,
an
expression
for
the
deflection
of
the
term n was chosen to be 15. Then, an expression for the deflection of the solar panel under
panel
under
an obtained.
external load can be obtained.
an solar
external
load
can be
Plate-1:
Plate-1:
1
iπ
q
15 15
2
i
(0.791349 + 0.0544 ∑ 1 2 2 cos
cos 2iπy − 0.132467y
2
（D0.791349
 0.0544 i=124i 2 πcos cos2
w（
2 i y  0.132467 y
1 x, y）
1
D1
2
i
i 1 4πx
+0.132467y) × (1 − cos
+ 1.196593x2 )
 x 2a
2
w1 ( x, y) =q

 0.132467 y）（1  cos
Plate-2:
w2 ( x, y) =
2a
 1.196593 x ）
(4)
(4)
15
1
q
iπ
(1.78621 − 0.012 ∑ 2 2 cos
cos 2iπy − 0.042y2 + 0.042y)
D1
2
4i
π
i =1
× 1 + 1.806462 ( x − a) + 0.544749 ( x − a)2 − 0.159476( x − a)3
(5)
15
q
1
iπ
(5.723103 − 0.00431 ∑ 2 2 cos
cos 2iπy + 0.015087y2
D1
2
4i
π
i =1
+0.015087y) × 1 + 0.8363989 ( x − 2a) + 0.020779 ( x − 2a)2
(6)
Plate-3:
w3 ( x, y) =
Plate‐3:
15
q
1
i
15
q
1 cos i cos2i y  0.015087 y 22
）

（

w（
x
,
y
5.723103
0.00431
3
2 2 cos
）

（

w（
x
,
y
5.723103
0.00431
D1
2 cos2i y  0.015087 y
i 2 2
3
i 1 4
D1
2
i 1 4i 

 0.015087 y） 1  0.8363989（x  2a） 0.020779（x  2a）22
 0.015087 y） 1  0.8363989（x  2a） 0.020779（x  2a）


Micromachines 2024, 15, 996
(6)
(6)
5 of 15
The result in Figure 3 was obtained by MATLAB programming. It was found that,
The result in Figure 3 was obtained by MATLAB programming. It was found that,
underThe
a uniform
load
q, the3 deflection
of the
was symmetrical
about
y = 0.5b
andthat,
the
result in
Figure
obtained
bypanel
MATLAB
programming.
It was
found
under a uniform
load
q, the was
deflection
of the
panel
was symmetrical
about
y = 0.5b
and the
deflection
reached
its
maximum
at
y
=
0.5b.
Overall,
the
difference
in
the
deflection
of
the
under a uniform
load
q, the deflection
theOverall,
panel was
y = 0.5b and
the
deflection
reached
its maximum
at y =of
0.5b.
thesymmetrical
difference inabout
the deflection
of the
solar
panels
in
the
y
direction
was
quite
small.
Therefore,
the
deflection
of
the
three-plate
deflection
reached
its
maximum
at
y
=
0.5b.
Overall,
the
difference
in
the
deflection
of
the
solar panels in the y direction was quite small. Therefore, the deflection of the three‐plate
structure
at yin
= 0.5bywas
taken was
as the
object
of study
to simplify
the plate of
structure.
solar panels
direction
quite
small.
Therefore,
the deflection
the three-plate
structure
at y = the
0.5b was
taken as the
object
of study
to simplify
the plate structure.
structure at y = 0.5b was taken as the object of study to simplify the plate structure.
Figure 3.
3. Deflection
Deflectionof
ofthe
thethree-plate
three-plate structure
structure under
under aa uniform
uniform load
load q.
q.
Figure
Figure 3. Deflection of the three‐plate structure under a uniform load q.
The deflection of each plate was at the maximum at its end. To further simplify the
The deflection of each plate was at the maximum at its end. To further simplify the
The
deflection
of each
plate was atofthe
maximum
its end. To studies,
further simplify
the
study
and
to facilitate
the application
the
results toatsubsequent
the average
study and to facilitate the application of the results to subsequent studies, the average
deflection
platethe
wasapplication
calculatedof
and
rotation
angle of each
plate was
derived
study
and of
to each
facilitate
thethe
results
to subsequent
studies,
the average
deflection of each plate was calculated and the rotation angle of each2 plate was
derived2
2, 5
from the average
When theand
uniform
load q angle
was 1 of
N/m
3 N/m
N/m ,
deflection
of each deflection.
plate was calculated
the rotation
each, plate
was
derived
from the average
deflection. When the uniform load q was 1 N/m22, 3 N/m22, 5 N/m22, or 10
2
or 102 the
N/m
, respectively,
the
deflections
at each
of qthe
three
plates
were, calculated,
as
from
average
deflection.
When
the uniform
load
was
1 N/m
, 3 N/m
5 N/m , or 10
N/m 2, respectively, the deflections at each of the three plates were calculated, as shown in
shown
in
Figure
4.
The
resulting
angles
of
rotation
are
listed
in
Table
1,
at
the
end
of
N/m , respectively, the deflections at each of the three plates were calculated, as shown in
Figure 4. The resulting angles of rotation are listed in Table 1, at the end of this subsection.
this subsection.
Figure
4. The resulting angles of rotation are listed in Table 1, at the end of this subsection.
Figure
Figure 4.
4. Deflection
Deflectioncurves
curvesfor
forthree-plate
three-plate structures.
structures.
Figure
4.
Deflection
curves
for
three‐plate
structures.
Table 1. Structural dynamics parameters of solar panel with 3 plates.
Rectangular plate size: a × b
Density: ρ
Elastic modulus: E
Poisson’s ratio
Torsion spring positions
First-order resonant frequency: f
q = 1 N/m2
q = 2 N/m2
Rotation angle
q = 3 N/m2
q = 5 N/m2
q = 10 N/m2
Plate-1
Plate-2
Plate-3
7.1830◦
14.1470◦
20.7107◦
32.2165◦
51.5687◦
1×1m×m
1700 kg/m3
1.5 × 1011 Pa
µ1 = 0.3; µ2 = 0.15
y1 = 0.25 m; y2 = 0.75 m
0.3754 Hz
16.4648◦
30.5870◦
41.5614◦
55.9132◦
71.3064◦
20.4489◦
36.7132◦
48.2042◦
61.7917◦
74.9871◦
Micromachines 2024, 15, 996
6 of 15
2.1.2. First-Order Resonant Frequency of the Solar Panel
Since the solar panel was characterized by a large span, a light mass, and large
flexure, it was easy to produce violent and long-lasting vibrations after receiving small
perturbations, and these low-frequency vibrations were mainly manifested as transverse
bending and vibrations perpendicular to the panel surface.
In the selected model, the stiffness coefficient of the torsion springs between the plates
was much larger than the bending stiffness coefficient of the plates. In analyzing the
resonant frequency, when the panel was unfolded, if the stiffness of the locking structure
was large enough, the solar panel composed of multiple plates could be approximated as a
single flexural rectangular plate. Therefore, in this subsection, the three-plate model was
simplified to a single plate for the analysis. The side length b in the y direction equaled
1m and the side length 3a in the x direction equaled 3m. The density of the material, the
form of the external force applied to the panel, and the magnitude of the external force
(uniformly distributed load q) were the same as in the previous subsection.
The vibration mode function of the solar panel was chosen as follows:
# "
(2k − 1)πy
(2k − 1)πx
(2k − 1)πy
p
+ c3 cos p
× 1 − cos
(7)
W ( x, y) = c1 + c2 sin
2a
2a 2 − µ
2a 2 − µ
If the order of the vibration mode is determined as k = 1, the first-order mode function
of the panel can be determined using the Ritz method. Subsequently, by applying the law
of conservation of energy to the system, a series of frequency values can be calculated, and
the smallest positive real root can be identified as the first-order resonant frequency ω of
the panel. According to the above method, the first mode frequency of the solar panel
was obtained as 0.3754 Hz with the help of the numerical calculation software. So far,
the parameters and vibration characteristics of the three-segment solar panel are listed in
Table 1.
2.2. Photovoltaic Power-Generation Model of a Typical Solar Panel
Micromachines 2024, 15, x FOR PEER REVIEWA photovoltaic module consists of solar cells connected in series and in parallel.
7 of 15A
solar cell can usually be represented as a current source and a diode in parallel [23,24], as
shown in Figure 5.
Rs
IL
IVD
ISC
D1
RL
Rsh
Figure
5. 5.
Equivalent
circuit
ofof
photovoltaic
cell.
Figure
Equivalent
circuit
photovoltaic
cell.
And
Andthere
thereisisa apractical
practicalmathematical
mathematicalmodel
modelfor
forengineering
engineeringuse,
use,asasfollows
follows[25]:
[25]:
VV

 LL

 C1 e C2CV2ocVoc − 1  − 1
(8)
IL =
1 1 
I L Isc I sc1−
1  C1 e
(8)
 

  Vm   
−
C1 = 1 − IIscm e CV2mVoc


I 
C1   1 hm e C 2Voc i−1
Vm
C2 = V − 1 I scln 1 − IIm
oc
sc
(9)
(9)
(10)
1
Equation (8) depicts the characteristic curve of a solar cell
  I m  ◦ for a standard irradiance of


V
2
m
Sref = 1000 W/m and a standard
(10)
 1 ln of1 Tref =25 C. However, the coefficients
C2  temperature

C1 , C2 , V oc , and Isc will vary due to
the
influence
of
the
external
sunlight
intensity
and
V
I


sc 
 oc   
Equation (8) depicts the characteristic curve of a solar cell for a standard irradiance
of Sref = 1000 W/m2 and a standard temperature of Tref = 25 °C. However, the coefficients C1,
C2, Voc, and Isc will vary due to the influence of the external sunlight intensity and the
ambient temperature. When the irradiance and reference temperature change, the formula
becomes inapplicable and needs to be adjusted to describe the new curve. The improving
Micromachines 2024, 15, 996
7 of 15
the ambient temperature. When the irradiance and reference temperature change, the
formula becomes inapplicable and needs to be adjusted to describe the new curve. The
improving method is to derive Isc ′ , V oc ′ , Im ′ , and V m ′ under general working conditions
(irradiance S and temperature T) from the Isc , V oc , Im , and V m under the standard sunlight
intensity Sref and the standard temperature Tref . Then, Equation (8) can still be utilized
to perform engineering calculations of the output characteristics under non-standard
working conditions.
First, the temperature difference ∆T and the relative irradiance difference ∆S between
the general and standard working conditions were calculated as follows:
∆T = T − Tre f
(11)
S
−1
Sre f
(12)
∆S =
Then, Isc ′ , V oc ′ , Im ′ , and V m ′ under general working conditions were calculated using
the following formulae.
S
′
Isc
= Isc
(1 + g∆T )
(13)
Sre f
′
Voc
= Voc (1 − k∆T ) ln(e + h∆S)
(14)
S
′ = I
Im
m S (1 + g∆T )
re f
(15)
Vm′ = Vm (1 − k∆T ) ln(e + h∆S)
(16)
The projection process assumed that the basic shape of the output curve was unchanged; the typical values of coefficients g, h, and k were 0.0025/◦ C, 0.5/(W/m2 ), and
0.00288/◦ C, respectively.
By replacing Isc , V oc , Im , and V m under the standard conditions with the obtained Isc ′ ,
V oc ′ , Im ′ , and V m ′ under the new conditions, C1 ′ and C2 ′ under the new conditions can be
obtained with Equations (9) and (10), thus solving the problem of calculating the output
characteristics under any irradiance and temperature with Equation (8).
3. Structural Dynamics and Photovoltaic Power Modeling of a Typical Solar Panel
3.1. Parameter Validation of Photovoltaic Power-Generation Model under Dynamic Conditions
In order to obtain the output characteristic curves of the solar panel and to ascertain
the influence of external environmental factors on its output characteristics, this section
utilized MATLAB/Simulink(R2020a) to conduct an output simulation of the solar cell. This
was based on the research content of the previous sections, and the correctness of the model
was verified experimentally.
The data for the selected solar panel were provided by the manufacturer, and the solar
panel’s parameters under standard conditions are shown in Table 2.
Table 2. Solar panel parameters.
Parameters
Names
Value
Short-circuit current
Open-circuit voltage
Maximum power point current
Maximum power point voltage
Isc
Voc
Tm
Vm
0.32 A
22.3 V
0.28 A
17.90 V
3.1.1. Simulation of Photovoltaic Power-Generation Model
Figure 6 shows the simulation model for a single solar panel. Inside the dashed box
lies the external measurement circuit used to capture the panel’s output current, voltage,
and power. The input ‘S’ in the model represents the solar irradiance, measured in units of
W/m2 . The input ‘T’ signifies the ambient temperature, measured in ◦ C. And the input ‘Vb’
Maximum power point voltage
Vm
17.90 V
3.1.1. Simulation of Photovoltaic Power-Generation Model
Micromachines 2024, 15, 996
Figure 6 shows the simulation model for a single solar panel. Inside the dashed box
8 of 15
lies the external measurement circuit used to capture the panel’s output current, voltage,
and power. The input ‘S’ in the model represents the solar irradiance, measured in units
of W/m2. The input ‘T’ signifies the ambient temperature, measured in °C. And the input
denotes
the the
vibration,
which
cancan
be adjusted
based
on different
vibration
characteristics,
‘Vb’
denotes
vibration,
which
be adjusted
based
on different
vibration
characterincluding
the vibration
form, form,
frequency
(Hz), and
vibration
angle (rad).
is important
istics,
including
the vibration
frequency
(Hz),
and vibration
angleIt(rad).
It is im-to
note that
‘Vb’ is
in ‘Vb’
the form
angle.
portant
to note
that
is in of
theanform
of an angle.
Figure
Figure6.6.Single
Singlesolar
solarpanel
panelsimulation.
simulation.
ItItisisworth
worthmentioning
mentioningthat,
that,ininpast
pastresearch,
research,ititwas
wasoften
oftentacitly
tacitlyassumed
assumedthat
thatthe
the
angle
angleofofincidence
incidenceofofsunlight
sunlightand
andthe
theoutput
outputcurrent
currentconform
conformtotothe
thecosine
cosinetheorem.
theorem.InIn
practice,
practice,however,
however,when
whenthe
theangle
angleofofincidence
incidenceexceeds
exceeds55°,
55◦the
, thevalue
valueofofthe
theoutput
outputcurrent
current
◦ , there
gradually
when itit exceeds
exceedsabout
about85
85°,
thereisisno
nooutput
outgraduallydeviates
deviatesfrom
fromthe
the cosine
cosine value, and when
put
power
from
solarcell,
cell,although
althoughtheoretically,
theoretically,there
thereshould
should still be 7.5%.
power
from
thethe
solar
7.5%.The
Theoutput
output
power
angle of
of incidence
incidenceof
ofthe
thesun
sunisisknown
knownas
powercurve
curveofofaareal
realsolar
solarcell
cell as
as aa function of the angle
asthethe
Kelly
cosine.
the of
sake
of this
rigor,
thischaracterizes
paper characterizes
the Kelly
cosine
Kelly
cosine.
For For
the sake
rigor,
paper
the Kelly cosine
relationship
between the output current and the angle of incidence θ based on an empirical formula
proposed in the literature [26]:
IL = max( IL0 cos θ − a·u(|θ |−θth )·(|θ |−θth ), 0)
(17)
The threshold angle θth was set to 55◦ , and when the angle of incidence was less than
the threshold, the experimental data conformed to the standard cosine law. When the
angle of incidence exceeded the threshold, the experimental data could be more accurately
modelled using the empirical Equation (17).
The model was built in Simulink according to the empirical formula, and the output
results were basically consistent with the theoretical data, as shown in Figure 7.
To delve into the impact of generating multiple localized poles on the maximum
output power of the solar panel under dynamic conditions, two sets of simulations were
conducted. These analyses aimed to compare and analyze the effects on the MPP of two
significant parameters influencing mechanical vibrations: the frequency and the vibration
angle. This investigation was crucial due to the multi-polar characteristics of the solar
panel’s output power under dynamic conditions.
From Figure 8, it is clear that the same number of local extremes occurred at the
same frequency of mechanical vibration. As the mechanical vibration angle increased, the
output power showed a sudden rise-and-fall characteristic with voltage escalation, and this
tendency became more pronounced with a higher vibration angle. Moreover, the maximum
power output showed a decreasing trend as the vibration angle increased.
Micromachines 2024, 15, 996
the threshold, the experimental data conformed to the standard cosine law. When the angle of incidence exceeded the threshold, the experimental data could be more accurately
modelled using the empirical Equation (17).
The model was built in Simulink according to the empirical formula, and the output
9 of 15
results were basically consistent with the theoretical data, as shown in Figure 7.
(a)
Micromachines 2024, 15, x FOR PEER REVIEW
10 of 15
(b)
operates at the MPP, it is generally expected that the power loss will decrease as the freFigure
7. Kelly
cosine
applied
to PV
its simulation
result.
(a) Kelly
cosine
simulation.
Figure
7. Kelly
cosine
applied
tosystems
PV systems
and
its
result.
Kelly
cosine
simulation.
quency
increases.
Nevertheless,
thisand
pattern
didsimulation
not hold
true
in(a)
the
low-frequency
range.
(b) Kelly
cosine
value
as aas
function
of the
angle.
(b) Kelly
cosine
value
a function
of the
angle.
To delve into the impact of generating multiple localized poles on the maximum output power of the solar panel under dynamic conditions, two sets of simulations were conducted. These analyses aimed to compare and analyze the effects on the MPP of two significant parameters influencing mechanical vibrations: the frequency and the vibration
angle. This investigation was crucial due to the multi-polar characteristics of the solar
panel’s output power under dynamic conditions.
From Figure 8, it is clear that the same number of local extremes occurred at the same
frequency of mechanical vibration. As the mechanical vibration angle increased, the output power showed a sudden rise-and-fall characteristic with voltage escalation, and this
tendency became more pronounced with a higher vibration angle. Moreover, the maximum power output showed a decreasing trend as the vibration angle increased.
In Figure 9, it was observed that, for the same vibration angle, the number of local
extreme points increased with an increase in the vibration frequency. However, the output
power curve
(a) consistently remained near the static output
(b) power curve. The maximum
output power in this case showed fluctuating changes. Assuming that the instrument
Figure8.8.PV
PVproperties
propertiesofofthe
thesame
samevibration
vibrationfrequency
frequencywith
withdifferent
differentvibration
vibrationangles.
angles.(a)
(a)The
TheI-V
I−V
Figure
curves. (b) The P−V curves.
curves. (b) The P-V curves.
In Figure 9, it was observed that, for the same vibration angle, the number of local
extreme points increased with an increase in the vibration frequency. However, the output
power curve consistently remained near the static output power curve. The maximum
output power in this case showed fluctuating changes. Assuming that the instrument
operates at the MPP, it is generally expected that the power loss will decrease as the
frequency increases. Nevertheless, this pattern did not hold true in the low-frequency range.
(a)
(b)
(a)
Micromachines 2024, 15, 996
(b)
Figure 8. PV properties of the same vibration frequency with different vibration angles. (a)10
The
I−V
of 15
curves. (b) The P−V curves.
(a)
(b)
The
IFigure9.9.PV
PVproperties
propertiesofofthe
thesame
samevibration
vibrationangle
anglewith
withdifferent
different
vibration
frequencies.
Figure
vibration
frequencies.
(a)(a)
The
I-V
V curves. (b) The P-V curves.
curves. (b) The P-V curves.
3.1.2.Validation
ValidationofofPhotovoltaic
PhotovoltaicPower-Generation
Power-GenerationModel
Model
3.1.2.
solarpanel
panelphotovoltaic
photovoltaiccharacterization
characterizationexperiment
experimentwas
wascarried
carriedout
outtotoverify
verifythe
the
AAsolar
correctness
of
the
simulation
model.
The
equipment
mainly
included
an
oscilloscope,
a
correctness of the simulation model.
mainly included an oscilloscope,
Micromachines 2024, 15, x FOR PEER REVIEW
11 of 15
a signal
generator,
and
a solar
panel.
Figure
10 illusa resistance
resistancebox,
box,aasolar
solarpower
powermeter,
meter,
a signal
generator,
and
a solar
panel.
Figure
10
illustrates
a schematic
diagram
of the
connection
of the
experimental
setup.
trates a schematic
diagram
of the
connection
of the
experimental
setup.
Figure10.
10.Schematic
Schematicdiagram
diagramofofthe
theexperimental
experimentalsetup.
setup.
Figure
Additionally,
mechanism
waswas
employed
to simulate
the vibration
of the of
solar
Additionally,a motion
a motion
mechanism
employed
to simulate
the vibration
the
panel
itsand
mechanical
sketchsketch
is presented
in Figure
11. The
β1 andβ1βand
the
solar and
panel
its mechanical
is presented
in Figure
11.angles
The angles
β2 satisfy
2 satisfy
following
system
of equations:
the following
system
of equations:
(
2
rcosβ1 )1 2 = ll22
 rsin
β 1 )21+
(d − dr sin
2 (e +e rcos
(18)

d−r sin β 1
(18)
cosβ
2=
dl  rsin
cos  2 

1
l
Taking e as 350 mm, r as 80 mm, and l as 600 mm, the relationship between the angles
β1 andTaking
β2 cane be
represented
as 350
mm, r asas
80follows:
mm, and l as 600 mm, the relationship between the angles
β1 and β2 can be represented as follows:
2
(19)
β 2 = 8sinβ 1 + 64(sin β 1 )2 − 560 cos β 1 + 2311 2
2
(19)
2  8sin1  64  sin 1   560cos 1  2311
According to the calculated results, the variation interval of the angle between the
◦ 45.7800◦ ]. In addition to this, the actual solar
solar panel
and the
ground
was [26.7440
According
to the
calculated
results, ,the
variation interval of the angle between the
incidence
angle,
was used
the simulation
model,
to to
bethis,
calculated
basedsolar
on
solar panel
andwhich
the ground
wasin[26.7440°,
45.7800°].
In needed
addition
the actual


incidence angle, which was used in the simulation model, needed to be calculated based
on the local latitude of the experiment, the solar time angle, the solar declination angle,
and the panel azimuth angle [27,28].
B
50°
β1 and β2 can be represented as follows:


2  8sin1  64  sin 1   560cos 1  2311
Micromachines 2024, 15, 996
2
2
(19)
According to the calculated results, the variation interval of the angle between11the
of 15
solar panel and the ground was [26.7440°, 45.7800°]. In addition to this, the actual solar
incidence angle, which was used in the simulation model, needed to be calculated based
onthe
thelocal
local
latitude
experiment,
solar
time
angle,
solar
declination
angle,
latitude
ofof
thethe
experiment,
thethe
solar
time
angle,
thethe
solar
declination
angle,
and
and
the
panel
azimuth
angle
[27,28].
the panel azimuth angle [27,28].
50°
B
45°
r
A
40°
l
β2
1
35°
e
30°
2
d
25°
C
(a)
0
5
10
β1 [rad]
15
(b)
Figure
11.11.
Mechanical
sketch
andand
calculations
of the
mechanism.
(a) Mechanical
sketch.
(b)
Figure
Mechanical
sketch
calculations
of motion
the motion
mechanism.
(a) Mechanical
sketch.
Calculations
of theofangle
between
the solar
panelpanel
and the
(b) Calculations
the angle
between
the solar
andground.
the ground.
Static
respectively,
Staticand
anddynamic
dynamicexperiments
experimentswere
wereconducted,
conducted,
respectively,and
andthe
theexperimental
experimental
results
resultswere
werecompared
comparedwith
withthe
thesimulation
simulationresults
resultsasasfollows:
follows:
ItItcan
canbebeseen
seenfrom
fromFigures
Figures1212and
and1313that
thatthe
thetrends
trendsininthe
theexperimental
experimentalcurves
curves
Micromachines 2024, 15, x FOR PEER
REVIEW
12
of 15
closely
aligned
the
closely
alignedwith
withthe
thesimulation
simulationcurves,
curves,providing
providingevidence
evidenceofofthe
thefeasibility
feasibilityofof
the
simulation
discrepancies
simulationmodel.
model.However,
However,there
therewere
weresome
some
discrepanciesbetween
betweencertain
certainexperimental
experimental
datapoints
pointsand
andthe
thesimulation
simulationresults,
results,indicating
indicatinga adegree
degreeofofdeviation.
deviation.Upon
Uponanalysis,
analysis,
data
potential
sources
of
error
in
the
experimental
results
included
the
following:
(1)
potential
sources
of
error
in
the
experimental
results
included
the
following:
(1)
Natural
sunlight is unstable and the irradiation is in real-time fluctuation; hence, thereNatural
existed a
sunlight
is unstable
and thethe
irradiation
is in real-time
hence, there
existed
large error
when reading
output voltage
under afluctuation;
specific irradiance.
(2) In
orderato
large
error
when
reading
the
output
voltage
under
a
specific
irradiance.
(2)
In
order
to
read the test data under a specific irradiance, the time span of the experiment was large,
read
the
test
data
under
a
specific
irradiance,
the
time
span
of
the
experiment
was
large,
and the solar time angle changed slowly during the process, while the solar incident angle
and
the
solar timeatangle
changed
slowly
during
the process,
while
the solar
incident
angle
was
calculated
a certain
moment
in the
experimental
time,
which
affected
the accuracy
was
calculated
at
a
certain
moment
in
the
experimental
time,
which
affected
the
accuracy
of the data results. (3) Compared with the theoretical results, the open-circuit voltage obof the data results. (3) Compared with the theoretical results, the open-circuit voltage
tained from the static experiments was large because the internal resistance of the solar
obtained from the static experiments was large because the internal resistance of the solar
panel varied with changes in the light intensity, cell temperature, and output voltage. In
panel varied with changes in the light intensity, cell temperature, and output voltage. In
addition, the factory parameters of solar panels provided by the manufacturer are the data
addition, the factory parameters of solar panels provided by the manufacturer are the data
for the same batch of panels, and the actual parameters of the panels may not be comfor the same batch of panels, and the actual parameters of the panels may not be completely
pletely consistent with the standard parameters.
consistent with the standard parameters.
Figure
Static
experimental
results.
Figure
12.12.
Static
experimental
results.
Micromachines 2024, 15, 996
12 of 15
Figure 12. Static experimental results.
Figure
Dynamic
experimental
results.
Figure
13.13.
Dynamic
experimental
results.
3.2.
Solar
Panel
under
Dynamic
Conditions
3.2.Power-Generation
Power‐GenerationCharacterization
Characterizationofofa Typical
a Typical
Solar
Panel
under
Dynamic
Conditions
From
the
above
experimental
results,
the
simulation
model
was
constructed
From the above experimental results, the simulation model was constructedwith
witha a
certain
single
solar
panel
certaincorrectness
correctnessand
andfeasibility.
feasibility.InInthis
thissubsection,
subsection,the
themodel
modelofofthis
this
single
solar
panel
was
utilized
toto
form
a solar
cellcell
array
in in
series,
which
was
used
to simulate
thethe
output
of aof
was
utilized
form
a solar
array
series,
which
was
used
to simulate
output
solar
panel
with
a
three-plate
structure
under
vibration
conditions.
a solar panel with a three-plate structure under vibration conditions.
When
Whensolar
solarpanels
panelsare
areconnected
connectedininseries,
series,the
thevibration
vibrationcondition
conditionofofeach
eachplate
plateis is
generally
different.
In
general,
the
amplitude
of
the
plate
solidly
connected
to
the
spacecraft
generally different. In general, the amplitude of the plate solidly connected to the spacebody
the smallest,
and the
farther
away
from
thethe
body
have
larger
vibration
craftisbody
is the smallest,
andplates
the plates
farther
away
from
body
have
larger
vibration
amplitudes.
From
the
output
characteristics
of
a
single
panel,
it
can
be
hypothesized
amplitudes. From the output characteristics of a single panel, it can be hypothesizedthat,
that,
under vibration conditions, the output power of the solar array when multiple panels are
connected in series also exhibits multipolar characteristics. The vibration frequency of
the solar cell wing and the average rotation angle of each plate under different uniform
loads were obtained through the kinetic calculation of the solar cell wing in Section 2. By
inputting each set of data into the simulation model separately, the output characteristics
of the solar cell wing under specific vibration conditions were obtained.
According to Figure 14, there were few cases of localized extreme points in the P-V
curve when the external uniform load was small. As the external load increased, the
vibration of the solar panel became more intense, the indentation of the curve was more
pronounced, and its loss of output power increased. When the vibration frequency and the
maximal deflection angle were 0.3754 Hz and 74.9871◦ , the maximum output power of the
selected solar PV wing was reduced by 5.53%, which is quite considerable. The magnitude
of maximum power reduction seemed to decrease with an increasing load, as shown in
Figure 15.
It is noteworthy that the fill factor (FF) is an important performance indicator when
analyzing the performance of solar cells. It is the ratio of the maximum output power of the
battery to the product of the maximum value of the short-circuit current and open-circuit
voltage, as shown in Equation (20). The FF reflects the ability and efficiency of a solar cell
to use light energy during operation. It is a key parameter that is used to measure the
performance of solar cells, with a typical range of 0.5 to 0.9, and the closer it is to 1, the
better the performance. Still referring to Figure 15, obviously, as the external load increased
to 10 N/m2 , the FF of the solar panel exhibited a decline from its original value of 0.8031
to 0.7587, which indicates that the overall energy conversion efficiency of the solar panel
decreased significantly under vibration conditions.
FF =
PM
VOC × ISC
(20)
Micromachines 2024, 15, 996
bration
of theofsolar
panelpanel
became
moremore
intense,
the indentation
of theofcurve
was more
bration
the solar
became
intense,
the indentation
the curve
was more
pronounced,
and
its
loss
of
output
power
increased.
When
the
vibration
frequency
and and
pronounced, and its loss of output power increased. When the vibration frequency
the maximal
deflection
angleangle
werewere
0.3754
Hz and
the maximum
output
power
the maximal
deflection
0.3754
Hz 74.9871°,
and 74.9871°,
the maximum
output
power
of the
solarsolar
PV wing
was reduced
by 5.53%,
which
is quite
considerable.
The The
ofselected
the selected
PV wing
was reduced
by 5.53%,
which
is quite
considerable.
magnitude
of maximum
power
reduction
seemed
to decrease
with with
an increasing
load,13
asof 15as
magnitude
of maximum
power
reduction
seemed
to decrease
an increasing
load,
shown
in
Figure
15.
shown in Figure 15.
(a) (a)
(b) (b)
Figure
14. Power-generation
characterization.
(a) The
I−V
curves.
(b) The
P−V
curves.
Figure
Power-generation
characterization.
The
curves.
The
P−V
curves.
Figure
14.14.
Power-generation
characterization.
(a)(a)
The
I-VI−V
curves.
(b)(b)
The
P-V
curves.
Figure
Loss
of of
output
power
of of
the
solar
panel
under
a uniform
load.
Figure
15. 15.
Loss
of
output
power
of the
solar
panel
under
a uniform
load.
Figure
15.
Loss
output
power
the
solar
panel
under
a uniform
load.
4. Conclusions
It is noteworthy
that the
(FF) is
an isimportant
performance
indicator
whenwhen
It is noteworthy
thatfill
thefactor
fill factor
(FF)
an important
performance
indicator
This
paper
analyzedofthe
mechanical
characteristics
of output
the solar
panelofof a
analyzing
the
performance
solar
cells.cells.
It isoscillation
the
of
theofmaximum
power
analyzing
the performance
of solar
It isratio
the ratio
the maximum
output
power of
which
is a typical
application
scenario
of
PV
modules. By
integrating
a solar
thespacecraft,
battery
to
the
product
of
the
maximum
value
of
the
short-circuit
current
and
openthe battery to the product of the maximum value of the short-circuit current and opencell
simulation
model,
this
study
established
the
output
behavior
of
solar
arrays
under
circuit
voltage,
as shown
in Equation
(20). (20).
The FF
the ability
and efficiency
of
a of a
circuit
voltage,
as shown
in Equation
Thereflects
FF reflects
the ability
and efficiency
mechanical
vibration
conditions
and
experimentally
verified
its
correctness
using
a
PV
solarsolar
cell tocell
usetolight
energy
during
operation.
It is aItkey
that is
used
to measure
use light
energy
during
operation.
is aparameter
key parameter
that
is used
to measure
platform.
Whilecells,
the impact
of mechanical
the output
characteristics
of
thetesting
performance
of solar
with with
a typical
rangerange
ofvibration
0.5ofto0.5
0.9,on
it is to
the performance
of solar cells,
a typical
toand
0.9,the
andcloser
the closer
it 1,
is the
to 1, the
the solar panel may seem like a minor detail, it holds significant importance for spacecraft
betterbetter
the performance.
Still referring
to Figure
15, obviously,
as the
load load
in- inthe performance.
Still referring
to Figure
15, obviously,
as external
the external
operations. And2 as solar
arrays become even larger, the seemingly acceptable power loss
2
creased
to
10
N/m
,
the
FF
of
the
solar
panel
exhibited
a
decline
from
its
original
value
of of
creased to 10 N/m , the FF of the solar panel exhibited a decline from its original value
on each solar panel will add up to a huge waste.
By examining the power-generation quality of solar arrays and evaluating the output
voltage, power, and other parameters affected by mechanical vibration, we can identify
performance improvement indices after the stabilization of the structure. This offers new
insights for optimizing spacecraft structure and power supply systems, ultimately forming
an evaluation model for enhancing the power supply efficiency and providing guidelines
for structural optimization.
However, it is important to note that this study only considered the vibration characteristics under a uniform load in the mechanical analysis of a solar cell wing. In actual
scenarios, spacecraft motion involves attitude adjustments, re-orbiting motion, and the
extension and retraction of the battery wing, resulting in a highly complex and variable
force situation. Further research and improvement efforts are required to comprehensively
address these challenges.
Author Contributions: Conceptualization, X.S. and Y.W.; methodology, Y.W.; validation, X.S., Q.Y.
and J.H.; investigation, X.S. and Q.Y.; data curation, C.Z. and J.S.; writing—original draft preparation,
Micromachines 2024, 15, 996
14 of 15
X.S.; writing—review and editing, Y.W.; supervision, Y.W. and C.Z.; funding acquisition, Y.W., C.Z.
and J.S. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by the Industry University Research Cooperation Fund of the
Eighth Research Institute of China Aerospace Science and Technology Corporation under grant
number SAST2023-041 and the Fund of Prospective Layout of Scientific Research for NUAA (Nanjing
University of Aeronautics and Astronautics).
Data Availability Statement: Data is contained within the article.
Conflicts of Interest: The authors declare no conflicts of interest.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
Xia, L.; Ma, S.; Tao, P.; Pei, W.; Liu, Y.; Tao, L.; Wu, Y. A wind-solar hybrid energy harvesting approach based on wind-induced
vibration structure applied in smart agriculture. Micromachines 2022, 14, 58. [CrossRef] [PubMed]
Dubey, S.; Sarvaiya, J.N.; Seshadri, B. Temperature Dependent Photovoltaic (PV) Efficiency and Its Effect on PV Production in the
World—A Review. Energy Procedia 2013, 33, 311–321. [CrossRef]
Glick, A.; Smith, S.E.; Ali, N.; Bossuyt, J.; Recktenwald, G.; Calaf, M.; Cal, R.B. Influence of flow direction and turbulence intensity
on heat transfer of utility-scale photovoltaic solar farms. Sol. Energy 2020, 207, 173–182. [CrossRef]
Huang, L.; Song, Z.; Dong, Q.; Song, Y.; Zhao, X.; Qi, J.; Shi, L. Surface temperature and power generation efficiency of PV arrays
with various row spacings: A full-scale outdoor experimental study. Appl. Energy 2024, 367, 123362. [CrossRef]
Cuce, E.; Cuce, P.M.; Bali, T. An experimental analysis of illumination intensity and temperature dependency of photovoltaic cell
parameters. Appl. Energy 2013, 111, 374–382. [CrossRef]
Hasan, K.; Yousuf, S.B.; Tushar, M.S.H.K.; Das, B.K.; Das, P.; Islam, S. Effects of different environmental and operational factors on
the PV performance: A comprehensive review. Energy Sci. Eng. 2021, 10, 656–675. [CrossRef]
Mohammed, H.; Kumar, M.; Gupta, R. Bypass diode effect on temperature distribution in crystalline silicon photovoltaic module
under partial shading. Sol. Energy 2020, 208, 182–194. [CrossRef]
Vieira, R.G.; de Araújo, F.M.U.; Dhimish, M.; Guerra, M.I.S. A comprehensive review on bypass diode application on photovoltaic
modules. Energies 2020, 13, 2472. [CrossRef]
Abdulmawjood, K.; Alsadi, S.; Refaat, S.S.; Morsi, W.G. Characteristic study of solar photovoltaic array under different partial
shading conditions. IEEE Access 2022, 10, 6856–6866. [CrossRef]
Bakker, K.; Weeber, A.; Theelen, M. Reliability implications of partial shading on CIGS photovoltaic devices: A literature review.
J. Mater. Res. 2019, 34, 3977–3987. [CrossRef]
Osmani, K.; Haddad, A.; Alkhedher, M.; Lemenand, T.; Castanier, B.; Ramadan, M. Anovel MPPT-Based ithium-Ion battery solar
charger for operation under fluctuating irradiance conditions. Sustainability 2023, 15, 9839. [CrossRef]
Ali, A.I.M.; Mohamed, H.R.A. Improved P&O MPPT algorithm with efficient open-circuit voltage estimation for two-stage
grid-integrated PV system under realistic solar radiation. Int. J. Electr. Power Energy Syst. 2022, 137, 107805.
Schmid, J.; Drapalik, M.; Kancsar, E.; Schlosser, V.; Klinger, G. A study of power quality loss in PV modules caused by wind
induced vibration located in Vienna. Sol. Energy 2011, 85, 1530–1536. [CrossRef]
Saygin, D.; Kempener, R.; Wagner, N.; Ayuso, M.; Gielen, D. The implications for renewable energy innovation of doubling the
share of renewables in the global energy mix between 2010 and 2030. Energies 2015, 8, 5828–5865. [CrossRef]
Zhang, Y.; Yuan, C. Effects of vibration on output characteristics of shipboard PV modules. Reg. Stud. Mar. Sci. 2021, 47, 101989.
[CrossRef]
Vidović, V.; Krajačić, G.; Matak, N.; Stunjek, G.; Mimica, M. Review of the potentials for implementation of floating solar panels
on lakes and water reservoirs. Renew. Sustain. Energy Rev. 2023, 178, 113237. [CrossRef]
Shen, X.; Wang, X.; Yu, M.; Wang, H.; Xu, F.; Zhang, Q. Crumpled Nanogenerator Based on Elastic Modulus Design for Fully
Self-Powered High-Speed Rail Speed Measurement. ACS Appl. Nano Mater. 2024, 7, 15478–15487. [CrossRef]
Wu, Y.; Yuan, Q.; Ren, K.; Shen, X.; Shen, H.; Badel, A.; Qiu, J. Bidirectional energy-controlled piezoelectric shunt damping
technology and its vibration attenuation performance. Int. J. Mech. Syst. Dyn. 2024, 4, 63–76. [CrossRef]
Patel, M.R. Spacecraft Power Systems, 1st ed.; CRC Press: Boca Raton, FL, USA, 2004; 732p.
Xu, R.; Li, D.; Jiang, J. An online learning-based fuzzy control method for vibration control of smart solar panel. J. Intell. Mater.
Syst. Struct. 2015, 26, 2547–2555. [CrossRef]
Li, D.; Liu, W. Vibration control for the solar panels of spacecraft: Innovation methods and potential approaches. Int. J. Mech. Syst.
Dyn. 2023, 3, 300–330. [CrossRef]
Li, D.; Deng, Z.; Chen, G. Free vibration of functionally graded sandwich plates in thermal environments. Int. J. Mech. Syst. Dyn.
2023, 3, 39–47. [CrossRef]
Tsai, H.L.; Tu, C.S.; Su, Y.J. Development of generalized photovoltaic model using MATLAB/SIMULINK. Lect. Notes Eng. Comput.
Sci. 2008, 2008, 2173.
Gow, J.; Manning, C. Development of a photovoltaic array model for use in power-electronics simulation studies. IEE Proc. Electr.
Power Appl. 1999, 146, 193–200. [CrossRef]
Micromachines 2024, 15, 996
25.
26.
27.
28.
15 of 15
Singer, S.; Rozenshtein, B.; Surazi, S. Characterization of PV array output using a small number of measured parameters. Sol.
Energy 1984, 32, 603–607. [CrossRef]
Zhang, Q.; Zhang, Y. Design and verification of an integrated panoramic sun sensor atop a small spherical satellite. Sensors 2022,
22, 8130. [CrossRef]
Wu, Y.; Li, S.; Fan, K.; Ji, H.; Qiu, J. Investigation of an ultra-low frequency piezoelectric energy harvester with high frequency
up-conversion factor caused by internal resonance mechanism. Mech. Syst. Signal Process. 2022, 162, 108038. [CrossRef]
Stanciu, C.; Stanciu, D. Optimum tilt angle for flat plate collectors all over the world—A declination dependence formula and
comparisons of three solar radiation models. Energy Convers. Manag. 2014, 81, 133–143. [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.