Improving the Fractional Order PID Controller Performance with an
Energy Storage System for Photovoltaics
TAREK A. BOGHDADY, ALI J. ALAMER, M. A. MOUSTAFA HASSAN, A. A. SEIF
Department of Electrical Engineering, Faculty of Engineering
Cairo University
Giza, Egypt
EGYPT
Abstract: - A PhotoVoltaic (PV) dependent maximum power point tracking controller is used, modeled, and
assessed. It includes a study of system components and their modelling. The model is then tested and validated
using more than one method. This article focuses on increasing energy extraction in grid-connected PV and
isolated systems, damping system oscillations, and reducing its settling time. Tuning the PID controller and the
fractional order PID controller is a challenging task that can be carried out by trial and error, Ziegler-Nichols
method, or by optimization techniques. In this article; genetic algorithms and whale optimization algorithms are
being used here to obtain desired controller response by minimizing the objective function. The objective
function is the integral square error. A PV is proposed to use a Fractional Order PID (FOPID) controller then
compared to its conventional PID controller. The results show that the output power has a faster response and
eliminates oscillations around the maximum power point under steady-state conditions. The results confirm that
the proposed controller with an energy storage system has improved energy extraction. All simulations were
carried out using MATLAB/SIMULINK.
Key-Words: - Genetic Algorithm, FOPID, Photovoltaics, Renewable Energy, Whale Optimization Algorithm.
Received: April 12, 2021. Revised: January 23, 2022. Accepted: February 12, 2022. Published: March 3, 2022.
1 Introduction
As the world faces a challenge to overcome the
energy crisis. The decreasing deposits of non-
renewable energy resources such as coal, natural
gas, fossil fuels, etc. have raised awareness of such
crisis. So, it has become a necessity to develop new
ways to replace traditional energy sources. Solar
energy is a renewable, inexhaustible, and ultimate
source of energy. If used properly, it can fulfil
numerous energy needs of the world. The power
from the sun intercepted by the earth is
approximately 1.8 x 1011 MW [1-5]. This amount of
energy is thousands of times larger than the current
consumption rate. Thus clarifies the importance of
renewable energy in general and solar energy in
particular. Solar energy is the source of all energies
on Earth. Fossil fuel is a storage of this energy over
a large period. Also, the wind is moved by
temperature difference caused by solar irradiance.
The PhotoVoltaic (PV) system consists of
interconnected components designed to achieve the
specific target of delivering desired electricity from
a small device to the load. PV systems are
categorized by the main categories of grid-
connected, stand-alone systems and hybrid systems,
which comprise of different energy sources such as
PV arrays, diesel generators, and wind generators.
In grid-connected and stand-alone systems, storage
elements such as batteries, fuel cells, or
supercapacitors may be adapted to store energy
during daytime. The systems are modelled using an
energy storage element such as a battery storage,
supercapacitor, and then both; then without energy
storage. While the PV panels may seem like a good
source of electricity, their conversion efficiency is
not very high; with high cost and low efficiency
(from 9-17%) [6]. Therefore, if the load is coupled
directly to the PV array, the PV array must usually
be oversized to supply required load power. This
leads to an oversized expensive system. Thus, PV
arrays should be operated at the Maximum Power
Point (MPP) which changes with different solar
irradiances and load variations. Several Maximum
PowerPoint Tracking (MPPT) techniques have been
developed for PV systems [7]. The main problem is
how to obtain optimal operating points (voltage &
current) automatically at maximum PV output
power under variable atmospheric conditions.
This paper is organized as follows: System
modeling including PV and DC-DC converter a
battery is introduced in Section 2 to select one of
them for the PI and FOPI tunning problem a
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comparison is carried out in Section 3 results and
discussion are stated. Finally, Section 4 summarizes
the simulation results.
2 System Modelling
2.1 PV Modelling
Fig. 1: A simple ideal equivalent circuit of a PV
cell.
From Figure 1 and by applying Kirchhoff’s current
law,
(1)
Where ID is the diode internal diffusion current, IPh
is the photocurrent or light generated current, which
is proportional to the radiation and surface
temperature. The output current and voltage of the
solar cell are represented by IPV and VPV,
respectively. The diode internal diffusion current is
modelled by Equation (2) [8].
(2)
Where q is the electron charge, 1.6×10-19 C, A is
the diode ideality factor and takes the value between
1 and 2, K is Boltzmann’s constant, 1.38×10-23 J/K.
TC is the cell’s operating temperature in kelvin and
IS is the cell saturation current, which varies with
temperature according to Equation (4), as stated in
[4]. Equation (3) calculates IPh the photocurrent
related to the cell’s operating temperature and solar
intensity.
(3)
Where ISC is the short-circuit current known from
the datasheet, Kt is the cell’s short circuit
temperature coefficient (Amperes/ K), TRef is the
cell reference temperature in kelvin, TRef = 298 K
(25 Co), G is the solar irradiance in W/m2, and GRef
represents the reference solar irradiance W/m2, GRef
= 1 kW/m2 [4]. The short circuit current is measured
under the standard test condition at a reference
temperature of 25 Co and solar irradiance 1 kW/m2.
(4)
In Equation (4), IRS is the cell’s reverse saturation
current in Ampere at TRef, and the solar irradiance 1
kW/m2. Eg is the band-gap energy of the cell’s
semiconductor used. The cell’s reverse saturation
current at reference temperature can be obtained by
Equation (5) [8].
(5)
Where VOC is the open-circuit voltage at reference
temperature TRef.
To account for the losses that occur inside a solar
cell, Rs (series resistance) and Rsh (parallel
resistance) are to be included in this model as shown
in Figure 2.
Fig. 2: An exact equivalent circuit for a PV cell.
Hence the PV cell output current IPV, in Figure 2 is
given by Equation (6) [8].
(6)
The shunt resistance, RSh, represents the shunt
leakage current to the ground due to p-n junction
non-ideal ties and impurities near the junction. The
series resistance RS is due to semiconductor-material
bulk resistance, the metal contact particularly that of
the front grid, and the transverse flow of current in
the solar emitter to the front grid.
In general, the variation of RSh does not affect the
PV cell short circuit current, ISC, but it reduces the
PV cell open-circuit voltage. Without leakage
current to the ground, RSh can be assumed to be
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infinite. On the other hand, a small variation in RS
leads to a reduction in the short-circuit current but
does not affect the open-circuit voltage therefore the
maximum power changes significantly according to
[9].
As mentioned above, the small variation in RS has a
significant effect on the PV panel output power. On
the other hand, the PV efficiency is insensitive to
variation in RSh, which can be assumed to approach
infinity without leakage current. Therefore, RSh can
be neglected to give appropriate model with suitable
complexity.
By neglecting the shunt resistance, as shown in
Figure 3, Equation (7) can be written as [4]:
(7)
According to reference [10], the value of RS can also
be calculated by Equation (8).
(8)
By using the method of panel modelling that was
introduced in [11] with the related equivalent circuit
as shown in Figure 2, a PV panel must be selected
first. For this work, PWX 500 (49 Watt-peak) panel
is chosen. The specifications of the PV panel from
its datasheet including the electrical ones are shown
in Table A.1.
Fig. 3: An appropriate equivalent circuit model.
The next step is to implement Equations (1) through
(8) as a MATLAB/SIMULINK model using
previously specified parameters. Its inputs are solar
irradiance (G) and cell temperature (T), and its
outputs are the related panel voltage and current.
By looking under this mask, the PV current source
is modelled by a controlled current source whose
value is calculated from equations depending on the
current solar irradiance (G). The inputs for this
subsystem are solar irradiance and temperatures.
The outputs are the PV panel output current and
voltage as Simulink signals and two physical
modelling ports to connect the panel with the DC-
DC converter.
2.2 DC-DC Converter Modelling
According to [12] the worst case is found at lowest
resistance for DC-DC boost converters to stay at the
continuous current mode (CCM). Hence, The MPP
capture will only be possible for load resistance
(RL) values higher than or equal to the resistance at
maximum power point (RMPP), the minimum used
load resistance in the simulations equals 10 Ω. Table
1 shows the specification values, from the datasheet
values for the PV module in Table A.1, used to
design the DC-DC boost converter parameters.
The converter duty ratio, D, can be calculated
from Equation (9) as discussed in [14]
(9)
Where D denotes the duty ratio. Vd and Vo
denote the input and output voltages of the boost
converter, respectively. From the previous equation,
the increase in duty ratio D will increase the value
of the output voltage, Vo.
Additionally; the change in duty ratio results in
an input and output converter current change. The
filter inductor and capacitor to operate the converter
in the continuous conduction mode can be
calculated by the following equations [14]:
(10)
(11)
Where Io is the output current, fs is the
switching frequency, denotes the variation, L is
the inductor value and C is the capacitor value.
Table 1. Specification values for the design of DC-
DC boost converter parameters.
Specification
Value
Input voltage
17 V
Input current
2.88 A.
Output voltage (Vo)
21.5 V
Voltage ripples (ΔVo)
5 %
Current ripples (ΔIL)
5 %
Output power of the PV @ 1000 w/m2
49 W.
Switching Frequency (fs)
10 kHz
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Hence, we can calculate the values of IL, ΔVo, ΔIL
as follows:
Assuming 100% converter efficiency:
Using Equation 11:
Using Equation 12:
Using Equation 13:
2.3 Battery Sizing
For models with batteries, a constant power load is
used. The load is 40 Watts at 12 V. For this load, the
battery sizing is done as follows [13]:
Total AH needed = 160/0.8 = 200 AH
3 PID and Fractional Order PID
(FOPID) Controller Testing
To test the system, three solar irradiance schemes
are used. The first is a uniform step change, where
radiation starts at maximum radiation (1000 W/m2)
and drops to 600 W/m2 at 0.3 sec., then increases to
800 W/m2 at 0.9 sec., and fixed. This is shown in
Figure 4. The second is the droop and raise scheme
where the radiation starts at 0 sec. with a value of
600 W/m2 and decreases linearly to 400 W/m2 at 0.4
sec. Then it increases to 700 W/m2 and increases
linearly to 1000 W/m2 at 0.6 sec. Then it drops to
600 W/m2 and increases linearly to 800 W/m2 at 1
sec. This is shown in Figure 5. The third and last is
that based on the medium-high ramp change
according to British standards (B.S.) EN50530 [14].
This is shown in Figure 6. The optimization
algorithms will be tested first.
Fig. 4: Uniforms step-change Radiation Scheme.
Fig. 5: Droop/Raise Radiation Scheme.
Fig. 6: B.S. EN50530 Radiation Change.
3.1 Testing/Comparing Whale Optimization
Algorithm (WOA) with Other Algorithms
Improving controller performance by tracing the
reference signal “(i.e. reducing the error between
measured and reference signals)” of an industrial
process is an important task by using traditional PID
controller, but finding optimum value of PID control
parameters is a very difficult issue. Most PID tuning
techniques use conventional methods such as
frequency response which requires considerable
technical experience to apply those formulas. Due to
their difficulties, PID controller parameters are
rarely tuned optimally.
The aim here is to test WOA [15] and carry out a
comparison between its performance with PSO, GA,
and Linearized Biogeography-Based Optimization
(LBBO) [16-19]. The squared error integral criteria
are the objective function to be minimized in the
step response of a process which is cascaded with a
PID controller as shown in Figure 7 by tuning
proportional gain (Kp), integral gain (Ki), and
differential gain (Kd) using MATLAB/SIMULINK.
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PID Plant
Input Error
E(s) Controller
output U(s) Output
Fig. 7: Block Diagram of Tested Systems
Table 2 shows seven transfer functions of
benchmark systems of different orders that will be
used here for testing WOA performance. The tuned
gains obtained by using WOA algorithm are given
in Table 2, while Table 3 presents tuned gains that
have been obtained using three algorithms.
For the first plant, values of Kp, Ki, and Kd founded
by WOA, GA, and LBBO are nearly the same,
while Particle Swarm Optimization (PSO) solution
is drifted by about 40% of the values found by other
algorithms. This is reflected in the objective
function values as shown in Table 4 where PSO has
the worst objective function value. The results of
three algorithms (WOA, GA, and LBBO) applied to
plants 2, 4, and 6 are nearly the same optimized
value. For plant 4 the PSO algorithm result has an
unstable solution, while solutions found by the three
other algorithms are almost the same. Also, it is
obvious that results obtained by WOA in plants 3, 5,
and 7 is much better than all other algorithms, the
WOA has approximately a reduction from 21% to
45% in the optimized value.
Table 4 (the lowest value in each row is shown in
black boldface) indicates that WOA produces better
results with a lower number of the objective
function evaluation. The unit step response for the
test plants using the four optimization algorithms
PSO, GA, LBBO, and WOA for tuning the PID
parameters are shown in Figure 8 through Figure 14.
In all cases, WOA has the fastest settling time.
Table 2. Tuned Values Obtained by WOA
Pla
nt
No.
Transfer Function
WOA
Kp
Ki
Kd
1
1.416
0
1.124
2
30
0
19.02
3
25
0
0.416
4
0.325
0.097
0.488
5
9.7.32
6.928
0.175
6
15
0.942
13.78
7
10
2.523
2.923
Table 3. Tuned Values for LBBO, GA, PSO.
Pl
an
t
N
o.
LBBO
GA
PSO
Kp
Ki
Kd
Kp
Ki
Kd
Kp
Ki
Kd
1
1.4
0.0
1.0
1.0
0
1.0
0.6
0
0.6
2
29
0
19
25
0
12
4.4
0
14
3
24
1
1.0
25
0.3
10
0.2
0
0.0
4
0.3
0.1
0.4
0.2
0.1
0.4
0.1
0.5
16
5
4.2
3.3
0.1
1
1
0
0.4
0.2
0.2
6
15
0.8
13
14
1
13
3.7
0.1
13
7
6.1
1.1
5.0
4.0
1
3.0
0.7
1.1
3.6
Table 4. Objective Function Values Obtained by
WOA, LBBO, Genetic Algorithm, and PSO.
Plant
No.
Best Min. Objective Function
WOA
LBBO
GA
PSO
1
1.1102
1.3132
1.9746
1.9520
2
0.6007
0.6497
0.6505
2.2400
3
0.0099
0.0105
0.216
0.1469
4
1.9101
1.9392
2.1500
21.7600
5
0.0206
0.0239
0.3661
0.3697
6
0.8737
0.9078
0.9513
2.5600
7
0.2153
0.2952
0.5374
1.5330
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Fig. 8: Plant 1 - Output Response.
Fig. 9: Plant 2 - Output Response.
Fig. 10: Plant 3 - Output Response.
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Fig. 11: Plant 4 – Output Response
Fig. 12: Plant 5 - Output Response.
Fig. 13: Plant 6 - Output Response.
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Fig. 14: Plant 7 - Output Response
.
In the next section, the obtained power curve using
PI and FOPI tuned by WOA. The input solar
irradiances are stated before, uniform step change,
droop/raise, and EN50530 radiations, shown in
Figures 4 to 6, those curves will test the controller
performance. The outputs are shown in Figures 15
to 20. FOPI has a faster response than PI (i.e. lower
rising and settling time) with lower oscillations.
Also, captured obtained energy by FOPI is higher in
the three tested radiations. It is also noticed that the
cell reached its rated power (49 W) in all cases for
FOPI controller while PI approximately reached it
only in uniform step-change radiation. FOPI has a
lower overshoot in all cases.
Fig. 15: PV Output Power Uniform Step-Change
Solar Irradiance using PI.
Fig. 16: PV Output Power Uniform Step-Change
Solar Irradiance using FOPI.
Fig. 17: PV Output Power Droop/Raise Solar
Irradiance using PI.
Fig. 18: PV Output Power Droop/Raise Solar
Irradiance using FOPI.
Fig. 19: PV Output Power with EN50530 Solar
Irradiance using PI.
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Fig. 20: PV Output Power with EN50530 Solar
Irradiance using FOPI.
4 Conclusions
Each PV system part discussed in this article, is
modelled using MATLAB/SIMULINK software
such as PV and DC-DC converter and the battery.
Further, a comparison between optimization
algorithms is carried out; showing that WOA was
the best tool for FOPID tuning problem. All output
Tables and Figures assured that FOPID has better
performance; lower variations and faster settling
also in capturing MPP.
Appendix A
Table A.1 Specifications of Simulated PV Module,
PWX-500 [20]
Parameter
Value
Pmax (W)
49
IMPP (A)
2.88
VMPP (V)
17
ISC (A)
3.11
VOC (V)
21.8
RS (Ω(
0.55
RP (Ω(
50000
Normal Operating Cell
Temperature (NOCT) (CO)
45
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