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Orthogonally Adapted Harris Hawk Optimization for
Parameter Estimation of Photovoltaic Models
Shan Jiaoa, Guoshuang Chongb, Changcheng Huanga*#, Hanqing Huc, Mingjing Wangd, Ali Asghar
Heidarie,f, Huiling Chena*#, Xuehua Zhaog*#
aDepartment of Computer Science and Artificial Intelligence, Wenzhou University, Wenzhou 325035, China
js_1307@163.com, cchuang@126.com,chenhuiling.jlu@gmail.com
bChina Industrial Control Systems Cyber Emergency Response Team, Beijing 100040, China
chonggsh@126.com
cLaboratory of Big Data Decision making for Green Development, Beijing Information Science and Technology University,
Beijing, China,100192
hanqinghu@bistu.edu.cn
d Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam
mingjingwang@duytan.edu.vn
eSchool of Surveying and Geospatial Engineering, College of Engineering, University of Tehran, Tehran, Iran
as_heidari@ut.ac.ir
fDepartment of Computer Science, School of Computing, National University of Singapore, Singapore, Singapore
aliasgha@comp.nus.edu.sg, t0917038@u.nus.edu
gSchool of Digital Media, Shenzhen Institute of Information Technology, Shenzhen 518172, China
lcrlc@sina.com
# These authors contributed equally to this work
Corresponding Author: Changcheng Huang (cchuang@126.com), Huiling Chen
(chenhuiling.jlu@gmail.com) and Xuehua Zhao (lcrlc@sina.com)
Abstract
Extracting parameters and constructing high-precision models of photovoltaic modules through actual
current-voltage data is required for simulation, control, and optimization of a photovoltaic system.
Because of the application of such problems, the identification of unknown parameters accurately and
reliably remains a challenging task. In this paper, we propose an enhanced Harris Hawk Optimization
(EHHO), which combines orthogonal learning (OL) and general opposition-based learning (GOBL),
to estimate the parameters of solar cells and photovoltaic modules effectively and accurately. In EHHO,
OL helps to improve the speed of the HHO method and the accuracy of the solution. At the same
time, the GOBL mechanism can increase both diversity of the population and the HHO’s exploitation
performance. In addition to this, these two mechanisms defend the equilibrium between the
exploitation and exploration rates. The results show that accuracy, reliability, and other aspects of this
method are better than most existing methods. Thus, EHHO can be used as an effective method for
parameter estimation of solar cells and photovoltaic modules.
Keywords: Parameters Estimation; Photovoltaic Models; Harris Hawk Optimization; Orthogonal
Learning; General Opposition-based Learning
Nomenclature
HHO
Harris hawks optimization
Boltzmann constant
EHHO
enhanced HHO
Kelvin temperature
OL
orthogonal learning
number of parallel solar cells
GOBL
general opposition-based learning
number of series solar cells
SDM
single diode model
short circuit current
DDM
double diode model
irradiance
PV
photovoltaic
temperature coefficient
RMSE
root mean square error
short circuit current at SCT
RTC France
name of solar cell
irradiance at STC
Photowatt-PWP
201
name of photovoltaic module
Kelvin temperature at STC
IAE
absolute error
a random hawk
RE
elative error
positions of prey
SM55
name of Mono-crystalline PV module
average position of hawks
ST40
name of Thin-film PV module
upper boundary
STC
standard test conditions
lower boundary
output current
escaping energy of prey
photo-generated current
initial escaping energy
,,
diode current
maximum iterations
shunt resistor current
current iteration
,,
reverse saturated current
escape probability
electron charge
force of random jump
output voltage
a randomly generated vector
series resistor
dimension of problem
parallel resistor
function of Levy-flight
,,
diode ideality factor
N
population size
1 Introduction
It is an urgent task to develop new alternative energy technologies due to the over-use and shortage
of fossil fuels. In response to the energy crisis, renewable energy, which can be a substitute for fossil
fuels, is booming in power generation. Among all kinds of renewable energy, the most commonly used
for power generation is solar energy for its features, such as sustainability, clean, green, and so on [1, 2].
In recent years, the solar photovoltaic system, mainly consisting of solar cells, has attracted more and
more attention. It can directly convert solar energy into electricity. Usually, we use the equivalent circuit
model to simulate the characteristics of a photovoltaic system in different environments. In all the
proposed models, the most widely used models are single diode model (SDM) and double diode model
(DDM), both of them can be constructed by mathematical model and parameter identification [3].
However, photovoltaic systems are usually exposed to the outdoor environment, so photovoltaic arrays
are inevitably prone to deterioration. Also, the actual behavior of the photovoltaic systems mainly
depends on its unknown parameters, which become unstable and volatile as the model faces various
problems. That is why it is so necessary to estimate unknown parameters for the simulation, control,
and optimization of photovoltaic systems.
Researchers have proposed many methods, including analytical method and numerical method for
extracting the parameters of solar cells and PV modules [4-6]. In the analytical method, PV parameters
are estimated utilizing some information in the commodity manual data provided by the manufacturer,
such as short circuit current, maximum power, open-circuit voltage, and by fitting at some regions of
the I-V curves and so on. Nevertheless, the numerical method, which obtains parameters that minimize
the error between the experimental data and the estimated data, uses all I-V points. In other words, the
essence of the numerical method is curve fitting. Among the methods of numerical extraction,
metaheuristic algorithms stand out, and many of them or their variants have already been applied to
solve the problem of parameter estimation in PV models [7-13].
For the sake of unknown parameter estimation, a biogeography-based optimization algorithm
with mutation strategies (BBO-M) was proposed by Niu et al.[14]. Hultmann et al.[15] applied free
search differential evolution (FSDE) algorithm to parameters identification of solar cell model, and
proposed an improved free search differential evolution (IFSDE) algorithm to deal with this problem.
Askarzadeh et al. proposed harmony search-based parameter estimation methods.[16], and five different
variations of bacterial foraging algorithm[17] are presented for SDM and DDM. Oliva et al.[18]
proposed an artificial bee colony (ABC) algorithm to solve problems, while Wu et al.[19] presented an
improved ant lion optimizer (IALO) to enhance the accuracy of solar cell model parameter
identification. Muhsend et al.[20] came up with an improved mutated particle swarm optimization to
extract model parameters of the PV module. In 2017, a particle swarm optimization (PSO) technique
with binary constraints[21] and a water cycle algorithm based on Evaporation Rate (ER-WCA)[22] had
been developed for parameter estimation. Yu et al.[23]improved the teaching-learning optimization
algorithm, which is called SATLBO, to identify different PV modules’ unknown parameters extraction.
To estimate the unknown parameters of SDM, a new optimization method based on cat swarm
optimization (CSO) was proposed by Guo et al. [24]. Merchaoui et al. [25] raised a mutated particle
swarm optimization, while Xiong et al. [26] put forward an improved whale optimization algorithm,
referred to as IWOA, to identify the optimal parameters of different photovoltaic models. Besides,
another improved version of the whale optimization algorithm was presented by Abd et al [27]. Kang
et al. [28] combined cuckoo search algorithm with three strategies to form a new optimization
technology called ImCSA, and used it to solve the parameter estimation problem of PV model. Wang
et al. [29] studied the parameter estimation of JAYA algorithm based on elite opposition-based learning
(EO-Jaya) in the photovoltaic cell model.
Harris Hawks Optimization (HHO) [30] is a new algorithm, whose main inspiration comes from
the chasing progression of the Harris hawks. Through the simulation of Harris hawks’ pursuit of prey,
sudden attack and several different attack strategies, the algorithm have established corresponding
mathematical models and been applied six well-known benchmark engineering problems, including
three-bar truss design problem, tension/compression spring design, pressure vessel design problem,
welded beam design problem, rolling element design problem and multi-plate disc clutch brake problem.
However, like any other natural heuristic algorithms [31-48], HHO is also prone to fall into the local
optimum and problem of slow convergence. Hence, there is much room for additional enhancement
in HHO [12, 13, 49, 50].
In this work, we proposed an improved HHO algorithm that introduced general opposition-based
learning (GOBL) and orthogonal learning (OL) based on the quantized crossover operator, called
EHHO, which can be applied to extract parameters of a photovoltaic system. The introduction of OL
in the search process of the original HHO favors the faster convergence and the improvement of the
accuracy. Moreover, the GOBL mechanism is added at the conclusion of each iteration to increase
population diversity. In other words, EHHO enhances the poise of exploration and exploitation drifts
of the original HHO by introducing the mechanism of OL and GOBL. To validate the properties of
EHHO, we compared EHHO with several other existing algorithms on SDM, DDM, and PV module.
Moreover, according to the results, EHHO shows excellent efficacy compared with other previous
approaches.
To summarize, the main contributions of this paper are as follows:
• The results and performance of the recently proposed HHO are evaluated deeply for the
first time to understand the suitability of this method for parameter identifications of
photovoltaic systems.
• To further enrich the performance, quality, and reliably of results, a novel improved HHO
is proposed using the GOBL and OL concepts based on the quantized crossover scheme
to realize the optimal performance of solar photovoltaic cells.
• The diversity of agents is boosted using the GOBL mechanism at the end of each
iteration, while OL aims to augment the convergence rate and accuracy of results. A
better sense of balance between exploratory and exploitative tendencies is realized in this
work.
• The efficiency of the new variant is validated and compared to previous well-established
methods in the literature using a comprehensive study on different SDM, DDM, and PV
modules.
The structure of this paper is as follows. Section 2 provides the problem formulation of SDM,
DDM, and PV module model. The proposed EHHO is presented in detail in Section 3. Besides, the
experiments are displayed in Section 4. Finally, Section 5 concludes this paper.
2 Problem definition
2.1. Solar cell model
2.1.1. Single diode model
In various fields, the Single diode model has been proverbially put into use, especially in describing
the characteristics of solar cells. Figure 1 shows the structure of the SDM, including photo-generated
current, diode current, and shunt resistor current. Moreover, the output current of SDM is given by
Eq. (1):
where represents the output current, stands for photo-generated current, is on behalf of
diode current which can be obtained by Eq.(2) according to Shockley equation, and means shunt
resistor current whose calculation method is given by Eq.(3) as follows:
In the two formulas above, is series resistor while is a parallel resistor. Also, is the
output voltage, represents the reverse saturated current of SDM, means the diode ideality
factor, is Boltzmann constant , expresses the charge of the
electron, and is the Kelvin temperature of the battery. So, Eq.(1) can be
transformed into the following forms:
According to Eq.(4), SDM contains five parameters,.
Iph IdRsh
IL
Ish
+
_
RS
VL
Fig.1. Equivalent-circuit diagram of single diode model
2.1.2. Double diode model
Based on SDM, the double diode model takes the effect of composite current loss into account,
which makes DDM more accurate than a single diode model. As shown in Figure 2, there is a shunt
resistor for shunting the photo-generated current and two diodes in parallel with the current source in
DDM, one of that is used as a rectifier. The other is utilized to simulate recombination current. The
mathematical expression of the output current in DDM is as followed:
where and are diffusion current and saturation current, and represent diffusion
and recombination diode ideality factors, respectively. From Eq.(5), we can see that there are seven
parameters ,, in the DDM.
Iph Id1 Rsh
IL
Ish
+
_
RS
Id2 VL
Fig.2. Equivalent-circuit diagram of double diode model
2.2. PV module model
The photovoltaic module model is composed of multiple solar cells in series or in parallel, whose
structure is shown in Figure 3. The PV module based on SDM and DDM can be expressed by Eq.(6)
and Eq.(7), respectively.
In the two formulas above, denotes the number of parallel solar cells while is the number
of series ones, and represent the output current and the output voltage of PV module,
respectively. We can easily find that the two models of PV module share the same unknown parameters,
respectively, with SDM and DDM.
Iph
Rsh
IL
+
_
RS
Ns
Np
Ish
VL
Fig.3. Equivalent-circuit diagram of PV module model
2.3. Objective function
In the identification, the output voltage moreover, the output current are the actual data
measured in experiments. All we have to do is finding the optimal value of unknown parameters so that
the error between the calculated current according to the model and measured current is as small as
possible. Therefore, the root mean square of the error shown in Eq.(8) is served as an objective function,
which is a non-linear transcendental one and is difficult to solve. So, this paper is mainly devoted to
seeking out the vector which makes the objective function reach the minimum value.
In Eq.(8), means the number of measured current data, and is the set of unknown
parameters. Also, the objective function of the SDM and DDM can be written as Eq.(9) and Eq.(10),
respectively.
3 The proposed algorithm
3.1 Harris hawk optimization
Harris hawks hunt prey by cooperating in nature. Their predation mainly depends on sudden
attacks. Besides, the Hawks can make different chasing strategies responding to the dynamic
characteristics of the environment and escape patterns of prey. In HHO, the algorithm will be described
by exploration, exploitation, and the transformation of those two states. The main logic of HHO is
demonstrated in Figure 4.
Fig. 4. The logic of HHO
3.1.1 Exploration
Harris hawks track and detect prey with their eyes, but sometimes it can be challenging to spot
prey. Hence, they will observe and capture prey judging by two strategies, of which the opportunity
adopted by Harris hawks is equal. In the first strategy, Harris hawks generate a new solution based on
random positions and other individuals, while another one is to obtain a new solution according to the
current optimal position and the mean of each agent. The mathematical expression is as followed:
where is the location of the hawk in the next iteration while means that in the
current iteration. Also, is a random individual selected in the current population,
represents the position of the prey(usually is a rabbit) which also means the current optimal individual,
and are random numbers inside which followed uniform distribution,
and are the upper and lower boundary variables. is on behalf of the average position of
hawks in the current population, which can be written as Eq.(12):
In Eq.(12), means the total number of hawks and is the location of Hawks in the
iteration.
3.1.2 Transition from exploration to exploitation
One of the most crucial parameters is the escaping energy of prey during the transition from
exploration to exploitation in HHO, whose expression is as followed:
In the above formula, represents the escaping energy of the prey, is the maximum
number of iterations, and means the current number of iterations, varies between and
in each iteration, which is the initial state of the escaping energy. In HHO, the algorithm is in the
exploration and hawks will search for prey in different areas when ; and hawks will seek around
the current solution when , which is called exploitation. The transition within 3 runs of HHO
is shown in Figure 5.
Fig. 5. Variation of transition rule within 3 runs with 250 iterations
3.1.3 Exploitation
At this stage, hawks capture the prey previously found by a surprise attack. However, the prey
often tries to escape so that Harris hawks take different strategies to chase it in the actual situation.
Unsurprisingly, there are four possible strategies in HHO to simulate the hawks’ behavior.
Assuming that is the escape probability of the prey before being an attack, when , it
indicates that the prey can escape successfully from the capture, but unsuccessfully when . No
matter what case it is, hawks always encircle the prey softly or hard according to the situation. is
involved in determining whether hawks have a soft besiege or not. Specifically, it besieges softly when
; Otherwise, it can be a hard one.
3.1.3.1 Soft besiege
When and , the prey has enough energy as ever to attempt to escape from the
encirclement of the hawks by random jumps, although it still fails in the end. In this process, Harris
hawks soft besiege the prey and dissipates its energy; then, the hawks make a surprise attack. The
mathematical model of this behavior is given by Eq.(14), in which represents the difference
of position vector between the prey and hawk, and its expression is as Eq.(15); is written as Eq.(16),
and it stands for the force of random jump when the prey is escaping. In Eq. (16), is a random
number inside .
3.1.3.2 Hard besiege
In the case of and , the escaping of the prey is at a low value. Although the
hawks hardly encircle the prey, they finally complete the raid. Eq.(17) is made use of updating the
locations of the current individuals in this circumstance.
3.1.3.3 Soft besiege with progressive rapid dives
When and , the prey can escape successfully with enough energy, and at this
moment, the Hawks build a soft siege before attacking. To simulate the escape behavior with zigzag
deception of the prey and the irregular dive of hawks, Levy-flight is introduced into HHO. Indeed,
previous studies have confirmed that Levy-flight is the optimal search strategy for foragers under non-
destructive foraging conditions, and it has been detected in monkeys, sharks, and other animals.
It is assumed that the hawks decide the next action, according to Eq.(18):
Then, the hawks perform a dive based on Levy-flight depend on Eq.(19). Furthermore, we
compare the possible result with the previous one to choose the better one. The mathematical
expression is shown as Eq.(20):
In Eq.(19), means the dimension of the problem, is a randomly generated vector with the
size of , indicates the function of Levy-flight whose specific is as following:
，
where and are random numbers that are normally distributed within the interval of，
is a constant coefficient, and in HHO.
3.1.3.4 Hard besiege with progressive rapid dives
When and , the Hawks have formed a hard encirclement before attacking,
and at the same time, the energy possessed by the prey is insufficient to support its escape. The situation
is similar to the soft besiege in terms of prey, but the Hawks are continually trying to narrow the average
distance to the prey. So, the following mathematical models are used for the description of hawks’
behavior:
where and can be written as followed:
Besides, both in Eq.(20) and Eq.(22) is the objective function, and means the fitness of
a vector. For instance, is the fitness of . Figure 6 shows the imagination of this step.
Fig. 6. Illustration of hard besiege phase with progressive gestures.
3.2 Orthogonal learning
OL is an effective strategy that has applied to many nature-inspired algorithms [51-53]. The
orthogonal experimental design is a multi-factor and multi-level optimization experimental design
method, which takes advantage of the orthogonal table to select representative experiments and to
analyze the experimental results. Through the orthogonal experiment, the factor and the level of the
optimal solution can be predicted. So, this paper introduces orthogonal design into HHO in order to
improve accuracy.
Usually, is used to indicate an orthogonal table, which is for the problem of factors
and levels of each factor. In orthogonal experiments, the states of factors are known as “level”.
in is the number of combinations under and . In general, is odd and which
is a positive integer. Meanwhile, the expression of is as Eq.(25) and the construction method of
is as followed where represents the factor in the combination. For instance,
is expressed mainly as Eq.(26), which means nine representative experiments can replace 81
comprehensive experiments for a problem with four factors and three levels. Each row in the
orthogonal table represents a set of representative experimental schemes, and each column is a
representative factor.
Algorithm 1: Construction of Orthogonal Table
Begin
Construct the basic columns as follows:
for to do
;
for to do
;
end for
Construct the non-basic columns as follows:
for to do
;
for to do
for to do
end for
Increment by one for all and
End
The orthogonal crossover operator was first proposed by Zhang and Leung [54], and Leung and
Wang [55] introduced quantization technology into orthogonal crossover operators to form
quantization orthogonal crossover operator. In this method, the continuous space is first quantized and
then selected orthogonal table is used to generate a certain number of candidate solutions. In this paper,
the quantization orthogonal crossover operator and extremum difference analysis are proposed to
improve the performance of HHO.
In the quantization orthogonal crossover operator, it is assumed that two hawks
and are two candidate solutions, and they can define a
search space , where and are as followed:
Secondly, each dimension of the space will be quantified into levels. The levels of the
dimension can be calculated according to Eq.(28):
Theoretically, each dimension can be regarded as a factor, but the dimension of the problem to
be solved is often greater than the number of factors. To solve this problem, different integers
are randomly generated first, and then
is divided into parts, each of which is considered as a factor. The expression is as
followed:
Quantization orthogonal crossover operator takes as a factor and defines the following
levels for it:
Then, offspring candidate solutions are obtained according to . Also, each of them
is evaluated one after another.
Finally, the objective function values of candidate solutions are analyzed to determine the
optimal level combination of each factor and constitute a new solution by extremum difference analysis.
The formula is as follows:
In Eq.(31), is the effect of the level of factor on the results. If there is the
level of factor in the group , ; Otherwise, . So far, new individuals
have been obtained through orthogonal learning.
3.3 General opposition-based Learning
Tizhoosh [56] proposed Opposition-based learning mechanism (OBL) in 2005, and Wang [57] put
forward one of the variants of OBL called general opposition-based learning (GOBL). So far, OBL
has been used in most of the papers [58-62], which sufficiently proved that this mechanism could
effectively enhance the performance of the swarm intelligence algorithm. The basic idea is that the
opposite solution (OP) of a candidate solution may be closer to the optimal solution of the problem
than itself. So, evaluating the candidate solution and its dual solution simultaneously can accelerate the
convergence speed of the algorithm and increase the probability to find the optimal solution.
In GOBL, the calculation formula of the general OP is as follows:
In Eq. (32), is the general OP of and . In high dimensional space, the general
OP solution of is shown in Eq. (32), where
，
, and is a random number in .
During the GOBL, the general OP
will be chosen whether to replace according to its
fitness value. Specifically, if
is superior to ,
will replace as the candidate solution.
3.4 Proposed enhanced HHO
The method proposed in this paper introduces OL with quantization orthogonal crossover
operators and GOBL based on the Harris Hawks Optimization algorithm. Orthogonal learning is an
efficient local search technology that can improve the accuracy of the algorithm. At the same time,
GOBL is capable of increasing population diversity when searching the whole, effective space that is
influential in avoiding the local optimal. By introducing these two mechanisms into HHO, not only a
better balance between the relationship of exploration and exploitation of the algorithm can be attained,
but also we can improve the optimization ability of HHO.
Firstly, orthogonal learning is performed for each updated individual. It is necessary to solve the
problem of how to obtain parent individuals and how to use quantization orthogonal crossover
operators to generate new excellent individuals when OL is introduced into HHO. In previous studies,
there are various ways to deal with these two problems in different algorithms. Of course, this paper
finally adopts the following method after some experiments. In each iteration, individuals in the
population are updated by using the search method of HHO itself, and the optimal candidate
in the current iteration is obtained. Then, and are used as parent individuals to generate
sub-individuals by using quantization orthogonal crossover operator with , and one
new candidate solution is got through extremum difference analysis of sub-individuals. Then, choose
the optimal one from new sub-individuals, called . At last, the best solution
among individuals obtaining from the above operation is selected to determine whether it is superior
to . If so, , instead of , is called the optimal solution in the current population.
Afterward, at the end of each iteration, GOBL is conducted for an updated population. The
process is as followed. Suppose that , the current population, it represents as Eq.(33), which
contains individuals, and is the current iteration number, is the dimension of the problem.
The general reverse population of is expressed as , which can be calculated according to
Eq.(32). After getting , the optimal individuals in the set are selected to enter
the next generation according to their fitness.
Based on the above statement, we call the improve HHO algorithm as EHHO, whose pseudocode
is as followed, and the flowchart is shown in Figure 7. Also, the values of the critical parameters in the
method are listed in Table 1.
Algorithm 2: The pseudo-code of EHHO
Initialize the population size N and maximum number of iterations T;
Initialize the random population (i=1,2,…,N);
While (stopping condition is not met) do
Calculate the fitness values of hawks;
Set as the location of the rabbit (best location);
for (each hawk()) do
Update the initial energy and jump strength ; ;
Update the using Eq.(13);
if () then
if () then
Update the location vector using Eq.(11);
else if () then
Update the location vector using Eq.(11);
if () then
if () then
Update the location vector using Eq.(14);
else if () then
Update the location vector using Eq.(17);
else if () then
Update the location vector using Eq.(20);
else if () then
Update the location vector using Eq.(22);
end if
Perform Orthogonal Learning operation for ;
end for
Update the population via General Opposition-based Learning;
end while
Return .
Fig.7. Flowchart of EHHO
Table 1. Values of parameters in EHHO
,,,,,
1.5
6
5
4 Simulation results
For the sake of verification about the performance of the proposed method, EHHO has primarily
applied to parameters estimation of three models, which are SDM, DDM, and PV models. The two
sets of data used in the experiment are related to RTC France photovoltaic cell and Photowatt-PWP
201 photovoltaic module. RTC France is a solar cell with 57 mm diameter, the data of which is measured
under the condition of and ; the data about Photowatt-PWP 201 photovoltaic
module which is composed of 36 pieces of polysilicon batteries in series is measured under and
. In the performance verification of other innovations [63-65], these two sets of data have
extensively been put into use. Parameter identification experiments in SDM and DDM of RTC France
photovoltaic are carried out in this study, and unknown parameters of the Photowatt-PWP 201
photovoltaic module are identified.
One of the main steps in mathematical modeling and providing solvers is the way we implement
it [66-69]. In this paper, EHHO is implemented in MATLAB R2014a, and 30 identification experiments
are carried out independently on each issue. Moreover, the size of the population is 30. Moreover, each
experiment terminates until reaching the maximum number of iterations that are set to 2000 for all
three models. The search range of unknown parameters for each model is shown in Table 2.
Figure 8 shows the convergence curves of EHHO, which state the average RMSE of 30
independent runs. Visibly, the convergence speed of EHHO is faster than that of HHO in the three
models.
Table 2. Bounds of parameters for different PV models
Parameters
Single diode/double diode
PV module
Lower bound
Upper bound
Lower bound
Upper bound
(A)
0
1
0
2
()
0
1
0
50
0
0.5
0
2
0
100
0
2000
n
1
2
1
50
()
0
1
0
50
()
0
1
0
50
n1
1
2
1
50
n2
1
2
1
50
Fig.8. Convergence graphs of EHHO for three models
4.1 Results for SDM
The I-V and P-V characteristics of estimated SDM identified by EHHO are shown in Figure 9; it
is clear that, within the entire range of measured voltage, current simulated by EHHO is highly in
coincidence with the measured data. Meanwhile, the absolute error (IAE) and relative error (RE) of the
current value between simulated data and measured data at each voltage measurement point are
presented in Figure 10. Table 3 further displays the values of current and power obtained from a
simulation experiment, as well as the relevant IAE and RE, whose definitions are shown in Eqs.(35)
and (36). Besides, all values of IAE are less than 2.509E-03, and the RE is within the range of [-
1.99793E-02，1.47E-01], indicating that SDM simulated by EHHO can accurately describe the actual
characteristics of solar cells.
The results, which contain values of five parameters and RMSE, obtained by EHHO are revealed
in Table 3, compared with other algorithms’ which contain CPSO[70], PS[71], LMSA[72], ABC[18],
ABSO[63], GOTLBO[73] and GOFPANM[74]. From Table 4, it can make out that EHHO, as well as
GOFPANM, got the best RSME with the value of 9.8602E−04. Based on the above results, we can
conclude that EHHO could be used as an effective method to identify SDM.
Fig.9. Comparisons between the experimental data and estimated model acquired by EHHO for single diode model (a)
I-V characteristic; (b) P-V characteristic.
Fig.10. Error index values of the experimental and the simulated current data for single diode model (a) IAE values;
(b) RE values.
Table 3. IAE of EHHO on SDM
Item
Measured data
Simulated current data
Simulated power data
1
-0.2057
0.7640
0.764085245
8.52453E-05
-0.157172335
1.75350E-05
2
-0.1291
0.7620
0.762661271
0.000661271
-0.098459570
8.53701E-05
3
-0.0588
0.7605
0.761354083
0.000854083
-0.044767620
5.02201E-05
4
0.0057
0.7605
0.760153310
0.000346690
0.004332874
1.97614E-06
5
0.0646
0.7600
0.759055022
0.000944978
0.049034954
6.10456E-05
6
0.1185
0.7590
0.758042608
0.000957392
0.089828049
0.000113451
7
0.1678
0.7570
0.757092320
9.23203E-05
0.127040091
1.54913E-05
8
0.2132
0.7570
0.756142386
0.000857614
0.161209557
0.000182843
9
0.2545
0.7555
0.755088179
0.000411821
0.192169942
0.000104808
10
0.2924
0.7540
0.753665373
0.000334627
0.220371755
9.78448E-05
11
0.3269
0.7505
0.751392511
0.000892511
0.245630212
0.000291762
12
0.3585
0.7465
0.747355258
0.000855258
0.267926860
0.000306610
13
0.3873
0.7385
0.740118270
0.001618270
0.286647806
0.000626756
14
0.4137
0.7280
0.727382699
0.000617301
0.300918223
0.000255377
15
0.4373
0.7065
0.706972438
0.000472438
0.309159047
0.000206597
16
0.4590
0.6755
0.675279287
0.000220713
0.309953193
0.000101307
17
0.4784
0.6320
0.630757008
0.001242992
0.301754153
0.000594647
18
0.4960
0.5730
0.571927077
0.001072923
0.283675830
0.000532170
19
0.5119
0.4990
0.499606109
0.000606109
0.255748367
0.000310267
20
0.5265
0.4130
0.413648577
0.000648577
0.217785976
0.000341476
21
0.5398
0.3165
0.317510666
0.001010666
0.171392257
0.000545557
22
0.5521
0.2120
0.212156131
0.000156131
0.117131400
8.61997E-05
23
0.5633
0.1035
0.102252737
0.001247263
0.057598967
0.000702583
24
0.5736
-0.0100
-0.008716633
0.001283367
-0.004999861
0.000736139
25
0.5833
-0.1230
-0.125507552
0.002507552
-0.073208555
0.001462655
26
0.5900
-0.2100
-0.208474053
0.001525947
-0.122999692
0.000900308
Sum of IAE
NA
NA
NA
0.021524060
NA
0.008730999
Table 4. Comparison among different algorithms on SDM
Item
CPSO
PS
LMSA
ABC
ABSO
GOTLBO
GOFPANM
EHHO
(A)
0.7607
0.7617
0.76078
0.7608
0.7608
0.760780
0.76077755
0.760775
()
0.4
0.998
0.31849
0.3251
0.30623
0.331552
0.3230208
3.23E-01
0.0354
0.0313
0.03643
0.0364
0.03659
0.036265
0.0363771
0.036375
59.012
64.1026
53.32644
53.6433
52.2903
54.11543
53.7185203
53.74282
n
1.5033
1.6
1.47976
1.4817
1.47878
1.483820
1.4811836
1.481238
RMSE
1.3900E-03
1.4940E-02
9.8640E-04
9.8620E-04
9.9124E-04
9.87442E-04
9.8602E-04
9.8602E−04
4.2 Results for DDM
In the case of DDM, Figure 11 presents the comparisons of characteristics of the estimated model
acquired by EHHO and actual measured data, and Figure 12 vividly shows the IAE and RE between
the simulated and experimental current. Noticeably, the curves of I-V and P-V of the simulated model
are both in good agreement with the experimental data. Concurrently, the values of current, power,
IAE, and RE acquired in the simulation are exhibited explicitly in Table 5. It can be clearly seen that at
each voltage measurement point, the value of IAE is less than 2.545E-03, and the range of RE is
，, which shows that DDM obtained by EHHO can accurately describe
the actual characteristics of solar cells.
Table 6 exhibits the results between EHHO and other algorithms, including PS[71], SA[75],
HS[16], GGHS[16], IGHS[16], ABC[18]. It is evident that the best RMSE among all algorithms is
9.83606E-04, which is obtained by EHHO. Therefore, EHHO proposed in this paper can also identify
parameters of DDM.
Fig.11. Comparisons between the experimental data and estimated model acquired by EHHO for double diode model
(a) I-V characteristic; (b) P-V characteristic.
Fig.12. Error index values of the experimental and the simulated current data for double diode model (a) IAE values;
(b) RE values.
Table 5. IAE of EHHO on DDM
Item
Measured data
Simulated current data
Simulated power data
1
-0.2057
0.7640
0.763964292
3.57076E-05
-0.157147455
7.34505E-06
2
-0.1291
0.7620
0.762588807
0.000588807
-0.098450215
7.60150E-05
3
-0.0588
0.7605
0.761325972
0.000825972
-0.044765967
4.85672E-05
4
0.0057
0.7605
0.760165458
0.000334542
0.004332943
1.90689E-06
5
0.0646
0.7600
0.759102738
0.000897262
0.049038037
5.79631E-05
6
0.1185
0.7590
0.758120165
0.000879835
0.089837240
0.000104260
7
0.1678
0.7570
0.757191768
0.000191768
0.127056779
3.21786E-05
8
0.2132
0.7570
0.756252410
0.000747590
0.161233014
0.000159386
9
0.2545
0.7555
0.755193105
0.000306895
0.192196645
7.81048E-05
10
0.2924
0.7540
0.753746053
0.000253947
0.220395346
7.42541E-05
11
0.3269
0.7505
0.751429777
0.000929777
0.245642394
0.000303944
12
0.3585
0.7465
0.747334983
0.000834983
0.267919591
0.000299341
13
0.3873
0.7385
0.740039060
0.001539060
0.286617128
0.000596078
14
0.4137
0.7280
0.727259414
0.000740586
0.300867219
0.000306381
15
0.4373
0.7065
0.706837179
0.000337179
0.309099898
0.000147448
16
0.4590
0.6755
0.675168306
0.000331694
0.309902253
0.000152247
17
0.4784
0.6320
0.630697258
0.001302742
0.301725568
0.000623232
18
0.4960
0.5730
0.571926880
0.001073120
0.283675732
0.000532268
19
0.5119
0.4990
0.499654048
0.000654048
0.255772907
0.000334807
20
0.5265
0.4130
0.413716813
0.000716813
0.217821902
0.000377402
21
0.5398
0.3165
0.317571453
0.001071453
0.171425070
0.000578370
22
0.5521
0.2120
0.212185807
0.000185807
0.117147784
0.000102584
23
0.5633
0.1035
0.102243122
0.001256878
0.057593550
0.000708000
24
0.5736
-0.0100
-0.008738829
0.001261171
-0.005012592
0.000723408
25
0.5833
-0.1230
-0.125545333
0.002545333
-0.073230593
0.001484693
26
0.5900
-0.2100
-0.208465587
0.001534413
-0.122994696
0.000905304
Sum of IAE
NA
NA
NA
0.021377384
NA
0.008815488
Table 6. Comparison among different algorithms based on DDM
Item
PS
SA
HS
GGHS
IGHS
ABC
EHHO
(A)
0.7602
0.7623
0.76176
0.76056
0.7608
0.7608
0.760769017
()
0.9889
0.4767
0.12545
0.37014
0.9731
0.0407
5.86184E-01
()
0.0001
0.01
0.25470
0.13504
0.1679
0.2874
2.40965E-01
0.032
0.0345
0.03545
0.03562
0.0369
0.0364
0.036598831
81.3008
43.1034
46.82696
62.7899
53.8368
53.7804
55.63943956
n1
1.6
1.5172
1.49439
1.49638
1.9213
1.4495
1.968451449
n2
1.192
2
1.49989
1.92998
1.4281
1.4885
1.456910409
RMSE
1.58E-02
1.664E-02
1.26E-03
1.07E-03
9.8635E-04
9.861E-04
9.83606E-04
4.3 Results for PV module
From Figure 13, which reveals the characteristics of the Photowatt-PWP 201 module model
estimated by EHHO, it can be distinctly observed that consistency of I-V and P-V curves between
simulated data and measured data are both in a high degree. Figure 14 also vividly shows the IAE and
RE of the experimental and measured current about the PV module model. Simultaneously, the data
obtained from the experiment, including the value of current and power as well as their IAR and RE,
are given in Table 7. It is explicit that the value of IAE is less than 4.8275E-03, and RE is within the
range of，. From this, we can see that the accuracy of parameters
identified by EHHO for the PV module model can be guaranteed.
As to further verify the performance of EHHO, six algorithms, which are Newton[76]，CPSO[70],
PS[71], SA[75], the method in literature[77] and CARO[78] are used to make a comparison. The results
are written in Table 8, which shows intuitively that EHHO gets the best results in which the RSME
value is 2.42508E-03, followed by CARO with the RSME of 2.427E-03. In short, the proposed method,
EHHO, is superior to other competitors and can be applied to estimate parameters of the PV module
model.
Fig.13. Comparisons between the experimental data and estimated model acquired by EHHO for PV module model
(a) I-V characteristic; (b) P-V characteristic.
Fig.14. Error index values of the experimental and the simulated current data for PV module model (a) IAE values; (b)
RE values.
Table 7. IAE of EHHO on PV module model
Item
Measured data
Simulated current data
Simulated power data
1
0.1248
1.0315
1.029107232
0.002392768
0.128432583
0.000298617
2
1.8093
1.0300
1.027373491
0.002626509
1.858826857
0.004752143
3
3.3511
1.0260
1.025738134
0.000261866
3.43735106
0.000877540
4
4.7622
1.0220
1.024106966
0.002106966
4.877002194
0.010033794
5
6.0538
1.0180
1.022294575
0.004294575
6.188766896
0.025998496
6
7.2364
1.0155
1.019935770
0.004435770
7.380663204
0.032099004
7
8.3189
1.0140
1.016369688
0.002369688
8.455077797
0.019713197
8
9.3097
1.0100
1.010503194
0.000503194
9.407481585
0.004684585
9
10.2163
1.0035
1.000635300
0.002864700
10.22279042
0.029266631
10
11.0449
0.9880
0.984552854
0.003447146
10.87428782
0.038073381
11
11.8018
0.9630
0.959523457
0.003476543
11.32410393
0.041029470
12
12.4929
0.9255
0.922837637
0.002662363
11.52891832
0.033260630
13
13.1231
0.8725
0.872595956
9.59561E-05
11.45116399
0.001259242
14
13.6983
0.8075
0.807269160
0.000230840
11.05821513
0.003162116
15
14.2221
0.7265
0.728331317
0.001831317
10.35840082
0.026045172
16
14.6995
0.6345
0.637134105
0.002634105
9.365552779
0.038720029
17
15.1346
0.5345
0.536211403
0.001711403
8.115345097
0.025901397
18
15.5311
0.4275
0.429512100
0.002012100
6.670795383
0.031250133
19
15.8929
0.3185
0.318777562
0.000277562
5.066299912
0.004411262
20
16.2229
0.2085
0.207394099
0.001105901
3.364533728
0.017940922
21
16.5241
0.1010
0.096172461
0.004827539
1.589163368
0.079770732
22
16.7987
-0.0080
-0.008321607
0.000321607
-0.139792176
0.005402576
23
17.0499
-0.1110
-0.110935045
6.49555E-05
-1.891431416
0.001107484
24
17.2793
-0.2090
-0.209249308
0.000249308
-3.615681566
0.004307866
25
17.4885
-0.3030
-0.300870560
0.002129440
-5.261774788
0.037240712
Sum of IAE
NA
NA
NA
0.048934120
NA
0.516607133
Table 8. Comparative results for PV module model
Item
Newton
CPSO
PS
SA
Method in[77]
CARO
EHHO
(A)
1.0318
1.0286
1.0313
1.0331
1.0310
1.03185
1.030498656
()
3.2875
8.3010
3.1756
3.6642
3.8236
3.28401
3.488188406
1.2057
1.0755
1.2053
1.1989
1.0920
1.20556
1.201110352
555.5556
1850.1000
714.2857
833.3333
689.6600
841.3213
984.4964824
n
48.4500
52.2430
48.2889
48.8211
48.9300
48.40363
48.64931708
RMSE
7.8050E-01
3.5000E-03
1.1800E-02
2.7000E-03
1.02E-01
2.427E-03
2.42508E-03
4.4 Tests for different PV cell and modules
To further prove EHHO’s validity and reliability, it is used to identify parameters of both SDM
and DDM of PVM 752 GaAs thin-film cell and two different PV modules, which are Thin-film ST40
[79] and Mono-crystalline SM55 [80]. The experimental data of PVM 752 GaAs thin-film cell include
44 pairs I-V points, which are extracted at T= 25°C and full irradiation(1000W/m2) from the literature
[81]. About the other two types of photovoltaic solar modules, ST40 is composed of a monolithic
structure of series connected Copper Indium Diselenide (CIS) based solar cells, while SM55 contains
36 PowerMax® monocrystalline silicon solar cells in series. The data used in the experiments is
extracted from the product manual of each photovoltaic module, including the data at different
irradiance levels and temperature levels. In this portion of the simulation experiment, the size of the
population is set to 30, and the maximum iterations are 2000.
4.4.1 Results for PVM 752 GaAs thin-film cell
In this case, EHHO is applied to identify parameters of SDM and DDM of PVM 752 thin-film
cells. The range of unknown parameters is shown in Table 2. The experimental results of the cell
achieved by EHHO are given in Tables 9-11. Table 9 and Table 10 are specifically display simulated
current data and error metrics for each measurement, including IAE and RE. From these two tables, it
can be concluded that parameter identification for this cell is more difficult to optimization algorithm
due to a large number of I-V pairs about PVM 752 thin film compared to simulation data from EHHO
on RTC France solar cell. However, from Table 11, the approach presented here still has advantages.
Table 11 shows the parameter estimation results gained by several different algorithms on SDM and
DDM. According to the results given in the table, EHHO has the optimal RMSE on the single diode
model and ranks first among those comparison methods. For DDM, EHHO only in second place, but
still better than IJAYA[1], MLBSA[82], and HHO.
Table 9. Error metrics for each measurement in SDM obtained by EHHO for PVM 752 thin-film cells
Item
Measured data
Simulated current data
1
-0.1659
0.1001
0.1001
0.104470745
0.004370745
-0.04184
2
-0.1281
0.1000
0.1000
0.104093245
0.004093245
-0.03932
3
-0.0888
0.0999
0.0999
0.103700745
0.003800745
-0.03665
4
-0.0490
0.0999
0.0999
0.103302745
0.003402745
-0.03294
5
-0.0102
0.0999
0.0999
0.102914745
0.003014745
-0.02929
6
0.0275
0.0998
0.0998
0.102538245
0.002738245
-0.0267
7
0.0695
0.0999
0.0999
0.102117745
0.002217745
-0.02172
8
0.1061
0.0998
0.0998
0.101752244
0.001952244
-0.01919
9
0.1460
0.0998
0.0998
0.101353242
0.001553242
-0.01533
10
0.1828
0.0997
0.0997
0.100985737
0.001285737
-0.01273
11
0.2230
0.0997
0.0997
0.100583727
0.000883727
-0.00879
12
0.2600
0.0996
0.0996
0.100214206
0.000614206
-0.00613
13
0.3001
0.0997
0.0997
0.099812654
0.000112654
-0.00113
14
0.3406
0.0996
0.0996
0.099408036
0.000191964
0.001931
15
0.3789
0.0995
0.0995
0.099025285
0.000474715
0.004794
16
0.4168
0.0994
0.0994
0.098646242
0.000753758
0.007641
17
0.4583
0.0994
0.0994
0.098229887
0.001170113
0.011912
18
0.4949
0.0993
0.0993
0.09786174
0.00143826
0.014697
19
0.5370
0.0993
0.0993
0.097433836
0.001866164
0.019153
20
0.5753
0.0992
0.0992
0.097037067
0.002162933
0.02229
21
0.6123
0.0990
0.0990
0.09663828
0.00236172
0.024439
22
0.6546
0.0988
0.0988
0.096138805
0.002661195
0.027681
23
0.6918
0.0983
0.0983
0.095617179
0.002682821
0.028058
24
0.7318
0.0977
0.0977
0.09485902
0.00284098
0.029949
25
0.7702
0.0963
0.0963
0.09372319
0.00257681
0.027494
26
0.8053
0.0937
0.0937
0.091971708
0.001728292
0.018792
27
0.8329
0.0900
0.0900
0.089742893
0.000257107
0.002865
28
0.8550
0.0855
0.0855
0.087125764
0.001625764
-0.01866
29
0.8738
0.0799
0.0799
0.084152797
0.004252797
-0.05054
30
0.8887
0.0743
0.0743
0.081196289
0.006896289
-0.08493
31
0.9016
0.0683
0.0683
0.07819008
0.00989008
-0.12649
32
0.9141
0.0618
0.0618
0.074789998
0.012989998
-0.17369
33
0.9248
0.0555
0.0555
0.071506657
0.016006657
-0.22385
34
0.9344
0.0493
0.0493
0.068250953
0.018950953
-0.27767
35
0.9445
0.0422
0.0422
0.064507819
0.022307819
-0.34582
36
0.9533
0.0357
0.0357
0.060923504
0.025223504
-0.41402
37
0.9618
0.0291
0.0291
0.057178535
0.028078535
-0.49107
38
0.9702
0.0222
0.0222
0.053226554
0.031026554
-0.58291
39
0.9778
0.0157
0.0157
0.049409382
0.033709382
-0.68225
40
0.9852
0.0092
0.0092
0.045438583
0.036238583
-0.79753
41
0.9926
0.0026
0.0026
0.041171529
0.038571529
-0.93685
42
0.9999
0.0040
−0.0040
0.036650047
0.040650047
-1.10914
43
1.0046
0.0085
−0.0085
0.033692695
0.042192695
-1.25228
44
1.0089
0.0124
−0.0124
0.030713565
0.043113565
-1.40373
Table 10. Error metrics for each measurement in DDM obtained by EHHO for PVM 752 thin-film cells
Item
Measured data
Simulated current data
1
-0.1659
0.1001
0.1001
0.104470745
0.004370745
-0.041837
2
-0.1281
0.1000
0.1000
0.104093245
0.004093245
-0.0393229
3
-0.0888
0.0999
0.0999
0.103700745
0.003800745
-0.0366511
4
-0.0490
0.0999
0.0999
0.103302745
0.003402745
-0.0329395
5
-0.0102
0.0999
0.0999
0.102914745
0.003014745
-0.0292936
6
0.0275
0.0998
0.0998
0.102538245
0.002738245
-0.0267046
7
0.0695
0.0999
0.0999
0.102117745
0.002217745
-0.0217175
8
0.1061
0.0998
0.0998
0.101752244
0.001952244
-0.0191862
9
0.1460
0.0998
0.0998
0.101353242
0.001553242
-0.015325
10
0.1828
0.0997
0.0997
0.100985737
0.001285737
-0.0127319
11
0.2230
0.0997
0.0997
0.100583727
0.000883727
-0.008786
12
0.2600
0.0996
0.0996
0.100214205
0.000614205
-0.0061289
13
0.3001
0.0997
0.0997
0.099812654
0.000112654
-0.0011287
14
0.3406
0.0996
0.0996
0.099408036
0.000191964
0.00193107
15
0.3789
0.0995
0.0995
0.099025285
0.000474715
0.00479388
16
0.4168
0.0994
0.0994
0.098646242
0.000753758
0.00764102
17
0.4583
0.0994
0.0994
0.098229887
0.001170113
0.01191198
18
0.4949
0.0993
0.0993
0.09786174
0.00143826
0.01469686
19
0.5370
0.0993
0.0993
0.097433836
0.001866164
0.01915314
20
0.5753
0.0992
0.0992
0.097037067
0.002162933
0.02228976
21
0.6123
0.0990
0.0990
0.09663828
0.00236172
0.02443876
22
0.6546
0.0988
0.0988
0.096138805
0.002661195
0.02768075
23
0.6918
0.0983
0.0983
0.095617179
0.002682821
0.02805794
24
0.7318
0.0977
0.0977
0.09485902
0.00284098
0.0299495
25
0.7702
0.0963
0.0963
0.09372319
0.00257681
0.02749384
26
0.8053
0.0937
0.0937
0.091971708
0.001728292
0.01879156
27
0.8329
0.0900
0.0900
0.089742893
0.000257107
0.00286492
28
0.8550
0.0855
0.0855
0.087125764
0.001625764
-0.01866
29
0.8738
0.0799
0.0799
0.084152797
0.004252797
-0.0505366
30
0.8887
0.0743
0.0743
0.081196289
0.006896289
-0.0849335
31
0.9016
0.0683
0.0683
0.07819008
0.00989008
-0.1264877
32
0.9141
0.0618
0.0618
0.074789998
0.012989998
-0.1736863
33
0.9248
0.0555
0.0555
0.071506657
0.016006657
-0.2238485
34
0.9344
0.0493
0.0493
0.068250953
0.018950953
-0.2776658
35
0.9445
0.0422
0.0422
0.064507819
0.022307819
-0.3458157
36
0.9533
0.0357
0.0357
0.060923504
0.025223504
-0.4140193
37
0.9618
0.0291
0.0291
0.057178535
0.028078535
-0.4910678
38
0.9702
0.0222
0.0222
0.053226554
0.031026554
-0.5829149
39
0.9778
0.0157
0.0157
0.049409382
0.033709382
-0.6822466
40
0.9852
0.0092
0.0092
0.045438583
0.036238583
-0.7975289
41
0.9926
0.0026
0.0026
0.041171529
0.038571529
-0.9368496
42
0.9999
0.0040
−0.0040
0.036650047
0.040650047
-1.1091404
43
1.0046
0.0085
−0.0085
0.033692695
0.042192695
-1.2522802
44
1.0089
0.0124
−0.0124
0.030713565
0.043113565
-1.4037304
Table 11. Parameters of PVM 752 thin-film cell for SDM and DDM achieved by different algorithms
Parameters
Algorithm
IJAYA
MLBSA
ELPSO[81]
HHO
EHHO
SDM
0.103253
0.103191
0.115016
0.103709
0.103312
1.13E-10
1.2E-10
0
2.11E-10
6.76E-11
0.5
0.463325
0.159052
0.49929
0.5
100
99.3907
14.429507
99.85768
100
1.88746
1.892766
1.768590
1.946415
1.84087
RMSE
2.297 E−3
2.384 E−3
2.5400E−2
2.394 E−3
2.261E−3
DDM
0.104029
0.102856
0.103192
0.103732
0.103312
0
9.86E-11
1.775000e−10
1.1E-10
2.61E-41
3.61E-10
1.03E-11
1.000000E−12
1.1E-10
6.76E-11
0.454924
0.378473
0.500000
0.487458
0.5
95.14813
80.33082
100.000000
97.4917
100
1.979059
1.915306
2.000000
1.950305
1.00599
2
1.769797
1.571052
1.950305
1.84087
RMSE
2.543 E−3
3.52 E−3
2.075 E−3
2.431 E−3
2.261 E−3
4.4.2 Results for ST40 and SM55
In this case, the range of unknown parameters is different. The initial range of photo-generated
current is determined according to short circuit current and temperature coefficient for
which are provided on the datasheet of each module at standard test conditions (STC). The
mathematical expression of is as followed:
In the formula, G and T represent the light irradiation and temperature respectively under non-
standard test conditions. So, the initial range of is , and
.
For two different kinds of PV modules, the optimal parameters of both SDM and DDM at
multiple irradiances extracted by EHHO are shown in Table 12 and Table 13, while parameters at
different temperatures are represented in Table 14 and Table 15. Also, Figures 15 and 16 show the I-V
characteristics of SDM and DDM of two PV modules at diverse irradiance. Besides, Figures 17 and 18
show that at different temperatures. As we can see from the figures, I-V characteristic curves of
estimated data are highly consistent with the measured data, no matter it is under different irradiance
or temperature for ST40 and SM55. As a rule, it is particularly important to identify parameters at low
irradiance because PV modules often encounter insufficient illumination in the actual situation. It can
be seen from results that EHHO can also identify the model with high accuracy even at low irradiance.
Moreover, the accuracy of the model increases with the increase of irradiance in general. In conclusion,
the proposed method in the paper, EHHO, has high accuracy for parameters estimation of PV modules
with different types, no matter at what irradiance and temperature. Moreover, it is simple to implement,
which can be regarded as a promising technology.
Table 12. Estimated optimal parameters using EHHO for ST40 at dissimilar irradiance and temperature
of 25 °C
Parameters
Irradiance
200
400
600
800
1000
SDM
2.675559278
2.136805848
1.602512168
1.065650909
0.531975801
1.55259E-06
1.32069E-06
2.03684E-06
2.76134E-06
2.78327E-06
1.112248048
1.114892467
1.068975948
1.010626318
0.818666022
360.955073
349.7586609
386.2108419
399.982988
360.6024468
1.75217988
1.734278379
1.788727519
1.833491292
1.840869338
RMSE
7.37682E-04
8.9812E-04
1.07509E-03
1.00894E-03
6.20739E-04
DDM
2.674676288
2.136878001
1.603369502
1.067555024
0.533206103
1.84845E-05
1.14837E-05
8.66155E-07
1.62629E-05
1.41507E-05
1.54098E-06
3.28044E-07
1.88239E-06
1.55719E-06
4.18959E-07
1.107432148
1.157232598
1.104221244
1.098946228
1.400880086
382.274102
363.6313361
373.2824169
365.6204254
354.6115388
3.638558789
2.473874016
1.703835742
3.760479325
2.72917543
1.75184816
1.593365882
2.053482631
1.756776765
1.605839498
RMSE
8.96644E-04
1.02061E-03
8.3643E-04
6.57051E-04
4.5715E-04
Fig.15. Estimated and measured (I-V) characteristics for Thin-film ST40 at diverse irradiance for SDM
and DDM
Table 13. Estimated optimal parameters using EHHO for SM55 at dissimilar irradiance and temperature
of 25 °C
Parameters
Irradiance
200
400
600
800
1000
SDM
3.447166885
2.756516817
2.069589648
1.380332822
0.692329283
2.13482E-07
2.54517E-07
1.90665E-07
2.30578E-07
1.03024E-07
0.322287764
0.307385909
0.318222203
0.293065599
0.378232658
591.5679331
608.0428674
489.4689111
501.1391674
430.6210501
1.414177975
1.428836091
1.404797558
1.422191603
1.350520126
RMSE
1.76534E-03
1.98588E-03
1.03554E-03
1.37423E-03
5.40104E-04
DDM
3.449592252
2.757909604
2.070694923
1.382350598
0.69209036
1.23313E-07
2.08842E-07
1.66704E-07
5.23521E-08
1.00568E-07
4.74128E-06
2.65006E-06
0
5.06867E-05
1.43857E-06
0.333784318
0.316961622
0.325875043
0.439965892
0.361873021
515.8892481
541.5105888
456.8131279
466.1999807
438.9004842
1.370646855
1.411895118
1.393377698
1.302145894
1.34871169
2.446219048
4
1.000488039
3.52767008
2.538927208
RMSE
1.29432E-03
1.36512E-03
8.4193E-04
6.10952E-04
5.30767E-04
Fig.16 Estimated and measured (I-V) characteristics for Mono-crystalline SM55 at diverse irradiance
for SDM and DDM
Table 14. The optimal extracted parameters for ST40 found by EHHO at different temperatures and
irradiance of 1000
Parameters
Temperature
25℃
40℃
50℃
70℃
SDM
2.669498426
2.682081008
2.691425588
2.692226947
2.35084E-06
4.91957E-06
1.92904E-05
8.78845E-05
1.085302456
1.139788157
1.147113011
1.125566558
480.5976343
342.9885094
301.7906005
369.9570862
1.803479581
1.704344562
1.722151378
1.727997565
RMSE
2.10946E-03
1.47997E-03
1.82941E-03
7.78103E-04
DDM
2.675062623
2.680655057
2.688730045
2.69269009
1.5449E-06
5.58422E-06
3.2067E-06
1.24761E-05
3.11949E-06
3.26347E-06
6.76119E-05
7.45726E-05
1.109540286
1.128578398
1.180214856
1.127413285
372.2765894
369.8219781
366.7858843
360.8358502
1.752196532
1.721045399
1.527758372
1.811927829
2.94191856
2.83019683
2.213610244
1.716074045
RMSE
8.04955E-04
1.33126E-03
1.67754E-03
7.85785E-04
Fig.17. Estimated and measured (I-V) characteristics for Thin-film ST40 at different temperatures for
SDM and DDM
Table 15. The optimal extracted parameters for SM55 found by the EHHO at different temperatures
and irradiance of 1000
Parameters
Temperature
25℃
40℃
60℃
SDM
3.447400365
3.467596400
3.493519670
2.11999E-07
1.30215E-06
7.30958E-06
0.322327728
0.308302101
0.316507475
579.6986592
609.6666228
529.8494052
1.413593939
1.430044326
1.411117007
RMSE
1.6933E-03
3.87440E-03
3.80568E-03
DDM
3.448827075
3.467300869
3.493656737
8.38835E-08
1.44235E-06
7.39927E-06
6.46202E-07
1.28441E-05
6.00786E-11
0.334324184
0.303856077
0.316260572
531.3402211
649.625709
527.0570722
1.347094432
1.440005932
1.412625557
1.778668985
4
1.001298645
RMSE
1.33235E-03
4.06173E-03
3.80961E-03
Fig.18. Estimated and measured (I-V) characteristics for Mono-crystalline SM55 at different
temperatures for SDM and DDM
5 Conclusions and future directions
To effectively estimate the unknown parameters of photovoltaic models, the new method, which
combines HHO with two mechanisms of OL and GOBL, was proposed in this paper. EHHO is
devoted to comprehensively verify unknown parameters identification problems in SDM, DDM of
different types of solar cell and different PV module models, and also compared with several existing
competitors. The results of the competition and statistical rates show that EHHO proposed in this
paper can reliably and efficiently complete parameter identification, which reveals that the exploration
and exploitation capability of HHO has been significantly improved. As a consequence, EHHO is
considered as a new tool for estimating unknown parameters in different solar cells and PV modules.
Although the experimental results have proved the efficiency of EHHO for parameter
identification of PV cell and module, some aspects still have space to be further explored. Firstly, the
proposed method can be applied to extract unknown parameters of other typical solar cell modules to
give full play to its capabilities. Secondly, the performance of the method proposed in this paper can be
further improved based on other optimization mechanisms, especially time efficiency. In addition,
EHHO is a stochastic optimizer, which is based on the distribution of the population. Therefore, it is
still possible that we face the problem of stagnation when dealing with some more difficult datasets.
Moreover, the developed EHHO achieves good results for parameter identification of photovoltaic
cells and modules, and its application to other practical engineering problems is also one of the future
research directions.
Acknowledgment
This research is supported by the National Key R&D Program of China (2017YFB1400400), National
Natural Science Foundation of China (U1809209, 71803136), Key R&D Program Projects in Zhejiang
Province (2019C01041), Guangdong Natural Science Foundation (2018A030313339), Scientific
Research Team Project of Shenzhen Institute of Information Technology (SZIIT2019KJ022).
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