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Application of successive discretization algorithm for determining photovoltaic cells parameters
Energy Conversion and Management 196 (2019) 545–556
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Application of successive discretization algorithm for determining photovoltaic cells parameters
T
⁎
Daniel T. Cotfasa, , Adrian M. Deaconub, Petru A. Cotfasa a b
Department of Electronics and Computers, Transilvania University of Brasov, Eroilor 29, 500036 Brasov, Romania Department of Mathematics and Computer Sciences, Transilvania University of Brasov, Eroilor 29, 500036 Brasov, Romania
A R T I C LE I N FO
A B S T R A C T
Keywords: Photovoltaic cells and panels Parameters determining Algorithms
The extraction of the photovoltaic cells and panels parameters still represents a hot topic, despite the numerous methods proposed by researchers in scientific literature. In order to optimize the efficiency of photovoltaic cells and panels and to predict the energy they generate, it is useful to accurately calculate their parameters function of temperature and irradiance. This paper proposes the Successive discretization algorithm developed to extract five parameters of photovoltaic cells and panels using the current voltage characteristic and the one diode model. Three widely used standard datasets, one for the photovoltaic cell and two for photovoltaic panels, are utilized. Other three datasets are analyzed, one measured in laboratory conditions for the monocrystalline photovoltaic cell and two under natural sunlight for the monocrystalline photovoltaic panel. The proposed algorithm proves its performance through comparison with other over twenty accepted methods used in specialized literature. The comparison is made for five parameters and the root mean square error. The obtained results demonstrate that the method presented is one of the best and it increases the determination accuracy of important photovoltaic cells’ parameters. Thus, the Successive discretization algorithm is a very good candidate as tool for extracting the photovoltaic cells and panels parameters.
1. Introduction The European Commission, EC, has established roadmaps for the next years to improve living standards taking into account the climate changes and its influencing factors. So, the EC aims to reduce the greenhouse gas emissions by 20% until 2020 in comparison with 1990 [1], to double this reduction in greenhouse gas emissions until 2030, to achieve a percentage of 27% from energy consumption as renewable energy [2], and the ratio must reach 85%–90% by 2050 [3]. EC also encourages the member states to follow Denmark, New Zeeland, Carinthia, Vancouver and others in implementing projects that aim to achieve 100% renewable energy being used in all sectors until 2050 or even sooner [4]. Solar energy converted in electric and thermal energy is becoming increasingly important and therefore it represents a main sector of renewable energy. Now, it is almost double in comparison with the one obtained from wind [5] and it is estimated that it will reach 69% from all renewable types by 2050 [6]. Accurate values of the photovoltaic cells or panels’ parameters function of temperature and irradiance are essential for estimating the generated energy of photovoltaic farms and at the same time for the
⁎
improvement of photovoltaic cells performance. The five parameters of the photovoltaic cells: photogenerated current Iph, reverse saturation current Io, ideality factor series of diode n, and parasitic resistances: series Rs and shunt Rsh are determined using the current-voltage characteristic I-V [7] and the equivalent circuits of the photovoltaic cells for one diode model [8]. There are many methods to determine the parameters of photovoltaic cells: some of them are used to determine all five parameters, others only one or more, but not all five [8]. These methods are based on statistical functions, non-linear regression, a mixed approach between theoretical and graphical analysis of the I-V characteristic [8], and metaheuristic methods, such as: Genetic Algorithm [9], Genetic algorithm with convex combination crossover [10], Particle Swarm Optimization [11], Harmony Search [12], Simulated annealing [13], Cat Swarm Optimization [14], Artificial Bee Colony [15], Chaotic Improved Artificial Bee Colony [16], Imperialist Competitive Algorithm [17], Teaching Learning Based Optimization [18], Water Cycle Algorithm [19], Jaya Algorithm [20], Flower Pollination Algorithm [21], Cuckoo Search [22], Bacterial Foraging Algorithm [23], Whale Optimization Algorithm [24], Moth-Flame Optimization [25], Gravitational Search Algorithm [26], Penalty-Based Differential Evolution [27],
Corresponding author. E-mail addresses: [email protected] (D.T. Cotfas), [email protected] (A.M. Deaconu), [email protected] (P.A. Cotfas).
https://doi.org/10.1016/j.enconman.2019.06.037 Received 9 April 2019; Received in revised form 9 June 2019; Accepted 16 June 2019 0196-8904/ © 2019 Published by Elsevier Ltd.
Energy Conversion and Management 196 (2019) 545–556
D.T. Cotfas, et al.
Fig. 1. Equivalent circuits of photovoltaic cells: a) One diode model; b) Two diodes model.
successive discretization. The implemented SDA algorithm can be utilized to find the global optimum for complex problems with numerous local minima. The other contributions are: comparison of the SDA performance with that of other algorithms from the best ones to some less performant, using the same I,V data from [33] for a photovoltaic cell and two PV panels and using the SDA algorithm to calculate the parameters of one photovoltaic cell, and one PV panel at different temperature and irradiance values. The results are compared with the ones obtained using two implemented algorithms – genetic algorithm and analytical five point method. The results obtained demonstrate that the SDA algorithm is the best in most cases under study or one of the best algorithms in the others. Thus, the SDA algorithm proposed is one step forward in solving both challenges: determining accurately and relatively fast the parameters of the photovoltaic cells. The paper is structured in four sections. The first is the introduction, the second briefly describes the photovoltaic models, the experimental set up and the SDA algorithm. The results obtained for the five photovoltaic cells and panels parameters are presented and discussed in the third section. In the first stage the SDA is applied for RTC France photovoltaic cell and for two panels: PWP201 and STM6-40, both made of 36 photovoltaic cells connected in series. The second stage is dedicated to using the SDA algorithm for a monocrystalline photovoltaic cell, mSi and monocrystalline photovoltaic panel, mSiPV. The last section presents the conclusions and future work.
Table 1 The monocrystalline photovoltaic panel parameters. No. cells
Isc [A]
Voc [V]
Imax [A]
Vmax [V]
Pmax [W]
18
1.62
10.4
1.47
8.19
12
Symbiotic Organisms Search Algorithm [28], Improved Cuckoo Search Algorithm [29], etc. The analytical five point method developed by Chan et al. is the most often used amongst the analytical methods. The PV parameters are calculated using the equations obtained for Isc, Voc, and Pmax points and the slopes around Isc and Voc for determining the Rsh and Rs [30]. The method is fast, but the accuracy is dependent on the correctness of the chosen points and the noise of the measurement [31]. The non-linear regression methods, such as: Newton–Raphson, Levenberg–Marquardt, and others are efficient methods, all the points on the I-V characteristic being taken into account to determine the photovoltaic parameters, but the results are dependent on the initial chosen values of the parameters and the number of I-V points [32]. The metaheuristic methods are inspired by natural phenomena. They are suitable to determine the PV parameters because the problem which has to be solved is non-linear multi-modal. The issue arising in this context is the necessary time they require to achieve the convergent, which increases significantly if the tuning of control parameters is incorrectly chosen [33]. Combining two or more metaheuristic algorithms is another way to improve the accuracy of parameters determination, for example Hybrid Adaptive Nelder-Mead Simplex Algorithm [31] and Teaching–learning–based artificial bee colony [34]. The discretization techniques can be used to solve different problems [35], among which the optimizing ones [36]. The novelty of the paper consists in implementing and using the Successive Discretization Algorithm (SDA) for the first time to determine the parameters of photovoltaic cells and panels using the one diode model. The solution obtained by the discretization algorithm is improved by using the
2. Models and methods The photovoltaic cells and panels can be analyzed from two perspectives: in static and dynamic regime [37]. Most methods and algorithms are developed to determine the parameters of the photovoltaic cells in static regime. The equivalent circuit in static regime used for the photovoltaic cells derives from the models used.
Fig. 2. The measurement systems: a) Solar simulator; (b) Outdoor measurement system. 546
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Table 2 Parameters range of the photovoltaic cell and panels. Parameters
Iph [A] Io [μA] n Rs [Ω] Rsh [Ω]
RTC France photovoltaic cell
PWP201 panel
STM6-40 panel
Lower value
Upper value
Lower value
Upper value
Lower value
Upper value
0 0 1 0 0
1 1 2 0.5 100
0 0 1 0 0
2 50 50 2 2000
0 0 1 0 0
2 50 60 0.36 1000
where VT represents the thermal voltage, VT = kT/q, k is the Boltzmann constant, T is the temperature and q is the elementary electrical charge. The mathematical model for the photovoltaic panel with the photovoltaic cells connected in series is described in Eq. (2), [26].
I = Iph − Io ⎛e ⎝
Fig. 3. The discretization of a 2D continuous function f on an interval [a, b].
The mathematical model of photovoltaic cells based on the equivalent circuit is function of the conduction mechanisms. The mathematical model using the one diode model – the conduction mechanism being diffusion [38], with parasitic resistance, is described by Eq.1 and the equivalent circuit is presented in Fig. 1a. V + IRs nVT
− 1⎞ − ⎠
V + IRs Rsh
− 1⎞ − ⎠
V + Ns IRs Ns Rsh
(2)
where Ns represents the number of photovoltaic cells connected in series. The two diodes model which takes into account other conduction mechanisms, such as recombination [39], is described by Eq. (3) and the equivalent circuit in Fig. 1b.
2.1. Models
I = Iph − Io ⎛e ⎝
V + Ns IRs nNs VT
V + IRs V + IRs V + IRs I = Iph − Iod ⎛e nd VT − 1⎞ − Ior ⎛e nr VT − 1⎞ − Rsh ⎝ ⎠ ⎝ ⎠
(3)
where Iod and Ior represent the reverse saturation current for diffusion and for recombination, nd and nr represent the ideality factor of diode for the same two mechanisms.
(1)
Fig. 4. Flowchart of Successive Discretization Algorithm. 547
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photovoltaic panel. The mainly parameters of the photovoltaic panel measured at Standard Test Conditions are presented in Table 1. The RELab system is used to measure the I-V characteristics in lab conditions and a system based on NI c-RIO embedded controller is used for outside conditions. The RELab system is described in more detail in paper [42]. The second system is based on NI cRIO 9074 platform, developed around a 400 MHz processor and a FPGA Spartan-3 chip. The used I/O modules are: NI 9215 and NI 9225 for the PV voltage and current measurements; NI 9211 and NI 9213 for the temperature measurements based on thermocouples. More details about the performances of the system are presented in paper [43]. The temperature of photovoltaic cells and irradiance are measured simultaneously with the I-V characteristic. The temperature of the photovoltaic cell is measured on the back of the cell using a temperature sensor with ± 0.5 °C accuracy. The temperature of the panel is measured using three thermocouples mounted on the back of the panel. The back panel temperature is considered the average of the three temperatures. The solar irradiance is measured using a SPN1 pyranometer, mounted in the same plane with the PV panels. The Solar simulator, A4-LA200-2, AAA class, is used to illuminate the photovoltaic cells, see Fig. 2a. The homogeneous illuminated area is 5 cm × 5 cm. The photovoltaic mSi panel is measured under natural sunlight conditions, see Fig. 2b, on a clear sky day.
Table 3 (I,V) points of RTC photovoltaic cell and the current calculated using SDA algorithm. Measured data
Current calculated
Error
V [V]
I [A]
Ie [A]
Rerr
−0.2057 −0.1291 −0.0588 0.0057 0.0646 0.1185 0.1678 0.2132 0.2545 0.2924 0.3269 0.3585 0.3873 0.4137 0.4373 0.4590 0.4784 0.4960 0.5119 0.5265 0.5398 0.5521 0.5633 0.5736 0.5833 0.5900
0.7640 0.7620 0.7605 0.7605 0.7600 0.7590 0.7570 0.7570 0.7555 0.7540 0.7505 0.7465 0.7385 0.7280 0.7065 0.6755 0.6320 0.5730 0.4990 0.4130 0.3165 0.2120 0.1035 −0.0100 −0.1230 −0.2100
0.764078 0.762656 0.761352 0.760153 0.759057 0.758046 0.757098 0.756149 0.755096 0.753673 0.75140 0.747361 0.740121 0.727381 0.706965 0.675268 0.630743 0.571913 0.499596 0.413644 0.317512 0.212163 0.102263 −0.00870953 −0.125507 −0.208485
−0.000078 −0.000656 −0.000852 0.000347 0.000943 0.000954 −0.000098 0.000851 0.000404 0.000327 −0.0009 −0.000861 −0.001621 0.000619 −0.000465 0.000232 0.001257 0.001087 −0.000596 −0.000644 −0.001012 −0.000163 0.001237 −0.00129047 0.002507 −0.001515
2.3. Successive discretization algorithm The algorithm developed to extract the photovoltaic cells and panel parameters based on one diode or two diodes models described by Eqs. (3) and (4) used the successive discretization method.
There are other mechanisms, such as: thermionic, tunneling processes and the space-charge limited current mechanism which can be considered [8]. The one diode model is the most widely used model to characterize the photovoltaic cells because it is simpler than the double diodes model, the results obtained are good, and for thin film photovoltaic cells it is the most accurate model [40].
2.3.1. Discretization process The idea behind the algorithm for determining the solar cell parameters is discretization. In Mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step towards making them suitable for numerical evaluation or implementation on digital computers. Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce this amount of error to a level considered negligible for the modeling purpose. A 2-dimensional example is considered in order to illustrate the idea of discretization. The discretization can also be applied in 4, 5, 6 and even 8 dimensions, but the basic idea remains the same. Let f:[a,b] → R be a 2D continuous function on the interval [a, b],
2.2. Experiment setup The I-V characteristics of the photovoltaic cells can be measured using different techniques: capacitor, MOSFET and electronic load [41]. The capacitor technique is used in this paper to measure the I-V characteristics for mSi commercial photovoltaic cell and mSiPV Table 4 Comparison of different the five parameters extraction techniques for RTC. Algorithm
Iph [A]
Io [μA]
n
Rs [Ω]
Rsh [Ω]
RMSE
Rank
SDA CWOA [24] CSO [14] NM-MPSO [48] STLBO [46] ABC-DE [47] ImCSA[29] ISCE [33] EHA-NMS [31] Rcr-IJADE [50] BMO [49] SOS [28] MABC [51] ABC [16] CPSO [52] 5P [42] PS [53] SA [13] GA [56]
0.760773 0.76077 0.76078 0.76078 0.76078 0.76077 0.760776 0.76077553 0.76077553 0.76077553 0.76077 0.7608 0.760779 0.7608 0.7607 0.7612 0.7617 0.762 0.7619
0.3244462 0.3239 0.3230 0.32306 0.32302 0.32302 0.323021 0.32302083 0.32302080 0.3230208 0.32479 0.3579 0.321323 0.3251 0.4000 0.1966 0.9980 0.4798 0.8087
1.48164 1.4812 1.48118 1.48120 1.48114 1.47986 1.481781 1.48118360 1.48118359 1.4811836 1.48173 1.4916 1.481385 1.4817 1.5033 1.43 1.6000 1.5172 1.5751
0.03636 0.03636 0.03638 0.03638 0.03638 0.03637 0.036377 0.03637709 0.03637709 0.03637709 0.03636 0.0359 0.036389 0.0364 0.0354 0.042 0.0313 0.0345 0.0299
53.842702 53.7987 53.7185 53.7222 53.7187 53.7185 53.718524 53.71852771 53.71852139 53.718525 53.8716 53.7835 53.39999 53.6433 59.012 95.28 64.1026 43.1034 42.3729
9.8598E–04 9.8602E–04 9.8602E–04 9.8602E–04 9.8602E–04 9.8602E–04 9.8602E–04 9.860219E–04 9.860219E–04 9.860219E–04 9.8608E–04 9.8609E–04 9.861E–04 9.862E–04 1.39E–03 8.674E–03 0.01494 0.019 0.01908
1 2 2 2 2 2 2 3 3 3 4 5 6 7 8 9 10 11 12
548
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Fig. 5. Data of RTC France photovoltaic cell: a) I-V characteristics measured and calculated using SDA algorithm; b) comparison between the absolute current errors obtained using the SDA and CWOA algorithms.
Considering Eq. (1), Eq. (6) becomes:
where a < b. Instead of the infinite number points of the function f is restricted only to d > 0 representatives. In order to do that, d values vi (i = 1, …, d) are considered from the interval J = [a, b] so that:
FI1, V (Iph, Io, n, Rs , Rsh) = 0
When solving the solar cell Eq. (7) the goal is to find the photovoltaic cell parameters Iph, Io, n, Rs and Rsh that minimize the root mean square error (RMSE), see Eq. (8) for the given pairs of current and voltage (I, V).
(4)
a < v1 < v2 < . .. < vd < b The set of d representatives of the function points is
{Pj (vj, f (vj ))|j = 1, ...,d}
(5)
Moreover, the set of line segments |PjPj+1|, j = 0, …, d form a linear approximation of the function f, where P0(a, f(a)) and Pd+1(b, f(b)), see Fig. 3. The process of discretization for the single-diode model with 5 cell parameters is presented. Eq. (1) is rewritten as follows in order to obtain the objective function:
FI1, V (Iph, Io, n, Rs , Rsh) = Iph − Io ⎛e ⎝
(7)
V + IRs nVT
− 1⎞ − ⎠
V + IRs −I Rsh
RMSE (Iph, Io, n, Rs , Rsh) =
∑(I , V ) (FI1, V (Iph, Io, n, Rs , Rsh ))2 p
(8)
where p is the number of the (I,V) pairs. The intervals J1 = [a1, b1], J2 = [a2, b2], J3 = [a3, b3], J4 = [a4, b4] and J5 = [a5, b5] are considered the definition domains for the parameters Iph, Io, n, Rs and, respectively, Rsh. The values ai and bi will be exactly set in section 2.2. It is observed that the function FI,V1 is continuous on the 5-
(6) 549
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the inequalities:
Table 5 (I,V) points of PWP201 photovoltaic panel and the current calculated using SDA algorithm. Measured data
Current calculated
Error
V [V]
I [A]
Ie [A]
Rerr
0.1248 1.8093 3.3511 4.7622 6.0538 7.2364 8.3189 9.3097 10.2163 11.0449 11.8018 12.4929 13.1231 13.6983 14.2221 14.6995 15.1346 15.5311 15.8929 16.2229 16.5241 16.7987 17.0499 17.2793 17.4885
1.0315 1.0300 1.0260 1.0220 1.0180 1.0155 1.0140 1.0100 1.0035 0.9880 0.9630 0.9255 0.8725 0.8075 0.7265 0.6345 0.5345 0.4275 0.3185 0.2085 0.1010 −0.0080 −0.1110 −0.2090 −0.3030
1.029121304 1.027382551 1.025742673 1.024107493 1.022291675 1.019930167 1.01636231 1.010495202 1.000628010 0.984547554 0.959521101 0.922838546 0.872599672 0.807274453 0.728336716 0.637138148 0.536213013 0.429511044 0.318773970 0.207388828 0.096166399 −0.008326038 −0.110936938 −0.209247415 −0.300863286
0.002378700 0.002617450 0.000257327 −0.002107490 −0.004291680 −0.004430170 −0.002362310 −0.000495202 0.002871990 0.003452450 0.003478900 0.002661450 −0.000099700 0.000225547 −0.001836720 −0.002638150 −0.001713010 −0.002011040 −0.000273970 0.001111170 0.004833600 0.000326038 −0.000063100 0.000247415 −0.002136710
ai < v1i < v2i < . .. < v dii < bi
In order to get a uniform distribution of the points in the interval Ji the values v ij (j = 1, …, di) are considered as follows:
v ij = ai + jli
Current calculated
Error
(10)
where li is given by the following equation:
li =
bi − ai di + 1
(11)
In this case, it is easily observed that:
v1i
= ai + li and v dii = bi − li
(12)
The following set of quintets of photovoltaic cell parameters is considered:
G = {(v 1j1 , v j22 , v j33 , v j44 , v 5j5 )|ji = 1, ...,di , i = 1, ...,5}
RMSE (v k11 , v k22 , v k33 , v k44 , v k55 ) = min RMSE (g )
I [A]
Ie [A]
Rerr
0 0.118 2.237 5.434 7.26 9.68 11.59 12.6 13.37 14.09 14.88 15.59 16.4 16.71 16.98 17.13 17.32 17.91 19.08 21.02
1.663 1.663 1.661 1.653 1.65 1.645 1.64 1.636 1.629 1.619 1.597 1.581 1.542 1.524 1.5 1.485 1.465 1.388 1.118 0
1.663492501 1.663285403 1.659563333 1.653896827 1.650531876 1.645380036 1.639180671 1.633666529 1.627250289 1.618293153 1.603092126 1.58161313 1.542378468 1.521256585 1.499248941 1.485328855 1.465704792 1.387610684 1.118237171 −2.68E−08
0.002378700 0.002617450 0.000257327 −0.002107490 −0.004291680 −0.004430170 −0.002362310 −0.000495202 0.002871990 0.003452450 0.003478900 0.002661450 −0.000099700 0.000225547 −0.001836720 −0.002638150 −0.001713010 −0.002011040 −0.000273970 0.001111170
(14)
g∈G
v k11 ,
v k22 ,
v k33 ,
v k44 ,
v k55
The quintet ( ) is called an approximate solution of Eq. (7) out of d1 × d2 × d3 × d4 × d5 quintets. 2.3.2. Refining the solution by successive discretization The method to obtain the best approximate solution for Eq. (7) is improved in order to obtain a better solution (a refined one). The best s ≥ 1 quintets are considered gj = (v 1 j , v 2j , v 3 j , v 4 j , v 5 j ) , k1
k2
k3
k4
k5
(j = 1, …, s) in the set G, having the smallest RMSE errors. For each quintet gj the search of approximate solutions is refined for Eq. (7) in the neighborhood of gj :
J1j × J2j × J3j × J4j × J5j V [V]
(13)
The quintet (v k11 , v k22 , v k33 , v k44 , v k55 )is calculated so that:
Table 6 (I,V) points of STM6-40 photovoltaic panel and the current calculated using SDA algorithm Measured data
(9)
(15)
where
Ji j = [v i j − li , v i j + li], i = 1, ...,5 ki
ki
(16)
The process of discretization on each 5D interval from Eq. (15) (j = 1, …, s) is performed, resulting in s solutions for each interval. From the resulting s2 quintets are selected the best s solutions for Eq. (7) and applied again the method of refining. After r > 0 refining iterations is stopped and the best found quintet is taken (with the smallest RMSE error) as the final solution for Eq. (7). There are some special cases when the bounds of the definition interval are powers, i.e., [a, b] = [10c, 10d], where c and d are integer values, c < d. In this cases representatives in the interval [a, b] are considered of all powers. For instance, if [a, b] = [10−12, 10−6] the representatives are uniformly distributed around 10−11, 10−10, 10−9, 10−8 and 10−7. So, instead of linear discretization, exponential discretization is considered, i.e., instead of linear discretization on the interval [a, b], the linear discretization is applied on the interval [c, d]. In the end the representatives in [a, b] are powers of 10 in the interval (c, d). The Successive Discretization Algorithm for 5-parameter identification of solar cells is presented in the flowchart from Fig. 4. In the first iteration of the algorithm an initial discretization is performed (t = 1) on the interval J1 × J2 × J3 × J4 × J5. When an iteration of the algorithm is finished “s” best approximation solutions are selected and then in the next iteration of the algorithm t = s discretizations are performed in the vicinity of these “s” solutions (“s” is given). “no_of_iterations” is also given as an input to the algorithm. The identification of the solar cell parameters using successive discretization can be easily adapted to the two diode model with 7 parameters. The only difference is that the discretization is done in 8-dimensional spaces, but in this paper is focused only on the one diode
dimensional interval J1 × J2 × J3 × J4 × J5. This interval is called the discretization 5D interval. The discretization is applied to the function FI,V1 on the 5D discretization interval. The discretization will be done in a 6-dimensional space. The function is defined on a 5D interval and its representation is 6D. This is similar to the 2D case presented in Fig. 3 where the function f is defined on a 1D interval [a, b]. The 5D discretization interval is J1 × J2 × J3 × J4 × J5 where Ji = [ai, bi], i = 1,2,3,4,5, Ji is a simple 1D interval. The solution is searched in this 5D interval. It represents the given boundaries of the search for the algorithm. Five positive integers d1, d2, d3, d4, d5 ∈ N* are considered. For each interval Ji (i = 1, …, 5) di values are taken v ij (j = 1, …, di), respecting 550
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Table 7 Comparison of different five parameters extraction techniques for PWP201 Algorithm
Iph [A]
Io [μA]
n
Rs [Ω]
Rsh [Ω]
RMSE
Rank
SDA ISCE [34] EHA-NMS [31] Rcr-IJADE [49] MPCOA [45] SOS [28] SA [13] FPA [55] ABC-DE [47] 5P [41] PS [53]
1.030517 1.0305143 1.0305143 1.0305143 1.03188 1.0303 1.0331 1.032091 1.0318 1.034 1.0313
3.4816148 3.48226304 3.48226292 3.4822629 3.37370 3.5616 3.6642 3.047538 3.2774 3.571 3.1756
48.598941 48.642835 48.642835 48.642835 48.50646 48.7291 48.8211 48.13128 48.3948 48.71 48.2889
1.201288 1.201271 1.201271 1.201271 1.20295 1.1991 1.1989 1.217583 1.2062 1.206 1.2053
981.599618 981.98228038 981.98222618 981.98216 849.6927 1017.7 833.3333 811.3721 845.2495 1123.00 714.2857
2.4250749E–03 2.425075E–03 2.425075E–03 2.425075E–03 2.4251E–03 2.4251E–03 2.7E–03 2.7425E–03 3.8855E–03 4.019E–03 0.0118
1 1 1 1 2 2 3 4 5 6 7
Table 8 Comparison of different five parameters extraction techniques for STM6-40 Algorithm
Iph [A]
Io [μA]
n
Rs [mΩ]
Rsh [Ω]
RMSE
Rank
SDA ISCE [34] EHA-NMS [31] Rcr-IJADE [49] TMP [53] 5P [41]
1.663949 1.66390478 1.66390478 1.66390480 1.6635 1.666
1.7025818 1.73865691 1.73865691 1.73865690 1.4142 2.654E-4
54.642016 54.73090512 54.73090512 54.73090512 53.9496 33.35
156.421 153.855765 153.855765 153.85577004 175.644 1.165E+3
570.227443 573.41858868 573.41858868 573.41858868 555.084 741.16
1.72964500E–03 1.72981371E–03 1.72981371E–03 1.72981371E–03 3.11854870E–03 7.17643E–02
1 2 2 2 3 4
between RMSE values obtained with the algorithms ranked on the first six places are very small. The second parameter taken into account to achieve a better image of algorithms performance is the number of iterations. The SDA algorithm performs 31 iterations (corresponding to a total of 31 discretizations). The analysis of the five parameters leads to the following results: Iph is the most stable parameter, its maximum variation from the optimal one, considered as the value given by SDA algorithm, is 0.16% if all algorithms are considered and 0.001% if only the first twelve algorithms are considered with RMSE under 0.001; Io has the highest variation, which is 307% when all algorithms are taken into consideration and 0.972% for the first twelve algorithms. It is the most sensitive parameter. The shunt resistance also has a high variation, the maximum being 76.96%, for series resistance 17.76% and for the ideality factor of diode 8%. If the first twelve algorithms are considered, the parameters variation is: Rsh – 0.829%, Rs – 0.110%, and n – 0.108%. The algorithms CPSO, 5P, PS, GA and SA have the RMSE higher than 0.001. The results obtained for the five parameters of the photovoltaic cells using these algorithms show their weakness. The high matching between the measured points of the I-V characteristic and calculated ones using SDA algorithm is shown in Fig. 5a. The comparison between absolute current errors obtained using the first two algorithms SDA and CWOA, Fig. 5b, shows that there isn’t any best algorithm for the values around the open circuit voltage.
model. 3. Results and discussion The important parameters from the one diode model obtained using the SDA algorithm, which was implemented in Visual C++ 2017, for the benchmark RTC France photovoltaic cell [44] and for benchmark two panels: PWP201[44] and STM6-40 [45] are compared with the ones obtained by some of the best metaheuristic algorithms, CWOA [24], CSO [14], NM-MPSO [48], STLBO [35], ABC-DE [47], ISCE [33], EHA-NMS [31], Rcr-IJADE [50], BMO [49], SOS [28], ImCSA[29], MABC [51], ABC [16], FPA [55], MPCOA [54] to prove the performance of the SDA algorithm. These results are also compared with the ones from less performant algorithms, CPSO [52], SA [13], GA [56] and PS [53] and the analytical five point method, 5P [30,42]. It is very important to identify the differences between the values of the photovoltaic cell parameters obtained with SDA and SA, PS and 5P, respectively. The lower and upper values for the RTC photovoltaic cell and PWP201 panel are chosen to be similar to [24,33,42] and for STM6-40 to be similar to [33,45], see Table 2. These values were chosen in order to make a fair comparison. 3.1. RTC photovoltaic cell The current-voltage (I,V) points taken into account to calculate the parameters of RTC photovoltaic cell are presented in Table 2, and they are used to this purpose in a lot of papers [13,14,16,24,31,33,46–53]. The I-V characteristic of the RTC is measured at 33 °C. The current calculated using the SDA algorithm and the relative error, Eq. (18), calculated by subtracting the calculated from the measured current values are also presented in Table 3.
R err = I − Ic
3.2. Results for photovoltaic panels In Table 5, the I-V data, measured and calculated, of PWP201 photovoltaic panel are listed and for STM6-40 in Table 6 respectively. These points are measured at 45 °C for PWP201 and 51 °C for STM6-40 [45]. Both of them have 36 photovoltaic cells connected in series. PWP201 panel is made of polycrystalline photovoltaic cells, and STM640 panel used monocrystalline photovoltaic cells. The parameters of the PWP201 photovoltaic panel extracted using some algorithms are listed in Table 7 and for STM6-40 in Table 8. By comparing RMSE obtained with SDA algorithm and data from other research, it is obvious that the SDA algorithm is the best. The number of discretizations used is 31. RMSE for SDA, ISCE, EHA-NMS and Rcr-IJADE are practically the same and the variation of the PWP201 photovoltaic panel parameters
(18)
The parameters of the RTC photovoltaic cell extracted using some algorithms are listed in Table 4. Firstly, the root mean square error, RMSE, is considered to prove the superior performance of the SDA algorithm against the ones obtained with the algorithms mentioned in Table 3. The ranking obtained using this parameter for comparison shows the superiority of the SDA algorithm. However, the difference 551
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Fig. 6. a) I-V characteristics of PWP201 photovoltaic panel measured and calculated using SDA algorithm; b) I-V characteristics of STM6-40 photovoltaic panel measured and calculated using SDA algorithm.
such as ABC-DE, 5P and PS the variation becomes significant. Rsh varies from the optimum value with more than 27%. RMSE for SDA, ISCE, EHA-NMS and Rcr-IJADE is the same. The STM6-40 photovoltaic panel parameters are practically the same for these three algorithms. The parameters obtained using TMP have: a small variation for Iph and n, under 1.5% in comparison with the optimal one, and a significant variation for Rsh, Io and Rs, 4%, 14% and 18% respectively. The 5P method has a very low performance for STM6-40 panel, overestimating the shunt and series resistance and underestimating the reverse saturation current and ideality factor of diode. This is due to the lack of points around the Voc where the first value of Rs is calculated. In such cases the 5P method cannot be used. The 5P also used the points around Isc to calculate Rsh, and if there is a lack of points around Isc the 5P algorithm has poor performance.
Table 9 Parameters range of the photovoltaic mSi cell and mSi PV panel. Parameters
Iph [A] Io [A] n Rs [Ω] Rsh [Ω]
mSi photovoltaic cell
mSi PV panel
Lower value
Upper value
Lower value
Upper value
0 10−12 1 0 0
1 10−6 2 0.2 500
0 10−12 18 0 0
2 10−6 36 2 5000
from the reference obtained using SDA algorithm is very small. The variation increases in the cases of MPCOP, SA and FPA, even if the RMSE increase is small. The highest variation is for the shunt resistance, followed by reverse saturation current. For less performant algorithms, 552
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Fig. 7. Data of the mSi photovoltaic cell: a) I-V characteristics measured and calculated using SDA algorithm; b) Comparison of the absolute current errors obtained for using the SDA, GA algorithms and 5P methods. Table 10 Comparison of different five parameters extraction techniques for mSi photovoltaic cell Algorithm
Iph [A]
Io [μA]
n
Rs [Ω]
Rsh [Ω]
RMSE
Rank
SDA GA 5P
0.425752 0.4256882 0.4255
0.5168535 0.8383311 0.30645567
1.679294 1.73926 1.618311
0.091316 0.0859435 0.10352224
99.136671 123.3659 145.222
5.63097253E–04 6.97414541E–04 2.25639649E–03
1 2 3
Table 11 Comparison of different five parameters extraction techniques for mSiPV photovoltaic panel Case
Algorithm
Iph [A]
Io [nA]
n
Rs [Ω]
Rsh [Ω]
RMSE
Rank
A
SDA GA 5P
1.224206 1.223082 1.224
0.4677 4.988143 0. 334
18.352944 20.5289 18.02
0.144083 0.02147292 0.134
1544.361724 1765.388 1242.91
2.765E–03 4.96271E–03 6.953433E–03
1 2 3
B
SDA GA [48] 5P [48]
1.627297 1.62905 1.6291
0.6451 3.133213 6.82
18.120725 19.50277 20.235
0.136626 0.0757812 0.101
2998.9 1130.687 804.71
4.111E–03 6.707233E–03 2.86E–02
1 2 3
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Fig. 8. Absolute current error comparison between SDA, GA and 5P: a) case A; b) case B.
The highly matching between the measured points of the I-V characteristic and calculated ones using SDA algorithm is shown in Fig. 6a for PWP201 and Fig. 6b for STM6-40.
(case B). The temperature of the photovoltaic panel is obtained by converting the temperature measured on the back of the photovoltaic panel according to King et al. and Du et al. [57,58]. The confidence intervals of the photovoltaic cell and panel parameters are presented in Table 9. The I-V characteristics of the mSi photovoltaic cell, compared with the ones calculated using SDA algorithm, are presented in Fig. 7a. The absolute current errors obtained using SDA and GA algorithms [56] and 5P method function of the voltage, are shown in Fig. 7.b. The distribution illustrates that the highest difference appears around the open circuit voltage, especially for the 5P method. In the first half the 5P method has very good results. The comparison of the parameters obtained using SDA and GA algorithms and 5P method, Table 10, shows a similar behaviour with the
3.3. Experimental results The performance of the SDA algorithm being proven superior by comparison with other algorithms, in all cases taken into consideration, it is applied to calculate the parameters for mSi commercial photovoltaic cell, sizes 3 × 3 cm and for mSi photovoltaic panel with 18 cells connected in series. The I-V characteristic for mSi photovoltaic cell is measured at 1000 W/m2, the temperature of the photovoltaic cell being 27 °C and for mSiPV photovoltaic panel at different temperatures and solar radiation: 35 °C and 700 W/m2 (case A); 40 °C and 983 W/m2 554
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one of the RTC photovoltaic cell. Iph is almost the same, while Io, Rsh and n have high variations. It can be observed that the SDA algorithm has the best results. The number of (I,V) points of the I-V characteristic measured for the mSiPV panel are 1001. Table 11 shows the optimal parameters in the two cases considered, A and B, comparatively. SDA algorithm gives the lowest value for RMSE. The distribution of the absolute current errors for the mSIPV panel in the two cases is shown in Fig. 8a for case A and in Fig. 8b for case B. The absolute errors are large around the open circuit voltage and also the 5P method gives wrong results around the open circuit voltage, the same as in mSi photovoltaic cell.
with convex combination crossover. Int J Ambient Energy 2018:1–8. [11] Ye M, Wang X, Xu Y. Parameter extraction of solar cells using particle swarm optimization. J Appl Phys 2009;105:094502–94508. [12] Askarzadeh A, Rezazadeh A. Parameter identification for solar cell models using harmony search-based algorithms. Sol Energy 2012;86:3241–9. [13] El-Naggar KM, AlRashidi MR, AlHajri MF, Al-Othman AK. Simulated annealing algorithm for photovoltaic parameters identification. Sol Energy 2012;86:266–74. [14] Guo L, Meng Z, Sun Y, Wang L. Parameter identification and sensitivity analysis of solar cell models with cat swarm optimization algorithm. Energy Convers Manage 2016;108:520–8. [15] Askarzadeh A, Rezazadeh A. Artificial bee swarm optimization algorithm for parameters identification of solar cell models. Appl Energy 2013;102:943–9. [16] Oliva D, Ewees AA, Aziz MAE, Hassanien AE, Peréz-Cisneros M. A Chaotic improved artificial bee colony for parameter estimation of photovoltaic cells. Energies 2017;10:865. [17] Fathy A, Rezk H. Parameter estimation of photovoltaic system using imperialist competitive algorithm. Renewable Energy 2017;111:307–20. [18] Chen X, Yu K, Du W, Zhao W, Liu G. Parameters identification of solar cell models using generalized oppositional teaching learning based optimization. Energy 2016;99:170–80. [19] Kler D, Sharma P, Banerjee A, Rana K, Kumar V. PV cell and module efficient parameters estimation using evaporation rate based water cycle algorithm. Swarm Evol Comput 2017;35:93–110. [20] Yu K, Liang J, Qu B, Chen X, Wang H. Parameters identification of photovoltaic models using an improved JAYA optimization algorithm. Energy Convers Manage 2017;150:742–53. [21] Ram JP, Babu TS, Dragicevic T, Rajasekar N. A new hybrid bee pollinator flower pollination algorithm for solar PV parameter estimation. Energy Convers Manage 2017;135:463–76. [22] Ma J, Ting T, Man KL, Zhang N, Guan S-U, Wong PW. Parameter estimation of photovoltaic models via cuckoo search. J Appl Math 2013;2013. [23] Awadallah MA. Variations of the bacterial foraging algorithm for the extraction of PV module parameters from nameplate data. Energy Convers Manage 2016;113:312–20. [24] Oliva D, El Aziz MA, Hassanien AE. Parameter estimation of photovoltaic cells using an improved chaotic whale optimization algorithm. Appl Energy 2017;200:141–54. [25] Allam D, Yousri D, Eteiba M. Parameters extraction of the three diode model for the multi-crystalline solar cell/module using Moth-Flame Optimization Algorithm. Energy Convers Manage 2016;123:535–48. [26] Sarjila R, Ravi K, Edward JB, Kumar KS, Prasad A. Parameter extraction of solar photovoltaic modules using gravitational search algorithm. J Electr Comput Eng 2016:2143572. [27] Ishaque K, Salam Z, Mekhilef S, Shamsudin A. Parameter extraction of solar photovoltaic modules using penalty-based differential evolution. Appl Energy 2012;99:297–308. [28] Xiong G, Zhang J, Yuan X, Shi D, He Y, Xiong G, et al. Application of symbiotic organisms search algorithm for parameter extraction of solar cell models. Appl Sci 2018;8:1–18. [29] Kang T, Yao J, Jin M, Yang S, Duong T, Kang T, et al. A novel improved cuckoo search algorithm for parameter estimation of photovoltaic (PV) models. Energies 2018;11:1–31. [30] Chan DSH, Phillips JR, Phang JCH. A comparative study of extraction methods for solar cell model parameters. Solid-State Electron 1986;29:329–37. [31] Chen Z, Wu L, Lin P, Wu Y, Cheng S. Parameters identification of photovoltaic models using hybrid adaptive Nelder-Mead simplex algorithm based on eagle strategy. Appl Energy 2016;182:47–57. [32] Dkhichi F, Oukarfi B, Fakkar A, Belbounaguia N. Parameter identification of solar cell model using Levenberg–Marquardt algorithm combined with simulated annealing. Sol Energy 2014;110:781–8. [33] Gao X, Cui Y, Hu J, Xu G, Wang Z, Qu J, et al. Parameter extraction of solar cell models using improved shuffled complex evolution algorithm. Energy Convers Manage 2018;157:460–79. [34] Chen Xu, Xu B, Mei C, Ding Y, Li K. Teaching–learning–based artificial bee colony for solar photovoltaic parameter estimation. Appl Energy 2018;212:1578–88. [35] Kotsiantis S, Kanellopoulos D. Discretization techniques: a recent survey. GESTS Int Trans Comput Sci Eng 2006;32:47–58. [36] Garcia S, Luengo J, Saez JA, Lopez V, Herrera F. A survey of discretization techniques: taxonomy and empirical analysis in supervised learning. IEEE Trans Knowl Data Eng 2013;25:734–50. [37] Cotfas DT, Cotfas PA, Kaplanis S. Methods and techniques to determine the dynamic parameters of solar cells: review. Renewable Sustainable Energy Rev 2016;61:213–21. [38] Pallarès J, Cabré R, Marsal LF. A compact equivalent circuit for the dark currentvoltage characteristics of nonideal solar cells. J Appl Phys 2006;100:084513-1-5. [39] Chen Y, Sun Y, Meng Z. An improved explicit double-diode model of solar cells: fitness verification and parameter extraction. Energy Convers Manage 2018;169:345–58. [40] Et-torabi K, Nassar-eddine I, Obbadi A, Errami Y, Rmaily R, Sahnoun S, et al. Parameters estimation of the single and double diode photovoltaic models using a Gauss-Seidel algorithm and analytical method: a comparative study. Energy Convers Manage 2017;148:1041–54. [41] Cotfas DT, Cotfas PA, Ursutiu D, Samoila C. Current-voltage characteristic raising techniques for solar cells. Comparisons and applications. In Optimization of Electrical and Electronic Equipment (OPTIM), 2010 12th International Conference on, Brasov, Romania, 20–22 May 2010. [42] Cotfas PA, Cotfas DT. Design and implementation of RELab system to study the
4. Conclusions The SDA algorithm is described and used to calculate the five important parameters of the photovoltaic cells and panels. The performance of the proposed SDA algorithm is proven through the results obtained in comparison with the other existent algorithms from research literature for all devices under test, RTC, PWP201 and STM6-40 panels. Also, the SDA algorithm has proven its performance in comparison with GA algorithm and 5P method implemented by the authors for experimental I-V characteristics measured for: mSi photovoltaic cell and mSIPV panel at different temperatures and irradiances. By analyzing the results obtained for the parameters for all devices under test, it can be concluded that: the photogenerated current is the most stable parameter, and the reverse saturation current and the shunt resistance have high variations. They can be either overestimated or underestimated, especially when less performance algorithms are used. The SDA algorithm performance (the best), the lowest RMSE and number of iterations, recommend it for the determination of the main photovoltaic cells and panels parameters under different conditions. The calculation of the Perovskite, multijunction and thin film photovoltaic cells and panels parameters using SDA algorithm and comparison with the results obtained by other accepted algorithms will be the target for future work. Another intended direction of further research is the calculation of the photovoltaic cells’ and panels’ parameters using two diodes models with SDA algorithm. Also, the hybrid algorithms, which use one or more algorithms, have a high potential to determine the parameters of the photovoltaic cells and panels with high accuracy and therefore will be considered in future work. Declaration of Competing Interest None. References [1] https://ec.europa.eu/energy/en/topics/renewable-energy (accessed on 18.03. 2019). [2] Knopf B, Nahmmacher P, Schmid E. The European renewable energy target for 2030 – an impact assessment of the electricity sector. Energy Policy 2015;85:50–60. [3] https://ec.europa.eu/energy/sites/ener/files/documents/2012_energy_roadmap_ 2050_en_0.pdf (accessed on 19.03.2019). [4] http://www.renewables100.org/en/programs/go-100-project/ (accessed on 21.03. 2019). [5] Sawin JL, Seyboth K, Sverrisson F. Renewables 2018 Global Status Report. Paris, France: REN21 Secretariat; 2018. [6] Ram M, Bogdanov D, Aghahosseini A, Oyewo AS, Gulagi A, Child M, et al. Global Energy System Based on 100% Renewable Energy-power Sector. Berlin: Study by Lappeenranta University of Technology and Energy Watch Grup; 2017. [7] Abbassi R, Abbassi A, Jemli M, Chebbi S. Identification of unknown parameters of solar cell models: a comprehensive overview of available approaches. Renewable Sustainable Energy Rev 2018;90:453–74. [8] Cotfas DT, Cotfas PA, Kaplanis S. Methods to determine the dc parameters of solar cells: a critical review. Renewable Sustainable Energy Rev 2013;28:588–96. [9] Sellami A, Bouaïcha M. Application of the genetic algorithms for identifying the electrical parameters of PV solar generators. In: Kosyachenko LA, editor. Solar Cellssilicon Wafer-based Technologies. InTech; 2011. p. 349–64. [10] Hamid N, Abounacer R, Idali Oumhand M, Feddaoui M, Agliz D. Parameters identification of photovoltaic solar cells and module using the genetic algorithm
555
Energy Conversion and Management 196 (2019) 545–556
D.T. Cotfas, et al.
[51] Jamadi M, Merrikh-Bayat F, Bigdeli M. Very accurate parameter estimation of single- and double-diode solar cell models using a modified artificial bee colony algorithm. Int J Energy Environ Eng 2016;7:13–25. [52] Huang W, Jiang C, Xue L, Song. Extracting solar cell model parameters based on chaos particle swarm algorithm. In: 2011 International conference on electric information and control engineering; 2011. p. 398–402. [53] AlHajri MF, El-Naggar KM, AlRashidi MR, Al-Othman AK. Optimal extraction of solar cell parameters using pattern search. Renewable Energy 2012;44:238–45. [54] Yuan X, Xiang Y, He Y. Parameter extraction of solar cell models using mutative scale parallel chaos optimization algorithm. Sol Energy 2014;108:238–51. [55] AlRashidi MR, AlHajri MF, El-Naggar KM, Al-Othman AK. A new estimation approach for determining the I-V characteristics of solar cells. Sol Energy 2011;85:1543–50. [56] Cotfas DT, Cotfas PA, Cataron A. Using the genetic algorithm to determine the parameters of photovoltaic cells and panels. In: International Symposium on Electronics and Telecommunications 2018, Timisoara, Romania, 8–9 November, 2018. [57] King DL, Boyson WE, Kratochvill JA. Photovoltaic array performance model Sandia Report Sandia National Laboratories; 2004 [58] Du Y, Tao W, Liu Y, Jiang J, Huang H. Heat transfer modeling and temperature experiments of crystalline silicon photovoltaic modules. Sol Energy 2017;146:257–63.
solar and wind energy. Measurement 2016;93:94–101. [43] Mahmoudinezhad S, Rezania A, Cotfas PA, Cotfas DT, Rosendahl LA. Transient behavior of concentrated solar oxide thermoelectric generator. Energy 2019;168:823–32. [44] Easwarakhanthan T, Bottin J, Bouhouch I, Boutrit C. Nonlinear minimization algorithm for determining the solar cell parameters with microcomputers. Int J Solar Energy 1986;4:1–12. [45] Tong NT, Pora W. A parameter extraction technique exploiting intrinsic properties of solar cells. Appl Energy 2016;176:104–15. [46] Niu Q, Zhang H, Li K. An improved TLBO with elite strategy for parameters identification of PEM fuel cell and solar cell models. Int J Hydrogen Energy 2014;39:3837–54. [47] Hachana O, Hemsas KE, Tina GM, Ventura C. Comparison of different metaheuristic algorithms for parameter identification of photovoltaic cell/module. J Renewable Sustainable Energy 2013;5:053122. [48] Hamid NFA, Rahim NA, Selvaraj J. Solar cell parameters identification using hybrid Nelder-Mead and modified particle swarm optimization. J Renewable Sustainable Energy 2016;8. [49] Askarzadeh A, Rezazadeh A. Extraction of maximum power point in solar cells using bird mating optimizer-based parameters identification approach. Sol Energy 2013;90:123–33. [50] Gong W, Cai Z. Parameter extraction of solar cell models using repaired adaptive differential evolution. Sol Energy 2013;94:209–20.
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Application of successive discretization algorithm for determining photovoltaic cells parameters
T
⁎
Daniel T. Cotfasa, , Adrian M. Deaconub, Petru A. Cotfasa a b
Department of Electronics and Computers, Transilvania University of Brasov, Eroilor 29, 500036 Brasov, Romania Department of Mathematics and Computer Sciences, Transilvania University of Brasov, Eroilor 29, 500036 Brasov, Romania
A R T I C LE I N FO
A B S T R A C T
Keywords: Photovoltaic cells and panels Parameters determining Algorithms
The extraction of the photovoltaic cells and panels parameters still represents a hot topic, despite the numerous methods proposed by researchers in scientific literature. In order to optimize the efficiency of photovoltaic cells and panels and to predict the energy they generate, it is useful to accurately calculate their parameters function of temperature and irradiance. This paper proposes the Successive discretization algorithm developed to extract five parameters of photovoltaic cells and panels using the current voltage characteristic and the one diode model. Three widely used standard datasets, one for the photovoltaic cell and two for photovoltaic panels, are utilized. Other three datasets are analyzed, one measured in laboratory conditions for the monocrystalline photovoltaic cell and two under natural sunlight for the monocrystalline photovoltaic panel. The proposed algorithm proves its performance through comparison with other over twenty accepted methods used in specialized literature. The comparison is made for five parameters and the root mean square error. The obtained results demonstrate that the method presented is one of the best and it increases the determination accuracy of important photovoltaic cells’ parameters. Thus, the Successive discretization algorithm is a very good candidate as tool for extracting the photovoltaic cells and panels parameters.
1. Introduction The European Commission, EC, has established roadmaps for the next years to improve living standards taking into account the climate changes and its influencing factors. So, the EC aims to reduce the greenhouse gas emissions by 20% until 2020 in comparison with 1990 [1], to double this reduction in greenhouse gas emissions until 2030, to achieve a percentage of 27% from energy consumption as renewable energy [2], and the ratio must reach 85%–90% by 2050 [3]. EC also encourages the member states to follow Denmark, New Zeeland, Carinthia, Vancouver and others in implementing projects that aim to achieve 100% renewable energy being used in all sectors until 2050 or even sooner [4]. Solar energy converted in electric and thermal energy is becoming increasingly important and therefore it represents a main sector of renewable energy. Now, it is almost double in comparison with the one obtained from wind [5] and it is estimated that it will reach 69% from all renewable types by 2050 [6]. Accurate values of the photovoltaic cells or panels’ parameters function of temperature and irradiance are essential for estimating the generated energy of photovoltaic farms and at the same time for the
⁎
improvement of photovoltaic cells performance. The five parameters of the photovoltaic cells: photogenerated current Iph, reverse saturation current Io, ideality factor series of diode n, and parasitic resistances: series Rs and shunt Rsh are determined using the current-voltage characteristic I-V [7] and the equivalent circuits of the photovoltaic cells for one diode model [8]. There are many methods to determine the parameters of photovoltaic cells: some of them are used to determine all five parameters, others only one or more, but not all five [8]. These methods are based on statistical functions, non-linear regression, a mixed approach between theoretical and graphical analysis of the I-V characteristic [8], and metaheuristic methods, such as: Genetic Algorithm [9], Genetic algorithm with convex combination crossover [10], Particle Swarm Optimization [11], Harmony Search [12], Simulated annealing [13], Cat Swarm Optimization [14], Artificial Bee Colony [15], Chaotic Improved Artificial Bee Colony [16], Imperialist Competitive Algorithm [17], Teaching Learning Based Optimization [18], Water Cycle Algorithm [19], Jaya Algorithm [20], Flower Pollination Algorithm [21], Cuckoo Search [22], Bacterial Foraging Algorithm [23], Whale Optimization Algorithm [24], Moth-Flame Optimization [25], Gravitational Search Algorithm [26], Penalty-Based Differential Evolution [27],
Corresponding author. E-mail addresses: [email protected] (D.T. Cotfas), [email protected] (A.M. Deaconu), [email protected] (P.A. Cotfas).
https://doi.org/10.1016/j.enconman.2019.06.037 Received 9 April 2019; Received in revised form 9 June 2019; Accepted 16 June 2019 0196-8904/ © 2019 Published by Elsevier Ltd.
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Fig. 1. Equivalent circuits of photovoltaic cells: a) One diode model; b) Two diodes model.
successive discretization. The implemented SDA algorithm can be utilized to find the global optimum for complex problems with numerous local minima. The other contributions are: comparison of the SDA performance with that of other algorithms from the best ones to some less performant, using the same I,V data from [33] for a photovoltaic cell and two PV panels and using the SDA algorithm to calculate the parameters of one photovoltaic cell, and one PV panel at different temperature and irradiance values. The results are compared with the ones obtained using two implemented algorithms – genetic algorithm and analytical five point method. The results obtained demonstrate that the SDA algorithm is the best in most cases under study or one of the best algorithms in the others. Thus, the SDA algorithm proposed is one step forward in solving both challenges: determining accurately and relatively fast the parameters of the photovoltaic cells. The paper is structured in four sections. The first is the introduction, the second briefly describes the photovoltaic models, the experimental set up and the SDA algorithm. The results obtained for the five photovoltaic cells and panels parameters are presented and discussed in the third section. In the first stage the SDA is applied for RTC France photovoltaic cell and for two panels: PWP201 and STM6-40, both made of 36 photovoltaic cells connected in series. The second stage is dedicated to using the SDA algorithm for a monocrystalline photovoltaic cell, mSi and monocrystalline photovoltaic panel, mSiPV. The last section presents the conclusions and future work.
Table 1 The monocrystalline photovoltaic panel parameters. No. cells
Isc [A]
Voc [V]
Imax [A]
Vmax [V]
Pmax [W]
18
1.62
10.4
1.47
8.19
12
Symbiotic Organisms Search Algorithm [28], Improved Cuckoo Search Algorithm [29], etc. The analytical five point method developed by Chan et al. is the most often used amongst the analytical methods. The PV parameters are calculated using the equations obtained for Isc, Voc, and Pmax points and the slopes around Isc and Voc for determining the Rsh and Rs [30]. The method is fast, but the accuracy is dependent on the correctness of the chosen points and the noise of the measurement [31]. The non-linear regression methods, such as: Newton–Raphson, Levenberg–Marquardt, and others are efficient methods, all the points on the I-V characteristic being taken into account to determine the photovoltaic parameters, but the results are dependent on the initial chosen values of the parameters and the number of I-V points [32]. The metaheuristic methods are inspired by natural phenomena. They are suitable to determine the PV parameters because the problem which has to be solved is non-linear multi-modal. The issue arising in this context is the necessary time they require to achieve the convergent, which increases significantly if the tuning of control parameters is incorrectly chosen [33]. Combining two or more metaheuristic algorithms is another way to improve the accuracy of parameters determination, for example Hybrid Adaptive Nelder-Mead Simplex Algorithm [31] and Teaching–learning–based artificial bee colony [34]. The discretization techniques can be used to solve different problems [35], among which the optimizing ones [36]. The novelty of the paper consists in implementing and using the Successive Discretization Algorithm (SDA) for the first time to determine the parameters of photovoltaic cells and panels using the one diode model. The solution obtained by the discretization algorithm is improved by using the
2. Models and methods The photovoltaic cells and panels can be analyzed from two perspectives: in static and dynamic regime [37]. Most methods and algorithms are developed to determine the parameters of the photovoltaic cells in static regime. The equivalent circuit in static regime used for the photovoltaic cells derives from the models used.
Fig. 2. The measurement systems: a) Solar simulator; (b) Outdoor measurement system. 546
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Table 2 Parameters range of the photovoltaic cell and panels. Parameters
Iph [A] Io [μA] n Rs [Ω] Rsh [Ω]
RTC France photovoltaic cell
PWP201 panel
STM6-40 panel
Lower value
Upper value
Lower value
Upper value
Lower value
Upper value
0 0 1 0 0
1 1 2 0.5 100
0 0 1 0 0
2 50 50 2 2000
0 0 1 0 0
2 50 60 0.36 1000
where VT represents the thermal voltage, VT = kT/q, k is the Boltzmann constant, T is the temperature and q is the elementary electrical charge. The mathematical model for the photovoltaic panel with the photovoltaic cells connected in series is described in Eq. (2), [26].
I = Iph − Io ⎛e ⎝
Fig. 3. The discretization of a 2D continuous function f on an interval [a, b].
The mathematical model of photovoltaic cells based on the equivalent circuit is function of the conduction mechanisms. The mathematical model using the one diode model – the conduction mechanism being diffusion [38], with parasitic resistance, is described by Eq.1 and the equivalent circuit is presented in Fig. 1a. V + IRs nVT
− 1⎞ − ⎠
V + IRs Rsh
− 1⎞ − ⎠
V + Ns IRs Ns Rsh
(2)
where Ns represents the number of photovoltaic cells connected in series. The two diodes model which takes into account other conduction mechanisms, such as recombination [39], is described by Eq. (3) and the equivalent circuit in Fig. 1b.
2.1. Models
I = Iph − Io ⎛e ⎝
V + Ns IRs nNs VT
V + IRs V + IRs V + IRs I = Iph − Iod ⎛e nd VT − 1⎞ − Ior ⎛e nr VT − 1⎞ − Rsh ⎝ ⎠ ⎝ ⎠
(3)
where Iod and Ior represent the reverse saturation current for diffusion and for recombination, nd and nr represent the ideality factor of diode for the same two mechanisms.
(1)
Fig. 4. Flowchart of Successive Discretization Algorithm. 547
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photovoltaic panel. The mainly parameters of the photovoltaic panel measured at Standard Test Conditions are presented in Table 1. The RELab system is used to measure the I-V characteristics in lab conditions and a system based on NI c-RIO embedded controller is used for outside conditions. The RELab system is described in more detail in paper [42]. The second system is based on NI cRIO 9074 platform, developed around a 400 MHz processor and a FPGA Spartan-3 chip. The used I/O modules are: NI 9215 and NI 9225 for the PV voltage and current measurements; NI 9211 and NI 9213 for the temperature measurements based on thermocouples. More details about the performances of the system are presented in paper [43]. The temperature of photovoltaic cells and irradiance are measured simultaneously with the I-V characteristic. The temperature of the photovoltaic cell is measured on the back of the cell using a temperature sensor with ± 0.5 °C accuracy. The temperature of the panel is measured using three thermocouples mounted on the back of the panel. The back panel temperature is considered the average of the three temperatures. The solar irradiance is measured using a SPN1 pyranometer, mounted in the same plane with the PV panels. The Solar simulator, A4-LA200-2, AAA class, is used to illuminate the photovoltaic cells, see Fig. 2a. The homogeneous illuminated area is 5 cm × 5 cm. The photovoltaic mSi panel is measured under natural sunlight conditions, see Fig. 2b, on a clear sky day.
Table 3 (I,V) points of RTC photovoltaic cell and the current calculated using SDA algorithm. Measured data
Current calculated
Error
V [V]
I [A]
Ie [A]
Rerr
−0.2057 −0.1291 −0.0588 0.0057 0.0646 0.1185 0.1678 0.2132 0.2545 0.2924 0.3269 0.3585 0.3873 0.4137 0.4373 0.4590 0.4784 0.4960 0.5119 0.5265 0.5398 0.5521 0.5633 0.5736 0.5833 0.5900
0.7640 0.7620 0.7605 0.7605 0.7600 0.7590 0.7570 0.7570 0.7555 0.7540 0.7505 0.7465 0.7385 0.7280 0.7065 0.6755 0.6320 0.5730 0.4990 0.4130 0.3165 0.2120 0.1035 −0.0100 −0.1230 −0.2100
0.764078 0.762656 0.761352 0.760153 0.759057 0.758046 0.757098 0.756149 0.755096 0.753673 0.75140 0.747361 0.740121 0.727381 0.706965 0.675268 0.630743 0.571913 0.499596 0.413644 0.317512 0.212163 0.102263 −0.00870953 −0.125507 −0.208485
−0.000078 −0.000656 −0.000852 0.000347 0.000943 0.000954 −0.000098 0.000851 0.000404 0.000327 −0.0009 −0.000861 −0.001621 0.000619 −0.000465 0.000232 0.001257 0.001087 −0.000596 −0.000644 −0.001012 −0.000163 0.001237 −0.00129047 0.002507 −0.001515
2.3. Successive discretization algorithm The algorithm developed to extract the photovoltaic cells and panel parameters based on one diode or two diodes models described by Eqs. (3) and (4) used the successive discretization method.
There are other mechanisms, such as: thermionic, tunneling processes and the space-charge limited current mechanism which can be considered [8]. The one diode model is the most widely used model to characterize the photovoltaic cells because it is simpler than the double diodes model, the results obtained are good, and for thin film photovoltaic cells it is the most accurate model [40].
2.3.1. Discretization process The idea behind the algorithm for determining the solar cell parameters is discretization. In Mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step towards making them suitable for numerical evaluation or implementation on digital computers. Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce this amount of error to a level considered negligible for the modeling purpose. A 2-dimensional example is considered in order to illustrate the idea of discretization. The discretization can also be applied in 4, 5, 6 and even 8 dimensions, but the basic idea remains the same. Let f:[a,b] → R be a 2D continuous function on the interval [a, b],
2.2. Experiment setup The I-V characteristics of the photovoltaic cells can be measured using different techniques: capacitor, MOSFET and electronic load [41]. The capacitor technique is used in this paper to measure the I-V characteristics for mSi commercial photovoltaic cell and mSiPV Table 4 Comparison of different the five parameters extraction techniques for RTC. Algorithm
Iph [A]
Io [μA]
n
Rs [Ω]
Rsh [Ω]
RMSE
Rank
SDA CWOA [24] CSO [14] NM-MPSO [48] STLBO [46] ABC-DE [47] ImCSA[29] ISCE [33] EHA-NMS [31] Rcr-IJADE [50] BMO [49] SOS [28] MABC [51] ABC [16] CPSO [52] 5P [42] PS [53] SA [13] GA [56]
0.760773 0.76077 0.76078 0.76078 0.76078 0.76077 0.760776 0.76077553 0.76077553 0.76077553 0.76077 0.7608 0.760779 0.7608 0.7607 0.7612 0.7617 0.762 0.7619
0.3244462 0.3239 0.3230 0.32306 0.32302 0.32302 0.323021 0.32302083 0.32302080 0.3230208 0.32479 0.3579 0.321323 0.3251 0.4000 0.1966 0.9980 0.4798 0.8087
1.48164 1.4812 1.48118 1.48120 1.48114 1.47986 1.481781 1.48118360 1.48118359 1.4811836 1.48173 1.4916 1.481385 1.4817 1.5033 1.43 1.6000 1.5172 1.5751
0.03636 0.03636 0.03638 0.03638 0.03638 0.03637 0.036377 0.03637709 0.03637709 0.03637709 0.03636 0.0359 0.036389 0.0364 0.0354 0.042 0.0313 0.0345 0.0299
53.842702 53.7987 53.7185 53.7222 53.7187 53.7185 53.718524 53.71852771 53.71852139 53.718525 53.8716 53.7835 53.39999 53.6433 59.012 95.28 64.1026 43.1034 42.3729
9.8598E–04 9.8602E–04 9.8602E–04 9.8602E–04 9.8602E–04 9.8602E–04 9.8602E–04 9.860219E–04 9.860219E–04 9.860219E–04 9.8608E–04 9.8609E–04 9.861E–04 9.862E–04 1.39E–03 8.674E–03 0.01494 0.019 0.01908
1 2 2 2 2 2 2 3 3 3 4 5 6 7 8 9 10 11 12
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Fig. 5. Data of RTC France photovoltaic cell: a) I-V characteristics measured and calculated using SDA algorithm; b) comparison between the absolute current errors obtained using the SDA and CWOA algorithms.
Considering Eq. (1), Eq. (6) becomes:
where a < b. Instead of the infinite number points of the function f is restricted only to d > 0 representatives. In order to do that, d values vi (i = 1, …, d) are considered from the interval J = [a, b] so that:
FI1, V (Iph, Io, n, Rs , Rsh) = 0
When solving the solar cell Eq. (7) the goal is to find the photovoltaic cell parameters Iph, Io, n, Rs and Rsh that minimize the root mean square error (RMSE), see Eq. (8) for the given pairs of current and voltage (I, V).
(4)
a < v1 < v2 < . .. < vd < b The set of d representatives of the function points is
{Pj (vj, f (vj ))|j = 1, ...,d}
(5)
Moreover, the set of line segments |PjPj+1|, j = 0, …, d form a linear approximation of the function f, where P0(a, f(a)) and Pd+1(b, f(b)), see Fig. 3. The process of discretization for the single-diode model with 5 cell parameters is presented. Eq. (1) is rewritten as follows in order to obtain the objective function:
FI1, V (Iph, Io, n, Rs , Rsh) = Iph − Io ⎛e ⎝
(7)
V + IRs nVT
− 1⎞ − ⎠
V + IRs −I Rsh
RMSE (Iph, Io, n, Rs , Rsh) =
∑(I , V ) (FI1, V (Iph, Io, n, Rs , Rsh ))2 p
(8)
where p is the number of the (I,V) pairs. The intervals J1 = [a1, b1], J2 = [a2, b2], J3 = [a3, b3], J4 = [a4, b4] and J5 = [a5, b5] are considered the definition domains for the parameters Iph, Io, n, Rs and, respectively, Rsh. The values ai and bi will be exactly set in section 2.2. It is observed that the function FI,V1 is continuous on the 5-
(6) 549
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the inequalities:
Table 5 (I,V) points of PWP201 photovoltaic panel and the current calculated using SDA algorithm. Measured data
Current calculated
Error
V [V]
I [A]
Ie [A]
Rerr
0.1248 1.8093 3.3511 4.7622 6.0538 7.2364 8.3189 9.3097 10.2163 11.0449 11.8018 12.4929 13.1231 13.6983 14.2221 14.6995 15.1346 15.5311 15.8929 16.2229 16.5241 16.7987 17.0499 17.2793 17.4885
1.0315 1.0300 1.0260 1.0220 1.0180 1.0155 1.0140 1.0100 1.0035 0.9880 0.9630 0.9255 0.8725 0.8075 0.7265 0.6345 0.5345 0.4275 0.3185 0.2085 0.1010 −0.0080 −0.1110 −0.2090 −0.3030
1.029121304 1.027382551 1.025742673 1.024107493 1.022291675 1.019930167 1.01636231 1.010495202 1.000628010 0.984547554 0.959521101 0.922838546 0.872599672 0.807274453 0.728336716 0.637138148 0.536213013 0.429511044 0.318773970 0.207388828 0.096166399 −0.008326038 −0.110936938 −0.209247415 −0.300863286
0.002378700 0.002617450 0.000257327 −0.002107490 −0.004291680 −0.004430170 −0.002362310 −0.000495202 0.002871990 0.003452450 0.003478900 0.002661450 −0.000099700 0.000225547 −0.001836720 −0.002638150 −0.001713010 −0.002011040 −0.000273970 0.001111170 0.004833600 0.000326038 −0.000063100 0.000247415 −0.002136710
ai < v1i < v2i < . .. < v dii < bi
In order to get a uniform distribution of the points in the interval Ji the values v ij (j = 1, …, di) are considered as follows:
v ij = ai + jli
Current calculated
Error
(10)
where li is given by the following equation:
li =
bi − ai di + 1
(11)
In this case, it is easily observed that:
v1i
= ai + li and v dii = bi − li
(12)
The following set of quintets of photovoltaic cell parameters is considered:
G = {(v 1j1 , v j22 , v j33 , v j44 , v 5j5 )|ji = 1, ...,di , i = 1, ...,5}
RMSE (v k11 , v k22 , v k33 , v k44 , v k55 ) = min RMSE (g )
I [A]
Ie [A]
Rerr
0 0.118 2.237 5.434 7.26 9.68 11.59 12.6 13.37 14.09 14.88 15.59 16.4 16.71 16.98 17.13 17.32 17.91 19.08 21.02
1.663 1.663 1.661 1.653 1.65 1.645 1.64 1.636 1.629 1.619 1.597 1.581 1.542 1.524 1.5 1.485 1.465 1.388 1.118 0
1.663492501 1.663285403 1.659563333 1.653896827 1.650531876 1.645380036 1.639180671 1.633666529 1.627250289 1.618293153 1.603092126 1.58161313 1.542378468 1.521256585 1.499248941 1.485328855 1.465704792 1.387610684 1.118237171 −2.68E−08
0.002378700 0.002617450 0.000257327 −0.002107490 −0.004291680 −0.004430170 −0.002362310 −0.000495202 0.002871990 0.003452450 0.003478900 0.002661450 −0.000099700 0.000225547 −0.001836720 −0.002638150 −0.001713010 −0.002011040 −0.000273970 0.001111170
(14)
g∈G
v k11 ,
v k22 ,
v k33 ,
v k44 ,
v k55
The quintet ( ) is called an approximate solution of Eq. (7) out of d1 × d2 × d3 × d4 × d5 quintets. 2.3.2. Refining the solution by successive discretization The method to obtain the best approximate solution for Eq. (7) is improved in order to obtain a better solution (a refined one). The best s ≥ 1 quintets are considered gj = (v 1 j , v 2j , v 3 j , v 4 j , v 5 j ) , k1
k2
k3
k4
k5
(j = 1, …, s) in the set G, having the smallest RMSE errors. For each quintet gj the search of approximate solutions is refined for Eq. (7) in the neighborhood of gj :
J1j × J2j × J3j × J4j × J5j V [V]
(13)
The quintet (v k11 , v k22 , v k33 , v k44 , v k55 )is calculated so that:
Table 6 (I,V) points of STM6-40 photovoltaic panel and the current calculated using SDA algorithm Measured data
(9)
(15)
where
Ji j = [v i j − li , v i j + li], i = 1, ...,5 ki
ki
(16)
The process of discretization on each 5D interval from Eq. (15) (j = 1, …, s) is performed, resulting in s solutions for each interval. From the resulting s2 quintets are selected the best s solutions for Eq. (7) and applied again the method of refining. After r > 0 refining iterations is stopped and the best found quintet is taken (with the smallest RMSE error) as the final solution for Eq. (7). There are some special cases when the bounds of the definition interval are powers, i.e., [a, b] = [10c, 10d], where c and d are integer values, c < d. In this cases representatives in the interval [a, b] are considered of all powers. For instance, if [a, b] = [10−12, 10−6] the representatives are uniformly distributed around 10−11, 10−10, 10−9, 10−8 and 10−7. So, instead of linear discretization, exponential discretization is considered, i.e., instead of linear discretization on the interval [a, b], the linear discretization is applied on the interval [c, d]. In the end the representatives in [a, b] are powers of 10 in the interval (c, d). The Successive Discretization Algorithm for 5-parameter identification of solar cells is presented in the flowchart from Fig. 4. In the first iteration of the algorithm an initial discretization is performed (t = 1) on the interval J1 × J2 × J3 × J4 × J5. When an iteration of the algorithm is finished “s” best approximation solutions are selected and then in the next iteration of the algorithm t = s discretizations are performed in the vicinity of these “s” solutions (“s” is given). “no_of_iterations” is also given as an input to the algorithm. The identification of the solar cell parameters using successive discretization can be easily adapted to the two diode model with 7 parameters. The only difference is that the discretization is done in 8-dimensional spaces, but in this paper is focused only on the one diode
dimensional interval J1 × J2 × J3 × J4 × J5. This interval is called the discretization 5D interval. The discretization is applied to the function FI,V1 on the 5D discretization interval. The discretization will be done in a 6-dimensional space. The function is defined on a 5D interval and its representation is 6D. This is similar to the 2D case presented in Fig. 3 where the function f is defined on a 1D interval [a, b]. The 5D discretization interval is J1 × J2 × J3 × J4 × J5 where Ji = [ai, bi], i = 1,2,3,4,5, Ji is a simple 1D interval. The solution is searched in this 5D interval. It represents the given boundaries of the search for the algorithm. Five positive integers d1, d2, d3, d4, d5 ∈ N* are considered. For each interval Ji (i = 1, …, 5) di values are taken v ij (j = 1, …, di), respecting 550
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Table 7 Comparison of different five parameters extraction techniques for PWP201 Algorithm
Iph [A]
Io [μA]
n
Rs [Ω]
Rsh [Ω]
RMSE
Rank
SDA ISCE [34] EHA-NMS [31] Rcr-IJADE [49] MPCOA [45] SOS [28] SA [13] FPA [55] ABC-DE [47] 5P [41] PS [53]
1.030517 1.0305143 1.0305143 1.0305143 1.03188 1.0303 1.0331 1.032091 1.0318 1.034 1.0313
3.4816148 3.48226304 3.48226292 3.4822629 3.37370 3.5616 3.6642 3.047538 3.2774 3.571 3.1756
48.598941 48.642835 48.642835 48.642835 48.50646 48.7291 48.8211 48.13128 48.3948 48.71 48.2889
1.201288 1.201271 1.201271 1.201271 1.20295 1.1991 1.1989 1.217583 1.2062 1.206 1.2053
981.599618 981.98228038 981.98222618 981.98216 849.6927 1017.7 833.3333 811.3721 845.2495 1123.00 714.2857
2.4250749E–03 2.425075E–03 2.425075E–03 2.425075E–03 2.4251E–03 2.4251E–03 2.7E–03 2.7425E–03 3.8855E–03 4.019E–03 0.0118
1 1 1 1 2 2 3 4 5 6 7
Table 8 Comparison of different five parameters extraction techniques for STM6-40 Algorithm
Iph [A]
Io [μA]
n
Rs [mΩ]
Rsh [Ω]
RMSE
Rank
SDA ISCE [34] EHA-NMS [31] Rcr-IJADE [49] TMP [53] 5P [41]
1.663949 1.66390478 1.66390478 1.66390480 1.6635 1.666
1.7025818 1.73865691 1.73865691 1.73865690 1.4142 2.654E-4
54.642016 54.73090512 54.73090512 54.73090512 53.9496 33.35
156.421 153.855765 153.855765 153.85577004 175.644 1.165E+3
570.227443 573.41858868 573.41858868 573.41858868 555.084 741.16
1.72964500E–03 1.72981371E–03 1.72981371E–03 1.72981371E–03 3.11854870E–03 7.17643E–02
1 2 2 2 3 4
between RMSE values obtained with the algorithms ranked on the first six places are very small. The second parameter taken into account to achieve a better image of algorithms performance is the number of iterations. The SDA algorithm performs 31 iterations (corresponding to a total of 31 discretizations). The analysis of the five parameters leads to the following results: Iph is the most stable parameter, its maximum variation from the optimal one, considered as the value given by SDA algorithm, is 0.16% if all algorithms are considered and 0.001% if only the first twelve algorithms are considered with RMSE under 0.001; Io has the highest variation, which is 307% when all algorithms are taken into consideration and 0.972% for the first twelve algorithms. It is the most sensitive parameter. The shunt resistance also has a high variation, the maximum being 76.96%, for series resistance 17.76% and for the ideality factor of diode 8%. If the first twelve algorithms are considered, the parameters variation is: Rsh – 0.829%, Rs – 0.110%, and n – 0.108%. The algorithms CPSO, 5P, PS, GA and SA have the RMSE higher than 0.001. The results obtained for the five parameters of the photovoltaic cells using these algorithms show their weakness. The high matching between the measured points of the I-V characteristic and calculated ones using SDA algorithm is shown in Fig. 5a. The comparison between absolute current errors obtained using the first two algorithms SDA and CWOA, Fig. 5b, shows that there isn’t any best algorithm for the values around the open circuit voltage.
model. 3. Results and discussion The important parameters from the one diode model obtained using the SDA algorithm, which was implemented in Visual C++ 2017, for the benchmark RTC France photovoltaic cell [44] and for benchmark two panels: PWP201[44] and STM6-40 [45] are compared with the ones obtained by some of the best metaheuristic algorithms, CWOA [24], CSO [14], NM-MPSO [48], STLBO [35], ABC-DE [47], ISCE [33], EHA-NMS [31], Rcr-IJADE [50], BMO [49], SOS [28], ImCSA[29], MABC [51], ABC [16], FPA [55], MPCOA [54] to prove the performance of the SDA algorithm. These results are also compared with the ones from less performant algorithms, CPSO [52], SA [13], GA [56] and PS [53] and the analytical five point method, 5P [30,42]. It is very important to identify the differences between the values of the photovoltaic cell parameters obtained with SDA and SA, PS and 5P, respectively. The lower and upper values for the RTC photovoltaic cell and PWP201 panel are chosen to be similar to [24,33,42] and for STM6-40 to be similar to [33,45], see Table 2. These values were chosen in order to make a fair comparison. 3.1. RTC photovoltaic cell The current-voltage (I,V) points taken into account to calculate the parameters of RTC photovoltaic cell are presented in Table 2, and they are used to this purpose in a lot of papers [13,14,16,24,31,33,46–53]. The I-V characteristic of the RTC is measured at 33 °C. The current calculated using the SDA algorithm and the relative error, Eq. (18), calculated by subtracting the calculated from the measured current values are also presented in Table 3.
R err = I − Ic
3.2. Results for photovoltaic panels In Table 5, the I-V data, measured and calculated, of PWP201 photovoltaic panel are listed and for STM6-40 in Table 6 respectively. These points are measured at 45 °C for PWP201 and 51 °C for STM6-40 [45]. Both of them have 36 photovoltaic cells connected in series. PWP201 panel is made of polycrystalline photovoltaic cells, and STM640 panel used monocrystalline photovoltaic cells. The parameters of the PWP201 photovoltaic panel extracted using some algorithms are listed in Table 7 and for STM6-40 in Table 8. By comparing RMSE obtained with SDA algorithm and data from other research, it is obvious that the SDA algorithm is the best. The number of discretizations used is 31. RMSE for SDA, ISCE, EHA-NMS and Rcr-IJADE are practically the same and the variation of the PWP201 photovoltaic panel parameters
(18)
The parameters of the RTC photovoltaic cell extracted using some algorithms are listed in Table 4. Firstly, the root mean square error, RMSE, is considered to prove the superior performance of the SDA algorithm against the ones obtained with the algorithms mentioned in Table 3. The ranking obtained using this parameter for comparison shows the superiority of the SDA algorithm. However, the difference 551
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Fig. 6. a) I-V characteristics of PWP201 photovoltaic panel measured and calculated using SDA algorithm; b) I-V characteristics of STM6-40 photovoltaic panel measured and calculated using SDA algorithm.
such as ABC-DE, 5P and PS the variation becomes significant. Rsh varies from the optimum value with more than 27%. RMSE for SDA, ISCE, EHA-NMS and Rcr-IJADE is the same. The STM6-40 photovoltaic panel parameters are practically the same for these three algorithms. The parameters obtained using TMP have: a small variation for Iph and n, under 1.5% in comparison with the optimal one, and a significant variation for Rsh, Io and Rs, 4%, 14% and 18% respectively. The 5P method has a very low performance for STM6-40 panel, overestimating the shunt and series resistance and underestimating the reverse saturation current and ideality factor of diode. This is due to the lack of points around the Voc where the first value of Rs is calculated. In such cases the 5P method cannot be used. The 5P also used the points around Isc to calculate Rsh, and if there is a lack of points around Isc the 5P algorithm has poor performance.
Table 9 Parameters range of the photovoltaic mSi cell and mSi PV panel. Parameters
Iph [A] Io [A] n Rs [Ω] Rsh [Ω]
mSi photovoltaic cell
mSi PV panel
Lower value
Upper value
Lower value
Upper value
0 10−12 1 0 0
1 10−6 2 0.2 500
0 10−12 18 0 0
2 10−6 36 2 5000
from the reference obtained using SDA algorithm is very small. The variation increases in the cases of MPCOP, SA and FPA, even if the RMSE increase is small. The highest variation is for the shunt resistance, followed by reverse saturation current. For less performant algorithms, 552
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Fig. 7. Data of the mSi photovoltaic cell: a) I-V characteristics measured and calculated using SDA algorithm; b) Comparison of the absolute current errors obtained for using the SDA, GA algorithms and 5P methods. Table 10 Comparison of different five parameters extraction techniques for mSi photovoltaic cell Algorithm
Iph [A]
Io [μA]
n
Rs [Ω]
Rsh [Ω]
RMSE
Rank
SDA GA 5P
0.425752 0.4256882 0.4255
0.5168535 0.8383311 0.30645567
1.679294 1.73926 1.618311
0.091316 0.0859435 0.10352224
99.136671 123.3659 145.222
5.63097253E–04 6.97414541E–04 2.25639649E–03
1 2 3
Table 11 Comparison of different five parameters extraction techniques for mSiPV photovoltaic panel Case
Algorithm
Iph [A]
Io [nA]
n
Rs [Ω]
Rsh [Ω]
RMSE
Rank
A
SDA GA 5P
1.224206 1.223082 1.224
0.4677 4.988143 0. 334
18.352944 20.5289 18.02
0.144083 0.02147292 0.134
1544.361724 1765.388 1242.91
2.765E–03 4.96271E–03 6.953433E–03
1 2 3
B
SDA GA [48] 5P [48]
1.627297 1.62905 1.6291
0.6451 3.133213 6.82
18.120725 19.50277 20.235
0.136626 0.0757812 0.101
2998.9 1130.687 804.71
4.111E–03 6.707233E–03 2.86E–02
1 2 3
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Fig. 8. Absolute current error comparison between SDA, GA and 5P: a) case A; b) case B.
The highly matching between the measured points of the I-V characteristic and calculated ones using SDA algorithm is shown in Fig. 6a for PWP201 and Fig. 6b for STM6-40.
(case B). The temperature of the photovoltaic panel is obtained by converting the temperature measured on the back of the photovoltaic panel according to King et al. and Du et al. [57,58]. The confidence intervals of the photovoltaic cell and panel parameters are presented in Table 9. The I-V characteristics of the mSi photovoltaic cell, compared with the ones calculated using SDA algorithm, are presented in Fig. 7a. The absolute current errors obtained using SDA and GA algorithms [56] and 5P method function of the voltage, are shown in Fig. 7.b. The distribution illustrates that the highest difference appears around the open circuit voltage, especially for the 5P method. In the first half the 5P method has very good results. The comparison of the parameters obtained using SDA and GA algorithms and 5P method, Table 10, shows a similar behaviour with the
3.3. Experimental results The performance of the SDA algorithm being proven superior by comparison with other algorithms, in all cases taken into consideration, it is applied to calculate the parameters for mSi commercial photovoltaic cell, sizes 3 × 3 cm and for mSi photovoltaic panel with 18 cells connected in series. The I-V characteristic for mSi photovoltaic cell is measured at 1000 W/m2, the temperature of the photovoltaic cell being 27 °C and for mSiPV photovoltaic panel at different temperatures and solar radiation: 35 °C and 700 W/m2 (case A); 40 °C and 983 W/m2 554
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one of the RTC photovoltaic cell. Iph is almost the same, while Io, Rsh and n have high variations. It can be observed that the SDA algorithm has the best results. The number of (I,V) points of the I-V characteristic measured for the mSiPV panel are 1001. Table 11 shows the optimal parameters in the two cases considered, A and B, comparatively. SDA algorithm gives the lowest value for RMSE. The distribution of the absolute current errors for the mSIPV panel in the two cases is shown in Fig. 8a for case A and in Fig. 8b for case B. The absolute errors are large around the open circuit voltage and also the 5P method gives wrong results around the open circuit voltage, the same as in mSi photovoltaic cell.
with convex combination crossover. Int J Ambient Energy 2018:1–8. [11] Ye M, Wang X, Xu Y. Parameter extraction of solar cells using particle swarm optimization. J Appl Phys 2009;105:094502–94508. [12] Askarzadeh A, Rezazadeh A. Parameter identification for solar cell models using harmony search-based algorithms. Sol Energy 2012;86:3241–9. [13] El-Naggar KM, AlRashidi MR, AlHajri MF, Al-Othman AK. Simulated annealing algorithm for photovoltaic parameters identification. Sol Energy 2012;86:266–74. [14] Guo L, Meng Z, Sun Y, Wang L. Parameter identification and sensitivity analysis of solar cell models with cat swarm optimization algorithm. Energy Convers Manage 2016;108:520–8. [15] Askarzadeh A, Rezazadeh A. Artificial bee swarm optimization algorithm for parameters identification of solar cell models. Appl Energy 2013;102:943–9. [16] Oliva D, Ewees AA, Aziz MAE, Hassanien AE, Peréz-Cisneros M. A Chaotic improved artificial bee colony for parameter estimation of photovoltaic cells. Energies 2017;10:865. [17] Fathy A, Rezk H. Parameter estimation of photovoltaic system using imperialist competitive algorithm. Renewable Energy 2017;111:307–20. [18] Chen X, Yu K, Du W, Zhao W, Liu G. Parameters identification of solar cell models using generalized oppositional teaching learning based optimization. Energy 2016;99:170–80. [19] Kler D, Sharma P, Banerjee A, Rana K, Kumar V. PV cell and module efficient parameters estimation using evaporation rate based water cycle algorithm. Swarm Evol Comput 2017;35:93–110. [20] Yu K, Liang J, Qu B, Chen X, Wang H. Parameters identification of photovoltaic models using an improved JAYA optimization algorithm. Energy Convers Manage 2017;150:742–53. [21] Ram JP, Babu TS, Dragicevic T, Rajasekar N. A new hybrid bee pollinator flower pollination algorithm for solar PV parameter estimation. Energy Convers Manage 2017;135:463–76. [22] Ma J, Ting T, Man KL, Zhang N, Guan S-U, Wong PW. Parameter estimation of photovoltaic models via cuckoo search. J Appl Math 2013;2013. [23] Awadallah MA. Variations of the bacterial foraging algorithm for the extraction of PV module parameters from nameplate data. Energy Convers Manage 2016;113:312–20. [24] Oliva D, El Aziz MA, Hassanien AE. Parameter estimation of photovoltaic cells using an improved chaotic whale optimization algorithm. Appl Energy 2017;200:141–54. [25] Allam D, Yousri D, Eteiba M. Parameters extraction of the three diode model for the multi-crystalline solar cell/module using Moth-Flame Optimization Algorithm. Energy Convers Manage 2016;123:535–48. [26] Sarjila R, Ravi K, Edward JB, Kumar KS, Prasad A. Parameter extraction of solar photovoltaic modules using gravitational search algorithm. J Electr Comput Eng 2016:2143572. [27] Ishaque K, Salam Z, Mekhilef S, Shamsudin A. Parameter extraction of solar photovoltaic modules using penalty-based differential evolution. Appl Energy 2012;99:297–308. [28] Xiong G, Zhang J, Yuan X, Shi D, He Y, Xiong G, et al. Application of symbiotic organisms search algorithm for parameter extraction of solar cell models. Appl Sci 2018;8:1–18. [29] Kang T, Yao J, Jin M, Yang S, Duong T, Kang T, et al. A novel improved cuckoo search algorithm for parameter estimation of photovoltaic (PV) models. Energies 2018;11:1–31. [30] Chan DSH, Phillips JR, Phang JCH. A comparative study of extraction methods for solar cell model parameters. Solid-State Electron 1986;29:329–37. [31] Chen Z, Wu L, Lin P, Wu Y, Cheng S. Parameters identification of photovoltaic models using hybrid adaptive Nelder-Mead simplex algorithm based on eagle strategy. Appl Energy 2016;182:47–57. [32] Dkhichi F, Oukarfi B, Fakkar A, Belbounaguia N. Parameter identification of solar cell model using Levenberg–Marquardt algorithm combined with simulated annealing. Sol Energy 2014;110:781–8. [33] Gao X, Cui Y, Hu J, Xu G, Wang Z, Qu J, et al. Parameter extraction of solar cell models using improved shuffled complex evolution algorithm. Energy Convers Manage 2018;157:460–79. [34] Chen Xu, Xu B, Mei C, Ding Y, Li K. Teaching–learning–based artificial bee colony for solar photovoltaic parameter estimation. Appl Energy 2018;212:1578–88. [35] Kotsiantis S, Kanellopoulos D. Discretization techniques: a recent survey. GESTS Int Trans Comput Sci Eng 2006;32:47–58. [36] Garcia S, Luengo J, Saez JA, Lopez V, Herrera F. A survey of discretization techniques: taxonomy and empirical analysis in supervised learning. IEEE Trans Knowl Data Eng 2013;25:734–50. [37] Cotfas DT, Cotfas PA, Kaplanis S. Methods and techniques to determine the dynamic parameters of solar cells: review. Renewable Sustainable Energy Rev 2016;61:213–21. [38] Pallarès J, Cabré R, Marsal LF. A compact equivalent circuit for the dark currentvoltage characteristics of nonideal solar cells. J Appl Phys 2006;100:084513-1-5. [39] Chen Y, Sun Y, Meng Z. An improved explicit double-diode model of solar cells: fitness verification and parameter extraction. Energy Convers Manage 2018;169:345–58. [40] Et-torabi K, Nassar-eddine I, Obbadi A, Errami Y, Rmaily R, Sahnoun S, et al. Parameters estimation of the single and double diode photovoltaic models using a Gauss-Seidel algorithm and analytical method: a comparative study. Energy Convers Manage 2017;148:1041–54. [41] Cotfas DT, Cotfas PA, Ursutiu D, Samoila C. Current-voltage characteristic raising techniques for solar cells. Comparisons and applications. In Optimization of Electrical and Electronic Equipment (OPTIM), 2010 12th International Conference on, Brasov, Romania, 20–22 May 2010. [42] Cotfas PA, Cotfas DT. Design and implementation of RELab system to study the
4. Conclusions The SDA algorithm is described and used to calculate the five important parameters of the photovoltaic cells and panels. The performance of the proposed SDA algorithm is proven through the results obtained in comparison with the other existent algorithms from research literature for all devices under test, RTC, PWP201 and STM6-40 panels. Also, the SDA algorithm has proven its performance in comparison with GA algorithm and 5P method implemented by the authors for experimental I-V characteristics measured for: mSi photovoltaic cell and mSIPV panel at different temperatures and irradiances. By analyzing the results obtained for the parameters for all devices under test, it can be concluded that: the photogenerated current is the most stable parameter, and the reverse saturation current and the shunt resistance have high variations. They can be either overestimated or underestimated, especially when less performance algorithms are used. The SDA algorithm performance (the best), the lowest RMSE and number of iterations, recommend it for the determination of the main photovoltaic cells and panels parameters under different conditions. The calculation of the Perovskite, multijunction and thin film photovoltaic cells and panels parameters using SDA algorithm and comparison with the results obtained by other accepted algorithms will be the target for future work. Another intended direction of further research is the calculation of the photovoltaic cells’ and panels’ parameters using two diodes models with SDA algorithm. Also, the hybrid algorithms, which use one or more algorithms, have a high potential to determine the parameters of the photovoltaic cells and panels with high accuracy and therefore will be considered in future work. Declaration of Competing Interest None. References [1] https://ec.europa.eu/energy/en/topics/renewable-energy (accessed on 18.03. 2019). [2] Knopf B, Nahmmacher P, Schmid E. The European renewable energy target for 2030 – an impact assessment of the electricity sector. Energy Policy 2015;85:50–60. [3] https://ec.europa.eu/energy/sites/ener/files/documents/2012_energy_roadmap_ 2050_en_0.pdf (accessed on 19.03.2019). [4] http://www.renewables100.org/en/programs/go-100-project/ (accessed on 21.03. 2019). [5] Sawin JL, Seyboth K, Sverrisson F. Renewables 2018 Global Status Report. Paris, France: REN21 Secretariat; 2018. [6] Ram M, Bogdanov D, Aghahosseini A, Oyewo AS, Gulagi A, Child M, et al. Global Energy System Based on 100% Renewable Energy-power Sector. Berlin: Study by Lappeenranta University of Technology and Energy Watch Grup; 2017. [7] Abbassi R, Abbassi A, Jemli M, Chebbi S. Identification of unknown parameters of solar cell models: a comprehensive overview of available approaches. Renewable Sustainable Energy Rev 2018;90:453–74. [8] Cotfas DT, Cotfas PA, Kaplanis S. Methods to determine the dc parameters of solar cells: a critical review. Renewable Sustainable Energy Rev 2013;28:588–96. [9] Sellami A, Bouaïcha M. Application of the genetic algorithms for identifying the electrical parameters of PV solar generators. In: Kosyachenko LA, editor. Solar Cellssilicon Wafer-based Technologies. InTech; 2011. p. 349–64. [10] Hamid N, Abounacer R, Idali Oumhand M, Feddaoui M, Agliz D. Parameters identification of photovoltaic solar cells and module using the genetic algorithm
555
Energy Conversion and Management 196 (2019) 545–556
D.T. Cotfas, et al.
[51] Jamadi M, Merrikh-Bayat F, Bigdeli M. Very accurate parameter estimation of single- and double-diode solar cell models using a modified artificial bee colony algorithm. Int J Energy Environ Eng 2016;7:13–25. [52] Huang W, Jiang C, Xue L, Song. Extracting solar cell model parameters based on chaos particle swarm algorithm. In: 2011 International conference on electric information and control engineering; 2011. p. 398–402. [53] AlHajri MF, El-Naggar KM, AlRashidi MR, Al-Othman AK. Optimal extraction of solar cell parameters using pattern search. Renewable Energy 2012;44:238–45. [54] Yuan X, Xiang Y, He Y. Parameter extraction of solar cell models using mutative scale parallel chaos optimization algorithm. Sol Energy 2014;108:238–51. [55] AlRashidi MR, AlHajri MF, El-Naggar KM, Al-Othman AK. A new estimation approach for determining the I-V characteristics of solar cells. Sol Energy 2011;85:1543–50. [56] Cotfas DT, Cotfas PA, Cataron A. Using the genetic algorithm to determine the parameters of photovoltaic cells and panels. In: International Symposium on Electronics and Telecommunications 2018, Timisoara, Romania, 8–9 November, 2018. [57] King DL, Boyson WE, Kratochvill JA. Photovoltaic array performance model Sandia Report Sandia National Laboratories; 2004 [58] Du Y, Tao W, Liu Y, Jiang J, Huang H. Heat transfer modeling and temperature experiments of crystalline silicon photovoltaic modules. Sol Energy 2017;146:257–63.
solar and wind energy. Measurement 2016;93:94–101. [43] Mahmoudinezhad S, Rezania A, Cotfas PA, Cotfas DT, Rosendahl LA. Transient behavior of concentrated solar oxide thermoelectric generator. Energy 2019;168:823–32. [44] Easwarakhanthan T, Bottin J, Bouhouch I, Boutrit C. Nonlinear minimization algorithm for determining the solar cell parameters with microcomputers. Int J Solar Energy 1986;4:1–12. [45] Tong NT, Pora W. A parameter extraction technique exploiting intrinsic properties of solar cells. Appl Energy 2016;176:104–15. [46] Niu Q, Zhang H, Li K. An improved TLBO with elite strategy for parameters identification of PEM fuel cell and solar cell models. Int J Hydrogen Energy 2014;39:3837–54. [47] Hachana O, Hemsas KE, Tina GM, Ventura C. Comparison of different metaheuristic algorithms for parameter identification of photovoltaic cell/module. J Renewable Sustainable Energy 2013;5:053122. [48] Hamid NFA, Rahim NA, Selvaraj J. Solar cell parameters identification using hybrid Nelder-Mead and modified particle swarm optimization. J Renewable Sustainable Energy 2016;8. [49] Askarzadeh A, Rezazadeh A. Extraction of maximum power point in solar cells using bird mating optimizer-based parameters identification approach. Sol Energy 2013;90:123–33. [50] Gong W, Cai Z. Parameter extraction of solar cell models using repaired adaptive differential evolution. Sol Energy 2013;94:209–20.
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