# An Improved Generalized Method for Evaluation of Parameters, Modeling, and Simulation of Photovoltaic Modules

Authors: Vandana Jha; Uday Shankar Triar
Journal: International Journal of Photoenergy
 (2017)

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## Abstract

This paper proposes an improved generalized method for evaluation of parameters, modeling, and simulation of photovoltaic modules. A new concept “Level of Improvement” has been proposed for evaluating unknown parameters of the nonlinearI-V equation of the single-diode model of PV module at any environmental condition, taking the manufacturer-specified data at Standard Test Conditions as inputs. The main contribution of the new concept is the improvement in the accuracy of values of evaluated parameters up to various levels and is based on mathematical equations of PV modules. The proposed evaluating method is implemented by MATLAB programming and, for demonstration, by using the values of parameters of the I-V equation obtained from programming results, a PV module model is build with MATLAB. The parameters evaluated by the proposed technique are validated with the datasheet values of six different commercially available PV modules (thin film, monocrystalline, and polycrystalline) at Standard Test Conditions and Nominal Operating Cell Temperature Conditions. The module output characteristics generated by the proposed method are validated with experimental data of FS-270 PV module. The effects of variation of ideality factor and resistances on output characteristics are also studied. The superiority of the proposed technique is proved.


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## Body

## 1. Introduction

Photovoltaics (PV) is a method of converting sunlight directly into electricity using semiconducting materials that exhibit the photovoltaic effect. A PV cell is the fundamental PV device. A PV cell is a specialized semiconductor diode that converts sunlight into direct current (DC) electricity. A PV module is a collection of PV cells wired in series/parallel combinations as required to meet current and voltage requirements. A PV panel includes one or more PV modules assembled as a prewired, field-installable unit. A PV array is the complete power-generating unit, consisting of any number of PV panels to form large PV systems.PV cells are expensive, and the characteristics of PV devices are highly dependent on environmental conditions [1]. Therefore, to ensure the maximum use of the available solar energy by a PV power system, it is important to study its behaviour through modeling, before implementing it in reality. The mathematical model of the PV device is very useful in studying various PV technologies and in designing several PV systems along with their components for application in practical systems.The equivalent circuit of the ideal PV cell is represented basically by (i) single-diode model [1–7] and (ii) two-diode model [8–12]. In single-diode model, the effect of the recombination loss of carriers in the depletion region is not considered, whereas in two-diode model, an additional diode is included to consider this effect. Many more sophisticated models have been developed so far to include the effects that are not considered by the earlier models, thus claiming more accuracy. The single-diode model is simple and accurate and is perfect for designers who are looking for a model for the modeling of PV devices where the intended result is achieved without great effort [4, 13].The parameter identification of the single-diode model with five parameters has been enormously researched [1–7]. The information provided in the datasheet cannot saturate the necessary restrictions to calculate the five unknown parameters. Consequently, some presumptions and approximations are generally needed to initiate the parameter extraction. For example, in [4], the evaluation of both photovoltaic current (Ipv) and diode reverse saturation current (I0) is based on approximation, due to which the deviation at open and short conditions becomes inevitable, and the diode ideality constant (a) has been assumed on the basis of PV cell technology. Since this constant affects the curvature of the I-V curves and its correct estimation improves the model accuracy, it must be evaluated accurately without assumption. A separate DC circuit is constructed to determine ideality constant in [1], thus increasing the cost of the evaluation procedure. Some other techniques adopted by the authors to estimate the ideality constant are curve fitting [6, 14], iteration [7], trial and error [14], and the concept of the minimum sum of squares [15]. But for evaluating ideality constant by any of the above methods, manufacturer-specified output characteristics are required. Thus, these processes can be cumbersome. Also, in some PV module datasheets, output characteristics are not given [16]. Therefore, evaluating ideality constant of such modules becomes very difficult.Apart from the evaluation ofIpv, I0, and a, extensive studies have been conducted to determine the series resistance (Rs) and parallel resistance (Rp). Some authors neglect Rp to simplify the model as the value of this resistance is generally high [14–19], and sometimes, the Rs is neglected, as its value is very low [20, 21]. The neglect of Rs and Rp has significant impact on the model accuracy. Several algorithms have been proposed to determine both Rs and Rp through iterative techniques [4, 12]. If the initialisation of the variables and the convergence conditions are not proper, then these iterative techniques require many iterations and, sometimes, may not converge. Curve fitting method can be utilized in the current density-voltage curves to estimate both Rs and Rp [22]. In [23], Rs and Rp are evaluated by using additional parameters which can be extracted from the current versus voltage curve of a PV module. These methods are quite poor, inaccurate, and tedious mainly because Rs and Rp are adjusted separately, which is not a good practice, if an accurate model is required. Moreover, these methods are applicable only if the manufacturer-specified output characteristics are provided. Differential evolution (DE) can be used to extract the excess seven parameters of a double-diode PV module model utilising only the information provided in the datasheets [8, 10]. An explicit modeling method based on Lambert W-function for PV arrays that has been used in [24] to find the values of parameters is intricate and time-consuming. Artificial intelligence (AI) such as fuzzy logic [25] and artificial neural network (ANN) [26, 27] and genetic algorithms such as particle swarm optimization (PSO) have also been proposed to model the I-V curves [28]. However, they are not widely adopted due to high computation burden. In [2], a comprehensive parameter identification method is proposed to enhance model accuracy while keeping the parameterization procedure in a simple form. Despite the accurate results, the approach requires extensive computation. A circuit-based piecewise linear PV device model has been developed and demonstrated using PSCAD/EMTDC for parameter identification [29]. As this method is based on trial and error for different values of irradiance and temperature and is based on a large number of approximations and assumptions, this method becomes complicated and less accurate.Hence, to overcome the above drawbacks, an improved generalized method for evaluation of parameters, modeling, and simulation of photovoltaic modules has been proposed in this paper. A new concept “Level of Improvement” has been proposed for evaluating unknown parameters of the nonlinearI-V equation of the single-diode model of PV module including series and parallel resistances at any environmental condition (STC and NOCTC in this paper), taking the manufacturer-specified data at Standard Test Conditions as inputs. The new concept helps in improving the accuracy of the values of evaluated parameters up to various levels. The proposed evaluating method is based on mathematical equations of PV modules, thus making the method fast, simple, and accurate. The method for evaluating a is based on the property that at the maximum power point, dP/dV=0. The proposed evaluating method is implemented by MATLAB programming and by using the values of parameters of the I-V equation obtained from programming results, a PV module model is build with MATLAB. The parameters evaluated by the proposed technique are validated with the datasheet values of six different commercially available PV modules for irradiance and temperature at Standard Test Conditions and Nominal Operating Cell Temperature Conditions. The module output characteristics generated by the proposed method are validated with experimental data of FS-270 PV module at different environmental conditions (varying irradiance and temperature). The effects of variation of ideality factor and resistances on the output characteristics are also studied. The superiority of the proposed technique over three popular existing techniques [4, 6, 12] is proved.


## 2. Mathematical Equation and Modeling of Photovoltaic Modules

Practical modules are composed of various PV cells connected in series or parallel. Figure1 shows the single-diode equivalent circuit of a practical PV module.Figure 1
Single-diode equivalent circuit of a practical PV module.The mathematical equation that describes theI-V characteristic of a practical PV module is
(1)I=Ipv−I0expV+RsIVta−1−V+RsIRp,(2)I=Ipv,cellNp−I0,cellNpexpV+RsIVta−1−V+RsIRp,(3)Vt=NskTq.The equation that mathematically describes theP-V characteristic of a practical PV module is
(4)P=VIpv−I0expV+RsIVta−1−V+RsIRp.For the observation of the characteristics of the PV module, it is required to evaluate all the parameters of (1) [6]. Datasheets generally give information about the parameters, characteristics, and performances of PV modules with respect to the Standard Test Condition (STC), which is taken as 1000 W/m2 solar irradiance and 25°C module temperature [30–36]. Some datasheets also give information about PV modules at Nominal Operating Cell Temperature Condition (NOCTC), which is generally taken as 800 W/m2 irradiance and 20°C ambient temperature. The parameters basically present in all PV module datasheets at STC and sometimes at NOCTC are as follows: Pmpp, Vmpp, Impp, Voc, Isc, KPmppT, KVmppT, KImppT, KVocT, and KIscT for any environmental condition. Table 1 shows the values of the parameters of seven different commercially available photovoltaic modules at STC provided by the manufacturers [30–36].Table 1
Parameters of seven different commercially available PV modules at STC (25°C and 1000 W/m2) provided by the manufacturers [30–36].
Parameter
Thin film
Monocrystalline
Multicrystalline
Shell ST40
FS-270
Shell SQ 150-PC
HIT-N240SE10
KD140GX-LFBS
KD260GX-LFB2
KU265-6MCA
P
m
p
p
S
T
CW
40
70
150
240
140
260
265
V
m
p
p
S
T
CV
16.6
65.5
34
43.7
17.7
31.0
31.0
I
m
p
p
S
T
CA
2.41
1.07
4.4
5.51
7.91
8.39
8.55
V
o
c
S
T
C V
23.3
88.0
43.4
52.4
22.1
38.3
38.3
I
s
c
S
T
C A
2.68
1.23
4.8
5.85
8.68
9.09
9.26
K
P
m
p
p
T
−0.6%/°C
−0.25%/°C
−0.52%/°C
−0.30%/°C
−0.46%/°C
−0.45%/°C
−0.45%/°C
K
V
m
p
p
T
−100 mV/°C
−0.34%/°C
−167 mV/°C
−0.131 V/°C
−0.52%/°C
−0.48%/°C
−0.48%/°C
K
I
m
p
p
T
−2.50 mA/°C
0.023%/°C
−2.38 mA/°C
2.105 mA/°C
0.0066%/°C
0.02%/°C
0.02%/°C
K
V
o
c
T
−100 mV/°C
−0.25%/°C
−161 mV/°C
−0.131 V/°C
−0.36%/°C
−0.36%/°C
−0.36%/°C
K
I
s
c
T
0.35 mA/°C
0.04%/°C
1.4 mA/°C
1.76 mA/°C
0.060%/°C
0.06%/°C
0.06%/°C
N
s
36
116
72
72
36
60
60
N
p
1
1
1
1
1
1
1Some of the parameters of (4) can be found in the manufacturer’s datasheets. The remaining parameters such as Ipv, a, I0, Rs, and Rp have to be evaluated. They are usually not specified by the manufacturers because they cannot be measured and are unique for every module [19].


## 3. Basic Evaluation of the Parameters

The proposed work attempts to evaluate the five unknown parameters (Ipv,a,I0,Rs, and Rp) of a PV module at different environmental conditions.


### 3.1. Evaluation of Photovoltaic Current

The parallel resistanceRp is generally very high, so the last term of (1) can be eliminated for the further work.
(5)I=Ipv−I0expV+RsIVta−1.By applying short-circuit condition (I=Isc,V=0) to (5), (6) can be derived as follows:
(6)Isc=Ipv−I0expIscRsVta−1.Since (I0expIscRs/Vta−1≈0), (6) can be written as
(7)Ipv≈Isc.The photovoltaic current of a PV module is approximately equal to the short-circuit current at any environmental condition.


### 3.2. Evaluation of Diode Ideality Constant

By applying open-circuit condition (I=0,V=Voc) to (5), (8) can be derived as follows:
(8)0=Ipv−I0expVocVta−1.By rearranging (8), (9) is obtained as follows:
(9)I0=IpvexpVoc/Vta−1.In the proposed method, the maximum power point is considered for evaluating “a” using the property dP/dV=0 at the maximum power point.
(10)P=VI,(11)1V⋅dPdV=dIdV+IV.By applying maximum power condition to (11), (12) is obtained as follows:
(12)dIdVmpp+ImppVmpp=0.By differentiating (5) and applying maximum power condition, (13) can be obtained as follows:
(13)dIdVmpp=−I0/VtexpVmpp+ImppRs/Vtaa+I0/VtRsexpVmpp+ImppRs/Vta.By substituting (13) in (12), (14) can be obtained as follows:
(14)aImpp+ImppRs−VmppI0/VtexpVmpp+ImppRs/Vtaa+I0/VtRsexpVmpp+ImppRs/VtaVmpp=0.For (14) to be valid, the numerator of (14) must be zero as the denominator is finite.
(15)aImpp+ImppRs−VmppI0VtexpVmpp+ImppRsVta=0.By applying maximum power point condition to (5) and rearranging terms, (16) and (17) can be obtained as follows:
(16)expVmpp+ImppRsVta=Ipv−Impp+I0I0,(17)Rs=VtaImpplnIpv−Impp+I0I0−VmppImpp.By combining (16) and (17), (18) can be obtained as follows:
(18)aImpp+Ipv−Impp+I0alnIpv−Impp+I0I0−2VmppVt=0.Substituting (7) and (9) in (18), (19) is obtained as follows:
(19)aImpp+Isc−Impp+IscexpVoc/Vta−1alnIsc−Impp+Isc/expVoc/Vta−1Isc/expVoc/Vta−1−2VmppVt=0.When the data specified in the manufacturer’s datasheets are used, the left hand side of (19) becomes a function of “a” and it is denoted by fa.
(20)fa=aImpp+Isc−Impp+IscexpVoc/Vta−1alnIsc−Impp+Isc/expVoc/Vta−1Isc/expVoc/Vta−1−2VmppVt.Thus, “a” is found by solving fa=0 with the help of MATLAB programming.


### 3.3. Evaluation of Diode Reverse Saturation Current

By (9), the diode reverse saturation current at any environmental condition can be obtained.


### 3.4. Evaluation of Series and Parallel Resistances

Equating the maximum power calculated by theP-V model of (4) (Pmppcal) to the power at the MPP from the datasheet (Pmpp) at any environmental condition, a relation between Rs and Rp will be obtained, that is,
(21)Pmppcal=Pmpp,(22)Pmppcal=VmppImpp=VmppIpv−I0expVmpp+RsImppVta−1−Vmpp+RsImppRp=Pmpp,(23)Rp=VmppVmpp+ImppRsVmppIpv−VmppI0expVmpp+ImppRs/Vta+VmppI0−Pmpp.According to (23), for any value of Rs, there will be a value of Rp that satisfies (21). It is required to find an only pair of Rs and Rp for the desired environmental condition that satisfies the model accurately.


#### 3.4.1. Proposed Algorithm

In this work,Rs and Rp in (23) are calculated through the following steps:
(a)
Eliminating the last term of (4), as the value of the parallel resistance is high, the power calculated by the P-V model excluding Rp, PwithoutRP is obtained as follows:(24)PwithoutRp=VIpv−I0expV+Rsmax′IwithoutRpVta−1.(b)
Iterations are performed on (24) where Rsmax′ is slowly incremented starting from Rsmax′=0, till the maximum of power calculated by the P-V model excluding Rp (maxPwithoutRp) becomes approximately equal to the power at the MPP from the datasheet (Pmpp).(c)
PuttingRs=Rsmax in (23), the value of Rp obtained is negative. Iterations are performed again on (23), where Rs is slowly decremented starting from Rs=Rsmax, until a positive value for Rp is obtained. Hence, the maximum value of parallel resistance Rpmax is obtained and the corresponding value of series resistance is the maximum value of series resistance, Rsmax.(d)
PuttingRs=0 in (23), the minimum value of parallel resistance Rpmin is obtained as follows:(25)Rpmin=VmppIpv−I0expVmpp/Vta+I0−Pmpp/Vmpp.The value ofRpmin obtained by the proposed method is more accurate as compared to other methods [4, 12].(e)
InitialisingRs=Rsmax and Rp=Rpmax, iterations are performed again on (4) where Rs is slowly decremented and the corresponding value of Rp is obtained, till the maximum of power calculated by the P-V model (maxP) becomes approximately equal to the power at the MPP from the datasheet (Pmpp), while Rp>Rpmin. Thus, Rs and Rp are obtained.


## 3.1. Evaluation of Photovoltaic Current

The parallel resistanceRp is generally very high, so the last term of (1) can be eliminated for the further work.
(5)I=Ipv−I0expV+RsIVta−1.By applying short-circuit condition (I=Isc,V=0) to (5), (6) can be derived as follows:
(6)Isc=Ipv−I0expIscRsVta−1.Since (I0expIscRs/Vta−1≈0), (6) can be written as
(7)Ipv≈Isc.The photovoltaic current of a PV module is approximately equal to the short-circuit current at any environmental condition.


## 3.2. Evaluation of Diode Ideality Constant

By applying open-circuit condition (I=0,V=Voc) to (5), (8) can be derived as follows:
(8)0=Ipv−I0expVocVta−1.By rearranging (8), (9) is obtained as follows:
(9)I0=IpvexpVoc/Vta−1.In the proposed method, the maximum power point is considered for evaluating “a” using the property dP/dV=0 at the maximum power point.
(10)P=VI,(11)1V⋅dPdV=dIdV+IV.By applying maximum power condition to (11), (12) is obtained as follows:
(12)dIdVmpp+ImppVmpp=0.By differentiating (5) and applying maximum power condition, (13) can be obtained as follows:
(13)dIdVmpp=−I0/VtexpVmpp+ImppRs/Vtaa+I0/VtRsexpVmpp+ImppRs/Vta.By substituting (13) in (12), (14) can be obtained as follows:
(14)aImpp+ImppRs−VmppI0/VtexpVmpp+ImppRs/Vtaa+I0/VtRsexpVmpp+ImppRs/VtaVmpp=0.For (14) to be valid, the numerator of (14) must be zero as the denominator is finite.
(15)aImpp+ImppRs−VmppI0VtexpVmpp+ImppRsVta=0.By applying maximum power point condition to (5) and rearranging terms, (16) and (17) can be obtained as follows:
(16)expVmpp+ImppRsVta=Ipv−Impp+I0I0,(17)Rs=VtaImpplnIpv−Impp+I0I0−VmppImpp.By combining (16) and (17), (18) can be obtained as follows:
(18)aImpp+Ipv−Impp+I0alnIpv−Impp+I0I0−2VmppVt=0.Substituting (7) and (9) in (18), (19) is obtained as follows:
(19)aImpp+Isc−Impp+IscexpVoc/Vta−1alnIsc−Impp+Isc/expVoc/Vta−1Isc/expVoc/Vta−1−2VmppVt=0.When the data specified in the manufacturer’s datasheets are used, the left hand side of (19) becomes a function of “a” and it is denoted by fa.
(20)fa=aImpp+Isc−Impp+IscexpVoc/Vta−1alnIsc−Impp+Isc/expVoc/Vta−1Isc/expVoc/Vta−1−2VmppVt.Thus, “a” is found by solving fa=0 with the help of MATLAB programming.


## 3.3. Evaluation of Diode Reverse Saturation Current

By (9), the diode reverse saturation current at any environmental condition can be obtained.


## 3.4. Evaluation of Series and Parallel Resistances

Equating the maximum power calculated by theP-V model of (4) (Pmppcal) to the power at the MPP from the datasheet (Pmpp) at any environmental condition, a relation between Rs and Rp will be obtained, that is,
(21)Pmppcal=Pmpp,(22)Pmppcal=VmppImpp=VmppIpv−I0expVmpp+RsImppVta−1−Vmpp+RsImppRp=Pmpp,(23)Rp=VmppVmpp+ImppRsVmppIpv−VmppI0expVmpp+ImppRs/Vta+VmppI0−Pmpp.According to (23), for any value of Rs, there will be a value of Rp that satisfies (21). It is required to find an only pair of Rs and Rp for the desired environmental condition that satisfies the model accurately.


### 3.4.1. Proposed Algorithm

In this work,Rs and Rp in (23) are calculated through the following steps:
(a)
Eliminating the last term of (4), as the value of the parallel resistance is high, the power calculated by the P-V model excluding Rp, PwithoutRP is obtained as follows:(24)PwithoutRp=VIpv−I0expV+Rsmax′IwithoutRpVta−1.(b)
Iterations are performed on (24) where Rsmax′ is slowly incremented starting from Rsmax′=0, till the maximum of power calculated by the P-V model excluding Rp (maxPwithoutRp) becomes approximately equal to the power at the MPP from the datasheet (Pmpp).(c)
PuttingRs=Rsmax in (23), the value of Rp obtained is negative. Iterations are performed again on (23), where Rs is slowly decremented starting from Rs=Rsmax, until a positive value for Rp is obtained. Hence, the maximum value of parallel resistance Rpmax is obtained and the corresponding value of series resistance is the maximum value of series resistance, Rsmax.(d)
PuttingRs=0 in (23), the minimum value of parallel resistance Rpmin is obtained as follows:(25)Rpmin=VmppIpv−I0expVmpp/Vta+I0−Pmpp/Vmpp.The value ofRpmin obtained by the proposed method is more accurate as compared to other methods [4, 12].(e)
InitialisingRs=Rsmax and Rp=Rpmax, iterations are performed again on (4) where Rs is slowly decremented and the corresponding value of Rp is obtained, till the maximum of power calculated by the P-V model (maxP) becomes approximately equal to the power at the MPP from the datasheet (Pmpp), while Rp>Rpmin. Thus, Rs and Rp are obtained.


## 3.4.1. Proposed Algorithm

In this work,Rs and Rp in (23) are calculated through the following steps:
(a)
Eliminating the last term of (4), as the value of the parallel resistance is high, the power calculated by the P-V model excluding Rp, PwithoutRP is obtained as follows:(24)PwithoutRp=VIpv−I0expV+Rsmax′IwithoutRpVta−1.(b)
Iterations are performed on (24) where Rsmax′ is slowly incremented starting from Rsmax′=0, till the maximum of power calculated by the P-V model excluding Rp (maxPwithoutRp) becomes approximately equal to the power at the MPP from the datasheet (Pmpp).(c)
PuttingRs=Rsmax in (23), the value of Rp obtained is negative. Iterations are performed again on (23), where Rs is slowly decremented starting from Rs=Rsmax, until a positive value for Rp is obtained. Hence, the maximum value of parallel resistance Rpmax is obtained and the corresponding value of series resistance is the maximum value of series resistance, Rsmax.(d)
PuttingRs=0 in (23), the minimum value of parallel resistance Rpmin is obtained as follows:(25)Rpmin=VmppIpv−I0expVmpp/Vta+I0−Pmpp/Vmpp.The value ofRpmin obtained by the proposed method is more accurate as compared to other methods [4, 12].(e)
InitialisingRs=Rsmax and Rp=Rpmax, iterations are performed again on (4) where Rs is slowly decremented and the corresponding value of Rp is obtained, till the maximum of power calculated by the P-V model (maxP) becomes approximately equal to the power at the MPP from the datasheet (Pmpp), while Rp>Rpmin. Thus, Rs and Rp are obtained.


## 4. Improving the Model

The accuracy of the values of evaluated parameters (Ipv,a,I0,Rs, and Rp) is improved up to various levels by using “Level of Improvement.”


### 4.1. First Level of Improvement

#### 4.1.1. Evaluation of Photovoltaic Current

By applying short-circuit condition (I=Isc,V=0) to (5), (26) can be derived as follows:
(26)Isc=Ipv1−I01expIscRsVta1−1−RsIscRp.Since (I01expIscRs/Vta1−1≈0), (26) can be written as
(27)Ipv1≈Isc1+RsRp.By (27), the first level of improved value of photovoltaic current of a PV module (Ipv1) at any environmental condition can be obtained.


#### 4.1.2. Evaluation of Diode Ideality Constant

By applying open-circuit condition (I=0,V=Voc) to (5), (28) can be derived as follows:
(28)0=Ipv1−I01expVocVta1−1−VocRp.By rearranging (28), (29) is obtained as follows:
(29)I01=Ipv1−Voc/RpexpVoc/Vta1−1.By applying the same procedure as is applied on (5) to find the diode ideality constant on (1), an improved and more accurate value of diode ideality constant can be obtained, as the values of both Rs and Rp are taken into account.The functionfa1, where a1 is the first level of improved value of diode ideality constant, becomes
(30)fa1=RpIsc1+RsRp−Impp+Isc1+Rs/Rp−Voc/RpexpVoc/Vta1−1−Vmpp+RsImpp−Vmpp+RsImpp−Vta1Vmpp−RpImpp−RsImpp.Thus, “a1” is found by solving fa1=0 with the help of MATLAB programming.


#### 4.1.3. Evaluation of Diode Reverse Saturation Current

By (29), the first level of improved value of diode reverse saturation current (I01) at any environmental condition can be obtained.


#### 4.1.4. Evaluation of Series and Parallel Resistances

By applying the same procedure as is applied to evaluateRs and Rp, a first level of improved values of series resistance (Rs1) and parallel resistance (Rp1) for the desired environmental condition can be found.


### 4.2. Second Level of Improvement

Again by puttingRs=Rs1 and Rp=Rp1 in (27) and (30), a second level of improved value of photovoltaic current (Ipv2) and diode ideality constant (a2) can be obtained. As a consequence, a second level of improved value of diode reverse saturation current (I02), series resistance (Rs2), and parallel resistance (Rp2) can be obtained. The improvement can go up to ith level (Ipvi,ai,I0i,Rsi, and Rpi), till the errors between the values of parameters evaluated by the proposed method and the values of parameters provided in manufacturer’s datasheet become minimum.It should also be noted that the value of diode ideality constant should not vary much in any level of improvement. This is possible only when the value of parallel resistance is high. In the proposed method, initialising the series and parallel resistances by their corresponding maximum values in every level of improvement helps in getting higher value of parallel resistance and also reduces the required number of iterations. In this paper, only the first level of improvement has been applied.


## 4.1. First Level of Improvement

### 4.1.1. Evaluation of Photovoltaic Current

By applying short-circuit condition (I=Isc,V=0) to (5), (26) can be derived as follows:
(26)Isc=Ipv1−I01expIscRsVta1−1−RsIscRp.Since (I01expIscRs/Vta1−1≈0), (26) can be written as
(27)Ipv1≈Isc1+RsRp.By (27), the first level of improved value of photovoltaic current of a PV module (Ipv1) at any environmental condition can be obtained.


### 4.1.2. Evaluation of Diode Ideality Constant

By applying open-circuit condition (I=0,V=Voc) to (5), (28) can be derived as follows:
(28)0=Ipv1−I01expVocVta1−1−VocRp.By rearranging (28), (29) is obtained as follows:
(29)I01=Ipv1−Voc/RpexpVoc/Vta1−1.By applying the same procedure as is applied on (5) to find the diode ideality constant on (1), an improved and more accurate value of diode ideality constant can be obtained, as the values of both Rs and Rp are taken into account.The functionfa1, where a1 is the first level of improved value of diode ideality constant, becomes
(30)fa1=RpIsc1+RsRp−Impp+Isc1+Rs/Rp−Voc/RpexpVoc/Vta1−1−Vmpp+RsImpp−Vmpp+RsImpp−Vta1Vmpp−RpImpp−RsImpp.Thus, “a1” is found by solving fa1=0 with the help of MATLAB programming.


### 4.1.3. Evaluation of Diode Reverse Saturation Current

By (29), the first level of improved value of diode reverse saturation current (I01) at any environmental condition can be obtained.


### 4.1.4. Evaluation of Series and Parallel Resistances

By applying the same procedure as is applied to evaluateRs and Rp, a first level of improved values of series resistance (Rs1) and parallel resistance (Rp1) for the desired environmental condition can be found.


## 4.1.1. Evaluation of Photovoltaic Current

By applying short-circuit condition (I=Isc,V=0) to (5), (26) can be derived as follows:
(26)Isc=Ipv1−I01expIscRsVta1−1−RsIscRp.Since (I01expIscRs/Vta1−1≈0), (26) can be written as
(27)Ipv1≈Isc1+RsRp.By (27), the first level of improved value of photovoltaic current of a PV module (Ipv1) at any environmental condition can be obtained.


## 4.1.2. Evaluation of Diode Ideality Constant

By applying open-circuit condition (I=0,V=Voc) to (5), (28) can be derived as follows:
(28)0=Ipv1−I01expVocVta1−1−VocRp.By rearranging (28), (29) is obtained as follows:
(29)I01=Ipv1−Voc/RpexpVoc/Vta1−1.By applying the same procedure as is applied on (5) to find the diode ideality constant on (1), an improved and more accurate value of diode ideality constant can be obtained, as the values of both Rs and Rp are taken into account.The functionfa1, where a1 is the first level of improved value of diode ideality constant, becomes
(30)fa1=RpIsc1+RsRp−Impp+Isc1+Rs/Rp−Voc/RpexpVoc/Vta1−1−Vmpp+RsImpp−Vmpp+RsImpp−Vta1Vmpp−RpImpp−RsImpp.Thus, “a1” is found by solving fa1=0 with the help of MATLAB programming.


## 4.1.3. Evaluation of Diode Reverse Saturation Current

By (29), the first level of improved value of diode reverse saturation current (I01) at any environmental condition can be obtained.


## 4.1.4. Evaluation of Series and Parallel Resistances

By applying the same procedure as is applied to evaluateRs and Rp, a first level of improved values of series resistance (Rs1) and parallel resistance (Rp1) for the desired environmental condition can be found.


## 4.2. Second Level of Improvement

Again by puttingRs=Rs1 and Rp=Rp1 in (27) and (30), a second level of improved value of photovoltaic current (Ipv2) and diode ideality constant (a2) can be obtained. As a consequence, a second level of improved value of diode reverse saturation current (I02), series resistance (Rs2), and parallel resistance (Rp2) can be obtained. The improvement can go up to ith level (Ipvi,ai,I0i,Rsi, and Rpi), till the errors between the values of parameters evaluated by the proposed method and the values of parameters provided in manufacturer’s datasheet become minimum.It should also be noted that the value of diode ideality constant should not vary much in any level of improvement. This is possible only when the value of parallel resistance is high. In the proposed method, initialising the series and parallel resistances by their corresponding maximum values in every level of improvement helps in getting higher value of parallel resistance and also reduces the required number of iterations. In this paper, only the first level of improvement has been applied.


## 5. Dependence of Parameters of the Characteristic Equation of the PV Module on Irradiance and Temperature

TheVoc, Isc, Vmpp, Impp, and Pmpp of the PV module depend on both solar irradiance and temperature and can be calculated using the following equations:
(31)Voc=1+KVocGlogGGSTCVocSTC+KVocTΔT,(32)Isc=GGSTCIscSTC+KIscT∆T,(33)Vmpp=1+KVmppGlogGGSTCVmppSTC+KVmppTΔT,(34)Impp=GGSTCImppSTC+KImppT∆T,(35)Pmpp=GGSTC1+KPmppGlogGGSTCPmppSTC+KPmppTΔT,(36)∆T=T−TSTC.


## 6. Evaluation of Parameters of the Characteristic Equation of the PV Module at STC

Applying STC to the above procedure and using the manufacturer-specified data at Standard Test Condition, five unknown parameters (IpvSTC,aSTC,I0STC,RsSTC, and RpSTC) of a PV module at STC can be evaluated.The flowchart for evaluation of five unknown parameters (IpvSTC,aSTC,I0STC,RsSTC, and RpSTC) of a PV module at STC is presented in Figure 2.Figure 2
Flowchart for evaluation of five unknown parameters (IpvSTC,aSTC,I0STC,RsSTC, and RpSTC) of a PV module at STC.Table2 shows the first level of improved values of the parameters of seven different commercially available PV modules for the proposed model and evaluated parameters of seven different commercially available PV modules for the Rs model [6], Rs and Rp model [4], and two-diode model [12] at STC.Table 2
First level of improved values of the parameters of seven different commercially available PV modules for the proposed model and evaluated parameters of seven different commercially available PV modules for theRs model, Rs and Rp model, and two-diode model at STC.
Model
Parameter
Thin film
Monocrystalline
Multicrystalline
Shell ST40
FS-270
Shell SQ 150-PC
HIT-N240SE10
KD140GX-LFBS
KD260GX-LFB2
KU265-6MCA
Proposed model
I
p
v
S
T
C A
2.68
1.23
4.80
5.85
8.68
9.09
9.26
a
S
T
C
1.61421
3.10639
1.56173
1.41612
1.83175
1.63175
1.62276
I
0
S
T
C A
4.4733 × 10−7
9.1598 × 10−5
1.4356 × 10−6
1.2024 × 10−8
1.8772 × 10−5
2.2181 × 10−6
2.0768 × 10−6
R
s
S
T
C Ω
1.35870
3.45266
0.45588
0.25077
0.03751
0.10264
0.10303
R
p
S
T
C Ω
139391.478
1791835.933
2123.494
405454010.026
485618.624
2135142.579
710599.657
I
p
v
S
T
C
1 A
2.680026122
1.230002370
4.801030482
5.849999758
8.680000670
9.089999904
9.260001342
a
S
T
C
1
1.61343
3.10212
1.51490
1.41164
1.83159
1.63119
1.62237
I
0
S
T
C
1 A
4.4395 × 10−7
9.0404 × 10−5
8.9866 × 10−7
1.1284 × 10−8
1.8751 × 10−5
2.2066 × 10−6
2.0692 × 10−6
R
s
S
T
C
1 Ω
1.35915
3.46672
0.48855
0.25478
0.03754
0.10286
0.10315
R
p
S
T
C
1 Ω
69436.135
295078.680
1219.87237
330003.053
185988.578
324974.945
161649.074
a
S
T
C
2
1.61267
3.09829
1.46550
1.41036
1.83143
1.63088
1.62201
R
s model
I
p
v
S
T
C A
2.68
1.23000
4.80
5.85
8.68
9.09
9.26
a
S
T
C
1.22997
1.10571
1.05938
1.06008
1.06425
1.10663
1.10663
I
0
S
T
C A
3.4150 × 10−9
3.10874 × 10−12
1.1570 × 10−9
1.4537 × 10−11
1.5423 × 10−9
1.6151 × 10−9
1.6453 × 10−9
R
s
S
T
C Ω
1.69665
14.74644
1.02961
0.56631
0.25480
0.34877
0.34138
R
s and Rp model
I
p
v
S
T
C A
2.68
1.23000
4.80
5.85
8.68
9.09
9.26
a
S
T
C
1.22997
1.10571
1.05938
1.06008
1.06425
1.10663
1.10663
I
0
S
T
C A
3.4150 × 10−9
3.10874 × 10−12
1.1570 × 10−9
1.4537 × 10−11
1.5423 × 10−9
1.6151 × 10−9
1.6453 × 10−9
R
s
S
T
C
Ω
1.54000
11.61000
0.83200
0.43700
0.17800
0.27400
0.26900
R
p
S
T
C Ω
266.37955
762.36643
262.75328
445.13305
53.88571
154.59042
154.15067
Two-diode model
I
p
v
S
T
C A
2.68
1.23000
4.80
5.85
8.68
9.09
9.26
a
1
S
T
C
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
a
2
S
T
C
1.20000
1.20000
1.20000
1.20000
1.20000
1.20000
1.20000
I
0
S
T
C A
3.0748 × 10−11
1.847458 × 10−13
3.1059 × 10−10
2.9186 × 10−12
3.6446 × 10−10
1.4740 × 10−10
1.5016 × 10−10
R
s
S
T
C Ω
1.71000
12.18000
0.90000
0.50000
0.20000
0.31000
0.30000
R
p
S
T
C Ω
204.84910
726.87888
274.79210
458.96360
55.92360
130.55740
124.58140


## 7. Evaluation of Parameters of the Characteristic Equation of the PV Module at NOCTC

Using the manufacturer-specified data at STC and NOCTC in (31), (33), and (35) irradiance coefficients of Voc, Vmpp, and Pmpp of the PV module can be calculated.Five unknown parameters of a PV module at NOCTC can be evaluated using the following equations:(37)IpvNOCTC1≈GNOCTCGSTCIscSTC+KIscT∆T1+RsNOCTCRpNOCTC,faNOCTC1=RpNOCTCGNOCTCGSTCIscSTC+KIscT∆T1+RsNOCTCRpNOCTC−GNOCTCGSTCImppSTC+KImppT∆T+GNOCTC/GSTCIscSTC+KIscT∆T1+RsNOCTC/RpNOCTC−1+KVocGlogGNOCTC/GSTCVocSTC+KVocT∆T/RpNOCTCexp1+KVocGlogGNOCTC/GSTCVocSTC+KVocT∆T/VtNOCTCaNOCTC1−1−1+KVmppGlogGNOCTCGSTCVmppSTC+KVmppT∆T+RsNOCTCGNOCTCGSTCImppSTC+KImppT∆T−1+KVmppGlogGNOCTCGSTCVmppSTC+KVmppT∆T+RsNOCTCGNOCTCGSTCImppSTC+KImppT∆T−VtNOCTCaNOCTC11+KVmppGlogGNOCTC/GSTCVmppSTC+KVmppT∆T−RpNOCTCGNOCTCGSTCImppSTC+KImppT∆T−RsNOCTCGNOCTCGSTCImppSTC+KImppT∆T,I0NOCTC1=GNOCTC/GSTCIscSTC+KIscT∆T1+RsNOCTC/RpNOCTC−1+KVocGlogGNOCTC/GSTCVocSTC+KVocT∆T/RpNOCTCexp1+KVocGlogGNOCTC/GSTCVocSTC+KVocT∆T/VtNOCTCaNOCTC1−1.By using the above equations and by applying the same procedure as is applied to findRsSTC and RpSTC, RsNOCTC and RpNOCTC can be obtained.
(38)RpNOCTC1=xy,x=1+KVmppGlogGNOCTCGSTCVmppSTC+KVmppT∆T1+KVmppGlogGNOCTCGSTCVmppSTC+KVmppT∆T+GNOCTCGSTCImppSTC+KImppT∆TRsNOCTC1,y=1+KVmppGlogGNOCTCGSTCVmppSTC+KVmppT∆TGNOCTCGSTCIscSTC+KIscT∆T1+RsNOCTCRpNOCTC−1+KVmppGlogGNOCTCGSTCVmppSTC+KVmppT∆TGNOCTC/GSTCIscSTC+KIscT∆T1+RsNOCTC/RpNOCTC−1+KVocGlogGNOCTC/GSTCVocSTC+KVocT∆T/RpNOCTCexp1+KVocGlogGNOCTC/GSTCVocSTC+KVocT∆T/VtNOCTCaNOCTC1−1exp1+KVmppGlogGNOCTC/GSTCVmppSTC+KVmppT∆T+GNOCTC/GSTCImppSTC+KImppT∆TRsNOCTC1VtNOCTCaNOCTC1+1+KVmppGlogGNOCTC/GSTCVmppSTC+KVmppT∆TGNOCTC/GSTCIscSTC+KIscT∆T1+RsNOCTC/RpNOCTC−1+KVocGlogGNOCTC/GSTCVocSTC+KVocT∆T/RpNOCTCexp1+KVocGlogGNOCTC/GSTCVocSTC+KVocT∆T/VtNOCTCaNOCTC1−1−GNOCTCGSTC1+KPmppGlogGNOCTCGSTCPmppSTC+KPmppT∆T.Table3 shows the evaluated irradiance coefficients of Voc, Vmpp, and Pmpp of seven different commercially available PV modules and evaluated parameters of seven different commercially available PV modules at NOCTC for the proposed model.Table 3
Evaluated irradiance coefficients ofVoc, Vmpp, and Pmpp of seven different commercially available PV modules and evaluated parameters of seven different commercially available PV modules at NOCTC (45°C and 1000 W/m2) for the proposed model.
Model
Parameter
Thin film
Monocrystalline
Multicrystalline
Shell ST40
FS-270
Shell SQ 150-PC
HIT-N240SE10
KD140GX-LFBS
KD260GX-LFB2
KU265-6MCA
Proposed model
K
V
o
c
G
0.084956
0.096490
0.046921
0.045882
0.067477
0.055781
0.055781
K
V
m
p
p
G
−0.093363
−0.022004
−0.074512
0.012071
−0.039787
0.019829
0.019829
K
P
m
p
p
G
0.012262
0.058966
−0.046283
−0.023365
0.030671
0.053982
0.044600
I
p
v
N
O
C
T
CA
2.150160000
0.991872000
3.863519999
4.706751999
7.027328000
7.359264000
7.496895999
a
N
O
C
T
C
1.75584
2.86595
1.67257
1.21016
1.74535
1.50015
1.49252
I
0
N
O
C
T
CA
1.5043 × 10−5
1.2555 × 10−4
2.4789 × 10−5
4.6055 × 10−9
5.6770 × 10−5
4.8923 × 10−6
4.6342 × 10−6
R
s
N
O
C
T
CΩ
1.21772
2.40133
0.24346
0.33097
0.02807
0.15316
0.14925
R
p
N
O
C
T
CΩ
1881.9879
14942.890
4073.0256
3790.1388
469.1091
552.7469
644.3563It is evident from the above equations that all the parameters of the characteristic equation of the PV module are subject to vary with irradiance and temperature, which is a truism. This increases the accuracy of the proposed method of evaluating parameters manifold as compared to existing methods in this area where the parameters are assumed constant [4, 6, 12]. The proposed model output characteristics very accurately match the experimental output characteristics and manufacturer’s datasheets which is authenticated from the results shown in the later sections.


## 8. Curves and Inferences

As shown in Figures3(a) and 3(b), I-V and P-V curves of the FS-270 PV module are generated for eight values of aSTC and the corresponding values of RsSTC and RpSTC. It can be noted that as the value of aSTC calculated by the proposed method is utilised, the value of VocSTC obtained from the proposed model becomes closest to the datasheet value. I-V and P-V curves of the FS-270 PV module are plotted again for six values of RsSTC and the corresponding values of aSTC and RpSTC (see Figures 4(a) and 4(b)). As the value of RsSTC calculated by the proposed method is utilised, the value of VocSTC obtained from the proposed model becomes closest to the datasheet value. Similar is the case with RpSTC. The accuracy of the model is highest, when the values of aSTC, RsSTC, and RpSTC calculated by the proposed technique are utilised for modeling. The proposed method of evaluating the unknown parameters is far more accurate as compared to other methods mentioned in literature.Figure 3
Characteristics for eight values ofaSTC and the corresponding values of RsSTC and RpSTC of the FS-270 PV module at 25°C and 1000 W/m2.
(a)
I-V characteristics
(b)
P-V characteristicsFigure 4
Characteristics for six values ofRsSTC and the corresponding values of aSTC and RpSTC of the FS-270 PV module at 25°C and 1000 W/m2.
(a)
I-V characteristics
(b)
P-V characteristics


## 9. Validation of the Model

The current and power output obtained from the proposed model is validated against measured current and power output data, respectively, for the FS-270 PV module provided by the National Institute of Technology Patna (NITP), India.Figures5(a) and 5(b) show the mathematical I-V and P-V characteristics of the FS-270 PV module [31] plotted with the experimental data by varying irradiance from 200 W/m2 to 1200 W/m2 at 25°C. Figures 6(a) and 6(b) show the I-V and P-V curves by varying temperature from 0°C to 75°C at 1000 W/m2. The experimental points are represented by circular markers in the curves. As the model is not perfect, some points are not exactly matched, although it is sufficiently accurate for majority points.Figure 5
Characteristics and experimental data of the FS-270 PV module at different irradiances (temperature constant = 25°C).
(a)
I-V characteristics
(b)
P-V characteristicsFigure 6
Characteristics and experimental data of the FS-270 PV module at different temperatures (irradiance constant = 1000 W/m2).
(a)
I-V characteristics
(b)
P-V characteristicsThe relative errors of the proposed model with respect to the experimental data for the FS-270 PV module at 25°C and 1000 W/m2 are shown in Figure 7. The model proposed in this paper is compared with the models proposed in [4, 6, 12]. The relative errors obtained by all the models are plotted on the same graphs. The model proposed in this paper is superior, because the values of relative errors obtained by the proposed model are very small as compared to other models.Figure 7
Relative errors of the model proposed in this paper (curve 1), in [4] (curve 2), in [12] (curve 3), and in [6] (curve 4) for the FS-270 PV module at 25°C and 1000 W/m2.Tables4–9 show the comparison of Rs model, Rs and Rp model, two-diode model, and proposed model output with manufacturer’s datasheets for six different commercially available PV modules [30–36].Table 4
Comparison ofRs model, Rs and Rp model, two-diode model, and proposed model output with manufacturer’s datasheet for Shell ST40 PV module [30].
Environmental conditions
Parameter
Datasheet value
R
s model value
R
s and Rp model value
Two-diode model value
Proposed model value
ErrorRs (%)
ErrorRs and Rp (%)
Error two-diode (%)
Error proposed (%)
STC(25°C and 1000 W/m2)
P
m
p
p W
40
29.0224
40.0055
40.0060
40.0000
−27.4589
−0.0012
0.0150
0.0000
V
m
p
p V
16.6
14.9120
16.7760
16.6000
16.6029
−10.1686
1.0602
0.0000
0.0174
I
m
p
p A
2.41
1.9462
2.3847
2.4100
2.4098
−19.2448
−1.0497
0.0000
−0.0082
V
o
c V
23.3
21.0254
23.2618
23.2460
23.2993
−9.7622
−0.1639
−0.2317
−0.0030
I
s
c A
2.68
2.1590
2.6646
2.6578
2.6800
−19.4402
−0.5746
−0.8283
0.0000
NOCTC(47°C and 800 W/m2)
P
m
p
p W
27.7
19.5088
28.2982
28.5162
27.7024
−29.5711
2.1595
2.9465
0.0086
V
m
p
p V
14.7
12.6790
14.9120
14.9120
14.6933
−13.7482
1.4421
1.4421
−0.0455
V
o
c V
20.7
18.8271
20.7820
20.8002
20.7009
−9.0478
0.3961
0.4840
0.0043
I
s
c A
2.2
1.7502
2.1378
2.1324
2.1982
−20.4454
−2.8272
−3.0727
−0.0818Table 5
Comparison ofRs model, Rs and Rp model, two-diode model, and proposed model output with manufacturer’s datasheet for Shell SQ 150-PC PV module [32].
Environmental conditions
Parameter
Datasheet value
R
s model value
R
s and Rp model value
Two-diode model value
Proposed model value
ErrorRs (%)
ErrorRs and Rp (%)
Error two-diode (%)
Error proposed (%)
STC(25°C and 1000 W/m2)
P
m
p
p W
150
111.6218
149.5998
149.6000
150.0000
−25.5854
−0.2668
−0.2666
0.0000
V
m
p
p V
34
30.3800
34.2860
34.0000
34.0115
−10.6470
0.8411
0.0000
0.0338
I
m
p
p A
4.4
3.6742
4.3633
4.4000
4.3998
−16.4954
−0.8340
0.0000
−0.0045
V
o
c V
43.4
39.7399
43.3302
43.3012
43.3972
−8.4334
−0.1608
−0.2276
−0.0064
I
s
c A
4.8
3.9475
4.7848
4.7843
4.8000
−17.7604
−0.3166
−0.3270
0.0000
NOCTC(46°C and 800 W/m2)
P
m
p
p W
108
85.7801
107.7001
107.8484
107.9950
−20.5739
−0.2776
−0.1403
−0.0046
V
m
p
p V
31
26.3800
30.8140
30.8140
30.9800
−14.9032
−0.5999
−0.5999
−0.0645
V
o
c V
39.6
36.4498
39.4670
39.4310
39.5964
−7.9550
−0.3358
−0.4267
−0.0090
I
s
c A
3.9
3.3635
3.8513
3.8509
3.8985
−13.7564
−1.2487
−1.2589
−0.0384Table 6
Comparison ofRs model, Rs and Rp model, two-diode model, and proposed model output with manufacturer’s datasheet for HIT-N240SE10 PV module [33].
Environmental conditions
Parameter
Datasheet value
R
s model value
R
s and Rp model value
Two-diode model value
Proposed model value
ErrorRs (%)
ErrorRs and Rp (%)
Error two-diode (%)
Error proposed (%)
STC(25°C and 1000 W/m2)
P
m
p
p W
240
186.2506
240.7864
240.7870
240.0000
−22.3955
0.3276
0.3279
0.0000
V
m
p
p V
43.7
40.8720
44.0160
43.7000
43.7008
−6.4713
0.7231
0.0000
0.0018
I
m
p
p A
5.51
4.5569
5.4704
5.5100
5.5096
−17.2976
−0.7186
0.0000
−0.0072
V
o
c V
52.4
49.3547
52.3588
52.3470
52.3988
−5.8116
−0.0786
−0.1011
−0.0022
I
s
c A
5.85
4.8447
5.8443
5.8436
5.8500
−17.1846
−0.0974
−0.1094
0.0000
NOCTC(44°C and 800 W/m2)
P
m
p
p W
182
149.8145
180.5653
180.9046
181.9713
−17.6843
−0.7882
−0.6018
−0.0157
V
m
p
p V
41.1
37.8720
41.3960
41.3960
41.0970
−7.8540
0.7201
0.7201
−0.0072
I
m
p
p A
4.44
3.6484
4.3619
4.3701
4.4396
−17.8288
−1.7590
−1.5743
−0.0090
V
o
c V
49.4
45.5426
49.3914
49.3938
49.3992
−7.8085
−0.0174
−0.0125
−0.0016
I
s
c A
4.71
3.8068
4.7021
4.7016
4.7098
−19.1762
−0.1677
−0.1783
−0.0042Table 7
Comparison ofRs model, Rs and Rp model, two-diode model, and proposed model output with manufacturer’s datasheet for KD140GX-LFBS PV module [34].
Environmental conditions
Parameter
Datasheet value
R
s model value
R
s and Rp model value
Two-diode model value
Proposed model value
ErrorRs (%)
ErrorRs and Rp (%)
Error two-diode (%)
Error proposed (%)
STC(25°C and 1000 W/m2)
P
m
p
p W
140
112.2240
139.9971
140.0070
140.0000
−19.8399
−0.0020
0.0050
0.0000
V
m
p
p V
17.7
15.9120
17.9010
17.7000
17.7007
−10.1016
1.1355
0.0000
0.0039
I
m
p
p A
7.91
7.0528
7.8206
7.9100
7.9091
−10.8369
−1.1302
0.0000
−0.0113
V
o
c V
22.1
20.3590
22.0514
22.0396
22.1000
−7.8778
−0.2199
−0.2733
0.0000
I
s
c A
8.68
7.6673
8.6514
8.6491
8.6800
−11.6670
−0.3294
−0.3559
0.0000
NOCTC(45°C and 800 W/m2)
P
m
p
p W
101
79.3291
101.7994
101.9756
100.9936
−21.4563
0.7914
0.9659
−0.0063
V
m
p
p V
16.0
14.6910
16.1330
16.1330
15.9880
−8.1812
0.8312
0.8312
−0.0750
I
m
p
p A
6.33
5.5852
6.3100
6.3209
6.3278
−11.7661
−0.3159
−0.1437
−0.0347
V
o
c V
20.2
18.0738
20.2155
20.2034
20.1983
−10.5257
0.0767
0.0168
−0.0084
I
s
c A
7.03
6.4273
7.0042
7.0023
7.0293
−8.5732
−0.3669
−0.3940
−0.0099Table 8
Comparison ofRs model, Rs and Rp model, two-diode model, and proposed model output with manufacturer’s datasheet for KD260GX-LFB2 PV module [35].
Environmental conditions
Parameter
Datasheet value
R
s model value
R
s and Rp model value
Two-diode model value
Proposed model value
ErrorRs (%)
ErrorRs and Rp (%)
Error two-diode (%)
Error proposed (%)
STC(25°C and 1000 W/m2)
P
m
p
p W
260
208.7319
260.0899
260.0900
260.0000
−19.7185
0.0345
0.0346
0.0000
V
m
p
p V
31.0
27.5760
31.0230
31.0000
31.0022
−11.0451
0.0741
0.0000
0.0070
I
m
p
p A
8.39
7.5693
8.3838
8.3900
8.3895
−9.7818
−0.0738
0.0000
−0.0059
V
o
c V
38.3
35.2913
38.2471
38.2249
38.2994
−7.8556
−0.1381
−0.1960
−0.0015
I
s
c A
9.09
8.0652
9.0739
9.0685
9.0900
−11.2739
−0.1771
−0.2365
0.0000
NOCTC(45°C and 800 W/m2)
P
m
p
p W
187
142.6852
189.3933
189.8687
186.9801
−23.6977
1.2798
1.5340
−0.0106
V
m
p
p V
27.9
24.7590
28.3420
28.3420
27.8997
−11.2580
1.5842
1.5842
−0.0010
I
m
p
p A
6.71
5.8309
6.6824
6.6992
6.7096
−13.1013
−0.4113
−0.1609
−0.0059
V
o
c V
35.1
33.8348
35.0769
35.0657
35.0982
−3.6045
−0.0658
−0.0977
−0.0051
I
s
c A
7.36
6.1593
7.3462
7.3418
7.3597
−16.3138
−0.1875
−0.2472
−0.0040Table 9
Comparison ofRs model, Rs and Rp model, two-diode model, and proposed model output with manufacturer’s datasheet for KU265-6MCA PV module [36].
Environmental conditions
Parameter
Datasheet value
R
s model value
R
s and Rp model value
Two-diode model value
Proposed model value
ErrorRs (%)
ErrorRs and Rp (%)
Error two-diode (%)
Error proposed (%)
STC(25°C and 1000 W/m2)
P
m
p
p W
265
213.0696
265.0499
265.0500
265.0000
−19.5963
0.0188
0.0188
0.0000
V
m
p
p V
31.0
27.5760
31.0230
31.0000
31.0030
−11.0451
0.0741
0.0000
0.0096
I
m
p
p A
8.55
7.7266
8.5437
8.5500
8.5490
−9.6304
−0.0736
0.0000
−0.0116
V
o
c V
38.3
35.2946
38.2529
38.2234
38.3000
−7.8469
−0.1229
−0.1999
0.0000
I
s
c A
9.26
8.2312
9.2439
9.2378
9.2600
−11.1101
−0.1738
−0.2397
0.0000
NOCTC(45°C and 800 W/m2)
P
m
p
p W
191
147.6087
193.0154
193.4476
190.9769
−22.7179
1.0551
1.2814
−0.0120
V
m
p
p V
27.9
24.5590
28.3420
28.3420
27.8975
−11.9749
1.5842
1.5842
−0.0089
I
m
p
p A
6.85
5.5069
6.8102
6.8255
6.8496
−19.6072
−0.5810
−0.3576
−0.0058
V
o
c V
35.1
33.6348
35.0778
35.0639
35.0975
−4.1743
−0.0632
−0.1028
−0.0071
I
s
c A
7.49
6.9605
7.4838
7.4789
7.4902
−7.0694
−0.0827
−0.1481
0.0026In the proposed method, it has been assumed that all the PV cells of a module are perfectly made but this is not the case in reality. Still the proposed method confirms to be very accurate as can be clearly seen from Figure7 and Tables 4–9. This is because the properties of the PV cells having the same parameters, manufactured from the same producer, do not vary much and hence can be assumed identical.


## 10. Simulation of the PV Module

The PV module model can be simulated in any circuit simulator by implementing (1) and (4) using basic math blocks. Figures 8 and 9 show a PV module model, where these two equations are implemented in MATLAB/Simulink. In the complete model, irradiance and temperature along with the evaluated and manufacturer-specified parameters of the PV module are the inputs while the outputs are current and power. The proposed model is most generalized as compared to the models proposed in previous works.Figure 8
Detailed implementation of PV module model in MATLAB/Simulink that solves theI-V equation.Figure 9
PV module model with subsystems in MATLAB/Simulink that solves theI-V equation.


## 11. Conclusion

In this paper, an improved generalized method for evaluation of parameters, modeling, and simulation of photovoltaic modules is proposed.The proposed PV module modeling method surpasses the other methods already published, as it has the following novelties:(i)
A new concept “Level of Improvement” has been proposed for evaluating unknown parameters of the nonlinearI-V equation of the single-diode model of PV module at any environmental condition (STC and NOCTC in this paper), taking the manufacturer-specified data at Standard Test Conditions as inputs.(ii)
The new method of evaluation of unknown parameters is based on mathematical equations of PV modules. By implementing simple set of equations using any software, the parameters can be determined numerically simply by feeding few manufacturer-specified data as input to the program.(iii)
In this paper, for the first time, the effects of varying ideality factor and resistances on the output curves have been observed. It has been inferred that as the values ofaSTC, RsSTC, and RpSTC calculated by the proposed method are utilised, the value of VocSTC obtained from the proposed model becomes closest to the datasheet value. The accuracy of the model is maximum, when the values of aSTC, RsSTC, and RpSTC calculated by the proposed technique are utilised for modeling.(iv)
A most generalized PV module model is build with MATLAB/Simulink by using the values of parameters of theI-V equation obtained from programming results in order to show the practical use of the proposed model.(v)
The proposed method proves to be more accurate in modeling commercially available PV modules as compared to other methods available in literature.


## 12. Future Work

Under partial shading condition (PSC) on a PV array, the irradiance and temperature of the PV modules undergoing shading change. The parameters of the shaded and unshaded PV modules of the array can be easily evaluated by employing the proposed method. By applying a suitable maximum power point tracking (MPPT) technique, maximum power point (MPP) of a PV array under PSC can be reached and hence MPPT can be ensured which will be the subject of our further investigations.



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*Source: 2532109-2017-12-03.xml*