Mathematical modelling of SPV array by considering the parasitic effects

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In the field of Solar systems, it is necessary for every engineer to start with the solar photovoltaic module (SPVM) design, this paper provides a complete mathematical design specification of the SPVM. The design and development of the SPVM are done to extract its electrical characteristics that are subjective to solar irradiance (G) and temperature (T). This paper model the SPVM with the datasheet of IB Solar-36 series and these modules are connected in parallel to form the Solar Photovoltaic Array (SPVA) is considered for the result verifications. To match the simulation performance of the system accurately with the practical model, this paper uses a novel approach for formulating the equations to find the exact values of shunt resistance (Rsh) and series resistance (Rs) called parasitic effects. The inverse slope method is used to formulate the Parasitic effects from the datasheets, which will extract the exact performance curves of the SPVA. These design principles can be applied to simulate the behavior of any large scale SPVA’s which are present in the system. The simulation and experimental verification using IB Solar-36 polycrystalline modules with varying T and G values for the SPVA are presented.


At present, conventional energy resources produce a massive amount of energy across the world. But in approaching years these resources get condensed day by day and also due to their harmful environmental effects alternate resources should be used. For this purpose, Non-conventional sources of energy (NCSE) during recent years gets increased significantly with the increase in load demand in the power system. The major part of NCSE is solar photovoltaic systems, which convert sunlight emitted from the sun into Electrical Energy. Out of the enormous Solar energy availability at the surface of the earth, if we are able to extract a small amount of available energy then our power demand from conventional sources will be reduced as well as environmental pollution also gets controlled. For this reason SPV power plants are widely encouraged to install throughout the globe.

An SPV power plant comprises of many components like SPVA’s, charge controller, DC-DC converter, filter elements, DC-Bus, DC-AC converter, and AC bus. The heart of SPV power plant is an SPVM comprises of PV cells. PV cell is used to extract electrical energy from the sunlight, these cells are arranged into clusters and forms as SPVM’s and SPVA’s [1]. To increase the input current and voltage these modules should be connected into series and parallel respectively [2]. In order to maximize the SPV power plant yielding a proper mathematical design and optimization of these SPVM and SPVA is required. The efficiency of an SPVA is subjected to the environmental conditions, loads connected to it and the parasitic effects. The parasitic effects must be considered prominently in SPVA design, without considering these effects the efficiency will be poor. Some of the papers in literature just estimated these values without any proper calculation and many papers not even considered these effects [3,4,5]. The parameters are commonly given in a datasheet, it gives the entire parameters which will state the performance and qualities of SPVA’s concerning these STC [6, 7].

From the literature, we will able to get the basic design of SPVM like single-diode model, two-diode model, and multi diode models. Out of all the available models single-diode model is extensively used by the researchers for simulating the large SPVA’s [8]. In this study also Single-diode model is selected for simulating the SPVA in view of its accuracy and reduced the complexity.

Many researchers used different modelling techniques of SPVM or SPVA, in [9, 10] experimental data was used to model the SPVM or SPVA, in this paper for finding the parameters they used characteristic plot which was obtained by doing the experiments at different conditions.

In [11] a technique called curve-fitting method is used to obtain the parameters accurately, this method is having a drawback that it is explicit to that SPVM only, so a generalized technique is to be needed.

For a generalized technique, innovative approaches are to be used to determine the parameters of SPVM by formulating the equations and under different operating conditions like OC, SC and MPPT conditions. It is not that easy to formulate the equations which are transcendental in nature [12,13,14]. Numerous iterative techniques were developed in literature in order to find the solution for obtaining the parameters.

The estimation of Rsh is explained in [15] which is used to obtain the other parameters, but without proper approximation of Rsh ridiculous values of output power will be obtained which will not be matched with the practical values.

In addition to this, in [16] selection of Rs is done by taking its value starting from zero and up to an acceptable margin without deviating the obtained values compared with MPP values, and also by varying the ideality factor between the minimum and maximum values from 1 to 2 respectively. The above discussed methods for finding Rsh and Rs are trial and error techniques for obtaining the output powers, but in order to match the practical values with the simulation values exactly, this technique takes more time and more values to be assumed for simulations. But these techniques are widely used in the literature in most papers.

In [17, 18] parameter extraction of single diode model based on both the datasheet information and I–V inverse slope method is used. But they used the estimation model for finding Rs and Rsh starting from zero to acceptable margins.

In [19, 20] analytical extraction technique is used which is similar to [17] and [18], but in addition to the datasheet information, it also needs the entire I–V slope curve data for estimating the parasitic resistance.

In [21] another analytical method Lambert W technique is used to extract the parasitic resistance which is only limited to STC conditions. But this method is complex and needs more computational iterations.

For extracting the parasitic resistance, [22] uses the data-driven I–V feature extraction technique which is basically an inverse slope method which measures the slope of the I–V curve, and uses time series algorithms for extracting the parameters which are a bit complex.

So a simple and exact solution is required to match the practical and simulated values directly without using these trial and error techniques in order to reduce the number of iterations for finding the parameters, this can be achieved from the Inverse slope method applied to the datasheet of a specific SPVM to find the parameters at OC and SC conditions for widely varying environmental conditions. This paper uses the Inverse slope technique which was existed in the literature, and clearly explains the process of obtaining the parasitic resistances from the datasheets of specific SPVM’s in a novel approach with clear mathematical formulations.

This paper also examines the simulation design and mathematical design of SPVA by considering the parasitic effects of Rsh and Rs to match the simulation and experimental outputs. It shows a short preface of how the SPVA behaves and works with different environmental concerns. In MPPT methods like P&O where power oscillations are heavier is due to many constraints, out of which improper matching of these parasitic effects is playing a vital role. So for these specific purposes, in order to improve the efficiency of the PV plants accurate matching of parameters is necessary which is discussed in this paper. This paper also provides the best way to get the parameters of the I–V conditions from practical information acquired in the datasheets.

This paper organizes as follows: Sect. 2 provides the functioning of PV cell, Sect. 3 provides the Ideal diode model of SPVM, Sect. 3 gives the Practical diode model of SPVM, Sect. 4 gives the simulation and experimental results, Sect. 5 concludes the paper.

Functioning of PV cell

The Basic structure of the PV cell, SPVM and SPVA are shown in Fig. 1. Solar cells are equipped with semiconductor materials, known as a semiconductor diode. These cells are having P–N junctions in it, which are kept out to sunlight to get the electrical energy. There are numerous cell manufacturers are available in the PV market, out of which mono-crystalline and poly-crystalline cells are mostly employed.

Fig. 1

Hierarchy of poly-crystalline SPVA

The working of a PV cell is based on the fundamental principle of the photoelectric effect. The photoelectric effect can be defined as a phenomenon in which an electron is ejected from the conductive band as a result of the matter absorbing sunlight from a certain wavelength and gets conducted as shown in Fig. 2. [23].

Fig. 2

Functioning of solar cell [23]

The sun radiates energy at an amount of 3.8 × 1023 (kW/s), nearly 60% of this is 1.08 × 1014 (kW/s) know which strikes the surface of the earth. The remaining energy is reflected into space and also absorbed by this atmosphere. As per the world, energy resources report the annual average Energy received by the Globe is 3400 × 103 (EJ/y). Worldwide solar-based power generation has seen exponential progress, stretching around 227 (GW) toward the finish of 2015, and delivering 1% of all power utilized universally as shown in Fig. 3. [24].

Fig. 3

Comparative primary energy consumption over the past 15 years [24]

The Equivalent Ideal PV diode model is shown in Fig. 4. The ideal diode modelling basic equations from semiconductor theory is represented in (1) and (2) taken from [25], which provides the I–V graph shown in Fig. 5 of SPVM is obtained from (1).

$$I = I_{PV - cell} - I_{d}$$


$$I_{d} = I_{o - cell} \left[ {e^{{\frac{qV}{akT}}} - 1} \right]$$
Fig. 4

Equivalent circuit of PV with Ideal diode model

Fig. 5

Ideal Diode PV model I–V characteristic

Practical diode model

The Practical PV diode equivalent model is shown in Fig. 6 [25], it contains a current source, a diode, series resistance (Rse) and shunt resistance (Rsh). The assessment of SPVA is commonly done by using standard spectral distributions generally, we determine by standard test conditions (STC).

Fig. 6

Equivalent circuit of SPVA with a practical single diode model (SDM)

The STC is specified as follows:

  • Solar Irradiation = 1000 (W/m2);

  • Spectral Density = 1.5 AM;

  • Cell operating Temperature = 25 °C.

The real-world SPVA’s consist of many SPVM’s, that are attached and the observation of characteristics at the terminals involves further parameters to be integrated into the basic equation as given below.

Photo current (Iph)

Iph is given by (3) [25], and the Matlab/Simulink subsystem for Iph is shown in Fig. 7.

$$I_{ph} = \left[ {I_{sc} + k_{i} *(T - 298)} \right]*\frac{G}{1000}$$
Fig. 7

Iph control block of SPVA for practical SDM

Saturation current (Io)

Io is given by (4) [25], which depends on each cell temperature, and the Matlab/Simulink subsystem is shown in Fig. 8.

$$I_{o} = I_{rs} *\left( {\frac{T}{{T_{n} }}} \right)^{3} *e\left[ {\frac{{q*E_{go} *\left( {\frac{1}{{T_{n} }} - \frac{1}{T}} \right)}}{a*k}} \right]$$
Fig. 8

Io control block of SPVA for practical SDM

Reverse saturation current (Irs)

Irs is given by (5) [25], and the Matlab/Simulink subsystem for Irs is shown in Fig. 9.

$$I_{rs} = \frac{{I_{sc} }}{{e^{{\left( {\frac{{q*V_{oc} }}{{a*N_{s} *K*T}}} \right)}} - 1}}$$
Fig. 9

Irs control block of SPVA for practical SDM

Shunt current (Ish)

Ish is given by (6) [25], and the Matlab/Simulink subsystem for Ish is shown in Fig. 10.

$$I_{sh} = \frac{{V + \left( {I*R_{s} } \right)}}{{R_{sh} }}$$
Fig. 10

Ish control block of SPVA for practical SDM

Output current (I)

The output current (I) is given by (7) [25], and the Matlab/Simulink subsystem for I is shown in Fig. 11, the connections of internal control blocks and the complete block of SPVA are shown in Figs. 12 and 13 respectively.

$$I = I_{ph} - I_{o} \left[ {e^{{\frac{{q*\left( {V + I*R_{s} } \right)}}{{a*N_{s} *K*T}}}} - 1} \right] - I_{sh}$$
Fig. 11

Output current control block of SPVA for practical SDM

Fig. 12

Internal control blocks of SPVA for practical SDM

Fig. 13

Complete block of SPVA for practical SDM at STC

SPVA simulation results

An SPVM of 40 (W) is modeled first and two 40 (W) modules are connected in parallel to form the SPVA in this paper. The electrical characteristics of the SPVM and SPVA will provide information on the system operations shown in Figs. 14 and 15 respectively. In order to design the SPVM in Matlab/Simulink platform, a 40 (W) Module from IB Solar-36 is taken into consideration and the module was designed based on the parameter values available in the datasheet. The Key specifications of IB Solar-36 SPVA at 25 °C, 1.5AM, 1000 (W/m2) is given in Table 1.

Fig. 14

Simulated electrical characteristics of 40 (W) SPVM at STC

Fig. 15

Simulated electrical characteristics of SPVA at STC

Table 1 IB Solar-36 SPVA data sheet

Simulated characteristics of SPVM and SPVA

As of Table 1, the electrical characteristics of SPVA are obtained by regulating the equivalent circuit model are detailed in Table 2.

Table 2 Electrical characteristics of SPVA considered for simulation

Effects of temperature variation

The SPVA has the two major controlling blocks namely Iph and Io. The Io varies cubic times the temperature, as given by (3) and (4). Due to the variation in temperature, the I–V and P–V plots are plotted as displayed in Fig. 16.

Fig. 16

Simulated electrical characteristics of SPVA for constant G = 1000 (W/m2) at different T

Effects of irradiance variation

In the SPVA, the effects of Irradiation variation mainly effects on Iph, the relation between irradiance and Iph is detailed in (3). Due to the variation in Irradiation, the I–V and P–V plots are plotted and exhibited in Fig. 17.

Fig. 17

Simulated electrical characteristics of SPVA for constant T = 25 °C at different G

Parasitic effects of SPVA

Parasitic effects impact the reduction of SPVA efficiency by dissipating the power, Rs and Rsh are the utmost parasitic resistances in an SPVA. Figure 6 shows the inclusion of Rs and Rsh on the SDM of SPVA [26]. Mostly in literature frequent method is trial and error method for matching the Simulated and Practical values, but in some papers Inverse slope method is just mentioned but not explained clearly and not formulated in order to get the parameters. In this paper Inverse slope method is approached in a novel way by clearly explaining and formulating the equations in order to get the parasitic effects by using the datasheets of the SPVM. The vital influence of parasitic resistance is the reduction of fill factor (FF), in most cases and for typical values of the Rs and Rsh.

Inverse slope method for finding Rs

The solar cell series resistance has 3 causes: firstly, current movement through the solar cell emitter and base; secondly the metal and silicon touching base resistance; and finally, the resistance of the front and back metallic contacts. The foremost effect of this is to decrease the fill factor and the Isc for extremely high values of Rs. I–V plot slope at Voc point will provide a straightforward method for estimating Rsh for an SDM. In order to get more efficiency from SPVA Rs should as low as possible from normally ranges from 0.01 to 10 (Ω).

$$R_{s} = - \left( {\frac{dV}{dI} - \frac{{V_{T} }}{{I_{sc} }}} \right)$$

whereVT should be calculated by using (9) as follows

$$V_{T} = \frac{{N_{s} *k*a*T_{n} }}{q}$$

From the IB Solar-36 datasheet, I–V plot is taken to calculate the Rs, Fig. 18 shows the I–V characteristics of the IB Solar-36 module with Zoom out a portion for Rs calculation. In this paper from the IB solar datasheet, the dimensions and angles are measured accurately using the Microsoft Visio 2013 software.

Fig. 18

Zoom out view of Portion on X-axis from 35 to 40 V and on Y-axis from 0 to 1 A taken from the IB solar Datasheet

From the Zoom out portion in Fig. 18,

  1. a.

    The angle measured in the I–V plot at the OC is = 96.5°,

  2. b.

    Y-axis scale: 1/2.75 (A/inch),

  3. c.

    X-axis scale: 5/3.1 (V/inch).

Substituting these values to solve for dV/dI and VT as follows,

$$\frac{dV}{dI}_{\text{at OC }} = \tan (90 - 96.5)*\frac{5}{6.425}*\frac{2.75}{1} = - 0.243\;({\text{V}}/{\text{A}})$$
$$V_{T} = \frac{{36*1*1.38*10^{ - 23} *298}}{{1.6*10^{ - 19} }} = 0.925\;({\text{V}})$$
$$\therefore\,R_{s} = - \left( { - 0.243 - \frac{0.925}{2.38}} \right) = 0.631\;(\varOmega )$$

Inverse slope method for finding Rsh

The existence of shunt resistance Rsh, typically results in significant power losses owing to production failures rather than the bad construction of the solar cell. Low Rsh creates power dissipations in solar cells by offering an unusual pathway for the light-produced current. Such alteration will decrease the quantity of electricity that flows through the junction of the solar cell and decreases the voltage. The influence of an Rsh is predominantly severe at low sunny levels, subsequently, there will be a reduced amount of light-produced current. The deduction of this current to the shunt has a greater effect, also at reduced voltages where the Rsh is large. In order to obtain more efficiency from the SPVA the value of Rsh should be larger, normally ranges from 200 to 1000 (Ω). I–V plot slope at Isc point will provide a straightforward method for estimating Rsh for an SDM.

$$R_{sh} = - \frac{dV}{dI}_{\text{at sc}}$$

I–V plot is taken to calculate the Rsh, Fig. 19 shows the I–V characteristics of the IB Solar-36 module with Zoom out a portion for Rsh calculation. For measuring the small angles precisely from the datasheet Angle Center tool is used in the Microsoft Visio 2013 software.

Fig. 19

Zoom out view of portion on Y-axis from 0 to 1 A and 8 to 9 A and on X-axis from 0 to 5 V taken from the IB solar datasheet

From the Zoom out portion in Fig. 19,

  1. a.

    The angle measured in the I–V plot at the OC is = − 0.55°,

  2. b.

    Y-axis scale: 1/2.18 (A/inch),

  3. c.

    X-axis scale: 5/2.825 (V/inch).

Substituting these values to solve for dV/dI and VT as follows,

$$\frac{dI}{dV}_{\text{at sc}} = \tan ( - 0.55)*\frac{1}{2.18}*\frac{5.65}{5} = - 4.976*10^{ - 3} \;({\text{A}}/{\text{V}})$$

The inverse of the slope of the I–V plot at SC gives the Rsh,

$$R_{sh} = \frac{dV}{dI}_{\text{at sc}} \, = - \frac{ 1}{{ - 4. 9 7 6 * 1 0^{ - 3} }} = 200.96\;(\varOmega )$$

For the taken IB Solar-36 module, by using the above expressions Rs and Rsh are calculated and are used to match the output of the simulation and experimental outcomes. The output power is extracted by doing simulation and experimental environment for various values of G and T as displayed in Table 3. It is clear that by properly matching the Parasitic effects of the simulation model with the practical model gives the accurate output power from the SPVA’s in both cases with slight deviation only.

Table 3 IB Solar-36 SPVA simulation and experimental output values powers

The experimental model with two polycrystalline Modules of IB Solar-36 is connected in parallel to form as SPVA is shown in Fig. 20. For the taken datasheet of 40 (W) module, the parasitic effects are calculated as Rs = 0.631 (Ω) and Rsh = 200.96 (Ω) which will result in the proper matching of output powers for both simulation and experimental verifications which is not considered in many papers of literature.

Fig. 20

Experimental setup on IB Solar-36 SPVA


This study clearly presents a novel approach to design the SPVM or SPVA with proper formulations for parasitic effects. These parasitic effects are calculated by taking the I–V plot from the datasheet using the Inverse slope method. The simulation and experimental verification for different values of T, G and with proper parasitic values are done by considering the IB Solar-36 modules. It is observed that the output power obtained from the experimentations is matched accurately with the simulation outcomes. Simulation plots are clearly matched with the curves which were given in the datasheet. This design can be taken as reference and it can be used to explore the study of all kinds of SPVM which is existing in the market. This paper is used as a part of further research-oriented studies to design and development of efficient MPPT techniques, helps to improve the efficiency of SPV power plants operating under partial shading conditions and for designing of SPV Grid tied power plants.


IPV-cell :

Incident light produced current (A)

Io-cell :

Saturation of diode leakage current (A)


Electron charge = 1.6*10−19 (C)

Id :

Equation for the Shockley diode


Boltzmann constant = 1.38 × 10−23 (J/K)


P–N junction temperature (K)


Diode ideality factor usually ≥ 1

Isc :

Short circuit current (A)

Ki :

Isc of SPVM at STC = 0.0032


Irradiance in W/m2 = 1000 (W/m2)

Tn :

Nominal temperature (K) = 298 (K)

Ego :

Energy bandgap of semiconductor = 1.1 (eV)

Voc :

Open circuit voltage (V)

Ns :

Number of cells allied in series


Voltage across the terminals of diode (V)

Rs :

Series resistance (Ω)

Rsh :

Shunt resistance (Ω)


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Kotla, R.W., Yarlagadda, S.R. Mathematical modelling of SPV array by considering the parasitic effects. SN Appl. Sci. 2, 50 (2020).

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  • SPVM
  • SPVA
  • Shunt resistance
  • Series resistance