Background

Mathematical modeling of PV module is being continuously updated to enable researchers to have a better understanding of its working. The models differ depending on the types of software researchers used such as C-programming, Excel, Matlab, Simulink or the toolboxes they developed.

A function in Matlab environment has been developed to calculate the current output from data of voltage, solar irradiation and temperature in the study of (Walker 2001) and (Gonzalez-Longatt 2005). Here, the effect of temperature, solar irradiation, and diode quality factor and series resistance is evaluated. A difficulty of this method is to require readers programming skills so it is not easy to follow. Another method which is the combination between Matlab m-file and C-language programming is even more difficult to clarify (Gow and Manning 1999).

Among other authors, a proposed model is based on solar cell and array’s mathematical equations and built with common blocks in Simulink environment in (Salmi et al. 2012), (Panwar and Saini 2012), (Savita Nema and Agnihotri 2010), and (Sudeepika and Khan 2014). In these studies, the effect of environmental conditions (solar insolation and temperature), and physical parameters (diode’s quality factor, series resistance Rs, shunt resistance Rsh, and saturation current, etc.) is investigated. One disadvantage of these papers is lack of presenting simulation procedure so it causes difficulties for readers to follow and simulate by themselves later. This disadvantage is filled in by (Jena et al. 2014), (Pandiarajan and Muthu 2011). A step-by-step procedure for simulating PV module with subsystem blocks with user-friendly icons and dialog in the same approach with Tarak Salmi and Savita Nema is developed by Jena, Pandiarajan and Muthu et al. However, the biggest gap of the studies mentioned above is shortage of considering the effect of partially shading condition on solar PV panel’s operation.

In other researches, authors used empirical data and Lookup Table or Curve Fitting Tool (CFtool) to build P–V and I–V characteristics of solar module (Banu and Istrate 2012). The disadvantage of this method is that it is quite challenging or even unable to collect sufficient data if no experimental system be available so that modeling curves cannot be built and modeled.

From the work of (Ibbini et al. 2014) and (Venkateswarlu and Raju 2013), a solar cell block which has already been built in Simscape/Simulink environment is employed. With this block, the input parameters such as short circuit current, open circuit voltage, etc. is provided by manufacturers. The negative point of this approach is that some parameters including saturation current, temperature, and so on cannot be evaluated.

Solar model developed with Tag tools in Simulink environment is recorded in the research of (Varshney and Tariq 2014), (Mohammed 2011), etc. In these papers, only two aspects (solar irradiation and temperature) are investigated without providing step-by-step simulation procedure.

In overall, although having advantages and disadvantages, different methods have similar gaps as follows:

  • The proposed models are not totally sufficient to study all parameters which can significantly affect to I–V and P–V characteristics of solar PV array, including physical parameters such as saturation current, ideality factor, series and shunt resistance, etc. and environmental working conditions (solar insolation, temperature and especially shading effect).

  • Lack of presenting step-by-step simulation procedure and this causes difficulties for readers and researchers to follow and do simulation by themselves.

Therefore, the study proposes a robust model built with Tag tools in Simulink environment. The proposed model shows strength in investigating all parameters’ influence on solar PV array’s operation. In addition, a unique step-by-step modeling procedure shown allows readers to follow and simulate by themselves to do research.

Methods

Mathematical equivalent circuit for photovoltaic array

The equivalent circuit of a PV cell is shown in Fig. 1. The current source Iph represents the cell photocurrent. Rsh and Rs are the intrinsic shunt and series resistances of the cell, respectively. Usually the value of Rsh is very large and that of Rs is very small, hence they may be neglected to simplify the analysis (Pandiarajan and Muthu 2011). Practically, PV cells are grouped in larger units called PV modules and these modules are connected in series or parallel to create PV arrays which are used to generate electricity in PV generation systems. The equivalent circuit for PV array is shown in Fig. 2.

Fig. 1
figure 1

PV cell equivalent circuit (Salmi et al. 2012)

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Fig. 2
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Equivalent circuit of solar array (Tu and Su 2008)

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The voltage–current characteristic equation of a solar cell is provided as (Tu and Su 2008; Salmi et al. 2012):  Module photo-current Iph:

$$I_{ph} = [I_{sc} + K_{i} (T - 298)] \times Ir/1000$$
(1)

Here, Iph: photo-current (A); Isc: short circuit current (A) ; Ki: short-circuit current of cell at 25 °C and 1000 W/m2; T: operating temperature (K); Ir: solar irradiation (W/m2).

Module reverse saturation current Irs:

$$I_{rs} = I_{sc} /[\exp (qV_{OC} /N_{S} knT) - 1]$$
(2)

Here, q: electron charge, = 1.6 × 10−19C; Voc: open circuit voltage (V); Ns: number of cells connected in series; n: the ideality factor of the diode; k: Boltzmann’s constant, = 1.3805 × 10−23 J/K.

The module saturation current I0 varies with the cell temperature, which is given by:

$$I_{0} = I_{rs} \left[ {\frac{T}{{T_{r} }}} \right]^{3} \exp \left[ {\frac{{q \times E_{g0} }}{nk}\left( {\frac{1}{T} - \frac{1}{{T_{r} }}} \right)} \right]$$
(3)

Here, Tr: nominal temperature = 298.15 K; Eg0: band gap energy of the semiconductor, = 1.1 eV; The current output of PV module is:

$$\begin{aligned}I &= N_{P} \times I_{ph} - N_{P} \times I_{0} \\ & \quad \times \left[ {\exp \left( {\frac{{V/N_{S} + I \times R_{s} /N_{P} }}{{n \times V_{t} }}} \right) - 1} \right] - I_{sh} \end{aligned}$$
(4)

With

$$V_{t} = \frac{k \times T}{q}$$
(5)

and

$$I_{sh} = \frac{{V \times N_{P} /N_{S} + I \times R_{S} }}{{R_{sh} }}$$
(6)

Here: Np: number of PV modules connected in parallel; Rs: series resistance (Ω); Rsh: shunt resistance (Ω); Vt: diode thermal voltage (V).

Reference model

The 100 W solar power module is taken as the reference module for simulation and the detailed parameters of module is given in Table 1.

Table 1 Electrical characteristics data of DS-100 M PV module
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Step by step procedure for modeling of photovoltaic arrays with tags

A mathematical model of PV array including fundamental components of diode, current source, series resistor and parallel resistor is modeled with Tags in Simulink environment (http://mathwork.com). The simulation of solar module is based on equations given in the section above and done in the following steps.

Step 1

Provide input parameters for modeling:

Tr is reference temperature = 298.15 K; n is ideality factor = 1.2; k is Boltzmann constant = 1.3805 × 10−23 J/K; q is electron charge = 1.6 × 10−19; Isc is PV module short circuit current at 25 °C and 1000 W/m2 = 6.11 A; Voc is PV module open circuit voltage at 25 °C and 1000 W/m2 = 0.6 V; Eg0 is the band gap energy for silicon = 1.1 eV. Rs is series resistor, normally the value of this one is very small, = 0.0001 Ω; Rsh is shunt resistor, the value of this is so large, = 1000 Ω (Fig. 3).

Fig. 3
figure 3

Input parameters for simulation model

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Step 2

Module photon-current is given in Eq. (1) and modeled as Fig. 4 (Ir0 = 1000 W/m2).

$$ I_{ph} = [I_{sc} + K_{i} (T - 298)] \times Ir/1000 $$
(7)

Step 3

Fig. 4
figure 4

Modeled circuit for Eq. (1)

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Module reverse saturation current is given in Eq. (2) and modeled as Fig. 5.

Fig. 5
figure 5

Modeled circuit for Eq. (2) and (5)

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Step 4

Module saturation current I0 is given in Eq. (3) and modeled as Fig. 6.

Fig. 6
figure 6

Modeled circuit for Eq. (3)

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Step 5

Modeled circuit for Eq. (6) (Fig. 7).

Fig. 7
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Modeled circuit for Eq. (6)

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Step 6

Modeled circuit for Eq. (4) (Fig. 8).

Fig. 8
figure 8

a Modeled circuit for Eq. (4). b Solar subsystem simulation model

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Step 7

The solar module simulation procedure is shown from Fig. 3 to Fig. 8b. The solar PV array includes six modules and each module has six solar cells connected in series. Therefore, the proposed model of solar PV array is given in Fig. 9.

Fig. 9
figure 9

Simulation model of solar PV array

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Experimental test

In order to validate the Matlab/Simulink model, the PV test system of Fig. 10 is installed. It consists of a rheostat, a solar irradiation meter, two digital multi-meters and a solar system of two DS-100M panels connected in series, each panel has the key specifications listed in Table 1.

Fig. 10
figure 10

Setup of the experimental solar system

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Result and discussion

Simulation scenario

With the developed model, the PV array characteristics are estimated as follows.

  1. (i)

    I–V and P–V characteristics under varying irradiation with constant temperature are given in Fig. 11a and b. Here, the solar irradiation changes with values of 100, 500 and 1000 W/m2 while temperature keeps constant at 25 °C.

    Fig. 11
    figure 11

    a Output I–V curve with varying irradiation. b Output P–V curve with varying irradiation

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    Summary when the irradiation increases, the current and voltage output increase. This results in rise in power output in this operating condition.

  2. (ii)

    I–V and P–V characteristics under varying temperature and constant irradiation are obtained in Fig. 12a and b. Here, the temperature varies from 25 to 50 and 75 °C respectively whereas the irradiation level keeps constant at 1000 W/m2.

    Fig. 12
    figure 12

    a Output I–V curve with varying temperature. b Output P–V curve with varying temperature

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    Summary when the operating temperature increases, the current output raises marginally but the voltage output decreases drastically. This leads to net reduction in power output with rise in temperature.

  3. (iii)

    I–V and P–V characteristics under varying shunt/parallel resistance Rsh, constant temperature and irradiation are shown in Fig. 13a and b. In this case, Rsh changes with three values of 0.1, 1 and 1000 Ω, respectively.

    Fig. 13
    figure 13

    a Output I–V curve with varying Rsh. b Output P–V curve with varying Rsh

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    Summary when Rsh varies between 1000 and 1 Ω, the current output and voltage output decreases slightly and this results in slight net reduction in power output. However, a significant decrease in current, voltage and power output is recorded when the value of Rsh is 0.1 Ω.

  4. (iv)

    I–V and P–V characteristics under varying Ns and Np are obtained in Fig. 14a and b. In practice, PV cells are connected in series into PV module and these PV modules then are connected in series or parallel to form PV array for generating more electricity from sunlight. The reference model is 36-series-connected-cell array so two cases are studied: two modules are connected in series and two modules are connected in parallel.

    Fig. 14
    figure 14

    a I–V curve with varying Ns and Np. b P–V curve with varying Ns and Np

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    Summary

    • With two modules connected in series (Ns = 72, Np = 1), the value of current output is similar to that of it in case of one module (Ns = 36, Np = 1) but the voltage output doubles so the power output doubles.

    • In term of two modules connected in parallel (Ns = 36, Np = 2), the value of voltage output is similar to that of it in case of one module (Ns = 36, Np = 1) but the current output doubles so the power output doubles. Similar value of power output is experienced in both cases of two modules despite different ways in module connection (parallel or series).

    The proposed model has advantages not only in studying effect of physical parameters such as series resistance Rs, shunt resistance Rsh, etc. but also in investigating impact of environmental condition like varying temperature, solar irradiation and especially shading effect. In this study, the evaluation of shading effect on solar PV array’s operation is carried out through following cases. The simulation results are given in Fig. 15a and b.

    Fig. 15
    figure 15

    a I–V curves under partial shading condition. b P–V curves under partial shading condition

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    Case

    Description

    1

    No shaded PV module (full irradiation on solar PV array): 1000 W/m2

    2

    One shaded module (receives irradiation of 500 W/m2), others receive full irradiation of 1000 W/m2

    3

    Two shaded modules (receive irradiation of 500 W/m2), others receive full irradiation of 1000 W/m2

    4

    Two shaded modules (receive irradiation of 500 and 250 W/m2), others receive irradiation of 1000 W/m2

    Summary

    • The power output of PV array reduces noticeably when it works under partial shading condition.

    • The I–V curve experiences multiple steps whereas the P–V curve gives many local peaks along with the maximum power point (the global peak). In addition, more shaded modules are higher number of power output peaks is shown.

Experimental results and validation

The Matlab/Simulink model is evaluated for the experimental test system (two DS-100M panels are connected in series). The results are shown in Fig. 16. On the other hand, the empirical results with a solar irradiation of 520 W/m2 and operating temperature of 40 °C are given in Fig. 17. The I–V and P–V simulation and experimental results show a good agreement in terms of short circuit current, open circuit voltage and maximum power output.

Fig. 16
figure 16

Solar Matlab/SIMULINK results

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Fig. 17
figure 17

Solar system experimental results

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Conclusion

A step-by-step procedure for simulating a PV array with Tag tools, with user-friendly icons and dialogs in Matlab/Simulink block libraries is shown. This modeling procedure serves as an aid to help people to closer understand of I–V and P–V operating curves of PV module. In addition, it can be considered as a robust tool to predict the behavior of any solar PV cells, modules and arrays under varying environmental conditions (temperature, irradiation and partially shading condition) and physical parameters (series resistance, shunt resistance, ideality factor and so on). This research is the first step to study a hybrid system where a PV power generation connecting to other renewable energy production sources like wind or biomass energy systems.