Harvesting of renewable solar energy has grown rapidly over the past decade due to the availability of cheap and affordable modules and deep-cycle energy storage systems [1]. Although the installation of solar photovoltaic systems faces different challenges, the solar power has the highest potential in the world as a major source of clean energy [2] [3]. Some of these challenges include diverse environmental factors such as varying solar irradiance and temperatures, dust and shades, low solar cell efficiency and high installation costs [4]. These drawbacks have attracted numerous research works for tracking the optimum power generated by a photovoltaic module at various environmental conditions in order to improve its efficiency [5] - [10]. Photovoltaic systems should be optimized to work at the maximum power for any solar irradiation level and ambient temperature. Modeling and simulation of the photovoltaic systems gives a better understanding of the maximum power point using characteristic curves [11].

A single diode model of a solar system has been studied for decades since it offers an elaborate, simple and reliable analysis of the current-voltage characteristics of solar cells [12] [13] [14]. The model requires extremely thorough and careful computation of I_ph, I_o, n, R_s and R_sh parameters that are based on the equivalent circuit analysis using Schottky’s diode equation [15].

Several techniques based on soft computing have been studied for unknown parameters determination using evolutionary algorithms [16]. These methods are strongly convergent and have less computing time. However, due to their stochastic nature, their efficiency depends on the choice of control parameters and search ranges which require high computational power [17] [18].

Analytical and numerical methods are the traditional techniques for estimating all parameters of a single diode photovoltaic model [19] [20]. The analytical approach relies heavily on the availability of data for short-circuit current (I_sc), maximum power point voltage (V_mpp), maximum power point current (I_mpp), open-circuit voltage (V_oc) and IV-curve intersection slopes [21] [22] [23]. Several authors have used information provided from the manufacturer’s datasheets for I_sc, V_oc, I_mpp, V_mpp to determine the photovoltaic parameters using nonlinear least square (NLS) algorithm [24], normalized root mean-square deviation (NRMSD) [25] [26], Newton-Raphson algorithm [27] [28] and Lambert W Function [29] [30] [31] [32] [33]. These approaches offer quick, robust and faster ways of extracting the parameters of a single-diode photovoltaic model. The precision of these approaches is based primarily on the accuracy of the data provided for the three crucial points on the I-V curve [20]. Standard photovoltaic module datasheet values for I_sc, V_oc, I_mpp and V_mpp are approximated to one or two decimal places while P_mpp values are given in the nearest tenth for all modules with the same power rating. However, each solar module has its individual values for each of these points at STC [34]. Therefore datasheet values can introduce significant errors in simulating the I-V or P-V curves. Numerical methods, on the contrary, are based on iterative algorithms for fitting simulated I-V curves into the experimental data [20] [35] [36]. This paper presents combined analytical and numerical approaches for determining the ideality factor (n), diode saturation current (I_o), photocurrent (I_ph), series (R_s) and shunt (R_sh) resistances for a single diode photovoltaic model. The analytical approach gives a straightforward, simple and rapid way of extracting ideality factor and saturation current by approximating their optimum values using three critical points from either the data sheet and/or the experimental data. The numerical approach gives the precise values of the ideality factor and saturation current in the proximity of optimal ideality factor (n_o) and optimal saturation current (I_opt) respectively. Further, series (R_s) and shunt (R_sh) resistances that are not provided in the manufacturer’s datasheet are determined using iterative algorithms. Finally, the photo current is explicitly determined using the extracted n, I_o, R_s, R_sh and datasheet or experimental values. A comparison of simulated I-V and P-V curves from datasheet and experimental data values is also presented for different environmental conditions.

Figure 1 shows a single diode equivalent circuit that can be evaluated using Equation (1). A current source is connected in series to R_s and in parallel to the Shockley’s diode [37] and shunt resistor R_sh.

The equivalent circuit can be presented mathematically by

I = I p h − I o ( exp ( q ( V + I R s ) n K T ) − 1 ) − V + I R s R s h . (1)

where; T = 298.15 K, q is the charge of an electron = 1.602176634 × 10⁻¹⁹ C and k is the Boltzmann’s Constant = 1.380649 × 10⁻²³ m²∙s⁻²∙kg∙K⁻¹.

The critical points of I-V and P-V curves for a photovoltaic system are the short circuit (SC), the maximum power point (MPP) and open-circuit (OC). These points play an important role since a quick look at their values gives a clear picture of the photovoltaic module characteristic and performance. We can use Equation (1) and rearrange it at each point as follows:

1) At short circuit, I = I_sc, V = 0;

We can rewrite Equation (1) as

I s c = I p h − I o ( exp I s c R s n N s V t − 1 ) − I s c R s R s h . (2)

or

I p h = I s c + I o ( exp I s c R s n N s V t − 1 ) + I s c R s R s h . (3)

where V t = k T / q = 0.025692607 is the thermal voltage.

2) At Open Circuit, I = 0, V = V_oc;

Equation (1) can be rearranged as

I p h = I o ( exp V o c n N s V t − 1 ) + V o c R s h . (4)

3) At Maximum Power Point, I = I_mpp, V = V_mpp;

We can similarly replace I = I_mpp, V = V_mpp in equation to obtain

I m p p = I p h − I o ( exp V m p p + I m p p R s n N s V t − 1 ) − V m p p + I m p p R s R s h . (5)

The above equations can be used to evaluate and determine the five unknown parameters using the experimental or manufacturer’s data as discussed in the following sections.

The transcendental Equation (1) has five unknown parameters that must be determined in order to have a model that represents the experimental data. These parameters include photocurrent (I_ph), ideality factor (n), saturation current (I_o), series (R_s) and shunt (R_sh) resistances that can be derived using I_sc, I_mpp, V_oc and V_mpp.

The following sections 4.1 to 4.3, addresses a detailed mathematical derivation of I_ph, I_o and n equations, outlining the disadvantages and benefits of each method. Section 4.4 discusses analytical approaches for R_s and R_sh determination.

The previous sections have demonstrated a simplified approach of obtaining the

model parameters. The approach heavily depended on the values of I_sc, I_mpp, V_oc and V_mpp; the main cardinal points. This can be considered as a rough understanding of the solution of Equation (1). We have thus applied Newton-Raphson technique to iteratively solve Equation (1) in order to consider all points in characterization of photovoltaic modules using I-V and P-V curves. The Newton-Raphson technique is based on estimation of a given function f(I) = 0 [54].

We can rearrange Equation (1) as,

f ( I ) = I p h − I o [ exp ( V + I R s n N s V t ) − 1 ] − V + I R s R s h − I = 0 . (44)

Differentiating Equation (44) with. respect to I leads to

∂ ( f ( I ) ) ∂ I = − I o R s n N s V t [ exp ( V + I R s n N s V t ) ] − R s R s h − 1 . (45)

Therefore, we can apply Equations (44) and (45) to implement the Newton-Raphson approach and obtain

I j + 1 = I j − f ( I j ) ∂ ( f ( I j ) ) ∂ I = I j − I p h − I o [ exp ( V + I j R s n N s V t ) − 1 ] − V + I j R s R s h − I j − I o R s n N s V t [ exp ( V + I j R s n N s V t ) ] − R s R s h − 1 (46)

where, j represents the number of iterative process.

Equations (44)-(46) require initial values of n and R_s. In this paper, we introduce the new approach presented in section 4.4 that easily determines the trio values of n, R_s and R_sh. These values are also used to calculate I_o and I_ph. Implementation of all five parameter values in Equations (44)-(46) reduces the number of iterative processes needed to obtain the most suitable I and V values, thereby increasing the computation speed.

The identified values of I and V are applied in the power equation given by

P = I p h V − I o V [ exp ( V + I R s n N s V t ) − 1 ] − V 2 R s h − V I R s R s h . (47)

The values of simulated parameters listed in Table 5 & Table 6 for Solinc 120 W and KC200GT, have been used to plot the I-V and P-V curves at standard test condition.

Figure 5 shows the current-voltage relationship for Solinc 120 W, in which the zoomed parts illustrate the short circuit, maximum power and open circuit points for all the four approaches. The curves converge at maximum power point but vary significantly at the point of short circuit and at the point of open circuit. The second approach gives the best replica of these cardinal points.

Figure 6 shows the power-voltage relationship for Solinc 120 W solar module, in which the zoomed sections show the areas surrounding the maximum power and the open circuit points. The zoomed sections show the differences between the four approaches, which is a reflection of the values of errors given in Table 4.

The current-voltage relationship for KC200GT is shown in Figure 7, in which the zoomed sections represent the short circuit, maximum power and open circuit points. The curves converge at maximum power point but vary greatly at the point of the short circuit and at the point of the open circuit. Again, the second method offers the best possible representation of these cardinal points.

Figure 8 displays the power-voltage relationship for KC200GT solar module, in which the zoomed parts represent the areas surrounding the maximum power and the open circuit points. The zoomed sections display the differences between the four methods, which is a representation of error values given in Table 5. According to the I-V and P-V plots of Figures 5-8, the four analytical approaches give satisfactory parameters values for a single diode model that matches the experimental data and manufacturer’s profile.

The effects of the actual solar irradiation (s_a) and module’s surface temperature T on five-model parameters have been recently studied at open circuit, closed circuit and maximum power points [38] [51].

Starting with the cardinal points, the short circuit current can be determined using

I s c ( s a , T ) = s a s S T C [ I s c S T C + K I s c ( T − T S T C ) ] (48)

where, K I s c is the temperature coefficient of I_sc in A/˚C that is usually provided in manufacturer’s data sheet. The maximum power-point current can be determined using

I m p p ( s a , T ) = s a s S T C [ I m p p S T C + K I s m p p ( T − T S T C ) ] (49)

where, K I m p p is the temperature coefficient of I_mpp in A/˚C. Unlike K I s c , the manufacturer’s data sheet does not include K I m p p . This can be evaluated by simply applying the data at both STC and NOCT.

Manufacturers of solar modules with ISO/IEC International Standard provide, nominal operation cell temperature (NOCT) values at 800 W/m² at 20˚C [55] [56]. This can be used to relate the module temperature (T) with the actual irradiance and the air temperature (T_a) using

T = T a + [ T N O C T − 20 ] s a 800 (50)

For maximum power-point voltage a more robust analytical approach has been recently used by [38], in which

V m p p ( s a , T ) = V m p p S T C + K v m p p ( T − T S T C ) + α v m p p ( s a − s S T C ) + β v m p p ( s a − s S T C ) 2 (51)

where, α V m p p and β V m p p are solar radiation coefficients at maximum-power point. Similarly, the short-circuit voltage can be evaluated as

V o c ( s a , T ) = V o c S T C + K v o c ( T − T S T C ) + α V o c ( s a − s S T C ) + β V o c ( s a − s S T C ) 2 (52)

where α V o c , β V o c are solar radiation coefficients at open circuit point.

Equations (51) and (52) are quadratic polynomials that require a careful extraction of the second-degree polynomial coefficients. In this paper, we have introduced a simpler approach for determining V_oc (s_a, T) and V_mpp (s_a, T) using

V o c ( s a , T ) = n ( s a , T ) N s V t ( T ) [ ln ( I s c ( s a , T ) ) − ln ( I o ( s a , T ) ) ] (53)

and,

V m p p ( s a , T ) = V o c ( s a , T ) − n o N s V t ( T ) × [ ln ( I s c ( s a , T ) I s c ( s a , T ) − I m p p ( s a , T ) ) ] (54)

The saturation current dependence on module temperature can be achieved by rewriting Equation (20) as

I o ( s a , T ) = I o S T C [ T T S T C ] 3 exp − q E g n ( s a , T ) N s ( 1 T S T C − 1 T ) . (55)

The temperature-dependent saturation current of Equation (55) can be compared with I_o of Equation (56), in which Equation (14) has been written as

I o = I s c ( s a , T ) exp ( V o c ( s a , T ) n ( s a , T ) N s V t ( T ) ) . (56)

Finally, the R_s and R_sh pair dependency on irradiance and temperature have been evaluated using Equation (27), by replacing its I_ph, I_o, I_mpp and V_mpp with values extracted using Equations ((7), (56), (49) and (54)) respectively.

The Kyocera KC200GT high-performance multi-crystal photovoltaic module with IEC standard has been used to demonstrate the effects of irradiance and temperature on main parameters of a single diode model as shown in Table 7 & Table 8. The datasheet module offers nominal operating cell temperature data at 47˚C and 800 W/m² for the three cardinal points that have been used as starting

conditions to evaluate other parameters at various irradiances. In Table 7, the simulated I_sc (s_a, T), I_mpp (s_a, T), V_mpp (s_a, T) and V_oc (s_a, T) data have been applied in approach 2 discussed in section 5.1 to extracted I_ph (s_a, T), I_o (s_a, T), n (s_a, T) R_s (s_a, T) and R_sh (s_a, T) and plot I-V and P-V curves at various irradiances.

The values of α V m p p , α V o c , β V m p p and β V o c presented in the supplementary data published by [38], have been applied to determine V_mpp and V_oc of equations (51) and (52). The simulated values of V_oc using Equation (53) and V_mpp using Equation (54) for irradiances chosen at 200, 400, 600, 800 and 1000 W/m², have been compared with the values obtained based on the [38] method as shown in Table 7. The data obtained using the new approaches are consistent with data simulated using [38] method. Figure 9 & Figure 10 illustrate the I-V and P-V curves at irradiances of 200, 400, 600, 800 and 1000 W/m², and air temperature of 20˚C and NOCT of 47˚C, while Figure 11 & Figure 12 show I-V and P-V curves at various temperatures.

In this report, we have considered photovoltaic systems operating at STC and various weather conditions and have presented two algorithms for extracting their five-model parameters based on a single-diode analogous circuit. The first algorithm plays an important role in deriving the unknown parameters to give a rough idea of their values that are used as preliminary data for the second algorithm based on Newton-Raphson numerical analysis method. This is a deviation from conventional methods, which assume initial arbitrary values.

In an attempt to establish the most comprehensive and simple procedure of arriving at the best five-model parameters, we categorized four approaches based

of saturation-current relationships at the cardinal points of I-V plot. We considered the effect of shunt and series resistance on the saturation current at each key cardinal point or combination of two points, and found that dependence on the actual resistance induces a slight deviation of the saturation current as opposed to extreme resistance. When applied to the Newton-Raphson numerical analysis method, all four approaches provided satisfactory output current and voltage values that match the experimental data or data presented in manufacturer’s datasheet.

Beginning with the numerical values of the five-model parameters at STC, we simulated the five-model parameters at various irradiances and temperatures. We have presented new approaches to obtaining the V_oc and V_mpp at various irradiances and temperatures.

The authors would like to thank the Department of Physics and the Board of Graduate School; University of Nairobi (Kenya) for providing all the facilities and financial support respectively for this research.

The authors declare no conflicts of interest regarding the publication of this paper.

Ndegwa, R., Ayieta, E., Simiyu, J. and Odero, N. (2020) A Simplified Simulation Procedure and Analysis of a Photovoltaic Solar System Using a Single Diode Model. Journal of Power and Energy Engineering, 8, 65-93. https://doi.org/10.4236/jpee.2020.89006