A Simplified Simulation Procedure and Analysis of a Photovoltaic Solar System Using a Single Diode Model

Robinson Ndegwa, Elijah Ayieta, Justus Simiyu, Nicodemus Odero

Department of Physics, University of Nairobi, Nairobi, Kenya.

DOI: 10.4236/jpee.2020.89006                

Abstract

A single diode model for a photovoltaic solar module is the most ideal and quick way of analyzing the module characteristics before implementing them in a solar plant. Solar modules manufacturers provide information for three critical points that are essential in I-V, P-V or P-I curves. In this study, we propose four separate simulation procedures to estimate the five-model parameters of an analogous single diode equivalent circuit by utilizing three cardinal points of the photovoltaic module I-V curve, described from experimental data using a solar simulator and manufacturer’s datasheet. The main objective is to extract and use the five unknown parameters of a single diode model to describe the photovoltaic system using I-V ad P-V plots under different environmental conditions. The most influential parameters that greatly alter the cardinal points defined at short circuit point (SCP), the maximum power point (MPP) and the open circuit point(OCP) are the ideality factor (n) and the diode saturation current (I_o). For a quick and fast convergence, we have determined the optimal ideality factor (n_o) and optimal saturation current (I_oopt) as the primary parameters by first assuming the optimal values of R_sh, R_s and I_ph at standard test conditions (STC). Further, we evaluated the effects of I_ph, R_s and R_sh on I-V and P-V curves by considering the values of n below n_o. We have evaluated different iterative procedures of determining R_sh and R_s at open-circuit, short-circuit point and the maximum-power points. These procedures have been classified into four approaches that guarantees positive shunt and series resistance for n ≤ n_o. These approaches have been categorized by deriving the saturation current as a dependent variable at each cardinal point with or without R_s and R_sh pair. The values obtained for the five parameters have been used to simulate the photovoltaic solar module characteristic curves with great precision at different air temperatures and irradiances, considering the effect of Nominal Operating Cell Temperature (NOCT).

Keywords

Five-Parameter Model, Ideality Factor, Saturation Current, Series and Shunt Resistances

1. Introduction

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.

2. A Single Diode Model

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

$I=I_{ph}-I_o\left(\exp \left(\frac{q\left(V+I{R}_s\right)}{nKT}\right)-1\right)-\frac{V+I{R}_s}{R_{sh}}$. (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.

Figure 1. A single diode equivalent circuit.

3. Evaluation of a Single Diode Model at Three Critical Points in I-V and P-V Curves

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

$I_{sc}=I_{ph}-I_o\left(\exp \frac{I_{sc}{R}_s}{n{N}_s{V}_t}-1\right)-\frac{I_{sc}{R}_s}{R_{sh}}$. (2)

or

$I_{ph}=I_{sc}+I_o\left(\exp \frac{I_{sc}{R}_s}{n{N}_s{V}_t}-1\right)+\frac{I_{sc}{R}_s}{R_{sh}}$. (3)

where $V_t=kT/q=0.025692607$ is the thermal voltage.

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

Equation (1) can be rearranged as

$I_{ph}=I_o\left(\exp \frac{V_{oc}}{n{N}_s{V}_t}-1\right)+\frac{V_{oc}}{R_{sh}}$. (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

$I_{mpp}=I_{ph}-I_o\left(\exp \frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}-1\right)-\frac{V_{mpp}+I_{mpp}{R}_s}{R_{sh}}$. (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.

4. Determination of Unknown Parameters for a Single Diode Photovoltaic Model

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.

4.1. Photocurrent (I_ph) Analysis

The photocurrent (I_ph) can be determined from Equations ((2), (3)) or by rewriting Equation (5) as

$I_{ph}=I_{mpp}+I_o\left(\exp \frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}-1\right)+\frac{V_{mpp}+I_{mpp}{R}_s}{R_{sh}}$. (6)

However, I_ph depends on the solar irradiance and module surface temperature (T). Therefore, the relationship between I_ph, T and actual irradiance (s_a) can be deduced using temperature coefficient of short circuit current (K_I) as discussed in [27] [38] [39]. Thus,

$I_{ph}=\frac{s_a}{s_{STC}}\left[I_{p{h}_{STC}}+K_I\left(T-T_{STC}\right)\right]$. (7)

4.2. Saturation Current (I_o)

The saturation current can be evaluated using Equation (1) at the three critical points by maintaining constant temperature at the standard test condition, where T_STC = 25˚C.

4.2.1. Saturation Current (I_o) at Each Critical Point

1) At the short circuit, Equation (2) can be rearranged to give

$I_o=\frac{I_{ph}{R}_{sh}-I_{sc}{R}_{sh}-I_{sc}{R}_s}{R_{sh}\left(\exp \left(\frac{I_{sc}R{}_s}{n{N}_s{V}_t}\right)-1\right)}$. (8)

2) At maximum power point, Equation (5) can be reorganized to obtain

$I_o=\frac{I_{ph}{R}_{sh}-I_{mpp}{R}_{sh}-V_{mpp}-I_{mpp}{R}_s}{R_{sh}\left(\exp \left(\frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}\right)-1\right)}$. (9)

Setting the boundary condition of R_s ≈ 0, R_sh ≈ ∞ and I_ph ≈ I_sc, Equation (9) yields

$I_o=\frac{I_{sc}-I_{mpp}}{\exp \frac{V_{mpp}}{n{N}_s{V}_t}-1}$. (10)

3) At the open circuit, Equation (4) can be rearranged to give

$I_o=\frac{I_{ph}{R}_{sh}-V_{oc}}{R_{sh}\left(\exp \left(\frac{V_{oc}}{n{N}_s{V}_t}\right)-1\right)}$. (11)

Similarly, setting boundary condition of R_sh ≈ ∞ and I_ph ≈ I_sc, Equation (11) yields

$I_o=\frac{I_{sc}}{\exp \left(\frac{V_{oc}}{n{N}_s{V}_t}\right)-1}$. (12)

4.2.2. Saturation Current (I_o) Calculation by Combining Two Out of Three Critical Point Equations

The saturation current can also be calculated by combining two of either Equations ((3), (4) or (6)). Subtracting Equations ((3) and (4)) eliminates I_ph as reported by [27] [40] and [41]. This reduces to

${\left[I_o\right]}_{I_{sc},V_{oc}}=\frac{I_{sc}{R}_{sh}+I_{sc}{R}_s-V_{oc}}{R_{sh}\left(\exp \left(\frac{V_{oc}}{n{N}_s{V}_t}\right)-\exp \left(\frac{I_{sc}{R}_s}{n{N}_s{V}_t}\right)\right)}$. (13)

Once more, taking R_s ≈ 0 and R_sh ≈ ∞, we can rewrite Equation (13) as

${\left[I_{o_{opt}}\right]}_{I_{sc},V_{oc}}=\frac{I_{sc}}{\exp \left(\frac{V_{oc}}{n{N}_s{V}_t}\right)}$. (14)

Similarly, Equation (3) can be merged with Equation (6) at I_sc and P_mpp to obtain

${\left[I_o\right]}_{I_{sc},P_{mpp}}=\frac{V_{mpp}+I_{mpp}{R}_{sh}+I_{mpp}{R}_s-I_{sc}{R}_s-I_{sc}{R}_{sh}}{R_{sh}\left(\exp \left(\frac{I_{sc}{R}_s}{n{N}_s{V}_t}\right)-\exp \left(\frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}\right)\right)}$. (15)

Returning to R_s ≈ 0 and R_sh ≈ ∞, Equation (15) reduces to

${\left[I_{o_{opt}}\right]}_{I_{sc},P_{mpp}}=\frac{I_{sc}-I_{mpp}}{\exp \left(\frac{V_{mpp}}{n{N}_s{V}_t}\right)}$. (16)

Finally, considering Equations ((4) and (6)) at V_oc and P_mpp, we can deduce the saturation current as

${\left[I_o\right]}_{V_{oc},P_{mpp}}=\frac{V_{mpp}-V_{oc}+I_{mpp}{R}_{sh}+I_{mpp}{R}_s}{R_{sh}\left(\exp \left(\frac{V_{oc}}{n{N}_s{V}_t}\right)-\exp \left(\frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}\right)\right)}$. (17)

Again, assuming R_s ≈ 0 and R_sh ≈ ∞, we can rewrite Equation (17) as

${\left[I_{o_{opt}}\right]}_{V_{oc},P_{mpp}}=\frac{I_{mpp}}{\exp \left(\frac{V_{oc}}{n{N}_s{V}_t}\right)-\exp \left(\frac{V_{mpp}}{n{N}_s{V}_t}\right)}$. (18)

4.2.3. Dependence of the Saturation Current on Temperature

The dark saturation current has been reported to be independent of irradiance and has been regarded as the reverse saturation current which is the reverse current in a solar cell caused by diffusion of minority carriers from the neutral regions to the depletion region in the absence of irradiation [42] [43]. However, the dark saturation current strongly depends on the parameters of the temperature, the cross-sectional area of semiconductor and the concentration of the intrinsic carrier [43] [44]. The intrinsic carrier concentration number also depends on the state conduction and valence band densities and the semiconductor energy band-gap (E_g) [44]. Therefore, as discussed by [45] [46], saturation current density can be derived as

$J_o=qA{N}_V{N}_C\left[\frac{1}{N_A}\sqrt{\frac{D_{n-}}{{\tau }_{n-}}}+\frac{1}{N_D}\sqrt{\frac{D_{p+}}{{\tau }_{p+}}}\right]\exp \left(\frac{-E_g}{kT}\right)$. (19)

where N_V, is the effective density of states in the valence band, N_C is the effective density of states in the conduction band, N_A is acceptor impurities concentration, N_D is donor impurities concentration, τ_n− is electron (minority carrier) lifetime, τ_p+ is hole (minority carrier) lifetime, A is cross-sectional area of solar cell, E_g is the energy band-gap, D_n− is electron diffusion coefficient and D_p+ is hole diffusion coefficient.

Applying Equation (1) to a solar module as explained by [47], we can obtain

$I_o=I_{o_{STC}}{\left[\frac{T}{T_{STC}}\right]}^3\exp \frac{-q{E}_g}{n{N}_s}\left(\frac{1}{T_{STC}}-\frac{1}{T}\right)$. (20)

The saturation current can be calculated using Equations (8) to (18) at constant temperature of 25˚C to obtain $I_{o_{STC}}$. This requires careful analysis of these equations to determine the one that produces the best results in the replication of the experimental data. However, these equations depend on I_ph, R_s, R_sh and n, which are unknown parameters that must be determined first.

4.3. Ideality Factor (n)

The ideality factor can be evaluated as a function of series ad shunt resistances or by considering their extreme values. Considering the approach we introduced in our previous work [48], in which the ideality factor was evaluated in the optimum ideality-factor neighborhood, in this paper we discuss further extraction of n for 0 ≤ n ≤ n_o.

4.3.1. Ideality Factor (n) Dependence on R_s and R_sh

The exponential term exp(I_scR_s/nN_sV_t), in the denominators of Equations ((13) and (15)) can be omitted, as it has insignificant value compared to the other exponential terms in the respective denominators. Therefore, Equations ((13) and (15)) can be written as

$I_o=\frac{I_{sc}{R}_{sh}+I_{sc}{R}_s-V_{oc}}{R_{sh}\exp \left(\frac{V_{oc}}{n{N}_s{V}_t}\right)}$. (21)

and

$I_o=\frac{I_{sc}{R}_s+I_{sc}{R}_{sh}-V_{mpp}-I_{mpp}{R}_{sh}-I_{mpp}{R}_s}{R_{sh}\exp \left(\frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}\right)}$. (22)

Equating Equations ((21) and (22)) and solving for n gives

$n=\frac{V_{oc}-V_{mpp}-I_{mpp}{R}_s}{N_s{V}_t\left[\ln \left(\frac{I_{sc}{R}_{sh}+I_{sc}{R}_s-V_{oc}}{I_{sc}{R}_{sh}+I_{sc}{R}_s-I_{mpp}{R}_{sh}-I_{mpp}{R}_s-V_{mpp}}\right)\right]}$. (23)

4.3.2. Ideality Factor (n) Dependence on Extremum Values of R_s and R_sh

The ideality factor can also be derived simply by first removing the exponential terms using logarithm and subtracting Equations (4) and (5) to obtain

$n=\frac{V_{oc}-V_{mpp}-I_{mpp}{R}_s}{N_s{V}_t\left[\ln \left(\frac{I_{ph}+I_o-\frac{V_{oc}}{R_{sh}}}{I_{ph}+I_o-I_{mpp}-\frac{V_{mpp}+I_{mpp}{R}_s}{R_{sh}}}\right)\right]}$. (24)

In Equation (24) the ideality factor relates with I_ph, I_o, R_s and R_sh at both maximum power point (I_mpp, V_mpp) and open circuit point (V_oc). For initial estimates, the R_s and R_sh values can be ignored in both the numerator and denominator. They have very small and very large values, where R_s ≈ 0 and R_sh ≈ ∞, respectively. This makes it possible to introduce the short circuit point into Equation (24) since the photocurrent relationship given in Equation (3) reduces to I_ph ≈ I_sc. Therefore, the ideality factor can be evaluated with respect to saturation current and the three crucial points as

$n=\frac{V_{oc}-V_{mpp}}{N_s{V}_t\left[\ln \left(\frac{I_{sc}+I_o}{I_{sc}+I_o-I_{mpp}}\right)\right]}$. (25)

This assumption gives n_o in terms of I_sc, I_mpp, V_oc and V_mpp only. Hence,

$n_o=\frac{V_{oc}-V_{mpp}}{N_s{V}_t\left[\ln \left(\frac{I_{sc}}{I_{sc}-I_{mpp}}\right)\right]}$. (26)

4.4. Shunt Resistance (R_sh) and Series Resistance (R_s)

The values of shunt and series resistance can be evaluated using the equations derived using I_sc, I_mpp, V_mpp and V_oc through an iterative process. Using I_mpp and V_mpp the relationship between R_sh and R_s can be evaluated by rearranging Equation (5) to obtain

$R_{sh}=\frac{V_{mpp}+I_{mpp}{R}_s}{I_{ph}-I_{mpp}-I_o\left(\exp \left(\frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}\right)-1\right)}$. (27)

The combination of Equations (3) and (4) gives a relation between R_sh and R_s in terms of I_sc and V_oc given by

$R_{sh}=\frac{V_{oc}-I_{sc}{R}_s}{I_{sc}+I_o\exp \left(\frac{I_{sc}{R}_s}{n{N}_s{V}_t}\right)-I_o\exp \left(\frac{V_{oc}}{n{N}_s{V}_t}\right)}$. (28)

Similarly, the combination of Equations ((3) and (5)) gives a relation between R_sh and R_s in terms of I_sc, I_mpp and V_mpp to obtain

$R_{sh}=\frac{V_{mpp}+I_{mpp}{R}_s-I_{sc}{R}_s}{I_{sc}-I_{mpp}-I_o\exp \left(\frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}\right)+I_o\exp \left(\frac{I_{sc}{R}_s}{n{N}_s{V}_t}\right)}$. (29)

Again, merging Equations ((4) and (5)) yields

$R_{sh}=\frac{V_{oc}-V_{mpp}-I_{mpp}{R}_s}{I_{mpp}+I_o\exp \left(\frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}\right)-I_o\exp \left(\frac{V_{oc}}{n{N}_s{V}_t}\right)}$. (30)

Shunt resistance (R_sh) and series resistance (R_s) can also be analyzed using the vanishing slope of the output power at maximum power point of Equation (1) and derivatives at short circuit and open circuit points with respect to V [12] [27] [49] [50] [51].

The derivative of Equation (1) with respect to V gives

$\frac{\partial I}{\partial V}=-\frac{I_o}{n{N}_s{V}_t}\left\{\left(1+\frac{\partial I}{\partial V}{R}_s\right)\exp \left(\frac{V+I{R}_s}{n{N}_s{V}_t}\right)\right\}-\frac{1}{R_{sh}}\left(1+\frac{\partial I}{\partial V}{R}_s\right)$. (31)

The derivative at short circuit point gives

${\left[\frac{\partial I}{\partial V}\right]}_{I=I_{sc}}=-\frac{1}{R_{sh}}$. (32)

and at open circuit point

${\left[\frac{\partial I}{\partial V}\right]}_{V=V_{oc}}=-\frac{1}{R_s}$. (33)

At maximum power point the power derivative with respect to voltage can be evaluated as

$\left[\frac{\partial P}{\partial V}\right]=\left(\frac{\partial I}{\partial V}\right)V+I=0$. (34)

At maximum power point the power derivative with respect to voltage can be evaluated as

$-\frac{I_{mpp}}{V_{mpp}}=-\frac{I_o}{n{N}_s{V}_t}\left\{\left(1-\frac{I_{mpp}}{V_{mpp}}{R}_s\right)\exp \left(\frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}\right)\right\}-\frac{1}{R_{sh}}\left(1+\frac{I_{mpp}}{V_{mpp}}{R}_s\right)$. (35)

By rearranging Equation (35) we can obtain

$R_{sh}=\frac{V_{mpp}-I_{mpp}{R}_s}{I_{mpp}-\frac{I_o}{n{N}_s{V}_t}\left(V_{mpp}+I_{mpp}{R}_s\right)\exp \left(\frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}\right)}$. (36)

4.4.1. Evaluation and Analysis of R_sh and R_s Pairs

This paper presents a simplified analytical approach for evaluating and analyzing R_sh and R_s pairs. Considering Equations ((27)-(30) and (36)), there are only three unknown parameters, i.e., R_s, the ideality factor and saturation current that appears on the right hand side of each equation. The saturation current has been derived in Equations ((14), (16) and (18)) with respect to ideality factor. A simple mathematical analysis can be done by replacing I_o in Equations ((27)-(30) and (36)) using Equation (14) to remain with ideality factor as the only unknown parameter. Comparing Equations ((27)-(30) and (36)), only Equation (30) that gives positive values of R_sh and R_s pairs after replacing I_o with Equation (14). Therefore, eliminating I_o of Equation (30) using (14) gives

$R_{sh}=\frac{V_{oc}-V_{mpp}-I_{mpp}{R}_s}{I_{mpp}-I_{sc}+I_{sc}\exp \left(\frac{V_{mpp}+I_{mpp}{R}_s-V_{oc}}{n{N}_s{V}_t}\right)}$. (37)

Equation (37) can be analyzed using an iterative approach to obtain the [R_s, R_sh] pairs by selecting the values of n that are less than n_o. As introduced by [52], the R_s and R_sh limits can be calculated using

$R_{s{h}_{\max }}=\frac{V_{oc}-V_{mpp}}{I_{mpp}}$. (38)

and

$R_{s{h}_{\min }}=\frac{V_{mpp}}{I_{sc}-I_{mpp}}-R_{s_{max}}$. (39)

Limiting the ideality factor selection within the 0 ≤ n ≤ n_o range and setting R_s and R_sh limits given by Equations ((38) and (39)) respectively, makes the process fast and robust. In addition, the ideality factor is selected in order to get an R_s and R_sh pair that guarantees the simulated maximum power (P_mpp(sim)) matches maximum power obtained experimentally, where $P_{mpp}\left(expt\right)=I_{mpp}\ast V_{mpp}$. The P_mpp(sim) relationship can be derived by replacing I = I_mpp and V = V_mpp in Equation (1) to yield

$P_{mpp}\left(sim\right)=V_{mpp}\left(I_{ph}-I_o\left(\exp \frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}-1\right)-\frac{V_{mpp}+I_{mpp}{R}_s}{R_{sh}}\right)=P_{mpp}\left(expt\right)$. (40)

The value of I_ph in Equation (40) can be replaced using Equation (3) by letting the term $I_o\exp \left(I_{sc}{R}_s/n{N}_s{V}_t\right)\approx 0$, since it has insignificant value compared to the other terms. Thus we can rewrite,

$I_{ph}=I_{sc}+\frac{I_{sc}{R}_s}{R_{sh}}$. (41)

Further, the saturation current in Equation (40) can be replaced by Equation (16) to obtain

$\begin{array}{c}{P}_{mpp}\left(sim\right)\\ =V_{mpp}\left(\left(I_{sc}+\frac{I_{sc}{R}_s}{R_{sh}}\right)-\left(\frac{I_{sc}-I_{mpp}}{\exp \left(\frac{V_{mpp}}{n{N}_s{V}_t}\right)}\right)\left(\exp \frac{V_{mpp}+I_{mpp}{R}_s}{n{N}_s{V}_t}-1\right)-\frac{V_{mpp}+I_{mpp}{R}_s}{R_{sh}}\right)\\ =P_{mpp}\left(expt\right)\end{array}$ (42)

Both Equations (37) and (42) can be solved simultaneous by arbitrarily selecting ideality factor below n_o and by increasing the values of R_s from zero to $R_{s_{\max }}$ using computer software. This process is repeated until the value of simulated maximum power (P_mpp(sim)) matches maximum power obtained experimentally P_mpp(expt) or has an error margin of less than 0.5% [53].

The percentage error in power can be expressed as

$P_{mpp}Error=\Delta P_{mpp}\%=\frac{P_{mpp}-I_{mpp}{R}_s}{I_{mpp}{V}_{mpp}}$. (43)

4.4.2. Extraction of Ideality Factor n, R_s and R_sh Using an Iterative Computational Process

The values of n, R_s and R_sh can be extracted using an iterative process using Equation (37) and verified using both Equations ((42) and (43)). The ideality factor is arbitrarily chosen starting from n ≤ n₀ [52] in steps of −0.001 and applied in Equation (37) to obtain [R_s, R_sh] pair. Subsequently, in the iterative stage, these n, R_s, and R_sh values are used in Equation (42) until the trio that satisfies $P_{mpp}\left(sim\right)\equiv P_{mpp}\left(expt\right)$ is achieved.

4.4.3. Simulation of R_sh and R_s Pairs

Table 1 provides a summary of data from the KC200GT datasheet profile and experimental data for Solinc 120 W measured using Gsola XJCM-10A solar simulator that has been used to simulate R_sh and R_s pairs. Figure 2 illustrates the R_sh and R_s relationship given by Equation (37) and has been sketched using data presented in Table 1.

The optimum ideality factors and optimum saturation current are also listed in Table 1. The optimum ideality factors values have been determined using Equation (26). Similarly, the optimal saturation current has been determined using Equation (14). These optimal values set the limit for both ideality factor and saturation current but they do not give the best results when plotting the I-V and P-V curves. This leads to further analysis of ideality factors near the optimal values and their respective saturation current.

Table 2 gives a summary of ideality factor n, R_s, R_sh, I_o and I_ph parameters for Solinc 120 W and Kyocera KC200GT solar modules. The main aim has been to use the available information for I_sc, I_mpp, V_oc and V_mpp to arrive at most appropriate values listed in Table 2. These values have been used as the starting figures for I-V relationship as demonstrated in the following section.

5. Improved Analysis of Current-Voltage Relationship for Five-Parameter Model Using Newton-Raphson Technique

The previous sections have demonstrated a simplified approach of obtaining the

Table 1. Solinc 120 W and Kyocera KC200GT photovoltaic modules data at STC.

Figure 2. A graph of R_sh versus R_s at STC for (a) Solinc 120 W from solar simulator values and (b) for KC200GT from datasheet values.

Table 2. Extracted parameters for Solinc 120 W and Kyocera KC200GT photovoltaic modules using equations (37) and (42).

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,

$f\left(I\right)=I_{ph}-I_o\left[\exp \left(\frac{V+I{R}_s}{n{N}_s{V}_t}\right)-1\right]-\frac{V+I{R}_s}{R_{sh}}-I=0$. (44)

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

$\frac{\partial \left(f\left(I\right)\right)}{\partial I}=-\frac{I_o{R}_s}{n{N}_s{V}_t}\left[\exp \left(\frac{V+I{R}_s}{n{N}_s{V}_t}\right)\right]-\frac{R_s}{R_{sh}}-1$. (45)

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

$I_{j+1}=I_j-\frac{f\left(I_j\right)}{\frac{\partial \left(f(I_j)\right)}{\partial I}}=I_j-\frac{I_{ph}-I_o\left[\exp \left(\frac{V+I_j{R}_s}{n{N}_s{V}_t}\right)-1\right]-\frac{V+I_j{R}_s}{R_{sh}}-I_j}{-\frac{I_o{R}_s}{n{N}_s{V}_t}\left[\exp \left(\frac{V+I_j{R}_s}{n{N}_s{V}_t}\right)\right]-\frac{R_s}{R_{sh}}-1}$ (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

$P=I_{ph}V-I_{o}V\left[\exp \left(\frac{V+I{R}_s}{n{N}_s{V}_t}\right)-1\right]-\frac{V^2}{R_{sh}}-VI\frac{R_s}{R_{sh}}$. (47)

5.1. Analysis of Different Approaches for Extracting Five-Model Parameters

All the five parameters in Table 2 for Solinc 120 W and KC200GT modules that have been deduced using the new simplified simulation procedure are applied in solving Equation (46) to obtain a good approximation of the output current. This approach offers a simple and very efficient calculation procedure for all current and voltage values starting from I = 0 to I = I_sc and V = 0 to V = V_oc. It offers very precise solutions for all the points needed to plot the I-V and P-V or P-I curves.

In this work, we have classified four most suitable data extraction approaches for determining I and V using Newton-Raphson method. These categories depend on the choice of saturation current equations discussed in section 4.2 and 4.3.1. Category 1 is based on saturation currents that are dependent on R_s and R_sh resistances at open and short circuit points (SCDR-OS) where I_o is calculated using Equations ((11), (13) and (21)). Category 2 is based on saturation current that is independent of R_s and R_sh resistances at open and short circuit points (SCIR-OS), where I_o is calculated using Equations ((12) and (14)). Categories 3 and 4 are based on saturation currents at both open circuit and maximum power points that are dependent on R_s and R_sh resistances (SCDR-OMP) and saturation currents that are independent of R_s and R_sh resistances (SCIR-OMP), where the I_os are calculated using Equations ((17) and (18)) respectively. The use of saturation current defined by Equations ((9), (10), (15), (16) and (22)) does, however, provide unsatisfactory data for I and V.

These procedures can be implemented using the algorithm shown in Figure 4 which outlines all the steps required to retrieve the data for plotting the I-V and P-V curves as follows.

· The process starts with input values of I_sc, I_mpp, V_mpp, V_oc, N_s and V_t from Table 1.

· Followed by setting the number of iterations, NiMax for current approximation and NvMax for voltage resolution plus precision description for R_s increment defined by Rsinc.

· The algorithm presented in Figure 3 is then applied to obtain n, R_s and R_sh values.

· These n, R_s and R_sh are used to calculate I_o and I_ph for the first iteration of determining the current and voltage data.

· The process is repeated severally for each iteration with an increment of R_s (R_s = R_s + Rsinc) until NiMax and NvMax are reached by solving Equations (44)-(46).

· The P-error is then evaluated to determine the most suitable values for n, R_s, R_sh, I_o and I_ph which give the best current, voltage and power data.

· If the error in power is greater than 0.5%, the process is repeated by inputting a new value of ideality actor.

· Finally, the process ends by plotting I-V and P-V curves and the cardinal point markers if the error in power is less or equal to 0.5%.

5.2. Evaluation of Extracted Parameters for Solinc 120 W and KC200GT Modules Using the Four Approaches

Table 3 & Table 4 display the five-model parameter data for Solinc 120 W and

Figure 3. An algorithm for evaluating the n, R_s and R_sh using I_sc, I_mpp, V_mpp and V_oc.

Table 3. Simulated data for Solinc 120, where I_o has been calculated using approach 1 of Equations ((11), (13) or (21)), approach 2 of Equations ((12) or (14)), approach 3 Equation (17) and approach 4 Equation (18).

Table 4. Simulated data for KC200GT, where Io has been calculated using approach 1 of equations (11), (13) or (21), approach 2 of equations (12) or (14), approach 3 Equation (17) and approach 4 Equation (18).

KC200GT solar modules and their simulated output power and errors. The following sections discuss the most realistic results of the five-model parameter that provide more practical data for each approach that fits experimental results. Four approaches are listed here since they give small percentage error.

Approach 1

The data shown in Table 3 & Table 4 in rows 2 - 3 summarizes the Solinc 120W and KC200GT parameters that have been extracted from the first category of data procedure. These data are based on open and short circuit points, where I_o is determined using Equations ((11), (13) or (21)). However, this category only gives a satisfying [R_s, R_sh] pair from Equations ((27) and (29)).

Approach 2

The data shown in Table 3 & Table 4 in rows 4 - 8, give category 2 data for Solinc 120 W and KC200GT where I_o has been calculated using either Equations (12) or (14) that are independent of R_s, R_sh pair. This category gives satisfactory

Figure 4. An algorithm for calculating current (I) using Newton-Raphson technique and plotting I-V and P-V curves.

[R_s, R_sh] pair from Equations ((27), (28) (29), (30) and (36)).

Approach 3

Again, the data shown in Table 3 & Table 4 in rows 9 - 12, represent category 3 where I_o is determined using Equation (17). This category gives satisfactory [R_s, R_sh] pair only from Equations ((27), (28), (29) and (36)).

Approach 4

Finally, rows 13 - 16 of Table 3 & Table 4 give category 4 data for Solinc 120 W and KC200GT, where I_o is determined using Equation (18) that is independent of R_s, R_sh pair. This category also gives satisfactory [R_s, R_sh] pair only from Equations ((27), (28), (29) and (36)).

It is clear from Table 3 & Table 4 that the saturation current (I_o) calculated using Equations ((12) or (14)) in approach 2 gives acceptable ∆P_mpp errors when applied to all R_sh Equations ((27)-(30), or (36)).

Approach 1 gives satisfactory data when applied in Equations ((27) and (29)). Other approaches, 3 and 4 give reasonable results when applied in Equations ((27), (28), (29) and (36)). Evidently, Equations ((27) and (29)) that have been derived at maximum power point and combination of maximum power point and short circuit equations, respectively, give the best results of ∆P_mpp error for all saturation current equations at constant ideality factor. Equation (28) gives very large and very small values of R_sh when applied in approaches 2 and 4. All the four approaches give satisfactory data when compared with other methods reported by different authors as demonstrated in Table 5 & Table 6.

These data are summarized in Table 5 for R_sh of Equation (27) that gives the smallest error for each approach. Approach 1 data and data from [52] have similar results while approaches 2 and 3 have different R_sh values for the same R_s = 0.233 Ω. However, approach 4 has R_s = 0.2395 Ω that gives the least ∆P_mpp error and very large value of R_sh = 18,565 Ω. Analytical approach reported by [41] provides satisfactory parameters that are closer to the values obtained using the four approaches.

The values of five-model parameters obtained using approaches discussed in the works of [41] [52] have been compared with the values of parameters for

Table 5. Comparison of calculated parameters with other approaches in literature for Solinc 120 W.

Table 6. Comparison of calculated parameters with other approaches in literature for KC200GT.

Solinc 120 W using approaches 1 - 4 of Table 3.

These data are summarized in Table 5 for R_sh of Equation (27) that gives the smallest error for each approach. Approach 1 data and data from [52] have similar results while approaches 2 and 3 have different R_sh values for the same R_s = 0.233 Ω. However, approach 4 has R_s = 0.2395 Ω that gives the least ∆P_mpp error and very large value of R_sh = 18565 Ω. Analytical approach reported by [41] provides satisfactory parameters that are closer to the values obtained using the four approaches.

The results obtained using analytical methods published by [38] [40] [41] [50] [51] [52] have been compared with the KC200GT parameters values in Table 4 and summarized in Table 6. The values of R_s obtained using the four methods discussed in this work are consistent with their analysis within ±0.02 Ω. All parameter values shown in Table 4 provide satisfactory results for output power within the error margin given by the manufacturer of +10% or −5%. A typical way of testing the 5-parameter model is through I-V and P-V plots.

6. I-V and P-V Characterization for Solinc 120 W and KC200GT Photovoltaic Modules Based on the Four Approaches at STC

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.

Figure 5. A graph of current versus voltage for Solinc 120 W at STC.

Figure 6. A graph of power versus s voltage for Solinc 120 W at STC.

Figure 7. A graph of current versus s voltage for KC200GT at STC.

Figure 8. A graph of power versus voltage for KC200GT at STC.

7. I-V and P-V Characterization at Ambient Temperature, NOCT and Actual Irradiance

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

$I_{sc}\left(s_a,T\right)=\frac{s_a}{s_{STC}}\left[I_{sc}{}_{{}_{STC}}+K_{I_{sc}}\left(T-T_{STC}\right)\right]$ (48)

where, $K_{I_{sc}}$ 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

$I_{mpp}\left(s_a,T\right)=\frac{s_a}{s_{STC}}\left[I_{mpp}{}_{{}_{STC}}+K_{I_{smpp}}\left(T-T_{STC}\right)\right]$ (49)

where, $K_{I_{mpp}}$ is the temperature coefficient of I_mpp in A/˚C. Unlike $K_{I_{sc}}$, the manufacturer’s data sheet does not include $K_{I_{mpp}}$. 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

$T=T_a+\frac{\left[T_{NOCT}-20\right]s_a}{800}$ (50)

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

$V_{mpp}\left(s_a,T\right)=V_{mp{p}_{STC}}+K_{v_{mpp}}\left(T-T_{STC}\right)+{\alpha }_{v_{mpp}}\left(s_a-s_{STC}\right)+{\beta }_{v_{mpp}}{\left(s_a-s_{STC}\right)}^2$ (51)

where, ${\alpha }_{V_{mpp}}$ and ${\beta }_{V_{mpp}}$ are solar radiation coefficients at maximum-power point. Similarly, the short-circuit voltage can be evaluated as

$V_{oc}\left(s_a,T\right)=V_{o{c}_{STC}}+K_{v_{oc}}\left(T-T_{STC}\right)+{\alpha }_{V_{oc}}\left(s_a-s_{STC}\right)+{\beta }_{V_{oc}}{\left(s_a-s_{STC}\right)}^2$ (52)

where ${\alpha }_{V_{oc}}$, ${\beta }_{V_{oc}}$ 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

$V_{oc}\left(s_a,T\right)=n\left(s_a,T\right)N_s{V}_t\left(T\right)\left[\ln \left(I_{sc}\left(s_a,T\right)\right)-\ln \left(I_o\left(s_a,T\right)\right)\right]$ (53)

and,

$V_{mpp}\left(s_a,T\right)=V_{oc}\left(s_a,T\right)-n_o{N}_s{V}_t\left(T\right)\times \left[\ln \left(\frac{I_{sc}\left(s_a,T\right)}{I_{sc}\left(s_a,T\right)-I_{mpp}\left(s_a,T\right)}\right)\right]$ (54)

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

$I_o\left(s_a,T\right)=I_{o_{STC}}{\left[\frac{T}{T_{STC}}\right]}^3\exp \frac{-q{E}_g}{n\left(s_a,T\right)N_s}\left(\frac{1}{T_{STC}}-\frac{1}{T}\right)$. (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

$I_o=\frac{I_{sc}\left(s_a,T\right)}{\exp \left(\frac{V_{oc}\left(s_a,T\right)}{n\left(s_a,T\right)N_s{V}_t\left(T\right)}\right)}$. (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

Table 7. Simulated data and extracted model parameters values for KC200GT photovoltaic module at air temperature of 20˚C and NOCT of 47˚C and different irradiance levels.

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 ${\alpha }_{V_{mpp}}$, ${\alpha }_{V_{oc}}$, ${\beta }_{V_{mpp}}$ and ${\beta }_{V_{oc}}$ 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.

8. Conclusions

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

Figure 9. A graph of current versus voltage for KC200GT at air temperature of 20˚C and NOCT of 47˚C for various irradiances.

Figure 10. A graph of current versus voltage for KC200GT at air temperature of 20˚C and NOCT of 47˚C for various irradiances.

Figure 11. A graph of current versus voltage for KC200GT showing various temperatures curves at 1000 W/m².

Table 8. Simulated data and extracted model parameters values for KC200GT photovoltaic module at arbitrary air and NOCT temperatures for 1000 W/m².

Figure 12. A graph of power versus voltage for KC200GT showing various temperatures curves at 1000 W/m².

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.

Acknowledgements

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.

Conflicts of Interest

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

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