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

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 (Io). For a quick and fast convergence, we 
have determined the optimal ideality factor (no) and optimal saturation current (Ioopt) as the 
primary parameters by first assuming the optimal values of Rsh, Rs and Iph at 
standard test conditions (STC). Further, we evaluated the effects of Iph, Rs and Rsh on I-V and 
P-V curves by considering the values of n below no. We have 
evaluated different iterative procedures of determining Rsh and Rs 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 ≤ no. These 
approaches have been categorized by deriving the saturation current as a 
dependent variable at each cardinal point with or without Rs and Rsh 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).

rimental 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

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].
tances 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 .

A Single Diode Model
The equivalent circuit can be presented mathematically by 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 m 2 •s −2 •kg•K −1 .

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 where 0.025692607 t V kT q = = is the thermal voltage.
2) At Open Circuit, I = 0, V = V oc ; Equation (1) 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.

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.

Photocurrent (Iph) Analysis
The photocurrent (I ph ) can be determined from Equations ((2), (3)) or by re- 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]

Saturation Current (Io)
The saturation current can be evaluated using Equation (1) 2) At maximum power point, Equation (5) Setting the boundary condition of R s ≈ 0, R sh ≈ ∞ and I ph ≈ I sc , Equation (9) Similarly, setting boundary condition of R sh ≈ ∞ and I ph ≈ I sc , Equation (11)

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 Journal of Power and Energy Engineering 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 The saturation current can be calculated using Equations (8) to (18)  . 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.

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 .

Ideality Factor (n) Dependence on Rs and Rsh
The exponential term exp(I sc R s /nN s V 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) Equating Equations ( (21) and (22)

Ideality Factor (n) Dependence on Extremum Values of Rs and Rsh
The ideality factor can also be derived simply by first removing the exponential terms using logarithm and subtracting Equations (4) and (5) 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 This assumption gives n o in terms of I sc , I mpp , V oc and V mpp only. Hence,

Shunt Resistance (Rsh) and Series Resistance (Rs)
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) 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 Similarly, the combination of Equations ( (3) and (5) Again, merging Equations ( (4) and (5) 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 The derivative at short circuit point gives and at open circuit point At maximum power point the power derivative with respect to voltage can be evaluated as By rearranging Equation (35)

Evaluation and Analysis of Rsh and Rs 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) 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 and 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 The value of I ph in Equation (40) can be replaced using Equation (3) Further, the saturation current in Equation (40) 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 max s R 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

Extraction of Ideality Factor n, Rs and Rsh 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 ≤ n 0 [52] in steps of −0.001 and applied in Equation (37) 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.

Simulation of Rsh and Rs Pairs
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.

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 Differentiating Equation (44) V I R  V I R  I  I  I  f I  nN V  R  I  I  I where, j represents the number of iterative process.

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) 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 o s 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 Journal of Power and Energy Engineering 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%. 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 .

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  (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)).
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  Solinc 120 W using approaches 1 -4 of Table 3.
These data are summarized in Table 5 for R sh of Equation (27) [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.

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.   Table   4.
The current-voltage relationship for KC200GT is shown in Figure 7 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.

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 Equations (51) and (52) The saturation current dependence on module temperature can be achieved by rewriting Equation (20) The temperature-dependent saturation current of Equation (55) 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/m 2 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.  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/m 2 , 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.

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 Journal of Power and Energy Engineering     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.