DOI: 10.26190/unsworks/32001
title: Data for "The Intrinsic Adjusted Single-Diode Model: Solid State Physics meets Accurate Yield Simulation"

description: Data supporting the SOLMAT article "The Intrinsic Adjusted Single-Diode Model: Solid State Physics meets Accurate Yield Simulation". This article describes how an "intrinsic-adjusted" single diode model can provide more accurate fits to modern, high efficiency solar cells. This repository includes csv and excel data supporting all figures and python code for key equations. Key items include - IV curves generated using the 3D device simulator Quokka3 - Fitting parameters and results for 4 variations of the single-diode model to the simulated IV data - IV curves measured at UNSW using a Sinton IV tester - Fitting parameters and results for the intrinsic-adjusted single-diode model to the measured data - python code for generating intrinsic recombination currents and hence intrinsic-adjusted single diode model IV data
description: Taken from the article: FITTING METHODS AND APPLICATION For numeric fitting methods and for application in yield modelling the key challenge is to generate the current for a given voltage, irradiance and temperature. For the fitting process this allows the calculation of errors, and hence the creation of objective functions. For yield modelling it allows the generation of both current_voltage and power_voltage curves, along with determination of key parameters such as the open-circuit voltage, maximum power point current and voltage and short circuit current. Single Diode i_from_v The implicit solution of the single-diode equation is a well understood problem and fundamental to almost all yield simulation software. For this work we make use of the open-source library pvlib-python [9], and particularly the bishop88_i_from_v function [27] The inputs to this function are the diode parameters from equations 4-8 for a given temperature and irradiance. Intrinsic Adjusted i_from_v The creation of an intrinsic adjusted i_from_v function is a little more complicated. In addition to the diode parameters from equations 4-8 two additional parameters are required: the concentration of dopants in the silicon bulk and the volume of silicon in the cell. It was also found to improve the fitting accuracy if the series resistance was split into internal and external components such that: R_S=R_(S.int)+R_(S.ext) (14) There are 4 main steps to fit the adjusted model to the data. First, the voltage at each data point is adjusted for the external series resistance 𝑅𝑆.𝑒𝑥𝑡. This allows the intrinsic recombination currents to be expressed as explicit functions of the adjusted voltage V_adj. Secondly, the recombination current due to intrinsic processes (Auger and radiative recombination) is calculated for each V_adj. The intrinsic recombination current is then added to the adjusted curve. Thirdly the contribution from the standard single-diode term is calculated using the bishop88_i_from_v process, with the 𝑅𝑆 term replaced by the internal series resistance 𝑅𝑆.𝑖𝑛𝑡. 𝐼𝑝ℎ, 𝐼0, n, 𝑅𝑆.𝑖𝑛𝑡 and 𝑅𝑆𝐻 were the fitting parameters. In the final step the diode and intrinsic recombination currents are combined. The final equation is then: I=I_ph-I_rad (V_adj )-I_Auger (V_adj )-I_0 [e^((V_adj+I.R_(S.int))/(nV_th ))-1]-(V_adj+I.R_(S.int))/R_SH (15) Where: V_adj=IR_(S.ext) (16) I_rad=A.w.R_rad (17) I_Auger=A.w.R_Auger (18) Where A is the cell area and w is the cell width. If a single series resistance value is used to simplify the process the equations are identical with R_(S.int) set to 0. The key challenge in calculating the intrinsic recombination current for a given adjusted voltage is determining the electron and hole densities. Once the intrinsic carrier density, electron density, hole density and quasi-fermi level splitting (n_i^2,n,p,∆ϕ) have been determined equations 11, 12, 17 and 18 can be used to calculate the recombination currents. When performing a fitting process V_adj can be calculated for a given voltage using the measured/simulated current. However, when performing yield simulations it is much simpler to determine the current for a given V_adj and then subsequently back out the voltage term. This still allows accurate determination of all relevant cell parameters. Fitting Process and Model Summary Table 1 presents the four models used in this study. The “Datasheet” approach largely attempted to mimic the description of the single-diode model within PVSyst [8], where datasheet values were used as the input. 𝑅𝑆 was determined based on the relative loss of efficiency under an irradiance of 200 W/m2 c.f. 1,000 W/m2 at 25oC. The temperature dependence of saturation currents, photocurrent and gamma were all calculated as per PVSyst. At present it is not possible to perform a similar fit for an adjusted model. The issue is that the residual single-diode curve often does not contain the two points that many analytical methods require to determine a good fit. There is no data point at open-circuit, and the maximum power point for the device does not align with the point that would give the maximum I-V product of the residual curve. All other models attempted to fit the entire I-V curve in a three-step process using the SciPy Python package scipy.minimize with a weighted RMSE as an objective. Weights of 20 and 100 were applied to the short-circuit current and maximum power point, respectively, and weights of 1 were assigned to all other values. Weighting of the open-circuit voltage was unnecessary since small errors in VOC translate into large errors in current at that point. These errors at each point were calculated as the difference between the target current and the output of the appropriate i_from_v function. In the first step of the fitting process, the R_S parameter was determined to minimize the combined weighted RMSE at both STC and low-light conditions. For method 4 where both internal and external series resistances were being considered, a nested minimization method was used to determine the most appropriate values for 𝑅𝑆.𝑒𝑥𝑡 and 𝑅𝑆.𝑖𝑛𝑡, while maintaining the conditions of equation 14. For methods 2 and 3 the fitting process was identical except that 𝑅𝑆.𝑖𝑛𝑡 was fixed at 0. A further nested optimization function then determined normalized parameters for n,I_o and R_SH to minimize the weighted RMSE at STC for a given R_S value. I_ph was determined iteratively based on the other values and I_SC. Finally, temperature coefficients μ_n and α_SC were determined to minimize the weighted RMSE at 50oC, 1000 W/m2. Table 1:Key features of the four models used in this study Method Fit to Include intrinsic terms RS 1: “DATASHEET” ISC, VOC, VMP, IMP, ΔTISC, ΔTPMP No Single value 2: SINGLE-DIODE I-V Curves No Single value 3: ADJUSTED (SINGLE RS) I-V Curves Yes Single value 4: ADJUSTED (RS.INT + RS.EXT) I-V Curves Yes External and Internal The authors acknowledge that it is highly likely that more advanced fitting methods would produce improved fits. Nonetheless, we contend that the current approach is sufficient to demonstrate the advantage of the “adjusted” model. The primary metrics used in this paper are the normalized root-mean-square error (n-RMSE) and the error in the maximum power (PMAX Error). Normalization is carried out by dividing the RMSE (calculated between the data and the modelled current at each voltage) by the short-circuit current. To prevent errors near open-circuit voltage having an outsized impact on the results, any negative currents were replaced with 0 before calculating RMSE.