[Paper Review] On Calculating the Current-Voltage Characteristic of Multi-Diode Models for Organic Solar Cells

K. Roberts, S. R. Valluri|arXiv (Cornell University)|Dec 16, 2015

Organic Electronics and Photovoltaics9 references16 citations

TL;DR

This paper presents overflow-resistant analytical formulations for calculating the current-voltage (I-V) characteristics of one- and two-diode models in organic solar cells, particularly those with S-shaped I-V curves. By reformulating the Lambert W function usage through a stable iterative algorithm, the method enables accurate, high-precision computation in standard floating-point environments like Fortran, C, or microcontrollers without requiring arbitrary-precision arithmetic libraries.

ABSTRACT

We provide an alternative formulation of the exact calculation of the current-voltage characteristic of solar cells which have been modeled with a lumped parameters equivalent circuit with one or two diodes. Such models, for instance, are suitable for describing organic solar cells whose current-voltage characteristic curve has an inflection point, also known as an S-shaped anomaly. Our formulation avoids the risk of numerical overflow in the calculation. It is suitable for implementation in Fortran, C or on micro-controllers.

Motivation & Objective

• To address numerical overflow issues in computing the exact I-V characteristic of multi-diode solar cell models using standard floating-point arithmetic.
• To enable robust implementation of one- and two-diode models in embedded systems and field-deployable solar energy monitoring tools.
• To eliminate reliance on specialized arbitrary-precision arithmetic libraries or symbolic math packages in solar cell modeling.
• To provide a numerically stable method for computing the inverse I-V function V=f(I) for S-shaped I-V curves common in organic photovoltaics.
• To support accurate parameter fitting and performance analysis of organic solar cells, especially during annealing processes that alter the S-curve anomaly.

Proposed method

• Reformulates the exact analytical solution V=f(I) for one- and two-diode models to avoid intermediate terms that cause arithmetic overflow in standard floating-point arithmetic.
• Introduces a stable iterative algorithm for computing the function y=g(x)=log(W(exp(x))), which replaces direct evaluation of the Lambert W function.
• Uses Halley’s method with cubic convergence to refine an initial estimate of g(x), ensuring high accuracy with minimal computational overhead.
• Employs only x, y, and exp(y) in calculations, avoiding direct computation of exp(x) to prevent overflow.
• Applies a piecewise initial estimate strategy: y₀=x for x≤−e, y₀=log(x) for x≥e, and linear interpolation in between.
• Implements error control via absolute error testing, especially critical near x=1 where y=0, to ensure convergence without numerical instability.

Experimental results

Research questions

• RQ1 Can the exact I-V characteristic of multi-diode solar cell models be computed without numerical overflow in standard floating-point environments?
• RQ2 How can the Lambert W function be evaluated robustly in embedded systems or microcontrollers without arbitrary-precision libraries?
• RQ3 What is the most computationally efficient and numerically stable method to compute V=f(I) for organic solar cells with S-shaped I-V curves?
• RQ4 Can the two-diode model’s complex parameter space be explored more effectively using overflow-resistant analytical solutions?
• RQ5 Does the proposed method allow for real-time or field-based solar cell characterization without specialized software?

Key findings

• The proposed algorithm computes y=g(x)=log(W(exp(x))) with absolute error below 10−10 after just two iterations, achieving high precision in standard double-precision arithmetic.
• The method avoids arithmetic overflow entirely by not computing exp(x), instead relying on exp(y) in iterative refinement, which remains numerically stable.
• The algorithm achieves cubic convergence using Halley’s method, requiring only two to three iterations to reach machine precision for most practical applications.
• The method is suitable for implementation in C, Fortran, or microcontrollers without needing a built-in Lambert W function or arbitrary-precision arithmetic.
• The approach enables accurate modeling of S-shaped I-V curves in organic solar cells, supporting analysis of annealing effects and performance degradation.
• The algorithm remains stable even when yₙ becomes very negative (e.g., yₙ < -100), where underflow might otherwise occur, by detecting such cases and terminating safely.