Computation of the lambert W function in photovoltaic modeling

Recently, the Lambert W function has emerged as a valuable mathematical tool in photovoltaic (PV) modeling and other scientific fields. This increasing interest is because it can be used to reformulate the implicit equations of the single-diode PV model into explicit form. However, the computation of the Lambert W function itself is still not clear in the literature; some studies use the iterative built-in functions in MATLAB or other computational platforms, while others adopt their own approximation formulae. This paper takes a deeper look at the ways the Lambert W function is evaluated in PV models and carries out a comparative study to assess the most commonly used methods in terms of accuracy, computational cost, and application range. These alternatives are implemented in a modern computer and a typical microcontroller to evaluate their performance in both simulations and embedded applications. The analysis concludes that some series expansions are good options for PV modeling applications, requiring less execution time than the built-in MATLAB lambertw function and exhibiting negligible approximation error

A Method for the Analytical Extraction of the Single-Diode PV Model Parameters

Determination of PV model parameters usually requires time consuming iterative procedures, prone to initialization and convergence difficulties. In this paper, a set of analytical expressions is introduced to determine the five parameters of the single-diode model for crystalline PV modules at any operating conditions, in a simple and straightforward manner. The derivation of these equations is based on a newly found relation between the diode ideality factor and the open circuit voltage, which is explicitly formulated using the temperature coefficients. The proposed extraction method is robust, cost-efficient, and easy-to-implement, as it relies only on datasheet information, while it is based on a solid theoretical background. Its accuracy and computational efficiency is verified and compared to other methods available in the literature through both simulation and outdoor measurements

An Explicit PV String Model Based on the Lambert W Function and Simplified MPP Expressions for Operation Under Partial Shading

In this paper, a reformulation of the widely used one-diode model of the photovoltaic (PV) cell is introduced, employing the Lambert W function. This leads to an efficient PV string model, where the terminal voltage is expressed as an explicit function of the current, resulting in significantly reduced calculation times and improved robustness of simulation. The model is experimentally validated and then used for studying the operation of PV strings under partial shading conditions. Various shading patterns are investigated to outline the effect on the string I-V and P-V characteristics. Simplified formulae are then derived to calculate the maximum power points of a PV string operating under any number of irradiance levels, without resorting to detailed modeling and simulation. Both the explicit model and the simplified expressions are intended for application in shading loss and energy yield calculations

Simple PV Performance Equations Theoretically Well Founded on the Single-Diode Model

There are several photovoltaic (PV) performance models in the literature, but most of them either employ complex and tedious calculations or require additional measurements apart from datasheet information. In this paper, a new set of performance equations to evaluate the short-circuit current, open-circuit voltage, and maximum power point at any operating conditions is introduced. The proposed expressions are simple functions of the irradiance and temperature, while they are generally applicable to any crystalline PV module and require only datasheet information as input data. This is achieved by introducing new formulas to determine the irradiance and temperature coefficients that are not provided in the datasheet, thus avoiding empirical constants or additional measurements. The novelty of the performance equations is their solid theoretical background, as they are in excellent agreement with the single-diode PV model, combined with simple and easy application. The proposed PV model is validated and compared with other methods found in the literature through simulations in MATLAB and outdoor measurements on commercial PV modules

Computation of the Lambert W function in photovoltaic modeling

Recently, the Lambert W function has emerged as a valuable mathematical tool in photovoltaic (PV) modeling and other scientific fields. This increasing interest is because it can be used to reformulate the implicit equations of the single-diode PV model into explicit form. However, the computation of the Lambert W function itself is still not clear in the literature; some studies use the iterative built-in functions in MATLAB or other computational platforms, while others adopt their own approximation formulae. This paper takes a deeper look at the ways the Lambert W function is evaluated in PV models and carries out a comparative study to assess the most commonly used methods in terms of accuracy, computational cost, and application range. These alternatives are implemented in a modern computer and a typical microcontroller to evaluate their performance in both simulations and embedded applications. The analysis concludes that some series expansions are good options for PV modeling applications, requiring less execution time than the built-in MATLAB lambertw function and exhibiting negligible approximation error.</p

Partial Shading Analysis of Multistring PV Arrays and Derivation of Simplified MPP Expressions

In this paper, the electrical response of a partially shaded photovoltaic (PV) array, comprising several strings connected in parallel, is investigated. The PV array is simulated by employing an enhanced version of the widely used single-diode model, reformulated in an explicit manner employing the Lambert W function. The multiple maximum power points (MPPs) that appear on the P-V characteristic of the array in partial shading conditions are analyzed, in terms of their number and properties. Simplified empirical expressions are then derived to calculate the voltage, current, and power for each local MPP, at any irradiance level and temperature, using only datasheet information, in a most simple and straightforward manner, without resorting to detailed modeling and simulations. The derived formulae are validated using both simulation and experimental results

Power reserves control for PV systems with real-time MPP estimation via curve fitting

In order for a photovoltaic (PV) system to provide a full range of ancillary services to the gird, including frequency response, it has to maintain active power reserves. In this paper, a new control scheme for the dc/dc converter of a two-stage PV system is introduced, which permits operation at a reduced power level, estimating the available power (maximum power point-MPP) at the same time. This control scheme is capable of regulating the output power to any given reference, from near-zero to 100% of the available power. The proposed MPP estimation algorithm applies curve fitting on voltage and current measurements obtained during operation to determine the MPP in real time. This is the first method in the literature to use the nonsimplified single-diode model for the determination of the MPP and the five model parameters while operating at a curtailed power level. The developed estimation technique exhibits very good accuracy and robustness in the presence of noise and rapidly changing environmental conditions. The effectiveness of the control scheme is validated through simulation and experimental tests using a 2-kW PV array and a dc/dc converter prototype at constant and varying irradiance conditions

Energy models for photovoltaic systems under partial shading conditions: a comprehensive review

The partial shading phenomenon and its implications on the electrical response and energy yield of photovoltaic (PV) systems have received increased attention in the last years. In order to study, foresee and mitigate such effects, several energy models are proposed in the bibliography, presenting different degrees of complexity, accuracy and applicability. This study presents an overview of the state of the art in the development of models for PV systems under partial shading conditions. Alternative modelling approaches are analysed, highlighting their advantages and shortcomings and models available in the literature are reviewed and classified according to important attributes, related to their accuracy and implementability. Current research trends, as well as topics that warrant further investigation, are identified and discussed

Direct MPP Calculation in Terms of the Single-Diode PV Model Parameters

In this paper, new expressions are introduced for the determination of the maximum power point (MPP) of photovoltaic (PV) systems as explicit functions of the five parameters of the single-diode model employing the Lambert W function. These equations provide the voltage and current at MPP in a direct and straightforward manner, thus dispensing with any need for iterative solution. They are initially derived for a PV system operating under uniform conditions, and subsequently extended for mismatched conditions at the PV string level. The novelty of these formulae lies in their solid theoretical foundation, which supports their validity in the general case and offers a well-founded symbolic formulation for the MPP evaluation problem. Extended simulations and experimental validation are performed to verify the accuracy and computational efficiency of the proposed equations compared with other methods available in the literature