Lambert W-function Research Articles

Parameters identification and optimization of photovoltaic panels under real conditions using Lambert W-function

This paper proposes a new approach based on Lambert W-function to extract the electrical parameters of photovoltaic (PV) panels. This approach can extract the optimal electrical characteristics of the PV panel under variable conditions of irradiation and temperature. Three benchmarking panels (shell SP70 monocrystalline silicon, shell ST40 thin film, and KC200GT Polycrystalline Silicon) are demonstrated and analyzed considering the electrical characteristics provided by the manufacturers. A comprehensive assessment is carried out under different weather condition to validate the capability and the robustness of the proposed approach. Furthermore, the simulated output characteristics of the three modules Photovoltaic are almost comparable and reproduce faithfully the manufacturer’s experimental data The novelty of this study is the using a new hybrid analytical and numerical method that straight forward and effective given value of Root mean square error less than those obtained by others methods that indicate the estimated results are very close to the experimental values provided by the manufacturers.

Photovoltaic mono and bifacial module/string electrical model parameters identification and validation based on a new differential evolution bee colony optimizer

Well estimating the electrical model parameters of the photovoltaic (PV) module/string serves to develop an accurate simulator and a fault diagnosis tool. Several based evolutionary techniques were proposed to identify the unknown circuit equivalent PV generator (PVG) parameters. Whereas most of them have not been examined to various real operating conditions of solar irradiance and PV cells temperature. That requires larger search range than the adopted one in the literature. Enlarging the search range imposes more computational time and high exploration and exploitation features. Hence, a novel hybrid differential evolution and artificial bee colony intelligence (nDEBCO) approach is proposed. In terms of convergence quality, CPU execution time, number of function evaluations (NFE), and error standard deviation (StD). The newly developed approach permits to accurately identify the PV module/string unknown parameters with suitable implementation complexity. Mono-facial CLS 220P PV string has been utilized employing an adequate experimental setup with online implementation. 1080 I-V curves have been measured and estimated, where the overall RMSE ± StD is below 0.02 ± 1e−16. The nDEBCO outperforms the present-day published works, for common case studies in the literature with two based root mean square error (RMSE) objective functions namely Lambert W function (LWF) and classic. It yields 7.73006268e− 4 of RMSE, 7.8785e− 18 of StD, and 2150 NFE under ODM with LWF for RTC France PV cell. Bifacial PV module has been evaluated and the electrical parameters have been extracted within less than 1.36 s of CPU run time and not>8.0299631e− 3 ± 6.9096e− 16 of RMSE ± StD for front and rear faces. Additionally, the parameter identification procedure has been well validated to simulate the real partial shading scenarios of the studied PV string with a RMSE less than 0.045 and 0.397% of power maximum point absolute error.

New Hydrodynamic Solutions for Line-driven Winds of Hot Massive Stars Using the Lambert W-function

Abstract Hot massive stars present strong stellar winds that are driven by absorption, scattering, and reemission of photons by the ions of the atmosphere (line-driven winds). A better comprehension of this phenomenon, and a more accurate calculation of hydrodynamics and radiative acceleration, is Required to reduce the number of free parameters in spectral fitting and to determine accurate wind parameters such as mass-loss rates and velocity profiles. We use the non-LTE model-atmosphere code CMFGEN to numerically solve the radiative transfer equation in the stellar atmosphere and to calculate the radiative acceleration g rad(r). Under the assumption that the radiative acceleration depends only on the radial coordinate, we solve analytically the equation of motion by means of the Lambert W-function. An iterative procedure between the solution of the radiative transfer and the equation of motion is executed in order to obtain a final self-consistent velocity field that is no longer based on any β-law. We apply the Lambert-procedure to three O supergiant stars (ζ Puppis, HD 165763, and α Cam) and discuss the Lambert solutions for the velocity profiles. It is found that, even without recalculation of the mass-loss rate, the Lambert-procedure allows the calculation of consistent velocity profiles that reduce the number of free parameters when a spectral fitting using CMFGEN is performed. Synthetic spectra calculated from our Lambert solutions show significant differences compared to the initial β-law CMFGEN models. The results indicate the importance of consistent velocity profile calculation in the CMFGEN code and its use in a fitting procedure and interpretation of observed spectra.

A generalized theoretical approach for solar cells fill factors by using Shockley diode model and Lambert W-function: A review comparing theory and experimental data

Solar Cells (SCs) energy-conversion technologies have been widely studied from their physical fundamentals to potential commercial applications. In particular, SCs Fill Factors (FFs) are a key factor for evaluating the transport efficiency of the photo-generated current and consequently the potential photovoltaic of the device. However, FF dependence on other solar cell relevant electronic properties is not entirely clear, blearing the physical meaning of this factor. In this context, this work reports the derivation of a self-consistent and generalized analytical equation by using simple Shockley diode equation and Lambert W-function that explicitly relates solar cells FFs with simple key electronic parameters. The photo-generated (JL)-to-reverse saturation (J0) current density ratio (JL/J0) was the key parameter considered for this approach because of its considerable limiting impact on FF magnitude. The accuracy of this equation was tested by an exhaustive contrast with a wide variety of experimental data for different solar cells technologies.

Computing Approximate Equilibria in Weighted Congestion Games via Best-Responses

We present a deterministic polynomial-time algorithm for computing [Formula: see text]-approximate (pure) Nash equilibria in (proportional sharing) weighted congestion games with polynomial cost functions of degree at most [Formula: see text]. This is an exponential improvement of the approximation factor with respect to the previously best deterministic algorithm. An appealing additional feature of the algorithm is that it only uses best-improvement steps in the actual game, as opposed to the previously best algorithms, that first had to transform the game itself. Our algorithm is an adaptation of the seminal algorithm by Caragiannis at al. [Caragiannis I, Fanelli A, Gravin N, Skopalik A (2011) Efficient computation of approximate pure Nash equilibria in congestion games. Ostrovsky R, ed. Proc. 52nd Annual Symp. Foundations Comput. Sci. (FOCS) (IEEE Computer Society, Los Alamitos, CA), 532–541; Caragiannis I, Fanelli A, Gravin N, Skopalik A (2015) Approximate pure Nash equilibria in weighted congestion games: Existence, efficient computation, and structure. ACM Trans. Econom. Comput. 3(1):2:1–2:32.], but we utilize an approximate potential function directly on the original game instead of an exact one on a modified game. A critical component of our analysis, which is of independent interest, is the derivation of a novel bound of [Formula: see text] for the price of anarchy (PoA) of [Formula: see text]-approximate equilibria in weighted congestion games, where [Formula: see text] is the Lambert-W function. More specifically, we show that this PoA is exactly equal to [Formula: see text], where [Formula: see text] is the unique positive solution of the equation [Formula: see text]. Our upper bound is derived via a smoothness-like argument, and thus holds even for mixed Nash and correlated equilibria, whereas our lower bound is simple enough to apply even to singleton congestion games.

Lambert-W function-based parameter extraction for FDSOI MOSFETs down to deep cryogenic temperatures

The applicability of the Lambert-W function-based parameter extraction methodology is demonstrated for 28 nm FDSOI MOSFETs down to deep cryogenic temperatures (4.2 K). The Lambert-W function enables to accurately model the inversion charge and drain current MOSFET characteristics from weak to strong inversion, while using the classical mobility law down to liquid helium temperature. The main MOSFET parameters were extracted versus temperature and gate length, showing the temperature independence of short channel effects and the strong mobility degradation at short channel length due to increased neutral defect scattering.

Estimation of irradiance and temperature of solar photovoltaic system operating at maximum power point

ABSTRACT This paper presents a novel method for estimating irradiance and temperature simultaneously for a photovoltaic (PV) system operating at maximum power point (MPP), utilising only datasheet information. This is the first estimation approach that properly estimates the irradiance and temperature of PV systems running at MPP with low irradiance levels, hence demonstrating its applicability over the entire irradiance range. The uniqueness of the equations lies in their firm theoretical foundation, which is demonstrated by their excellent agreement with the PV single-diode model (SDM). Extraction and translation of PV system parameters is carried out using the recently introduced Lambert-W function in PV modelling. In the context of rapidly changing irradiance and temperature situations, the proposed method is compared to current techniques available in the literature, and the results show that the proposed estimation method outperforms the state of the art.

EXCEL VBA-BASED USER DEFINED FUNCTIONS FOR HIGHLY PRECISE COLEBROOK’S PIPE FLOW FRICTION APPROXIMATIONS: A COMPARATIVE OVERVIEW

This review paper gives Excel functions for highly precise Colebrook’s pipe flow friction approximations developed by users. All shown codes are implemented as User Defined Functions – UDFs written in Visual Basic for Applications – VBA, a common programming language for MS Excel spreadsheet solver. Accuracy of the friction factor computed using nine to date the most accurate explicit approximations is compared with the sufficiently accurate solution obtained through an iterative scheme which gives satisfying results after sufficient number of iterations. The codes are given for the presented approximations, for the used iterative scheme and for the Colebrook equation expressed through the Lambert W-function (including its cognate Wright ω-function). The developed code for the principal branch of the Lambert W-function has additional and more general application for solving different problems from variety branches of engineering and physics. The approach from this review paper automates computational processes and speeds up manual tasks.

A SPICE Compact Model for Ambipolar 2-D-Material FETs Aiming at Circuit Design

We report a charge-based analytic and explicit compact model for field-effect transistors (FETs) based on 2-D materials (2DMs), for the simulation of 2DM-based analog and digital circuits. The device electrostatics is handled by invoking 2-D density of states and Fermi-Dirac statistics that are later combined with the Lambert-W function and Halley's correction to eventually obtain explicit expressions for the electron and hole charges, which are exploited in the calculation of drift-diffusion currents for both carriers. Furthermore, the charge model is extended to obtain characteristics of 2DM-based negative capacitance FETs. The model is benchmarked against experimental MoS <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> FET measurements and experimental ambipolar characteristics of narrowband-gap materials, such as black phosphorus. Its soundness for SPICE circuit-level simulations is also demonstrated.

Analytical Modeling of Current-Voltage Photovoltaic Performance: An Easy Approach to Solar Panel Behavior

In this paper, we propose very simple analytical methodologies for modeling the behavior of photovoltaic (solar cells/panels) using a one-diode/two-resistor (1-D/2-R) equivalent circuit. A value of a = 1 for the ideality factor is shown to be very reasonable for the different photovoltaic technologies studied here. The solutions to the analytical equations of this model are simplified using easy mathematical expressions defined for the Lambert W-function. The definition of these mathematical expressions was based on a large dataset related to solar cells and panels obtained from the available academic literature. These simplified approaches were successfully used to extract the parameters from explicit methods for analyzing the behavior of solar cells/panels, where the exact solutions depend on the Lambert W-function. Finally, a case study was carried out that consisted of fitting the aforementioned models to the behavior (that is, the I-V curve) of two solar panels from the UPMSat-1 satellite. The results show a fairly high level of accuracy for the proposed methodologies.

Analytical solution for nonlinear hydrologic routing with general power-law storage function

A new analytical solution for nonlinear storage routing with a general power-law storage function is presented. The resulting implicit solution, expressed in terms of the incomplete beta function, is valid for inflow hydrographs that may be represented by a series of pulses of arbitrary widths. The new solution thus extends the existing analytical solutions reported in the literature which are valid only for specific exponents in the power-law storage function. An explicit solution expressed in terms of the Lambert-W function is also presented for reservoirs for reservoirs with linear storage-elevation relationship and a single orifice or submerged sluice gate outlet. To support practical design applications and to demonstrate the utility of the new solution, we provide approximate explicit solutions and four simple applications. The results show that, being exact, the new solution is not subject to numerical errors or instabilities and may hence be used in a variety of modeling, design, optimization, forecasting, and control applications involving nonlinear routing with power-law storage functions.

Simplified Lambert W-Function Math Equations When Applied to Photovoltaic Systems Modeling

In this article, simplified and easy-to-work-with equations for the Lambert W-function are derived. This function is widely used to solve equations related to photovoltaic systems. More specifically, this mathematical function represents a useful tool when modeling solar cells/panels performance (that is, the current-voltage curve) by analytical approaches. However, the Lambert W-function has a complex solving process which might represent an unaffordable mathematical challenge for a great number of professionals/technicians in the photovoltaic industrial sector. Simple approximations for the Lambert W-function on both of its branches (positive and negative) are proposed in this article. The results of the present article show a simple but accurate way for photovoltaic systems modeling, even when these systems comprise a maximum power point tracking subsystem.

Optimal Estimation of the Roughness Coefficient and Friction Factor of a Pipeline

Abstract This work addresses the estimation of two interrelated parameters of the fluid flow in pipes. First, a numerical and experimental evaluation of some proposed methods to compute the friction factor in turbulent regime is presented. Special attention is given to an explicit solution obtained through the Lambert W-function. Subsequently, a method to estimate the roughness coefficient using nonlinear optimization techniques is proposed, which then allows determining the friction factor from it. Numerical tests were performed for a wide range of operating points of a pipeline. In order to validate the proposed approach, experimental analysis was carried out on a pipeline pilot-plant. The results show the applicability and effectiveness of the proposed method.

Parameters identification of PV model using improved slime mould optimizer and Lambert W-function

The characterization of PV unit can be made using one-, two-, and triple-diode electrical model. Each model has its own merits in terms of number of unknown parameters to be extracted and burden of calculation, and etc. In most applications, one-diode model (1-DM) is sufficed for the purpose of simulation and analysis in steady-state and dynamic conditions. This paper cares of extracting the unknown five parameters of the 1-DM exploiting the real I-V dataset points. New effort of employing the slime mould algorithm (SMA) and its improved version (ImSMA) is addressed to attain the same goal. Lambert W-function or omega function is used for accurate calculus of PV current. Two benchmarking test cases widely used in the literature are demonstrated and analysed to appraise the performance of SMA/ImSMA complete with subsequent analysis and discussions. The best ImSMA’s results of root mean squared current errors are 7.73006e−4 A and 1.3798e−2 A for RTC solar cell and STP6-120/36; respectively. Various scenarios under varied conditions are demonstrated utilizing the cropped best values of the model’s parameters In addition to that, performance measures are made to validate the cropped results along with comparisons to other recent competing algorithms. It can be concluded that the validations in consort with established outcomes signify the ImSMA in recognizing the PV unidentified 1-DM parameters.

Direct Solution of Various Micrometeorological Problems via the Lambert-W Function

For a family of relatively common problems in micrometeorology, whose forms previously required iterative or indirect treatment, analytical solution is possible via the Lambert-W function. Use of this function reduces computational time and the need for approximations, while offering a clearer view of the character and solution of such problems. This note provides a generalized form for such problems, and demonstrates its use for three applications: internal boundary-layer growth due to a change in surface roughness; flow displacement over a canopy; and determination of friction velocity and roughness length over the sea.

Kinetics of the parallel-consecutive bimolecular reaction: a solution to the inverse problem involving the Lambert-W function.

For the parallel-consecutive bimolecular reaction mechanism, a solution to the inverse kinetic problem can be approached directly using a characteristic equation specified in terms of the Lambert-W function, similar to the logarithmic and reciprocal plot-treatments for simple first and second order reaction kinetics, respectively.

Implicit modelling of series-parallel photovoltaic arrays using double-diode model and its solution

As defined, a solar photovoltaic array is internally a power-limited nonlinear current source. They are commonly used to convert electromagnetic radiation from the sun, into electric power. Accurate models of arrays in series–parallel configuration, represent each sub-module with the double-diode model. Nevertheless, most of those models imply a high computational cost because they require the Lambert W-function to obtain approximated explicit equations for the sub-module’s voltage and current. This paper proposes a model of series–parallel photovoltaic arrays, operating under homogeneous and non-homogeneous irradiance conditions, where each sub-module is represented by the implicit expression derived from the double-diode model. The array is divided into strings and a system of implicit nonlinear equations is obtained for each string with the sub-modules’ voltages and string’s current as unknowns. The corresponding systems of implicit equations are solved by using the Trust-Region Dogleg method to obtain all the electrical variables of the array. The results from the proposed model are compared with those obtained with the equivalent electrical circuit of the array, used as a reference. Simulation data for small, medium, and large arrays show that the proposed method yields data akin to those of the reference method obtaining RMSE values below 0.104 A for current-voltage curves in homogeneous and non-homogeneous conditions. Likewise, the model is validated against several experimental conditions, showing a remarkable agreement with RMSE values below 0.161 A for current-voltage curves in each condition. Therefore, the proposed model can be used to represent accurately series-parallel arrays operating in homogeneous and non-homogeneous conditions.

Suitability for coding of the Colebrook’s flow friction relation expressed by symbolic regression approximations of the Wright-ω function

This article analyses a form of the empirical Colebrook’s pipe flow friction equation given originally by the Lambert W-function and recently also by the Wright ω-function. These special functions are used to explicitly express the unknown flow friction factor of the Colebrook equation, which is in its classical formulation given implicitly. Explicit approximations of the Colebrook equation based on approximations of the Wright ω-function given by an asymptotic expansion and symbolic regression were analyzed in respect of speed and accuracy. Numerical experiments on 8 million Sobol’s quasi-Monte points clearly show that also both approaches lead to approximately the same complexity in terms of speed of execution in computers. However, the relative error of the developed symbolic regression-based approximations is reduced significantly, in comparison with the classical basic asymptotic expansion. These numerical results indicate promising results of artificial intelligence (symbolic regression) for developing fast and accurate explicit approximations.

Design Optimization of Resource Allocation in OFDMA-Based Cognitive Radio-Enabled Internet of Vehicles (IoVs).

Joint optimal subcarrier and transmit power allocation with QoS guarantee for enhanced packet transmission over Cognitive Radio (CR)-Internet of Vehicles (IoVs) is a challenge. This open issue is considered in this paper. A novel SNBS-based wireless radio resource scheduling scheme in OFDMA CR-IoV network systems is proposed. This novel scheduler is termed the SNBS OFDMA-based overlay CR-Assisted Vehicular NETwork (SNO-CRAVNET) scheduling scheme. It is proposed for efficient joint transmit power and subcarrier allocation for dynamic spectral resource access in cellular OFDMA-based overlay CRAVNs in clusters. The objectives of the optimization model applied in this study include (1) maximization of the overall system throughput of the CR-IoV system, (2) avoiding harmful interference of transmissions of the shared channels’ licensed owners (or primary users (PUs)), (3) guaranteeing the proportional fairness and minimum data-rate requirement of each CR vehicular secondary user (CRV-SU), and (4) ensuring efficient transmit power allocation amongst CRV-SUs. Furthermore, a novel approach which uses Lambert-W function characteristics is introduced. Closed-form analytical solutions were obtained by applying time-sharing variable transformation. Finally, a low-complexity algorithm was developed. This algorithm overcame the iterative processes associated with searching for the optimal solution numerically through iterative programming methods. Theoretical analysis and simulation results demonstrated that, under similar conditions, the proposed solutions outperformed the reference scheduler schemes. In comparison to other scheduling schemes that are fairness-considerate, the SNO-CRAVNET scheme achieved a significantly higher overall average throughput gain. Similarly, the proposed time-sharing SNO-CRAVNET allocation based on the reformulated convex optimization problem is shown to be capable of achieving up to 99.987% for the average of the total theoretical capacity.

Numerical solutions of hyperbolic systems of conservation laws combining unsteady friction and viscoelastic pipes

Abstract The main contribution of the paper is to incorporate pipe-wall viscoelastic and unsteady friction in the derivation of the water-hammer solutions of non-conservative hyperbolic systems with conserved quantities as variables. The system is solved using the Godunov finite volume scheme to obtain numerical solutions. This results in the appearance of a new term in the mass conservation equation of the classical governing system. This new numerical algorithm implements the Godunov approach to one-dimensional hyperbolic systems of conservation laws on a finite volume stencil. The viscoelastic pipe-wall response in the mass conservation part of the source term has been modeled using generalized Kelvin–Voigt theory. For the momentum part of the source term a fast, robust and accurate numerical scheme linked to the Lambert W-function for calculating the friction factor has been used. A case study has been used to illustrate the influence of the various formulations; a comparison between the classical solution, the numerical solution including quasi-steady friction, the numerical solution incorporating the viscoelastic effects, and measurements are presented. The inclusion of viscoelastic effects results in better agreement between the measured and solved values.