In this article, simplified and easy-to-work-with equa-Q1 5 tions for the Lambert W-function are derived. This function is 6 widely used to solve equations related to photovoltaic systems. More 7 specifically, this mathematical function represents a useful tool 8 when modeling solar cells/panels performance (that is, the current-9 voltage curve) by analytical approaches. However, the Lambert 10 W-function has a complex solving process which might represent 11 an unaffordable mathematical challenge for a great number of pro-12 fessionals/technicians in the photovoltaic industrial sector. Simple 13 approximations for the Lambert W-function on both of its branches 14 (positive and negative) are proposed in this article. The results of 15 the present article show a simple but accurate way for photovoltaic 16 systems modeling, even when these systems comprise a maximum 17 power point tracking subsystem. 18 Index Terms-1-diode/2-resistor model, I-V curve, Lambert W-19 function, maximum power point tracking (MPPT), photovoltaic 20 systems performance, solar cell, solar panel. 21 I. INTRODUCTION 22 R ENEWABLE energy plays a very important role in reduc-23 ing fossil resources consumption [1], which is a present 24 and urgent need for mankind due to problems, such as global 25 warming, climate change and air pollution [2]. Among the 26 different renewable energy sources, solar energy is probably the 27 most relevant, as it is clean, safe, and unlimited [3], [4].

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In the present work, the effect of the friction forces at bearings on cup anemometer performance is studied. The study is based on the classical analytical approach to cup anemometer performance (2-cup model), used in the analysis by Schrenk (1929) and Wyngaard (1981). The friction torque dependence on temperature was modelled using exponential functions fitted to the experimental results from RISØ report #1348 by Pedersen (2003). Results indicate a logical poorer performance (in terms of a lower rotation speed at the same wind velocity), with an increase of the friction. However, this decrease of the performance is affected by the aerodynamic characteristics of the cups. More precisely, results indicate that the effect of the friction is modified depending on the ratio between the maximum value of the aerodynamic drag coefficient (at 0° yaw angle) and the minimum one (at 180° yaw angle). This reveals as a possible way to increase the efficiency of the cup anemometer rotors. Besides, if the friction torque is included in the equations, a noticeable deviation of the rotation rate (0.5-1% with regard to the expected rotation rate without considering friction) is found for low temperatures.

The most relevant results from the IDR/UPM Institute research on solar cells/panels and Li-ion batteries performance are reviewed in this paper. The aim of this work is to present the possibilities of the mathematical procedures developed for space applications, with a view to extend their use to other industrial sectors. The research at the IDR/UPM Institute has been driven by selecting simple tools and procedures to model the behavior of solar cell/panel, since modeling these photovoltaic devices is normally carried out by 1-Diode/2-Resistor equivalent circuit models, which might represent an unaffordable mathematical challenge for many professionals and technicians in the renewable energy field. Concerning Li-ion batteries, the large experience accumulated during the maintenance of the UPMSat-2 battery resulted in the successful development of new mathematical models to study the performance of batteries. The models developed are based on the amount of discharged energy, measured in W•h, instead of other more traditional parameters such as the State Of Charge (SOC), measured in A•h.