In this paper, nineteen models were used to estimate the monthly average hourly global solar irradiation from the daily global irradiation value; at the “Cirque de Mafate” which is an isolated high mountain and rugged relief site in Reunion Island. These models are divided into three groups; the first depends on solar parameters like hour angle or solar time, the second implies that the estimation function follows a Gaussian distribution, and the third is a simplified form of the first. The main target is to find, for the site, the best model to estimate the abovementioned monthly average hourly irradiation. The measured data used to validate the models are from an in situ weather station. The following statistical criteria; normalized mean bias error, normalized absolute mean bias error, normalized root mean square, t-statistical test, correlation coefficient, relative standard error and Nash-Sutcliffe Equation were used to evaluate the performance for each model. To rank and compare the nineteen models by the abovementioned seven criteria, the Multi-Criteria Decision Making (MCDM) approach has been used and especially the Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS). The basic principle of TOPSIS is to define the ideal model and the worst model by the set of the statistical criteria’s value for all models. Then the Euclidian distance to the ideal model and/or the worst model is calculated. The best model is the one that is nearest the ideal model and farthest the worst model. To use the TOPSIS, a normalized weight, that indicates the importance or priority, for each statistical criterion has been calculated by objective and subjective way. As result, it was found that the best model came from the first group and it is the Collares-Pereira and Rabl model modified by Gueymard (CPRG) and in second position is the Gueymard model.

The aim of this paper is to apply all the existing pyranometric methods to estimate the sunshine duration from global solar irradiation in order to find the most suitable method for a tropical site in its original form. Then, in a second step, one of these methods will be optimized to effectively fit tropical sites. Five methods in the literature (Step algorithm, Carpentras Algorithm, Slob and Monna Algorithm, Slob and Monna 2 Algorithm, and linear algorithm) were applied with eleven years of global and diffuse solar radiation data. As a result, with regard to its original form, the step algorithm is in the first rank. But in the second step, after improving its main coefficients, the Carpentras Algorithm was found to be the best algorithm for tropical sites in the southern hemisphere.

We study a class of dynamic unilateral contact problems with sub-differential friction law, and thermal effects, for time depending long memory visco-elastic materials, with or without the clamped condition. We describe the mechanical problem, derive its variational formulation, and after specifying the assumptions on the data and operators, we prove an existence and uniqueness of weak solution on displacement and temperature fields.

In non-interconnected areas, the efficient use of renewable energies requires an optimal management of power consumption. We study the case of the "Cirque de Mafate" on Reunion Island. A model based on mixed integer linear programming has been developed as optimization tool. In order to minimize energy losses by maximizing the use of electrical appliances or the use of solar energy and the energy stored in the battery during the day, this paper aims to find the optimal balance between the production system and the consumption of the microgrid by considering uncertainties of demand as well as the production resources. The loads are the electrical appliances of a group of three houses. It consists to model, optimize and simulate the stochastic operation of an autonomous microgrid by mutualizing production and storage resources. The uncertainty propagation method is developed to take into account uncertainty of solar radiation as well as electrical consumption of the users in the optimization model. Indeed, to estimate the probability density function of the electricity consumption, non- parametric methods for the estimation of probability density functions are applied. These same methods were used to estimate the probability density functions of solar radiation for the solar resource forecast. Indeed, the prediction of the intermittent resource and the combination of generation sources taking into account the uncertainty of the users' demand is the key for the proper operation of a microgrid in autonomous mode. The results show the performance of the system to achieve self-consumption for three days if the solar forecast is pessimistic and allow us to evaluate the performance of the system under random constraints. Indeed, they allow us to make decisions in order to advise users on the use of electrical appliances by taking into account their behavior and the last minute modification of their time slots of use of electrical appliances.

In off-grid areas, successful use of intermittent renewable energy sources requires optimal management of power consumption. Our experimental site is the “Cirque de Mafate” on Reunion Island. Our laboratory has developed a mixed integer linear programming model which minimizes the electricity consumption of a cluster of houses. This model is deterministic. Our study focuses on the stochastic part, it aims to model, optimize and simulate the stochastic operation of an autonomous microgrid by mutualizing production and storage resources. A study for the solar resource forecasting is performed, using nonparametric methods for the estimation of probability density functions. Indeed, the prediction of the intermittent resource and the combination of production sources are the keys to the good functioning of a microgrid in autonomous mode. Three neighbouring houses are concerned. The experimentation at the scale of a house has already been performed and the results proved the performance of the system under random constraints. This paper presents the results of an experimentation for two houses. One of the strategies adopted is to aim for auto-consumption for three days if the solar forecast is pessimistic, a part of the energy is then reserved at the battery level for the following two days. The results allow to assess the performance of the system in front of random constraints and to make decisions.

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

We study a class of non-clamped dynamical problems for visco-elastic materials, the contact condition is modeled by a normal compliance, with friction, damage and heat exchange. The weak formulation leads to a general system defined by a second-order quasi-variational evolution inequality on the displacement field coupled with a nonlinear evolutional inequality on temperature field and a parabolic variational inequality on the damage field. We present and establish an existence and uniqueness result of different fields, by using general results on evolution variational inequalities, with monotone operators and fixed point methods. Then, we present a fully discrete numerical scheme of approximation and derive an error estimate. Finally, various numerical computations are developed.

. This paper presents an operational model of an electrical power supply in order to meet the load of a cluster of houses in a remote mountainous area. In outlying areas, an isolated power network represents the most economical solution. However, the implementation of a cluster of houses in an electrical microgrid requires optimal management of the power supply-demand in order to reach the users’ requirements. Our case study is located in the “Cirque de Mafate” in Reunion Island. To build the model, the different types of individual consumption and the available energy production in situ are described. Energy management is achieved through a large mixed integer linear programming system. The model allows the production to fit the consumption by minimizing losses. Numerical calculations have been performed in order to determine an optimal solution that minimizes the use of the battery energy storage system and also satisfies the comfort of the inhabitants. The use of fractional derivative is introduced in the battery storage model. Simulations show that this emerging technology may lead to a technical solution that meets the above requirements of battery energy use and consumer satisfaction. It is also shown that a most effective and efficient use of energy resources is required in order to achieve sustainable management of electrical energy.

In this paper, we study a class of dynamic thermal problems involving a frictional normal compliance adhesive contact model and a non-clamped condition for visco-elastic materials. The variational formulation of the problem leads to a general system defined by a second-order quasi-variational evolution inequality on the displacement field coupled with the two first-order evolution equations describing the evolution of the temperature and adhesion. We establish an existence and uniqueness result of the solution on displacement, temperature and adhesion. Then we provide a fully discrete numerical scheme of approximation and derive an error estimate. Finally, various numerical computations are included.

We study a class of dynamic sub-differential contact problems with friction, and thermale ects, for time depending long memory visco-elastic materials, with or without the clamped condition.We describe the mechanical problem, derive its variational formulation, and after specifyingthe assumptions on the data and operators, we prove an existence and uniqueness of weak solutionon displacement and temperature fields. Then we present a fully discrete scheme for numerical approximationsof the different solutions, and provide analysis of error order estimates. Finally variousnumerical computations in dimension two will be given.

A duality approximation method combined with an adaptive finite element method is applied to solve a free boundary problem with periodic boundary conditions in lubrication theory.

We study a class of evolutional variational inequalities of parabolic type where we establish a general existence and uniqueness result. Then we apply the abstract result to solve a dynamic thermal subdifferential contact problem with friction, for time depending nonlinear long memory visco-elastic materials, with or without the clamped condition, which can be put into a general model of system defined by a second order evolution inequality, coupled with a first order evolution equation. We present and state an existence and uniqueness of weak solution, by using fixed point methods, monotonicity and convexity.

We study a class of dynamic thermal sub-differential contact problems with friction, for long memory viscoelastic materials, without the clamped condition, which can be put into a general model of system defined by a second order evolution inequality, coupled with a first order evolution equation. After statement of an existence and uniqueness result, we present a fully discrete scheme for numerical approximations and analysis of error order estimate.

In this chapter we study a dynamic frictional contact problem with normal compliance and non-clamped contact conditions, for thermo-viscoelastic materials. The weak formulation of the problem leads to a general system defined by a second order quasivariational evolution inequality coupled with a first order evolution equation. We state and prove an existence and uniqueness result, by using arguments on parabolic variational inequalities, monotone operators and fixed point. Then, we provide a numerical scheme of approximations and various numerical computations.

A mathematical problem modeling dynamical behavior of viscoelastic materials is studied. The problem is related to highly nonlinear and non-smooth phenomena like contact, friction and a class of dynamic contact problems with friction and thermal effects. The applied approach leads to a system involving variational inequalities and a fully discrete scheme for numerical approximations and the analysis of error estimate order is provided. Numerical computations are given so as to illustrate the mathematical model.

We study a class of dynamic thermal sub-differential contact problems with friction, for long memory visco-elastic materials, without the clamped condition, which can be put into a general model of system defined by a second order evolution inequality, coupled with a first order evolution equation. We present and establish an existence and uniqueness result, by using general results on first order evolution inequality, with monotone operators and fixed point methods. Finally a fully discrete scheme for numerical approximations is provided, and corresponding various numerical computations in dimension two will be given.