# Volume 30 - 2019 - MADEV health and energy (2017)

Special issue dedicated to MADEV 2017, papers on wealth and energy Editors: Françoise Lamnabhi-Lagarrigue, Noura Yousfi, and Nabil Gmati

### 1. Global stability of a fractional order HIV infection model with cure of infected cells in eclipse stage

Modeling by fractional order differential equations has more advantages to describe the dynamics of phenomena with memory which exists in many biological systems. In this paper, we propose a fractional order model for human immunodeficiency virus (HIV) infection by including a class of infected cells that are not yet producing virus, i.e., cells in the eclipse stage. We first prove the positivity and bound-edness of solutions in order to ensure the well-posedness of the proposed model. By constructing appropriate Lyapunov functionals, the global stability of the disease-free equilibrium and the chronic infection equilibrium is established. Numerical simulations are presented in order to validate our theoretical results.

### 2. A mathematical model on the effect of growth hormone on glucose homeostasis

Extending an existing model devoted to the interaction between β-Cell Mass, Insulin, Glucose, Receptor Dynamics and Free Fatty Acids in glucose regulatory system simulation, this paper proposes a mathematical model introducing the effect of growth hormone on the glucose homeostasis alongside the other variables. Stability analysis is carried out and pragmatic explanation of the equilibrium points is emphasized. Finally, simulation illustrated how β-Cell Mass, Insulin, Glucose, Receptor Dynamics, Free Fatty Acids and Growth Hormone may vary with different values of some parameters in the model.

### 3. Dynamics of an HBV infection model with cell-to-cell transmission and CTL immune response

In this work, we propose a mathematical model to describe the dynamics of the hepatitis B virus (HBV) infection by taking into account the cure of infected cells, the export of precursor cytotoxic T lympho-cytes (CTL) cells from the thymus and both modes of transmission that are the virus-to-cell infection and the cell-to-cell transmission. The local stability of the disease-free equilibrium and the chronic infection equilibrium is obtained via characteristic equations. Furthermore, the global stability of both equilibria is established by using two techniques, the direct Lyapunov method for the disease-free equilibrium and the geometrical approach for the chronic infection equilibrium.

### 4. Application des modèles mathématiques pour l’optimisation de l’énergie dans un système PV

This paper proposes an approach used to optimize the energy for a stand-alone photovoltaic (PV) system in isolated regions. The intended objective is house energy comfort. The aim is to present the impact of flow energy of housing on the system reliability. The operation of stand-alone PV system is represented by a simulation program. This later describes the principle of energy equilibrium among diverse sub-systems, using different mathematical models of different parts of renewable energy system. The recommended models were implemented via Matlab-Simulink software with real input data. The reliability is achieved by reducing the loss power supply probability criteria, with improvement of the battery life cycle during the operating years of the PV system.

### 5. Photovoltaic Hybrid Systems for remote villages

Electricity access in remote areas of Sub-Saharan Africa is limited due to high costs of grid extension to areas characterised by low population and low energy densities. Photovoltaic hybrid systems can be computed using an energy balance equation involving one unknown. For hypothetical village with an average daily energy demand of 153.6 kWh/d, the monthly energy output of photovoltaic modules at Garoua, Cameroon, enabled the evaluation of feasible photovoltaic hybrid (PVHS) options. An option with a renewable energy fraction of 0.557 having lower initial investments is suggested for electrification of more remote villages in Sub-Saharan African countries which have high solar radiation levels. This option comprises a 23.56 kWp PV array, a 15 kWp PV inverter, a 25 kW bi-directional inverter, a battery bank of capacity 324.48 kWh and a 25 kW diesel generator with an operating time of 1309 h/yr or 3.59h/d. The size of the PV array determined is smaller compared to the sizes of PV arrays which have been evaluated in the range 30-45 kWp using HOMER software for medium villages in Senegal.

### 6. Optimization of absorption systems: case of the refrigerators and heat pumps

The new thermo-ecological performance optimization of absorption is investigated by taking the ecological coefficient of performance ECOP as an objective function. ECOP has been expressed in terms of the temperatures of the working fluid in the main components of the system. The maximum of ECOP and the corresponding optimal temperatures of the working fluid and other optimal performance design parameters such as coefficient of performance, specific cooling load of absorption refrigerators, specific heating load of absorption heat pumps, specific entropy generation rate and the distributions of the heat exchanger areas have been derived analytically. The obtained results may provide a general theoretical tool for the ecological design of absorption refrigerators and heat pumps.

### 7. Nonlinear Control for Isolated DC MicroGrids

The access to electricity for isolated communities is capital for improving life in those societies. The use of renewable energy sources (renewables) and energy storage systems is the key for clean energy supply in remote areas without the main grid. The standalone operation of renew-ables represents a challenge for operation and reliability, therefore the DC MicroGrid concept is seen as a powerful solution allowing renewables integration and reliable operation of the system is simple way. This paper proposes a distributed nonlinear control strategy for an isolated MicroGrid composed of renewables and different timescale storage systems to supply a DC load. The simulations results show the behavior of the proposed MicroGrid and a comparison with classical linear control is done to highlight the control performance.

### 9. Optimal control of a parabolic solar collector

The aim of this paper is to study an optimal control problem for a parabolic solar collector. We consider a bilinear distributed model, where the control models the velocity of the heat-transfer fluid. We prove the existence of an optimal control, and we derive a necessary optimality condition. Then we give an algorithm for the computation of the optimal control. The obtained results are illustrated by simulations of the collector model, using data of Ain Beni Mathar solar plant in Morocco.

### 10. Challenges of mastering the energy sector and sustainable solutions for development in Africa

The African continent is currently experiencing a period of sustained economic and population growth that requires massive investment in the energy sector to effectively meet the energy needs in the context of sustainable development. At the same time, COP21 agreements now call all the states to use clean energy. Yet a great energy potential is available, but the electrification rate of the continent currently accounts for only 3% of the world's energy production. Also, the demand for both quality and quantity energy requires the mastery of applied mathematical tools to efficiently solve problems arising in the energy system. In this article, the major problems affecting the energy sector in Africa are identified, some solutions to the challenges are recalled and some new ones are proposed, with emphasis given to applied mathematics tools as well as energy policy. As case studied, a new control strategy of Static Synchronous Series Compensator (SSSC) devices-which are modern power quality Flexible Alternating Current Transmission Systems (FACTS)-is proposed for Power Flow Control.

### 11. Chronic myeloid leukemia model with periodic pulsed treatment

In this work we develop a mathematical model of chronic myeloid leukemia including treatment with instantaneous effects. Our analysis focuses on the values of growth rate γ which give either stability or instability of the disease free equilibrium. If the growth rate γ of sensitive leukemic stem cells is less than some threshold γ * , we obtain the stability of disease free equilibrium which means that the disease is eradicated for any period of treatment τ 0. Otherwise, for γ great than γ * , the period of treatment must be less than some specific value τ * 0. In the critical case when the period of treatment is equal to τ * 0 , we observe a persistence of the tumor, which means that the disease is viable.

### 12. Modeling and simulation of Power Electronic based Electrotechnical Systems for Renewable Energy, Transportation and Industrial Applications: A Brief Review

In this paper, a completed review of recent researches about modern power converter based electrotechnical systems (ETSs) has been carried out. In particular, power electronics (PEs) based ETSs have been investigated. The literature review consists of a standard classification of PEs-based ETSs, along with a survey on strengths and weaknesses of these devices impact on renewable energy sources.