Analysis of Lead-Acid batteries with a third-order dynamical model

Electrical energy storage is a really important issue nowadays. As electricity is not easy to be directly stored, it can be stored in other forms and converted back to electricity when needed. As a consequence, storage technologies for electricity can be classified by the form of storage, and in particular we focus on electrochemical energy storage systems, better known as electrochemical batteries. Largely the more widespread batteries are the Lead-Acid ones, in the two main types known as flooded and valve-regulated. Batteries need to be present in many important applications such as in renewable energy systems and in motor vehicles. Consequently, in order to simulate these complex electrical systems, reliable battery models are needed. Although there exist some models developed by experts of chemistry, they are too complex and not expressed in terms of electrical networks. Thus, they are not convenient for a practical use by electrical engineers, who need to interface these models with other electrical systems models, usually described by means of electrical circuits. There are many techniques available in literature by which a battery can be modeled. Starting from the Thevenin based electrical model, it can be adapted to be more reliable for Lead-Acid battery type, with the addition of a parasitic reaction branch and a parallel network. The third-order formulation of this model can be chosen, being a trustworthy general-purpose model, characterized by a good ratio between accuracy and complexity. Considering the equivalent circuit network, all the useful equations describing the battery model are discussed, and then implemented one by one in Matlab/Simulink. The model has been finally validated, and then used to simulate the battery behaviour in different typical conditions.

Pricing in stochastic-local volatility models with default

In recent years is becoming increasingly important to handle credit risk. Credit risk is the risk associated with the possibility of bankruptcy. More precisely, if a derivative provides for a payment at cert time T but before that time the counterparty defaults, at maturity the payment cannot be effectively performed, so the owner of the contract loses it entirely or a part of it. It means that the payoff of the derivative, and consequently its price, depends on the underlying of the basic derivative and on the risk of bankruptcy of the counterparty. To value and to hedge credit risk in a consistent way, one needs to develop a quantitative model. We have studied analytical approximation formulas and numerical methods such as Monte Carlo method in order to calculate the price of a bond. We have illustrated how to obtain fast and accurate pricing approximations by expanding the drift and diffusion as a Taylor series and we have compared the second and third order approximation of the Bond and Call price with an accurate Monte Carlo simulation. We have analysed JDCEV model with constant or stochastic interest rate. We have provided numerical examples that illustrate the effectiveness and versatility of our methods. We have used Wolfram Mathematica and Matlab.

Equazioni stocastiche backward nel mercato delle emissioni

In questa tesi viene esposto il modello EU ETS (European Union Emission Trading Scheme) per la riduzione delle emissoni di gas serra, il quale viene formalizzato matematicamente da un sistema di FBSDE (Forward Backward Stochastic Differential Equation). Da questo sistema si ricava un'equazione differenziale non lineare con condizione al tempo finale non continua che viene studiata attraverso la teoria delle soluzioni viscosità. Inoltre il modello viene implementato numericamente per ottenere alcune simulazioni dei processi coinvolti.

Geometric properties of 2-dimensional minimal surfaces in a sub-Riemannian manifold which models the Visual Cortex

In this paper we study the notion of degree forsubmanifolds embedded in an equiregular sub-Riemannian manifold and we provide the definition of their associated area functional. In this setting we prove that the Hausdorff dimension of a submanifold coincides with its degree, as stated by Gromov. Using these general definitions we compute the first variation for surfaces embedded in low dimensional manifolds and we obtain the partial differential equation associated to minimal surfaces. These minimal surfaces have several applications in the neurogeometry of the visual cortex.

Mathematical models for cellular aggregation: the chemotactic instability and clustering formation

In this thesis we present a mathematical formulation of the interaction between microorganisms such as bacteria or amoebae and chemicals, often produced by the organisms themselves. This interaction is called chemotaxis and leads to cellular aggregation. We derive some models to describe chemotaxis. The first is the pioneristic Keller-Segel parabolic-parabolic model and it is derived by two different frameworks: a macroscopic perspective and a microscopic perspective, in which we start with a stochastic differential equation and we perform a mean-field approximation. This parabolic model may be generalized by the introduction of a degenerate diffusion parameter, which depends on the density itself via a power law. Then we derive a model for chemotaxis based on Cattaneo's law of heat propagation with finite speed, which is a hyperbolic model. The last model proposed here is a hydrodynamic model, which takes into account the inertia of the system by a friction force. In the limit of strong friction, the model reduces to the parabolic model, whereas in the limit of weak friction, we recover a hyperbolic model. Finally, we analyze the instability condition, which is the condition that leads to aggregation, and we describe the different kinds of aggregates we may obtain: the parabolic models lead to clusters or peaks whereas the hyperbolic models lead to the formation of network patterns or filaments. Moreover, we discuss the analogy between bacterial colonies and self gravitating systems by comparing the chemotactic collapse and the gravitational collapse (Jeans instability).

Fractional Order Modeling and Control of a Flexible Satellite

Researchers have engrossed fractional-order modeling because of its ability to capture phenomena that are nearly impossible to describe owing to its long-term memory and inherited properties. Motivated by the research in fractional modeling, a fractional-order prototype for a flexible satellite whose dynamics are governed by fractional differential equations is proposed for the first time. These relations are derived using fractional attitude dynamic description of rigid body simultaneously coupled with the fractional Lagrange equation that governs the vibration of the appendages. Two attitude controls are designed in the presence of the faults and uncertainties of the system. The first is the fractional-order feedback linearization controller, in which the stability of the internal dynamics of the system is proved. The second is the fractional-order sliding mode control, whose asymptotic stability is demonstrated using the quadratic Lyapunov function. Several nonlinear simulations are implemented to analyze the performance of the proposed controllers.