47 resultados para Gravitational potential energy


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In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

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Transmission system operators and distribution system operators are experiencing new challenges in terms of reliability, power quality, and cost efficiency. Although the potential of energy storages to face those challenges is recognized, the economic implications are still obscure, which introduce the risk into the business models. This thesis aims to investigate the technical and economic value indicators of lithium-ion battery energy storage systems (BESS) in grid-scale applications. In order to do that, a comprehensive performance lithium-ion BESS model with degradation effects estimation is developed. The model development process implies literature review on lifetime modelling, use, and modification of previous study progress, building the additional system parts and integrating it into a complete tool. The constructed model is capable of describing the dynamic behavior of the BESS voltage, state of charge, temperature and capacity loss. Five control strategies for BESS unit providing primary frequency regulation are implemented, in addition to the model. The questions related to BESS dimensioning and the end of life (EoL) criterion are addressed. Simulations are performed with one-month real frequency data acquired from Fingrid. The lifetime and cost-benefit analysis of the simulation results allow to compare and determine the preferable control strategy. Finally, the study performs the sensitivity analysis of economic profitability with variable size, EoL and system price. The research reports that BESS can be profitable in certain cases and presents the recommendations.