Supercapacitor modelling, analysis and design

This thesis develops tools for the analysis and design of electrochemical energy storage devices known as supercapacitors. Supercapacitors are characterised by low internal resistances, high capacitances as well as limited degradation and temperature influence. They are typically used for high power applications, and have been successfully implemented within hybrid power sources and for wind turbine fault ride through amongst other applications. Methods from systems and control theory are employed in this thesis to form a reliable supercapacitor design methodology, based upon high fidelity modelling, for the construction of cheaper supercapacitors with improved performance. The first part of the thesis considers supercapacitor model development and analysis. A model is introduced that describes the electrochemistry of the supercapacitor energy storage mechanism. This model is called a physics based model and is described by a set of partial differential equations (PDEs). A computationally efficient implementation of this model, based upon spectral collocation spatial discretisation of the PDEs, is proposed. This allows the model to be effectively implemented and analysed as a nonlinear finite dimensional system. The extent to which model information can be extracted from current-voltage data is considered using an observability and parameter estimation analysis. A mathematical transformation from the electrochemical PDEs to an equivalent circuit representation is described, linking the two main supercapacitor modeling approaches in a quantitative manner. A framework for supercapacitor design is proposed that relates energy storage properties, including the resistance, capacitance, dissipated energies and maximum safe voltages, to the electrochemical parameters, such as the diffusion coefficient of the electrolyte and the electrode's conductivity. This relationship is obtained via the solution of a convex optimisation problem and so can be easily obtained. Using this design approach reduces the number of physical experiments needed to be carried out, making the design process faster, cheaper and more detailed. The problem of configuring the layers of supercapacitors with structured electrodes is considered, and it is shown that destroying the single phase nature of the electrodes radically alters their frequency response. These results bring a fresh perspective to the supercapacitor design problem, by adding rigorous mathematical modelling and analysis. The second part of the thesis develops tools for the local stability analysis of nonlinear systems. A Lyapunov function is proposed for nonlinear systems that can be represented as the feedback interconnection of a linear system with a nonlinearity which is decentralised, sector bounded and slope restricted. A main result is to relax strict positivity of many of the terms of this Lyapunov function by utislising information about the nonlinearity. These results are extended to systems that evolve along a vector field which is a rational function of the states and the nonlinearity, with positivity of many terms of a polynomial Lyapunov function again being relaxed. Relaxing the strict positivity conditions of the terms of these Lyapunov functions results in less conservative stability certificates for the systems. Verification of the stability conditions is expressed as the solution to a convex semi-definite program, and so can be efficiently computed. The supercapacitor model was a particular example of the class of nonlinear systems considered, with the inspiration behind this work being the desire to obtain less conservative bounds for the supercapacitor resistance.