Ross Drummond

Low-order mathematical modelling of electric double layer supercapacitors using spectral methods

Abstract This work investigates two physics-based models that simulate the non-linear partial differential algebraic equations describing an electric double layer supercapacitor. In one model the linear dependence between electrolyte concentration and conductivity is accounted for, while in the other model it is not. A spectral element method is used to discretise the model equations and it is found that the error convergence rate with respect to the number of elements is faster compared to a finite difference method. The increased accuracy of the spectral element approach means that, for a similar level of solution accuracy, the model simulation computing time is approximately 50% of that of the finite difference method. This suggests that the spectral element model could be used for control and state estimation purposes. For a typical supercapacitor charging profile, the numerical solutions from both models closely match experimental voltage and current data. However, when the electrolyte is dilute or where there is a long charging time, a noticeable difference between the numerical solutions of the two models is observed. Electrical impedance spectroscopy simulations show that the capacitance of the two models rapidly decreases when the frequency of the perturbation current exceeds an upper threshold.

On observer performance for an electrochemical supercapacitor model

In this paper, the implementation of an observer with an electrochemical supercapacitor model is discussed. Using Lie derivatives, the model was shown to be locally nonlinear observable. When the two transference numbers, related to ion mobility, were selected to be equal, the model became linear and the states relating to ionic concentration became unobservable. These unobservable states were shown to not satisfy the weaker detectability condition. The energy storage characteristic of supercapacitors meant that they incorporated integrator dynamics and this led to random walk trajectories with a Kalman filter implementation. This problem could be corrected by using a more suitable observer or by introducing a control system such that the model becomes asymptotically stable. The comments on observer performance made in this paper could be applied to more complex electrochemical models, such as those for Lithium ion batteries, as they use many similar physical relationships.

Circuit synthesis of electrochemical supercapacitor models

This paper is concerned with the synthesis of RC electrical circuits from physics-based supercapacitor models describing conservation and diffusion relationships. The proposed synthesis procedure uses model discretisation, linearisation, balanced model order reduction and passive network synthesis to form the circuits. Circuits with different topologies are synthesized from several physical models. This work will give greater understanding to the physical interpretation of electrical circuits and will enable the development of more generalised circuits, since the synthesized impedance functions are generated by considering the physics, not from experimental fitting which may ignore certain dynamics.

Parameter estimation of an electrochemical supercapacitor model

Parameter estimation of a supercapacitor model that uses PDEs to describe electrochemical physical phenomena is studied. The model parameters were identified from time domain charge/discharge data and also from frequency domain data in the form of electrical impedance spectroscopy at different open cell voltages. The identified physical model could fairly accurately simulate the experimental data, however, this accuracy was found to drop off at high frequencies. The performance of the physical model was compared to a second order circuit, with the two models having similar levels of accuracy. The circuit was found to be simpler to implement but, unlike the physical model, could not supply physical information about the internal state of the supercapacitor.

A Lyapunov function for a PDE model of a supercapacitor

In this paper, a global positive-definite Lyapunov function is constructed analytically for the set of non-linear partial differential algebraic equations that describe a physics-based model of a supercapacitor. The model uses a set of conservation and diffusion equations to describe ion transport in the electrodes and separator. Global stability of the model equations can be shown through Lyapunov analysis. Local asymptotic stability of the equations can also be shown when the concentration is constrained to some set. This analysis can be used to study the stability of more complex electrochemical devices, such as lithium ion batteries, which are based on similar physical relationships. It can also be used as the basis for an advanced PDE boundary charging control system whose performance will be independent of model discretisation.

The Aizerman and Kalman conjectures using symmetry

Abstract Using a decomposition of a Lurie system in terms of symmetric and skew-symmetric matrices, this paper presents reformulations of the classical conjectures of Aizerman and Kalman which give valid conditions for absolute stability. Under this decomposition, it is shown that a restatement of the Aizerman conjecture implies stability while the re-stated Kalman conjecture implies contraction.

Gradient filter methods for predictive control with simple constraints

We present a framework for developing computationally efficient, first-order optimisation solvers using gradient filtering techniques. These solvers are similar in form to Nesterovs fast gradient method, however they are differentiated by the increased number of historical iterations used in the gradient filter. The filter tap weights are selected offline by solving a system of algebraic equations such that bounds on the convergence rates of the filters are minimised. This approach was found to accelerate the convergence rate of the fast gradient method and could compute the solution of a test case quadratic programming problem, based on MPC of an atomic force microscope, in a shorter time than both the fast gradient and interior point methods.