Ultimate bound analysis and optimisation with application to energy systems stability

Research Doctorate - Doctor of Philosophy (PhD)This thesis studies steady state behaviour of control systems subject to persistent perturbations. When non-vanishing perturbations act on an otherwise asymptotically stable system, convergence to the equilibrium point may not be achievable. Instead, it is desirable to have the system trajectories ultimately lie inside a tight ultimate bound set when starting within a region of attraction in the state space. This behaviour proves stability beyond local results and is considered as practical stability of systems under the effect of persistent perturbations. This study adopts ultimate bounds as the main instrument to analyse steady state behaviour of systems under persistent perturbations and makes contributions in two distinct directions. The first direction is the application of ultimate bounds to analyse regional (beyond linearised local) stability properties of frequency control systems in islanded microgrids of inverters which are inherently nonlinear and operate under the effect of persistent unknown load demand variations. The existing works on the frequency control problem have predominantly derived local results to address stability of the network in the vicinity of an equilibrium point. The second direction of the thesis is the minimisation of componentwise ultimate bounds by feedback control design in discrete-time linear systems under persistent perturbations with constant bounds, which serves as a certificate of improved practical stability of the perturbed system. Existing related work has studied the attenuation of disturbances by minimising state norms. A distinct contribution of the present thesis is its emphasis on state componentwise attenuation of persistent disturbances through ultimate boundedness. The analysis of ultimate boundedness in the framework of the frequency control problem in inverter-based microgrids with primary and secondary control loops led us to show that (i) frequency regulation can be ensured without assuming time-scale separation and, (ii) ultimate boundedness of the trajectories starting inside a region of attraction is guaranteed under a condition on the power mismatch between demand and generation at each inverter bus. This result departs from conventional studies which rely on time-scale separation between primary and secondary control loops to show local stability of equilibria. By way of contrast, we derive an estimate of the region of attraction from which a quantifiable ultimate bound set for the state trajectories can be determined by recursive iterations of a nonlinear mapping, leading to closed-form expressions for some classes of microgrids. For the problem of feedback design to minimise state ultimate bounds in discrete-time linear systems subject to persistent bounded perturbations, we derive structural conditions on the system matrices to guarantee that a target eigenstructure, which is shown to achieve the lowest possible ultimate-bounds for one or more state components, can be assigned by state feedback. When the required structural conditions are satisfied, an eigenvector assignment procedure can be applied to simultaneously minimise multiple ultimate bounds in systems with multiple inputs, or one ultimate bound in systems with a single input. These results are applied to the voltage control problem under load variations in inverter based microgrids. The results on the minimisation of ultimate bounds are then extended to perturbed discrete-time switched linear systems to achieve closed-loop stability under arbitrary switching and minimum ultimate bounds for specific state components. Previous results derived an iterative algorithm that computes the required feedback matrices, and established conditions under which this procedure is possible. Based on these conditions, ultimate bound minimisation of some state components is achieved by exploiting available degrees of freedom in the iterative algorithm and assigning suitable eigenstructure to the subsystems