New control methods for multi-time-scale linear systems with smart grid applications
Creators: name: Kodra, Kliti
Publisher: name: No Publisher Supplied
Year: 2017

description: Power systems within smart grid architectures are generally large scale and have a tendency to exhibit multiple time-scales when modeled in their entirety due to the presence of physical components of different nature and parasitic parameters associated with them. Research in current literature primarily focuses on studying power system architectures based on a two time-scale decomposition. In this dissertation, we use singular perturbation theory to investigate time-scale decomposition and related anomalies and propose new control methods by considering the presence of multiple time-scales.   We start with an open-loop study of a simplified model of an islanded microgrid in singularly perturbed form with highly oscillatory and highly damped modes. Simulation results and analytical analysis conclude that the model does not contain any slow time-scales even though the eigenvalue distribution of the model tells otherwise. While the singular perturbation parameter is very small, the classical two time-scale decomposition in this case is not effective. On the other hand, the modes corresponding to the fastest time-scales provide a very accurate approximation of the original model. The results obtained via singular perturbation methods are also corroborated by using the balancing realization technique. Namely, only the states corresponding to the fastest modes are dominant.  Motivated by the structure of the state-space input matrix of the previous problem, we consider a new class of singularly perturbed systems where individual inputs control slow and fast modes independently. We study the linear quadratic regulator optimal control problem for three cases that are common in real physical systems, namely when the inputs are completely decoupled or independent, when weak coupling is present between the inputs, and when the fast subsystem is weakly controlled. We obtain the zero-order approximation solution of the continuous algebraic Riccati equations for each case in terms of simplified sub-problems which avoid possible ill-conditioning. As a follow-up, parallel recursive algorithms based on fixed-point methods are proposed to improve the error of the approximations leading to the accurate solution of Riccati equations and the cost functional in a few iterations of the algorithm. These results are further extended to the stochastic case. The linear-quadratic Gaussian control problem is investigated and its solution is also obtained very accurately in an iterative fashion.  Lastly, implicit singularly perturbed systems with multiple time-scales are considered. The Schur decomposition is utilized to transform the control matrix into an upper quasi-triangular form where the time-scales are explicitly ordered and a singularly perturbed model is obtained after perturbation parameters are evaluated and extracted. The standard multi-time-scale system is then decoupled into individual time-scales by sequentially applying an invariant transformation. Multi-time-scale control of the Schur-decomposed system is then considered. Control based on the eigenvalue placement method is initially proposed, where the individual decoupled states are fed back sequentially instead of the whole state vector. Furthermore, we design a combined optimal control-eigenvalue placement scheme, where linear-quadratic control is applied to the fastest subsystem and eigenvalue assignment is used for the rest of the states.