Problems in the design and operation of uncertain complex engineering systems

In this dissertation, we consider two problems. The first one is a general approach to the optimal design of uncertain dynamical systems, where the uncertainty is represented by a random parameter. The problem is formulated using two types of performance criteria, that result in two different optimal design methods. However, both of them are difficult to solve analytically for most uncertain complex dynamical systems. A numerical scheme is developed for the optimal design that involves two steps. First, in order to obtain a numerical algorithm for the optimal solution, we apply randomized algorithms for average performance synthesis to approximate the optimal solution. Second, using the properties of the Perron-Frobenius operator we develop an efficient computation approach for calculating the stationary distribution for the uncertain dynamical systems and the average performance criteria. The proposed approach is demonstrated through numerical examples. The second problem is a novel approach for evaluating the short-term Loss of Load Probability (LOLP) in power systems that include wind generation resources that vary stochastically in time. We firstly introduce a mathematical model for calculating the short-term LOLP, and then a novel quantitative measure of its behavior when converging to its steady-state level is derived. In addition, the corresponding empirical formulas are offered which can be used in practice to estimate the convergence time of LOLP under different conditions. Finally, an application of the outcomes of the analytical work in estimation of the dynamic behavior of short-term LOLP with an actual wind generation profile is presented to show the significance of the developed measures.