Upgrading of the decision-making process for system development

The planning horizon for a system development planner is long and a system planner must take into account potential changes to generation capacity and demand during that horizon. The uncertainties that affect system development are greater than those that apply, for example, in system operations. Thus, it is important to build up a credible set of operating states that is informed by various uncertainties and adequately represents the range of conditions that might reasonably be expected to arise. The next step in the system development process is to assess these operating states for system operability. If a power system is not operable on some (probable) operating states, then it identifies a potential need for investment in the system. However, given the number and range of uncertainties relevant to system development, pragmatic approaches must also be developed allowing their assessment. In this report, a comprehensive system development approach is presented

Dependence in Stochastic Simulation Models

There is a growing need for the ability to model and generate samples of dependent random variables as primitive inputs to stochastic models. We consider the case where this dependence is modeled in terms of a partially-specified finite-dimensional random vector. A random vector sampler is commonly required to match a given set of distributions for each of its components (the marginal distributions) and values of their pairwise covariances. The NORTA method, which produces samples via a transformation of a joint-normal random vector sample, is considered the state-of-the-art method for matching this specification. We begin by showing that the NORTA method has certain flaws in its design which limit its applicability. A covariance matrix is said to be feasible for a given set of marginal distributions if a random vector exists with these properties. We develop a computational tool that can establish the feasibility of (almost) any covariance matrix for a fixed set of marginals. This tool is used to rigorously establish that there are feasible combinations of marginals and covariance matrices that the NORTA method cannot match. We further determine that as the dimension of the random vector increases, this problem rapidly becomes acute, in the sense that NORTA becomes increasingly likely to fail to match feasible specifications. As part of this analysis, we propose a random matrix sampling technique that is possibly of wider interest. We extend our study along two natural paths. First, we investigate whether NORTA can be modified to approximately match a desired covariance matrix that the original NORTA procedure fails to match. Results show that simple, elegant modifications to the NORTA procedure can help it achieve close approximations to the desired covariance matrix, and these modifications perform well with increasing dimension. Second, the feasibility testing procedure suggests a random vector sampling technique that can exactly match (almost) any given feasible set of marginals and covariances, i.e., be free of the limitations of NORTA. We develop a strong characterization of the computational effort needed by this new sampling technique. This technique is computationally competitive with NORTA in low to moderate dimensions, while matching the desired covariances exactly

Dependence in probabilistic modeling, Dempster-Shafer theory, and probability bounds analysis.

This report summarizes methods to incorporate information (or lack of information) about inter-variable dependence into risk assessments that use Dempster-Shafer theory or probability bounds analysis to address epistemic and aleatory uncertainty. The report reviews techniques for simulating correlated variates for a given correlation measure and dependence model, computation of bounds on distribution functions under a specified dependence model, formulation of parametric and empirical dependence models, and bounding approaches that can be used when information about the intervariable dependence is incomplete. The report also reviews several of the most pervasive and dangerous myths among risk analysts about dependence in probabilistic models

Numerical Solution of Systems with Stochastic Uncertainties: A General Purpose Framework for Stochastic Finite Elements

This work develops numerical techniques for the simulation of systems with stochastic parameters, modelled by stochastic partial differential equations (SPDEs). After treating the theory of linear and nonlinear elliptic SPDEs, discretisation techniques are presented. The spatial discretisation is performed by existing simulation software and the stochastic discretisation is carried out by directly integrating statistics or by expansions in tensor products of finite element shape functions times stochastic functions. Monte Carlo and Smolyak integration techniques are employed for the direct integration of statistics, whereas the discretisation by series expansions is realised either by orthogonal projections or by Galerkin methods, which yield large systems of coupled block equations. For the solution of linear SPDEs, efficient representations of the linear block equations are developed and used in iterative solvers. Due to the size of the equations, a parallel solver is supplied. The solution of nonlinear SPDEs is performed by approximate and by quasi-Newton methods. An adaptive refinement of the stochastic ansatz-spaces is implemented based on the solution of dual problems. The numerical techniques described in this thesis are implemented in a general purpose software for stochastic finite elements that allows to introduce stochastic uncertainties into existing simulation codes and that permits to propagate the input uncertainties to the system response.Inhalt der Arbeit ist die numerische Simulation von Systemen mit stochastischen Parametern, die durch stochastische partielle Differentialgleichungen (SPDGLn) beschrieben werden. Es werden die Theorie linearer und nichtlinearer elliptischer SPDGLn sowie Diskretisierungsverfahren beschrieben. Für die räumliche Diskretisierung wird eine existierende Simulationssoftware verwendet, während die stochastische Diskretisierung durch die direkte numerische Integration von Statistiken unter Verwendung von Monte Carlo- und Smolyak-Quadraturverfahren oder durch Reihenentwicklungen in Tensorprodukten finiter Elemente und stochastischer Ansatzfunktionen erfolgt. Die Reihenentwicklung wird dabei durch orthogonale Projektionen oder durch Galerkinverfahren gewonnen. Bei der Anwendung stochastischer Galerkinvervahren entstehen große Systeme gekoppelter Blockgleichungssysteme, welche hier durch iterative Verfahren gelöst werden. Zur Lösung linearer SPDGln werden effiziente Darstellungen der Gleichungssysteme und iterative Löser entwickelt. Aufgrund der Größe der entstehenden Gleichungssysteme wird ein paralleler Löser bereitgestellt. Die Lösung nichtlinearer SPDGLn geschieht durch approximative und Quasi-Newtonverfahren. Ein duales Verfahren ermöglicht die adaptive Verfeinerung der Lösung. Diese Verfahren werden in einer Allzwecksoftware für stochastische finite Elemente implementiert, die es erlaubt, existierende Simulationscodes um stochastische Unsicherheiten zu erweitern