We introduce the Skipping Sampler, a novel algorithm to efficiently sample from the restriction of an arbitrary probability density to an arbitrary measurable set. Such conditional densities can arise in the study of risk and reliability and are often of complex nature, for example having multiple isolated modes and non-convex or disconnected support. The sampler can be seen as an instance of the Metropolis-Hastings algorithm with a particular proposal structure, and we establish sufficient conditions under which the Strong Law of Large Numbers and the Central Limit Theorem hold. We give theoretical and numerical evidence of improved performance relative to the Random Walk Metropolis algorithm.
In many stochastic systems, however, the relationship between exogenous inputs and system behaviour is implicit and non-linear. Examples include markets or distributed systems, where complexity arises from the actions of multiple players, or systems involving complex optimisation procedures such as security constrained optimal power flow @cite_21 . In the present work (and unlike @cite_36 ) we therefore do not assume that the set @math is known but, rather, that any sampling procedure may only test pointwise whether @math . As illustrated in Figure , the areas of zero density induced by conditioning on @math can create multiple isolated modes. We therefore now briefly recall the literature on sampling from rare events and from multimodal targets.
@cite_36 @cite_21
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The frequency stability of power systems is increasingly challenged by various types of disturbance. In particular, the increasing penetration of renewable energy sources is increasing the variability of power generation while reducing system inertia against disturbances. In this paper we explore how this could give rise to rate of change of frequency (RoCoF) violations. Correlated and non -Gaussian power disturbances, such as may arise from renewable generation, have been shown to be significant in power system security analysis. We therefore introduce ghost sampling which, given any unconditional distribution of disturbances, efficiently produces samples conditional on a violation occurring. Our goal is to address questions such as “which generator is most likely to be disconnected due to a RoCoF violation?” or “what is the probability of having simultaneous RoCoF violations, given that a violation occurs?” Preface. Chapter 1: Market Overview in Electric Power Systems. Chapter 2: Short-Term Load Forecasting. Chapter 3: Electricity Price Forecasting. Chapter 4: Price-Based Unit Commitment. Chapter 5: Arbitrage in Electricity Markets. Chapter 6: Market Power Analysis Based on Game Theory. Chapter 7: Generation Asset Valuation and Risk Analysis. Chapter 8: Security-Constrained Unit Commitment. Chapter 9: Ancillary Services Auction Market Design. Chapter 10: Transmission Congestion Management and Pricing. Appendix A: List of Symbols. Appendix B: Mathematical Derivation. Appendix C: RTS Load Data. Appendix D: Example Systems Data. Appendix E: Game Theory Concepts. Appendix F: Congestion Charges Calculation. References. Index.