Transitional Markov-Chain Monte-Carlo

The Bayesian updating expressed in equation (2.30) needs a normalizing factor P pD|Iq, that can be very complex to obtain or not treatable. An effective stochastic simulation algorithm, called Transitional Markov Chain Monte Carlo (TMCMC) [28], has been

used in this analysis. This algorithm allows the generation of samples from the complex shaped unknown posterior distribution through an iterative approach. In this algorithm, m intermediate distributions Pi are introduced:

Pi9P pD|θ, IqβiP pθ|Iq (2.31)

where the contribution of the likelihood is scaled down by an exponent βi, with

0 “ β0 ă .. ă βi ă .. ă βm “ 1, thus the first distribution is the prior PDF, and

the last is the posterior distribution. The value of these exponents βi is automatically

selected to ensure that the dispersion of the samples at each step meet a prescribed target. For additional information, the reader is reminded to [28]. These intermediate distributions show a more gradual change in the shape from one step to the next when compared with the shape variation from the prior to the posterior.

In the first step, samples are generated from the prior PDF using direct Monte-Carlo. Then, samples from the Pi`1 distribution are generated using Markov chains with the

Metropolis-Hasting algorithm [55], starting from selected samples taken from the Pidis-

tribution, and βi is updated. This step is repeated until the distribution characterized

by βi“ 1is reached. By using the Metropolis-Hasting algorithm, samples are generated

from the posterior PDF without the necessity of ever computing the normalization con- stant. By employing intermediate distributions, it is easier for the updating procedure to generate samples also from posterior showing very complex distribution, e.g., very peaked or multi-modal. The Bayesian model updating procedure solved using TMCMC requires many model evaluations and the overall computational time for the detection can result very high. To reduce computational time, a surrogate model approach can be adopted.

Chapter 3

Power Grid Reliability,

3.1

Abstract

In this chapter we introduce some of the basic concepts for the analysis of complex sys- tems and critical infrastructures. In particular, resilience-related concepts such as risk, safety, reliability and vulnerability are reviewed and discusses. Quantitative metric are reviewed and load-flow solvers used to analyse power grid systems are introduced. It is worth pointing out that, although definitions, metrics and analysis tools introduced here come from the power grid literature, this shouldn’t be regarded as a lost of generality. In fact, many of the resilience-related ideas and load-flow analysis methods naturally translates to wider class of complex systems and critical infrastructures which power grids well-represent.

Table 3.1 presents two definitions of complex systems and critical infrastructures. It is fair to argue that power grid is an extremely complex systems and it has a crucial role in the society, i.e. is a critical infrastructure. Power grid includes many highly inter- connected components which are operated to provide a service/utility to the customers and society as a whole. Maximising the system productivity, minimizing the costs while assuring a safe and reliable delivery of electric power is of uttermost import. This requires robust uncertainty quantification and decision-making frameworks, capable of outperforming existing experience-based methods while accounting for uncertainty affecting system operational conditions, components health states and interactions with the external environment. Power grids are generally complex, heterogeneous, with

Complex System

A complex system can be defined as an ensemble of components, functionally and physically interconnected as in a network struc- ture. Often heterogeneous and showing complex dynamic and complex structural behaviours [167].

Critical infrastruc- ture

An infrastructure is called critical if its incapacity or destruction has a significant impact on health, safety, security, economics and social well-being (Council Directive 2008/114/EC) [167]. Another definition is ’infrastructure whose unavailability or destruction would have a extensive impact on economy, Infrastructure Gov- ernment services and, in general, on everyday life, with severe consequences for a nation [34].

Table 3.1: Definition of Complex Systems and Critical Infrastructure

many highly interconnected components connected in a network-like fashion. Also, even a partial disruption of the grid would cause significant damage to the society and economy of the affected state. The power grid is probably the most representative example of a complex system and critical infrastructure and, thus has been selected in this work as the reference system for the testing and verification of novel methods for generalised quantification of uncertainty. In this chapter, an in-depth review of probabilistic methods for power grid safety analysis and a concise introduction to safety-related terminology is presented. In this chapter we introduce some of the basic concepts for the analysis of complex systems and critical infrastructures safety and reliability and with particular regard to power grid systems. Two literature definitions of complex systems and critical infrastructures are presented within Table 3.1.