This thesis presents a linearized technique to determine a risk-based index for dynamic security. The method is an extension to an existing technique in which the risk of steady state security is calculated using the mean and variance of load uncertainty. The proposed method is applied to calculate the risk indices for the New England 39 bus test system. The results obtained from the proposed method are validated against those estimated by Monte Carlo simulation. Both approaches produce virtually the same results for small load deviations.

This thesis presents a linearized technique to determine a risk-based index for dynamic security. The method is an extension to an existing technique in which the risk of steady state security is calculated using the mean and variance of load uncertainty. The proposed method is applied to calculate the risk indices for the New England 39 bus test system. The results obtained from the proposed method are validated against those estimated by Monte Carlo simulation. Both approaches produce virtually the same results for small load deviations.

This paper is a compilation of results in combinatorics utilizing the probabilistic method. Below is a brief description of the results highlighted in each chapter. Chapter 1 provides basic definitions, lemmas, and theorems from graph theory, asymptotic analysis, and probability which will be used throughout the paper. Chapter 2 introduces the independent domination number. It is then shown that in the random graph model G(n,p) with probability tending to one, the independent domination number is one of two values. Also, the the number of independent dominating sets of given cardinality is analyzed statistically. Chapter 3 introduces the tree domination number. It is then shown that in the random graph model G(n,p) with probability tending to one, the tree domination number is one of two values. Additional related domination parameters are also discussed. Chapter 4 introduces a generalized rook polynomial first studied by J. Goldman et al. Central and local limit theorems are then proven for certain classes of the generalized rook polynomial. Special cases include known central and local limit theorems for the Stirling numbers of the first and second kind and additionally new limit theorems for the Lah numbers and certain classes of known generalized Stirling numbers. Chapter 5 introduces the Kneser Graph. The exact expected value and variance of the distance between [n] and a vertex chosen uniformly at random is given. An asymptotic formula for the expectation is found.

With the increasing amount of renewable and difficult-to-forecast generation units, Transmission System Operators (TSO) face new challenges to operate the grid properly. The deterministic N-1 criterion is currently used to assess grid reliability. This criterion states that the loss of an active component should not trigger the violation of operational constraints. It has been chosen in the conventional context of electricity in which large units produced power transmitted to the consumer through the transmission and the distribution systems. The renewable energy sources, the distributed generations, and the liberalization of the electricity market led to a revolution in power systems. Renewable energies present intrinsic variability and limited predictability. Those variables are thus subject to forecasting errors. Distributed generations changed the structure of the power system to include smaller productions dispersed in the grid. The competitive electricity market led the consumers to react to electricity prices. The load is thus now subject to higher forecasting errors. An increasing share of the power system’s variables is thus now subject to errors that are likely to affect the operations and the planning of the grid. The N-1 criterion reaches limitations in considering the new characteristics of the power system. To account for this evolution, TSOs have to make a shift in paradigm. They must go from the N-1 criterion to a reliability-based approach, with risk management and integration of errors on forecasted values. This amounts to going from deterministic to probabilistic approaches capable of quantifying this risk.The purpose of this research project is to develop the basis of an industrial tool. The method considers thus the barriers to the use of a probabilistic method for grid planning. TSOs are indeed reluctant to give up the N-1 criterion. They fear that a probabilistic method would be too difficult to understand and apply and that the related computational time would be too long for the operational planning.The method proposed in this research project aims at overcoming those barriers. The method sets the basis of a decision support tool for the planners to make sound decisions. It is thus not a black-box and the planners are included in the assessment. The method is based on the current work of the planners and widens it to englobe probabilistic considerations. It offers thus a smooth evolution from a deterministic to a probabilistic method which will ease the industrial development of the tool using it. The method is called Discrete Forecast ERrors Scenarios method: DIFERS. It has been developed to be consistent with the operational planning in terms of time constraints and available information. It relies on three evolutions from the deterministic N-1 criterion: 1. Include possible variations from the best estimate of the forecasts.2. Enlarge the contingency list to higher N-k events.3. Consider the impact and the probability of the events to compute their risk.The contingency list evolves thus toward a risk-based classification of events. The planners’ work aims then at proposing actions to decrease the risk to an acceptable level. The first step of DIFERS is performed off-line to relax the time constraints. It aims at evaluating the contingency list for a set of situations. It performs an assessment on the most probable events, considering a larger group than the N-1 criterion. The assessment focuses on N-1 and nearby N-2 events. The nearby events have been selected based on a distance criterion defined in terms of number of components in the smallest path from one component to another. Some random N-3 and N-4 events are also analyzed to assess the evolution of the risk with regard to the number of failing components. Continuous variables are represented by their probability density functions (PDFs), which represent the variation range for the set of situations considered. Those PDFs are discretized to limit the computational time. The assessment of the contingencies is performed on each combination of those discrete points. The second step uses the contingency list developed in step 1 to assess the risk related to a specific situation: a grid plan. The PDFs used in this step represent the forecasting errors on the continuous variables for the grid plan considered. They are also discretized and each combination is tested with the events of the contingency list. At the end of the assessment, indicators are computed and provided to the planner. The planner can then propose actions to be tested by the tool to see their impact on the reliability indicators. The assessment stops once the reliability target is met.The final step aims at updating the information computed in step 2 with newly acquired forecasts. As real-time is closer, those forecasts are more reliable. The method has been tested on a plausible scenario and on a simplified version of the Belgian grid. The load and the offshore wind production have been considered as input variables for this implementations. The results show that there is an interest in evolving toward a risk-based assessment to capture the new characteristics of the evolving context of electricity supply. The implementation of the DIFERS method should continue on several scenarios. It should integrate all continuous variables such as solar and onshore productions. Moreover, all real-life considerations on the input variables, such as correlations, should be included to represent the power system as best as possible.This research project has been conducted in collaboration with the Belgian TSO Elia and it has been financed by the Doctiris grant of Innoviris. / Doctorat en Sciences de l'ingénieur et technologie / info:eu-repo/semantics/nonPublished

Consider the random set system (Formula presented.), where (Formula presented.) and Ajselected with probabilityp=pn}. A set H⊆[n] is said to be a hitting set for (Formula presented.). The second moment method is used to exhibit the sharp concentration of the minimal size of H for a variety of values of p.

Probabilistic models and fuzzy set models describe different aspects of uncertainty. Probabilistic models primarily describe random variability in parameters. In engineering system safety, examples are variability in material properties, geometrical dimensions, or wind loads. In contrast, fuzzy set models of uncertainty primarily describe vagueness, such as vagueness in the definition of safety. When there is only limited information about variability, it is possible to use probabilistic models by making suitable assumptions on the statistics of the variability. However, it has been repeatedly shown that this can entail serious errors. Fuzzy set models, which require little data, appear to be well suited to use with designing for uncertainty, when little is known about the uncertainty. Several studies have compared fuzzy set and probabilistic methods in analysis of safety of systems under uncertainty. However, no study has compared the two approaches systematically as a function of the amount of available information. Such a comparison, in the context of design against failure, is the objective of this dissertation. First, the theoretical foundations of probability and possibility theories are compared. We show that a major difference between probability and possibility is in the axioms about the union of events. Because of this difference, probability and possibility calculi are fundamentally different and one cannot simulate possibility calculus using probabilistic models. We also show that possibility-based methods tend to be more conservative than probability-based methods in systems that fail only if many unfavorable events occur simultaneously. Based on these theoretical observations, two design problems are formulated to demonstrate the strength and weakness of probabilistic and fuzzy set methods. We consider the design of tuned damper system and the design and construction of domino stacks. These problems contain narrow failure zones in their uncertain variables and are tailored to demonstrate the pitfalls of probabilistic methods when little information is available for uncertain variables. Using these design problems we demonstrate that probabilistic methods are better than possibility-based methods if sufficient information is available. Just as importantly, we show possibility-based methods can be better if little information is available. Our conclusion is that when there is little information available about uncertainties, a hybrid method should be used to ensure a safe design. / Ph. D.

In today’s Nordic construction industry, it is difficult for new and innovative building solutions to be introduced due to prescriptive and inflexible regulations. Trading products and services cross-border is something that could loosen the tough market, but this is not possible due to the lack of common international frameworks that is performance based with the possibility to perform fire safety engineering. This is something that the Nordic Innovation project group called Fire Safety Engineering for Innovative and Sustainable Building Solutions wants to change. By introducing a new probabilistic method to verify fire safety in buildings, with the intention to become a Nordic standard, so will hopefully parts of these problems be resolved. The fourth work package of the project includes field testing of the new method which this thesis is a part of. The idea is to asses and improve the new probabilistic approach by comparing it to an existing non-probabilistic method and introduce ameliorating recommendations. Comparison of the probabilistic method is performed against a Swedish verification process that’s based on the General recommendations on analytical design of fire safety strategy (BBRAD) by verifying fire safety in a car park, that is located below an office building, with both verification methods. The two performance-based analyses treat deviations from a prescriptive solution, performed with the Boverket’s Building Regulations (BBR), and the results of these verifications is compared. The requirements that is verified are; escape in event of fire, protection against the outbreak of fire, protection against the development and spread of fire and smoke in buildings, protection against spread of fire between buildings, possibility of rescue responses and ensuring fire resistance in the structural members. Fire safety designs and approaches for treatment of the deviations are compared and analysed which concludes in the improvement recommendation that’s been presented.  Questions that has been answered during the work process is: How do the methods treat the possibility of a fire safety design without sprinkler? What is the main difference between the two verification methods? Which improvements could be done to the new Probabilistic method?  The recommendations of improvement that has been presented is based on the work process of the probabilistic approach and the comparison with the Swedish verification process. Development of the following areas is advocated: Treatment of critical levels for evacuation scenarios  Form a common Nordic statistical database Improved guidance of how to complete the validation analysis The thesis does not include all parts that’s required in a fire safety design but will merely focus on the deviations of the pre-accepted solution. The verification is only performed on the car park, i.e. the office part of the building is not included. / I dagens nordiska byggbransch är det svårt för nya och innovativa byggnadslösningar att införas på grund av de preskriptiva och fyrkantiga regelverk som finns. Handel av produkter och tjänster över gränserna är något som kan luckra upp den tuffa marknaden, men det är svårt på grund av bristen utav gemensamma internationella regelverk som är funktionsbaserade med möjlighet till fire safety engeinnering. Det är något som ett nordiskt innovationsprojekt kallat Fire Safety Engineering for Innovative and Sustainable Building Solutions vill förändra. Genom att införa en ny probabilistisk metod för att verifiera brandsäkerheten i byggnader, med avsikten att skapa en nordisk standard, kan förhoppningsvis delar av dessa problem lösas. Det fjärde arbetspaketet inom projektet består av att testa den nya metoden, vilket denna avhandling är en del av. Tanken är att bedöma och ta fram förbättringsförslag till den nya probabilistiska metoden genom att jämföra den med en befintlig scenariobaserad metod och presentera förbättringsrekommendationer. Jämförelse av probabilistiska metoden utförs mot en svensk verifieringsprocess som baseras på Boverkets allmänna råd om analytisk dimensionering av byggnaders brandskydd (BBRAD) genom att verifiera brandsäkerheten i ett parkeringsgarage, som ligger under en kontorsbyggnad, med båda verifieringsmetoderna. De två funktionsbaserade analyserna behandlar avvikelser från en förenklad dimensionering, som är utförd enligt Boverkets Byggregler (BBR), och resultaten av dessa verifikationer jämförs. De krav som verifieras är; utrymning i händelse av brand, skydd mot uppkomst av brand, skydd mot utveckling och spridning av brand och rök i byggnader, skydd mot brandspridning mellan byggnader, möjlighet till räddningsinsats och att säkerställa bärförmåga vid brand. Brandskyddets utformning och metodernas behandling av avvikelserna jämförs och analyseras vilket konkluderar i de rekommendationer för förbättring som presenteras. Frågor som har besvarats under arbetsprocessen är: Hur behandlar metoderna möjligheten att dimensionera brandsäkerheten utan sprinklersystem? Vad är den stora skillnaden mellan de två verifieringsmetoderna? Vilka förbättringar kan göras på den nya probabilistiska metoden? Rekommendationerna till förbättring som har tagits fram är baserad på arbetsprocessen i den probabilistiska metoden och jämförelsen med den svenska verifieringsprocessen. Utveckling av följande områden förespråkas: Behandling av kritiska nivåer i utrymningsscenarion Uppställning av en gemensam statistiskdatabas för de nordiska länderna Förbättrad förklaring om hur man utför valideringarna av analysen Avhandlingen omfattar inte alla delar som behövs vid bandskyddsprojektering utan fokusera endast på avvikelserna från den förenklade dimensioneringen. Verifikationen är endast utförd på parkeringsgaraget, det vill säga kontorsdelen av byggnaden behandlas inte. / Fire Safety Engineering for Innovative and Sustainable Building Solutions

By using existing radio network infrastructure, a user can be positioned even where GPS and other positioning technologies lack coverage. The LTE Positioning Protocol (LPP) supports user Reference Signal Time Difference (RSTD) reports based on the Time of Arrival (TOA) for a Positioning Reference Signal (PRS). In the current reporting format, only one RSTD for each base station is considered, but for indoor environments this is easily biased due to fading and multipath issues, resulting in a Non-Line of Sight (NLOS) bias. With a rich User Equipment (UE) feedback that can represent the multipath channel for each Base Station (BS), the positioning accuracy can be increased. This thesis develops and evaluates a UE reporting format representing multiple TDOA candidates, and a probabilistic positioning algorithm, in terms of positioning accuracy and amount of data reported. By modeling time measurements as Gaussian Mixture (GM), the time information can be compressed with arbitrary resolution and used in a Maximum-Likelihood (ML) estimation to find the position. Results were obtained through simulation in a radio network simulator and post-processing of simulation data in Matlab. The results suggest that several TOA candidates improve the positioning accuracy, but that the largest improvement comes from a noise based threshold by increasing LOS detectability reducing the NLOS bias, while suppressing noise. The results also suggest that the accuracy for the method can be further improved by combining multiple time measurement occasions.

O presente trabalho é um esforço de apresentar, organizado em forma de survey, um conjunto de resultados que ilustram a aplicação de um certo método probabilístico. Embora não apresentemos resultados novos na área, acreditamos que a apresentação sistemática destes resultados pode servir para a compreensão de uma ferramenta útil para quem usa dos métodos probabilísticos na sua pesquisa em combinatória. Os resultados de que falaremos tem aparecido na última década na literatura especializada e foram usados na investigação de problemas que resitiram a outras aproximações mais clássicas. Em vez de teorizar sobre o método a apresentar, nós adotaremos a estratégia de apresentar três problemas, usando-os como exemplos práticos da aplicação do método em questão. Surpeendentemente, apesar da dificuldade que apresentaram para ser resolvidos, estes problemas compartilham a caraterística de poder ser formulados muito intuitivamente, como veremos no Capítulo 1. Devemos advertir que embora os problemas que conduzem nossa exposição pertençam a áreas tão diferentes quanto teoria de números, geometria e combinatória, nosso intuito é fazer énfase no que de comum tem as suas soluções e não das posteriores implicações que estes problemas tenham nas suas respectivas áreas. Ocasionalmente comentaremos sim, outras possíveis aplicações das ferramentas usadas para solucionar estes problemas de motivação. Os problemas de que trataremos tem-se caracterizado por aguardar várias décadas em espera de solução: O primeiro, da teoria de números, surgiu na pesquisa de séries de Fourier que Sidon realizava a princípios de século e foi proposto por ele a Erdös em 1932. Embora tenham havido, desde 1950, diversos avanços na pesquisa deste problema, o resultado de que falaremos data de 1981. Já o segundo problema, da geometria, é uma conjectura formulada em 1951 por Heilbronn e refutada finalmente em 1982. O último problema, de combinatória, é uma conjectura de Erdös e Hanani de 1963, que foi tratada em diversos casos particulares até ser finalmente resolvida em toda sua generalidade em 1985. / The following work is an effort to present, in survey form, a collection of results that illustrate the application of a certain probabilistic method in combinatorics. We do not present new results in the area; however, we do believe that the systematic presentation of these results can help those who use probabilistic methods comprenhend this useful technique. The results we refer to have appeared over the last decade in the research literature and were used in the investigation of problems which have resisted other, more classical, approaches. Instead of theorizing about the method, we adopted the strategy of presenting three problems, using them as practical examples of the application of the method in question. Surpisingly, despite the difficulty of solutions to these problems, they share the characteristic of being able to be formulated very intuitively, as we will see in Chapter One. We should warn the reader that despite the fact that the problems which drive our discussion belong to such different fields as number theory, geometry and combinatorics, our goal is to place emphasis on what their solutions have in common and not on the subsequent implications that these problems have in their respective fields. Occasionally, we will comment on other potential applications of the tools utilized to solve these problems. The problems which we are discussing can be characterized by the decades-long wait for their solution: the first, from number theory, arose from the research in Fourier series conducted by Sidon at the beginning of the century and was proposed by him to Erdös in 1932. Since 1950, there have been diverse advances in the understanding of this problem, but the result we talk of comes from 1981. The second problem, from geometry, is a conjecture formulated in 1951 by Heilbronn and finally refuted in 1982. The last problem, from combinatorics, is a conjecture formulated by Erdös and Hanani in 1963 that was treated in several particular cases but was only solved in its entirety in 1985.

O presente trabalho é um esforço de apresentar, organizado em forma de survey, um conjunto de resultados que ilustram a aplicação de um certo método probabilístico. Embora não apresentemos resultados novos na área, acreditamos que a apresentação sistemática destes resultados pode servir para a compreensão de uma ferramenta útil para quem usa dos métodos probabilísticos na sua pesquisa em combinatória. Os resultados de que falaremos tem aparecido na última década na literatura especializada e foram usados na investigação de problemas que resitiram a outras aproximações mais clássicas. Em vez de teorizar sobre o método a apresentar, nós adotaremos a estratégia de apresentar três problemas, usando-os como exemplos práticos da aplicação do método em questão. Surpeendentemente, apesar da dificuldade que apresentaram para ser resolvidos, estes problemas compartilham a caraterística de poder ser formulados muito intuitivamente, como veremos no Capítulo 1. Devemos advertir que embora os problemas que conduzem nossa exposição pertençam a áreas tão diferentes quanto teoria de números, geometria e combinatória, nosso intuito é fazer énfase no que de comum tem as suas soluções e não das posteriores implicações que estes problemas tenham nas suas respectivas áreas. Ocasionalmente comentaremos sim, outras possíveis aplicações das ferramentas usadas para solucionar estes problemas de motivação. Os problemas de que trataremos tem-se caracterizado por aguardar várias décadas em espera de solução: O primeiro, da teoria de números, surgiu na pesquisa de séries de Fourier que Sidon realizava a princípios de século e foi proposto por ele a Erdös em 1932. Embora tenham havido, desde 1950, diversos avanços na pesquisa deste problema, o resultado de que falaremos data de 1981. Já o segundo problema, da geometria, é uma conjectura formulada em 1951 por Heilbronn e refutada finalmente em 1982. O último problema, de combinatória, é uma conjectura de Erdös e Hanani de 1963, que foi tratada em diversos casos particulares até ser finalmente resolvida em toda sua generalidade em 1985. / The following work is an effort to present, in survey form, a collection of results that illustrate the application of a certain probabilistic method in combinatorics. We do not present new results in the area; however, we do believe that the systematic presentation of these results can help those who use probabilistic methods comprenhend this useful technique. The results we refer to have appeared over the last decade in the research literature and were used in the investigation of problems which have resisted other, more classical, approaches. Instead of theorizing about the method, we adopted the strategy of presenting three problems, using them as practical examples of the application of the method in question. Surpisingly, despite the difficulty of solutions to these problems, they share the characteristic of being able to be formulated very intuitively, as we will see in Chapter One. We should warn the reader that despite the fact that the problems which drive our discussion belong to such different fields as number theory, geometry and combinatorics, our goal is to place emphasis on what their solutions have in common and not on the subsequent implications that these problems have in their respective fields. Occasionally, we will comment on other potential applications of the tools utilized to solve these problems. The problems which we are discussing can be characterized by the decades-long wait for their solution: the first, from number theory, arose from the research in Fourier series conducted by Sidon at the beginning of the century and was proposed by him to Erdös in 1932. Since 1950, there have been diverse advances in the understanding of this problem, but the result we talk of comes from 1981. The second problem, from geometry, is a conjecture formulated in 1951 by Heilbronn and finally refuted in 1982. The last problem, from combinatorics, is a conjecture formulated by Erdös and Hanani in 1963 that was treated in several particular cases but was only solved in its entirety in 1985.