arXiv subjects

Bert Zwart

Publications and source records attributed to Bert Zwart.

At least 55 records · Page 3Linked to original sources

arXiv2020

Analyzing large frequency disruptions in power systems using large deviations theory

We propose a method for determining the most likely cause, in terms of conventional generator outages and renewable fluctuations, of power system frequency reaching a predetermined level that is deemed unacceptable to the system operator. Our parsimonious model of system frequency incorporates primary and secondary control mechanisms, and supposes that conventional outages occur according to a Poisson process and renewable fluctuations follow a diffusion process. We utilize a large deviations theory based approach that outputs the most likely cause of a large excursion of frequency from its desired level. These results yield the insight that current levels of renewable power generation do not significantly increase system vulnerability in terms of frequency deviations relative to conventional failures. However, for a large range of model parameters it is possible that such vulnerabilities may arise as renewable penetration increases.

eess.SY↗

arXiv2020

Ranking transmission lines by overload probability using the empirical rate function

We develop a non-parametric procedure for ranking transmission lines in a power system according to the probability that they will overload due to stochastic renewable generation or demand-side load fluctuations, and compare this procedure to several benchmark approaches. Using the IEEE 39-bus test network we provide evidence that our approach, which statistically estimates the rate function for each line, is highly promising relative to alternative methods which count overload events or use incorrect parametric assumptions.

eess.SY↗

arXiv2020

A Fluid Model of an Electric Vehicle Charging Network

We develop and analyze a measure-valued fluid model keeping track of parking and charging requirements of electric vehicles in a local distribution grid. We show how this model arises as an accumulation point of an appropriately scaled sequence of stochastic network models. The invariant point of the fluid model encodes the electrical characteristics of the network and the stochastic behavior of its users, and it is characterized, when it exists, by the solution of a so-called Alternating Current Optimal Power Flow (ACOPF) problem.

math.PR↗

arXiv2020

Computing first passage times for Markov-modulated fluid models using numerical PDE problem solvers

A popular method to compute first-passage probabilities in continuous-time Markov chains is by numerically inverting their Laplace transforms. Past decades, the scientific computing community has developed excellent numerical methods for solving problems governed by partial differential equations (PDEs), making the availability of a Laplace transform not necessary here for computational purposes. In this study we demonstrate that numerical PDE problem solvers are suitable for computing first passage times, and can be very efficient for this purpose. By doing extensive computational experiments, we show that modern PDE problem solvers can outperform numerical Laplace transform inversion, even if a transform is available. When the Laplace transform is explicit (e.g. does not require the computation of an eigensystem), numerical transform inversion remains the primary method of choice.

math.NA↗

arXiv2020

Large Fluctuations in Locational Marginal Prices

This paper investigates large fluctuations of Locational Marginal Prices (LMPs) in wholesale energy markets caused by volatile renewable generation profiles. Specifically, we study events of the form $\mathbb{P} \Big ( \mathbf{LMP} \notin \prod_{i=1}^n [α_i^-, α_i^+] \Big),$ where $\mathbf{LMP}$ is the vector of LMPs at the $n$ power grid nodes, and $\boldsymbolα^-,\boldsymbolα^+\in\mathbb{R}^n$ are vectors of price thresholds specifying undesirable price occurrences. By exploiting the structure of the supply-demand matching mechanism in power grids, we look at LMPs as deterministic piecewise affine, possibly discontinuous functions of the stochastic input process, modeling uncontrollable renewable generation. We utilize techniques from large deviations theory to identify the most likely ways for extreme price spikes to happen, and to rank the nodes of the power grid in terms of their likelihood of experiencing a price spike. Our results are derived in the case of Gaussian fluctuations and are validated numerically on the IEEE 14-bus test case.

math.OC↗

arXiv2020

Optimization of stochastic lossy transport networks and applications to power grids

Motivated by developments in renewable energy and smart grids, we formulate a stylized mathematical model of a transport network with stochastic load fluctuations. Using an affine control rule, we explore the trade-off between the number of controllable resources in a lossy transport network and the performance gain they yield in terms of expected power losses. Our results are explicit and reveal the interaction between the level of flexibility, the intrinsic load uncertainty and the network structure.

math.OC↗

arXiv2020

Consistency of the PLFit estimator for power-law data

We prove the consistency of the Power-Law Fit PLFit method proposed by Clauset et al.(2009) to estimate the power-law exponent in data coming from a distribution function with regularly-varying tail. In the complex systems community, PLFit has emerged as the method of choice to estimate the power-law exponent. Yet, its mathematical properties are still poorly understood. The difficulty in PLFit is that it is a minimum-distance estimator. It first chooses a threshold that minimizes the Kolmogorov-Smirnov distance between the data points larger than the threshold and the Pareto tail, and then applies the Hill estimator to this restricted data. Since the number of order statistics used is random, the general theory of consistency of power-law exponents from extreme value theory does not apply. Our proof consists in first showing that the Hill estimator is consistent for general intermediate sequences for the number of order statistics used, even when that number is random. Here, we call a sequence intermediate when it grows to infinity, while remaining much smaller than the sample size. The second, and most involved, step is to prove that the optimizer in PLFit is with high probability an intermediate sequence, unless the distribution has a Pareto tail above a certain value. For the latter special case, we give a separate proof.

math.ST↗

arXiv2019

Sample-path large deviations for Lévy processes and random walks with Weibull increments

We study sample-path large deviations for Lévy processes and random walks with heavy-tailed jump-size distributions that are of Weibull type. Our main results include an extended form of an LDP (large deviations principle) in the $J_1$ topology, and a full LDP in the $M_1'$ topology. The rate function can be represented as the solution to a quasi-variational problem. The sharpness and applicability of these results are illustrated by a counterexample proving the nonexistence of a full LDP in the $J_1$ topology, and by an application to a first passage problem.

math.PR↗

arXiv2019

Economies-of-scale in resource sharing systems: tutorial and partial review of the QED heavy-traffic regime

Multi-server queueing systems describe situations in which users require service from multiple parallel servers. Examples include check-in lines at airports, waiting rooms in hospitals, queues in contact centers, data buffers in wireless networks, and delayed service in cloud data centers. These are all situations with jobs (clients, patients, tasks) and servers (agents, beds, processors) that have large capacity levels, ranging from the order of tens (checkouts) to thousands (processors). This survey investigates how to design such systems to exploit resource pooling and economies-of-scale. In particular, we review the mathematics behind the Quality-and-Efficiency Driven (QED) regime, which lets the system operate close to full utilization, while the number of servers grows simultaneously large and delays remain manageable. Aimed at a broad audience, we describe in detail the mathematical concepts for the basic Markovian many-server system, and only provide sketches or references for more advanced settings related to e.g. load balancing, overdispersion, parameter uncertainty, general service requirements and queueing networks. While serving as a partial survey of a massive body of work, the tutorial is not aimed to be exhaustive.

math.PR↗

arXiv2019

Temperature Overloads in Power Grids Under Uncertainty: a Large Deviations Approach

The advent of renewable energy has huge implications for the design and control of power grids. Due to increasing supply-side uncertainty, traditional reliability constraints such as strict bounds on current, voltage and temperature in a transmission line have to be replaced by computationally demanding chance constraints. In this paper we use large deviations techniques to study the probability of current and temperature overloads in power grids with stochastic power injections, and develop corresponding safe capacity regions. In particular, we characterize the set of admissible power injections such that the probability of overloading of any line over a given time interval stays below a fixed target. We show how enforcing (stochastic) constraints on temperature, rather than on current, results in a less conservative approach and can thus lead to capacity gains.

math.OC↗

arXiv2019

Finite-time ruin probabilities under large-claim reinsurance treaties for heavy-tailed claim sizes

We investigate the probability that an insurance portfolio gets ruined within a finite time period under the assumption that the r largest claims are (partly) reinsured. We show that for regularly varying claim sizes the probability of ruin after reinsurance is also regularly varying in terms of the initial capital, and derive an explicit asymptotic expression for the latter. We establish this result by leveraging recent developments on sample-path large deviations for heavy tails. Our results allow, on the asymptotic level, for an explicit comparison between two well-known large-claim reinsurance contracts, namely LCR and ECOMOR. We finally assess the accuracy of the resulting approximations using state-of-the-art rare event simulation techniques.

math.PR↗

arXiv2019

A Stochastic Resource-Sharing Network for Electric Vehicle Charging

We consider a distribution grid used to charge electric vehicles such that voltage drops stay bounded. We model this as a class of resource-sharing networks, known as bandwidth-sharing networks in the communication network literature. We focus on resource-sharing networks that are driven by a class of greedy control rules that can be implemented in a decentralized fashion. For a large number of such control rules, we can characterize the performance of the system by a fluid approximation. This leads to a set of dynamic equations that take into account the stochastic behavior of EVs. We show that the invariant point of these equations is unique and can be computed by solving a specific ACOPF problem, which admits an exact convex relaxation. We illustrate our findings with a case study using the SCE 47-bus network and several special cases that allow for explicit computations.

math.OC↗

arXiv2018

Bounds and Limit Theorems for a Layered Queueing Model in Electric Vehicle Charging

The rise of electric vehicles (EVs) is unstoppable due to factors such as the decreasing cost of batteries and various policy decisions. These vehicles need to be charged and will therefore cause congestion in local distribution grids in the future. Motivated by this, we consider a charging station with finitely many parking spaces, in which electric vehicles arrive in order to get charged. An EV has a random parking time and a random charging time. Both the charging rate per vehicle and the charging rate possible for the station are assumed to be limited. Thus, the charging rate of uncharged EVs depends on the number of cars charging simultaneously. This model leads to a layered queueing network in which parking spaces with EV chargers have a dual role, of a server (to cars) and customers (to the grid). We are interested in the performance of the aforementioned model, focusing on the fraction of vehicles that get fully charged. To do so, we develop several bounds and asymptotic (fluid and diffusion) approximations for the vector process, which describes the total number of EVs and the number of not fully charged EVs in the charging station, and we compare these bounds and approximations with numerical outcomes.

math.PR↗

arXiv2018

Heavy traffic limit for the workload plateau process in a tandem queue with identical service times

We consider a two-node tandem queueing network in which the upstream queue is GI/GI/1 and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. To investigate the evolution of workload in the second queue, we introduce and study a process M, called the plateau process, which encodes most of the information in the workload process. We focus on the case of infinite-variance service times and show that under appropriate scaling, workload in the first queue converges, and although the workload in the second queue does not converge, the plateau process does converges to a limit that is a certain function of two independent Levy processes. Using excursion theory, we compare a time changed version of the limit to a limit process derived in previous work.

math.PR↗

arXiv2018

Emergent failures and cascades in power grids: a statistical physics perspective

We model power grids transporting electricity generated by intermittent renewable sources as complex networks, where line failures can emerge indirectly by noisy power input at the nodes. By combining concepts from statistical physics and the physics of power flows, and taking weather correlations into account, we rank line failures according to their likelihood and establish the most likely way such failures occur and propagate. Our insights are mathematically rigorous in a small-noise limit and are validated with data from the German transmission grid.

physics.soc-ph↗

arXiv2017

Electric vehicle charging: a queueing approach

The number of electric vehicles (EVs) is expected to increase. As a consequence, more EVs will need charging, potentially causing not only congestion at charging stations, but also in the distribution grid. Our goal is to illustrate how this gives rise to resource allocation and performance problems that are of interest to the Sigmetrics community.

math.OC↗

arXiv2017

Heavy-Traffic Analysis of Sojourn Time under the Foreground-Background Scheduling Policy

We consider the steady-state distribution of the sojourn time of a job entering an M/GI/1 queue with the foreground-background scheduling policy in heavy traffic. The growth rate of its mean, as well as the limiting distribution, are derived under broad conditions. Assumptions commonly used in extreme value theory play a key role in both the analysis and the results.

math.PR↗

arXiv2017

Sample Path Large Deviations for Lévy Processes and Random Walks with Regularly Varying Increments

Let $X$ be a Lévy process with regularly varying Lévy measure $ν$. We obtain sample-path large deviations for scaled processes $\bar X_n(t) \triangleq X(nt)/n$ and obtain a similar result for random walks. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the increment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connections with the classical large deviations framework. In that setting, we show that a weak large deviation principle (with logarithmic speed) holds, but a full large deviation principle does not hold.

math.PR↗