arXiv subjects

Bert Zwart

Publications and source records attributed to Bert Zwart.

At least 37 records · Page 2Linked to original sources

arXiv2023

Grid-Aware Real-Time Control and Balancing Between Microgrids

Due to the energy transition, lots of research has been conducted within the last decade on the topics of energy management systems or local energy trading approaches, often on the day-ahead or intraday level. A large majority of these approaches focuses on 15 or 60-minute time intervals for their operation, however, the question of how the planned solutions are realized within these time intervals is often left unanswered. Within this work, we aim to close this gap and propose a real-time balancing and control approach for a set of microgrids, which implements the day-ahead solutions. The approach is based on a three-step framework, in which the first step consists of ensuring the feasibility of devices within the microgrids. The second step focuses on the grid constraints of the connecting medium voltage grid using the DC power flow formulation due to the running time requirements of a real-time approach. The last step is to propagate the solution into the individual microgrids, where the allocated power needs to be distributed among the devices and households. Within a case study, we show that the proposed real-time control approach works as intended and is comparable to an optimal offline algorithm under some mild assumptions.

eess.SY↗

arXiv2023

Threshold-Based Algorithms for an Online Rolling Horizon Framework Under Uncertainty -- With an Application to Energy Management

Decision problems encountered in practice often possess a highly dynamic and uncertain nature. In particular fast changing forecasts for parameters (e.g., photovoltaic generation forecasts in the context of energy management) pose large challenges for the classical rolling horizon framework. Within this work, we propose an online scheduling algorithm for a rolling horizon framework, which directly uses short-term forecasts and observations of the uncertainty. The online scheduling algorithm is based on insights and results from combinatorial online optimization problems and makes use of key properties of robust optimization. Applied within a robust energy management approach, we show that the online scheduling algorithm is able to reduce the total electricity costs within a local microgrid by more than 85% compared to a classical rolling horizon framework and by more than 50% compared to a tailor-made dynamic, yet still offline rolling horizon framework. A detailed analysis provides insights into the working of the online scheduling algorithm under different underlying forecast error distributions.

eess.SY↗

arXiv2023

Large deviations asymptotics for unbounded additive functionals of diffusion processes

We study large deviations asymptotics for a class of unbounded additive functionals, interpreted as normalized accumulated areas, of one-dimensional Langevin diffusions with sub-linear gradient drifts. Our results provide parametric insights on the speed and the rate functions in terms of the growth rate of the drift and the growth rate of the additive functional. We find a critical value in terms of these growth parameters that dictates regions of sub-linear speed for our large deviations asymptotics. Our approach is based upon various constructions of independent interest, including a decomposition of the diffusion process in terms of alternating renewal cycles and a detailed analysis of the paths during a cycle using suitable time and spatial scales. The key to the sub-linear behavior is a heavy-tailed large deviations phenomenon arising from the principle of a single big jump coupled with the result that at each regeneration cycle the upper-tail asymptotic behavior of the accumulated area of the diffusion process is proven to be semi-exponential (i.e., of heavy-tailed Weibull type).

math.PR↗

arXiv2023

Sample-path large deviations for unbounded additive functionals of the reflected random walk

We prove a sample path large deviation principle (LDP) with sub-linear speed for unbounded functionals of certain Markov chains induced by the Lindley recursion. The LDP holds in the Skorokhod space $\mathbb{D}[0,T]$ equipped with the $M_1'$ topology. Our technique hinges on a suitable decomposition of the Markov chain in terms of regeneration cycles. Each regeneration cycle denotes the area accumulated during the busy period of the reflected random walk. We prove a large deviation principle for the area under the busy period of the MRW, and we show that it exhibits a heavy-tailed behavior.

math.PR↗

arXiv2023

Extreme values for the waiting time in large fork-join queues

We prove that the scaled maximum steady-state waiting time and the scaled maximum steady-state queue length among $N$ $GI/GI/1$-queues in the $N$-server fork-join queue, converge to a normally distributed random variable as $N\to\infty$. The maximum steady-state waiting time in this queueing system scales around $\frac{1}γ\log N$, where $γ$ is determined by the cumulant generating function $Λ$ of the service distribution and solves the Cramér-Lundberg equation with stochastic service times and deterministic inter-arrival times. This value $\frac{1}γ\log N$ is reached at a certain hitting time. The number of arrivals until that hitting time satisfies the central limit theorem, with standard deviation $\frac{σ_A}{\sqrt{Λ'(γ)γ}}$. By using distributional Little's law, we can extend this result to the maximum queue length. Finally, we extend these results to a fork-join queue with different classes of servers.

math.PR↗

arXiv2023

Dynamic random intersection graph: Dynamic local convergence and giant structure

Random intersection graphs containing an underlying community structure are a popular choice for modelling real-world networks. Given the group memberships, the classical random intersection graph is obtained by connecting individuals when they share at least one group. We extend this approach and make the communities dynamic by letting them alternate between an active and inactive phase. We analyse the new model, delivering results on degree distribution, local convergence, giant component, and maximum group size, paying particular attention to the dynamic description of these properties. We also describe the connection between our model and the bipartite configuration model, which is of independent interest.

math.PR↗

arXiv2023

Dynamic Rolling Horizon-Based Robust Energy Management for Microgrids Under Uncertainty

Within the last few years, the trend towards more distributed, renewable energy sources has led to major changes and challenges in the electricity sector. To ensure a stable electricity distribution in this changing environment, we propose a robust energy management approach to deal with uncertainty occurring in microgrids. For this, we combine robust optimization with a rolling horizon framework to obtain an algorithm that is both, tractable and can deal with the considered uncertainty. The main contribution of this work lies within the development and testing of a dynamic scheduling tool, which identifies good starting time slots for the rolling horizon. Combining this scheduling tool with the rolling horizon framework results in a dynamic rolling horizon model, which better integrates uncertainty forecasts and realizations of uncertain parameters into the decision-making process. A case study reveals that the dynamic rolling horizon model outperforms the classical version by up to 57% in costs and increases the local use of PV by up to 11%.

eess.SY↗

arXiv2023

Efficient Scenario Generation for Heavy-tailed Chance Constrained Optimization

We consider a generic class of chance-constrained optimization problems with heavy-tailed (i.e., power-law type) risk factors. In this setting, we use the scenario approach to obtain a constant approximation to the optimal solution with a computational complexity that is uniform in the risk tolerance parameter. We additionally illustrate the efficiency of our algorithm in the context of solvency in insurance networks.

math.OC↗

arXiv2023

Large deviations for stochastic fluid networks with Weibullian tails

We consider a stochastic fluid network where the external input processes are compound Poisson with heavy-tailed Weibullian jumps. Our results comprise of large deviations estimates for the buffer content process in the vector-valued Skorokhod space which is endowed with the product $J_1$ topology. To illustrate our framework, we provide explicit results for a tandem queue. At the heart of our proof is a recent sample-path large deviations result, and a novel continuity result for the Skorokhod reflection map in the product $J_1$ topology.

math.PR↗

arXiv2022

Maximum waiting time in heavy-tailed fork-join queues

In this paper, we study the maximum waiting time $\max_{i\leq N}W_i(\cdot)$ in an $N$-server fork-join queue with heavy-tailed services as $N\to\infty$. The service times are the product of two random variables. One random variable has a regularly varying tail probability and is the same among all $N$ servers, and one random variable is Weibull distributed and is independent and identically distributed among all servers. This setup has the physical interpretation that if a job has a large size, then all the subtasks have large sizes, with some variability described by the Weibull-distributed part. We prove that after a temporal and spatial scaling, the maximum waiting time process converges in $D[0,T]$ to the supremum of an extremal process with negative drift. The temporal and spatial scaling are of order $\tilde{L}(b_N)b_N^{\fracβ{(β-1)}}$, where $β$ is the shape parameter in the regularly varying distribution, $\tilde{L}(x)$ is a slowly varying function, and $(b_N,N\geq 1)$ is a sequence for which holds that $\max_{i\leq N}A_i/b_N\overset{\mathbb{P}}{\longrightarrow}1$, as $N\to\infty$, where $A_i$ are i.i.d.\ Weibull-distributed random variables. Finally, we prove steady-state convergence.

math.PR↗

arXiv2022

A Classification Scheme for Local Energy Trading

The current trend towards more renewable and sustainable energy generation leads to an increased interest in new energy management systems and the concept of a smart grid. One important aspect of this is local energy trading, which is an extension of existing electricity markets by including prosumers, who are consumers also producing electricity. Prosumers having a surplus of energy may directly trade this surplus with other prosumers, which are currently in demand. In this paper, we present an overview of the literature in the area of local energy trading. In order to provide structure to the broad range of publications, we identify key characteristics, define the various settings, and cluster the considered literature along these characteristics. We identify three main research lines, each with a distinct setting and research question. We analyze and compare the settings, the used techniques, and the results and findings within each cluster and derive connections between the clusters. In addition, we identify important aspects, which up to now have to a large extent been neglected in the considered literature and highlight interesting research directions, and open problems for future work.

eess.SY↗

arXiv2022

Two more ways of spelling Gini Coefficient with Applications

In this paper, we draw attention to a promising yet slightly underestimated measure of variability - the Gini coefficient. We describe two new ways of defining and interpreting this parameter. Using our new representations, we compute the Gini index for a few probability distributions and describe it in more detail for the negative binomial distribution. We also suggest the latter as a tool to measure overdispersion in epidemiology.

math.ST↗

arXiv2022

Tail Asymptotics for the Delay in a Brownian Fork-Join Queue

In this paper, we study the tail behavior of $\max_{i\leq N}\sup_{s>0}\left(W_i(s)+W_A(s)-βs\right)$ as $N\to\infty$, with $(W_i,i\leq N)$ i.i.d. Brownian motions and $W_A$ an independent Brownian motion. This random variable can be seen as the maximum of $N$ mutually dependent Brownian queues, which in turn can be interpreted as the backlog in a Brownian fork-join queue. In previous work, we have shown that this random variable centers around $\frac{σ^2}{2β}\log N$. Here, we analyze the rare-event that this random variable reaches the value $(\frac{σ^2}{2β}+a)\log N$, with $a>0$. It turns out that its probability behaves roughly as a power law with $N$, where the exponent depends on $a$. However, there are three regimes, around a critical point $a^{\star}$; namely, $0 a^{\star}$. The latter regime exhibits a form of asymptotic independence, while the first regime reveals highly irregular behavior with a clear dependence structure among the $N$ suprema, with a nontrivial transition at $a=a^{\star}$.

math.PR↗

arXiv2022

Persistence of heavy-tailed sample averages: principle of infinitely many big jumps

We consider the sample average of a centered random walk in $\mathbb{R}^d$ with regularly varying step size distribution. For the first exit time from a compact convex set $A$ not containing the origin, we show that its tail is of lognormal type. Moreover, we show that the typical way for a large exit time to occur is by having a number of jumps growing logarithmically in the scaling parameter.

math.PR↗

arXiv2022

Conjectures on Symmetric Queues in Heavy Traffic

This note describes several open questions concerning scaling limits of queue-length processes of symmetric queues in heavy traffic, distinguishing between service-time distributions with finite and infinite variance.

math.PR↗

arXiv2021

Large fork-join queues with nearly deterministic arrival and service times

In this paper, we study an $N$ server fork-join queue with nearly deterministic arrival and service times. Specifically, we present a fluid limit for the maximum queue length as $N\to\infty$. This fluid limit depends on the initial number of tasks. In order to prove these results, we develop extreme value theory and diffusion approximations for the queue lengths.

math.PR↗

arXiv2021

Complete resource pooling of a load balancing policy for a network of battery swapping stations

To reduce carbon emission in the transportation sector, there is currently a steady move taking place to an electrified transportation system. This brings about various issues for which a promising solution involves the construction and operation of a battery swapping infrastructure rather than in-vehicle charging of batteries. In this paper, we study a closed Markovian queueing network that allows for spare batteries under a dynamic arrival policy. We propose a provisioning rule for the capacity levels and show that these lead to near-optimal resource utilization, while guaranteeing good quality-of-service levels for Electric Vehicle (EV) users. Key in the derivations is to prove a state-space collapse result, which in turn implies that performance levels are as good as if there would have been a single station with an aggregated number of resources, thus achieving complete resource pooling.

math.PR↗

arXiv2020

Emergence of scale-free blackout sizes in power grids

We model power grids as graphs with heavy-tailed sinks, which represent demand from cities, and study cascading failures on such graphs. Our analysis links the scale-free nature of blackout sizes to the scale-free nature of city sizes, contrasting previous studies suggesting that this nature is governed by self-organized criticality. Our results are based on a new mathematical framework combining the physics of power flow with rare event analysis for heavy-tailed distributions, and are validated using various synthetic networks and the German transmission grid.

physics.soc-ph↗