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We develop a discrete-time renewal framework in which renewal events evolve along multiple time coordinates and the sojourn mechanism is described by a general distribution on the multi-index lattice. The resulting processes, called multi-time renewal chains, are studied through multi-index convolution and the associated algebra of multivariate formal power series. This algebraic formulation gives explicit representations for multi-time renewal equations, constructive coefficient formulas, and practical inversion schemes. For computation, we combine FFT-based multidimensional convolution with Newton-type reciprocal iteration to evaluate renewal quantities on large grids. For asymptotics, we prove strong laws and central limit theorems under proportional growth of the observation horizon, including a general central limit theorem for additive functionals and a Gaussian limit for the renewal counting process in directions with a unique rate-determining coordinate. We also study fixed-horizon observations: the terminal age vector induces a genuinely multivariate right-censoring mechanism, leading to an exact nonparametric maximum likelihood estimator and its asymptotic normality. Appl

While renewable energy sources are the fastest-growing electricity generation technology globally, their competitiveness is still the subject of controversy. This paper presents an electricity system model for investment and dispatch to determine the cost-optimal shares of renewable energy sources. We compute and analyse renewable generation shares in market equilibrium for Germany and Texas, using annual data for 2015 to 2024, and in five-year intervals for 2030 to 2050. Furthermore, we identify the key drivers of the renewable competitiveness and quantify their contribution through parameter variations. Our results show that renewable generation achieves considerable market shares even without subsidies. In Germany, the increase in renewable generation is primarily driven by CO2 pricing, complemented by declining investment costs for renewable technologies. In Texas, solar PV is part of the cost-optimal system, even in the absence of CO2 pricing and despite low natural gas prices.

The renewal process is a key statistical model for describing a wide range of stochastic systems in Physics. This work investigates the behavior of the probability distribution of the number of renewals in renewal processes in the short-time limit, with a focus on cases where the number of renewals is large. We find that the specific details of the sojourn time distribution $φ(τ)$ in this limit can significantly modify the behavior in the large-number-of-renewals regime. We explore both non-equilibrium and equilibrium renewal processes, deriving results for various forms of $φ(τ)$. Using saddle point approximations, we analyze cases where $φ(τ)$ follows a power-series expansion, includes a cutoff, or exhibits non-analytic behavior near $τ= 0$. Additionally, we show how the short-time properties of $φ(τ)$ shape the decay of the number of renewals in equilibrium compared to non-equilibrium renewal processes. The probability of the number of renewals plays a crucial role in determining rare event behaviors, such as Laplace tails. The results obtained here are expected to help advance the development of a theoretical framework for rare events in transport processes in complex systems.

We investigate the impact of intermittent energy injections on a Brownian particle, modeled as stochastic renewals of its kinetic energy to a fixed value. Between renewals, the particle follows standard underdamped Langevin dynamics. For energy renewals occurring at a constant rate, we find non-Boltzmannian energy distributions that undergo a shape transition driven by the competition between the velocity relaxation timescale and the renewal timescale. In the limit of rapid renewals, the dynamics mimics one-dimensional run-and-tumble motion, while at finite renewal rates, the effective diffusion coefficient exhibits non-monotonic behavior. To quantify the system's departure from equilibrium, we derive a modified fluctuation-response relation and demonstrate the absence of a consistent effective temperature. The dissipation is characterized by deviations from equilibrium-like response, captured via the Harada-Sasa relation. Finally, we extend the analysis to non-Poissonian renewal processes and introduce a dimensionless conversion coefficient that quantifies the thermodynamic cost of diffusion.

The transformation of the energy system has raised concerns about the reliability of fully renewable energy systems. We address this question for a 2050 European energy system using an economically optimal adequacy assessment. Our results show that a cost-optimal, fully renewable European system can be as reliable as a fossil-based one, with an average loss of load of only 0.03% due to variability in renewable generation. Outages primarily affect industrial and service sectors, while household supply remains largely uninterrupted. Regional differences in supply security emerge, with outages concentrated in countries with a low Value of Lost Load (VoLL). We demonstrate that system reliability can be fully ensured at negligible additional cost (+0.17%) by modestly increasing hydrogen turbine (+10%) and battery capacities (+15%) beyond the cost-optimal levels. We conclude that well-designed renewable energy systems are stable, with hydrogen-based backup being a key enabler of reliability.

Hawkes processes are point processes with self-exciting and clustering properties that are popular in applications. In recent years, renewal Hawkes processes have gained attention, due to their versatility such as the capability of capturing dependence between clusters. In this paper, three classes of novel renewal Hawkes processes are introduced after incorporating exogenous and endogenous renewal factors. The relationship among five related Hawkes processes is studied. The expectations of the three classes of renewal Hawkes processes are derived by establishing a set of integral equations. A general common renewal equation for these expectations is derived and further discussions are provided. The special case of constant exogenous factor is discussed as well for the three proposed Hawkes processes. Finally, we apply our proposed models to study the optimization problems that arise in a periodic replacement policy for systems with cascading failures and provide numerical illustrations. The numerical solutions rely on computing the expectations of the proposed renewal Hawkes processes that can be efficiently obtained by using the direct Riemann integration method.

Renewal process is a point process where an inter-event time between successive renewals is an independent and identically distributed random variable. Alternating renewal process is a dichotomous process and a slight generalization of the renewal process, where the inter-event time distribution alternates between two distributions. We investigate statistical properties of the number of renewals and occupation times for one of the two states in alternating renewal processes. When both means of the inter-event times are finite, the alternating renewal process can reach an equilibrium. On the other hand, an alternating renewal process shows aging when one of the means diverges. We provide analytical calculations for the moments of the number of renewals, occupation time statistics, and the correlation function for several case studies in the inter-event-time distributions. We show anomalous fluctuations for the number of renewals and occupation times when the second moment of inter-event time diverges. When the mean inter-event time diverges, distributional limit theorems for the number of events and occupation times are shown analytically. These are known as the Mittag-Leffler distr

We present RENEW, a global path planner for Autonomous Surface Vehicle (ASV) in dynamic environments with external disturbances (e.g., water currents). RENEW introduces a unified risk- and energy-aware strategy that ensures safety by dynamically identifying non-navigable regions and enforcing adaptive safety constraints. Inspired by maritime contingency planning, it employs a best-effort strategy to maintain control under adverse conditions. The hierarchical architecture combines high-level constrained triangulation for topological diversity with low-level trajectory optimization within safe corridors. Validated with real-world ocean data, RENEW is the first framework to jointly address adaptive non-navigability and topological path diversity for robust maritime navigation.

We develop a discrete Markov renewal theory on a standard Borel state space, with vector-valued sojourn times and lower-rectangle observation on $\N^d$. The Markov renewal potential is a kernel-valued convolution resolvent and yields unified representations for semi-Markov transitions, first-passage laws, occupation measures and rewards. The semi-Markov field observed on the partially ordered lattice is generally not Markov. We identify its canonical Markovian augmentation through the backward recurrence vector and give a lumpability criterion for the exceptional cases in which the augmentation can be projected back to the original state space. The lower-rectangle order leads to a stratified inverse-renewal theory: the direction simplex is decomposed into rate-determining cells, with Gaussian limits on cells having a unique active coordinate and minima of correlated Gaussian fields on their interfaces. We establish functional inverse limits, critical-interface limits and logarithmic estimates for inverse deviations. Exact-time potentials are obtained from an operator-theoretic local theorem for Fourier--Laplace perturbations of Markov-additive kernels, while a regenerative theorem

Power Purchase Agreements (PPAs) are bilateral over-the-counter contracts central to renewable energy financing. While their capacity to stabilise revenues and hedge price risk is well recognised, their OTC structure exposes both parties to counterparty credit risk. This is a dimension yet to be explored in the literature, particularly given the dual price and volumetric uncertainty inherent in renewable sources. This paper develops a framework for the pricing and valuation of wind power PPAs and for quantifying this risk through Credit Valuation Adjustment (CVA) and Debit Valuation Adjustment (DVA). We model the joint dynamics of electricity spot prices and renewable output, incorporate default probabilities, and compute valuation adjustments that reflect the fair value of bilateral credit risk. The framework provides market participants with a transparent metric for PPA valuation under counterparty risk. While initiatives such as the European Investment Bank's pilot guarantee scheme aim to mitigate credit risk for certain offtakers, such interventions do not cover all PPA transactions. Rigorous internal credit risk assessment therefore remains indispensable for lenders, producers

Serving the energy demand with renewable energy is hindered by its limited availability near load centres (i.e. places where the energy demand is high). To address this challenge, the concept of Remote Renewable Energy Hubs (RREH) emerges as a promising solution. RREHs are energy hubs located in areas with abundant renewable energy sources, such as sun in the Sahara Desert or wind in Greenland. In these hubs, renewable energy sources are used to synthetise energy molecules. To produce specific energy molecules, a tailored hub configuration must be designed, which means choosing a set of technologies that are interacting with each other as well as defining how they are integrated in their local environment. The plurality of technologies that may be employed in RREHs results in a large diversity of hubs. In order to characterize this diversity, we propose in this paper a taxonomy for accurately defining these hubs. This taxonomy allows to better describe and compare designs of hubs as well as to identify new ones. Thus, it may guide policymakers and engineers in hub design, contributing to cost efficiency and/or improving local integration.

We consider renewal-type processes whose positive inter-renewal times may be dependent, non-identically distributed, and may have mixed distributions. We introduce a generalised intensity measure extending the classical hazard-rate representation to this setting. Under a two-sided comparison scheme for the inter-renewal laws and an additional renewal-measure domination condition \textnormal{(RD)}, we prove a Lorden-type bound for the forward recurrence time. This bound provides an explicit first-moment input for coupling constructions and, once the remaining coupling parameters are controlled, yields a total-variation estimate. We illustrate the result on exponential, mixed, Markov-modulated, and Pareto benchmarks. In the i.i.d.\ benchmarks, the bound has the correct renewal scale up to a universal factor; in the Markov-modulated benchmark, the explicit Lorden constant is verified while the final convergence consequence remains conditional on \textnormal{(RD)}; and in the Pareto case the construction identifies the natural second-moment threshold for finiteness of the Lorden input.

The purpose of this note is to prove the celebrated Discrete Renewal Theorem in a common special case. We use only very elementary methods from real analysis, rather than markov chain theory, complex analysis, or generating functions. Provided is an introduction to a 1d discrete renewal process via a board game example, our proof the discrete renewal theorem, as well as background and history of the Erdǒs-Feller-Pollard Theorem.

We replicate a renewal process at random times, which is equivalent to nesting two renewal processes, or considering a renewal process subject to stochastic resetting. We investigate the consequences on the statistical properties of the model of the intricate interplay between the two probability laws governing the distribution of time intervals between renewals, on the one hand, and of time intervals between resettings, on the other hand. In particular, the total number ${\mathcal N}_t$ of renewal events occurring within a specified observation time exhibits a remarkable range of behaviours, depending on the exponents characterising the power-law decays of the two probability distributions. Specifically, ${\mathcal N}_t$ can either grow linearly in time and have relatively negligible fluctuations, or grow subextensively over time while continuing to fluctuate. These behaviours highlight the dominance of the most regular process across all regions of the phase diagram. In the presence of Poissonian resetting, the statistics of ${\mathcal N}_t$ is described by a unique `dressed' renewal process, which is a deformation of the renewal process without resetting. We also discuss the rel

At present, electricity markets largely ignore the fact that renewable power producers impose significant externalities on non-renewable energy producers. This is because consumers are generally guaranteed electricity within certain load parameters. The intermittent nature of production by renewable energy producers implies that they rely on non-renewable producers so that the aggregate power delivered meets the promised quality of service. This implicit insurance provided by the non-renewable power sector to consumers is not currently priced and leads to an often ignored, hidden monetary transfer from non-renewable producers to renewable producers. As the fraction of energy supplied by renewable resources increases, these externalities also increase. In this paper, we quantify these externalities by developing the market clearing price of energy in the presence of renewable energy. We consider a day-ahead electricity market where renewable and non-renewable generators bid by proposing their asking price per unit of energy to an independent system operator (ISO). The ISO's problem is a multi-stage stochastic optimization problem to dispatch energy from each generator to minimize th

Consider a finite renewal process in the sense that interrenewal times are positive i.i.d. variables and the total number of renewals is a random variable, independent of interrenewal times. A finite point process can be obtained by probabilistic sampling of the finite renewal process, where each renewal is sampled with a fixed probability and independently of other renewals. The problem addressed in this work concerns statistical inference of the original distributions of the total number of renewals and interrenewal times from a sample of i.i.d. finite point processes obtained by sampling finite renewal processes. This problem is motivated by traffic measurements in the Internet in order to characterize flows of packets (which can be seen as finite renewal processes) and where the use of packet sampling is becoming prevalent due to increasing link speeds and limited storage and processing capacities.

We consider a renewal process which models a cumulative shock model that fails when the accumulation of shocks up-crosses a certain threshold. The ratio limit properties of the probabilities of non-failure after n cumulative shocks are studied. We establish that the ratio of survival probabilities converges to the probability that the renewal epoch equals zero. This limit holds for any renewal process, subject only to mild regularity conditions on the individual shock random variable. Precisions on the rates of convergence are provided depending on the support structure and the regularity of the distribution. Arguments are provided to highlight the coherence between this new results and the pre-existing results on the behavior of summands of i.i.d. real random variables.

World models are widely used in offline reinforcement learning (RL) to improve sample efficiency and generate experience beyond a fixed dataset. However, they are vulnerable to model exploitation where data coverage is thin. Prior work addresses this either by collecting more expert demonstrations, which is often expensive, unsafe, or unavailable, or by conservative algorithms that avoid uncertain regions, which limits generalization. We propose instead to repair exploitation directly using human preferences over imagined rollouts, leveraging the strong intuitive physics that allows humans to easily spot egregious dynamics hallucinations. We formalize this as Dynamics Learning from Human Feedback (DLHF), a Bradley-Terry preference loss over trajectory log-likelihoods under a learned dynamics model. Unfortunately, naive DLHF is sample inefficient, so we introduce RENEW, which uses epistemic uncertainty to focus finetuning where the model is most exploitable. We evaluate on several Jumanji and classic control environments and find that while naive DLHF requires an outsize preference budget, RENEW makes the framework practical by improving sample efficiency, limiting catastrophic forg

We present and establish large deviations principles for general multivariate renewal-reward processes associated with a classical discrete-time renewal process. A renewal-reward process describes a cumulative reward over time, supposing that a broad-sense multivariate reward is obtained at each occurrence of the event that is renewed under the renewal process. We consider both the standard model and a constrained model that is constructed conditioning on the event that one of the renewals occurs at a predetermined time. With a different interpretation of the time coordinate, the constrained renewal model includes several important models of statistical mechanics, such as the model of polymer pinning, the Poland-Scheraga model of DNA denaturation, the Wako-Saitô-Muñoz-Eaton model of protein folding, and the Tokar-Dreyssé model of strained epitaxy. We attack the problem of large deviations in constrained renewal models by an argument based on convexity and super-additivity. Then, we transfer results to standard renewal processes by resorting to conditioning. In the context of constrained renewal models, we also propose an explicit application of the general theory to deterministic r