Stochastic Problem Research Articles

Linear, Nonlinear, and Distributed-Parameter Observers Used for (Renewable) Energy Processes and Systems—An Overview

Full- and reduced-order observers have been used in many engineering applications, particularly for energy systems. Applications of observers to energy systems are twofold: (1) the use of observed variables of dynamic systems for the purpose of feedback control and (2) the use of observers in their own right to observe (estimate) state variables of particular energy processes and systems. In addition to the classical Luenberger-type observers, we will review some papers on functional, fractional, and disturbance observers, as well as sliding-mode observers used for energy systems. Observers have been applied to energy systems in both continuous and discrete time domains and in both deterministic and stochastic problem formulations to observe (estimate) state variables over either finite or infinite time (steady-state) intervals. This overview paper will provide a detailed overview of observers used for linear and linearized mathematical models of energy systems and review the most important and most recent papers on the use of observers for nonlinear lumped (concentrated)-parameter systems. The emphasis will be on applications of observers to renewable energy systems, such as fuel cells, batteries, solar cells, and wind turbines. In addition, we will present recent research results on the use of observers for distributed-parameter systems and comment on their actual and potential applications in energy processes and systems. Due to the large number of papers that have been published on this topic, we will concentrate our attention mostly on papers published in high-quality journals in recent years, mostly in the past decade.

Stochastic control with distributionally robust constraints for cyber–physical systems vulnerable to attacks

In this paper, we investigate the control of a cyber–physical system (CPS) while accounting for its vulnerability to external attacks. We formulate a constrained stochastic problem with a robust constraint to ensure robust operation against potential attacks. We seek to minimize the expected cost subject to a constraint limiting the worst-case expected damage an attacker can impose on the CPS. We present a dynamic programming decomposition to compute the optimal control strategy in this robust-constrained formulation and prove its recursive feasibility. We also illustrate the utility of our results by applying them to a numerical simulation.

Optimal stability results and nonlinear duality for L∞ entropy and L1 viscosity solutions

We give a new and rigorous duality relation between two central notions of weak solutions of nonlinear PDEs: entropy and viscosity solutions. It takes the form of the nonlinear dual inequality:(⋆)∫|Stu0−Stv0|φ0dx≤∫|u0−v0|Gtφ0dx,∀φ0≥0,∀u0,∀v0, where St is the entropy solution semigroup of the anisotropic degenerate parabolic equation∂tu+divF(u)=div(A(u)Du), and where we look for the smallest semigroup Gt satisfying (⋆). This amounts to finding an optimal weighted L1 contraction estimate for St. Our main result is that Gt is the viscosity solution semigroup of the Hamilton-Jacobi-Bellman equation∂tφ=supξ⁡{F′(ξ)⋅Dφ+tr(A(ξ)D2φ)}. Since weighted L1 contraction results are mainly used for possibly nonintegrable L∞ solutions u, the natural spaces behind this duality are L∞ for St and L1 for Gt. We therefore develop a corresponding L1 theory for viscosity solutions φ. But L1 itself is too large for well-posedness, and we rigorously identify the weakest L1 type Banach setting where we can have it – a subspace of L1 called Lint∞. A consequence of our results is a new domain of dependence like estimate for second order anisotropic degenerate parabolic PDEs. It is given in terms of a stochastic target problem and extends in a natural way recent results for first order hyperbolic PDEs by [N. Pogodaev, J. Differ. Equ., 2018].

General Procedure to Provide High-Probability Guarantees for Stochastic Saddle Point Problems

General Procedure to Provide High-Probability Guarantees for Stochastic Saddle Point Problems

Efficiently Realized Approximate Models of Random Functions in Stochastic Problems of the Theory of Particle Transfer

Efficiently Realized Approximate Models of Random Functions in Stochastic Problems of the Theory of Particle Transfer

Improvement the resiliency of distribution network with routing, scheduling, and prepositioning mobile energy resources with limited fuel

Mobile energy resources (MERs) are equipment which are used to improve the resiliency of distribution networks after natural disasters. An important point about using these resources is optimal routing and scheduling them to supply essential loads. In this paper, to make the problem more actual, a new formulation considering limited fuel for the trunks that move these energy resources is presented. Then, refueling vehicle is used to lessen the effect of limited fuel. Therefore, an optimization algorithm is presented for optimal allocating refueling vehicle, simultaneously optimal routing and scheduling of limited-fuel MERs. In the next stage, to achieve a better result and reducing the restoration time, a new formulation is presented for prepositioning MERs considering the stochastic occurrence of damages in various points of the network. Latin hypercube sampling method is used as scenario reduction method to implement the stochastic problem formulation. To illustrate the efficiency of the proposed method considering limited fuel for MERs, refueling vehicles, and MERs prepositioning, simulations are carried out on IEEE 33-bus test system and the results are compared with the previous studies’ results.

A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for p=2 to results in the Lp space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish.

Confidence Intervals for Piecewise Normal Distributions and Stochastic Variational Inequalities

In this paper, we first show how to obtain easy-to-compute confidence intervals for the center of a piecewise normal distribution given a sample from this distribution (assuming that the center belongs to a known affine set parallel to the common lineality space of all cones defining the piecewise normal distribution) by using certain skewed projectors on that space. We then extend this method to an asymptotic setting. Next, we apply this method to compute confidence intervals for the true solution of a stochastic variational inequality given a solution to a sample average approximation (SAA) problem for the general situation in which the asymptotic distribution of SAA solutions is piecewise normal. For stochastic complementarity problems, we obtain asymptotic normality of certain estimators of the true solution when the asymptotic distribution of the SAA solutions is piecewise normal. Funding: The research reported in this paper was supported the National Science Foundation [Grant DMS-1814894].

Consumer service level-oriented resilience optimisation for e-commerce supply chain considering hybrid strategies with blockchain adoption

ABSTRACT The booming development of customised e-commerce makes e-commerce supply chains experience a severe test of consumer service level (CSL). An efficient e-commerce supply chain resilience optimisation method addressing economic performance and service performance is proposed to improve customised services under disruption risks. The proposed method includes a hybrid strategy considering both a resistance strategy with blockchain (BCT) adoption and time-dependent recovery strategies simultaneously. A two-stage multi-period multi-product stochastic programming with BCT adoption is then proposed, which (1) implements BCT as the resistance strategy in the pre-disruption stage; (2) collaborates four recovery strategies in the post-disruption stage; (3) supports the e-tailer in making decisions and optimises his profit and order fulfillment time while taking product priorities into account. Using the actual data of Chinese e-commerce during the public health emergency in 2020, it is demonstrated that (1) the applicability of the model with BCT adoption in both supply chain resilience and CSL improvement; (2) the performance of the hybrid strategy in long-term disruption management; (3) the efficiency and robustness of the proposed solution approach for multi-objective high-dimensional stochastic programming problems. E-tailers can optimise decision-making under long-term disruptions by the proposed methodology in practices.

Efficient reliability analysis of stochastic dynamic first-passage problems by probability density evolution analysis with subset supported point selection

This study introduces a novel point selection procedure for the Probability Density Evolution Method (PDEM) to estimate time-dependent reliability and failure probabilities in dynamic systems under first-passage failure conditions. The method integrates and modifies features of the Subset simulation procedure to adaptively generate dependent sample sets suitable for a reliability analysis by a direct probability integration approach levering PDEM. Performance function assessments are used as weighting factors in the Subset supported Point Selection (S-PS), enhancing the ultimate failure probability estimation accuracy. The presented approach effectively identifies samples in the failure region, particularly benefiting for dynamic systems under stochastic excitation, tested with random dimensions up to 60. It also offers a computationally efficient structural reliability estimation procedure by analyzing full time–history responses. The proposed method provides deeper insights into rare failure events and mechanisms through the visualization of intermediate results. This research presents an advanced framework for estimating structural reliability and understanding critical events in dynamic systems.

Simulation Learning and Optimization: Methodology and Applications

Due to the rapid advancement of computational power and the Internet of Things (IoT) together with recent developments in digital-twin and cyber-physical systems, the integration of big data analytics techniques (BDAT) (e.g., data mining, machine learning (ML), deep learning, data farming, etc.) into traditional simulation methodologies, along with the enhancement of past simulation optimization techniques, has given rise to an emerging field known as simulation learning and optimization (SLO). Unlike simulation optimization, SLO is a data-driven fusion approach that integrates conventional simulation modeling methodologies, simulation optimization techniques, and learning-based big data analytics (BDA) to enhance the ability to address stochastic and dynamic decision-making problems in the real-world complex system and quickly find the best outcomes for the decision support in both real-time and nonreal-time analyses to achieve the data-driven digital twin capability. Although some literature in the past has mentioned similar applications and concepts, no paper provides a structural explanation and comprehensive review of relevant literature on SLO from both methodological and applied perspectives. Consequently, in the first part of this paper, we conduct a literature review on a novel SLO methodology emerging from the fusion of traditional simulation methodology, simulation optimization algorithms, and BDAT. In the second part of this paper, we review the applications of SLO in various contexts, with detailed discussions on manufacturing, maintenance, redundancy allocation, hybrid energy systems, humanitarian logistics, and healthcare systems. Lastly, we investigate potential research directions for SLO in this new era of big data and artificial intelligence (AI).

A novel modified artificial rabbit optimization for stochastic energy management of a grid-connected microgrid: A case study in China

The energy management (EM) solution of the microgrids (MGs) is a crucial task to attain most economic, reliable and sustainable operation state of the MGs. This paper aims to solve the optimal scheduling and stochastic EM problem of a smart MG without and with demand side response (DSR) including MT, FC, PV, WT, and a battery storage system (BSS). A study case in Wenzhou city of China is conducted to reduce the operation cost and maximize the utilization of renewable energy. The system uncertainties like loading, temperature, solar irradiance and wind speed are considered which were obtained from real meteorological data. The normal, lognormal, Weibull PDFs as well as Monte-Carlo and RBS methods are used for uncertainty modelling. A modified artificial rabbit optimization (MARO) is proposed for EM solution based on three strategies including fitness-distance balance, exploitation mechanism of PDO and quasi-opposite-based learning (QOBL) to boost exploration and exploitation phases of traditional ARO. The statistical and non-parametric tests are applied via benchmark functions to validate the performance of MARO. As per the obtained results, the MARO is superior for EM solution compared to other techniques and the operation cost is reduced from 252.0721€ct/day to 184.8435€ct/day with saving of 26.86 % considerably with application of DSR.

Task-driven Risk-bounded Hierarchical Reinforcement Learning Based on Iterative Refinement

Deep Reinforcement Learning (DRL) has garnered substantial acclaim for its versatility and widespread applications across diverse domains. Aligned with human-like learning, DRL is grounded in the fundamental principle of learning from interaction, wherein agents dynamically adjust behavior based on environmental feedback in the form of rewards. This iterative trial-and-error process, mirroring human learning, underscores the importance of observation, experimentation, and feedback in shaping understanding and behavior. DRL agents, trained to navigate complex surroundings, refine their knowledge through hierarchical and abstract representations, empowered by deep neural networks. These representations enable efficient handling of long-horizon tasks and flexible adaptation to novel situations, akin to the human ability to construct mental models for comprehending complex concepts and predicting outcomes. Hence, abstract representation building emerges as a critical aspect in the learning processes of both artificial agents and human learners, particularly in long-horizon tasks. Furthermore, human decision-making, deeply rooted in evolutionary history, exhibits a remarkable capacity to balance the tradeoff between risk and cost across various domains. This cognitive process involves assessing potential negative consequences, evaluating factors such as the likelihood of adverse outcomes, severity of potential harm, and overall uncertainty. Humans intuitively gauge inherent risks and adeptly weigh associated costs, extending beyond monetary expenses to include time, effort, and opportunity costs. The nuanced ability of humans to consider the tradeoff between risk and cost highlights the complexity and adaptability of human decision-making, a skill lacking in typical DRL agents. Principles like these derived from human-like learning present an avenue for inspiring advancements in DRL, fostering the development of more adaptive and intelligent artificial agents. Motivated by these observations and focusing on practical challenges in robotics, our efforts target risk-aware stochastic sequential decision-making problem which is crucial for tasks with extended time frames and varied strategies. A novel integration of model-based conditional planning with DRL is proposed, inspired by hierarchical techniques. This approach breaks down complex tasks into manageable subtasks(motion primitives), ensuring safety constraints and informed decision-making. Unlike existing methods, our approach addresses motion primitive improvement iteratively, employing diverse prioritization functions to guide the search process effectively. This risk-bounded planning algorithm seamlessly integrates conditional planning and motion primitive learning, prioritizing computational efforts for enhanced efficiency within specified time limits.

Real-Time Microgrid Energy Scheduling Using Meta-Reinforcement Learning

With the rapid development of renewable energy and the increasing maturity of energy storage technology, microgrids are quickly becoming popular worldwide. The stochastic scheduling problem of microgrids can increase operational costs and resource wastage. In order to reduce operational costs and optimize resource utilization efficiency, the real-time scheduling of microgrids becomes particularly important. After collecting extensive data, reinforcement learning (RL) can provide good strategies. However, it cannot make quick and rational decisions in different environments. As a method with generalization ability, meta-learning can compensate for this deficiency. Therefore, this paper introduces a microgrid scheduling strategy based on RL and meta-learning. This method can quickly adapt to different environments with a small amount of training data, enabling rapid energy scheduling policy generation in the early stages of microgrid operation. This paper first establishes a microgrid model, including components such as energy storage, load, and distributed generation (DG). Then, we use a meta-reinforcement learning framework to train the initial scheduling strategy, considering the various operational constraints of the microgrid. The experimental results show that the MAML-based RL strategy has advantages in improving energy utilization and reducing operational costs in the early stages of microgrid operation. This research provides a new intelligent solution for microgrids’ efficient, stable, and economical operation in their initial stages.

Are weaker stationarity concepts of stochastic MPCC problems significant in absence of SMPCC-LICQ?

Are weaker stationarity concepts of stochastic MPCC problems significant in absence of SMPCC-LICQ?

Matched queues with matching batch pair [formula omitted]

In this paper, we develop the matrix-analytic method to discuss an interesting but challenging bilateral stochastic matching problem: A matched queue with matching batch pair (m,n) and two types of impatient customers, where the two types of customers arrive according to two independent Poisson processes. Once m A-customers and n B-customers are matched as a group, the m+n customers immediately leave the system. We show that this matched queue can be expressed as a novel bidirectional level-dependent quasi-birth-and-death (QBD) process whose analysis has its own interests, and specifically, computing the maximal non-positive inverse matrices of bidirectional infinite sizes by using the RG-factorizations. Based on this, we can provide an effective matrix-analytic method to deal with this matched queue, including the system stability, the average stationary queue lengths, the average sojourn times, and the departure process. We believe that the methodology and results developed in this paper can be applicable to studying more general matched queueing systems, which are widely encountered in many practical areas.

Decomposition methods for multi-horizon stochastic programming

Multi-horizon stochastic programming includes short-term and long-term uncertainty in investment planning problems more efficiently than traditional multi-stage stochastic programming. In this paper, we exploit the block separable structure of multi-horizon stochastic linear programming, and establish that it can be decomposed by Benders decomposition and Lagrangean decomposition. In addition, we propose parallel Lagrangean decomposition with primal reduction that, (1) solves the scenario subproblems in parallel, (2) reduces the primal problem by keeping one copy for each scenario group at each stage, and (3) solves the reduced primal problem in parallel. We apply the parallel Lagrangean decomposition with primal reduction, Lagrangean decomposition and Benders decomposition to solve a stochastic energy system investment planning problem. The computational results show that: (a) the Lagrangean type decomposition algorithms have better convergence at the first iterations to Benders decomposition, and (b) parallel Lagrangean decomposition with primal reduction is very efficient for solving multi-horizon stochastic programming problems. Based on the computational results, the choice of algorithms for multi-horizon stochastic programming is discussed.

A disaggregated integer L-shaped method for stochastic vehicle routing problems with monotonic recourse

This paper proposes a new integer L-shaped method for solving two-stage stochastic integer programs whose first-stage solutions can decompose into disjoint components, each one having a monotonic recourse function. In a minimization problem, the monotonicity property stipulates that the recourse cost of a component must always be higher or equal to that of any of its subcomponents. The method exploits new types of optimality cuts and lower bounding functionals that are valid under this property. The stochastic vehicle routing problem is particularly well suited to be solved by this approach, as its solutions can be decomposed into a set of routes. We consider the variant with stochastic demands in which the recourse policy consists of performing a return trip to the depot whenever a vehicle does not have sufficient capacity to accommodate a newly realized customer demand. This work shows that this policy can lead to a non-monotonic recourse function, but that the monotonicity holds when the customer demands are modeled by several commonly used families of probability distributions. We also present new problem-specific lower bounds on the recourse that strengthen the lower bounding functionals and significantly speed up the solving process. Computational experiments on instances from the literature show that the new approach achieves state-of-the-art results.

Space‐time stochastic Galerkin boundary elements for acoustic scattering problems

SummaryAcoustic emission or scattering problems naturally involve uncertainties about the sound sources or boundary conditions. This article initiates the study of time domain boundary elements for such stochastic boundary problems for the acoustic wave equation. We present a space‐time stochastic Galerkin boundary element method which is applied to sound‐hard, sound‐soft and absorbing scatterers. Uncertainties in both the sources and the boundary conditions are considered using a polynomial chaos expansion. The numerical experiments illustrate the performance and convergence of the proposed method in model problems and present an application to a problem from traffic noise.

ODC and ROC Curves, Comparison Curves and Stochastic Dominance

SummaryWe discuss two novel approaches to inter‐distributional comparisons in the classical two‐sample problem. Our starting point is properly standardised and combined, very popular in several areas of statistics and data analysis, ordinal dominance and receiver characteristic curves, denoted by ODC and ROC, respectively. The proposed new curves are termed the comparison curves. Their estimates, being weighted rank processes on (0,1), form the basis of inference. These weighted processes are intuitive, well‐suited for visual inspection of data at hand and are also useful for constructing some formal inferential procedures. They can be applied to several variants of two‐sample problem. Their use can help improve some existing procedures both in terms of power and the ability to identify the sources of departures from the postulated model. To simplify interpretation of finite sample results, we restrict attention to values of the processes on a finite grid of points. This results in the so‐called bar plots (B‐plots), which readably summarise the information contained in the data. What is more, we show that B‐plots along with adjusted simultaneous acceptance regions provide principled information about where the model departs from the data. This leads to a framework that facilitates identification of regions with locally significant differences.We show an implementation of the considered techniques to a standard stochastic dominance testing problem. Some min‐type statistics are introduced and investigated. A simulation study compares two tests pertinent to the comparison curves to well‐established tests in the literature and demonstrates the strong and competitive performance of the former in many typical situations. Some real data applications illustrate simplicity and practical usefulness of the proposed approaches. A range of other applications of considered weighted processes is briefly discussed too.