Nonlinear Stochastic Model Research Articles

A two-stage stochastic programming model for selective maintenance optimization

The problem of selective maintenance exists in many multicomponent systems carrying out an alternating sequence of missions, with scheduled breaks where only a limited number of components can be maintained due to time limit. In this paper, a stochastic programming approach is proposed for determining an optimum maintenance plan to minimize maintenance costs and expected failure costs, while maximizing the probability of successful accomplishment of the next mission under uncertainties in future operating conditions. Traditionally, future operating conditions that affect failure time distribution when calculating reliability in selective maintenance models were assumed as deterministic. In this study, future operating conditions are assumed to be uncertain. The system is subject to several uncertain condition scenarios of exposure, conditional, usage, stress, etc. Each scenario is modeled with its associated occurrence probability. The presented model is a two-stage stochastic mixed-integer nonlinear programming model with fixed recourse, where the first stage is associated with maintenance decisions made before uncertainties are revealed, and the second stage is modeled as a recourse function which is related to the occurrence probability of system failure. A numerical example of a series-parallel system is used to demonstrate the effectiveness of the suggested model.

Stochastic Cooperative Bidding Strategy for Multiple Microgrids With Peer-to-Peer Energy Trading

It is a significant and challenging problem to coordinate multiple microgrids (MMGs) belonging to different entities and achieve their excellent energy-sharing performance to ensure the stability of electricity markets. This article studies a grid-oriented energy bidding problem for MMGs with peer-to-peer (P2P) energy trading under uncertainty. A stochastic Cartel game (SCG) based strategy is developed. A stochastic Cartel nonlinear programming model is formulated to characterize the joint energy bidding, the energy production, and P2P energy transactions while minimizing the total cost for MMGs under uncertainty. A diagonal quadratic approximation method is employed to linearize quadratic terms, and the SCG problem for MMGs is further decomposed into subproblems for individual MGs based on a surrogate Lagrangian relaxation method. The equivalence of problem transformation is proved and equilibrium solutions are derived in an iterative and distributed manner. Comparisons for different strategies, models, and solution algorithms are conducted to testify the rationality and validity of the proposed strategy.

A game-theoretic perspective to study a nonlinear stochastic parabolic model of population competition

In this paper, a game of competition between two populations is designed. Populations are considered as two players competing against each other to increase their sizes. Players have been provided with costly two-component strategies. Population densities are described by two coupled nonlinear stochastic second-order parabolic equations. We have added stochastic terms to the deterministic model to consider random perturbations. First, we have considered the stochastic payoff functions and proved the game has a unique stochastic Nash equilibrium. Then, we have shown the stochastic Nash equilibrium is also a Nash equilibrium for the game with deterministic payoff functions defined by expectation. Cooperative game is also studied and stochastic Pareto efficient strategies are obtained. Then, the existence and uniqueness analyses of the deterministic Nash equilibrium and deterministic Pareto efficient strategies for the game with deterministic payoff functions are provided. The explicit forms of Nash equilibriums and Pareto efficient strategies are presented. Results about the losses the players experience when they don't cooperate are given. Finally, some numerical experiments are included to illustrate what we have claimed in the statements of theorems and show the effects of the Nash equilibrium and Pareto efficient strategy on the payoffs of the players.

Information entropy, continuous improvement, and US energy performance: a novel stochastic-entropic analysis for ideal solutions (SEA-IS)

To address the gap in energy performance studies regarding the role of information entropy in feedback processes between input and output slacks, the paper develops a novel Stochastic-Entropic Analysis for Ideal Solutions (SEA-IS) model. The non-linear stochastic optimization model is then applied to assess the potential information gains that may arise from energy slacks minimization given the different optimal reduction quantiles in US states. The model relies on Beta distributed priors to model the odds-ratio of learning feedback and takes advantages of numerous strengths present in DEA and TOPSIS approaches for performance management. Machine learning methods are also employed to predict information gains in terms of contextual variables. We find that California is the only U.S. state that has indicate strong mutual information feedback and continuous improvements in efficiency. The results indicate there is ample scope for harnessing the power of information gains in improving energy efficiency, particularly in 37 U.S. states, which indicates scope for a public–private partnership to achieve this goal.

Applying interval stability concept to empirical model of middle Pleistocene transition.

Interval stability is a novel method for the study of complex dynamical systems, allowing for the estimation of their stability to strong perturbations. This method describes how large perturbation should be to disrupt the stable dynamical regime of the system (attractor). In our work, interval stability is used for the first time to study the properties of a real natural system: to analyze the stability of the earth's climate system during the last 2.6×106 years. The main abrupt shift in global climate during this period is the middle Pleistocene transition (MPT), which occurred about 1×106 years ago as a change of the periodicity of glacial cycles from 41 to 100 kyr. On the basis of the empirical nonlinear stochastic model proposed in our recent work, we demonstrate that the global climate stability to any perturbations decreases throughout the Pleistocene period (including the MPT), enhancing its response to fast (with a millennial scale or less) internal disturbances.

Data-driven stochastic model for cross-interacting processes with different time scales.

In this work, we propose a new data-driven method for modeling cross-interacting processes with different time scales represented by time series with different sampling steps. It is a generalization of a nonlinear stochastic model of an evolution operator based on neural networks and designed for the case of time series with a constant sampling step. The proposed model has a more complex structure. First, it describes each process by its own stochastic evolution operator with its own time step. Second, it takes into account possible nonlinear connections within each pair of processes in both directions. These connections are parameterized asymmetrically, depending on which process is faster and which process is slower. They make this model essentially different from the set of independent stochastic models constructed individually for each time scale. All evolution operators and connections are trained and optimized using the Bayesian framework, forming a multi-scale stochastic model. We demonstrate the performance of the model on two examples. The first example is a pair of coupled oscillators, with the couplings in both directions which can be turned on and off. Here, we show that inclusion of the connections into the model allows us to correctly reproduce observable effects related to coupling. The second example is a spatially distributed data generated by a global climate model running in the middle 19th century external conditions. In this case, the multi-scale model allows us to reproduce the coupling between the processes which exists in the observed data but is not captured by the model constructed individually for each process.

Dynamical analysis and design of computational methods for nonlinear stochastic leprosy epidemic model

This article presents the dynamical analysis of the stochastic leprosy epidemic model. Positivity and boundedness are the criteria used in the deterministic model. A primary technique is known as the Euler Maruyama used in the solution of the said model. The standard computational methods will evaluate the design stability and efficiency based on the chosen criteria. The traditional computational methods like the stochastic Euler and the stochastic Runge Kutta fail to restore the essential features of biological problems. However, our proposed approach, the stochastic non-standard finite difference (NSFD), is used and found to be efficient, cost-effective, and accommodates all the desired feasible properties. Our method achieves all-time convergence against the backdrop of other classical techniques that perform conditionally or fail over a long period. In the end, a comparison between this scheme and the existing ones reviews the novelty of our approach.

Mathematical Modeling of Proliferative Immune Response Initiated by Interactions Between Classical Antigen-Presenting Cells Under Joint Antagonistic IL-2 and IL-4 Signaling

During an adaptive immune response from pathogen invasion, multiple cytokines are produced by various immune cells interacting jointly at the cellular level to mediate several processes. For example, studies have shown that regulation of interleukin-4 (IL-4) correlates with interleukin-2 (IL-2) induced lymphocyte proliferation. This motivates the need to better understand and model the mechanisms driving the dynamic interplay of proliferation of lymphocytes with the complex interaction effects of cytokines during an immune response. To address this challenge, we adopt a hybrid computational approach comprising of continuous, discrete and stochastic non-linear model formulations to predict a system-level immune response as a function of multiple dependent signals and interacting agents including cytokines and targeted immune cells. We propose a hybrid ordinary differential equation-based (ODE) multicellular model system with a stochastic component of antigen microscopic states denoted as Multiscale Multicellular Quantitative Evaluator (MMQE) implemented using MATLAB. MMQE combines well-defined immune response network-based rules and ODE models to capture the complex dynamic interactions between the proliferation levels of different types of communicating lymphocyte agents mediated by joint regulation of IL-2 and IL-4 to predict the emergent global behavior of the system during an immune response. We model the activation of the immune system in terms of different activation protocols of helper T cells by the interplay of independent biological agents of classic antigen-presenting cells (APCs) and their joint activation which is confounded by the exposure time to external pathogens. MMQE quantifies the dynamics of lymphocyte proliferation during pathogen invasion as bivariate distributions of IL-2 and IL-4 concentration levels. Specifically, by varying activation agents such as dendritic cells (DC), B cells and their joint mechanism of activation, we quantify how lymphocyte activation and differentiation protocols boost the immune response against pathogen invasion mediated by a joint downregulation of IL-4 and upregulation of IL-2. We further compare our in-silico results to in-vivo and in-vitro experimental studies for validation. In general, MMQE combines intracellular and extracellular effects from multiple interacting systems into simpler dynamic behaviors for better interpretability. It can be used to aid engineering of anti-infection drugs or optimizing drug combination therapies against several diseases.

Reconstruction of Epidemiological Data in Hungary Using Stochastic Model Predictive Control

In this paper, we propose a model-based method for the reconstruction of not directly measured epidemiological data. To solve this task, we developed a generic optimization-based approach to compute unknown time-dependent quantities (such as states, inputs, and parameters) of discrete-time stochastic nonlinear models using a sequence of output measurements. The problem was reformulated as a stochastic nonlinear model predictive control computation, where the unknown inputs and parameters were searched as functions of the uncertain states, such that the model output followed the observations. The unknown data were approximated by Gaussian distributions. The predictive control problem was solved over a relatively long time window in three steps. First, we approximated the expected trajectories of the unknown quantities through a nonlinear deterministic problem. In the next step, we fixed the expected trajectories and computed the corresponding variances using closed-form expressions. Finally, the obtained mean and variance values were used as an initial guess to solve the stochastic problem. To reduce the estimated uncertainty of the computed states, a closed-loop input policy was considered during the optimization, where the state-dependent gain values were determined heuristically. The applicability of the approach is illustrated through the estimation of the epidemiological data of the COVID-19 pandemic in Hungary. To describe the epidemic spread, we used a slightly modified version of a previously published and validated compartmental model, in which the vaccination process was taken into account. The mean and the variance of the unknown data (e.g., the number of susceptible, infected, or recovered people) were estimated using only the daily number of hospitalized patients. The problem was reformulated as a finite-horizon predictive control problem, where the unknown time-dependent parameter, the daily transmission rate of the disease, was computed such that the expected value of the computed number of hospitalized patients fit the truly observed data as much as possible.

Multi-objective imperfect selective maintenance optimization for series-parallel systems with stochastic mission duration

To improve the probability that an engineering system successfully completes its next mission, it is crucial to implement timely maintenance activities, especially when maintenance time or maintenance resources are limited. Taking series-parallel system as the object of study, this paper develops a multi-objective imperfect selective maintenance optimization model. Among it, during the scheduled breaks, potential maintenance actions are implemented for the components, ranging from minimal repair to replacement. Considering that the level of maintenance actions is closely related to the maintenance cost, age reduction coefficient and hazard rate adjustment coefficient are taken into account. Moreover, improved hybrid hazard rate approach is adopted to describe the reliability improvement of the components, and the mission duration is regarded as a random variable. On this basis, a nonlinear stochastic optimization model is established with dual objectives to minimize the total maintenance cost and maximize the system reliability concurrently. The fast elitist non-dominated sorting genetic algorithm (NSGA-II) is adopted to solve the model. Numerical experiments are conducted to verify the effectiveness of the proposed approach. The results indicate that the proposed model can obtain better scheduling schemes for the maintenance resources, and more flexible maintenance plans are gained.

Optimal Price-maker Trading Strategy of Wind Power Producer Using Virtual Bidding

This paper proposes a stochastic optimization model for generating the optimal price-maker trading strategy for a wind power producer using virtual bidding, which is a kind of financial tool available in most electricity markets of the United States. In the proposed model, virtual bidding is used to improve the wind power producer's market power in the day-ahead (DA) market by trading at multiple buses, which are not limited to the locations of the wind units. The optimal joint wind power and virtual trading strategy is generated by solving a bi-level nonlinear stochastic optimization model. The upper-level problem maximizes the total expected profit of the wind power and virtual bidding while using the conditional value at risk (CVaR) for risk management. The lower-level problem represents the clearing process of the DA market. By using the Ka-rush-Kuhn-Tucker (KKT) conditions, duality theory, and big- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M$</tex> method, the bi-level nonlinear stochastic model is firstly transferred into an equivalent single-level stochastic mathematical program with the equilibrium constraints (MPEC) model and then a mixed-integer linear programming (MILP) model, which can be solved by existing commercial solvers. To reduce the computational cost of solving the proposed stochastic optimization model for large systems, a method of reducing the number of buses considered for virtual bidding is proposed to simplify the stochastic MPEC model by reducing its decision variables and constraints related to virtual bidding. Case studies are performed to show the effectiveness of the proposed model and the method of reducing the number of buses considered for virtual bidding. The impacts of the transmission limits, wind unit location, risk aversion parameters, wind power volatility, and wind and virtual capacities on the price-maker trading strategy are also studied through case studies.

Integration of Risk-Based Self-Maintenance Activities: Study Case for Crane Distributed System

In today's practice, the coordination of maintenance activities is steadily improved with the use of models. Among them, simulation-based models are attractive because they offer flexibility and can absorb system complexity with ease. Although we avoid state enumeration by using simulation-based models, they can sometimes become computationally demanding. To obtain efficient results, simulation-based models are often implemented on a digital platform. However, there is always concern about whether the implemented model performs adequately and delivers the expected results. In this paper, to address this concern, we propose to perform model calibration. The subject of the paper is calibration through comparison of a risk-based self-maintenance layer, which was created to be executed online without human intervention. The layer is intended for self-assessment of a critical system of overhead cranes operating in hazardous conditions. The case study of the paper is a critical system of overhead cranes operating at a continuous rate, which are part of a hot rolling mill system in a steel plant. The online self-maintenance layer is a nonlinear stochastic optimization model with bounded constraints that aims to evaluate a system-level risk indicator. The estimation of the risk indicator is based on Monte Carlo simulations. Given the features of the model, to evaluate the accuracy and robustness of the layer, only free derivative algorithms, Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Nelder-Mead Modified (NMm) in our case, are used in the comparison. The presented work shows in practice common challenges when facing complex nondeterministic polynomial (NP) problems with stochastic functions.

Employing NMPC scheme utilizing improved cubature Kalman filter for an AUV system

This paper deals with the diving-plane controller design problem for a class of torpedo-type Autonomous Underwater Vehicles (AUVs). Complex hydrodynamic characteristics of AUVs are truly onerous to model and control. AUVs exhibit peculiar nonlinear behaviour and have a high degree of coupling. As a result, they suffer from many practical limitations, such as a lack of hovering capability, limited payload-carrying capacity, poor pitch and roll motion stability, etc. To design an effective and reliable diving-plane control scheme for AUVs, this paper develops a new Stochastic Nonlinear Model Predictive Control (SNMPC) scheme relying on an improved Cubature Kalman Filtering (ICKF) algorithm. This scheme enables an AUV to reach the desired depth underwater and keep hovering at that depth. The ICKF algorithm estimates both the states and plant parameters to update the control law, which helps to reducethe effect of measurement noise and input disturbances. An in-depth simulation case study has been conducted in the presence of external disturbances, noises and plant-parameter variations to demonstrate the usefulness of the proposed ICKF-based NMPC scheme. The results are further analysed based on 'root mean square error' to show the controller's performance.

Numerical scheme and analytical solutions to the stochastic nonlinear advection diffusion dynamical model

Abstract In this study, we give the numerical scheme to the stochastic nonlinear advection diffusion equation. This models is considered with white noise (or random process) having same intensity by changing frequencies. Furthermore, the stability and consistency of proposed scheme are also discussed. Moreover, it is concerned about the analytical solutions, the Riccati equation mapping method is adopted. The different families of single (shock and singular) and mixed (complex solitary-shock, shock-singular, and double-singular) form solutions are obtained with the different choices of free parameters. The graphical behavior of solutions is also depicted in 3D and corresponding contours.

Agricultural water and land resources allocation considering carbon sink/source and water scarcity/degradation footprint

There is an urgent need for scientific management of agricultural water and land resources to cope with global warming and water shortages. Therefore, a stochastic multi-objective non-linear programming model was established under the society-economy-ecology framework in this study, which is capable of (1) considering the carbon sink function of farmland vegetation and the carbon emissions produced from the input of production materials; (2) dealing with the potential impact of agricultural water use on ecology by the indexes of water scarcity and degradation footprint (3) weighing the conflicts and contradictions among different developing targets with economic benefits, and water productivity as well as ecological considerations; (4) obtaining the planting structure, irrigation water allocation schemes and irrigation schedules. In order to verify the applicability and effectiveness of the model, it was applied to Huangyang Irrigation District in Shiyang River Basin, northwest China. After optimization, the net carbon sink increased by 2.55 × 104 t of which fertilizer contributed nearly 50% of carbon emission, while the water footprint reduced by 0.48 × 108 m3. To analyze the impact of the different allocation schemes of water and land resources on the ecological, economic and social subsystems as well as their interaction, coupling coordination degree (CCD) models were introduced to evaluate status quo and the optimization results of different models. The results showed that, compared with the single-objective models and the status quo, the proposed model has improved the value of CCD from 0.428 to 0.674. The proposed model can promote the harmonious and sustainable development of agricultural production and is equally applicable to agricultural management systems in the regions with similar conditions.

Nutrient requirement equations for Indian goat by multiple regression analysis and least cost ration formulation using a linear and non-linear stochastic model.

An elaborate multiple regression analysis was done to arrive a nutrient requirement equation for goat including dry matter intake, DMI (kg/day), total digestible nutrient, TDN (g/day) and crude protein, CP (g/day) based on animal body weight (BW) (kg) and average daily gain (ADG)(g/day). The derived equations were highly significant (p<0.001) and had high R2 (0.99) values. The estimated values of TDN, CP and DMI are compared with NRC (1981), Kearl (Nutrient Requirements of Ruminants in Developing Countries, All Graduate Theses and Dissertations, 1982), as well as ICAR (Livestock Management, 2013). The estimated total TDN and CP requirements at different body weights and ADG are close to the values of recommended feeding standards of Mandal et al. (Small Ruminant Res., 58, 2005, 201). The estimated DMI values are close to the values of ICAR (Livestock Management, 2013) but lower (26.5%-43.8%) as compared to NRC (1981). Regressed values are used to develop a linear programming (LP) model and a stochastic model (SM) for least-cost ration formulation for the Indian goat breed, whose average BW is about 45kg and ADG is 130 (g/day), and which is solved using LP simplex and Generalised Reduced Gradient (GRG) nonlinear of Microsoft Excel. The models satisfy the nutrient requirement calculated by regression equations with minimum specified level of variation (usually 5%-10%) in CP and TDN. Both methods adequately meet the nutritional requirements. Therefore, an electronic sheet is developed in Excel to calculate DMI, TDN and CP for different body weights, ADG and formulate the ration by LP and stochastic model.

Markov Chain Monte Carlo-based Bayesian method for nonlinear stochastic model updating

This paper proposes a Markov Chain Monte Carlo (MCMC)-based Bayesian method for nonlinear stochastic model updating by using the instantaneous characteristics of the structural dynamic responses. According to the discrete analytical mode decomposed method and Hilbert transform, the instantaneous characteristics of the mono-components are firstly extracted from the structural dynamic response and applied to the calculation of likelihood function. Then, the posterior probability density function associated with Bayesian theorem is derived under the assumption of Gaussian prior distribution by using instantaneous characteristics. Afterwards, to calculate the posterior probability density function and improve the sampling efficiency, the delayed rejection adaptive Metropolis-Hastings (DRAM) algorithm is implemented with the advantages of strong adaptive and fast convergence. In the process of Bayesian inference, the posterior samples generated by DRAM require vast quantities of finite element analysis to guarantee the accuracy. For reducing the computational cost, the response surface model is constructed to establish the mathematical regression model between the structural parameters and the theoretical dynamic responses. To validate the effectiveness and applicability of the proposed method, the numerical cases on a three-story nonlinear structure under earthquake excitation considering various noise level effects and an Iwan beam model with two types of excitations are simulated. In addition, an experimental validation on a ¼ scale, 3-story steel frame structure subjected to a series of earthquake excitations in the laboratory is also performed to further verify the proposed method. Both the numerical and experimental results demonstrate that the DRAM-based Bayesian method can be effectively used to update nonlinear stochastic models with a high accuracy.

Simulation of Rare Events in Stochastic Systems

The paper presents algorithms for simulation rare events in stochastic systems based on the theory of large deviations. Here, this approach is used in conjunction with the tools of optimal control theory to estimate the probability that some observed states in a stochastic system will exceed a given threshold by some upcoming time instant. Algorithms for obtaining controlled extremal trajectory (A-profile) of the system, along which the transition to a rare event (threshold) occurs most likely under the influence of disturbances that minimize the action functional, are presented. It is also shown how this minimization can be efficiently performed using numerical-analytical methods of optimal control for linear and nonlinear systems. These results are illustrated by an example for a precipitation-measured monsoon intraseasonal oscillation (MISO) described by a low-order nonlinear stochastic model.

Stochastic Chemotaxis Model with Fractional Derivative Driven by Multiplicative Noise

We introduce stochastic model of chemotaxis by fractional Derivative generalizing the deterministic Keller Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. In this work, we study of nonlinear stochastic chemotaxis model with Dirichlet boundary conditions, fractional Derivative and disturbed by multiplicative noise. The required results prove the existence and uniqueness of mild solution to time and space-fractional, for this we use analysis techniques and fractional calculus and semigroup theory, also studying the regularity properties of mild solution for this model.

Nonlinear degradation modeling and prognostics: A Box-Cox transformation perspective

Transforming nonlinear degradation paths into nearly linear ones has been widely used for nonlinear degradation modeling and prognostics. However, types of the current transformation functions are difficult to determine. This paper addresses issues in nonlinear stochastic degradation modeling and prognostics from a Box-Cox transformation (BCT) perspective. Specifically, the BCT is first used to transform the nonlinear degradation data into nearly linear data, and then the Wiener process with random drift is utilized to model the evolving process of the transformed data. To determine the model parameters, a two-stage estimation procedure is developed including offline stage and online stage. In the offline stage, the parameters are determined via maximum likelihood estimation method based on the historical degradation data and such estimated values are used to initialize the online stage. During the online stage, the Bayesian method is adopted to update the model parameters using the data of the degrading system in service, in which the hyperparameters are updated by the expectation maximization algorithm. A closed-form solution to remaining useful life with updated model parameters is further derived for prognostics. Finally, case studies for lithium-ion batteries and liquid coupling devices are provided to demonstrate the proposed approach.