Stochastic Load Research Articles

A Novel Method of Bridge Deflection Prediction Using Probabilistic Deep Learning and Measured Data.

The deflection control of the main girder in suspension bridges, as flexible structures, is critically important during their operation. To predict the vertical deflection of existing suspension bridge girders under the combined effects of stochastic traffic loads and environmental temperature, this paper proposes an integrated deflection interval prediction method based on a Convolutional Neural Network (CNN), Long Short-Term Memory (LSTM), a probability density estimation layer, and bridge monitoring data. A time-series training dataset consisting of environmental temperature, vehicle load, and deflection monitoring data was built based on bridge health monitoring data. The CNN-LSTM combined layer is used to capture both local features and long-term dependencies in the time series. A Gaussian distribution (GD) is adopted as the probability density function, and its parameters are estimated using the maximum likelihood method, which outputs the optimal deflection prediction and probability intervals. Furthermore, this paper proposes a method for identifying abnormal deflections of the main girder in existing suspension bridges and establishes warning thresholds. This study indicates that, for short time scales, the CNN-LSTM-GD model achieves a 47.22% improvement in Root Mean Squared Error (RMSE) and a 12.37% increase in the coefficient of determination (R2) compared to the LSTM model. When compared to the CNN-LSTM model, it shows an improvement of 28.30% in RMSE and 6.55% in R2. For long time scales, the CNN-LSTM-GD model shows a 54.40% improvement in RMSE and a 10.22% increase in R2 compared to the LSTM model. Compared to the CNN-LSTM model, it improves RMSE by 38.43% and R2 by 5.31%. This model is instrumental in more accurately identifying abnormal deflections and determining deflection thresholds, making it applicable to bridge deflection early-warning systems.

A New Stochastic Controller for Efficient Power Extraction from Small-Scale Wind Energy Conversion Systems under Random Load Consumption

This paper presents an innovative scheme to enhance the efficiency of power extraction from wind energy conversion systems (WECSs) under random loads. The study investigates how stochastic load consumption, modeled and predicted using a Markov chain process, impacts WECS efficiency. The suggested approach regulates the rectifier voltage rather than the rotor speed, making it a sensorless and reliable method for small-scale WECSs. Nonlinear WECS dynamics are represented using Takagi–Sugeno (TS) fuzzy modeling. Furthermore, the closed-loop system’s stochastic stability and recursive feasibility are guaranteed regardless of random load changes. The performance of the suggested controller is compared with the traditional perturb-and-observe (P&O) algorithm under varying wind speeds and random load variations. Simulation results show that the proposed approach outperforms the traditional P&O algorithm, demonstrating higher tracking efficiency, rapid convergence to the maximum power point (MPP), reduced steady-state oscillations, and lower error indices. Enhancing WECS efficiency under unpredictable load conditions is the primary contribution, with simulation results indicating that the tracking efficiency increases to 99.93%.

Flexibility-Oriented AC/DC Hybrid Grid Optimization Using Distributionally Robust Chance-Constrained Method

With the increasing integration of stochastic sources and loads, ensuring the flexibility of AC/DC hybrid distribution networks has become a pressing challenge. This paper aims to enhance the operational flexibility of AC/DC hybrid distribution networks by proposing a flexibility-oriented optimization framework that addresses the growing uncertainties. Notably, a comprehensive evaluation method for operational flexibility assessment is first established. Based on this, this paper further proposes a flexibility-oriented operation optimization model using the distributionally robust chance-constrained (DRCC) method. A customized solution method utilizing second-order cone relaxation and sample average approximation (SAA) is also introduced. The results of case studies indicate that the flexibility of AC/DC hybrid distribution networks is enhanced through sharing energy storage among multiple feeders, adaptive reactive power regulation using soft open points (SOPs) and static var compensators (SVCs), and power transfer between feeders via SOPs.

Study of a Corrective Function for Calculations of Durability under Stochastic Loading

Study of a Corrective Function for Calculations of Durability under Stochastic Loading

A tail probability density distribution model of stress amplitude under band-limited stochastic vibration loadings

Resonance of engineering structures under stochastic vibration loadings is a primary cause of fatigue failure. Considering that cyclic loadings below the fatigue limit of the material contribute little to fatigue damage, the portion of the stress amplitude distribution that exceeds the fatigue limit is in focus in this paper, and a tail probability density distribution model of stress amplitude under band-limited stochastic vibration loadings is proposed. First, the response stress power spectral density functions under band-limited loadings are simulated by using the normal distribution and the Weibull distribution, respectively, and then the frequency domain signals are converted into time domain signals by adopting the inverse fast Fourier transform method. A two-parameter Weibull model of tail probability density distribution of stress amplitude is proposed, and the relationship between model parameters and power spectral moment parameters is given. Finally, the comparative analysis of the model accuracy and the vibration fatigue test of aluminum alloy notch specimen are carried out. It is demonstrated that the fatigue life predicted by the presented model is highly coincident with the results of tests and the time domain method.

A joint time–space extrapolation approach within the Wiener path integral technique for efficient stochastic response determination of nonlinear systems

A joint time–space extrapolation approach within the Wiener path integral (WPI) technique is developed for determining, efficiently and accurately, the non-stationary stochastic response of diverse nonlinear dynamical systems. The approach can be construed as an extension of a recently developed space-domain extrapolation scheme to account also for the temporal dimension. Specifically, based on a variational principle, the WPI technique yields a boundary value problem (BVP) to be solved for determining a most probable path corresponding to specific final boundary conditions. Further, the most probable path is used for evaluating, approximately, a point of the system response joint probability density function (PDF) corresponding to a specific time instant. Remarkably, the BVP exhibits two unique features that are exploited in this paper for developing an efficient joint time–space extrapolation approach. First, the BVPs corresponding to two neighboring grid points in the spatial domain of the response PDF not only share the same equations, but also the boundary conditions differ only slightly. Second, information inherent in the time-history of an already determined most probable path can be used for evaluating points of the response PDF corresponding to arbitrary time instants, without the need for solving additional BVPs. In a nutshell, relying on the aforementioned unique and advantageous features of the WPI-based BVP, the complete non-stationary response joint PDF is determined, first, by calculating numerically a relatively small number of PDF points, and second, by extrapolating in the joint time–space domain at practically zero additional computational cost. Compared to a standard brute-force implementation of the WPI technique, the developed extrapolation approach reduces the associated computational cost by several orders of magnitude. Two numerical examples relating to an oscillator with asymmetric nonlinearities and fractional derivative elements, and to a nonlinear structure under combined stochastic and deterministic periodic loading are considered for demonstrating the reliability of the extrapolation approach. Juxtapositions with pertinent Monte Carlo simulation data are included as well.

Numerical assessment of transient flow and energy dissipation in a Pelton turbine during startup

The Pelton turbine, known for its high application water head, wide efficient operating range, and rapid start-stop capability, is ideal for addressing intermittent and stochastic load issues. This study numerically analyzes the transient two-phase flow and energy dissipation during the startup of a Pelton turbine. Dynamic mesh technology controlled nozzle opening changes, and momentum balance equations managed runner rotation. Findings showed that the runner speed initially increased rapidly and then more slowly, and flow rate matched the nozzle opening variations. Runner torque first rose linearly, then decreased, with the fastest decline during nozzle closing. Hydraulic efficiency peaked early in nozzle reduction but then dropped sharply. Strong vortices formed due to upstream inflow and downstream backflow impact in the distributor pipe. The jet needle and guide vane improved flow in the converging section of nozzle, but flow began to diffuse with increased stroke. Initially, the jet spread fully on the bucket surface, but later only affected the bucket tips. Pressure fluctuations in the water supply mechanism were primarily due to jet needle motion, with higher amplitude during movement and lower when stationary. These fluctuations propagated upstream, weakening over distance. Reynolds stress work and turbulent kinetic energy generation, respectively, dominated energy transmission and energy dissipation, with their maximum contribution exceeding 96% and 70%. High-energy clusters corresponded to jet impact positions, highlighting jet-bucket interference as crucial for energy transport. This study established a performance evaluation method for Pelton turbine startups, supporting further investigation into characteristic parameters, flow evolution, and energy dissipation patterns.

Unveiling Mispricing Risks: Nonlarge Homogeneous Portfolio Factor Copula Models for Enhanced Valuation of Subordinated Loan Securitization

AbstractThis paper presents an innovative factor copula model for collateralized loan obligation (CLO) tranche valuation, incorporating non‐Gaussian distributions and dynamic correlations without relying on the large homogeneous portfolio (LHP) assumption. Through numerical analyses and comparisons with LHP models, I find that non‐LHP models produce higher tranche spreads, especially for lower‐rated tranches. Sensitivity analysis reveals varying sensitivities to changes in the number of collaterals, risk‐free rate, average collateral ratings, recovery rates, and time to maturity. The non‐LHP one‐factor copula models, including stochastic correlation and random factor loading models, outperform LHP models in root mean squared errors when calibrated to market data. The results underscore the importance of considering model limitations in CLO tranche pricing and highlight potential mispricing of spread risk in higher‐rated tranches using LHP models. The proposed models contribute to a more comprehensive understanding of CLO tranche pricing by accounting for various factors and assumptions influencing fair premiums.

Optimal allocation of distributed energy resources to cater the stochastic E-vehicle loading and natural disruption in low voltage distribution grid

The everyday extreme uncertainties become the new normal for our world. Critical infrastructures like electrical power grid and transportation systems are in dire need of adaptability to dynamic changes. Moreover, stringent policies and strategies towards zero carbon emission require the heavy influx of renewable energy sources (RES) and adoption of electric transportation systems. In addition, the world has seen an increased frequency of extreme natural disasters. These events adversely impact the electrical grid, specifically the less hardened distribution grid. Hence, a resilient electrical network is the demand of the future to fulfill critical loads and charging of emergency electrical vehicles (EV). Therefore, this paper proposes a two-dimensional methodology in planning and operational phase for a resilient electric distribution grid. Initially stochastic modelling of EV load has been performed duly considering the geographical feature and commute pattern to form probability distribution functions. Thenceforth, the impact assessment of extreme natural events like earthquakes using damage state classification has been done to model the impact on distribution grid. The efficacy of the proposed methodology has been tested by simulating an urban Indian distribution grid with mapped EV on DigSILENT PowerFactory integrated with supervised learning tools on Python. Subsequently 24-h load profile before event and after event have been compared to analyze the impact.

Assessing Voltage Stability in Distribution Networks: A Methodology Considering Correlation among Stochastic Variables

Distributed photovoltaic (PV) output exhibits strong stochasticity and weak adjustability. After being integrated with the network, its interaction with stochastic loads increases the difficulty of assessing the distribution network’s static voltage stability (SVS). In response to this issue, this article presents a probabilistic assessment method for SVS in a distribution network with distributed PV that considers the bilateral uncertainties and correlations on the source and load sides. The probabilistic models for the uncertain variables are established, with the correlation between stochastic variables described using the Copula function. The three-point estimate method (3PEM) based on the Nataf transformation is used to generate correlated samples. Continuous power flow (CPF) calculations are then performed on these samples to obtain the system’s critical voltage stability state. The distribution curves of critical voltage and load margin index (LMI) are fitted using Cornish-Fisher series. Finally, the utility function is introduced to establish the degree of risk of voltage instability under different scenarios, and the SVS assessment of the distribution network is completed. The IEEE 33-node distribution system is utilized to test the method presented, and the results across various scenarios highlight the method’s effectiveness.

Velocity divergence—Generalized adaptive probability density evolution method

AbstractThis study proposes a novel velocity divergence‐generalized adaptive probability density evolution method (VD‐GAPDEM) for calculating the probability density function of the stochastic response process of stochastic structures under stochastic dynamic loads. First, based on the principle of probability conservation, the velocity divergence‐generalized adaptive probability density evolution equation (VD‐GAPDEE) is derived for a stochastic system that can effectively consider the shape and location changes of the joint transitional probability density of representative points (RPs) in the stochastic response process. Second, a novel VD‐GAPDEM is proposed to solve the VD‐GAPDEE directly using the point selection technique based on the generalized F discrepancy and the second‐order Runge–Kutta method with a smoothing kernel method (Runge–Kutta‐SKFAM). Furthermore, the differences and connections between VD‐GAPDEM and the existing probability density evolution method are analyzed. Additionally, the high computational efficiency and accuracy of the proposed VD‐GAPDEM are demonstrated through three typical examples of stochastic response analysis, involving stochastic systems subjected to stochastic dynamic loads.

Sustainable energy management in microgrids: a multi-objective approach for stochastic load and intermittent renewable energy resources

Sustainable energy management in microgrids: a multi-objective approach for stochastic load and intermittent renewable energy resources

Perturbed Utility Stochastic Traffic Assignment

This paper develops a fast algorithm for computing the equilibrium assignment with the perturbed utility route choice (PURC) model. Without compromise, this allows the significant advantages of the PURC model to be used in large-scale applications. We formulate the PURC equilibrium assignment problem as a convex minimization problem and find a closed-form stochastic network loading expression that allows us to formulate the Lagrangian dual of the assignment problem as an unconstrained optimization problem. To solve this dual problem, we formulate a quasi-Newton accelerated gradient descent algorithm (qN-AGD*). Our numerical evidence shows that qN-AGD* clearly outperforms a conventional primal algorithm and a plain accelerated gradient descent algorithm. qN-AGD* is fast with a runtime that scales about linearly with the problem size, indicating that solving the perturbed utility assignment problem is feasible also with very large networks. Funding: This work has been financed by the European Union—NextGenerationEU.

Low‐carbon demand response program for power systems considering uncertainty based on the data‐driven distributionally robust chance constrained optimization

AbstractThe demand response (DR) program is an effective solution to promote the low‐carbon operation of power systems with increasing penetration of renewable energy sources (RESs). This paper proposes a low‐carbon DR program for power systems to enhance both the environmental friendliness and uncertainty resilience of the system operation. The system operator aims to minimize both the system's operation cost and carbon trading cost. To handle the uncertainty associated with stochastic RES generation power and load consumption power, a data‐driven method named the two‐sided distributionally robust chance‐constrained (TS‐DRCC) approach is adopted to enhance the system's uncertainty resilience. A ladder‐type carbon trading scheme is utilized to calculate the carbon emission cost of the system. Comprehensive analyses of case studies have been conducted to validate that the proposed strategy can effectively reduce the total carbon emissions and total operation costs with good uncertainty resilience performance. The proposed low‐carbon DR program is verified to achieve 63.64% more carbon emission reduction compared with the conventional DR program. Besides this, the proposed low‐carbon DR program can also achieve 4.39% carbon‐intensive generation power reduction and 5.52% RES power consumption compared with the conventional DR program.

Results of wear endurance runs of wind turbine pitch bearings

Pitch bearings of wind turbines connect the rotor blade and the rotor hub. They have to cope with highly dynamic and stochastic loads and oscillating movements for lifetimes of 20 years and more. While several test rigs for pitch bearings are in operation, only very limited data is publicly accessible. Oscillating bearings typically suffer from wear, and this work covers the results of wear endurance tests of full-scale pitch bearings. Four bearings were tested approximately 120 days each under realistic conditions which included dynamic loads in five degrees of freedom. The bearings did not show any or very limited signs of raceway wear, in contrast to results of small-scale tests and short-term tests under worst-case conditions.

Time-frequency analysis of plate-shell coupled structures under moving stochastic load

Random vibration is of critical importance in understanding the dynamic behavior of engineering structures, especially the nonstationary responses for their coupled enclosers consisting of plates and shells subjected to moving stochastic loads. In this study, the pseudo excitation method (PEM) is integrated into the method of reverberation ray matrix (MRRM) to establish a full-domain stochastic dynamic system. With the help of rearrangement matrices and coupling spring connection technologies, the two plate-shell models of sequential and reinforced coupling are considered by employing the simple first-order shear deformation shell theory (S-FSDST) and Hamilton's principle. Then, the continuous movement of the excitation between each substructure is achieved through load coordinate normalization in nonhomogeneous equations, which are separated from the generalized solutions in the frequency domain. Therefore, it can be easily converted into a global pseudo-excitation and obtain the time-history responses relying on the inverse Laplace transform. After matching the present results with the finite element software and Monte Carlo simulation (MCS), the accuracy and efficiency of the proposed RRM-PEM are well-confirmed. Finally, by performing a series of numerical cases of power spectral density (PSD) and time-varying root mean square (RMS) under different loading velocities, frequency bands and system modal dampings, the time-frequency rules for different substructures of closed plate-shell coupled system under moving random loads are revealed for the first time.

Time-Variant Reliability for Systems with Non-monotonic Limit-State Functions

Most engineering time-variant reliability problems are the result of component degradation and stochastic loading. The resultant failure modes, and their resultant limit-state functions, produce limit-state surfaces with unpredictable temporal trajectories that may exhibit a combination of increasing and decreasing failure probabilities. In many cases the trajectories are monotonic so that failure increases predictably: in other cases, this is not so. In this paper we present the discrete-time set theory derivation for non-monotonic situations wherein the limit-state surface may recede to provide, what only appears to be, ever decreasing failure probability. The presence of both monotonic and non-monotonic limit-state functions can be easily detected by a parametric polar plot of the most-likely failure points in standard normal space. The polar plot reveals the temporal limit-state surfaces that need to be retained to represent the system limit-state surfaces at any time instant. The minimum set herein is called the extreme limit-state surface. The impact of the work is that the cumulative distribution function (cdf) can be provided with a minimum of failure and safe events. This in turn gives rise to several solution options such as the multi-normal integral method or a special Monte Carlo simulation that obviates the tedious marching-out routine. A series system and a parallel system show the efficacy of the theory.

Multi-agent cooperative optimal scheduling strategy of integrated energy system in urban area under extreme events

Because the global climate change intensifies as well as the natural disasters frequently occur, extreme events have caused serious impacts on the energy system in urban areas, and at the same time, they have brought great challenges to the supply and scheduling of urban energy systems. Therefore, in order to better integrate and manage various energy resources in urban areas, a Deep Q-Leaning Network-Quasi Upper Confidence Bound model is innovatively constructed using deep reinforcement learning technology to learn the state and behavior mapping relationship of energy system. Use deep learning to fit complex nonlinear models to optimize the entire energy system. Compare and verify the experiment with the real energy system. The improved Deep reinforcement learning algorithm is compared with Q-learning model, PDWoLF PHC algorithm model, Quasi Upper Confidence Bound algorithm model and deep Q-Leaning Network algorithm model. The results show that the research algorithm has the smallest instantaneous error value and absolute value of frequency deviation for area control, and the average value of the research algorithm in the absolute value of the frequency deviation is reduced by 45%–73% compared to other algorithms; over time, the unit output power of the research algorithm is able to flexibly track the stochastic square wave loads. Therefore, the proposed system strategies can provide feasible solutions to meet the challenges of extreme events and promote the sustainable development and safe operation of urban energy systems.

Dynamic serviceability and safety reliability analysis of aging PC girder bridges with non-prestressed reinforcement considering concrete shrinkage, creep and stochastic vehicle load flows

Prestressed concrete (PC) bridges’ service performance displays time-dependency due to several factors including concrete shrinkage, creep, resistance degradation, and stochastic vehicle load flows. These non-stationary vehicle load flows, combined with concrete shrinkage, creep, and resistance degradation, result in sequential alterations of internal forces and deformations in simply supported PC girder bridges, ultimately influencing their dynamic reliability. This study proposes a robust methodology that considers the impact of traffic loading, shrinkage, and creep to analyze the dynamic reliability of simply supported PC girder bridges through the development of a load probability model. A stochastic model for truck loads is generated using motion measurement data, which can be used to calculate bridge internal forces and subsequent quantitative analysis of structural deformation due to shrinkage and creep. Subsequently, the formula for the first exceedance probability under resistance degradation is presented and validated through Monte Carlo simulation. Reliability analysis is conducted for both serviceability and load carrying capacity limit states to assess the impact of shrinkage, creep, and increased mean vehicle load on structural reliability. The proposed method offers the theoretical foundation and practical guidance for assessing the structural health status of in-service bridges, thereby providing valuable insights into their long-term performance and maintenance requirements.

Updating Nonlinear Stochastic Dynamics of an Uncertain Nozzle Model Using Probabilistic Learning With Partial Observability and Incomplete Dataset

Abstract This article introduces a methodology for updating the nonlinear stochastic dynamics of a nozzle with uncertain computational model. The approach focuses on a high-dimensional nonlinear computational model constrained by a small target dataset. Challenges include the large number of degrees-of-freedom, geometric nonlinearities, material uncertainties, stochastic external loads, underobservability, and high computational costs. A detailed dynamic analysis of the nozzle is presented. An updated statistical surrogate model relating the observations of interest to the control parameters is constructed. Despite small training and target datasets and partial observability, the study successfully applies probabilistic learning on manifolds (PLoM) to address these challenges. PLoM captures geometric nonlinear effects and uncertainty propagation, improving conditional mean statistics compared to training data. The conditional confidence region demonstrates the ability of the methodology to accurately represent both observed and unobserved output variables, contributing to advancements in modeling complex systems.