Search Results (70)

The high penetration rate of renewable energy poses significant challenges to the planning and operation of power systems in regions with scarce data. In these regions, it is impossible to accurately simulate the complex nonlinear dependencies among hydro–wind–solar energy resources, which leads to huge operational risks and investment uncertainties. To bridge this gap, this study proposes a new data-driven framework that embeds the natural climate cycle (24 solar terms) into a physically consistent scenario generation process, surpassing the traditional linear approach. This framework introduces the Comprehensive Similarity Distance (CSD) indicator to quantify the curve similarity of power amplitude, pattern trend, and fluctuation position, thereby improving the K-means clustering. Compared with the K-means algorithm based on the standard Euclidean distance, the accuracy of the improved clustering pattern extraction is increased by 3.8%. By embedding the natural climate cycle and employing a two-stage dimensionality reduction architecture: time compression via improved clustering and feature fusion via Kernel PCA, the framework effectively captures cross-source dependencies and preserves climatic periodicity. Finally, combined with the simplified Vine Copula model, high-fidelity joint scenarios with a normalized root mean square error (NRMSE) of less than 3% can be generated. This study provides a reliable and computationally feasible tool for stochastic optimization and reliability analysis in the planning and operation of future power systems with high renewable energy grid integration. Full article

The objective of this study is to analyze spatial entropy flows that reveal the directional dynamics of climate change—patterns that remain obscured in traditional statistical analyses. This approach enables the identification of pathways for “climate information transport”, highlights associations with atmospheric circulation types, and allows for the localization of both sources and “informational voids”—regions where entropy is dissipated. The analytical framework is grounded in a quantitative assessment of long-term climate variability across Europe over the period 1901–2010, utilizing Shannon entropy as a measure of atmospheric system uncertainty and variability. The underlying assumption is that the variability of temperature and precipitation reflects the inherently dynamic character of climate as a nonlinear system prone to fluctuations. The study focuses on calculating entropy estimated within a 70-year moving window for each calendar month, using bivariate distributions of temperature and precipitation modeled with copula functions. Marginal distributions were selected based on the Akaike Information Criterion (AIC). To improve the accuracy of the estimation, a block bootstrap resampling technique was applied, along with numerical integration to compute the Shannon entropy values at each of the 4165 grid points with a spatial resolution of 0.5° × 0.5°. The results indicate that entropy and its derivative are complementary indicators of atmospheric system instability—entropy proving effective in long-term diagnostics, while its derivative provides insight into the short-term forecasting of abrupt changes. A lag analysis and Spearman rank correlation between entropy values and their potential supported the investigation of how circulation variability influences the occurrence of extreme precipitation events. Particularly noteworthy is the temporal derivative of entropy, which revealed strong nonlinear relationships between local dynamic conditions and climatic extremes. A spatial analysis of the information entropy field was also conducted, revealing distinct structures with varying degrees of climatic complexity on a continental scale. This field appears to be clearly structured, reflecting not only the directional patterns of change but also the potential sources of meteorological fluctuations. A field-theory-based spatial classification allows for the identification of transitional regions—areas with heightened susceptibility to shifts in local dynamics—as well as entropy source and sink regions. The study is embedded within the Fokker–Planck formalism, wherein the change in the stochastic distribution characterizes the rate of entropy production. In this context, regions of positive divergence are interpreted as active generators of variability, while sink regions function as stabilizing zones that dampen fluctuations. Full article

Considering the effects of complex correlations between variables and uncertainty of degradation processes in multivariate degradation systems, a system reliability assessment method that integrated Chatterjee correlation coefficient and stochastic process theory is proposed. First, due to temporal uncertainty and measurement error in the univariate degradation process, a general Wiener-process-based state space model is constructed to determine the marginal distributions. Secondly, the nonlinear and asymmetric correlation between variables is analyzed by the nonparametric Chatterjee correlation coefficient. The multivariate joint degradation model is constructed by combining the Vine copula technique. The copula structure selection is optimized based on the goodness-of-fit criterion for modeling the degradation dependency network. In order to verify the validity of the method, comparative experiments based on the C-MAPSS aero-engine degradation dataset are conducted. Compared with the independent model ignoring the correlation of the variables, Vine copula with Chatterjee coefficient shows the rationality of the system reliability assessment. The system reliability curve lies between the cases of complete independence and complete dependence of variables. Compared to the traditional Vine copula model with Kendall coefficient, the method in this paper shows a significant improvement in asymmetric correlation characterization, with a Vuong test value of 6.37. The assessment method given in this paper provided an improved paradigm for reliability assessments of complex systems. Full article

To address the challenges of high complexity in modeling the correlation of multi-dimensional stochastic variables and the difficulty of solving long-term scheduling models in continuous action spaces in multi-energy complementary systems, this paper proposes a long-term optimization scheduling method based on Deep Deterministic Policy Gradient (DDPG). First, an improved C-Vine Copula model is used to construct the multi-dimensional joint probability distribution of water, wind, and solar energy, and Latin Hypercube Sampling (LHS) is employed to generate a large number of water–wind–solar coupling scenarios, effectively reducing the model’s complexity. Then, a long-term optimization scheduling model is established with the goal of maximizing the absorption of clean energy, and it is converted into a Markov Decision Process (MDP). Next, the DDPG algorithm is employed with a noise dynamic adjustment mechanism to optimize the policy in continuous action spaces, yielding the optimal long-term scheduling strategy for the water–wind–solar multi-energy complementary system. Finally, using a water–wind–solar integrated energy base as a case study, comparative analysis demonstrates that the proposed method can improve the renewable energy absorption capacity and the system’s power generation efficiency by accurately quantifying the uncertainties of water, wind, and solar energy and precisely controlling the continuous action space during the scheduling process. Full article

Modeling mainshock–aftershock ground motions is essential for seismic risk assessment, especially in regions experiencing frequent earthquakes. Recent studies have often employed Copula-based joint distributions or machine learning techniques to simulate the statistical dependency between mainshock and aftershock parameters. While effective at capturing nonlinear correlations, these methods are typically black box in nature, data-dependent, and difficult to generalize across tectonic settings. More importantly, they tend to focus solely on marginal or joint parameter correlations, which implicitly treat mainshocks and aftershocks as independent stochastic processes, thereby overlooking their inherent spectral interaction. To address these limitations, this study proposes an explicit and parameterized modeling framework based on the evolutionary power spectral density (EPSD) of random ground motions. Using the magnitude difference between a mainshock and an aftershock as the control variable, we derive attenuation relationships for the amplitude, frequency content, and duration. A coherence function model is further developed from real seismic records, treating the mainshock–aftershock pair as a vector-valued stochastic process and thus enabling a more accurate representation of their spectral dependence. Coherence analysis shows that the function remains relatively stable between 0.3 and 0.6 across the 0–30 Rad/s frequency range. Validation results indicate that the simulated response spectra align closely with recorded spectra, achieving R² values exceeding 0.90 and 0.91. To demonstrate the model’s applicability, a case study is conducted on a representative frame structure to evaluate seismic vulnerability and economic loss. As the mainshock PGA increases from 0.2 g to 1.2 g, the structure progresses from slight damage to complete collapse, with loss rates saturating near 1.0 g. These findings underscore the engineering importance of incorporating mainshock–aftershock spectral interaction in seismic damage and risk modeling, offering a transparent and transferable tool for future seismic resilience assessments. Full article

The simulation of wind power, electricity load, and natural gas prices will allow commodity traders to see the future movement of prices in a more probabilistic manner. The ability to observe possible paths for wind power, electricity load, and natural gas prices enables traders to obtain valuable insights for placing their trades on electricity prices. Since the above processes involve a seasonality factor, the seasonality component was modeled using a truncated Fourier series, and the random component was modeled using stochastic differential equations (SDE). It is evident from the literature that all the above processes are mean-reverting processes; thus, three mean-reverting Ornstein–Uhlenbeck (OU) processes were considered the model for wind power, the electricity load, and natural gas prices. Industry experts believe there is a correlation between wind power, the electricity load, and natural gas prices. For example, when wind power is higher and the electricity load is lower, natural gas prices are relatively low. The novelty of this study is the incorporation of the correlation structure between processes into the mean-reverting OU process using a copula function. Thus, the study utilized a vine copula and integrated it into the simulation. The study was conducted for the Texas energy market and used daily time scales for the simulations, and it was able to conclude that the proposed novel mean-reverting OU process outperforms the classical mean-reverting process in the case of wind power and the electricity load. Full article

Weather derivative markets, particularly Chicago Mercantile Exchange (CME) Heating Degree Day (HDD) and Cooling Degree Day (CDD) futures, face challenges from complex temperature dynamics and spatially heterogeneous co-extremes that standard Gaussian models overlook. Using daily data from 13 major U.S. cities (2014–2024), we first construct a two-stage baseline model to extract standardized residuals isolating stochastic temperature deviations. We then estimate the Extreme Value Index (EVI) of HDD/CDD residuals, finding that the nonlinear degree-day transformation amplifies univariate tail risk, notably for warm-winter HDD events in northern cities. To assess multivariate extremes, we compute Tail Dependence Coefficient (TDC), revealing pronounced, geographically clustered tail dependence among HDD residuals and weaker dependence for CDD. Finally, we compare Gaussian, Student’s t, and Regular Vine Copula (R-Vine) copulas via joint VaR–ES backtesting. The R-Vine copula reproduces HDD portfolio tail risk, whereas elliptical copulas misestimate portfolio losses. These findings highlight the necessity of flexible dependence models, particularly R-Vine, to set margins, allocate capital, and hedge effectively in weather derivative markets. Full article

To address the impacts of source load temporal–spatial uncertainties on distribution network planning considering the global transition towards sustainable energy systems with high-penetration photovoltaic (PV) integration, this paper proposes a source–network–storage coordinated stochastic planning method. A temporal–spatial correlation probability model for PV output and load demand is constructed based on Copula theory. Scenario generation and efficient reduction are achieved through Monte Carlo sampling and K-means clustering, extracting representative daily scenarios that preserve the temporal–spatial characteristics. A coordinated planning model targeting the minimization of comprehensive costs is established to holistically optimize PV deployment, energy storage system (ESS) configuration, and network expansion schemes. Simulations on typical distribution network systems demonstrate that the proposed method, by integrating temporal–spatial correlation modeling and multi-element collaborative decision-making, significantly improves PV accommodation capacity and reduces planning costs while improving the overall economic efficiency of distribution network planning. This study provides a robust technical pathway for developing economically viable and resilient distribution networks capable of integrating large-scale renewable energy, thereby contributing to the decarbonization of the power sector and advancing the goals of sustainable energy development. Full article

Recent advancements in science and technology have resulted in products with enhanced reliability and extended lifespans across the aerospace and related sectors. Traditional statistical models struggle to assess their reliability accurately, prompting increased interest in predicting product lifespans during service. These products, characterized by intricate structures and diverse functionalities, exhibit complex, multistage, multiperformance, and nonlinear degradation processes. To address these challenges, this paper proposes a framework for multiperformance, multi-phase Wiener process modeling and reliability analysis. It introduces a two-phase nonlinear Wiener degradation model and identifies change points via the Schwarz information criterion (SIC). The analytical formula for remaining useful life (RUL) is obtained from the concept of the first hitting time (FHT), which considers the stochastic nature of the degradation amount at the change point. The Akaike information criterion (AIC) is then utilized, and an appropriate copula function is chosen to analyze the correlation between two performance indices, given an established complexity with parameters in the degradation model. A two-step method for estimating these uncertain parameters is presented in this paper. Validation through a turbine engine case study underscores its potential to advance reliability theory and engineering practices. Full article

This paper addresses the challenge of assessing photovoltaic (PV) hosting capacity in distribution networks while accounting for the uncertainty of PV output, a critical step toward achieving sustainable energy transitions. Traditional optimization methods for dealing with uncertainty, including robust optimization (RO) and stochastic optimization (SO), often result in overly conservative or optimistic assessments, hindering the efficient integration of renewable energy. To overcome these limitations, this paper proposes a novel distributionally robust chance-constrained (DRCC) assessment method based on Kullback–Leibler (KL) divergence. First, the time-segment adaptive bandwidth kernel density estimation (KDE) combined with Copula theory is employed to model the conditional probability density of PV forecasting errors, capturing temporal and output-dependent correlations. The KL divergence is then used to construct a fuzzy set for PV output, quantifying its uncertainty within specified confidence levels. Finally, the assessment results are derived by integrating the fuzzy set into the optimization model. Case studies demonstrate its effectiveness of the method. Key findings indicate that higher confidence levels reduce PV hosting capacities due to broader uncertainty ranges, while increased historical sample sizes enhance the accuracy of distribution estimates, thereby increasing assessed capacities. By balancing conservatism and optimism, this method enables safer and more efficient PV integration, directly supporting sustainability goals such as reducing fossil fuel dependence and lowering carbon emissions. The findings provide actionable insights for grid operators to maximize renewable energy utilization while maintaining grid stability, advancing global efforts toward sustainable energy infrastructure. Full article

Separation functions, so-called Tromp functions, are often used to quantitatively analyze the separation behavior in particle processing with respect to individual particle descriptors. However, since the separation behavior of particles is typically influenced by multiple particle descriptors, multivariate Tromp functions are required. This study focuses on methods that allow for the computation of multivariate parametric Tromp functions by means of statistical image analysis and copula-based modeling. The computations are exemplarily performed for the magnetic separation of Li-bearing minerals, including quartz, topaz, zinnwaldite, and muscovite, based on micro-computed tomography images and scanning electron microscopy with energy-dispersive X-ray spectroscopy analysis. In particular, the volume equivalent diameter, zinnwaldite fraction, flatness, and sphericity are examined as possible influencing particle descriptors. Moreover, to compute the Tromp functions, the probability distributions of these descriptors for concentrate and tailing should be used. In this study, 3D image data depicting particles in feed, concentrate, and tailings is available for the computation of Tromp functions. However, concentrate particles tend to be elongated, plate-like, and densely packed, making segmentation for extracting individual particles from image data extremely difficult. Thus, information on the concentrate could not be obtained from the available database. To remedy this, an indirect optimization approach is used to estimate the distribution of particle descriptors of the concentrate. It turned out that this approach can be successfully applied to analyze the influence of size, shape, and composition of particles on their separation behavior. Full article

Risk assessment of deep shale reservoirs is very important for subsurface energy development. However, due to complex geological environments and physicochemical interactions, shale reservoir fabric parameters exhibit variability. Moreover, the actual investigation and testing information is very limited, which is a typical small-sample problem. In this paper, the heterogeneity and statistical characteristics of deep shale reservoirs are clarified by the measured mechanical parameters. A deep learning method for deep shale reservoirs with limited survey data information is proposed. The variability of deep shale reservoirs is characterized by random field theory. A variable stiffness method and stochastic analysis method are developed to evaluate the risk of deep shale reservoirs. The detailed workflow of the stochastic risk assessment framework is presented. The frequency distribution and failure risk of deep shale reservoirs are calculated and analyzed. The risk assessment of deep shale reservoirs under different model parameters is discussed. The results show that a stochastic risk assessment framework of deep shale reservoirs, using a deep learning method and random field theory, is scientifically reasonable. The scatter plots of the elasticity modulus (EM), cohesive force (CF), and Poisson ratio (PR) distribute along the 45-degree line. The different distributed variables of EM, CF, and PR have a positive correlation. The statistical properties of the measurement data and deep learning data are approximately the same. The principal stress of deep shale follows the normal distribution with significance level 0.1. Under positive copula conditions, the maximum failure probability is 5.99%. Under negative copula conditions, the maximum failure probability is 4.58%. Different copula functions under positive and negative copula conditions have different failure probabilities. For the exponential correlation structure, the minimum failure probability is 3.46%, while the maximum failure probability is 6.19%. The mean failure probability of the EM, CF, and PR of deep shale reservoirs is 4.85%. Different random field-related structures and parameters have different influences on the failure risk. Full article

The work presented here marks a further advance in expert uncertainty quantification. In a recent probabilistic evaluation of ice sheet process contributions to sea level rise, tail dependence was elicited and propagated through an uncertainty analysis for the first time. The elicited correlations and tail dependencies concerned pairings of three processes: Accumulation, Discharge and Run-off, which operate on major ice sheets in the West and East Antarctic and in Greenland. The elicitation enumerated dependencies between these processes under selected global temperature change scenarios over different future time horizons. These expert judgments allowed us to populate a Paired Copula Bayesian network model to obtain the estimated contributions of these ice sheets for future sea level rise. Including positive central tendency dependence and tail dependence increases the fatness of the upper tails of projected sea level rise distributions, an amplification important for designing and evaluating possible mitigation strategies. Detailing and jointly computing distributional dependencies and tail dependencies can be crucial components of good practice for assessing the influence of uncertainties on extreme values when modelling stochastic multifactorial processes. Full article

Structural planes are one of the key factors controlling the stability of rock masses. A comprehensive understanding of the spatial distribution characteristics of structural planes is essential for accurately identifying key blocks, analyzing rock mass stability, and addressing various rock engineering challenges. This study compares the effectiveness of four stochastic structural plane generation methods—the Monte Carlo method, the Copula-based method, generative adversarial networks (GAN), and denoised diffusion models (DDPM)—in generating stochastic structural planes and capturing potential correlations between structural plane parameters. The Monte Carlo method employs the mean and variance of three parameters of the measured factual structural planes to generate data that follow the same distributions. The other three methods take the entire set of measured factual structural planes as the overall input to generate structural planes that exhibit the same probability distributions. Five sets of structural planes on four rock slopes in Norway are examined as an example. The validation and analysis were performed using histogram comparison, data feature comparison, scatter plot comparison, and linear regression analysis. The results show that the Monte Carlo method fails to capture the potential correlation between the dip direction and dip angle despite the best fit to the measured factual structural planes. The Copula-based method performs better with smaller datasets, and GAN and DDPM are better at capturing the correlation of measured factual structural planes in the case of large datasets. Therefore, in the case of a limited number of measured structural planes, it is advisable to employ the Copula-based method. In scenarios where the dataset is extensive, the deep generative model is recommended due to its ability to capture complex data structures. The results of this study can be utilized as a valuable point of reference for the accurate generation of stochastic structural planes within rock masses. Full article

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. Full article