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In engineering practice, some weighted-k-out-of-n: G systems often show multiple performance levels, resulting in multi-failure-mode mechanism. For these complex degradation systems, a novel reliability model with heterogeneous components is proposed on account of the number of working components, the total weight, and the sojourn time in some a stage, where different weights and thresholds are taken into account in different stages, and so do failure modes. After presenting the indispensable definitions of first passage time and the system lifetime, the system reliability indexes are derived by the Copula method to depict the dependence between components. Furthermore, maximum likelihood estimation of the Copula parameters is adopted and a Copula-based expression for component importance in each type is obtained. Finally, the reliability of the overall supercharging subsystem of an advanced fighter air energy system, as an example of weighted-k-out-of-n: G system with multi-failure modes is analyzed by the suggested model, which offers the comparison of independent and dependent cases. Meanwhile, sensitivity analysis is available to point out the effects on the system. [ABSTRACT FROM AUTHOR]

Baseflow is a critical hydrological variable linked to a variety of environmental factors. The construction of a dam may modify baseflow generation and adjust its environmental dependence via the regulation of discharge. However, its effects on the relationships among air temperature, water temperature, and baseflow remain unclear. Here, we use copula functions to explore the relationships among these variables in the Yangtze River and show how they have changed in response to the Three Gorges Dam (TGD). Because of the buffering effects of the baseflow, baseflow is negatively correlated to water temperature, and the relationship between baseflow and water temperature lessens as air temperature rises. Construction of the TGD attenuates the relationship between baseflow and water temperature when air temperature is low, and this attenuation is especially prominent in winter and spring. Our findings will enhance environmental protection in the Yangtze River and improve the ecological operations of the TGD. [ABSTRACT FROM AUTHOR]

In this study, we propose a new formula for spread option pricing with the dependence of two assets described by a copula function. The proposed method's advantage lies in its requirement of solely computing one-dimensional integrals. Any univariate stock price process, admitting an affine characteristic function, can be used in our formula to get an efficient numerical pricing procedure for a spread option. In the numerical analysis we present a comparison with the Monte Carlo simulation method to assess the performance of our approach, assuming that the univariate stock price follows three widely applied models: variance gamma, Heston's stochastic volatility and affine Heston–Nandi GARCH(1,1) models. [ABSTRACT FROM AUTHOR]

Groundwater serves as the primary source of freshwater in arid and semi-arid regions, rendering them resilient against precipitation scarcity. Various aspects of the drought phenomenon necessitate the application of multivariate analysis and composite drought indices. These composite indices afford a more comprehensive understanding of various drought types. However, few studies have considered the integration of meteorological and groundwater drought information. This research delves into determining a threshold for triggering the transition from meteorological drought to groundwater drought in an arid region (Minab plain). It introduces the Joint Deficit Meteo-Groundwater Drought Index, a result of combining two droughts based on copula function. Furthermore, trivariate drought frequency analysis (severity-duration-peak) is conducted based on three scenarios in three clusters of the Minab Plain. Drought risk is then assessed using conditional probability. The results indicate that the northern and central parts of the Minab Plain face a high risk of meteo-groundwater drought. With increasing drought duration and severity, the peak also rises, placing the entire region at risk of widespread drought. A comprehensive understanding of meteo-groundwater drought and an assessment of its characteristics provide insights into the signals of propagation from meteorological drought to groundwater, facilitating the management of groundwater and surface water resources to mitigate the effects of drought. [ABSTRACT FROM AUTHOR]

Quantitatively analyze the uncertainties in turbulence models is of great meaning for research and engineering. In this paper, we develop a Bayesian uncertainty analysis framework based on copulas. The proposed framework is applied to the parametric uncertainty quantification and calibration of the shear-stress transport turbulence model for the flow over a Gaussian bump. A copula function is used to establish the correlation between variables and construct the weight function, which enables greater consideration of Bayesian inference. Prior analysis is conducted based on Sobol indices with the nonintrusive polynomial chaos method, demonstrating the important effects of κ , a 1 , and β ∗ on the flow, both for the wall force coefficients and the Reynolds shear stress. Next, the wall force coefficients and Reynolds shear stress are used as training sets to verify the effectiveness of the method. The results show that modeling the relationship based on copulas is necessary and effective, and that the weight function adaptively balances the influence of physical quantities at different inference locations. The proposed method effectively corrects the computational deviations of the corresponding physical quantities. In addition, the confidence interval for the posterior sample is increased, enhancing the likelihood of obtaining results close to the actual values. • A copula-based Bayesian uncertainty quantification framework with adaptive weight function is proposed. • Gaussian bump is a recently proposed meaningful geometry shape, attracting wide concerns, but has not been studied enough by now. • Detailed and innovative prior and posterior analysis is conducted. • The physical mechanisms of the parameters for separated flow are analyzed. [ABSTRACT FROM AUTHOR]

After a transmission line is covered by ice in winter, ice-shedding and vibration occurs under special meteorological and external dynamic conditions, which leads to intense transmission line shaking. Transmission line ice-shedding and vibration often cause line flashover trips and outages. In January 2018, three 500 kV transmission lines, namely, the 500 kV Guanli line, the 500 kV Dushan line, and the 500 kV Guanqiao line, tripped and cut off due to ice-shedding and vibration in Anhui province, seriously threatening the safe operation of a large power grid. Current studies mainly focus on analyzing the influence factors and characteristics of line ice-shedding and investigating suppression measures, but they only analyze the correlation between each influencing factor and icing or shedding, and do not consider the coupling effects between multiple factors. In this paper, the key influencing factors and the probability distribution of transmission line ice-shedding were analyzed, and a multiple-factor coupling fault probability calculation model of line ice-shedding based on Copula function was proposed. The fault probability was calculated directly by considering multiple influence factors at the same time, which effectively overcame the error caused by multi-factor transformation in fuzzy membership degree and other methods. It provided an important decision-making basis for preventing and controlling transmission line ice-shedding faults. [ABSTRACT FROM AUTHOR]

In a high proportion of wind–photovoltaic–hydro hybrid power systems, fluctuation and dispersion make it difficult to accurately quantify the output characteristics. Therefore, in this study, a cloud model and copula correlation coefficient matrix were constructed for a hybrid power generation system based on the output data. Multiple backward cloud transformation based on the sampling-with-replacement method was proposed to calculate the improved entropy and hyperentropy to analyze the fluctuation range and dispersion degree quantitatively. A similarity index was proposed to evaluate the similarity between wind power, PV power, and hydropower. A suitable copula function was selected, and the Kendall and Spearman coefficients show the correlation relationships of the hybrid systems. The temporal and spatial characteristics of the hybrid systems were analyzed based on the two models. A typical example in Qinghai proved the effectiveness and applicability of the method. The results show that the correlation between photovoltaic power and hydropower is better and that, in summer, hydropower can be used to adjust the output of renewable energy. [ABSTRACT FROM AUTHOR]

Objectives: The traditional method uses a single monitoring model to conduct the early warning of dam displacement behavior. However, the single monitoring model may not reflect the real behavior of the dam due to its low accuracy, thus causing false alarms. Methods: In order to improve the reliability of early warning results, a new multi-model early warning method is proposed. The hydraulic-season-time (HST) model and autoregressive moving average (ARMA) model, which are the most common models in engineering, are taken as the basic model. First, the advantages and disadvantages of the two models in dealing with dam displacement monitoring data are analyzed. Then the one-dimensional residual distribution of the two models is discussed by kernel density estimation. On this basis, the frequency analysis of the joint residuals of HST-ARMA is carried out and it shows the joint residuals is weakly correlated in non-tail, but strongly correlated in tail. Finally, the Copula function is used to fit the joint residuals of HSTARMA, and the joint distribution function is obtained, which realizes the joint early warning of dam displacement behavior with multiple models. Results: It is verified by several measuring points on the wire alignment system of concrete dam. The case study shows that if a single HST model or ARMA model is used for early warning, the false alarm rate can reach 23.17%-27.94% due to the influence of modeling sequence features and model structure features. And if the HST-ARMA method is used, the false alarm rate can be reduced to 0.00%-0.63% due to combine the advantages of different models. Conclusions: The new method can combine the advantages of different models and avoid the disadvantages of the single model, thus effectively reducing the frequency of false alarms. The warning results of HST-ARMA method are more reliable and can more truly reflect the dam displacement behavior. It can provide reference for improving the dam safety management level. [ABSTRACT FROM AUTHOR]

This paper investigated reliability modeling for systems subject to dependent competing risks considering that variation of the failure threshold is not considered in most studies on competing failure reliability. Firstly, the variation of degradation quantity under shocks was analyzed, and the variation of the threshold was considered on this basis. Secondly, the cumulative degradation under the influence of the random shock process was analyzed. The attractive property of Copula functions is symmetry. Then, a linear Wiener process model was applied to model performance degradation failure, and a multi-performance degradation correlated-competition model based on a Copula function was constructed, which considered the correlated competition between multi-performance degradation failures. Lastly, a micromotor system was used to analyze the applicability of the proposed model for bivariate instances, demonstrating the rationality and effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions are presented for comparing the residual lifetimes and inactivity times of dependent random variables at different fixed times, with respect to hazard rate, mean residual life and likelihood ratio orders. The stochastic comparisons of residual lifetimes at different random times are discussed next. Several examples are presented to illustrate the established results. [ABSTRACT FROM AUTHOR]

The lack of effective failure correlation analysis is one main reason for the gap between the reliability models and the actual complex systems with mixed static and dynamic characteristics. Takagi and Sugeno (T-S) dynamic fault tree is one powerful tool to analyze the static and dynamic failure logic relationship but it assumes the failure probability of the event is independent. Therefore, this paper proposes a multi-dimensional T-S dynamic fault tree analysis method involving failure correlation. The method integrates the failure probability distribution function of basic events with multi-factors and the multi-dimensional copula function, and the important measure of this method is also deduced. The reliability model expression for systems with failure correlations, both in series and in parallel, is discussed and verified. Compare the proposed method with the assumption that the probability of a failure event is independent. This method solves the problem of a large error when ignoring the failure correlation between parts and the degree of the correlation between variables can be characterized. The reliability analysis can be conducted on complex systems affected both by multi-factors and failure correlations. The proposed method is applied to the reliability analysis of a hydraulic height adjustment system and the correctness and superiority of the method are verified. [ABSTRACT FROM AUTHOR]

This study introduces methodologies for constructing joint probability distribution functions utilizing the Copula function and neural networks, and evaluates their efficacy in marine and civil engineering projects. Through an analytical comparison of both models using a numerical example, it is revealed that the neural network model exhibits superior adaptability to large sample sizes. This adaptability is attributed to the neural network's ability to learn complex relationships within the data, which is especially beneficial when dealing with large datasets. The neural network model also demonstrates higher accuracy in constructing joint probability distribution functions compared to the Copula function model. In marine and civil engineering, the adaptability and accuracy of neural networks are of paramount importance due to the variable and complex nature of weather patterns. A practical engineering application is presented, wherein a joint probabilistic distribution neural network model of wind velocity and rain intensity is established for the Lanzhou–Xinjiang high-speed railroad in China. This model illustrates the promising application of neural networks in engineering projects where weather factors play a critical role. Subsequent to the construction of the joint probability distribution functions, a feature importance analysis is incorporated to quantify the contribution of different weather parameters such as wind velocity and rain intensity to the joint distribution function. This analysis provides an objective assessment of the relative importance of various weather factors and offers data-driven insights that are essential for engineering applications where weather conditions are a significant consideration. The study concludes by highlighting the potential benefits of neural network models in marine and civil engineering, suggesting areas for future exploration. [ABSTRACT FROM AUTHOR]

Hydrological droughts are closely related to the meteorological ones in the arid and semi-arid areas. It is of great significance to reveal the response relationship between the meteorological and hydrological droughts in drought warning and water resource management. In this study, the changing patterns of drought were determined in the Xiangjiang River Basin of western China using Copula functions. Daily precipitation data was collected from 40 meteorological stations, while the monthly runoff data was from three hydrological stations (Laobutou, Hengyang, and Xiangtan stations). The standardized precipitation index (SPI) and standardized runoff index (SRI) were adopted to describe meteorological and hydrological droughts. The Pearson correlation coefficient was used to determine the response time to drought. The run theory was used to identify, merge, and remove the drought events. The Mann-Kendall trend test and wavelet analysis were employed to reveal the trend and periodicity of drought in different seasons in the upper, middle, and lower reaches. Furthermore, seven widely used distribution functions were selected to fit the duration and severity sequences of meteorological and hydrological drought that were obtained from SPI and SRI. The optimal parameter estimation was obtained in each marginal distribution using the maximum likelihood. There were strong links between meteorological and hydrological droughts using three Archimedean Copula functions (Frank, Gumbel, and Clayton Copula) by coupling the response probability curve in a Bayesian network model. Finally, the model performance was evaluated using the Akaike information criterion (AIC) and ordinary least squares (OLS). The main results demonstrate: 1) The run theory [0.5, 0, −0.5] was achieved to better identify the drought events, with a matching rate of over 90% between meteorological and hydrological droughts. There was a two-month propagation time from the meteorological to hydrological drought in the whole watershed. The SPI and SRI effectively detected the drought events and better matched them with the historical drought statistics from the different periods (in 1963, 1986, and 2003). 2) The frequency of meteorological drought was higher than that of hydrological drought. The duration and severity of hydrological drought were greater than those of meteorological one, where the downstream area experienced the most severe hydrological drought, but the midstream is the most severely affected by meteorological drought. The difference was attributed to the higher evapotranspiration and high level of socio-economic development in the downstream. 3) There was a tendency toward wetness of the historical data in the basin, which was consistent with the remote sensing monitoring of terrestrial water storage anomaly. However, there was a distinct seasonal variation in the inter-annual wetness of the basin. The upper and middle reaches showed a drier trend in the autumn, while the middle reaches were in a trend toward dryness in the summer. Additionally, there were several outstanding oscillation periods of dry and wet, namely 3-5, 6-10, and 18-21 a. 4) The optimal joint distribution of meteorological-hydrological drought severity in the upstream and downstream area was the Gumbel Copula function, while the Frank Copula function was the optimal joint distribution for the rest. 5) The response probability of hydrological drought duration and severity increased with the increase of characteristic variables in the meteorological drought. There was a stable response probability of hydrological drought duration when the duration of meteorological drought exceeded a specific threshold. This study is of great significance in understanding the mechanism of drought in the Xiangjiang River Basin as well as guiding the drought prevention and fighting work in the region. [ABSTRACT FROM AUTHOR]

This paper examines the preservation of several aging classes of lifetime distributions in the formation of coherent and mixed systems with independent and identically distributed (i.i.d.) or identically distributed (i.d.) component lifetimes. The increasing mean inactivity time class and the decreasing mean time to failure class are developed for the lifetime of systems with possibly dependent and i.d. component lifetimes. The increasing likelihood ratio property is also discussed for the lifetime of a coherent system with i.i.d. component lifetimes. We present sufficient conditions satisfied by the signature of a coherent system with i.i.d. components with exponential distribution, under which the decreasing mean remaining lifetime, the increasing mean inactivity time, and the decreasing mean time to failure are all satisfied by the lifetime of the system. Illustrative examples are presented to support the established results. [ABSTRACT FROM AUTHOR]

In this article, we study an integral transformation of the copula function. We prove that under certain conditions, the studied integral transformation is also the copula function. Some properties of the integral transformation of a copula function are studied. Also we derive bounds for Kendall's tau and Spearman's rho. [ABSTRACT FROM AUTHOR]

The Copula function was used to analyze the annual compound events of heat and drought in North China, which can provide reference for agricultural water management and disaster prevention and mitigation in North China. Based on the daily maximum temperature and precipitation data of 36 meteorological stations in North China from 1960 to 2019, the year-by-year heat intensity and drought intensity were identified, the Copula function was introduced to construct a two-dimensional joint cumulative probability distribution function of heat intensity and drought intensity, and the return period of compound events of heat and drought in different grades were analyzed to assess the occurrence characteristics of the compound events. The results showed that when fitting the marginal distributions of annual number of heat days and drought intensity, the GEV function worked best at more stations; the most applied Copula function was the Symmetrised Joe-Clayton function when combining annual number of heat days and drought intensity in two dimensions; compared with high temperature intensity, drought intensity had a greater effect on the magnitude of the joint return period of compound events. North China is more prone to compound events with high heat intensity in the southwest and drought intensity in the south-central part of the country. The leading factors of compound events in North China vary from region to region, and different measures need to be taken to mitigate the damage caused by compound events in different regions. [ABSTRACT FROM AUTHOR]

One of the useful distributions in modeling mortality (or failure) data is the univariate Gompertz–Makeham distribution. To examine the relationship between the two variables, the extended bivariate Gompertz–Makeham distribution is introduced, and its properties are provided. Also, some reliability indices, including aging intensity and stress-strength reliability, are calculated for the proposed model. Here, a new copula function is constructed based on the extended bivariate Gompertz–Makeham distribution. Some of its features including dependency properties, such as dependence structure, some measures of dependence, and tail dependence, are studied. The estimation of the parameters of new copula is presented, and at the end, a simulation study and a performance analysis based on the real data are presented. So, by analyzing the mortality data due to COVID-19, the appropriateness of the proposed model is examined. [ABSTRACT FROM AUTHOR]

The study introduced the transformed copula models and a D-vine structure for three variables. The numerical applications of the models are evaluated using two different sets of real-life data that exhibit nearly no dependence, highly dependent, over, and under dispersed characteristics. We examined only the volatilities of the first data using the exponentiated weighted moving average (EWMA). The parameter estimates of the models were obtained based on the maximum likelihood estimation method for the bivariate copula models and the Dissmann algorithm for sequential top-down estimation for the D-vine structure. The results showed that the introduced copula models outperformed some existing copula models in terms of their fit statistics for both real-life and simulated data sets. In addition, the Gaussian copula model gave a better fit to the D-vine structure than some existing copula models and could be recommended for modeling a D-vine structure comprising of variables that are positively weak correlated and highly correlated. [ABSTRACT FROM AUTHOR]

In this paper, a functionality unit named as meta-action unit (MAU) is proposed to correlate the system function with the part actions and assess reliability of mechanical system. Firstly, the function of system is decomposed into multiple MAUs by function-movement-action (FMA). Then, the lifetime of MAU is fitted by Weibull distribution, and its parameters are estimated by support vector regression (SVR). In addition, taking the distributions of MAU as marginal distributions, the lifetime distribution of mechanical system is constructed by copula function to assess system reliability, and its parameters are estimated using the maximum likelihood estimator (MLE). Further, the reliability assessment accuracy based on MAU is compared with that from traditional method based on mechanical part failure modes. Finally, the reliability assessment of the indexing turntable (IT) is performed as an example to illustrate the feasibility and reasonability of the proposed method. [ABSTRACT FROM AUTHOR]

The probabilistic dependence in a multi-variate random field consists of two parts: the spatial dependence of a random quantity at different positions and the probabilistic dependence between different random variables at the same position. The classical model cannot capture the possible non-Gaussian dependence or nonlinear dependence between different random variables. To this end, in this paper, an approach by synthesizing the spatial correlation function and the multi-variate copula function (SCFVCF) is proposed. In this model, the correlation function model is adopted to quantify the spatial dependence involved in the random field of the prime variable, and the copula function model is adopted to quantify the dependence configuration between subordinate variables. The properties of such multi-variate random fields are then studied. To generate samples of such multi-variate random fields, the spectral representation method is incorporated with the conditional sampling method. As an example, to illustrate the application of SCFVCF, the random field of the constitutive parameters of concrete is adopted. The results demonstrate that the proposed method can capture the spatial dependence of compressive strength, and at the same time the dependence configuration between different parameters is consistent with the test complete compressive stress-strain curves. [ABSTRACT FROM AUTHOR]

In medicals sciences and reliability engineering, the failure of individuals or units (I/Us) occurs due to independent causes of failure. In general, the symmetry between dependent and independent causes of failure is essential to the nature of the problem at hand. In this study, we considered the accelerated dependent competing risks model when the lifetime of I/Us was modeled using a generalized half-logistic distribution. The data were obtained with respect to constant stress accelerated life tests (ALTs) with a type-II progressive censoring scheme. The dependence structure was formulated using the copula approach (symmetric Archimedean copula). The model parameters were estimated with the maximum likelihood method; only two dependent causes of failure and bivariate Pareto copula functions were proposed. The approximate confidence intervals were constructed using both the asymptotic normality distribution of MLEs and bootstrap techniques. Additionally, an estimator of the reliability of the system under a normal stress level was constructed. The results from the estimation methods were tested by performing a Monte Carlo simulation study. Finally, an analysis of data sets from two stress levels was performed for illustrative purposes. [ABSTRACT FROM AUTHOR]

The seismic fragility of water supply networks is an important research component for lifeline engineering in earthquake-prone regions. An accurate computational model was established to rapidly and efficiently assess pipeline damage under seismic excitation. In this study, the fragility of the pipeline network is deduced based on Copula function technology and the series-parallel model. This solves the low accuracy problem caused by simplifying elbow, tee, and other nodes and ignores the correlation between connected components in the fragility calculation of the buried pipeline network. Then, the shaking table tests of the elbow pipe, tee pipe, and scale pipeline network are performed, and the prototype pipeline network's finite element comparative calculation model is also established. Then, the optimal Copula function for the pipeline network, the Gumbel Copula function, is obtained by calculating the correlation between each component's seismic demand. It was found that the seismic demand of pipe and elbow components in the sand is higher than that in the clay, while the opposite is valid for the tee component. After that, a two-parameter log-normal distribution is employed to obtain the seismic fragility of three types of components in different soil properties. It was also found that the maximum failure probability can be obtained from the elbow leakage. The fragility of the elbow pipe is greater than that of the tee pipe. The fragility of components is then applied to the fragility calculation of the scale pipeline network by a series-parallel model, and the obtained results are compared with numerical simulations. It is demonstrated that the series-parallel combination calculation-based results have a high safety reserve for the pipeline design. At present, urban municipal pipeline networks face many challenges in earthquake-prone regions, such as old pipeline infrastructure and poor seismic performance, which make their nodes and push-on joints vulnerable to damage. Based on the above social needs, this paper focuses on the accurate calculation theory of the seismic fragility of the buried water supply network. A more reasonable calculation method of the reliability of the pipeline network is established by considering the influence of the fragility of connected pipe components. This method employs the failure probability of various components to accurately calculate the fragility of the large and complex pipeline network. This study can provide basic theories for the seismic fragility of pipeline networks, the calculation of seismic connectivity, and the construction of resilient pipeline networks. In addition, the conclusions proposed in the study, such as the fragility comparison of tees and elbows and the design concept of "strong nodes", can guide the design of pipeline network nodes, which has great practical significance and social benefits. [ABSTRACT FROM AUTHOR]

To provide the theoretical basis for the engineering application of metakaolin as aluminum-rich pozzolanic ash materials, the promoting effect of metakaolin on the macro- and micro-mechanical properties of composite cement-based composite materials was explored under high-temperature steam and standard curing conditions. Analysis techniques, which involved thermogravimetric and nanoindentation coupled with scanning electron microscopy–energy dispersive X-ray spectroscopy, were used. To analyze the experimental data, the ACI empirical formula and the copula function were used. The correlation among the hydration degree of the MK-cement cementation system, the Al/Si of the C-(A)-S-H phase, and the nanomechanical properties of the C-(A)-S-H gel phase was investigated. According to prior research, the macroscopic mechanical properties and the substitution rate of the MK-cement composite cementitious system can be improved under high-temperature curing conditions in a short period, in which the optimum substitution rate of metakaolin is 20%. The ACI empirical formula was used to demonstrate the functional relationship between the metakaolin replacement rate, curing time, and compressive strength. The MK-cement cementation system can eliminate the defect phase, reduce the CH phase content, and then increase the C-(A)-S-H gel phase content and bulk density. The micro-mechanical properties of the C-(A)-S-H gel phase rises due to its phase content and Al/Si ratio. Furthermore, the copula function verifies the dependence of the nanomechanical properties of C-(A)-S-H gel and Al/Si. [ABSTRACT FROM AUTHOR]

Late frost is one of the most destructive meteorological disasters in the Loess Plateau of China. A great threat can be posed to the sustainable production of apples, leading to great economic losses in the apple industry. Thus, it is a high demand to explore the occurrence of late frost events for the prevention of apple late frost disasters in the decision-making of the local apple industry. In this study, the late frost return periods were investigated on the duration and severity of late frost events using the Coupla functions. The reliability of the model was then verified to analyze the characteristics of apple late frost. The study area was taken as the apple-producing areas of Shaanxi Province in western China. The meteorological datasets were collected from the seven weather stations from 1971-2018. The daily minimum temperature (Tmin) of 0℃ was taken as the critical temperature for the occurrence of apple late frost events, in order to extract the two characteristic variables of duration and severity of late frost events. These characteristic variables of late frost events were first fitted by seven common distribution functions, respectively. Kolmogorov Smirnov (K-S) test was then carried out to verify the model. The joint distributions of late frost characteristic variables were constructed to evaluate the goodness-of-fit using six Copula functions. The occurrence probability and return period of late frost events were analyzed with the optimized Copula functions. The results showed that the severity of late frost risks generally increased from the southeast to northwest in the study area from 1971 to 2018. The optimal marginal distribution of late frost duration was in the log-normal distribution, while there was a great difference in the optimal marginal distributions of late frost severity. A significant positive correlation was found between the duration variables and the severity of late frost at each station. The goodness-of-fit test demonstrated that most of the stations presented the optimal distribution of Normal Copula function for late frost events. The joint cumulative probability increased significantly, as the severity and duration of late frost increased, but the increasing trend was much slower than before. A much more significant increase was observed in the co-occurrence return period under the same increase of univariate value, compared with the joint. The univariate return period was always between the joint and co-occurrence return period. Once the univariate return period was small enough, the optimal range of the actual univariate return period can be estimated, according to the joint and the co-occurrence return period. In general, a low probability was found in the late frost events with long duration and high severity at the weather stations in apple-producing areas in Shaanxi Province of China. However, the stations in the Yan’an area were more susceptible to the late frost events with high severity or long duration, as well as the late frost events with both high severity and long duration. Thus, more attention can be paid to late frost risks in the Yan’an area. This finding can provide a theoretical base to deal with the late frost disaster in apple production. [ABSTRACT FROM AUTHOR]

Human activity has been one of the most important factors inducing regional ecological risks in recent years. But, there are different visualization and response degree of ecological risks and human activities in different land scales. It is a high demand to scientifically analyze the correlation characteristics of regional Landscape Ecological Risk (LER) and Human Activity Intensity (HAI), in order to coordinate the territorial system of human-environment interaction for better regional high-quality development. The ever-increasing conflict can be found between ecological protection and economic and social development, due mainly to rapid urbanization. The Yangtze River Delta (YRD) is one of the typical regions with intensive human activities, remarkable economic development, and outstanding ecological status. Taking the YRD as the subject, a three-level scale of city, county, and grid was established to construct the LER and HAI assessment models, in order to characterize the spatial and temporal response at different scales. The data was also collected from the land use, nighttime lighting, and population spatial distribution in the four periods from 1990 to 2020. The Copula function, bivariate spatial autocorrelation, and coupled coordination degree models were used to reveal the spatial and temporal correlation between the LER and HAI. The results show that (1) the spatial distribution pattern of LER was high in the north of the study area, and low in the south under different scales. The high risk areas continued to decrease during 1990-2020, whereas, the low risk areas showed an increasing trend. There was the most outstanding reduction in the high risk areas at the city scale, with a decrease of 19.51 percentage points over 30 years. By contrast, there was the most significant increase in the low-medium risk and low risk areas at the county scale, indicating an increase of 16.29 percentage points over 30 years. (2) The spatial distribution pattern of HAI was high in the northeast of the study area during 1990-2020, while low in the southwest under different scales. All the regions with high HAI showed a significant increasing trend. There was a significant decreasing trend in the regions with low HAI. The high intensity region presented the largest increase at 13.42 percentage points on the grid scale, whereas, the low intensity region was the most significant decrease at 9.76 percentage points on the city scale. (3) The correlation between HAI and LER shared a positive correlation between 1990 and 2020, but the correlation between them was weakening, indicating the weak influence of regional HAI on LER. By contrast, the coupling and coordination between them showed an increasing trend from 0.3031 in 1990 to 0.3112 in 2020. It infers that the relationship between them was gradually shifting from conflict to coordination. Spatially, there was a continuous decreasing trend in the H-H clustering area. Furthermore, the L-L clustering area showed an increasing and then decreasing trend with an overall decrease. Both L-H and H-L increased significantly after 2010. The spatial correlation characteristics of regional LER and HAI were combined to manage and control the regional ecological environment. The finding can provide a theoretical basis for land management and landscape planning, together with the spatial guidelines for the regional ecological risk prevention and ecological restoration. [ABSTRACT FROM AUTHOR]

In multivariate longitudinal studies, several outcomes are repeatedly measured for each subject over time. The data structure of these studies creates two types of associations which should take into account by the model: association of outcomes at a given time point and association among repeated measurements over time for a specific outcome. In our approach, because of some advantageous arisen from features like flexibility of marginal distributions, a copula-based approach is used for joint modeling of multivariate outcomes at each time points, also a transition model is used for considering the association of longitudinal measurements over time. For the problem of incomplete data, missingness mechanism is assumed to be ignorable. Some simulation results are reported in different scenarios using the Gaussian, t and several commonly used copulas of the family of Archimedean copulas. Akaike information criterion (AIC) is used to select the best copula function. The proposed approach is also used for analyzing a real obesity data set. [ABSTRACT FROM AUTHOR]

Climate change impacts many aspects of life and requires innovative thinking on various issues. The electricity sector is affected in several ways, including changes in the production components and consumption patterns. One of the most important issues for Independent System Operators, a state-controlled organization responsible for ensuring the reliability, availability, and quality of electricity delivery in the country, is the response to climate change. This is reflected in the appropriate design of production units to cope with the increase in demand due to extreme heat and cold events and the development of models aimed at predicting the probability of such events. In our work, we address this challenge by proposing a novel probability model for peak demand as a function of wet temperature (henceforth simply temperature), which is a weighting of temperature and humidity. We study the relationship between peak demand and temperature using a new Archimedean copula family, shown to be effective for this purpose. This family, the Clayton generalized Gamma, is a multi-parameter copula function that comprises several members. Two new measures of fit, an economic measure and a conditional coverage measure, were introduced to select the most appropriate family member based on the empirical data of daily peak demand and minimum temperature in the winter. The Clayton Gamma copula showed the lowest cost measure and the best conditional coverage and was, therefore, proven to be the most appropriate member of the family. [ABSTRACT FROM AUTHOR]

This article introduces a flexible modelling strategy to extend the familiar mixed-effects models for analysing longitudinal responses in the multivariate setting. By initiating a flexible multivariate multimodal distribution, this strategy relaxes the imposed normality assumption of related random-effects. We use copulas to construct a multimodal form of elliptical distributions. It can deal with the multimodality of responses and the non-linearity of dependence structure. Moreover, the proposed model can flexibly accommodate clustered subject-effects for multiple longitudinal measurements. It is much useful when several subpopulations exist but cannot be directly identifiable. Since the implied marginal distribution is not in the closed form, to approximate the associated likelihood functions, we suggest a computational methodology based on the Gauss-Hermite quadrature that consequently enables us to implement standard optimization techniques. We conduct a simulation study to highlight the main properties of the theoretical part and make a comparison with regular mixture distributions. Results confirm that the new strategy deserves to receive attention in practice. We illustrate the usefulness of our model by the analysis of a real-life dataset taken from a low back pain study. [ABSTRACT FROM AUTHOR]

In power system planning, the growth of renewable energy generation leads to several challenges including system reliability due to its intermittency and uncertainty. To quantify the relatively reliable capacity of this generation, capacity credit is usually adopted for long-term power system planning. This paper proposes an evaluation of the capacity credit of renewable energy generation using stochastic models for resource availability. Six renewable energy generation types including wind, solar PV, small hydro, biomass, biogas, and waste were considered. The proposed models are based on the stochastic process using the Wiener process and other probability distribution functions to explain the randomness of the intermittency. Moreover, for solar PV—the generation of which depends on two key random variables, namely irradiance and temperature—a copula function is used to model their joint probabilistic behavior. These proposed models are used to simulate power outputs of renewable energy generations and then determine the capacity credit which is defined as the capacity of conventional generation that can maintain a similar level of system reliability. The proposed method is tested with Thailand's power system and the results show that the capacity credit depends on the time of day and the size of installed capacity of the considered renewable energy generation. [ABSTRACT FROM AUTHOR]

Many highly reliable products (systems) usually have multiple dependent degradation processes because of their complex structure. Therefore, it is important to investigate the multivariate degradation models along with the reliability assessment in the stochastic modeling. This article proposes a framework for bivariate degradation modeling based on hybrid stochastic processes for products with two performance characteristics (PCs), the dependence of which is captured by copula functions. Considering heterogeneities among product units, different random effects are introduced in marginal stochastic processes. Then two classes of reliability functions based on joint posterior distributions are derived. In addition, a simulation study indicates that the holistic Bayesian parameter estimation method is better than the two-step Bayesian parameter estimation method. Finally, this article concludes with a case application to demonstrate the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

The traditional DGA (Dissolved Gas Analysis) diagnosis method does not consider the dependence between fault characteristic gases and uses the relationship between gas ratio coding and fault type to make the decision. As a tool of the dependence mechanism between variables, a copula function can effectively analyze the correlation between variables when it cannot determine whether the linear correlation coefficient can correctly measure the correlation between variable relationships. In this paper, the edge variable of a copula function is selected from the fault characteristic gas of a transformer, and the distribution type of the edge variable is fitted at the same time. Then, Bayesian estimation with the Gaussian residual likelihood function is used to fit the parameters of a copula function and a copula function is selected to describe the optimal dependence of the fault characteristic gas of transformer. The relationship between a copula function and the state of transformer is studied. The results show that the copula function boundary with hydrocarbon gas as edge variable can divide the transformer as healthy or defective state. When the cumulative distribution probability (CDF) value of the dissolved gas in the oil in the copula function is close to 0.8, the fluctuation of its gas concentration leads to a sharp change in the probability. Therefore, the analysis of dissolved gas in oil based on a copula function can be used as a powerful technical solution for oil-immersed power transformer fault diagnosis. [ABSTRACT FROM AUTHOR]

Accurate prediction of wind speed plays an important role in increasing the power generation of wind turbines and realizing efficient use of wind energy. However, due to the large number of wind turbines in the wind farm and the complex wake effects between the units, the coupling degree and spatial correlation of the wind speed of the wind turbines are increased. Accordingly, this paper proposes a wind speed prediction model based on spatio-temporal dependency analysis. The proposed model first uses long short-term memory(LSTM) neural network to predict the wind speed of each wind turbine to obtain its residuals, which can extract the time correlation of the wind-speed series; Then by applying the Clayton Copula function to analyze the correlation between the residual series and wind-speed series to get the joint-distribution function. The joint-distribution function can be used to calculate the prediction error of the wind speed and complete the wind speed prediction. The validity of the method in this work is verified using the measured wind-speed data of a wind farm. Experimental results show that the method effectively solves the problem of finding spatio-temporal dependency of wind speed and significantly improves the prediction accuracy of wind speed. [ABSTRACT FROM AUTHOR]

An important aspect of the multidimensional perception of poverty phenomenon is the dependence among the underlying indicators. However, the commonly applied approaches to multidimensional poverty assessment do not capture this interdependence. In this paper we propose a new multidimensional poverty index accounting for the dependence and innovate over the weighting approach. The weighting method proposed here incorporates the copula-based rank dependence among well-being dimensions and contains necessary normative parameters. In particular, the latter includes the elasticity of substitution among dimensions and the belief-adjusting parameter, which specifies the direction of relation between the dependence and the weights. The results of poverty evaluation in the selected European countries suggest that multidimensional poverty is driven not only by the individual shortfalls, but also by the degree of interdependence among well-being indicators. Moreover, multidimensional poverty is relatively higher, if dimensional weights are in direct proportion to the dependence compared to the cases of inverse relation and equal weighting. Considering the novel weighting approach, this paper contributes to the literature on composite indicators by suggesting a channel to enclose the dependence structure in the multidimensional poverty index. [ABSTRACT FROM AUTHOR]

Rockfill strength parameters are important indicators of the stability of a rockfill dam slope. Therefore, a rational representation of the distribution characteristics of these parameters is extremely important to accurately assess the stability of the dynamic response and reliability of the three-dimensional (3D) slope of a rockfill dam. A joint distribution model of the rockfill strength parameters is constructed in this study using the copula function. In addition, a dynamic reliability analysis method for slope stability is proposed using a number-theoretical method and the generalized probability density evolution method (GPDEM). First, the method is validated by comparing the results for a numerical example with those obtained by Monte Carlo simulation (MCS). Second, the traditional and proposed joint distribution models are comparatively analyzed in terms of the dynamic response and reliability. Finally, the effect of the type of copula function on the dynamic reliability of the slope is investigated. The results show that the copula theory, the number-theoretical method, and the GPDEM provide an effective approach to analyze the dynamic response and reliability of slope stability. The randomness of rockfill strength parameters is sensitive to the safety factor of the slope stability of rockfill dams. The reliability analysis method can reasonably characterize the influence of strength parameter uncertainty on dam slope stability, which makes up for the deficiency of using a single safety measurement standard in the deterministic analysis method. The copula function can reasonably characterize the nonnormal distribution characteristics of rockfill strength parameters, improve the calculation accuracy of the slope stability reliability of rockfill dams, and provide an important scientific basis for the accurate safety assessment of dam slope stability. The type of copula function has an important impact on the dynamic reliability results for slope stability. Thus, it is extremely important to build a joint distribution model of rockfill strength parameters based on a rational selection of the optimal copula function. [ABSTRACT FROM AUTHOR]

The instability of seabed slope sediments is the main factor influencing the safety of marine resource development. Therefore, to ensure the safe operation of submarine pipelines under complex and uncertain seabed rock and soil conditions, a reliability model was developed to elucidate the trend of impact-related pipeline damage due to submarine slides. Then, a risk assessment of the damage process of submarine slides impacting pipelines was conducted, which is of great significance for the in-depth safety assessment of pipelines impacted by submarine slides. Based on the copula function, a joint probability distribution model considering the correlation among risk variables was established for rational correlation characterization. A probability analysis method of impact-related pipeline damage attributed to submarine slides based on the copula function was proposed. The Monte Carlo simulation (MCS) method was employed to simulate the random uncertainty in limited observation values and accurately determine the reliability of safe pipeline operation under the action of submarine slides. The conclusions were as follows: (1) Based on the copula function, a joint probability distribution model of risk variables with any marginal distribution function and related structure could be developed. (2) The copula function could reasonably characterize relevant nonnormal distribution characteristics of risk variables and could simulate samples conforming to the distribution pattern of the risk variables. (3) The failure probability calculated with the traditional independent normal distribution model was very low, which could result in a notable overestimation of the reliability of submarine pipelines. [ABSTRACT FROM AUTHOR]

In order to accurately calculate the reliability of mechanical components and systems with multiple correlated failure modes and to reduce the computational complexity of these calculations, the Copula function is used to represent related structures among failure modes. Based on a correlation analysis of the failure modes of parts of a system, a life distribution model of components is constructed using the Copula function. The type of Copula model was initially selected using a binary frequency histogram of the life empirical distribution between the two components. The unknown parameters in the Copula model were estimated using the maximum likelihood estimation method and the most suitable Copula model was determined by calculating the square Euclidean distance. The reliability of series, parallel, and series–parallel systems was analyzed based on the Copula function, where life was used as a variable to measure the correlation between components. Thus, a reliability model of a system with life correlations was established. Reliability calculation of a particular diesel crank and connecting rod mechanism was taken as a practical example to illustrate the feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

There are a variety of degradation models in the literature, each with a certain effect of random variation around the mean degradation path on the time-to-failure of the device being degraded. To assess the dependence that the random variation around the mean degradation path exerts on the resulting time-to-failure, this paper presents copula functions for time-to-failure-based degradation models with respect to two well-known degradation models, namely, the multiplicative degradation model and the additive degradation model. The implied copula functions for the case of the multiplicative degradation model have explicit forms. The implied copula functions are proved to be symmetric in the case of deterministic effect of degradation on failure, but the copulas obtained when failure is affected uncertainly by degradation are asymmetric. Necessary and sufficient conditions for the implicit copula functions to be symmetric are given. [ABSTRACT FROM AUTHOR]

This study is conducted to decompose inequality in four important micronutrient intakes, i.e. calcium, iodine, iron, and zinc due to socioeconomic factors during 2006–2016. For this purpose, data were taken from national-level surveys, i.e. HIES-2006 and HIICS-2016. We applied a recently proposed decomposition method based on copula function to decompose between year changes in the consumption of micronutrients into structure and composition effect. The results show that average calcium and iron intakes increased by 518.54 (mg) and 0.962 (mg), respectively, while average iodine and zinc intakes decreased by 2.009 (mg) and 3.411 (mg), respectively, during the decade. Estimates of structure effect show that calcium, iodine, and iron consumption increased on average, i.e. 525.316 (mg), 14.615 (mg), and 2.15 (mg), respectively, while zinc intake decreased by 2.735 (mg). The composition effect is negative for all the four micronutrients, implying that consumption of calcium, iodine, iron, and zinc decreased as 6.7766 (mg), 16.624 (mg), 1.189 (mg), and 0.677 (mg), respectively. The main factors of this change are household income, urbanization, provinces, and household size. The policymakers need to know the importance of socioeconomic factors to improve hidden hunger in terms of micronutrient intakes. [ABSTRACT FROM AUTHOR]

Quantifying the uncertainty of non-stationary flood frequency analysis is very crucial and beneficial for planning and design of water engineering projects, which is fundamentally challenging especially in the presence of high climate variability and reservoir regulation. This study proposed an integrated approach that combined the Generalized Additive Model for Location, Scale and Shape parameters (GAMLSS) method, the Copula function and the Bayesian Uncertainty Processor (BUP) technique to make reliable probabilistic interval estimations of design floods. The reliability and applicability of the proposed approach were assessed by flood datasets collected from two hydrological monitoring stations located in the Hanjiang River of China. The precipitation and the reservoir index were selected as the explanatory variables for modeling the time-varying parameters of marginal and joint distributions using long-term (1954–2018) observed datasets. First, the GAMLSS method was employed to model and fit the time-varying characteristics of parameters in marginal and joint distributions. Second, the Copula function was employed to execute the point estimations of non-stationary design floods. Finally, the BUP technique was employed to perform the interval estimations of design floods based on the point estimations obtained from the Copula function. The results demonstrated that the proposed approach can provide reliable probabilistic interval estimations of design floods meanwhile reducing the uncertainty of non-stationary flood frequency analysis. Consequently, the integrated approach is a promising way to offer an indication on how design values can be estimated in a high-dimensional problem. [ABSTRACT FROM AUTHOR]

The relationship between reliability and dependence of components within each element is discussed in complex systems. In the present text, a complex system is composed of n elements and each of its elements has two dependent components. In this setting, dependency among components is a crucial issue, since the reliability and mean residual life (MRL) functions of such systems are affected by its component dependence. In this article, copula functions are used to describe dependency among components. Some stochastic orderings are made between two complex systems with different degrees of dependence among the components. Some examples and illustrative graphs are presented, as well. [ABSTRACT FROM AUTHOR]

This paper presents a copula-based cloud analysis method for seismic fragility assessment with practical applications for nuclear power plant structures. Copula theory enables the flexible and efficient modeling of joint probability density function (PDF) for critical parameters, specifically the intensity measures (IM) and the engineering demand parameters (EDP). It decomposes the joint PDF into the product of marginal PDFs and copula density function, which are then separately fitted to the data. The study showcases the versatility of parameterized joint distribution models in estimating optimal distribution parameters derived from structural seismic analysis data. By utilizing copula functions, the method calculates conditional EDP distributions given IM, facilitating the construction of seismic fragility curves. The research compares various copula models' suitability for fitting fragility curves, employing methods like Akaike information criterion (AIC) for model selection. The study provides a comprehensive framework for seismic fragility assessment, from random sampling of structural parameters and seismic data to nonlinear finite element modeling and copula-based parametric cloud analysis. The application to prestressed concrete containment vessel (PCCV) structures exemplifies the method's adaptability. This innovative approach not only highlights the flexibility of copula-based analysis in assessing structural performance under different dynamic scenarios but also offers invaluable insights for performance-based earthquake engineering (PBEE) and safety assessments. • A novel copula-based cloud analysis approach is proposed, offering a flexible and precise framework for seismic fragility assessment in engineering structures. • The method's adaptability is demonstrated through its successful application to nuclear power plant structures, showcasing its potential across diverse engineering scenarios. • Various copula models are compared rigorously, providing valuable insights into the most effective copula parameterization for fitting seismic fragility curves. [ABSTRACT FROM AUTHOR]

A novel method for analyzing the reliability of structures under non-stationary stochastic seismic excitations, considering the combined effect of multiple structural demand extreme values, is proposed. The spectral representation method is employed to establish a non-stationary stochastic seismic excitation model, and based on the theory of first-passage probability, multiple integral formulas for seismic reliability under multidimensional limit states are derived. The extreme value distribution is established using the Gamma mixture model (GMM). The equations for estimating the model parameters are derived based on both fractional moments and moment-generating functions, while the determination of the number of gamma distribution components is guided by the probability distribution and statistical characteristics of the samples. The joint probability density function (JPDF) for multiple demand extreme values is established by incorporating copula functions to account for correlation, and the fitting accuracy of different copula functions is assessed. The proposed method is illustrated using reinforced concrete (RC) frame structures. The results demonstrate that the fitting accuracy of extreme value distribution can be enhanced by adjusting the number of gamma distribution components in the GMM, which exhibits high accuracy in fitting both the main and tail regions. The presence of correlation can induce variations in the JPDF, thereby exerting an influence on the failure probability. Consequently, disregarding correlation is not conducive to reliability analysis. [ABSTRACT FROM AUTHOR]

In hydropower engineering, monitoring the instability of rock slopes is a crucial undertaking. Current early warning models of rock slopes lack consideration of the overall stability and cannot reflect the stability differences of different structure forms. Addressing these issues, we have successfully developed an integrative early warning indicator for the overall stability of rock slopes, utilizing the copula function and the safety stability rate. Initially, we introduce the safety stability rate to quantify the impact of rock structure on stability. Subsequently, the ISSR-MDF model, which integrates the safety stability rate with the marginal distribution function, is proposed. On this basis, we established the early warning indicator using the copula function. The results show that the indicator can reflect the structural characteristics of rock slopes and the trend of changes in multi-point residuals. The early warning model can provide valuable references for the stability assessment of slopes in hydropower engineering. • Develop an early warning indicator for rock slopes in engineering geology. • Introduce the safety stability rate to quantify the impact of rock structure on stability. • Construct the ISSR-MDF model to alter the influence of each measurement point. • Establish a multidimensional residual joint distribution function using copula theory. • Case study demonstrates indicator's efficacy in overall stability assessment. [ABSTRACT FROM AUTHOR]

• The contribution of TGD and climate change on the drought propagation in the YRB was assessed. • TGD's regulation had effectively raised the drought propagation threshold and prolonged the drought propagation time. • TGD's regulation had consequently weakened the drought propagation process and led to a decrease in HD characteristics. The propagation process from meteorological drought (MD) to hydrological drought (HD) is crucial for drought monitoring and mitigation. While it's widely recognized that the Three Gorges Dam (TGD) and climate change have substantially altered the hydrological conditions in the Yangtze River Basin (YRB), the effects of these changes on the drought propagation process have remained unclear. Using natural runoff, observational runoff and precipitation data during 1988–2018, and employing the Run theory, Copula function, and Bayesian network model, we quantified the impacts of the TGD's regulation and climate variability on drought propagation in the YRB. The results showed that the TGD's regulation raised the drought propagation threshold (DPTH) and prolonged the drought propagation time (DPT) in the upper, middle, and lower reaches of the Yangtze River (URYR, MRYR, and LRYR). It consequently weakened the drought propagation process and led to a decrease in HD characteristics (duration, severity and frequency) in the YRB by −20.57 % ∼ −0.96 %, except for the severity and frequency in the LRYR. However, the effects of climate variability on DPTH and DPT were relatively complex. It led to a decrease in HD characteristics by −27.46 % ∼ −10.84 % in the URYR and MRYR, excluding the frequency in the MRYR. Conversely, it resulted in an increase of 9.87 % ∼ 42.37 % in HD characteristics in the LRYR. This study effectively underscored the crucial role of the TGD in mitigating HD events in the YRB and contributed valuable insights into understanding how human activities and climate variability interacted in shaping drought propagation process. [ABSTRACT FROM AUTHOR]

This paper proposes a distribution network fault emergency power supply recovery strategy based on 5G base station energy storage. This strategy introduces Theil's entropy and modified Gini coefficient to quantify the impact of power supply reliability in different regions on base station backup time, thereby establishing a more accurate base station's backup energy storage capacity model that can fully utilize the base station's energy storage resources to participate in the emergency power supply of distribution network faults. First, the optimal Copula function in each cycle is determined through the Akaike information criterion and squared Euclidean distance to establish a wind power-photovoltaic output scenario. Secondly, the traditional Gini coefficient is modified using the weighted average node degree and influence rate indicators. A comprehensive vulnerability model is established through the modified Gini coefficient and Theil's entropy indicators. Based on the comprehensive vulnerability model, a backup energy storage time model and a modified backup energy storage capacity model of the base station affected by power supply reliability are established. Thus, the callable energy storage capacity of base stations in different areas is obtained. Finally, a two-stage robust optimization model is introduced to minimize system operating costs to solve the volatility of 5G base station communications and wind-solar output, thereby establishing an emergency power supply recovery model for base station energy storage and wind-solar output. Simulated with the improved IEEE-33 node model, the results show that the proposed base station's energy storage model improves the utilization of the base station energy storage resources and, at the same time, effectively reduces the loss of load during the fault phase of the distribution network and improves the absorption of the PV output. • Modeling of 5G base station backup energy storage. Aiming at the shortcomings of existing studies that ignore the time-varying characteristics of base station's energy storage backup, based on the traditional base station energy storage capacity model in the paper [ 18 ], this paper establishes a distribution network vulnerability index to quantify the power supply reliability of the distribution network nodes by adopting the voltage Theil's entropy and the modified Gini coefficient. Based on the power supply reliability of power grid nodes and combined with load level weights, a model for the backup energy storage time of base stations affected by power supply reliability is established. • The model of 5G base station energy storage participating in power supply recovery In view of the impact of changes in communication volume on the emergency power supply output of base station energy storage in distribution network fault areas, this paper introduces a two-stage robust optimization model to deal with the uncertainty of communication volume. Based on the base station energy storage capacity model established in contribution (1), an objective function is established to minimize the system operating cost in the fault area, and the base station energy storage owned by mobile operators is used as an emergency power source to participate in power supply restoration. Establish sub-objective functions of the loss cost of base station energy storage charging and discharging, the subsidy cost of base station energy storage charging and discharging, and the base station energy storage operating cost to supplement the shortcomings of existing research. [ABSTRACT FROM AUTHOR]

Both Reservoir Inflow (RI) and Irrigation Water Demand (IWD) depend mainly on the basin-wide climate. Hence, various unpredictable (stochastic) weather and climate have induced significant variation in the water supply and demand processes in recent years. Meanwhile, the spatial similarity of climate conditions was also required to consider the relevancy between water supply and demand. More importantly, the smaller the basin is, the more considerable the relevancy is. However, the traditional one-dimensional stochastic dynamic programming cannot concern the relevancy between the multiple variables of water resources. In this study, a novel multi-dimensional stochastic dynamic programming was developed to deal with the relevancy and stochasticity in the streamflow and demand for the reservoir operations. Firstly, the one-dimensional distribution was selected to characterize the stochasticity of individual RI and IWD processes. Secondly, a Copula function of several variables was constructed using the marginal distribution. Specifically, the relevancy involved two aspects, i.e., the RI or IWD between the adjacent time intervals, and between RI and IWD at the same time. The stochasticity was referred to the individual RI, IWD, as well as the two-dimensional RI and IWD processes. Finally, the Copula function was integrated into the stochastic dynamic programming for the optimal dispatching model of the reservoir. A case study was set as an annual regulating reservoir responsible for agricultural irrigation. The results indicate that: 1) The Generalized Extreme Value (GEV) distribution was performed better on the RI and IWD in months. 2) There was remarkable relevancy of RI or IWD between a few adjacent time intervals. Moreover, there was a strong significance in April, May, June, July, September, and October. 3) Various types of Copula function were selected to model the dependent structure of RI/IWD variables at the adjacent time intervals. A Frank Copula was also employed to describe the negative relationship of dependent structure between RI and IWD at the same time. 4) The reservoir operation model considering the relevancy between RI and IWD was performed better than not. Specifically, the case study showed that the total water shortage was 10.37 × 108 m³, when considering the relevancy and stochasticity of multiple variables, which was less than before (10.49×108 m³). Nevertheless, there was a little difference between the optimal total water shortages by the two multi-dimensional stochastic dynamic programmings. This trend was attributed to a little difference of state transition probability between with and without considering the relevancy of RI and IWD. Consequently, the newly-developed multi-dimensional stochastic dynamic programming can be widely expected to extend for the multi-dimensional stochastic optimal scheduling in fields, such as the multi-energy supplementary model considering the hydropower, wind, and light energy. Moreover, a better fitting can be suitable for the current relevancy and stochasticity in multi-dimensional processes, where the wind speed and solar radiation are stochastic variables in this case. [ABSTRACT FROM AUTHOR]

Flash floods cause great harm to people's lives and property safety. Rainfall is one of the main causes of flash floods in small watersheds. The uncertainty of rainfall events results in inconsistency between the traditional single rainfall pattern and the actual rainfall process, which poses a great challenge for the early warning and forecasting of flash floods. To carry out the effective flash flood early warning and forecasting, this paper proposes a novel rainfall pattern by coupling total rainfall and peak rainfall intensity based on copula functions, i.e., the rainfall pattern of risk probability combination (RPRPC). On this basis, the Hydrologic Engineering Center-Hydrologic Modeling System (HEC-HMS) hydrological model is used to simulate the rainfall-runoff process, the trial algorithm is used to calculate the critical rainfall (CR), and the optimistic-general-pessimistic (O–G-P) early warning mode considering the decision maker's risk preference is proposed. The small watershed of Xinxian in Henan province, China, is taken as a case study for calculation. The results show that the RPRPC is feasible and closer to the actual rainfall process than the traditional rainfall pattern, Frank copula function is the best for determining the joint distribution function of total rainfall and peak rainfall intensity, and the HEC-HMS model can be applied to small watersheds in hilly areas. Additionally, both RPRPC and antecedent soil moisture condition (ASMC) have influence on CR, and the variation of RPRPC will change the influence of ASMC on CR. Finally, the effectiveness of O–G-P early warning mode is verified. [ABSTRACT FROM AUTHOR]

The variation and uncertainty of renewable energy potentially impact power system stability. This article proposes an efficient uncertainty evaluation algorithm to evaluate the influence of renewable energy uncertainty on power system dynamic performance. This algorithm applies probabilistic collocation method (PCM), which greatly reduces the simulation burden without compromising the result accuracy, to approximate the dynamic simulation results. The proposed algorithm also considers the correlation among adjacent renewable generators by using the Copula function, which is suitable to model nonlinear correlations. Additionally, this article extends the utilization of PCM. The actual historical data of renewable generation productions can be utilized directly, and kernel density estimation (KDE) is used to capture the nonparametric distribution of renewable generation production. To improve the accuracy of the approximation results, the proposed method adopts K-means clustering technique to select the approximation samples of input variables. The proposed probabilistic dynamic simulation algorithm is compared with Monte Carlo simulations (MCS) with the probabilistic results on the IEEE 39-bus system with multiple renewable generators, and the accuracy and efficiency of the proposed algorithm are validated. [ABSTRACT FROM AUTHOR]

• Uncertainty of heterogeneity of composites is characterized. • The polynomial chaos is extended to describe the random field of composite parameters with spatial distributed uncertainty. • The correlation between elastic parameters is represented by using the Copula function. Considering the heterogeneity and randomness in the elastic performance of composite materials, a novel method is proposed for vibration analysis of composites with spatially distributed uncertain parameters. Firstly, the polynomial chaos expansion is used to characterizing the material field, which is expressed as the product of the deterministic coefficient and the basis function of polynomial chaos, the coefficient is expanded by using orthogonal polynomial to represent the spatially distributed heterogeneous of composites. Secondly, the correlation of elastic parameters is expressed by using Copula function. Finally, according to the non-intrusive method, the probabilistic characteristics of the composite structure with spatially distributed uncertain parameters are obtained. The method is implemented for free vibration analysis of a composite panel, results show that the variability of the standard deviation of the dynamics performance is dependent on the correlation between the elastic parameters of composites. [ABSTRACT FROM AUTHOR]

The NPMLE of a distribution function from doubly truncated data was introduced in the seminal paper of Efron and Petrosian (J Am Stat Assoc 94:824–834, 1999). The consistency of the NPMLE depends however on the assumption of independent truncation. In this work we introduce an extension of the Efron–Petrosian NPMLE when the variable of interest and the truncation variables may be dependent. The proposed estimator is constructed on the basis of a copula function which represents the dependence structure between the variable of interest and the truncation variables. Two different iterative algorithms to compute the estimator in practice are introduced, and their performance is explored through an intensive Monte Carlo simulation study. We illustrate the use of the estimators on two real data examples. [ABSTRACT FROM AUTHOR]