Probability Density Evolution Research Articles

A Unified Stochastic Hybrid System Approach to Aggregate Modeling of Responsive Loads

Aggregate load modeling is of fundamental importance for systematic analysis and design of various demand response strategies. Instead of keeping track of the trajectories of individual loads, the aggregate modeling problem focuses on characterizing the density evolution of the load population. Most existing models are only applicable to thermostatically controlled loads (TCLs) with first-order linear dynamics. This paper develops a unified aggregate modeling approach that can be used for general TCLs as well as deferrable loads. We propose a deterministic hybrid system model to describe individual load dynamics under demand response rules, and develop a general stochastic hybrid system (SHS) model to capture the population dynamics. We also derive a set of partial differential equations (PDEs) that govern the probability density evolution of the SHS. Our results cannot be obtained using the existing SHS tools in the literature as the proposed SHS model involves both random and deterministic switchings with general switching surfaces in multidimensional domains. The derived PDE model includes many existing aggregate load modeling results as special cases and can be used in many other realistic modeling scenarios that have not been studied in the literature.

Critical slip surface and landslide volume of a soil slope under random earthquake ground motions

Finding the critical slip surface and estimating the landslide volume are of primary importance for slope seismic design. However, this may be difficult due to the uncertainty of ground motions. To address this problem, a new method for calculating uncertainties is recommended in this paper, especially for the critical slip surface and landslide volume under random earthquake ground motions. Firstly, a series of intensity–frequency nonstationary random earthquake ground motions were generated based on an improved orthogonal expansion method. A given number of potential slip surfaces were set in a soil slope. Subsequently, the factor of safety (FOS) of each slip surface for all ground motions was calculated and the minimum FOS curves were obtained. It was found that the critical slip surfaces and failure times are uncertain under different earthquakes. The Monte Carlo method was used to verify the accuracy of probability density evolution method (PDEM), and the results of the PDEM and the Monte Carlo method are consistent, meaning that the PEDM has higher computational efficiency. Moreover, the distributions of earthquake-triggered landslide volume and landslide depth were analyzed by considering equivalent extreme events. Both landslide volume and depth exhibit a normal distribution for a homogeneous soil slope. The framework of this study is meaningful for slope seismic design in engineering, for example, the location of critical slip surface can be used for slope reinforcement, and the distribution of sliding volume can be used for disaster assessment.

Performance-based seismic fragility analysis of retaining walls based on the probability density evolution method

Seismic fragility analysis is an efficient way to study the seismic behaviour and performance of structures under the excitation of earthquakes of varying intensity, and an essential part of the seismic risk assessment of structures. A recently developed dynamic reliability methodology, the probability density evolution method (PDEM), is proposed for the dynamic reliability and seismic fragility analysis of a retaining wall. The PDEM can obtain an instantaneous probability density function of the seismic responses and easily acquire the seismic reliability of the structural system. An important advantage of the PDEM is its high efficiency relative to that of the Monte Carlo simulation method, which is often used in the reliability and fragility analysis of structures. The present study uses a typical gravity retaining wall to illustrate stochastic seismic responses and fragility curves that can be obtained by the PDEM. The combined uncertainties of the seismic force and soil properties are explicitly and systematically modelled by stochastic ground motions and random variables respectively. The performance of the retaining wall is analysed for different acceptable levels of backfill settlement. Additionally, seismic fragility curves are constructed without assuming the distribution of the seismic response.

Stochastic Analysis of a Nonlinear Business Cycle Model with Correlated Random Income Disturbance

The economic cycle has always been an important feature of the evolution of an economic system. In the presence of many uncertain factors, it appears in the manner of very complex nonlinearity and randomness. Based on the theory of stochastic nonlinear dynamics, a nonlinear economic cycle model with correlated random income disturbance is established. The probability density evolution of the nonlinear economic cycle model under random disturbance is numerically analyzed by using a path integration method. The analysis shows the high saving rate reduces the investment and improves the probabilities of low income and low income change rate. In order to achieve a higher income, the saving rate should be controlled to some reasonable small value. The nonlinearity of the economic cycle model increases the probabilities of high income and high income change rate, which can lead to the increase of income in a probabilistic sense. The increase of random income interference enhances the uncertainty of income. Meanwhile, the increase of correlated random income disturbance can lead to a nonsymmetric distribution of the probability distributions of income and income change rate. In such cases, the income is more difficult to forecast and control.

PROBABILITY DENSITY EVOLUTION FOR TIME-VARYING RELIABILITY ASSESSMENT OF WING STRUCTURES

Reliability evaluation is a key factor in serviceability and safety analysis of air vehicles. Structural health monitoring methods have grown to a degree of maturity in many industries. However, there is a challenging interest to tie in SHM with reliability assessment. In this respect, consideration of stochastic structural dynamics with SHM data and random loadings opens a new chapter in failure prevention. The current study focuses on the stochastic behavior of structures as a way to relate SHM data with reliability. In this respect, uncertain factors such as atmospheric turbulence, structural parameters, and sensor outputs are considered in the process of reliability assessment. Firstly, an experimental evaluation is conducted using a simple cantilevered beam. Subsequently, system identification is weaved in with a probability density evolution equation for calculating the reliability of a wing structural component. Numerical simulations demonstrate that structural reliability of a typical WSC can be effectively evaluated. The proposed scheme paves the way for new SHM research topics such as online life prediction and reliability based failure prevention.

Multi-scale stochastic damage model for concrete and its application to RC shear wall structure

PurposeThis paper aims to present a multi-scale stochastic damage model (SDM) for concrete and apply it to the stochastic response analysis of reinforced concrete shear wall structures.Design/methodology/approachThe proposed SDM is constructed at two scales, i.e. the macro-scale and the micro-scale. The general framework of the SDM is established on the basis of the continuum damage mechanics (CDM) at the macro-scale, whereas the detailed damage evolution is determined through a parallel fiber buddle model at the micro-scale. The parallel buddle model is made up of micro-elements with stochastic fracture strains, and a one-dimensional random field is assumed for the fracture strain distribution. To represent the random field, a random functional method is adopted to quantify the stochastic damage evolution process with only two variables; thus, the numerical efficiency is greatly enhanced. Meanwhile, the probability density evolution method (PDEM) is introduced for the structural stochastic response analysis.FindingsBy combing the SDM and PDEM, the probabilistic analysis of a shear wall structure is performed. The mean value, standard deviation and the probability density function of the shear wall responses, e.g., shear capacity, accumulated energy consumption and damage evolution, are obtained.Originality/valueIt is noted that the proposed method can reflect the influences of randomness from material level to structural level, and is efficient for stochastic response determination of shear wall structures.

Seismic performance evaluation of high CFRD slopes subjected to near-fault ground motions based on generalized probability density evolution method

This research mainly addresses evaluating the seismic reliability and analyzing the random dynamic response of high concrete-faced rockfill dam (CFRD) slopes suffering from the near-fault earthquakes by establishing a novel method of generating the representative acceleration time histories coupled with the recently proposed generalized probability density evolution method (GPDEM). A probability evaluation is performed after generating two groups of near-fault ground motions, pulse-like and non-pulse-like earthquakes, coupling a series of statistical stochastic parameters with the spectral representation-random function method. A 242-m-high CFRD is selected as an example for finite element stochastic dynamic time series analysis, and second-order statistical values (including mean and standard deviation) and probability information of three indices the safety factor, the cumulative time of safety factor ≤ 1.0 (Fs ≤ 1.0) and the maximum cumulative slippage of the dam slopes are determined, respectively. The effective non-stationary near-fault earthquakes will be adopted to determine the stochastic earthquake loading for the dynamic response evaluation of dam slopes. The statistical and probabilistic results demonstrate that the seismic characteristics make a great difference to the seismic response of slope stability as well as the pulse properties. The stochastic near-fault earthquakes combined with the GPDEM can evaluate seismic performance and reliability effectively from the perspective of randomness, and the failure probability can be obtained directly.

Discussion on “Seismic reliability assessment of earth-rockfill dam slopes considering strain-softening of rockfill based on generalized probability density evolution method” by R. Pang et al. [Soil Dyn Earthq Eng 107 (2018) 96–107

This discussion is based on the paper by Pang et al. [1]. In this paper, the authors made important contribution to the understanding of seismic stability of rockfill dams by considering the effects of strain softening of materials and the uncertainty of earthquake excitations. It is well known that granular materials may exhibit significant softening behaviour depending on the magnitudes of confining stresses and shear strains, and such behaviour leads to a progressive propagation of failure zones in the relevant rockfill structure. This discussion presents some comments on the definition of safety factor, the representativeness of stochastic ground motions and the range of shear strains used in the reported dynamic response analyses, implying that some further clarification and / or improvement may be needed.

Robustness‐evaluation of a stochastic dynamic system and the instant equivalent extreme‐value event

AbstractMany theoretical methods for evaluating robustness have been developed in the last decades. Especially probability‐based robustness quantification promises to provide a rational basis for decision‐making. To allow for this type of robustness evaluation, the probability of failure (or the corresponding reliability index) of the structural system in the damaged and the undamaged state has to be determined. In this paper, the Probability Density Evolution Method (PDEM) is used to evaluate the dynamic reliability, taking into account the randomness which comes both from the excitation and from the stochastic system properties. The efficiency of the PDEM for robustness quantification is evaluated through a simple cargo crane example case. In addition, the instant equivalent extreme‐value event, which is quite different from the equivalent extreme‐value event for one variable, is applied to obtain the extreme value of reaction forces for each sample at instant time.

A Stochastic Semi-Physical Model of Seismic Ground Motions in Time Domain

A stochastic function model of seismic ground motions is presented in this paper. It is derived from the consideration of physical mechanisms of seismic ground motions. The model includes the randomness inherent in the seismic source, propagation path and local site. For logical selection of the seismic acceleration records, a cluster analysis method is employed. Statistical distributions of the random parameters associated with the proposed model are identified using the selected data. Superposition method of narrow-band wave groups is then adopted to simulate non-stationary seismic ground motions. In order to verify the feasibility of the proposed model, comparative studies of time histories and response spectra of the simulated seismic accelerations against those of the recorded seismic accelerations are carried out. Their probability density functions, moreover, are readily investigated by virtue of the probability density evolution method.

Response to the discussion on “Seismic reliability assessment of earth-rockfill dam slopes considering strain-softening of rockfill based on generalized probability density evolution method”

Response to the discussion on “Seismic reliability assessment of earth-rockfill dam slopes considering strain-softening of rockfill based on generalized probability density evolution method”

Exact-exchange optimized effective potential and memory effect in time-dependent density functional theory

The memory effect in time-dependent density functional theory (TDDFT) is important in simulating many time-dependent physical processes, and its implementation in real time has been a longstanding challenge, thus limiting most of TDDFT applications to either adiabatic or linear-response regime. In this paper, we conduct the non-adiabatic calculations for a one-dimensional two-electron Helium model in a triplet state using the recently formulated Sturm-Liouville-type time-local equation for the time-dependent optimized effective potential (TDOEP) with the exact exchange functional, and the results agree with the exact time-dependent Schrodinger equation solutions. It is also found that the time-dependent dipole moment and probability density calculated from the TDOEP approach are more accurate than those from the adiabatic time-dependent Krieger-Li-Iafrate (TDKLI) approximation and the adiabatic local spin density approximation. Specifically, the non-adiabatic and memory-dependent terms in the time-local TDOEP equation correctly describe the time-dependent structure of exchange-correlation potential and yield the probability density evolution. These findings should provide important insights toward future studies on memory effects in TDDFT.

Generation of artificial seismic wave and structure-TMD parameter optimization based on the new seismic code considering site soil properties

In this paper, parameters of site soils based on the new seismic code are adopted and the stochastic ground motion generation method based on the modified YX Hu-XY Zhou model is used to generate reasonable and reliable artificial seismic waves, which proved to have obvious statistical characteristics and can meet the requirements of seismic wave selection. The improved genetic algorithm and the probability density evolution method (PDEM) are combined to optimize the response and reliability of a TMD-structure under the obtained artificial wave excitations. Several examples were given to verify the correctness of the proposed methods, as well as several conclusions were drawn.

Seismic response analysis of nonlinear structures with uncertain parameters under stochastic ground motions

An orthogonal expansion of stochastic ground motion model is incorporated into the probability density evolution method (PDEM). In this regard, a new methodology for seismic response analysis of nonlinear structures with uncertain parameters under stochastic ground motions can be formulated. The fundamentals and numerical algorithms of the methodology are introduced. Then, a rotational quasi-symmetrical point strategy (RQ-SPS) is developed for selecting the representative points in the random-variate space, which is of paramount importance to the tradeoff of accuracy and efficiency for the proposed methodology. An eight storey shear frame structure under seismic excitations, which exhibits strong nonlinear mechanical behaviors, is investigated. The effectiveness of the RQ-SPS is first verified, where the results by Monte Carlo simulations are also provided for comparisons. Then, case studies are implemented, in which the randomness involved in both stochastic ground motions and structural parameters are taken into account. The computational results demonstrate that the effect of randomness in structural parameters cannot be ignored compared to that in stochastic ground motions. Some features of the PDF evolutionary process of response are also discussed.

Stochastic seismic response and stability reliability analysis of a vertical retaining wall in front of the pumping station of a nuclear power plant using the probability density evolution method

Vertical retaining walls are widely used to protect the water intake and drainage structures of nuclear power plants (NPPs); in particular, they are built as the inlet in front of the pumping station pool. It is of great significance to investigate the seismic behavior of the vertical retaining walls considering the uncertain nature of earthquakes according to the nuclear safety design requirements. The probability density evolution method (PDEM), which is a new and efficient methodology, is proposed here to study the stochastic seismic response and stability reliability of a vertical retaining wall in front of the pumping station pool of a NPP. Firstly, a set of representative acceleration time histories of non-stationary earthquake ground motions are generated by the spectral representation random function method according to the RG1.60 spectra for the NPP project design. Then, a series of deterministic stochastic seismic response analysis of a 26-m-high vertical retaining wall are performed. Finally, the probability information and seismic reliability of the vertical retaining wall under two seismic levels (SL-1 (operating-basis earthquake) and SL-2 (safety shutdown earthquake)) are obtained based on two physical parameters: the anti-sliding safety factor and anti-overturning safety factor. The results demonstrate that the proposed method of investigating the stochastic responses and seismic reliability of vertical retaining walls can provide more objective indices to evaluate the seismic safety during SL-1 and SL-2 seismic events. Furthermore, the proposed method can effectively investigate the ultimate seismic capacity of vertical retaining walls and other geo-structures in NPPs.

Utility of probability density evolution method for experimental reliability-based active vibration control

The utility of probability density evolution method for reliability-based active vibration control of a cantilevered flexible beam is experimentally investigated. In this respect, an optimal linear quadratic regulator (LQR) is utilized together with an observer to design an online full-state feedback controller. In order to design a well-performing controller and to simulate the controller performance, a system model is obtained via identification techniques. Reliability tests are consequently performed to verify the effectiveness of the presented reliability assessment method as a foundation for reliability-based control. Subsequently, a hybrid metaheuristic optimization scheme of continuous ant colony system is used for simultaneous integrated tuning of the LQR and observer parameters through all processes. Optimization of the controller and the observer is performed with the objective of system structural reliability. An optimized version of classical LQR controller is also developed for making comparisons between the two control methods. Finally, a comparison is made between the mean power consumption of the 2 controllers. It is realized that the reliability-based controller decreases operating costs without compromising system safety.

The instability probability density evolution of the bistable system driven by Gaussian colored noise and white noise

In this paper, we studied the instability probability density evolution of the bistable system driven by Gaussian colored noise and white noise. Firstly, the bistable system is linearized in the initial area by applying Ω expansion theory of the Green function. Next, the time-dependent non-stationary state solutionpx,t of the Fokker–Planck equation for the linearized system is obtained by using eigenvalue and eigenvector theory. Finally, the effect of the time t, the colored noise intensity α, the correlation time τ and the white noise intensity D on p(x,t) are analyzed. Numerical computation results show that: p(x,t) is a monotonic decreasing function of variable x and intensity α. In contrast,p(x,t) is a monotonic increasing function of correlation time τ. Moreover, p(x,t) appears a peak with the increasing of t when x or D are fixed values respectively.

Modified Factor for Segmented Pipes in Chinese Pipe Seismic Design Code Based on Probability Density Evolution Method

Buried pipes are important components of water and gas supply systems. A series of investigations indicate that buried pipes in China suffered serious damages in previous strong earthquakes. Therefore, seismic response analysis and design of buried pipes is an important research topic. In this study, a program is proposed to give the modified factors for segmented pipes in Chinese pipe seismic design code based on the probability Density Evolution Method (PDEM). A stochastic analysis method based on a finite element model of buried pipes and the Probability Density Evolution Method (PDEM) is proposed to determine probability density functions of seismic responses of buried segmented pipes. Then, seismic responses with different guaranteed rates can be derived and adopted for the design of buried pipes in preparation for earthquakes. Stochastic seismic responses of three segmented pipes buried in four sites are calculated using the proposed method. Joint deformations with guaranteed rates of 50%, 80%, 90%, 95%, and 99% are then summarized, and the modified factors for segmented pipes are derived to modify the Chinese code.

Stochastic Fourier spectrum model and probabilistic information analysis for wind speed process

From the view point of physical mechanism, stochastic dynamic excitations, such as earthquake and strong wind, subjecting on structures can be represented by a physical model with elemental random variables. In this article, a physical model for wind speed process is investigated to obtain the probability density function of wind speed process. Firstly, the stochastic Fourier spectrum, consisting of wave-number spectrum and phase spectrum, is introduced. The wave-number spectrum displays the energy distribution over frequency domain. The phase spectrum is viewed as the evolutionary result driven by characteristic velocity of air vortex. After that, the elemental variables are carefully selected in term of physical relation and the measurement data collected at a wind observation station is analyzed to gain the statistics of the elemental random variables. With the help of probability density evolution method (PDEM), the probability density information of wind speed process can be obtained. The comparison with the measurement data validates the effectiveness of the stochastic Fourier spectrum to simulate the wind speed process.

Probabilistic load flow calculation by using probability density evolution method

This paper presents a novel framework methodology based on the probability density evolution method (PDEM) for solving the probabilistic load flow (PLF) problem. By leveraging a constructed visual stochastic process, the joint probability density evolution equation of a system statement and random inputs is derived based on the principle of preservation of probability. The probability density function of the system statement can then be numerically solved by means of a TVD-based difference scheme. The proposed method is validated through case studies in which the active power and reactive power consumptions of buses are assumed to obey a normal distribution and Weibull distribution, respectively. The cumulative probability functions of the voltage magnitudes of buses and active power branches are computed using the PDEM with 100 samples. The mean, standard deviation, skewness, and kurtosis are also examined. The comparison to Monte Carlo of 10,000 simulations demonstrates the accuracy and efficiency of the proposed approach and verifies its suitability to solve the PLF problem.