Mohamed El-Beltagy

This work presents an improved modelling approach for wind turbine power curves (WTPCs) using fractional differential equations (FDE). Nine novel FDE-based models are presented for mathematically modelling commercial wind turbine modules’ power–velocity (P-V) characteristics. These models utilize Weibull and Gamma probability density functions to estimate the capacity factor (CF), where accuracy is measured using relative error (RE). Comparative analysis is performed for the WTPC mathematical models with a varying order of differentiation (α) from 0.5 to 1.5, utilizing the manufacturer data for 36 wind turbines with capacities ranging from 150 to 3400 kW. The shortcomings of conventional mathematical models in various meteorological scenarios can be overcome by applying the Riemann–Liouville fractional integral instead of the classical integer-order integrals. By altering the sequence of differentiation and comparing accuracy, the suggested model uses fractional derivatives to increase flexibility. By contrasting the model output with actual data obtained from the wind turbine datasheet and the historical data of a specific location, the models are validated. Their accuracy is assessed using the correlation coefficient (R) and the Mean Absolute Percentage Error (MAPE). The results demonstrate that the exponential model at α=0.9 gives the best accuracy of WTPCs, while the original linear model was the least accurate.

The fractional Brownian motion (FBM) is a common model for long and short-range dependent phenomena that appears in different fields, including physics, biology, and finance. In the current work, a new spectral technique named the fractional Wiener Hermite Expansion (FWHE) is developed to analyze stochastic models with FBM. The technique has a theoretical background in the literature and proof of convergence. A new complete orthogonal Hermite basis set is developed. Calculus derivations and statistical analysis are performed to handle the mixed multi-dimensional fractional and/or integer-order integrals that appear in the analysis. Formulas for the mean and variance are deduced and are found to be based on fractional integrals. Using the developed expansion with the statistical properties of the basis functionals will help to reduce the stochastic model to equivalent deterministic fractional models that can be analyzed numerically or analytically with the well-known techniques. A numerical algorithm is developed to be used in case there is no available analytical solution. The numerical algorithm is compared with the fractional Euler-Maruyama (EM) technique to verify the results. In comparison to sampling based techniques, FWHE provides an efficient analytical or numerical alternative. The applicability of FWHE is demonstrated by solving different examples with additive and multiplicative FBM.

Modelling of physical systems with stochastic variations is mandatory in many applications. In this work, a new stochastic space–time kinetic model for the nuclear reactor is developed. The model is an efficient alternative to existing techniques available in the literature. The main advantage is to avoid square root of the covariance matrix and hence reduces the computational cost. The model is constructed and derived in detail. The mean, statistical properties, and quantification of uncertainties due to noise are obtained. The computational complexity is compared to the existing models to validate the efficiency. The model is tested against several problems and has shown accuracy and efficiency compared with existing models in the literature.

In this paper, a mixed spectral technique is suggested for the analysis of stochastic models with parameters having random variations. The proposed mixed technique considers a Volterra-like expansions for all types of randomness. Particularly, the generalized polynomial chaos (gPC) expansion is used for the random parameters and the Wiener–Hermite functionals (WHF) technique is used for the noise. The statistical properties of the functionals enables to derive a deterministic system used to evaluate the solution statistical moments. The new mixed technique is shown to be efficient compared with the classical techniques and analytical solutions could be obtained in many cases. The suggested technique allows to separate the contributions of the different random sources and hence enables to evaluate variance components which are used to estimate the sensitivity indices. The technique is applied successfully to different models with additive and multiplicative noise and compared with the classical sampling techniques. The stochastic nuclear reactor model with random parameters is analyzed with the new technique.

In this paper, we introduce an estimation of the random Klinkenberg slip coefficient in the apparent permeability model using a chaos decomposition technique. The apparent permeability expression (Klinkenberg model) is used to describe natural gas transport in low-permeability media. In this process, the Klinkenberg factor is considered as a random parameter that depends on two random variables. The mean and variance (or standard deviation) of the two random variables can be estimated from the empirical data available in the literature. Therefore, the variation in the pressure is related directly to the random variation in the Klinkenberg factor. The polynomial chaos expansion is used to decompose the governing equation into a set of coupled deterministic equations that are solved and then used to compute the mean and variance of the solution. The algorithm of how to solve the deterministic coupled system is also presented. For verification, the model and its solution have been compared with the analytical solution of the basic steady-state version of the model. The comparison shows a very good agreement. The effects of a number of important parameters have been presented in graphs and discussed. It was found that the stochastic model works very well with small values of the liquid equivalent permeability, which meets the characteristics of low-permeability reservoirs. Also, the stochastic model works very well with small values of gas viscosity. On the other hand, the porosity seems to be not engaged well with the low-permeability model. The sensitivity of selection of random parameters is also investigated as well as the transient effect.

The point kinetic model is a system of differential equations that enables analysis of reactor dynamics without the need to solve coupled space-time system of partial differential equations (PDEs). The random variations, especially during the startup and shutdown, may become severe and hence should be accounted for in the reactor model. There are two well-known stochastic models for the point reactor that can be used to estimate the mean and variance of the neutron and precursor populations. In this paper, we reintroduce a new stochastic model for the point reactor, which we named the Langevin point kinetic model (LPK). The new LPK model combines the advantages, accuracy, and efficiency of the available models. The derivation of the LPK model is outlined in detail, and many test cases are analyzed to investigate the new model compared with the results in the literature.

The population models allow for a better understanding of the dynamical interactions with the environment and hence can provide a way for understanding the population changes. They are helpful in studying the biological invasions, environmental conservation and many other applications. These models become more complicated when accounting for the stochastic and/or random variations due to different sources. In the current work, a spectral technique is suggested to analyze the stochastic population model with random parameters. The model contains mixed sources of uncertainties, noise and uncertain parameters. The suggested algorithm uses the spectral decompositions for both types of randomness. The spectral techniques have the advantages of high rates of convergence. A deterministic system is derived using the statistical properties of the random bases. The classical analytical and/or numerical techniques can be used to analyze the deterministic system and obtain the solution statistics. The technique presented in the current work is applicable to many complex systems with both stochastic and random parameters. It has the advantage of separating the contributions due to different sources of uncertainty. Hence, the sensitivity index of any uncertain parameter can be evaluated. This is a clear advantage compared with other techniques used in the literature.

Analysis of fluids in porous media is of great importance in many applications. There are many mathematical models that can be used in the analysis. More realistic models should account for the stochastic variations of the model parameters due to the nature of the porous material and/or the properties of the fluid. In this paper, the standard porous media problem with random permeability is considered. Both the deterministic and stochastic problems are analyzed using the finite volume technique. The solution statistics of the stochastic problem are computed using both Polynomial Chaos Expansion (PCE) and the Karhunen-Loeve (KL) decomposition with an exponential correlation function. The results of both techniques are compared with the Monte Carlo sampling to verify the efficiency. Results have shown that PCE with first order polynomials provides higher accuracy for lower (less than 20%) permeability variance. For higher permeability variance, using higher-order PCE considerably improves the accuracy of the solution. The PCE is also combined with KL decomposition and faster convergence is achieved. The KL-PCE combination should carefully choose the number of KL decomposition terms based on the correlation length of the random permeability. The suggested techniques are successfully applied to the quarter-five spot problem.

The stochastic point reactor with random parameters is considered in this work. The hybrid uncertain variations—noise and random parameters—are analyzed with the spectral techniques for the efficiency and high rates of convergence. The proposed hybrid technique enables one to derive an equivalent deterministic system that can be solved to get the mean solution and deviations due to each uncertainty. The contributions of different sources uncertainties can be decomposed and quantified. The deviations in the thermal hydraulics are also computed in the current work. Two model reactors are tested with the proposed technique and the comparisons show the advantages and efficiency compared with the other techniques.

In the current work, the stochastic point reactor model is analyzed using the Wiener-Hermite expansion (WHE). The simplified stochastic point reactor model (Ayyoubzadeh and Vosoughi, 2014), at which no matrix square root, is considered. The stochastic system is reduced to a set of deterministic equations that are solved to get the mean and variance of the neutron and precursor populations. The well-known numerical deterministic techniques are used to get the solution without the need for the time-consuming sampling techniques. Estimations of the neutron and precursor groups fluctuations at the startup are quantified. Many cases are tested and compared with the results in the literature. The current technique is shown to be efficient, accurate and simple compared with the available techniques.