Functions In Laplace Domain Research Articles

Reaction-diffusion approach to electronic relaxation in solution: Simple derivation for delta function sink models

We propose a very simple method for finding the exact analytical solution for the problem of electronic relaxation in solution, modelled by a particle undergoing diffusive motion under any arbitrary potential in presence of a Dirac delta function sink. The diffusive motion of the molecule is modeled by Smoluchowski equation in one dimension. The exact analytical expression of survival probability and different rate constants have been derived for different potentials. The knowledge of Green’s function in Laplace domain without the sink is required for using our method. Our method is very simple in comparison to the earlier method.

АЛГОРИТМ ПОСТРОЕНИЯ РАЦИОНАЛЬНЫХ АППРОКСИМАЦИЙ ФУНКЦИЙ В ИЗОБРАЖЕНИЯХ ЛАПЛАСА С ПРИМЕНЕНИЕМ К ДИНАМИЧЕСКИМ ЗАДАЧАМ ПОРОУПРУГОСТИ

In this paper, the Vector Fitting method is used for fitting functions in Laplace domain by pole relocating. The used version of the method is described for one function and for vector functions. An algorithm based on the Vector Fitting method for obtaining strictly proper rational approximations of vector functions is presented. Algorithm produces approximations with increasing orders until the specified convergence condition is satisfied. Specific recommendations for selecting algorithm parameters are given. The proposed algorithm allows to adaptively select frequency samples that are used for obtaining rational approximations, without any previously known information about the nature of functions in Laplace domain. Time-domain solutions are obtained by analytical inversion of the rational approximations. The developed algorithm is tested in detail on the problem of wave propagation in one-dimensional poroelastic column with finite length. The model of linear isotropic fully saturated poroelasticity proposed by Biot is employed. For the general case, analytical solutions of the problem are presented in Laplace domain for displacements, stresses, pore pressures and flux. For the limiting value of the permeability of a poroelastic material, the corresponding analytical solutions are also given in time domain. Rational approximations of solutions in Laplace domain obtained with the proposed algorithm are presented in details for two values of the permeability. The convergence of the algorithm is investigated depending on the number of complex frequencies. It was found that with an increase in the order of approximation, the relative error changes similarly for all considered functions. It is shown that when parameters of the algorithm are chosen within the recommended ranges, the solutions in time domain are stable and the amplitude of the oscillations caused by truncation of the high frequency parts of the solutions remains moderate over the entire specified time interval. For the limiting value of the permeability, high accuracy of the obtained approximate results in comparison with analytical solutions in time domain is clearly demonstrated.

Dynamic condensation of non-classically damped structures using the method of Maclaurin expansion of the frequency response function in Laplace domain

As the number of degrees of freedom (DOFs) in structural dynamic problems becomes larger, the analyzing complexity and CPU usage of computers increase drastically. Condensation (or reduction) method is an efficient technique to reduce the size of the full model or the dimension of the structural matrices by eliminating the unimportant DOFs. After the first presentation of condensation method by Guyan in 1965 for undamped structures, which ignores the dynamic effects of the mass term, various forms of dynamic condensation methods were presented to overcome this issue. Moreover, researchers have tried to expand the dynamic condensation method to non-classically damped structures. Dynamic reduction of such systems is far more complicated than undamped systems. The proposed non-iterative method in this paper is introduced as 'Maclaurin Expansion of the frequency response function in Laplace Domain' (MELD) applied for dynamic reduction of non-classically damped structures. The present approach is implemented in four numerical examples of 2D bending-shear-axial frames with various numbers of stories and spans and also a floating raft isolation system. The results of natural frequencies and dynamic responses of models are compared with each other before and after the dynamic reduction. It is shown that the result accuracy has acceptable convergence in both cases. In addition, it is indicated that the result of the proposed method is more accurate than the results of some other existing condensation methods.

Design of an assemble‐type fractional‐order unit circuit and its application in Lorenz system

A new generalised Lorenz three-dimensional integer-order non-linear system is constructed, and integer-order derivatives are simply replaced by fractional-order ones. Transfer functions in Laplace domain have been calculated and compared for a set of fractional orders q i in (0, 1) with decrement of 0.025 for the maximum error Y = 1 dB/2 dB/3 dB. A novel basic fractional-order unit circuit named ‘Assemble Type’ is designed consulting to the four existing unit circuits, which are ‘Chain Type’, ‘Tree Type’, ‘Mixed Type’ and ‘New Type’. Meanwhile, it is compared with the four existing unit circuits in circuit theory and applications. The novel fractional-order unit circuit is applied to the circuit simulation of Lorenz system combined with the other four unit circuits, and circuit simulation shows that they have very resemble chaotic behaviours, actually, the scheme can be applied in any tiny values of q i , which shows that the fractional-order system has better effectiveness, flexibility and universality. Finally, using stability analytical method based on two stability theorems and necessary conditions to exhibit validity and feasibility of this novel-type design.

A semi-analytical model for the flow behavior of naturally fractured formations with multi-scale fracture networks

Summary This paper presents a semi-analytical model for the flow behavior of naturally fractured formations with multi-scale fracture networks. The model dynamically couples an analytical dual-porosity model with a numerical discrete fracture model. The small-scale fractures with the matrix are idealized as a dual-porosity continuum and an analytical flow solution is derived based on source functions in Laplace domain. The large-scale fractures are represented explicitly as the major fluid conduits and the flow is numerically modeled, also in Laplace domain. This approach allows us to include finer details of the fracture network characteristics while keeping the computational work manageable. For example, the large-scale fracture network may have complex geometry and varying conductivity, and the computations can be done at predetermined, discrete times, without any grids in the dual-porosity continuum. The validation of the semi-analytical model is demonstrated in comparison to the solution of ECLIPSE reservoir simulator. The simulation is fast, gridless and enables rapid model setup. On the basis of the model, we provide detailed analysis of the flow behavior of a horizontal production well in fractured reservoir with multi-scale fracture networks. The study has shown that the system may exhibit six flow regimes: large-scale fracture network linear flow, bilinear flow, small-scale fracture network linear flow, pseudosteady-state flow, interporosity flow and pseudoradial flow. During the first four flow periods, the large-scale fracture network behaves as if it only drains in the small-scale fracture network; that is, the effect of the matrix is negligibly small. The characteristics of the bilinear flow and the small-scale fracture network linear flow are predominantly determined by the dimensionless large-scale fracture conductivity. And low dimensionless fracture conductivity will generate large pressure drops in the large-scale fractures surrounding the wellbore. With the increasing of the interporosity flow parameter, flow exchange between the matrix and the small-scale fracture network will be advanced and may mask the pseudosteady-state flow period. The duration of flow exchange increases and the dip caused by the interporosity flow gets deeper with the decreasing of the storability ratio. Finally, an appropriate choice of the pseudosteady or transient dual-porosity model to idealize the small-scale fracture networks with the matrix depends entirely on a better understanding of the geological evidence supporting either model.

Permanent Magnet Synchronous Generator Stability Analysis and Control

In this paper a theoretical approach has been developed to address the stability problem of permanent magnet synchronous generator (PMSG). It is used because of great advantages such as reliability and effectiveness. The proposed technique is obtained through three stages. First stage is to apply linear approximation to the original system. The second stage is to obtain the transfer function in Laplace domain. The last stage is to separate the unstable zero from the original system. Once it's separated a suitable feedback will be designed to treat the instability phenomena. The proposed technique is simulated and tested using Mat Lab program. The results show that the developed approach proof to be a powerful tool for controlling the permanent magnet synchronous generator.

Transient behavior of complex fracture networks

Abstract Interacting with the pre-existing natural fractures, multi-stage hydraulic fracturing along horizontal wells creates complex fracture networks in the reservoirs. The flow behavior of these created networks is significantly different from that of traditional single or multiple planar fractures due to complex interplay of flow caused by the interconnected fractures. Therefore, modelling the flow behavior of such fracture networks is essential if one intends to evaluate well performance or characterize fracture properties. In this paper, we present a semi-analytical model for the transient behavior of complex fracture networks in which hydraulic and natural fractures can interconnect with arbitrary angles. To model the flow behavior more rigorously, the fractures within the network are represented explicitly rather than idealized as dual-porosity media with the matrix around them. This approach allows us to concentrate on the details of network properties, especially for the complex geometry and varying conductivity within the fracture. In the model, the transient responses are obtained from coupling a reservoir flow model and a fracture network flow model dynamically. Source functions in Laplace domain and superposition principle are adopted to model the flow behavior in reservoir analytically and the transient flow equation in the network is solved by finite difference method numerically. Furthermore, Star-Delta transformation commonly used in discrete fracture network simulation (DFN) is adopted to solve the interplay of flow between the interconnected fractures. The validation of the semi-analytical model is demonstrated in comparison to the solution of ECLIPSE reservoir simulator for orthogonal network. The parameter analysis indicates that the planar penetration coefficient of natural fractures and the varying fracture conductivity have a great effect on the transient responses and drainage pattern for orthogonal networks. Finally, a field example from the Barnett Shale shows the capacity and practical use of the semi-analytical model to evaluate well performance and stimulation effectiveness of complex fracture networks in unconventional reservoirs.

Research on fractional order controller based on best rational approximation approach

Introduce the basic concept of best rational approximation and constructing procedure of the best rational approximation function in Laplace domain. By an illustrative example the design method of FOC based on best rational approximation is discussed. The performance comparison of transfer functions between the FOC obtained by best rational approximation method and conventional PID shows that it is effective to apply the method in design of FOC, which can guarantee maximum absolute error of amplitude frequency characteristic of approximation function in a given error tolerance.

A new method for screening the contingencies before dynamic security assessment of a multimachine power system

AbstractThis paper proposes a novel method for identifying the faults that would impact the transient stability of a multimachine power system. In this method each synchronous machine's swing equation is formulated in the form of a closed loop transfer function in Laplace domain (s‐plane). The duration of a fault under consideration, and the changes of maximum electrical power output of the machine during and after the fault are included in the same transfer function. Then the narrowest possible range containing the real part of the ‘dominant’ root of each machine's characteristic equation is identified on the real axis of s‐plane. This is because it is the real part that decides whether the transient response would sustain or subside. If the average of the lower limits of the real parts of dominant roots for all the machines is in the left half but very close to the imaginary axis (poorly damped) or in the right half of the s‐plane (absence of damping), the system as a whole is unstable under the considered fault. The proposed method is extremely fast. An extensive digital simulation test and comparison of the proposed method with a conventional step‐by‐step (SBS) numerical integration method has been done on a 25‐machine practical system belonging to Bangladesh Power Development Board (BPDB), and good agreements observed for an array of fault scenarios. The faults for which the aforesaid average lower limit is in the left half plane but its distance from imaginary axis is marginally above or below an established minimum value, can be further investigated by a detailed time domain simulation (TDS) transient stability analysis technique. Thus the proposed method is useful for real‐time screening of the contingencies (e.g., faults) before being subjected to a full‐scale dynamic security analysis in a power system of any size with any number of machines. Copyright © 2006 John Wiley & Sons, Ltd.