Search results for: Asymptotic stability

Authors: Behrooz Rezaie, Zahra Rahmani Cherati, Mohammad Reza Jahed Motlagh, Mohammad Farrokhi

Abstract:

In this paper, a fuzzy controller is designed for stabilization of the Lorenz chaotic equations. A simple Mamdani inference method is used for this purpose. This method is very simple and applicable for complex chaotic systems and it can be implemented easily. The stability of close loop system is investigated by the Lyapunov stabilization criterion. A Lyapunov function is introduced and the global stability is proven. Finally, the effectiveness of this method is illustrated by simulation results and it is shown that the performance of the system is improved.

Keywords: Chaotic system, Fuzzy control, Lorenz equation.

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Authors: Siamak Feizi, Gunvor Baardvik

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Soil erosion has special consequences for landfills that are more serious than those found at conventional construction sites. Different potential heads between two sides of a landfill and the subsequent movement of water through pores within the soil body could trigger the soil erosion and construction instability. Such condition was encountered in a landfill project in the southern part of Norway. To check the risk of internal erosion due changes in the groundwater level (because of seasonal flooding in the river), a series of numerical simulations by means of Geo-Seep software were conducted. Output of this study provides a total picture of the landfill stability, possibilities of erosions and necessary measures to prevent or reduce the risk for the landfill operator.

Keywords: Erosion, seepage, landfill, stability.

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Authors: A. Chellil, A. Nour, S. Lecheb, H. Mechakra, A. Bouderba, H. Kebir

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In this paper, a coupled damage effect in the instability of a composite rotor is presented, under dynamic loading response in the harmonic analysis condition. The analysis of the stress which operates the rotor is done. Calculations of different energies and the virtual work of the aerodynamic loads from the rotor blade are developed. The use of the composite material for the rotor offers a good stability. Numerical calculations on the model developed prove that the damage effect has a negative effect on the stability of the rotor. The study of the composite rotor in transient system allowed determining the vibratory responses due to various excitations.

Keywords: Rotor, composite, damage, finite element, numerical.

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Authors: Mohammad Saleh Tavazoei, Mohammad Haeri

Abstract:

This paper presents comparison among methods of determination of the characteristic polynomial coefficients. First, the resultant systems from the methods are compared based on frequency criteria such as the closed loop bandwidth, gain and phase margins. Then the step responses of the resultant systems are compared on the basis of the transient behavior criteria including overshoot, rise time, settling time and error (via IAE, ITAE, ISE and ITSE integral indices). Also relative stability of the systems is compared together. Finally the best choices in regards to the above diverse criteria are presented.

Keywords: Characteristic Polynomial, Transient Response, Filters, Stability.

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Authors: S. Panda, N. P. Patidar, R. Singh

Abstract:

Power system stability enhancement by simultaneous tuning of a Power System Stabilizer (PSS) and a Static Var Compensator (SVC)-based controller is thoroughly investigated in this paper. The coordination among the proposed damping stabilizers and the SVC internal voltage regulators has also been taken into consideration. The design problem is formulated as an optimization problem with a time-domain simulation-based objective function and Real-Coded Genetic Algorithm (RCGA) is employed to search for optimal controller parameters. The proposed stabilizers are tested on a weakly connected power system with different disturbances and loading conditions. The nonlinear simulation results are presented to show the effectiveness and robustness of the proposed control schemes over a wide range of loading conditions and disturbances. Further, the proposed design approach is found to be robust and improves stability effectively even under small disturbance and unbalanced fault conditions.

Keywords: Real-Coded Genetic Algorithm (RCGA), Static Var Compensator (SVC), Power System Stabilizer (PSS), Low Frequency Oscillations, Power System Stability.

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Authors: Ahmed Soltan, Ahmed G. Radwan, Ahmed M. Soliman

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This work aims to generalize the integer order Sallen-Key filters into the fractional-order domain. The analysis in the case of two different fractional-order elements introduced where the general transfer function becomes four terms which is unusual in the conventional case. In addition, the effect of the transfer function parameters on the filter poles and hence the stability is introduced and closed forms for the filter critical frequencies are driven. Finally, different examples for the fractional order Sallen-Key filter design are presented with circuit simulations using ADS where a great matching between the numerical and simulation results is obtained.

Keywords: Analog Filter, Low-Pass Filter, Fractance, Sallen-Key, Stability.

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Authors: Elragig Aiman, Dreiwi Hanan, Townley Stuart, Elmabrook Idriss

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In this paper, we reconsider the analysis of the Oregonator model. We highlight an error in this analysis which leads to an incorrect depiction of the parameter region in which diffusion driven instability is possible. We believe that the cause of the oversight is the complexity of stability analyses based on eigenvalues and the dependence on parameters of matrix minors appearing in stability calculations. We regenerate the parameter space where Turing patterns can be seen, and we use the common Lyapunov function (CLF) approach, which is numerically reliable, to further confirm the dependence of the results on diffusion coefficients intensities.

Keywords: Diffusion driven instability, common Lyapunov function (CLF), turing pattern, positive-definite matrix.

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Authors: Amal Ezzidani, Abdelghani Ben Tahar, Mohamed Hanini

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A sequence of finite tandem queue is considered for this study. Each one has a single server, which operates under the egalitarian processor sharing discipline. External customers arrive at each queue according to a renewal input process and having a general service times distribution. Upon completing service, customers leave the current queue and enter to the next. Under mild assumptions, including critical data, we prove the existence and the uniqueness of the fluid solution. For asymptotic behavior, we provide necessary and sufficient conditions for the invariant state and the convergence to this invariant state. In the end, we establish the convergence of a correctly normalized state process to a fluid limit characterized by a system of algebraic and integral equations.

Keywords: Fluid Limit, fluid model, measure valued process, processor sharing, tandem queue.

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Authors: Yongkun Li, Tianwei Zhang, Shufa Bai

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By using the method of coincidence degree theory and constructing suitable Lyapunov functional, several sufficient conditions are established for the existence and global exponential stability of anti-periodic solutions for Cohen-Grossberg shunting inhibitory neural networks with delays. An example is given to illustrate our feasible results.

Keywords: Anti-periodic solution, coincidence degree, global exponential stability, Cohen-Grossberg shunting inhibitory cellular neural networks.

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Authors: Roslinda Nazar, Amin Noor, Khamisah Jafar, Ioan Pop

Abstract:

In this paper, the steady laminar three-dimensional boundary layer flow and heat transfer of a copper (Cu)-water nanofluid in the vicinity of a permeable shrinking flat surface in an otherwise quiescent fluid is studied. The nanofluid mathematical model in which the effect of the nanoparticle volume fraction is taken into account is considered. The governing nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations using a similarity transformation which is then solved numerically using the function bvp4c from Matlab. Dual solutions (upper and lower branch solutions) are found for the similarity boundary layer equations for a certain range of the suction parameter. A stability analysis has been performed to show which branch solutions are stable and physically realizable. The numerical results for the skin friction coefficient and the local Nusselt number as well as the velocity and temperature profiles are obtained, presented and discussed in detail for a range of various governing parameters.

Keywords: Heat Transfer, Nanofluid, Shrinking Surface, Stability Analysis, Three-Dimensional Flow.

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Authors: Mohamed Abdelwahed

Abstract:

The aeration process via injectors is used to combat the lack of oxygen in lakes due to eutrophication. A 3D numerical simulation of the resulting flow using a simplified model is presented. In order to generate the best dynamic in the fluid with respect to the aeration purpose, the optimization of the injectors location is considered. We propose to adapt to this problem the topological sensitivity analysis method which gives the variation of a criterion with respect to the creation of a small hole in the domain. The main idea is to derive the topological sensitivity analysis of the physical model with respect to the insertion of an injector in the fluid flow domain. We propose in this work a topological optimization algorithm based on the studied asymptotic expansion. Finally we present some numerical results, showing the efficiency of our approach

Keywords: Quasi Stokes equations, Numerical simulation, topological optimization, sensitivity analysis.

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Authors: J. Ritonja, R. Brezovnik, M. Petrun, B. Polajžer

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Many modern synchronous generators in power systems are extremely weakly damped. The reasons are cost optimization of the machine building and introduction of the additional control equipment into power systems. Oscillations of the synchronous generators and related stability problems of the power systems are harmful and can lead to failures in operation and to damages. The only useful solution to increase damping of the unwanted oscillations represents the implementation of the power system stabilizers. Power system stabilizers generate the additional control signal which changes synchronous generator field excitation voltage. Modern power system stabilizers are integrated into static excitation systems of the synchronous generators. Available commercial power system stabilizers are based on linear control theory. Due to the nonlinear dynamics of the synchronous generator, current stabilizers do not assure optimal damping of the synchronous generator’s oscillations in the entire operating range. For that reason the use of the robust power system stabilizers which are convenient for the entire operating range is reasonable. There are numerous robust techniques applicable for the power system stabilizers. In this paper the use of sliding mode control for synchronous generator stability improvement is studied. On the basis of the sliding mode theory, the robust power system stabilizer was developed. The main advantages of the sliding mode controller are simple realization of the control algorithm, robustness to parameter variations and elimination of disturbances. The advantage of the proposed sliding mode controller against conventional linear controller was tested for damping of the synchronous generator oscillations in the entire operating range. Obtained results show the improved damping in the entire operating range of the synchronous generator and the increase of the power system stability. The proposed study contributes to the progress in the development of the advanced stabilizer, which will replace conventional linear stabilizers and improve damping of the synchronous generators.

Keywords: Control theory, power system stabilizer, robust control, sliding mode control, stability, synchronous generator.

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Authors: Sh. Javadi, A. Alijani, A.H. Mazinan

Abstract:

This paper reviews the optimization available transmission capability (ATC) of power systems using a device of FACTS named SSSC equipped with energy storage devices. So that, emplacement and improvement of parameters of SSSC will be illustrated. Thus, voltage magnitude constraints of network buses, line transient stability constraints and voltage breakdown constraints are considered. To help the calculations, a comprehensive program in DELPHI is provided, which is able to simulate and trace the parameters of SSSC has been installed on a specific line. Furthermore, the provided program is able to compute ATC, TTC and maximum value of their enhancement after using SSSC.

Keywords: available transmission capability (ATC), total transmission capability (TTC), voltage constraints, stability constraints, FACTS, SSSC.

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Authors: Jinfeng Wang, Yuanhong Bi, Hong Li, Yang Liu, Meng Zhao

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In this paper, a new time discontinuous expanded mixed finite element method is proposed and analyzed for two-order convection-dominated diffusion problem. The proofs of the stability of the proposed scheme and the uniqueness of the discrete solution are given. Moreover, the error estimates of the scalar unknown, its gradient and its flux in the L1( ¯ J,L2( )-norm are obtained.

Keywords: Convection-dominated diffusion equation, expanded mixed method, time discontinuous scheme, stability, error estimates.

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Authors: Mohamad Mahdavi, Mojtaba Mahdavi

Abstract:

This paper tries to represent a new method for computing the reliability of a system which is arranged in series or parallel model. In this method we estimate life distribution function of whole structure using the asymptotic Extreme Value (EV) distribution of Type I, or Gumbel theory. We use EV distribution in minimal mode, for estimate the life distribution function of series structure and maximal mode for parallel system. All parameters also are estimated by Moments method. Reliability function and failure (hazard) rate and p-th percentile point of each function are determined. Other important indexes such as Mean Time to Failure (MTTF), Mean Time to repair (MTTR), for non-repairable and renewal systems in both of series and parallel structure will be computed.

Keywords: Reliability, extreme value, parallel, series, lifedistribution

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Authors: D. Mondal, A. Chakrabarti, A. Sengupta

Abstract:

This paper aims to select the optimal location and setting parameters of TCSC (Thyristor Controlled Series Compensator) controller using Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) to mitigate small signal oscillations in a multimachine power system. Though Power System Stabilizers (PSSs) are prime choice in this issue, installation of FACTS device has been suggested here in order to achieve appreciable damping of system oscillations. However, performance of any FACTS devices highly depends upon its parameters and suitable location in the power network. In this paper PSO as well as GA based techniques are used separately and compared their performances to investigate this problem. The results of small signal stability analysis have been represented employing eigenvalue as well as time domain response in face of two common power system disturbances e.g., varying load and transmission line outage. It has been revealed that the PSO based TCSC controller is more effective than GA based controller even during critical loading condition.

Keywords: Genetic Algorithm, Particle Swarm Optimization, Small Signal Stability, Thyristor Controlled Series Compensator.

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Authors: Young Wook Lee, Sun-Jae Kim, Soo-Yeon Lee, Moon-Kyu Song, Woo-Geun Lee Min-Koo Han

Abstract:

We have fabricated a-IGZO TFT and investigated the stability under positive DC and AC bias stress. The threshold voltage of a-IGZO TFT shifts positively under those biases, and that reduces on-current. For this reason, conventional shift-register circuit employing TFTs which stressed by positive bias will be unstable, may do not work properly. We have designed a new 6-transistor shift-register, which has less transistors than prior circuits. The TFTs of the proposed shift-register are not suffering from positive DC or AC stress, mainly kept unbiased. Despite the compact design, the stable output signal was verified through the SPICE simulation even under RC delay of clock signal.

Keywords: Indium Gallium Zinc Oxide (IGZO), Thin FilmTransistor (TFT), shift-register

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Authors: Negar Riazifar, Nigel G. Stocks

Abstract:

This paper proposes strategies in level crossing (LC) sampling and reconstruction that provide high fidelity signal reconstruction for speech signals; these strategies circumvent the problem of exponentially increasing number of samples as the bit-depth is increased and hence are highly efficient. Specifically, the results indicate that the distribution of the intervals between samples is one of the key factors in the quality of signal reconstruction; including samples with short intervals does not improve the accuracy of the signal reconstruction, whilst samples with large intervals lead to numerical instability. The proposed sampling method, termed reduced conventional level crossing (RCLC) sampling, exploits redundancy between samples to improve the efficiency of the sampling without compromising performance. A reconstruction technique is also proposed that enhances the numerical stability through linear interpolation of samples separated by large intervals. Interpolation is demonstrated to improve the accuracy of the signal reconstruction in addition to the numerical stability. We further demonstrate that the RCLC and interpolation methods can give useful levels of signal recovery even if the average sampling rate is less than the Nyquist rate.

Keywords: Level crossing sampling, numerical stability, speech processing, trigonometric polynomial.

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Authors: S. Panda, S. C. Swain, A. K. Baliarsingh, A. K. Mohanty, C. Ardil

Abstract:

Many real-world optimization problems involve multiple conflicting objectives and the use of evolutionary algorithms to solve the problems has attracted much attention recently. This paper investigates the application of multi-objective optimization technique for the design of a Thyristor Controlled Series Compensator (TCSC)-based controller to enhance the performance of a power system. The design objective is to improve both rotor angle stability and system voltage profile. A Genetic Algorithm (GA) based solution technique is applied to generate a Pareto set of global optimal solutions to the given multi-objective optimisation problem. Further, a fuzzy-based membership value assignment method is employed to choose the best compromise solution from the obtained Pareto solution set. Simulation results are presented to show the effectiveness and robustness of the proposed approach.

Keywords: Multi-objective optimisation, thyristor controlled series compensator, power system stability, genetic algorithm, pareto solution set, fuzzy ranking.

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Authors: Pardeep Singh, Sanjay Sharma, Parvinder S. Sandhu

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This work aims to reduce the read power consumption as well as to enhance the stability of the SRAM cell during the read operation. A new 10-transisor cell is proposed with a new read scheme to minimize the power consumption within the memory core. It has separate read and write ports, thus cell read stability is significantly improved. A 16Kb SRAM macro operating at 1V supply voltage is demonstrated in 65 nm CMOS process. Its read power consumption is reduced to 24% of the conventional design. The new cell also has lower leakage current due to its special bit-line pre-charge scheme. As a result, it is suitable for low-power mobile applications where power supply is restricted by the battery.

Keywords: A 16Kb 10T-SRAM, 4x Read-Power Reduction

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Authors: Ilya Popov, Steven Hulshoff

Abstract:

A numerical investigation of the effects of nanosecond barrier discharge on the stability of a two-dimensional free shear layer is performed. The computations are carried out using a compressible Navier-Stokes algorithm coupled with a thermodynamic model of the discharge. The results show that significant increases in the shear layer-s momentum thickness and Reynolds stresses occur due to actuation. Dependence on both frequency and amplitude of actuation are considered, and a comparison is made of the computed growth rates with those predicted by linear stability theory. Amplitude and frequency ranges for the efficient promotion of shear-layer instabilities are identified.

Keywords: NS-DBD, plasma, actuator, flow control, instability, shear layer

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Authors: Shinji Kajiwara

Abstract:

The basic ability of a vehicle is to “run”, “turn” and “stop”. The safeness and comfort during a drive on various road surfaces and speed depends on the performance of these basic abilities of the vehicle. Stability and maneuverability of a vehicle are vital in automotive engineering. The stability of a vehicle is the ability of the vehicle to revert back to a stable state during a drive when faced with crosswinds and irregular road conditions. Maneuverability of a vehicle is the ability of the vehicle to change direction during a drive swiftly based on the steering of the driver. The stability and maneuverability of a vehicle can also be defined as the driving stability of the vehicle. Since the fossil fueled vehicle is the main type of transportation today, the environmental factor in automotive engineering is also vital. By improving the fuel efficiency of the vehicle, the overall carbon emission will be reduced, thus reducing the effect of global warming and greenhouse gas on the Earth. Another main focus of the automotive engineering is the safety performance of the vehicle, especially with the worrying increase of vehicle collision every day. With better safety performance of a vehicle, every driver will be more confident driving every day. Next, let us focus on the “turn” ability of a vehicle. By improving this particular ability of the vehicle, the cornering limit of the vehicle can be improved, thus increasing the stability and maneuverability factor. In order to improve the cornering limit of the vehicle, a study to find the balance between the steering systems, the stability of the vehicle, higher lateral acceleration and the cornering limit detection must be conducted. The aim of this research is to study and develop a new suspension system that will boost the lateral acceleration of the vehicle and ultimately improving the cornering limit of the vehicle. This research will also study environmental factor and the stability factor of the new suspension system. The double wishbone suspension system is widely used in a four-wheel vehicle, especially for high cornering performance sports car and racing car. The double wishbone designs allow the engineer to carefully control the motion of the wheel by controlling such parameters as camber angle, caster angle, toe pattern, roll center height, scrub radius, scuff, and more. The development of the new suspension system will focus on the ability of the new suspension system to optimize the camber control and to improve the camber limit during a cornering motion. The research will be carried out using the CAE analysis tool. Using this analysis tool we will develop a JSAE Formula Machine equipped with the double wishbone system and also the new suspension system and conduct simulation and conduct studies on the performance of both suspension systems.

Keywords: Automobile, Camber Thrust, Cornering force, Suspension.

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Authors: W.S. Lim, M.V.C. Rao

Abstract:

In this paper, the processing of sonar signals has been carried out using Minimal Resource Allocation Network (MRAN) and a Probabilistic Neural Network (PNN) in differentiation of commonly encountered features in indoor environments. The stability-plasticity behaviors of both networks have been investigated. The experimental result shows that MRAN possesses lower network complexity but experiences higher plasticity than PNN. An enhanced version called parallel MRAN (pMRAN) is proposed to solve this problem and is proven to be stable in prediction and also outperformed the original MRAN.

Keywords: Ultrasonic sensing, target classification, minimalresource allocation network (MRAN), probabilistic neural network(PNN), stability-plasticity dilemma.

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Authors: Li Hongwei

Abstract:

In this paper, a predator-prey model with time delay and habitat complexity is investigated. By analyzing the characteristic equations, the local stability of each feasible equilibria of the system is discussed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By choosing the sum of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as  crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main theoretical results.

Keywords: Predator-prey system, delay, habitat complexity, HOPF bifurcation.

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Authors: Benalia Nadia, Bensiali Nadia, Mekki Mounira

Abstract:

In this paper, we propose an easily computable proximity index for predicting voltage collapse of a load bus using only measured values of the bus voltage and power; Using these measurements a polynomial of fourth order is obtained by using LES estimation algorithms. The sum of the absolute values of the polynomial coefficient gives an idea of the critical bus. We demonstrate the applicability of our proposed method on 6 bus test system. The results obtained verify its applicability, as well as its accuracy and the simplicity. From this indicator, it is allowed to predict the voltage instability or the proximity of a collapse. Results obtained by the PV curve are compared with corresponding values by QV curves and are observed to be in close agreement.

Keywords: least square method, Voltage Collapse, Voltage Stability, PV curve

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Authors: A. Chellil, I. Gahlouz, S. Lecheb, A. Nour, S. Chellil, H. Mechakra, H. Kebir

Abstract:

In this paper, the experimental study for the instability of a separator rotor is presented, under dynamic loading response in the harmonic analysis condition. The global measurement and analysis of vibration on the cement separator RC500 is carried, the points of measurement used are radial dots, vertical, horizontal and oblique. The measures of trends and spectral analysis for reconnaissance of the main anomalies, the main defects in the separator and manifestation, the results prove that the defects effect has a negative effect on the stability of the rotor. Experimentally the study of the rotor in transient system allowed to determine the vibratory responses due to the unbalances and various excitations.

Keywords: Rotor, experimental, defect, frequency, specter.

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Authors: S. S. Dol, M. S. Chan, S. F. Wong, J. S. Lim

Abstract:

Emulsion formation is unavoidable and can be detrimental to an oil field production. The presence of stable emulsions also reduces the quality of crude oil and causes more problems in the downstream refinery operations, such as corrosion and pipeline pressure drop. Hence, it is important to know the effects of emulsions in the pipeline. Light crude oil was used for the continuous phase in the W/O emulsions where the emulsions pass through a flow loop to test the pressure drop across the pipeline. The results obtained shows that pressure drop increases as water cut is increased until it peaks at the phase inversion of the W/O emulsion between 30% to 40% water cut. Emulsions produced by gradual constrictions show a lower stability as compared to sudden constrictions. Lower stability of emulsions in gradual constriction has the higher influence of pressure drop compared to a sudden sharp decrease in diameter in sudden constriction. Generally, sudden constriction experiences pressure drop of 0.013% to 0.067% higher than gradual constriction of the same ratio. Lower constriction ratio cases cause larger pressure drop ranging from 0.061% to 0.241%. Considering the higher profitability in lower emulsion stability and lower pressure drop at the developed flow region of different constrictions, an optimum design of constriction is found to be gradual constriction with a ratio of 0.5.

Keywords: Constriction, pressure drop, turbulence, water cut, water-in-oil emulsions.

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Authors: E. G. Bautista, J. M. Reyes, O. Bautista, J. C. Arcos

Abstract:

In this work, we analyze the deformation of surface waves in shallow flows conditions, propagating in a channel of slowly varying cross-section. Based on a singular perturbation technique, the main purpose is to predict the motion of waves by using a dimensionless formulation of the governing equations, considering that the longitudinal variation of the transversal section obey a power-law distribution. We show that the spatial distribution of the waves in the varying cross-section is a function of a kinematic parameter,κ , and two geometrical parameters εh and w ε . The above spatial behavior of the surface elevation is modeled by an ordinary differential equation. The use of single formulas to model the varying cross sections or transitions considered in this work can be a useful approximation to natural or artificial geometrical configurations.

Keywords: Surface waves, Asymptotic solution, Power law function, Non-dispersive waves.

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Authors: Yaping Ren, Yongkun Li

Abstract:

In this paper, by using the continuation theorem of coincidence degree theory, M-matrix theory and constructing some suitable Lyapunov functions, some sufficient conditions are obtained for the existence and global exponential stability of periodic solutions of recurrent neural networks with distributed delays and impulses on time scales. Without assuming the boundedness of the activation functions gj, hj , these results are less restrictive than those given in the earlier references.

Keywords: Recurrent neural networks, global exponential stability, periodic solutions, distributed delays, impulses, time scales.

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Authors: N.Karpagam, D.Devaraj

Abstract:

Static Var Compensator (SVC) is a shunt type FACTS device which is used in power system primarily for the purpose of voltage and reactive power control. In this paper, a fuzzy logic based supplementary controller for Static Var Compensator (SVC) is developed which is used for damping the rotor angle oscillations and to improve the transient stability of the power system. Generator speed and the electrical power are chosen as input signals for the Fuzzy Logic Controller (FLC). The effectiveness and feasibility of the proposed control is demonstrated with Single Machine Infinite Bus (SMIB) system and multimachine system (WSCC System) which show improvement over the use of a fixed parameter controller.

Keywords: FLC, SVC, Transient stability, SMIB, PIDcontroller.

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