Search results for: Power System oscillations.

Authors: R.M. Kainkwa

Abstract:

The African Great Lakes Region refers to the zone around lakes Victoria, Tanganyika, Albert, Edward, Kivu, and Malawi. The main source of electricity in this region is hydropower whose systems are generally characterized by relatively weak, isolated power schemes, poor maintenance and technical deficiencies with limited electricity infrastructures. Most of the hydro sources are rain fed, and as such there is normally a deficiency of water during the dry seasons and extended droughts. In such calamities fossil fuels sources, in particular petroleum products and natural gas, are normally used to rescue the situation but apart from them being nonrenewable, they also release huge amount of green house gases to our environment which in turn accelerates the global warming that has at present reached an amazing stage. Wind power is ample, renewable, widely distributed, clean, and free energy source that does not consume or pollute water. Wind generated electricity is one of the most practical and commercially viable option for grid quality and utility scale electricity production. However, the main shortcoming associated with electric wind power generation is fluctuation in its output both in space and time. Before making a decision to establish a wind park at a site, the wind speed features there should therefore be known thoroughly as well as local demand or transmission capacity. The main objective of this paper is to utilise monthly average wind speed data collected from one prospective site within the African Great Lakes Region to demonstrate that the available wind power there is high enough to generate electricity. The mean monthly values were calculated from records gathered on hourly basis for a period of 5 years (2001 to 2005) from a site in Tanzania. The documentations that were collected at a height of 2 m were projected to a height of 50 m which is the standard hub height of wind turbines. The overall monthly average wind speed was found to be 12.11 m/s whereas June to November was established to be the windy season as the wind speed during the session is above the overall monthly wind speed. The available wind power density corresponding to the overall mean monthly wind speed was evaluated to be 1072 W/m2, a potential that is worthwhile harvesting for the purpose of electric generation.

Keywords: Hydro power, windy season, available wind powerdensity.

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Authors: Sharareh Mirsaeidi Farahani , Gholamreza Chitsaz

Abstract:

Management Systems are powerful tools for businesses to manage quality , environmental and occupational health and safety requirements . where once these systems were considered as stand alone control mechanisms , industry is now opting to increase the efficiency of these documented systems through a more integrated approach . System integration offers a significant step forward, where there are similarities between system components , reducing duplication and adminstration costs and increasing efficiency . At first , this paper reviews integrated management system structure and its benefits. The second part of this paper focuses on the one example implementation of such a system at Imam Khomeini Hospital and in final part of the paper will be discuss outcomes of that proccess .

Keywords: environmental management , Integratedmanagement systems, occupational healt and safetymanagement , quality management.

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Authors: F. Rahnamay Roodposhti, M. Falah Shams, H. Kordlouie

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The aim of the article is extending and developing econometrics and network structure based methods which are able to distinguish price manipulation in Tehran stock exchange. The principal goal of the present study is to offer model for approximating price manipulation in Tehran stock exchange. In order to do so by applying separation method a sample consisting of 397 companies accepted at Tehran stock exchange were selected and information related to their price and volume of trades during years 2001 until 2009 were collected and then through performing runs test, skewness test and duration correlative test the selected companies were divided into 2 sets of manipulated and non manipulated companies. In the next stage by investigating cumulative return process and volume of trades in manipulated companies, the date of starting price manipulation was specified and in this way the logit model, artificial neural network, multiple discriminant analysis and by using information related to size of company, clarity of information, ratio of P/E and liquidity of stock one year prior price manipulation; a model for forecasting price manipulation of stocks of companies present in Tehran stock exchange were designed. At the end the power of forecasting models were studied by using data of test set. Whereas the power of forecasting logit model for test set was 92.1%, for artificial neural network was 94.1% and multi audit analysis model was 90.2%; therefore all of the 3 aforesaid models has high power to forecast price manipulation and there is no considerable difference among forecasting power of these 3 models.

Keywords: Price Manipulation, Liquidity, Size of Company, Floating Stock, Information Clarity

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Authors: A. El Ougli, I. Lagrat, I. Boumhidi

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In this paper we propose a robust adaptive fuzzy controller for a class of nonlinear system with unknown dynamic. The method is based on type-2 fuzzy logic system to approximate unknown non-linear function. The design of the on-line adaptive scheme of the proposed controller is based on Lyapunov technique. Simulation results are given to illustrate the effectiveness of the proposed approach.

Keywords: Fuzzy set type-2, Adaptive fuzzy control, Nonlinear system.

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Authors: Yong-Jun Kim, Jae-Hyun Ro, Chang-Bin Ha, Hyoung-Kyu Song

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Because current wireless communication requires high reliability in a limited bandwidth environment, this paper proposes the variable modulation scheme based on the codebook. The variable modulation scheme adjusts transmission power using the codebook in accordance with channel state. Also, if the codebook is composed of many bits, the reliability is more improved by the proposed scheme. The simulation results show that the performance of proposed scheme has better reliability than the the performance of conventional scheme.

Keywords: MIMO-OFDM, variable modulation, codebook, channel state.

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Authors: Wenyu Song, Bingxi Li, Zhongbin Fu, Baocheng Jiang

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Full - Scale Accelerated Loading System, one part of “the Eleventh - Five - Year National Grand Technology Infrastructure Program" is a facility to evaluate the performance and service life of different kinds of pavements subjected to traffic loading under full - controlled environment. While simulating the environments of frigid zone and permafrost zone, the accurate control of air temperature, road temperature and roadbed temperature are the key points and also aporias for the designment. In this paper, numerical simulations are used to determine the design parameters of the frozen soil simulation system. At first, a brief introduction of the Full - Scale Accelerate Loading System was given. Then, the temperature control method of frozen soil simulation system was proposed. Finally, by using finite element simulations, the optimal design of frozen soil simulation system was obtained. This proposed design, which was obtained by finite element simulations, provided significant referents to the ultimate design of the environment simulation system.

Keywords: China, finite element simulation, frozen soilsimulation system, preliminary design.

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Authors: Jing Hang Choo, Soon Nyean Cheong, Yee Lien Lee, Sze Hou Teh

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The advancement of smartphones, wireless networking and Near Field Communication (NFC) technology have opened up a new approach to indoor navigation. Although NFC technology has been used to support electronic commerce, access control, and ticketing, there is a lack of research work on building NFC-based indoor navigation system for smartphone users. This paper presents an indoor interactive navigation system (named I2Navi) based on NFC technology for users to navigate within a building with ease using their smartphones. The I2Navi system has been implemented at the Faculty of Engineering (FOE), Multimedia University (MMU) to enable students, parents, visitors who own NFC-enabled Android smartphones to navigate themselves within the faculty. An evaluation is carried out and the results show positive response to the proposed indoor navigation system using NFC and smartphone technologies.

Keywords: Near Field Communication, indoor navigation system, smartphones.

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Authors: Muhammad Marsudi, Dzuraidah Abdul Wahab, Che Hassan Che Haron

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Modeling of a manufacturing system enables one to identify the effects of key design parameters on the system performance and as a result to make correct decision. This paper proposes a manufacturing system modeling approach using a spreadsheet model based on queuing network theory, in which a static capacity planning model and stochastic queuing model are integrated. The model was used to improve the existing system utilization in relation to product design. The model incorporates few parameters such as utilization, cycle time, throughput, and batch size. The study also showed that the validity of developed model is good enough to apply and the maximum value of relative error is 10%, far below the limit value 32%. Therefore, the model developed in this study is a valuable alternative model in evaluating a manufacturing system

Keywords: Manufacturing system, product design, spreadsheet model, utilization.

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Authors: Akram Khaleghei Ghosheh Balagh, Viliam Makis

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In this paper, we investigate the residual life prediction problem for a partially observable system subject to two failure modes, namely a catastrophic failure and a failure due to the system degradation. The system is subject to condition monitoring and the degradation process is described by a hidden Markov model with unknown parameters. The parameter estimation procedure based on an EM algorithm is developed and the formulas for the conditional reliability function and the mean residual life are derived, illustrated by a numerical example.

Keywords: Partially observable system, hidden Markov model, competing risks, residual life prediction.

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Authors: M. Zareiee, A. Dideban

Abstract:

This paper addresses the problem of forbidden states in non safe Petri Nets. In the system, for preventing it from entering the forbidden states, some linear constraints can be assigned to them. Then these constraints can be enforced on the system using control places. But when the number of constraints in the system is large, a large number of control places must be added to the model of system. This concept complicates the model of system. There are some methods for reducing the number of constraints in safe Petri Nets. But there is no a systematic method for non safe Petri Nets. In this paper we propose a method for reducing the number of constraints in non safe Petri Nets which is based on solving an integer linear programming problem.

Keywords: discrete event system, Supervisory control, Petri Net, Constraint

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Authors: Parkpoom Ch., Narongkorn D.

Abstract:

This paper presents the system identification by physical-s law method and designs the controller for the Azimuth Angle Control of the Platform of the Multi-Launcher Rocket System (MLRS) by Root Locus technique. The plant mathematical model was approximated using MATLAB for simulation and analyze the system. The controller proposes the implementation of PID Controller using Programmable Logic Control (PLC) for control the plant. PID Controllers are widely applicable in industrial sectors and can be set up easily and operate optimally for enhanced productivity, improved quality and reduce maintenance requirement. The results from simulation and experiments show that the proposed a PID Controller to control the elevation angle that has superior control performance by the setting time less than 12 sec, the rise time less than 1.6 sec., and zero steady state. Furthermore, the system has a high over shoot that will be continue development.

Keywords: Azimuth angle control, PID Controller, The platform of Multi-Launcher Rocket System.

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Authors: Behrouz Fathi-Vajargah, Sara Ghasemalipour

Abstract:

This paper studies a random fuzzy queueing system that the interarrival times of customers arriving at the server and the service times are independent and identically distributed random fuzzy variables. We match the random fuzzy queueing system with the random fuzzy alternating renewal process and we do not use from α-pessimistic and α-optimistic values to estimate the average chance of the event ”random fuzzy queueing system is busy at time t”, we employ the fuzzy simulation method in practical applications. Some theorem is proved and finally we solve a numerical example with fuzzy simulation method.

Keywords: Random fuzzy variables, Fuzzy simulation, Queueing system, Interarrival times.

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Authors: Jiangyong Liu, Xiangxiang Xu, Bote Luo, Xiaoxue Luo, Jiang Zhu, Lingzhi Yi

Abstract:

To address the problems of non-linearity and high randomness of the original power load sequence causing the degradation of power load forecasting accuracy, a short-term load forecasting method is proposed. The method is based on the least square support vector machine (LSSVM) optimized by an improved sparrow search algorithm combined with the variational mode decomposition proposed in this paper. The application of the variational mode decomposition technique decomposes the raw power load data into a series of intrinsic mode functions components, which can reduce the complexity and instability of the raw data while overcoming modal confounding; the proposed improved sparrow search algorithm can solve the problem of difficult selection of learning parameters in the LSSVM. Finally, through comparison experiments, the results show that the method can effectively improve prediction accuracy.

Keywords: Load forecasting, variational mode decomposition, improved sparrow search algorithm, least square support vector machine.

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Authors: Nyein Nyein Soe, Thet Thet Han Yee, Soe Sandar Aung

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This paper is proposed the dynamic simulation of small power induction motor based on Mathematical modeling. The dynamic simulation is one of the key steps in the validation of the design process of the motor drive systems and it is needed for eliminating inadvertent design mistakes and the resulting error in the prototype construction and testing. This paper demonstrates the simulation of steady-state performance of induction motor by MATLAB Program Three phase 3 hp induction motor is modeled and simulated with SIMULINK model.

Keywords: Squirrel cage induction motor, modeling andsimulation, MATLAB software, torque, speed.

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Authors: Costin Badica, Maria Ganzha, Marcin Paprzycki

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Recently, a model multi-agent e-commerce system based on mobile buyer agents and transfer of strategy modules was proposed. In this paper a different approach to code mobility is introduced, where agent mobility is replaced by local agent creation supplemented by similar code mobility as in the original proposal. UML diagrams of agents involved in the new approach to mobility and the augmented system activity diagram are presented and discussed.

Keywords: Agent system, agent mobility, code mobility, e-commerce, UML formalization.

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Authors: Byung-Wan Jo, Kwang-Won Yoon, Ji-Sun Choi

Abstract:

Load managing method on road became necessary since overloaded vehicles occur damage on road facilities and existing systems for preventing this damage still show many problems.Accordingly, efficient managing system for preventing overloaded vehicles could be organized by using the road itself as a scale by applying genetic algorithm to analyze the load and the drive information of vehicles.Therefore, this paper organized Ubiquitous sensor network system for development of intelligent overload vehicle regulation system, also in this study, to use the behavior of road, the transformation was measured by installing underground box type indoor model and indoor experiment was held using genetic algorithm. And we examined wireless possibility of overloaded vehicle regulation system through experiment of the transmission and reception distance.If this system will apply to road and bridge, might be effective for economy and convenience through establishment of U-IT system..

Keywords: Overload Vehicle. Genetic Algorithm, EmbeddedSystem, Wim Sensor, overload vehicle regulation

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Authors: Jahedul Islam Chowdhury, Bao Kha Nguyen, David Thornhill, Roy Douglas, Stephen Glover

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Organic Rankine Cycle (ORC) is the most commonly used method for recovering energy from small sources of heat. The investigation of the ORC in supercritical condition is a new research area as it has a potential to generate high power and thermal efficiency in a waste heat recovery system. This paper presents a steady state ORC model in supercritical condition and its simulations with a real engine’s exhaust data. The key component of ORC, evaporator, is modelled using finite volume method, modelling of all other components of the waste heat recovery system such as pump, expander and condenser are also presented. The aim of this paper is to investigate the effects of mass flow rate and evaporator outlet temperature on the efficiency of the waste heat recovery process. Additionally, the necessity of maintaining an optimum evaporator outlet temperature is also investigated. Simulation results show that modification of mass flow rate is the key to changing the operating temperature at the evaporator outlet.

Keywords: Organic Rankine cycle, supercritical condition, steady state model, waste heat recovery.

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Authors: Barenten Suciu, Yoshiki Tsuji

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In this paper, a theoretical investigation on the dynamic characteristics of one degree of freedom vibration system equipped with inerter of variable inertance, is presented. Differential equation of movement was solved under proper initial conditions in the case of free undamped/damped vibration, considered in the absence/presence of the inerter in the mechanical system. Influence of inertance on the amplitude of vibration, phase angle, natural frequency, damping ratio, and logarithmic decrement was clarified. It was mainly found that the inerter decreases the natural frequency of the undamped system and also of the damped system if the damping ratio is below 0.707. On the other hand, the inerter increases the natural frequency of the damped system if the damping ratio exceeds 0.707. Results obtained in this work are useful for the adequate design of inerters.

Keywords: One degree of freedom vibration system, inerter, parallel connection, variable inertance, frequency control, damping.

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Authors: J. R. Wang, J. H. Yang, Y. Chiang, H. C. Chen, C. Shih, S. W. Chen, S. C. Chiang, T. Y. Yu

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In this study of ultimate response guideline (URG), Kuosheng BWR/6 nuclear power plant (NPP) TRACE model was established. The reactor depressurization, low pressure water injection, and containment venting are the main actions of URG. This research focuses to evaluate the efficiency of URG under Fukushima-like conditions. Additionally, the sensitivity study of URG was also performed in this research. The analysis results of TRACE present that URG can keep the peak cladding temperature (PCT) below 1088.7 K (the failure criteria) under Fukushima-like conditions. It implied that Kuosheng NPP was at the safe situation.

Keywords: BWR, TRACE, safety analysis, URG.

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Authors: Chingtham Tejbanta Singh, Shivashankar B. Nair

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This paper explores an application of an adaptive learning mechanism for robots based on the natural immune system. Most of the research carried out so far are based either on the innate or adaptive characteristics of the immune system, we present a combination of these to achieve behavior arbitration wherein a robot learns to detect vulnerable areas of a track and adapts to the required speed over such portions. The test bed comprises of two Lego robots deployed simultaneously on two predefined near concentric tracks with the outer robot capable of helping the inner one when it misaligns. The helper robot works in a damage-control mode by realigning itself to guide the other robot back onto its track. The panic-stricken robot records the conditions under which it was misaligned and learns to detect and adapt under similar conditions thereby making the overall system immune to such failures.

Keywords: Adaptive, AIS, Behavior Arbitration, ClonalSelection, Immune System, Innate, Robot, Self Healing.

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Authors: Sachin Kamble, Raghothama Anekal, Shivakumar Bhavi

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Thermal management of electronic components packaged inside an IP65 rated enclosure is of prime importance in industrial applications. Electrical enclosure protects the multiple board configurations such as inverter, power, controller board components, busbars, and various power dissipating components from harsh environments. Industrial environments often experience relatively warm ambient conditions, and the electronic components housed in the enclosure dissipate heat, due to which the enclosures and the components require thermal management as well as reduction of internal ambient temperatures. Design of Experiments based thermal simulation approach with MOSFET arrangement, Heat sink design, Enclosure Volume, Copper and Aluminum Spreader, Power density, and Printed Circuit Board (PCB) type were considered to optimize air temperature inside the IP65 enclosure to ensure conducive operating temperature for controller board and electronic components through the different modes of heat transfer viz. conduction, natural convection and radiation using Ansys ICEPAK. MOSFET’s with the parallel arrangement, IP65 enclosure molded heat sink with rectangular fins on both enclosures, specific enclosure volume to satisfy the power density, Copper spreader to conduct heat to the enclosure, optimized power density value and selecting Aluminum clad PCB which improves the heat transfer were the contributors towards achieving a conducive operating temperature inside the IP-65 rated Motor Inverter enclosure. A reduction of 52 ℃ was achieved in internal ambient temperature inside the IP65 enclosure between baseline and final design parameters, which met the operative temperature requirements of the electronic components inside the IP-65 rated Motor Inverter.

Keywords: Ansys ICEPAK, Aluminum Clad PCB, IP 65 enclosure, motor inverter, thermal simulation.

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Authors: Chokri Baccouch, Chayma Bahhar, Hedi Sakli, Nizar Sakli

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In this article, we present the combination of an antenna patch structure with a photovoltaic cell in one device for telecommunication applications in isolated environments. The radiating patch element of a patch antenna was replaced by a solar cell. DC current generation is the original feature of the solar cell, but now it was additionally able to receive and transmit electromagnetic waves. A mathematical model which serves in the minimization of power losses of the cell and therefore the improvement in conversion performance was studied. Simulation results of this antenna show a resonance at a frequency of 16.55 GHz in Ku-band with a gain of 4.24 dBi.

Keywords: Electric power collected, optical and electrical losses, optimization of the grid of collection, patch antenna, photovoltaic cell.

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Authors: D. P. Sharma, A. Chaturvedi, G.Purohit , R.Shivarudraswamy

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In today scenario, to meet enhanced demand imposed by domestic, commercial and industrial consumers, various operational & control activities of Radial Distribution Network (RDN) requires a focused attention. Irrespective of sub-domains research aspects of RDN like network reconfiguration, reactive power compensation and economic load scheduling etc, network performance parameters are usually estimated by an iterative process and is commonly known as load (power) flow algorithm. In this paper, a simple mechanism is presented to implement the load flow analysis (LFA) algorithm. The reported algorithm utilizes graph theory principles and is tested on a 69- bus RDN.

Keywords: Radial Distribution network, Graph, Load-flow, Array.

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Authors: Riza Muhida, Maisarah Ali, Puteri Shireen Jahn Kassim, Muhammad Abu Eusuf, Agus G.E. Sutjipto, Afzeri

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Over recent years, the number of building integrated photovoltaic (BIPV) installations for home systems have been increasing in Malaysia. The paper concerns an analysis - as part of current Research and Development (R&D) efforts - to integrate photovoltaics as an architectural feature of a detached house in the new satellite township of Putrajaya, Malaysia. The analysis was undertaken using calculation and simulation tools to optimize performance of BIPV home system. In this study, a the simulation analysis was undertaken for selected bungalow units based on a long term recorded weather data for city of Kuala Lumpur. The simulation and calculation was done with consideration of a PV panels' tilt and direction, shading effect and economical considerations. A simulation of the performance of a grid connected BIPV house in Kuala Lumpur was undertaken. This case study uses a 60 PV modules with power output of 2.7 kW giving an average of PV electricity output is 255 kWh/month..

Keywords: Building integrated photovoltaic, Malaysia, Simulation, panels' tilt and direction.

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Authors: Jamie Wong, Daisy Sang, Chang-Shyh Peng

Abstract:

Geofence is a virtual fence defined by a preset physical radius around a target location. Geofencing App provides location-based services which define the actionable operations upon the crossing of a geofence. Geofencing requires continual location tracking, which can consume noticeable amount of battery power. Additionally, location updates need to be frequent and accurate or order so that actions can be triggered within an expected time window after the mobile user navigate through the geofence. In this paper, we build an Android mobile geofencing Application to remotely and autonomously control a power switch.

Keywords: Location based service, geofence, autonomous, remote switch.

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Authors: A. D. Parekh, P. R. Tailor

Abstract:

Capacity and efficiency of any refrigerating system diminish rapidly as the difference between the evaporating and condensing temperature is increased by reduction in the evaporator temperature. The single stage vapour compression refrigeration system is limited to an evaporator temperature of -40 0C. Below temperature of -40 0C the either cascade refrigeration system or multi stage vapour compression system is employed. Present work describes thermal design of main three heat exchangers namely condenser (HTS), cascade condenser and evaporator (LTS) of R404A-R508B and R410A-R23 cascade refrigeration system. Heat transfer area of condenser (HTS), cascade condenser and evaporator (LTS) for both systems have been compared and the effect of condensing and evaporating temperature on heat-transfer area for both systems have been studied under same operating condition. The results shows that the required heat-transfer area of condenser and cascade condenser for R410A-R23 cascade system is lower than the R404A-R508B cascade system but heat transfer area of evaporator is similar for both the system. The heat transfer area of condenser and cascade condenser decreases with increase in condensing temperature (Tc), whereas the heat transfer area of cascade condenser and evaporator increases with increase in evaporating temperature (Te).

Keywords: Heat-transfer area, R410A, R404A, R508B, R23, Refrigeration system, Thermal design

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Authors: Soltani Amir, Wang Xuan

Abstract:

The advantage of using non-linear passive damping  system in vibration control of two adjacent structures is investigated  under their base excitation. The base excitation is El Centro  earthquake record acceleration. The damping system is considered as  an optimum and effective non-linear viscous damper that is  connected between two adjacent structures. A MATLAB program is  developed to produce the stiffness and damping matrices and to  determine a time history analysis of the dynamic motion of the  system. One structure is assumed to be flexible while the other has a  rule as laterally supporting structure with rigid frames. The response  of the structure has been calculated and the non-linear damping  coefficient is determined using optimum LQR algorithm in an  optimum vibration control system. The non-linear parameter of  damping system is estimated and it has shown a significant advantage  of application of this system device for vibration control of two  adjacent tall building.

Keywords: Structural Control, Active and passive damping, Vibration control, Seismic isolation.

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Authors: Saeed Mohammed Baneamoon, Rosalina Abdul Salam, Abdullah Zawawi Hj. Talib

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Designing a simulated system and training it to optimize its tasks in simulated environment helps the designers to avoid problems that may appear when designing the system directly in real world. These problems are: time consuming, high cost, high errors percentage and low efficiency and accuracy of the system. The proposed system will investigate and improve the efficiency and accuracy of a simulated robot to choose correct behavior to perform its task. In this paper, machine learning, which uses genetic algorithm, is adopted. This type of machine learning is called genetic-based machine learning in which a distributed classifier system is used to improve the efficiency and accuracy of the robot. Consequently, it helps the robot to achieve optimal action.

Keywords: Machine Learning, Genetic-Based MachineLearning, Learning Classifier System, Behaviors.

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Authors: Anjum Iqbal, Mohd Aizaini Maarof

Abstract:

Artificial Immune System (AIS) is relatively naive paradigm for intelligent computations. The inspiration for AIS is derived from natural Immune System (IS). Classically it is believed that IS strives to discriminate between self and non-self. Most of the existing AIS research is based on this approach. Danger Theory (DT) argues this approach and proposes that IS fights against danger producing elements and tolerates others. We, the computational researchers, are not concerned with the arguments among immunologists but try to extract from it novel abstractions for intelligent computation. This paper aims to follow DT inspiration for intelligent data processing. The approach may introduce new avenue in intelligent processing. The data used is system calls data that is potentially significant in intrusion detection applications.

Keywords: artificial immune system, danger theory, intelligent processing, system calls

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Authors: R .Lassoued, M. Guenfoud

Abstract:

An accurate procedure to determine free vibrations of beams and plates is presented. The natural frequencies are exact solutions of governing vibration equations witch load to a nonlinear homogeny system. The bilinear and linear structures considered simulate a bridge. The dynamic behavior of this one is analyzed by using the theory of the orthotropic plate simply supported on two sides and free on the two others. The plate can be excited by a convoy of constant or harmonic loads. The determination of the dynamic response of the structures considered requires knowledge of the free frequencies and the shape modes of vibrations. Our work is in this context. Indeed, we are interested to develop a self-consistent calculation of the Eigen frequencies. The formulation is based on the determination of the solution of the differential equations of vibrations. The boundary conditions corresponding to the shape modes permit to lead to a homogeneous system. Determination of the noncommonplace solutions of this system led to a nonlinear problem in Eigen frequencies. We thus, develop a computer code for the determination of the eigenvalues. It is based on a method of bisection with interpolation whose precision reaches 10 -12. Moreover, to determine the corresponding modes, the calculation algorithm that we develop uses the method of Gauss with a partial optimization of the "pivots" combined with an inverse power procedure. The Eigen frequencies of a plate simply supported along two opposite sides while considering the two other free sides are thus analyzed. The results could be generalized with the case of a beam by regarding it as a plate with low width. We give, in this paper, some examples of treated cases. The comparison with results presented in the literature is completely satisfactory.

Keywords: Free frequencies, beams, rectangular plates.

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