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Tesis sobre el tema "Asymptotic stability and transient"

Autor: Grafiati

Publicado: 4 de junio de 2021

Última modificación: 2 de febrero de 2022

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1

Araujo, Thiago Jefferson de. "M?todos num?ricos para resolu??o de equa??es diferenciais ordin?rias lineares baseados em interpola??o por spline." Universidade Federal do Rio Grande do Norte, 2012. http://repositorio.ufrn.br:8080/jspui/handle/123456789/17010.

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Made available in DSpace on 2014-12-17T15:26:38Z (GMT). No. of bitstreams: 1 ThiagoJA_DISSERT.pdf: 636679 bytes, checksum: 3497bfd29c2779ff70f7932b9a308a9c (MD5) Previous issue date: 2012-08-13<br>Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior<br>In this work we have elaborated a spline-based method of solution of inicial value problems involving ordinary differential equations, with emphasis on linear equations. The method can be seen as an alternative for the traditional solvers such as Runge-Kutta, and avoids root calculations in the linear time invariant case. The method is

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2

Townsend, Philip. "Asymptotic and transient behaviour of nonlinear control systems." Thesis, University of Bath, 2007. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.486479.

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In this thesis, the problem of controlling both transient and asymptotic behaviour of solutions of functional differential equations is addressed. The work begins, in Chapter 1, with an introduction to basic control theory principles that will be used throughout. This is followed by the introduction of a class of nonlinear operators in Chapter 2 and the development of suitable existence theories for the associated system classes of functional differential equations and inclusions in Chapter 3. A discussion is provided, in Chapter 2, describing diverse phenomena, such as delays and hysteresis,

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3

Nguyen, Toan. "Asymptotic stability of noncharacteristic viscous boundary layers." [Bloomington, Ind.] : Indiana University, 2009. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3378375.

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Thesis (Ph.D.)--Indiana University, Dept. of Mathematics, 2009.<br>Title from PDF t.p. (viewed on Jul 12, 2010). Source: Dissertation Abstracts International, Volume: 70-10, Section: B, page: 6259. Adviser: Kevin R. Zumbrun.

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4

Couturier, Nicolas. "Transient Stability During Asymmetrical Faults." Thesis, KTH, Elektroteknisk teori och konstruktion, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-160521.

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This research project has been conducted at RTE in order to study the transient stability after asymmetrical faults. When three-phase short-circuits occur in a network, almost all the electrical power is lost on the relevant line(s). Among all short-circuit types, it is the most drastic event and the issue has to be solved very quickly. But oddly, it is also the easiest problem to solve mathematically speaking. This comes from the fact that the system stays balanced, and equations can be simplified. However with line-to-ground faults this is no longer the case, and transient stability analysis

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5

Liu, Fang-Lan. "Some asymptotic stability results for the Boussinesq equation." Diss., Virginia Tech, 1993. http://hdl.handle.net/10919/40052.

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6

Kwasnicki, Wieslaw T. "High Speed Transient Stability, multiprocessing solutions." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape17/PQDD_0024/NQ32881.pdf.

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7

Collier, Nicholas Richard. "On asymptotic stability of prime ideals in noncommutative rings." Thesis, University of Warwick, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.403145.

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8

Genovese, de Oliveira Andrea. "Asymptotic and stability analysis of a tumour growth model." Thesis, University of Nottingham, 2017. http://eprints.nottingham.ac.uk/40473/.

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We investigate avascular tumour growth as a two-phase process consisting of cells and liquid. Initially, we simulate a continuum moving-boundary model formulated by Byrne, King, McElwain, Preziosi, (Applied Mathematics Letters, 2003, 16, 567-573) in one dimension and analyse the dependence of the tumour growth on the natural nutrient and cell concentration levels outside of the tumour along with its ability to model known biological dynamics of tumour growth. We investigate linear stability of time-dependent solution profiles in the moving-boundary formulation of a limit case (with negligible

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9

Karimishad, Amir. "Transient stability-constrained load dispatch, ancillary services allocation and transient stability assessment procedures for secure power system operation." University of Western Australia. Energy Systems Centre, 2008. http://theses.library.uwa.edu.au/adt-WU2009.0028.

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[Truncated abstract] The present thesis is devoted to the development of new methods for transient stability-constrained optimal power flow, probabilistic transient stability assessment and security-constrained ancillary services allocation. The key objective of the thesis is to develop novel dispatch and assessment methods for power systems operation in the new environment of electricity markets to ensure power systems security, particularly transient stability. A new method for economic dispatch together with nodal price calculations which includes transient stability constraints and, at the

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10

Niday, Thomas A. "Stability and transient effects in ultraviolet filaments." Diss., The University of Arizona, 2004. http://hdl.handle.net/10150/280637.

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Short, high intensity laser pulses induce nonlinear optical effects in the atmosphere that have the potential to make them propagate for long distances. Applications for long distance propagation of short pulses include active spectral remote sensing and laser lightning control. Much of the work in this field has been done with infrared pulses; however, it has been proposed that ultraviolet pulses have the advantage that longer pulse lengths can be used, thereby delivering more energy. Long pulse lengths lead to a simplified instantaneous model for the plasma response, which has been shown by

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11

Hiskens, Ian A. "Energy functions, transient stability and voltage behaviour /." Online version, 1990. http://bibpurl.oclc.org/web/30417.

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12

Vuuren, Jan H. van. "Permanence and asymptotic stability in diagonally convex reaction-diffusion systems." Thesis, University of Oxford, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.294385.

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13

Musaazi, M. K. "Generator transient stability improvement by optimal aim strategies." Thesis, Imperial College London, 1985. http://hdl.handle.net/10044/1/37795.

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14

Oztop, Celal. "Beforehand Obtaining A Safety Operation Condition By Using Daily Load Curves In Transient Stability And Graphical Software For Transient Stability Applications." Master's thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/2/12606457/index.pdf.

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ABSTRACT In this thesis, relationship between two most important transient stability indices, critical clearing time and generator rotor angle is examined for one machine-infinite bus system and then extended to the multimachine case and is observed to be linear. By using the linear relationship between critical clearing time and generator rotor angle and utilizing the daily load curve, a new preventive method is proposed. The aim of this method is to make all critical clearing times longer than the relay and circuit breaker combination operation time. In the proposed method, desired critic

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15

Li, Peng. "On guaranteed asymptotic stability of uncertain systems with incomplete state information." Thesis, University of Ottawa (Canada), 1986. http://hdl.handle.net/10393/4830.

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16

Jiriwibhakorn, Somchat. "Transient stability assessment of power systems using neural networks." Thesis, Imperial College London, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.249489.

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17

Anderson, Sharon Lee. "Reduced order power system models for transient stability studies." Thesis, This resource online, 1993. http://scholar.lib.vt.edu/theses/available/etd-09052009-040743/.

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18

MORAES, ADELIO JOSE DE. "CONTRIBUITIONS TO THE LOAD MODELLING FOR TRANSIENT STABILITY STUDIES." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1992. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=8361@1.

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UNIVERSIDADE FEDERAL DE UBERLÂNDIA<br>Este trabalho, relacionado com a área de modelagem de cargas para estudos de estabilidade transitória de sistemas elétricos, tem por objetivo discutir a adequação dos modelos clássicos normalmente utilizados, bem como o aprimoramento e desenvolvimento de representações para cargas com características específicas. Inicialmente, são enfocados os aspectos das representações clássicas de cargas e, através de desenvolvimentos teóricos e estudos computacionais, ressaltadas as adequações destas estratégias ao contexto atual. Reconhecendo algumas limitaç

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19

WENG, CHIYUAN. "Transient Stability Analysis of Power Systems with Energy Storage." Case Western Reserve University School of Graduate Studies / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=case1348453228.

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20

Hishida, Toshiaki. "Global existence, asymptotic behavior and stability of solutions to the convection problem /." Electronic version of summary, 1993. http://www.wul.waseda.ac.jp/gakui/gaiyo/1952.pdf.

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21

Al, Marhoon Hussain Hassan. "A Practical Method for Power Systems Transient Stability and Security." ScholarWorks@UNO, 2011. http://scholarworks.uno.edu/td/114.

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Stability analysis methods may be categorized by two major stability analysis methods: small-signal stability and transient stability analyses. Transient stability methods are further categorized into two major categories: numerical methods based on numerical integration, and direct methods. The purpose of this thesis is to study and investigate transient stability analysis using a combination of step-by-step and direct methods using Equal Area Criterion. The proposed method is extended for transient stability analysis of multi machine power systems. The proposed method calculates the potentia

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22

Gonzalez-Torres, Juan Carlos. "Transient stability of high voltage AC-DC electric transmission systems." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLS041.

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Les nouvelles politiques adoptées par les autorités nationales ont encouragé pendant les dernières années l'intégration à grande échelle des systèmes d'énergie renouvelable (RES). L'intégration à grande échelle des RES aura inévitablement des conséquences sur le réseau de transport d'électricité tel qu'il est conçu aujourd'hui, car le transport de l'électricité massif sur de longues distances pourrait amener les réseaux de transport à fonctionner près de leurs limites, réduisant ainsi leurs marges de sécurité. Des systèmes de transport d’électricité plus complexes seront donc nécessaires.Dans

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23

Zhou, Bowen. "The impact of electric vehicles on power system transient stability." Thesis, Queen's University Belfast, 2016. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.709884.

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The penetration of the electric vehicle (EV) has increased rapidly in recent years mainly as a consequence of advances in both transportation and electricity sectors and in response to global pressure to reduce carbon emissions and limit fossil fuel consumption. Large-scale EV integration in power systems has modified the nature of the traditional electric load such that it should be controllable. Moreover, uncertain power sources and demand pose challenges in electricity transmission grid, leading to significant impact on power system security and stability. Therefore, it is timely that a com

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24

Zadehkhost, Sajjad. "Efficient algorithms to expedite transient stability analysis of power systems." Thesis, University of British Columbia, 2015. http://hdl.handle.net/2429/52816.

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With rapid increase in complexity of modern power systems, there is a strong need for better computational tools to ensure the reliable operation of electrical grids. These tools need to be accurate, computationally efficient, and capable of using advanced measurement devices. In this context, transient stability assessment (TSA) is an important study that determines system’s dynamic security margins following a major disturbance. The TSA consists of a set of differential-algebraic equations (DAEs), which are typically solved using time-domain simulation (TDS) approach. While being very accura

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25

Mihirig, Ali Mohamed. "Transient stability analysis of multimachine power systems by catastrophe theory." Thesis, University of British Columbia, 1987. http://hdl.handle.net/2429/29022.

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Transient stability analysis is an important part of power system planning and operation. For large power systems, such analysis is very demanding in computation time. On-line transient stability assessment will be necessary for secure and reliable operation of power systems in the near future because systems are operated close to their maximum limits. In the last two decades, a vast amount of research work has been done in the area of fast transient stability assessment by direct methods. The major difficulties associated with direct methods are the limitations in the power system model, det

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26

Wang, C. "On asymptotic stability of stochastic differential equations with delay in infinite dimensional spaces." Thesis, University of Liverpool, 2017. http://livrepository.liverpool.ac.uk/3007651/.

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In most stochastic dynamical systems which describe process in engineering, physics and economics, stochastic components and random noise are often involved. Stochastic effects of these models are often used to capture the uncertainty about the operating systems. Motivated by the development of analysis and theory of stochastic processes, as well as the studies of natural sciences, the theory of stochastic differential equations in infinite dimensional spaces evolves gradually into a branch of modern analysis. In the analysis of such systems, we want to investigate their stabilities. This thes

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27

Sleep, Matthew David. "Analysis of Transient Seepage Through Levees." Diss., Virginia Tech, 2011. http://hdl.handle.net/10919/40356.

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Levees are a significant part of the United States flood protection infrastructure. It is estimated that over 100,000 miles of levees exist in the United States. Most of these levees were designed many years ago to protect farmland and rural areas. As growth continues in the United States, many of these levees are now protecting homes and other important structures. The American Society of Civil Engineers gave the levees in the United States a grade of D- in 2009. To bring flood protection up to modern standards there requires adequate methods of evaluating levees with respect to seepag

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28

Zhang, Yi. "Adaptive remedial action schemes for transient instability." Online access for everyone, 2007. http://www.dissertations.wsu.edu/Dissertations/Fall2007/y_zhang_112707.pdf.

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29

Rudraraju, Seetharama raju. "SMALL SIGNAL AND TRANSIENT STABILITY ANALYSIS OF MVDC SHIPBOARD POWER SYSTEM." MSSTATE, 2009. http://sun.library.msstate.edu/ETD-db/theses/available/etd-11052009-170217/.

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Recent developments in high power rated Voltage Source Converters (VSCs) have resulted in their successful application in Multi-Terminal HVDC (MTDC) transmission systems and also have potential in the Medium Voltage DC (MVDC) distribution systems. This work presents the findings of stability studies carried out on a zonal MVDC architecture for the shipboard power distribution system. The stability study is confined to rotor angle stability of the power system, i.e. the transient and small signal stability analysis. The MTDC ring structure similar to MVDC shipboard power system was implemented

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30

SILVA, KARLA GUEDES CARNEIRO DA. "POWER SYSTEM TRANSIENT STABILITY ASSESSMENT USING THE EXTENDED EQUAL AREA CRITERION." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1996. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=8752@1.

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CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO<br>Este trabalho analisa o desempenho do Critério das Áreas Iguais Estendido na avaliação da estabilidade transitória de sistemas de potência. O Critério das Áreas Iguais Estendido consiste em representar o sistema multimáquina por um sistema equivalente de duas máquinas, que é, em seguida, reduzido a um sistema máquina-barra infinita, no qual é aplicado o conhecido critério das áreas iguais. A evolução no tempo do sistema máquina-barra infinita resultante é aproximada por uma série de Taylor truncada. As medidas de estabil

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31

PESSANHA, JOSE EDUARDO ONODA. "VOLTAGE STABILITY PHENOMENON ANALYSIS: TRANSIENT AND LONG TERM TIME DOMAIN SIMULATION." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1997. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=8941@1.

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CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO<br>Este trabalho analisa o fenômeno da estabilidade de tensão no domínio do tempo incluindo fenômenos rápidos (transitórios) e lentos (longo-termo). As diferentes formas de estabilidade existentes nos sistemas elétricos de potência são definidas e classificadas de acordo com a variável de interesse e com a amplitude do distúrbio. Utilizando-se um simples sistema elétrico de potência, verifica-se através de formulações dinâmicas fatores relevantes ao fenômeno da estabilidade de tensão. Investiga-se a influência de cargas do t

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32

Al, Marhoon Hussain Hassan. "Adaptive Online Transient Stability Assessment of Power Systems for Operational Purposes." ScholarWorks@UNO, 2015. http://scholarworks.uno.edu/td/2036.

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Online stability assessment is an important problem that has not been solved completely yet. The purpose of this research is to tackle online transient stability assessment. Currently, most utility companies use step-by-step integration in order to set protective equipment so that they effectively work for critical contingencies. However, there are times an unforeseen contingency may occur which may cause the system to transit and the protective equipment to misoperate and does not isolate the disturbed part of the system. This research introduces a method that automatically determines a group

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33

Li, Meiyan. "Transient Stability Prediction based on Synchronized Phasor Measurements and Controlled Islanding." Diss., Virginia Tech, 2013. http://hdl.handle.net/10919/52623.

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Traditional methods for predicting transient stability of power systems such as the direct method, the time domain approach, and the energy function methods do not work well for online transient stability predictions problems. With the advent of Phasor Measurement Units (PMUs) in power systems, it is now possible to monitor the behavior of the system in real time and provide important information for transient stability assessment and enhancement. Techniques such as the rotor oscillation prediction method based on time series have made the prediction of system stability possible for real-time

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34

Emmanuel, Paul. "Investigation into transient stability of a nuclear power plant using DIgSILENT." Master's thesis, University of Cape Town, 2016. http://hdl.handle.net/11427/20322.

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The current electricity crisis, coupled with the lack of generation, has led to a major focus on continuity of supply. The Western Cape has been severely affected, as it only has a limited number of generation sources, namely Koeberg Nuclear Power Station (base supply) along with other power stations used for peak load running such as Ankerlig, Palmiet, etc. Koeberg Nuclear Power Station is located at the end of a long transmission line with no other base-load generation for 1500km, between the power station and the mass pool of coal-fired generation in Mpumalanga. Koeberg Nuclear Power Statio

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35

Khomari, Moloantoa. "The impact of wind generators on a Powe system's transient stability." Master's thesis, University of Cape Town, 2009. http://hdl.handle.net/11427/5193.

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This thesis discusses the investigations carried out on the different types of wind generators and how these would affect the transient stability of a hypothetical power network as presented in this report. Focus was on the transient responses of the conventional synchronous generator’s rotor angle and terminal voltage when connected to different types of wind generators. The three different wind generator technologies explored were the squirrel cage induction generator (SCIG), doubly-fed induction generator (DFIG) and the converter driven synchronous generator (CDSG).

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36

Olulope, Paul Kehinde. "Transient stability assessment of hybrid distributed generation using computational intelligence approaches." Doctoral thesis, University of Cape Town, 2014. http://hdl.handle.net/11427/9288.

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Includes bibliographical references.<br>Due to increasing integration of new technologies into the grid such as hybrid electric vehicles, distributed generations, power electronic interface circuits, advanced controllers etc., the present power system network is now more complex than in the past. Consequently, the recent rate of blackouts recorded in some parts of the world indicates that the power system is stressed. The real time/online monitoring and prediction of stability limit is needed to prevent future blackouts. In the last decade, Distributed Generators (DGs) among other technologies

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37

Fahmi, Nagi R. "Modelling of induction motors for system faults and transient stability studies." Thesis, Aston University, 1986. http://publications.aston.ac.uk/8049/.

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The research carried out in this thesis was mainly concerned with the effects of large induction motors and their transient performance in power systems. Computer packages using the three phase co-ordinate frame of reference were developed to simulate the induction motor transient performance. A technique using matrix algebra was developed to allow extension of the three phase co-ordinate method to analyse asymmetrical and symmetrical faults on both sides of the three phase delta-star transformer which is usually required when connecting large induction motors to the supply system. System simu

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38

Roberts, Lewis George Wilson. "Parametric analysis of transient stability in power systems using classical models." Thesis, University of Bristol, 2016. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.702145.

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This thesis aims to strengthen the bridge between mathematical and practical research into the transient stability of power systems. Literature that exploits the mathematical parallels between models for transient dynamics in power systems and the phenomenon of synchronisation in complex networks is explored. However, it is contended that research at the intersection of complex systems and power system stability can limit its applications to practical issues in power engineering. This thesis focuses on the measurement of transient stability in power systems in terms of a traditional stability

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39

Sirisup, Sirod. "Issues in low-dimensional modeling of unsteady flows : convergence, asymptotic stability and reconstruction procedures /." View online version; access limited to Brown University users, 2005. http://wwwlib.umi.com/dissertations/fullcit/3174674.

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40

Rogers, Stuart John. "An asymptotic linear stability theory for channel flow for quantitative comparison with numerical solutions." Thesis, Keele University, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.425968.

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We consider the linear stability of flows with two and three dimensional disturbances. Our main focus is the application of matched asymptotic expansions. The model used throughout is plane Poiseuille flow. The stability properties of this flow have been analysed before, both analytically and numerically. Previous applications of matched asymptotics have been developed from two distinct theories, one relating to the upper branch of the neutral curve and the other to the lower branch. Our new analytical approach, still based on matched asymptotic expansions, allows us to develop a single ration

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41

Kanchanaharuthai, Adirak. "Small-Signal Stability, Transient Stability and Voltage Regulation Enhancement of Power Systems with Distributed Renewable Energy Resources." Case Western Reserve University School of Graduate Studies / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=case1321988036.

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Parsons, Antony Cozart. "Automatic location of transient power quality disturbances /." Digital version accessible at:, 1999. http://wwwlib.umi.com/cr/utexas/main.

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43

Banu, Nurzahan. "Free convection in fluid-saturated porous media." Thesis, University of Bath, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.311254.

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44

Khorrami, Farshad. "Asymptotic perturbation and Lyapunov stability based approaches for control of flexible and rigid robot manipulators /." The Ohio State University, 1988. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487592050230916.

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45

Jiang, Hong. "A graphical interface for the BPA load flow and transient stability programs." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/mq23355.pdf.

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46

Chapman, Jeffrey W. (Jeffrey Wayne). "Power system control for large-disturbance stability : security, robustness and transient energy." Thesis, Massachusetts Institute of Technology, 1996. http://hdl.handle.net/1721.1/39393.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1996.<br>Includes bibliographical references (p. 223-228).<br>by Jeffrey Wayne Chapman.<br>Ph.D.

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47

Ferri, Matthew A. (Matthew Anthony). "Current distribution and stability criteria for superconducting cables in transient magnetic fields." Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/10406.

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48

Laury, John. "Transient Stability in Low Frequency Railways with Mixed Electronic and Rotational Generation." Licentiate thesis, Luleå tekniska universitet, Energivetenskap, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-26485.

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Transient stability concerns the ability of a power system to maintain synchronism after a large disturbance. Transient stability plays an important role in guaranteeing operational security and reliability and has been extensively studies for large 50 Hz and 60 Hz transmission systems. However, transient stability of low frequency railway grids has not been properly investigated.As low frequency railway grids operate at another frequency than the public grid, conversion of frequency is needed. This conversion is performed by Rotary Frequency Converters or by Static Frequency Converters. These

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49

Parsi-Feraidoonian, Raiomand. "Application of catastrophe theory to transient stability analysis of multimachine power systems." Thesis, University of British Columbia, 1990. http://hdl.handle.net/2429/29723.

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Transient stability analysis is an important part of power planning and operation. For large power systems, such analysis is very time consuming and expensive. Therefore, an online transient stability assessment will be required as these large power systems are operated close to their maximum limits. In this thesis swallowtail catastrophe is used to determine the transient stability regions. The bifurcation set represents the transient stability region in terms of power system transient parameters bounded by the transient stability limits. The system modelling is generalized in such, that the

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Brown, Thomas John. "The asymptotic stability of stochastic kernel operators." Diss., 1997. http://hdl.handle.net/10500/16068.

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A stochastic operator is a positive linear contraction, P : L1 --+ L1, such that llPfII2 = llfll1 for f > 0. It is called asymptotically stable if the iterates pn f of each density converge in the norm to a fixed density. Pf(x) = f K(x,y)f(y)dy, where K( ·, y) is a density, defines a stochastic kernel operator. A general probabilistic/ deterministic model for biological systems is considered. This leads to the LMT operator P f(x) = Jo - Bx H(Q(>.(x)) - Q(y)) dy, where -H'(x) = h(x) is a density. Several particular examples of cell cycle models are examined. An operator overlaps sup

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