Nonlinear analysis of dynamical complex networks

Copyright © 2013 Zidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Complex networks are composed of a large number of highly interconnected dynamical units and therefore exhibit very complicated dynamics. Examples of such complex networks include the Internet, that is, a network of routers or domains, the World Wide Web (WWW), that is, a network of websites, the brain, that is, a network of neurons, and an organization, that is, a network of people. Since the introduction of the small-world network principle, a great deal of research has been focused on the dependence of the asymptotic behavior of interconnected oscillatory agents on the structural properties of complex networks. It has been found out that the general structure of the interaction network may play a crucial role in the emergence of synchronization phenomena in various fields such as physics, technology, and the life sciences

Assessment of the worthwhileness of efficient driving in railway systems with high-receptivity power supplies

Eco-driving is one of the most important strategies for significantly reducing the energy consumption of railways with low investments. It consists of designing a way of driving a train to fulfil a target running time, consuming the minimum amount of energy. Most eco-driving energy savings come from the substitution of some braking periods with coasting periods. Nowadays, modern trains can use regenerative braking to recover the kinetic energy during deceleration phases. Therefore, if the receptivity of the railway system to regenerate energy is high, a question arises: is it worth designing eco-driving speed profiles? This paper assesses the energy benefits that eco-driving can provide in different scenarios to answer this question. Eco-driving is obtained by means of a multi-objective particle swarm optimization algorithm, combined with a detailed train simulator, to obtain realistic results. Eco-driving speed profiles are compared with a standard driving that performs the same running time. Real data from Spanish high-speed lines have been used to analyze the results in two case studies. Stretches fed by 1 × 25 kV and 2 × 25 kV AC power supply systems have been considered, as they present high receptivity to regenerate energy. Furthermore, the variations of the two most important factors that affect the regenerative energy usage have been studied: train motors efficiency ratio and catenary resistance. Results indicate that the greater the catenary resistance, the more advantageous eco-driving is. Similarly, the lower the motor efficiency, the greater the energy savings provided by efficient driving. Despite the differences observed in energy savings, the main conclusion is that eco-driving always provides significant energy savings, even in the case of the most receptive power supply network. Therefore, this paper has demonstrated that efforts in improving regenerated energy usage must not neglect the role of eco-driving in railway efficiency

Pantograph-Catenary Dynamic Models and their Implementation in Hardware-in-the-Loop Tests

Tesis por compendio[ES] Existe una extensa red de líneas ferroviarias electrificadas en todo el mundo. La mayoría de ellas utilizan líneas aéreas de contacto o catenarias para suministrar electricidad a los trenes. Las catenarias son estructuras de cables ubicadas sobre las vías ferroviarias, diseñadas para ser contactadas por los pantógrafos que se encuentran sobre la parte superior de los trenes. El correcto funcionamiento del sistema requiere un alto nivel de exigencia, especialmente a alta velocidad, cuando la continuidad del contacto se ve comprometida. La herramienta más empleada el sistema pantógrafo-catenaria es el uso de simulaciones numéricas. En particular, el Método de los Elementos Finitos (MEF) es la técnica más extendida para modelar y simular la interacción dinámica del pantógrafo con la catenaria. Después de la etapa de simulación, el pantógrafo y la catenaria tienen que ser testados mediante ensayos experimentales en vía. Sin embargo, existe una alternativa que puede reemplazar esos ensayos con una reducción significativa de costes. Dicha alternativa, llamada Hardware In the Loop (HIL), permite testar pantógrafos en el laboratorio mediante un banco de ensayos que emula la interacción con una catenaria virtual. Diferentes grupos de investigación han implementado HIL; sin embargo, en todos los intentos se han adoptado soluciones de compromiso, lo que demuestra el reto que supone la aplicación de HIL. Esta Tesis pretende avanzar en el campo de ensayos HIL, impulsando las capacidades de la técnica y solventando algunas de las limitaciones encontradas en la literatura. Para ello se proponen dos tipos diferentes de modelos de catenaria para su uso en ensayos HIL. El primero es un modelo analítico basado en un cable tensado con perfil geométrico periódico que proporciona la solución estacionaria del sistema. Este enfoque reduce la complejidad de la catenaria, pero mantiene las principales características que intervienen en la dinámica. El modelo ha demostrado ser útil para explicar el comportamiento fundamental de la catenaria, ayudando a comprender el fenómeno de interferencia entre dos pantógrafos. Este modelo analítico es adecuado para HIL debido a su bajo coste computacional. En el presente trabajo se propone un algoritmo iterativo para utilizar el modelo analítico en HIL. El hecho de que el modelo sea periódico permite la aplicación de una estrategia específica para compensar el retraso del lazo de control. Esta estrategia tiene un excelente rendimiento y precisión, validados al comparar ensayos HIL con simulaciones. La validación se realiza con un peso en el lugar del pantógrafo para eliminar las potenciales diferencias en el modelo. Si bien la precisión alcanzada es buena, el modelo analítico de catenaria carece de fidelidad, lo que ha motivado el desarrollo del siguiente modelo. El segundo modelo de catenaria para ensayos HIL es el Modelo Periódico de Elementos Finitos (MPEF), discretizado con el MEF para evitar adicionales simplificaciones topológicas y estructurales. En la formulación se incluye la condición de periodicidad y la dinámica se resuelve mediante análisis en frecuencia. Además, las no linealidades de la catenaria se consideran en la formulación. Un algoritmo iterativo, similar al utilizado para los ensayos HIL con catenaria analítica, es usado para realizar ensayos HIL con catenarias MPEF. La estrategia anterior de utilización de un peso se emplea para validar el sistema de ensayos, resultando tener una gran precisión. Los resultados son gratificantes debido a la sofisticación del modelo de catenaria, la precisión de los ensayos y la cancelación del retraso. Los ensayos realizados simulan la respuesta de catenarias realistas con la hipótesis simplificativa de periodicidad. Son adecuados para la dinámica de catenarias de vanos iguales en la zona central de cada cantón, sin embargo es necesario seguir realizando esfuerzos para eliminar la condición de periodicidad sin comprometer la precisión de los resultados.[CA] Existeix una extensa xarxa de línies ferroviàries electrificades a tot el món. La majoria d'elles utilitzen Línies Aèries de Contacte o catenàries per a subministrar electricitat als trens. Les catenàries són estructures de cables situades sobre les vies ferroviàries, dissenyades per a ser contactades pels pantògrafs que es troben sobre la part superior de la locomotora. El correcte funcionament del sistema requereix un alt nivell d'exigència, especialment a alta velocitat, quan la continuïtat del contacte es veu compromesa. L'eina majoritària per el sistema pantògraf-catenària és l'ús de simulacions numèriques. En particular, el Mètode dels Elements Finits (MEF) és la tècnica més usada per a modelar i simular la interacció dinàmica del pantògraf amb la catenària. Aquest mètode permet modelar catenàries. Després de l'etapa de simulació, el pantògraf i les catenàries han de ser testats en assajos experimentals en via. No obstant això, existeix una alternativa que pot reemplaçar eixos assajos amb una reducció significativa de costos. Aquesta alternativa, anomenada Hardware in the Loop (HIL), permet testar pantògrafs en el laboratori amb un banc d'assajos que emula la interacció amb una catenària virtual. Diferents grups d'investigació han implementat HIL; no obstant això, en tots els intents s'han adoptat solucions de compromís, la qual cosa demostra el repte que suposa l'aplicació de HIL. Aquesta Tesi pretén avançar en el camp dels assajos HIL, impulsant les capacitats de la tècnica i solucionant algunes de les limitacions trobades en la literatura. Aquesta Tesi proposa dos tipus diferents de models de catenària per al seu ús en assajos HIL. El primer és un model analític basat en un cable tens amb perfil geomètric periòdic que proporciona la solució estacionària del sistema. Aquest model redueix la complexitat de la catenària, però manté les principals característiques que intervenen en la dinàmica. El model ha demostrat ser útil per a explicar la dinàmica fonamental de la catenària, ajudant a comprendre el fenomen d'interferència entre dos pantògrafs. Aquest model analític és adequat per a realitzar assajos HIL a causa del seu baix cost computacional. Aquest treball proposa un algoritme iteratiu per a utilitzar el model analític en assajos HIL de pantògrafs. El fet que el model siga periòdic permet l'aplicació d'una estratègia específica per a compensar el retard del llaç de control. Aquesta estratègia té un excel·lent rendiment i precisió, validats en comparar assajos HIL amb simulacions numèriques. La validació es realitza amb una massa en el lloc del pantògraf per a eliminar les potencials diferències en el model. Si bé la precisió aconseguida és bona, el model analític de catenària manca de fidelitat, la qual cosa ha motivat el desenvolupament d'un model periòdic més avançat. El segon model de catenària per a assajos HIL és el Model Periòdic d'Elements Finits MPEF, discretitzat amb el MEF per a evitar simplificacions topològiques i estructurals addicionals. El model inclou la condició de periodicitat i la dinàmica es resol mitjançant anàlisi en freqüència. A més, les no linealitats de la catenària es consideren en la formulació. Un algoritme iteratiu, similar a l'utilitzat per als assajos HIL amb catenària analítica, és usat per a realitzar assajos HIL amb catenàries MPEF. L'estratègia anterior d'utilització d'una massa s'empra per a validar el sistema d'assajos, resultant tindre una gran precisió. Els resultats són gratificants a causa de la sofisticació del model de catenària, la precisió dels assajos i la cancel·lació del retard. Els assajos realitzats simulen la resposta de catenàries realistes amb la hipòtesi simplificativa de periodicitat. Són adequats per a la dinàmica de catenàries de vans iguals en la zona central dels seccionaments, no obstant això és necessari continuar fent esforços per a eliminar la condició de periodicitat sense comprometre la precisió dels res[EN] There is an extensive network of electrified railway lines over the world. Most of them use overhead contact lines or catenaries to provide the trains with electrical power. Catenaries consist of electrified wires placed over the rail track, designed to contact the pantograph placed on the roof of the train. The proper operation of the system is very demanding, especially at high speed, when the continuity of the contact is compromised. The most predominant tool for studying and designing the pantograph-catenary system is the use of numerical simulations. Notably, the Finite Element Method (FEM) is the most popular technique for modelling and simulating the dynamic interaction of the pantograph and the catenary. This method allows modelling catenaries with outstanding fidelity and without any loss of generality. After the simulation stage, the pantograph and the catenaries have to be assessed by in-line experimental tests. However, there is an alternative that can replace those tests with a significant reduction in costs. The alternative method, called Hardware In the Loop (HIL), allows testing pantographs in the laboratory with a test rig that emulates the interaction with a virtual catenary. Different research groups have implemented HIL; however, in every attempt, a compromise solution has been adopted, demonstrating the challenging nature of HIL. This Thesis aims to advance in the field of HIL tests, pushing forward the capabilities of the technique and solving some of the limitations found in the literature. This Thesis proposes two different kinds of catenary models for their use in HIL tests. The first is an analytical model based on a string of periodic geometric profile that accounts for the steady state. It reduces the complexity of the catenary but keeps the main features involved in the dynamic. The model has proven useful in explaining the fundamental dynamics of the catenary, helping understand the interference between two pantographs. This analytical model is suitable for HIL because of its low computational cost. An iterative algorithm is proposed to use the analytical model in HIL. The fact that the model is periodic permits a specific strategy to compensate the control loop delay. This strategy has excellent performance and accuracy, validated by comparing HIL tests with numerical simulations and getting an agreement. This agreement will not be possible if the pantograph model of the simulations is inaccurate. Therefore, the validation is carried out with a weight or mass model in place of the pantograph to eliminate potential differences. Even though the precision achieved is good, the analytical catenary model lacks fidelity, which has motivated the development of a more advanced periodic model. The second catenary model for HIL tests is the Periodic Finite Element Model (PFEM), discretised with FEM to avoid further topological and structural simplifications. The model includes the periodicity condition, and the dynamics are solved by frequency analysis. Furthermore, the catenary non-linearities are considered in the formulation. An iterative algorithm, similar to the one used for the HIL tests with the analytical catenary, is used to realise HIL tests with PFEM catenaries. The previous strategy with a mass model is used to validate the test, confirming great precision. The results are gratifying due to the sophistication of the model, the accuracy of the tests and the cancellation of the delay. The tests simulate the response of realistic catenaries with the simplifying periodicity hypothesis. They are adequate for the dynamic of equal-span catenary at the central zone of every section, but future efforts have to be made to get rid of the periodicity condition while keeping the accuracy of the results.The authors would like to acknowledge the financial support received from the State Research Agency of the Spanish Science and Innovation Ministry (PID2020- 113458RB-I00) and from the Valencian Regional Government (PROMETEO/2021/046) (PROMETEO/2016/007) and the Spanish Ministry of Economy, Industry and Competitiveness (TRA2017-84736-R), also the funds received jointly from the Regional Government of Valencia and the Euro- pean Social Fund, under Grant APOSTD/2019/205Gil Romero, J. (2022). Pantograph-Catenary Dynamic Models and their Implementation in Hardware-in-the-Loop Tests [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/191501Compendi

Optimal type-3 fuzzy system for solving singular multi-pantograph equations

In this study a new machine learning technique is presented to solve singular multi-pantograph differential equations (SMDEs). A new optimized type-3 fuzzy logic system (T3-FLS) by unscented Kalman filter (UKF) is proposed for solution estimation. The convergence and stability of presented algorithm are ensured by the suggested Lyapunov analysis. By two SMDEs the effectiveness and applicability of the suggested method is demonstrated. The statistical analysis show that the suggested method results in accurate and robust performance and the estimated solution is well converged to the exact solution. The proposed algorithm is simple and can be applied on various SMDEs with variable coefficients

Optimal Type-3 Fuzzy System for Solving Singular Multi-Pantograph Equations

In this study a new machine learning technique is presented to solve singular multi-pantograph differential equations (SMDEs). A new optimized type-3 fuzzy logic system (T3-FLS) by unscented Kalman filter (UKF) is proposed for solution estimation. The convergence and stability of presented algorithm are ensured by the suggested Lyapunov analysis. By two SMDEs the effectiveness and applicability of the suggested method is demonstrated. The statistical analysis show that the suggested method results in accurate and robust performance and the estimated solution is well converged to the exact solution. The proposed algorithm is simple and can be applied on various SMDEs with variable coefficients.publishedVersio

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Since the collocation method approximates ordinary differential equations, partial differential equations, and integral equations in physical space, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear differential equations. In this paper, we derive a Jacobi-Gauss-Lobatto collocation method (J-GL-C) to solve numerically nonlinear time-delayed Burgers-type equations. The proposed technique is implemented in two successive steps. In the first one, we apply nodes of the Jacobi-Gauss-Lobatto quadrature which depend upon the two general parameters , and the resulting equations together with the two-point boundary conditions constitute a system of ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve a system of ODEs of second order in time. We present numerical results which illustrate the accuracy and flexibility of these algorithms

Peak Estimation of Time Delay Systems using Occupation Measures

This work proposes a method to compute the maximum value obtained by a state function along trajectories of a Delay Differential Equation (DDE). An example of this task is finding the maximum number of infected people in an epidemic model with a nonzero incubation period. The variables of this peak estimation problem include the stopping time and the original history (restricted to a class of admissible histories). The original nonconvex DDE peak estimation problem is approximated by an infinite-dimensional Linear Program (LP) in occupation measures, inspired by existing measure-based methods in peak estimation and optimal control. This LP is approximated from above by a sequence of Semidefinite Programs (SDPs) through the moment-Sum of Squares (SOS) hierarchy. Effectiveness of this scheme in providing peak estimates for DDEs is demonstrated with provided examplesComment: 34 pages, 14 figures, 3 table

Open issues in devising software for the numerical solution of implicit delay differential equations

AbstractWe consider initial value problems for systems of implicit delay differential equations of the formMy′(t)=f(t,y(t),y(α1(t,y(t))),…,y(αm(t,y(t)))),where M is a constant square matrix (with arbitrary rank) and αi(t,y(t))⩽t for all t and i.For a numerical treatment of this kind of problems, a software tool has been recently developed [6]; this code is called RADAR5 and is based on a suitable extension to delay equations of the 3-stage Radau IIA Runge–Kutta method.The aim of this work is that of illustrating some important topics which are being investigated in order to increase the efficiency of the code. They are mainly relevant to(i)the error control strategies in relation to derivative discontinuities arising in the solutions of delay equations;(ii)the integration of problems with unbounded delays (like the pantograph equation);(iii)the applications to problems with special structure (as those arising from spatial discretization of evolutions PDEs with delays).Several numerical examples will also be shown in order to illustrate some of the topics discussed in the paper

Approximate Solutions of Multi-Pantograph Type Delay Differential Equations Using Multistage Optimal Homotopy Asymptotic Method

In this paper, a numerical procedure called multistage optimal homotopy asymptotic method (MOHAM) is introduced to solve multi-pantograph equations with time delay. It was shown that the MOHAM algorithm rapidly provides accurate convergent approximate solutions of the exact solution using only one term. A comparative study between the proposed method, the homotopy perturbation method (HPM) and the Taylor matrix method are presented. The obtained results revealed that the method is of higher accuracy, effective and easy to use

Fourier operational matrices of differentiation . . .

This paper introduces Fourier operational matrices of differentiation and transmission for solving high-order linear differential and difference equations with constant coefficients. Moreover, we extend our methods for generalized Pantograph equations with variable coefficients by using Legendre Gauss collocation nodes. In the case of numerical solution of Pantograph equation, an error problem is constructed by means of the residual function and this error problem is solved by using the mentioned collocation scheme. When the exact solution of the problem is not known, the absolute errors can be computed approximately by the numerical solution of the error problem. The reliability and efficiency of the presented approaches are demonstrated by several numerical examples, and also the results are compared with different methods