Hybrid analysis of nonlinear circuits: DAE models with indices zero and one

We extend in this paper some previous results concerning the differential-algebraic index of hybrid models of electrical and electronic circuits. Specifically, we present a comprehensive index characterization which holds without passivity requirements, in contrast to previous approaches, and which applies to nonlinear circuits composed of uncoupled, one-port devices. The index conditions, which are stated in terms of the forest structure of certain digraph minors, do not depend on the specific tree chosen in the formulation of the hybrid equations. Additionally, we show how to include memristors in hybrid circuit models; in this direction, we extend the index analysis to circuits including active memristors, which have been recently used in the design of nonlinear oscillators and chaotic circuits. We also discuss the extension of these results to circuits with controlled sources, making our framework of interest in the analysis of circuits with transistors, amplifiers, and other multiterminal devices

Quantication of the Impact of Uncertainty in Power Systems using Convex Optimization

University of Minnesota Ph.D. dissertation. June 2017. Major: Electrical Engineering. Advisor: Sairaj Dhople. 1 computer file (PDF); viii, 85 pages.Rampant integration of renewable resources (e.g., photovoltaic and wind-energy conversion systems) and uncontrollable and elastic loads (e.g., plug-in hybrid electric vehicles) are rapidly transforming power systems. In this environment, an analytic method to quantify the impact of parametric and input uncertainty will be critical to ensure the reliable operation of next-generation power systems. This task is analytically and computationally challenging since power-system dynamics are nonlinear in nature. In this thesis, we present analytic methods to quantify the impact of parametric and input uncertainties for two important applications in power systems: i) uncertainty propagation in power-system differential-algebraic equation model and power flow, and ii) robust stability assessment of power-system dynamics. For the first topic, an optimization-based method is presented to estimate maximum and minimum bounds on state variables while acknowleding worst-case parametric and input uncertainties in the model. The approach leverages a second-order Taylor-series expansion of the states around a nominal (known) solution. Maximum and minimum bounds are then estimated from either Semidefinite relaxation of Quadratically-Constrained Quadratic-Programming or Alternating Direction Method of Multipliers. For the second topic, an analytical method to quantify power systems stability margins while acknowleding uncertainty is presented within the framework of Lyapunov's direct method. It focuses on the algorithmic construction of Lyapunov functions and the estimation of the robust Region-Of-Attraction with Sum-of-Squares optimization problems which can be translated into semidefinite problems. For both topics, numerical case studies are presented for different test systems to demonstrate and validate the proposed methods

Multiphysics Simulation and Model-based System Testing of Automotive E-Powertrains

Programa Oficial de Doutoramento en Enxeñaría Naval e Industrial . 5015V01[Abstract] Model-Based System Testing emerges as a new paradigm for the development cycle that is currently gaining momentum, especially in the automotive industry. This novel approach is focused on combining computer simulation and real experimentation to shift the bulk of problem detection and redesign tasks towards the early stages of the developments. Along these lines, Model-Based System Testing is aimed at decreasing the amount of resources invested in these tasks and enabling the early identification of design flaws and operation problems before a full-vehicle prototype is available. The use of Model-Based System Testing, however, requires to implement some critical technologies, three of which will be discussed in this thesis. The first task addressed in this thesis is the design of a multiplatform framework to assess the description and resolution of the equations of motion of virtual models used in simulation. This framework enables the efficiency evaluation of different modelling and solution methods and implementations. In Model-Based System Testing contexts virtual models interact with physical components, therefore it is mandatory to guarantee their real-time capabilities, regardless of the software or hardware implementations. Second, estimation techniques based on Kalman Filters are of interest in Model- Based System Testing applications to evaluate parameters, inputs or states of a virtual model of a given system. These procedures can be combined with the use of Digital Twins, virtual counterparts of real systems, with which they exchange information in a two-way communication. The available measurements from the sensors located at a physical system can be fused with the results obtained from the simulation of the virtual model. Thus, this avenue improves the knowledge of the magnitudes that cannot be measured directly by these sensors. In turn, the outcomes obtained from the simulation of the virtual model could serve to make decisions and apply corrective actions onto the physical system. Third, co-simulation techniques are necessary when a system is split into several subsystems that are coordinated through the exchange of a reduced set of variables at discrete points in time. This is the case with a majority of Model-Based System Testing applications, in which physical and virtual components are coupled through a discrete-time communication gateway. The resulting cyber-physical applications are essentially an example of real-time co-simulation, in which all the subsystems need to achieve real-time performance. Due to the presence of physical components, which cannot iterate over their integration steps, explicit schemes are often mandatory. These, however, introduce errors associated with the inherent delays of a discrete communication interface. These errors can render co-simulation results inaccurate and even unstable unless they are eliminated. This thesis will address this correction by means of an energy-based procedure that considers the power exchange between subsystems. This research work concludes with an example of a cyber-physical application, in which real components are interfaced to a virtual environment, which requires the application of all the MBST technologies addressed in this thesis.[Resumen] Los ensayos de sistemas basados en modelos emergen como un nuevo paradigma de desarrollo que actualmente está ganando popularidad, especialmente en la industria automotriz. Este nuevo enfoque se centra en combinar la simulación por ordenador con la experimentación para desplazar la mayor parte de la detección de problemas y rediseños hacia las fases tempranas del desarrollo. De esta forma, los ensayos de sistemas basados en modelos se centran en disminuir la cantidad de recursos invertidos en estas tareas y habilitar la identificación temprana de errores de diseño y problemas durante la operación, incluso antes de que los prototipos del vehículo completo estén disponibles. Sin embargo, el uso de esta estrategia requiere implementar algunas tecnologías críticas, tres de las cuales serán tratadas en esta tesis. La primera tarea abordada en esta tesis es el diseño de un entorno multiplataforma para evaluar la descripción y resolución de las ecuaciones de la dinámica de los modelos virtuales usados en las simulaciones. Este marco permite una evaluación eficiente de las diferentes formas de modelar los sistemas y de los métodos de resolución e implementación. En este contexto de ensayos basados en modelos, los sistemas virtuales interactúan con los componentes de los sistemas físicos, por lo tanto es necesario garantizar sus capacidades de ejecución en tiempo real, independientemente de la plataforma de software y hardware utilizada. En segundo lugar, las técnicas de estimación basadas en filtros de Kalman son de gran interés en las aplicaciones que usan ensayos basados en modelos para evaluar los parámetros, entradas o estados de los modelos virtuales de un sistema dado. Estos procedimientos se pueden combinar con el uso de gemelos digitales, homólogos virtuales de un sistema físico, con el cual mantienen un flujo bidireccional de intercambio de información. Las medidas disponibles procedentes de los sensores instalados en un sistema físico se pueden combinar con los resultados obtenidos de la simulación del sistema virtual. De este modo, este enfoque mejora el conocimiento de las magnitudes que no pueden ser medidas directamente por los sensores. A su vez, los resultados de la simulación de los sistemas de los modelos virtuales pueden servir para tomar decisiones y aplicar medidas correctivas al sistema real. En tercer lugar, las técnicas de co-simulación son necesarias cuando un sistema se divide en varios subsistemas, coordinados a través del intercambio de un reducido número de variables en momentos puntuales. Este es el caso de la mayor parte de las aplicaciones que siguen la estrategia de ensayos basados en modelos, en los cuales los componentes físicos y virtuales se acoplan mediante una comunicación en tiempo discreto. Como resultado las aplicaciones ciberfísicas son en esencia un ejemplo de co-simulación en tiempo real, en la que todos los subsistemas necesitan cumplir los requisitos de ejecución en tiempo real. Debido a la presencia de componentes físicos, que no pueden reiterar sus pasos de integración, el uso de esquemas explícitos es frecuentemente necesario. Sin embargo, estos esquemas introducen errores asociados con los retrasos propios de una interfaz de tiempo discreto. Estos errores pueden dar lugar a resultados erróneos e incluso inestabilizar la co-simulación, si no son eliminados. Esta tesis aborda la corrección de la co-simulación a través de métodos energéticos basados en la potencia intercambiada por los subsistemas. Este trabajo de investigación concluye con un ejemplo de aplicación ciberfísica, en la que se conectan componentes reales a una simulación por ordenador. Esta aplicación requiere la aplicación de las tecnologías de ensayos basados en modelos presentadas a lo largo de esta tesis.[Resumo] Os ensaios de sistemas baseados en modelos xorden como un novo paradigma de desenvolvemento que actualmente está gañando popularidade, especialmente na industria automotriz. Este novo enfoque céntrase en combinar a simulación por ordenador coa experimentación para desprazar a maior parte da detección de problemas e redeseños cara as fases iniciais do ciclo de produto. Deste xeito, os ensaios de sistemas baseados en modelos fundaméntanse en diminuír a cantidade de recursos investidos nestas tarefas e habilitar a identificación temperá de erros de deseño e problemas durante a operación, aínda se os prototipos do vehículo completo non están dispoñibeis. Porén, o uso desta estratexia require implementar algunhas tecnoloxías críicas, tres das cales serán tratadas nesta tese. A primeira tarefa tratada nesta tese é o deseño dun entorno multiplataforma para avaliar a descripción e resolución das ecuacións da dinámica dos modelos virtuais empregados nas simulacións. Este entorno permite unha avaluación eficiente dos diferentes xeitos de modelar os sistemas e dos métodos de resolución e implementación. Neste contexto de ensaios baseados en modelos, os sistemas virtuais interactúan cos compoñentes dos sistemas físicos, polo tanto é necesario garantir as súas capacidades de execución en tempo real, independentemente da plataforma de hardware e software escollida. En segundo lugar, as técnicas de estimación baseadas en filtros de Kalman son de grande interese nas aplicacións que usan ensaios baseados en modelos para avaliar os seus parámetros, entradas ou estados dos modelos virtuais dun certo sistema. Estes procedementos pódense combinar co uso de xemelgos dixitais, homólogos virtuais dun sistema físico, co cal manteñen un fluxo bidireccional de intercambio de información. As medidas dispoñíbeis procedentes dos sensores instalados nun sistema físico pódense combinar cos resultados obtidos da simulación do sistema virtual. Deste xeito, este enfoque mellora o coñecemento das magnitudes que non poden ser medidas directamente polos sensores. Á súa vez, os resultados da simulación dos sistemas dos modelos virtuais poden servir para tomar decisións e aplicar medidas correctivas ao sistema real. En terceiro lugar, as técnicas de co-simulación son necesarias cando un sistema é dividido en varios subsistemas, coordinados a través do intercambio dun reducido número de variables en momentos puntuais. Este é o caso da maior parte das aplicacións que seguen a estratexia de ensaios baseados en modelos, nos cales os compoñentes físicos e virtuais se acoplan mediante unha comunicación en tempo discreto. Como resultado as aplicacións ciberfísicas son esencialmente un exemplo de co-simulación en tempo real, na que tódolos subsistemas necesitan cumprir os requisitos de execución en tempo real. Debido á presenza de compoñentes físicos, que non poden reiterar os seus pasos de integración, o uso de esquemas explícitos é polo xeral necesario. Con todo, estes esquemas introducen erros asociados cos atrasos derivados dunha interface de tempo discreto. Estes erros poden provocar resultados incorrectos e incluso inestabilizar a co-simulación, de non seren eliminados. Esta tese aborda a corrección da co-simulación a través de métodos enerxéticos baseados na potencia intercambiada polos subsistemas. Este traballo conclúe cun exemplo de aplicación ciberfísica, na que os compoñentes reais son conectados a un entorno virtual. Isto require o emprego de tódalas tecnoloxías de ensaios baseadas en modelos presentadas ao longo desta tese

SCEE 2008 book of abstracts : the 7th International Conference on Scientific Computing in Electrical Engineering (SCEE 2008), September 28 – October 3, 2008, Helsinki University of Technology, Espoo, Finland

This report contains abstracts of presentations given at the SCEE 2008 conference.reviewe

Robust stability assessment for future power systems

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2018.Cataloged from PDF version of thesis. "Due to the condition of the original material, there are unavoidable flaws in this reproduction. Some pages in the original document contain text that is illegible"--Disclaimer Notice page.Includes bibliographical references (pages 119-128).Loss of stability in electrical power systems may eventually lead to blackouts which, despite being rare, are extremely costly. However, ensuring system stability is a non-trivial task for several reasons. First, power grids, by nature, are complex nonlinear dynamical systems, so assessing and maintaining system stability is challenging mainly due to the co-existence of multiple equilibria and the lack of global stability. Second, the systems are subject to various sources of uncertainties. For example, the renewable energy injections may vary depending on the weather conditions. Unfortunately, existing security assessment may not be sufficient to verify system stability in the presence of such uncertainties. This thesis focuses on new scalable approaches for robust stability assessment applicable to three main types of stability, i.e., long-term voltage, transient, and small-signal stability. In the first part of this thesis, I develop a novel computationally tractable technique for constructing Optimal Power Flow (OPF) feasibility (convex) subsets. For any inner point of the subset, the power flow problem is guaranteed to have a feasible solution which satisfies all the operational constraints considered in the corresponding OPF. This inner approximation technique is developed based on Brouwer's fixed point theorem as the existence of a solution can be verified through a self-mapping condition. The self-mapping condition along with other operational constraints are incorporated in an optimization problem to find the largest feasible subsets. Such an optimization problem is nonlinear, but any feasible solution will correspond to a valid OPF feasibility estimation. Simulation results tested on several IEEE test cases up to 300 buses show that the estimation covers a substantial fraction of the true feasible set. Next, I introduce another inner approximation technique for estimating an attraction domain of a post-fault equilibrium based on contraction analysis. In particular, I construct a contraction region where the initial conditions are "forgotten", i.e., all trajectories starting from inside this region will exponentially converge to each other. An attraction basin is constructed by inscribing the largest ball in the contraction region. To verify contraction of a Differential-Algebraic Equation (DAE) system, I also show that one can rely on the analysis of extended virtual systems which are reducible to the original one. Moreover, the Jacobians of the synthetic systems can always be expressed in a linear form of state variables because any polynomial system has a quadratic representation. This makes the synthetic system analysis more appropriate for contraction region estimation in a large scale. In the final part of the thesis, I focus on small-signal stability assessment under load dynamic uncertainties. After introducing a generic impedance-based load model which can capture the uncertainty, I propose a new robust small signal (RSS) stability criterion. Semidefinite programming is used to find a structured Lyapunov matrix, and if it exists, the system is provably RSS stable. An important application of the criterion is to characterize operating regions which are safe from Hopf bifurcations. The robust stability assessment techniques developed in this thesis primarily address the needs of a system operator in electrical power systems. The results, however, can be naturally extended to other nonlinear dynamical systems that arise in different fields such as biology, biomedicine, economics, neuron networks, and optimization. As the robust assessment is based on sufficient conditions for stability, there is still room for development on reducing the inevitable conservatism. For example, for OPF feasibility region estimation, an important open question considers what tighter bounds on the nonlinear residual terms one can use instead of box type bounds. Also, for attraction basin problem, finding the optimal norms and metrics which result in the largest contraction domain is an interesting potential research question.by Hung Dinh Nguyen.Ph. D

Contraction analysis of nonlinear systems and its application

The thesis addresses various issues concerning the convergence properties of switched systems and differential algebraic equation (DAE) systems. Specifically, we focus on contraction analysis problem, as well as tackling problems related to stabilization and synchronization. We consider the contraction analysis of switched systems and DAE systems. To address this, a transformation is employed to convert the contraction analysis problem into a stabilization analysis problem. This transformation involves the introduction of virtual systems, which exhibit a strong connection with the Jacobian matrix of the vector field. Analyzing these systems poses a significant challenge due to the distinctive structure of their Jacobian matrices. Regarding the switched systems, a time-dependent switching law is established to guarantee uniform global exponential stability (UGES). As for the DAE system, we begin by embedding it into an ODE system. Subsequently, the UGES property is ensured by analyzing its matrix measure. As our first application, we utilize our approach to stabilize time-invariant switched systems and time-invariant DAE systems, respectively. This involves designing control laws to achieve system contractivity, thereby ensuring that the trajectory set encompasses the equilibrium point. In oursecond application, we propose the design of a time-varying observer by treating the system’s output as an algebraic equation of the DAE system. In our study on synchronization problems, we investigate two types of synchronization issues: the trajectory tracking of switched oscillators and the pinning state synchronization. In the case of switched oscillators, we devise a time-dependent switching law to ensure that these oscillators effectively follow the trajectory of a time-varying system. As for the pinning synchronization problem, we define solvable conditions and, building upon these conditions, we utilize contraction theory to design dynamic controllers that guarantee synchronization is achieved among the agents