October 16, 2014
This paper deals with the development and the analysis of asymptotic stable and consistent schemes in the joint quasi-neutral and fluid limits for the collisional Vlasov-Poisson system. In these limits, the classical explicit schemes suffer from time step restrictions due to the small plasma period and Knudsen number. To solve this problem, we propose a new scheme stable for choices of time steps independent from the small scales dynamics and with comparable computational cost with respect to stan- dard explicit schemes. In addition, this scheme reduces automatically to consistent discretizations of the underlying asymptotic systems. In this first work on this subject, we propose a first order in time scheme and we perform a relative linear stabilityanalysis to deal with such problems. The framework we propose permits to extend this approach to high order schemes in the next future. We finally show the capability of the method in dealing with small scales through numerical experiments.
Keywords: hybrid systems, nonlinear systems, Coulomb friction, Lyapunov methods, global asymptotic stability, PID control.
In this paper, we present a hybrid model formulation for a proportional-integral-derivative (PID) controlled single- mass motion system subject to Coulomb friction. Friction is a performance-limiting factor in many high-precision positioning systems in terms of the achievable setpoint accuracy and settling times. Many different control strate- gies have been developed for frictional systems (e.g., model-based compensation, see Armstrong-H´ elouvry et al. (1994), impulsive control, see van de Wouw and Leine (2012), or sliding mode control, see Bartolini et al. (2003)). PID control, however, is still used in the vast majority of industrial motion systems, since the integrator action is able to compensate for unknown static friction. However, it has performance limitations (e.g., long settling times, see Beerens et al. (2018)). The popularity of the PID controller for mechatronics applications in industry, how- ever, motivates the development of hybrid add-ons, such as reset control strategies, to complement the classical PID controller and improve its baseline performance.
All generators except g13, g19 and g20 are equipped with PSS using the rotor speed ω as input (a zero value for K p in Table 13 indicates the absence of PSS). ω is in per unit. Each PSS includes a washout filter and
two identical lead filters in cascade. The PSS phase compensation was chosen considering the maximum and minimum equivalent Th´evenin impedances seen by the machines of each group (units 7 and 18 for the round-rotor, units 4 and 12 for the salient-pole machines). The PSS transfer functions provide damping for oscillation frequencies from 0.2 Hz to more than 1 Hz.
While polynomial approximation methods for the analysis of infinite dimension systems is not a new idea (see for in- stance the convex optimization and sum-of squares frame- works developed in Papachristodoulou and Peet (2006); Peet (2014) or Ahmadi et al. (2014)), the novelty of this approach relies on the use of efficient integral inequalities that are able to give a measure of the conservatism as- sociated to the approximation. These inequalities can be interpreted as a Bessel inequality on Hilbert spaces. In previous work, e.g. Seuret and Gouaisbaut (2014, 2015), the efficiency of these inequalities for the stabilityanalysis of TDS has been shown. Indeed, one can also read in Seuret et al. (2015) a method based on a polynomial approximation of the distributed nature of the delay, using Legendre polynomials and their properties to construct Lyapunov-Krasovskii functionals. In the present paper, where a simple transport equation replaces the delay terms (an approach also studied in e.g. Bekiaris-Liberis and Krstic (2013)), an alternative use of this new method is proposed.
More generally, this system can be seen either as the control of the PDE by a finite dimensional dynamic control law generated by an ODE  or on the contrary the robustness of a linear closed loop system with a control signal conveyed by a damped string equation. On the first scenario, both the ODE and the PDE are stable and the stability of the coupled system is studied. The second case corresponds to an unstable but stabilizable ODE connected to a stable PDE. To sum up, this paper focuses on the stabilityanalysis of closed-loop coupled system (1) with a potentially unstable closed-loop ODE but a stable PDE. This differs significantly from the backstepping methodology of  which aims at designing an infinite dimensional control law ensuring the stability of a cascaded ODE-PDE system with a closed-loop stable ODE. A. Existence and regularity of solutions
Abstract: The objective of this contribution is to improve recent stability results for a system coupling ordinary differential equations to a vectorial transport partial differential equation by proposing a new structure of Lyapunov functional. Following the same process of most of the investigations in literature, that are based on an a priori selection of Lyapunov functionals and use the usual integral inequalities (Jensen, Wirtinger, Bessel...), we will present an efficient method to estimate the exponential decay rate of this coupled system leading to a tractable test expressed in terms of linear matrix inequalities. These LMI conditions stem from the new design of a candidate Lyapunov functional, but also the inherent properties of the Legendre polynomials, that are used to build a projection of the infinite dimensional part of the state of the system. Based on these polynomials and using the appropriate Bessel-Legendre inequality, we will prove an exponential stability result and in the end, we will show the efficiency of our approach on academic example.
for a given function h : [0, ∞) → R.
This suggests to search for stationary solutions of (1)–(4) as minimizers of a Lya- pounov functional using (8), (9) and (10). The natural energy (7) does not have enough structure to serve as a Lyapounov functional for the evolution problem. The idea is thus to combine (7) and (10) and to study the critical points of this Casimir- energy functional, under a mass constraint. It turns out that the critical points are also stationary solutions of (1)–(4). Moreover, the minimization property defines the framework that allows us to investigate the stability issue. We refer for instance the reader to [3, 6, 12, 18, 26] for similar reasonings, see also the overview . We proceed as follows. Let h : [0, ∞) → R be a strictly convex function. We set
∗ LAAS-CNRS, Universit´ e de Toulouse, CNRS, UPS, Toulouse,
Abstract: This paper addresses the stabilityanalysis of a system of ordinary differential equations coupled with a classic heat equation using a Lyapunov approach. Relying on recent developments in the area of time-delay systems, a new method to study the stability of such a class of coupled finite/infinite dimensional systems is presented here. It consists in a Lyapunov analysis of the infinite dimensional state of the system using an energy functional enriched by the mean value of the heat variable. The main technical step relies on the use an efficient Bessel-like integral inequality on Hilbert space leading to tractable conditions expressed in terms of linear matrix inequalities. The results are then illustrated on academic examples and demonstrate the potential of this new approach.
The idea of using Simulink to develop an educational pack- age for the analysis of Power System Dynamics first arose in 1997 at ULg and matured progressively, in particular within the context of a collaboration with the University of Bologna . On the lines of this original idea various modules simulat- ing power system components have been developed at ULg, as well as in the Electrical Energy Systems Lab of NTUA. These include generic models of traditional components (synchro- nous generator, AVR, governor, voltage-dependent loads, in- duction motor, SVC, etc.), as well as specific models for FACTS devices, steam and hydro power plants, wind parks
3.6 Observation of chaotic dynamics
This nonlinear system exhibits the double-scroll chaotic attractor. It should be pointed out that, with different parameters, system (3) can evolve to other complex dynamics such as single-scroll chaotic attractor and periodic orbit (Huang and Zhao, 2016). In this system, the capacitor C is a parameter of control and the value of C can be changed within a certain range. When the parameter C is changed, the chaos behaviour of this system can effectively be controlled. In the numerical simulation, the initial values of the system (3) are (0, 5, 4). Using MATLAB program, the numerical simulation have been completed. This nonlinear system exhibits the complex and abundant of the chaotic dynamics behaviours, the strange attractors (Tremori et al., 2016) are shown in Figure 6. These phase portrait are obtained by solving equations (3) by means of Runge-Kutta method for step size of 0.000001. At the beginning of simulation u 1 = u 2 = 0. Now is clear that the
E5A Ca 2+ (from left to right), with respect to the wild-type (HSE) pilus. The differences are then mapped on the structure of each system. The 297
mutation positions are shown with yellow sticks and pointed to by black arrows. (D) The distances between pairs of residues forming salt
Power systems are evolving towards massive penetration of renewable energy resources, including utility-scale and distributed photovoltaic (PV) plants, to leverage their economic and environmental advantages . Regarding the operation of PV-rich power systems, one of the main issues is related to frequency instability which corresponds to generation-load mismatch and may lead to cascading failures in the form of generation trip, load shedding, or even splitting of the system into islanded areas . Technically, a system with adequate frequency control ancillary services (FCAS) is more likely to regain a stable equilibrium point following frequency contingencies . However, the increase of PV penetration and the subsequent decline in synchronous generation drives power systems into low-inertia conditions which may result in higher rate of change of frequency (RoCoF) values following disturbances, also meaning faster frequency dynamics in general. It is therefore necessary to model the system adequately to be able to capture such fast frequency behaviour. For short-term frequency stabilityanalysis, electromagnetic transient (EMT) models or transient stability models have been mostly employed, which comprise numerous differential-algebraic equations (DAEs) to provide proper representation of power system components . Moreover, there are practical evidence that transmission- connected PV units are required by grid-codes to participate in primary frequency response while the frequency is beyond the normal frequency operating band , . Hence, it is also
a transmission line modeled by hyperbolic PDEs. Since we combine both dynamics, the coupling is defined at the boundaries of the hyperbolic PDEs. It results in what we call, coupled PDE-ODE with dynamic boundary condition. We aim then at studying the stability of such systems un- der the assumption that one wants the system to operate in free-flow zone to avoid congestion. The main contribution of this work is the modeling and the study of input- to-state stability properties of a communication network when operating at some optimal equilibrium point. This paper is organized as follows. In Section 2 we in- troduce the proposed model. Section 3 contains the well- posedness andstabilityanalysis, including the steady-state characterization. Section 4 provides a numerical example to illustrate the results. Finally, conclusions are given in Section 5.
proliferation of CIGs, faster dynamics will gain more prominence when analyzing future power system dynamic behavior compared to the phenomena within the time scale of several milliseconds to minutes. Focusing on the time scale of the electromechanical transients enabled several simplifications in power system modeling and representation, which significantly aided the characterization andanalysis of the related phenomena. A key aspect of these simplifications is the assumption that voltage and current waveforms are dominated by the fundamental frequency component of the system (50 or 60 Hz). As a consequence, the electrical network could be modeled considering steady-state voltage and current phasors, also known as a quasi-static phasor modeling approach. With this modelling approach, high- frequency dynamics and phenomena, such as the dynamics associated with the switching of power electronic converters, are only represented by either steady-state models or simplified dynamic models, meaning that fast phenomena, like switching, cannot be completely captured. Considering the CIG related time scales of operation mentioned previously, there is a need to extend the bandwidth of the phenomena to be examined and include faster dynamics within electromagnetic time scales when the faster dynamics is of importance and can affect overall system dynamics.
Stabilityanalysis of singularly perturbed switched and impulsive linear systems
Jihene Ben Rejeb, Irinel-Constantin Mor˘arescu, Antoine Girard and Jamal Daafouz
Abstract— This paper proposes a new methodology for stabilityanalysis of singularly perturbed linear systems whose dynamics is affected by switches and state jumps. The overall problem is formulated in the framework of hybrid singularly perturbed systems and we use Lyapunov-based techniques to investigate its stability. We emphasize that, beside the stability of slow and fast dynamics, we need a dwell-time condition to guarantee the overall singularly perturbed system is globally asymptotically stable. Furthermore, we characterize this dwell-time as the sum of one term related to the stabilization of systems evolving on one time-scale (slow dynamics) and one term of the order of the parameter defining the ratio between the time-scales. As highlighted in the paper the second term is required to compensate the effect of the jumps introduced in the state of the boundary layer system by the switches and impulses affecting the overall dynamics. Some numerical examples illustrates our results.
The next objective of data mining will be to determine the main weak points of the power system being studied (i.e., the relative frequency of different types of behaviors: stable, voltage collapse, cascading line tripping, loss of synchronism, frequency collapse, oscillations, dynamic instabilities...). This can be achieved using automatic un- supervised learning techniques, as illustrated in Section 5. Finally, further steps will consist of assessing how the different variables used to characterize the scenarios are interrelated, and will automatically identify those that are useful to detect various undesirable phenomena. In particular, supervised learning and correlation analysis may be used to screen automatically a large number of variables and scenarios relevant for each task. Notice that at this step the engineer can try out various possibili- ties, being limited only by his imagination and the in- formation contained in the database. In order to find efficient ways to predict system failures, he may try out different ways of decomposing the overall problem into subproblems, and test different sets of measurements and detection logics. Such detection rules may be defined either manually or automatically using supervised learn- ing methods.
More precisely, after a contingency inception and its clearance, SIME drives a time-domain program so as to accomplish the fol- lowing tasks: identify the critical and non critical machines and aggregate them into two groups; replace these groups by succes- sively a two-machine, then an OMIB equivalent system; assess transient stability of this OMIB, using the EAC . The various steps of the method are briefly described below and illustrated in Figs 1 drawn for the 3-machine system simulated in Section 5. For more details about SIME, see .
II. B OND G RAPH B ASIC P RINCIPLES
The BG formalism [1-3], introduced by H. Paynter in 1961 and formalised by Karnopp and Rosenberg in 1975, is now regularly used in many companies, particularly in automobile industry (PSA, Renault, Ford, Toyota, General Motors, …). This graphical method illustrates the energetic transfers in the system. The orientation of a half arrow indicates the power flow as expressed in Fig. 1. The primary characteristic of the BG is relative to its unified aspect through analogies summarised in Table 1. It obviously shows that this method can be applied to all physical domains by using generalised variables for :
Figure 2(a) obviously shows the stable behaviors detected by Theorem 1 with N = 0, with a quite fast convergence to the equilibrium. The illustration of the second case Figure 2(b) is consistent with Figure 1, since the solution of this system diverges. This is consistent with the fact that no solutions to the conditions of Theorem 1 can be found for any N ≤ 12. More interestingly, the last situation, presented in Figure 2(c), shows simulations results which are very slowly converging to the origin, with however a lightly damped oscillatory behavior of the state of the ODE and of the PDE close to the boundary x = 0. On the other side, the state function u(x, t) for sufficiently large values of x is clearly smooth and converges slowly to the origin. Actually, case (c) illustrates a situation where a very small diffusion coefficient γ induces a slow convergent behavior for which the conditions of Theorem 1 are only fulfilled for a large parameter N ≥ 5. This may indicate a
Index Terms—Lyapunov theory, nonlinear systems, power systems analysis, region of attraction, sum of squares.
I. I NTRODUCTION
Transient stability is one of the most important power systems analysis problems. From a physical viewpoint, transient stability can be defined as the ability of a power system to maintain its synchronism when subjected to large and transient disturbances . Widespread assessment tools are generally based on indirect methods which rely on the numerical integration of nonlinear differential equations describing system dynamics. However, this approach is not suited to synthesize controllers with a direct quantification of stability margin since it does not provide an analytical characterization of stability . Alternatively, direct methods are based on the estimation of the stability domain of the equilibrium point; they ensure that all the trajectories initiated in this domain converge to the equilibrium point . The main drawback of direct methods is that they rely on the identification of Lyapunov functions which are hard to determine. Indeed, there is not a systematic method for the construction of Lyapunov functions. Moreover, it has been shown that quadratic (i.e., energy type) Lyapunov functions