Marian Anghel

Stochastic Model for Power Grid Dynamics

We introduce a stochastic model that describes the quasi-static dynamics of an electric transmission network under perturbations introduced by random load fluctuations, random removing of system components from service, random repair times for the failed components, and random response times to implement optimal system corrections for removing line overloads in a damaged or stressed transmission network. We use a linear approximation to the network flow equations and apply linear programming techniques that optimize the dispatching of generators and loads in order to eliminate the network overloads associated with a damaged system. We also provide a simple model for the operators response to various contingency events that is not always optimal due to either failure of the state estimation system or due to the incorrect subjective assessment of the severity associated with these events. This further allows us to use a game theoretic framework for casting the optimization of the operators response into the choice of the optimal strategy which minimizes the operating cost. We use a simple strategy space which is the degree of tolerance to line overloads and which is an automatic control (optimization) parameter that can be adjusted to trade off automatic load shed without propagating cascades versus reduced load shed and an increased risk of propagating cascades. The tolerance parameter is chosen to describes a smooth transition from a risk averse to a risk taken strategy. We present numerical results comparing the responses of two power grid systems to optimization approaches with different factors of risk and select the best blackout controlling parameter

Synchronization of trajectories in canonical molecular-dynamics simulations: Observation, explanation, and exploitation

For two methods commonly used to achieve canonical-ensemble sampling in a molecular-dynamics simulation, the Langevin thermostat and the Andersen [H. C. Andersen, J. Chem. Phys. 72, 2384 (1980)] thermostat, we observe, as have others, synchronization of initially independent trajectories in the same potential basin when the same random number sequence is employed. For the first time, we derive the time dependence of this synchronization for a harmonic well and show that the rate of synchronization is proportional to the thermostat coupling strength at weak coupling and inversely proportional at strong coupling with a peak in between. Explanations for the synchronization and the coupling dependence are given for both thermostats. Observation of the effect for a realistic 97-atom system indicates that this phenomenon is quite general. We discuss some of the implications of this effect and propose that it can be exploited to develop new simulation techniques. We give three examples: efficient thermalization (a concept which was also noted by Fahy and Hamann [S. Fahy and D. R. Hamann, Phys. Rev. Lett. 69, 761 (1992)]), time-parallelization of a trajectory in an infrequent-event system, and detecting transitions in an infrequent-event system.

Impact of Time Delays on Power System Stability

The paper describes the impact of time-delays on small-signal angle stability of power systems. With this aim, the paper presents a power system model based on delay differential algebraic equations (DDAE) and describes a general technique for computing the spectrum of DDAE. The paper focuses in particular on delays due to the terminal voltage measurements and transducers of automatic voltage regulators and power system stabilizers of synchronous machines. The proposed technique is applied to a benchmark system, namely the IEEE 14-bus test system, as well as to a real-world system. Time domain simulations are also presented to confirm the results of the DDAE spectral analysis.

Algorithmic Construction of Lyapunov Functions for Power System Stability Analysis

We present a methodology for the algorithmic construction of Lyapunov functions for the transient stability analysis of classical power system models. The proposed methodology uses recent advances in the theory of positive polynomials, semidefinite programming, and sum of squares decomposition, which have been powerful tools for the analysis of systems with polynomial vector fields. In order to apply these techniques to power grid systems described by trigonometric nonlinearities we use an algebraic reformulation technique to recast the systems dynamics into a set of polynomial differential algebraic equations. We demonstrate the application of these techniques to the transient stability analysis of power systems by estimating the region of attraction of the stable operating point. An algorithm to compute the local stability Lyapunov function is described together with an optimization algorithm designed to improve this estimate.

Adaptive mesh refinement for multiscale nonequilibrium physics

In this paper, the authors demonstrate how to use adaptive mesh refinement (AMR) methods for the study of phase transition kinetics. In particular, they apply a block-structured AMR approach to investigate phase ordering in the time-dependent Ginzburg-Landau equations.

Stability analysis of power systems using network decomposition and local gain analysis

We introduce a method for analyzing large-scale power systems by decomposing them into coupled lower order subsystems. This reduces the computational complexity of the analysis and enables us to scale the Sum of Squares programming framework for nonlinear system analysis. The method constructs subsystem Lyapunov functions which are used to estimate the region of attraction pertaining to the equilibrium point of each isolated subsystem. Then a disturbance analysis framework uses the level sets defined by these Lyapunov functions to calculate the stability of pairwise interacting subsystems. This analysis is then used to infer the stability of the entire system when an external disturbance is applied. We demonstrate the application of these techniques to the transient stability analysis of power systems.

Consistency of support vector machines for forecasting the evolution of an unknown ergodic dynamical system from observations with unknown noise

We consider the problem of forecasting the next (observable) state of an unknown ergodic dynamical system from a noisy observation of the present state. Our main result shows, for example, that support vector machines (SVMs) using Gaussian RBF kernels can learn the best forecaster from a sequence of noisy observations if (a) the unknown observational noise process is bounded and has a summable α-mixing rate and (b) the unknown ergodic dynamical system is defined by a Lipschitz continuous function on some compact subset of R d and has a summable decay of correlations for Lipschitz continuous functions. In order to prove this result we first establish a general consistency result for SVMs and all stochastic processes that satisfy a mixing notion that is substantially weaker than α-mixing.

Dynamical System Analysis and Forecasting of Deformation Produced by an Earthquake Fault

Abstract — We present a method of constructing low-dimensional nonlinear models describing the main dynamical features of a discrete 2-D cellular fault zone, with many degrees of freedom, embedded in a 3-D elastic solid. A given fault system is characterized by a set of parameters that describe the dynamics, rheology, property disorder, and fault geometry. Depending on the location in the system parameter space, we show that the coarse dynamics of the fault can be confined to an attractor whose dimension is significantly smaller than the space in which the dynamics takes place. Our strategy of system reduction is to search for a few coherent structures that dominate the dynamics and to capture the interaction between these coherent structures. The identification of the basic interacting structures is obtained by applying the Proper Orthogonal Decomposition (POD) to the surface deformation fields that accompany strike-slip faulting accumulated over equal time intervals. We use a feed-forward artificial neural network (ANN) architecture for the identification of the system dynamics projected onto the subspace (model space) spanned by the most energetic coherent structures. The ANN is trained using a standard back-propagation algorithm to predict (map) the values of the observed model state at a future time, given the observed model state at the present time. This ANN provides an approximate, large-scale, dynamical model for the fault. The map can be evaluated once to provide a short-term predictions or iterated to obtain a prediction for the long-term fault dynamics.

On the effective dimension and dynamic complexity of earthquake faults

Abstract We measure the effective dimensionality of a driven, dissipative fault model as its dynamics explore a wide parameter range from a crack like model to a dislocation model. The dynamics of each fault model are probed by recording (a) the first and second order moments of the stresses and slips defined in the fault plane, and (b) the surface deformations that indirectly reflect the brittle processes of the fault and which are observable by InSAR and GPS techniques. In order to study the asymptotic attractors of the model we identify the coherent structures (dominant modes) present in the surface deformation fields and project the model dynamics onto the principal directions defined by these coherent structures. The projection is based on the Karhunen–Loeve procedure for the determination of an optimal set of basis functions based on second order statistics. We estimate the effective dimensionality of the dynamics by computing the number of modes needed to capture a certain fraction of the statistical variation of the surface deformation fields. We detect a sharp transition in the number of effective degrees of freedom as we vary the dynamic weakening toward larger and dynamically more significant values. This transition is also associated with a separation of the dynamics in slow and fast degrees of freedom and with the presence of multiple length and time scales in the dynamics. This conclusion is also supported by direct dimension estimates using the correlation dimension. We finally compute the significance of evidence for nonlinearity using the method of surrogate data on the correlation dimension statistics.

Adaptive learning in random linear nanoscale networks

While the top-down engineered CMOS technology favors regular and locally interconnected structures, emerging molecular and nanoscale bottom-up self-assembled devices will be built from vast numbers of simple, densely arranged components that exhibit high failure rates, are relatively slow, and connected in a disordered way. Such systems are not programmable by standard means. Here we provide a solution to the supervised learning problem of mapping a desired binary input to a desired binary output in an random nanoscale network of linear functions with given control nodes. The network model is inspired after self-assembled silver nanowires. Our results show that one- and two-control node random networks can implement linearly separable sets.