The loading margin to a saddle-node or fold bifurcation measures the proximity to voltage collapse blackouts of electric power transmission systems. Sensitivities of the loading margin can be used to select controls to avoid voltage collapse. We analytically justify the use of static models to compute loading margins and their sensitivities and explain how the results apply to underlying dynamic models. The relation between fold bifurcations of the static models and saddle-node bifurcations of the underlying dynamic models is clarified.

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Many applications of smart grid can be formulated as constrained optimization problems. Because of the discrete controls involved in power systems, these problems are essentially mixed-integer nonlinear programs. In this paper, we review the Trust-Tech-based methodology for solving mixed-integer nonlinear optimization. Specifically, we have developed a two-stage Trust-Tech-based methodology to systematically compute all the local optimal solutions for constrained mixed-integer nonlinear programming (MINLP) problems. In the first stage, for a given MINLP problem this methodology starts with the construction of a new, continuous, unconstrained problem through relaxation and the penalty function method. A corresponding dynamical system is then constructed to search for a set of local optimal solutions for the unconstrained problem. In the second stage, a reduced constrained NLP is defined for each local optimal solution by determining and fixing the values of integral variables of the MINLP problem. The Trust-Tech-based method is used to compute a set of local optimal solutions for these reduced NLP problems, from which the optimal solution of the original MINLP problem is determined. A numerical simulation of several testing problems is provided to illustrate the effectiveness of our proposed method.

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In this paper we develop optimal sensor siting methods along the edges of a large network of electrical oscillators to identify a parametric model for the network using dynamic measurements of electrical signals corrupted with Gaussian noise. We pose the identification problem as estimation of four essential parameters for each edge in the network, namely the real and imaginary components of the edge-weight, or equivalently, the resistance and reactance of the tie-line connecting any two oscillators, and the inertias of the oscillators connected by this edge. We then formulate the Cramer-Rao bounds for the estimates of these four unknown parameters using three fundamental outputs - namely, the magnitude, the phase angle and the frequency of the voltage phasor along each edge, and show that the bounds are functions of the sensor locations on the edges as well as of the contribution of each variable in the combined output. We finally state the condition for finding the optimal sensor location and the optimal signal combination to achieve the tightest Cramer-Rao bound. The problem is first addressed for open-loop networks and, thereafter, for networks where outputs measured at desired locations on the edges are fed back to the nodes to improve transient performance. We show that unlike the first case where the open-loop configuration allows us to optimize the bounds in a distributed fashion for each individual edge, for the latter situation the problem no longer has a decoupled structure under the influence of feedback, and must be carried out in a centralized fashion.

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Symmetrical three-phase circuits are fundamental models of power systems. Although the circuits have structural symmetry, asymmetric patterns of oscillations have been observed in real power systems. This paper describes an approach to understanding typical patterns of oscillations in the three-phase circuits using symmetry. In order to figure out oscillation patterns, we introduce a three LC ladder circuit which has a higher symmetry than the three-phase circuit. Using only the symmetries of the three LC ladder circuit, we classify periodic oscillations and construct a lattice of those modes. Further, extending the method to almost periodic oscillations, we decompose and characterize typical almost periodic oscillations by their symmetry. Finally, by observing a global phase space in the three LC ladder circuit, we confirm typical oscillations in the three-phase circuit.

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A power system is a nonlinear dynamical system. Understanding the dynamic behavior of a power system is necessary for its reliable operation. An equivalent mechanical model of a power system was proposed by K. Noda to help the physical interpretation of a power system behavior obtained by a mathematical model of a power system. The validity of the mechanical model was confirmed qualitatively. This study gives the quantitative validation of the mechanical model through experiments to assess the accuracy of the analogy between the mechanical and mathematical power system models. To this end, steady-state and transient stability of a power system, which are the local and global stability, are demonstrated. The cause of error of the mechanical model is discussed and the limitation on the application of the mechanical-mathematical model analogy is provided.

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We introduce a problem of control for power grids with the microgrid architecture. Microgrid is a novel architecture of power grid in which a cluster of distributed energy resources is locally and consistently operated. The objective of the control problem is to determine a controller that keeps the load-generation balance in a power grid and regulates the grid's behavior to satisfy safety specifications. We use a combination of continuous and discrete controllers in order to archive the objective for a rudimentary microgrid. The result is a hybrid controller for the continuous-time nonlinear model and is correct for the microgrid with a realistic setting of parameters.

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We propose a novel and simple discrete reaction-diffusion model that self-organizes striped and spotted spatial patterns. Through theoretical analyses, we show that the spatial frequency of the generated patterns is proportional to diffusion length. Through extensive numerical simulations, we demonstrate that the model can produce Turing-like striped and spotted spatial patterns.

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