Synchronization of Nonlinear Circuits in Dynamic Electrical Networks with General Topologies

Sufficient conditions are derived for global asymptotic synchronization in a system of identical nonlinear electrical circuits coupled through linear time-invariant (LTI) electrical networks. In particular, the conditions we derive apply to settings where: i) the nonlinear circuits are composed of a parallel combination of passive LTI circuit elements and a nonlinear voltage-dependent current source with finite gain; and ii) a collection of these circuits are coupled through either uniform or homogeneous LTI electrical networks. Uniform electrical networks have identical per-unit-length impedances. Homogeneous electrical networks are characterized by having the same effective impedance between any two terminals with the others open circuited. Synchronization in these networks is guaranteed by ensuring the stability of an equivalent coordinate-transformed differential system that emphasizes signal differences. The applicability of the synchronization conditions to this broad class of networks follows from leveraging recent results on structural and spectral properties of Kron reduction---a model-reduction procedure that isolates the interactions of the nonlinear circuits in the network. The validity of the analytical results is demonstrated with simulations in networks of coupled Chua's circuits

Output Impedance Diffusion into Lossy Power Lines

Output impedances are inherent elements of power sources in the electrical grids. In this paper, we give an answer to the following question: What is the effect of output impedances on the inductivity of the power network? To address this question, we propose a measure to evaluate the inductivity of a power grid, and we compute this measure for various types of output impedances. Following this computation, it turns out that network inductivity highly depends on the algebraic connectivity of the network. By exploiting the derived expressions of the proposed measure, one can tune the output impedances in order to enforce a desired level of inductivity on the power system. Furthermore, the results show that the more "connected" the network is, the more the output impedances diffuse into the network. Finally, using Kron reduction, we provide examples that demonstrate the utility and validity of the method

Voltage Stabilization in Microgrids via Quadratic Droop Control

We consider the problem of voltage stability and reactive power balancing in islanded small-scale electrical networks outfitted with DC/AC inverters ("microgrids"). A droop-like voltage feedback controller is proposed which is quadratic in the local voltage magnitude, allowing for the application of circuit-theoretic analysis techniques to the closed-loop system. The operating points of the closed-loop microgrid are in exact correspondence with the solutions of a reduced power flow equation, and we provide explicit solutions and small-signal stability analyses under several static and dynamic load models. Controller optimality is characterized as follows: we show a one-to-one correspondence between the high-voltage equilibrium of the microgrid under quadratic droop control, and the solution of an optimization problem which minimizes a trade-off between reactive power dissipation and voltage deviations. Power sharing performance of the controller is characterized as a function of the controller gains, network topology, and parameters. Perhaps surprisingly, proportional sharing of the total load between inverters is achieved in the low-gain limit, independent of the circuit topology or reactances. All results hold for arbitrary grid topologies, with arbitrary numbers of inverters and loads. Numerical results confirm the robustness of the controller to unmodeled dynamics.Comment: 14 pages, 8 figure

Towards Optimal Kron-based Reduction Of Networks (Opti-KRON) for the Electric Power Grid

For fast timescales or long prediction horizons, the AC optimal power flow (OPF) problem becomes a computational challenge for large-scale, realistic AC networks. To overcome this challenge, this paper presents a novel network reduction methodology that leverages an efficient mixed-integer linear programming (MILP) formulation of a Kron-based reduction that is optimal in the sense that it balances the degree of the reduction with resulting modeling errors in the reduced network. The method takes as inputs the full AC network and a pre-computed library of AC load flow data and uses the graph Laplacian to constraint nodal reductions to only be feasible for neighbors of non-reduced nodes. This results in a highly effective MILP formulation which is embedded within an iterative scheme to successively improve the Kron-based network reduction until convergence. The resulting optimal network reduction is, thus, grounded in the physics of the full network. The accuracy of the network reduction methodology is then explored for a 100+ node medium-voltage radial distribution feeder example across a wide range of operating conditions. It is finally shown that a network reduction of 25-85% can be achieved within seconds and with worst-case voltage magnitude deviation errors within any super node cluster of less than 0.01pu. These results illustrate that the proposed optimization-based approach to Kron reduction of networks is viable for larger networks and suitable for use within various power system applications