Motivated by recent interest for multiagent systems and smart grid architectures, we discuss the synchronization problem for the network-reduced model of a power system with nontrivial transfer conductances. Our key insight is to exploit the relationship between the power network model and a first-order model of coupled oscillators. Assuming overdamped generators (possibly due to local excitation controllers), a singular perturbation analysis shows the equivalence between the classic swing equations and a nonuniform Kuramoto model. Here, nonuniform Kuramoto oscillators are characterized by multiple time constants, nonhomogeneous coupling, and nonuniform phase shifts. Extending methods from transient stability, synchronization theory, and consensus protocols, we establish sufficient conditions for synchronization of nonuniform Kuramoto oscillators. These conditions reduce to necessary and sufficient tests for the standard Kuramoto model. Combining our singular perturbation and Kuramoto analyses, we derive ...

Synchronization and Transient Stability in Power Networks and Nonuniform Kuramoto Oscillators

Motivated by recent interest for multi-agent systems and smart power grid architectures, we discuss the synchronization problem for the network-reduced model of a power system with non-trivial transfer conductances. Our key insight is to exploit the relationship between the power network model and a first-order model of coupled oscillators. Assuming overdamped generators (possibly due to local excitation controllers), a singular perturbation analysis shows the equivalence between the classic swing equations and a non-uniform Kuramoto model. Here, non-uniform Kuramoto oscillators are characterized by multiple time constants, non-homogeneous coupling, and non-uniform phase shifts. Extending methods from transient stability, synchronization theory, and consensus protocols, we establish sufficient conditions for synchronization of non-uniform Kuramoto oscillators. These conditions reduce to and improve upon previously-available tests for the standard Kuramoto model. Combining our singular perturbation and Kuramoto analyses, we derive concise and purely algebraic conditions that relate synchronization and transient stability of a power network to the underlying system parameters and initial conditions.

Synchronization and Transient Stability in Power Networks and Non-Uniform Kuramoto Oscillators

Active research activity in power systems areas has focused on developing computational methods to solve load flow equations where a key question is the maximum number of solutions. Although several upper bounds exist, recent studies have hinted that much sharper upper bounds that depend on the topology of underlying power networks may exist. This paper provides a significant refinement of these observations. We also develop a geometric construction called adjacency polytope that accurately captures the topology of a power network and is immensely useful in the computation of the solution bound. Finally, we highlight the significant implications of the development of such solution bounds in numerically solving load flow equations.

On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations

The power flow equations, which relate power injections and voltage phasors, are at the heart of many electric power system computations. While Newton-based methods typically find the “high-voltage” solution to the power flow equations, which is of primary interest, there are potentially many “low-voltage” solutions that are useful for certain analyses. This paper addresses the number of solutions to the power flow equations. There exist upper bounds on the number of power flow solutions; however, there is only limited work regarding bounds that are functions of network topology. This paper empirically explores the relationship between the network topology, as characterized by the maximal cliques, and the number of power flow solutions. To facilitate this analysis, we use a numerical polynomial homotopy continuation approach that is guaranteed to find all complex solutions to the power flow equations. The number of solutions obtained from this approach upper bounds the number of real solutions. Testing with many small networks informs the development of upper bounds that are functions of the network topology. Initial results include empirically derived expressions for the maximum number of solutions for certain classes of network topologies.

Toward topologically based upper bounds on the number of power flow solutions

A large amount of research activity in power systems areas has focused on developing computational methods to solve load flow equations where a key question is the maximum number of isolated solutions.Though several concrete upper bounds exist, recent studies have hinted that much sharper upper bounds that depend the topology of underlying power networks may exist. This paper establishes such a topology dependent solution bound which is actually the best possible bound in the sense that it is always attainable. We also develop a geometric construction called adjacency polytope which accurately captures the topology of the underlying power network and is immensely useful in the computation of the solution bound. Finally we highlight the significant implications of the development of such solution bound in solving load flow equations.

New results are presented on the enumeration of critical points for classical power system models involving the power flow equations and the "swing" equations. It is shown how incorporation of electrical losses into these models restricts the numbers of physically meaningful solutions to the power flow equations.

J. Baillieul

Papers

The critical point analysis of electric power systems