Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators

By Florian Dörfler, Francesco Bullo. Proceedings of the 2010 American Control Conference, 2009

841 citations recorded by the source corpus.

Motivated by recent interest for multi-agent systems and smart 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 characterized by multiple time constants, non-homogeneous coupling, and non-uniform phase shifts. By extending methods from 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 classic Kuramoto model. By 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 network parameters and initial conditions.

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Citations in the corpus

Coupling functions:universal insights into dynamical interaction mechanisms (2017, background): “…et al., 2010; Cabral et al., 2014; Cumin and Unsworth, 2007), Josephsonjunction arrays (Filatrella et al., 2000; Wiesenfeld et al., 1996, 1998), power grids (Dorfler and Bullo, 2012; Filatrella et al., 2008), glassy states (Iatsenko et al., 2014) and laser arrays (Vladimirov et al., 2003).”
The Kuramoto model in complex networks (2015, methodology): “The powergrid model (343) with multiple time constants, nonhomogeneous coupling and nonuniform phase shifts was explored in [359].”
Kron Reduction of Graphs With Applications to Electrical Networks (2011, unknown): “In [45], we derived graph-theoretic conditions under which the reduced model (8) synchronizes.”
Oscillators that sync and swarm (2017, background): “For example, insights from biological synchronization have shed light on neutrino oscillations [17], phase locking in Josephson junction arrays [18], the dynamics of power grids [19, 20], and the unexpected wobbling of London’s Millennium Bridge on opening day [21].”
Synchronization in complex networks of phase oscillators: A survey (2014, background): “…to (Chopra and Spong, 2009, Theorem 3.1) and (Dörfler and Bullo, 2012b, Theorem 4.1) for a formal proof, and to (Schmidt et al., 2012, Lemma 3.5), (Dörfler and Bullo, 2012b, Theorem 4.1), and (Wang and Doyle, 2013, Theorem 4) for extensions to more general coupling functions, time-delays, and…”
Finite-Time Consensus for Switching Network Topologies with Disturbances (2013, background): “[1, 2, 3, 4]) and in the solution of several control problems involving synchronization or agreement between dynamical systems (see [5, 6, 7]).”
Conditions for stability of droop-controlled inverter-based microgrids (2014, background): “Although there has been progress in addressing this issue [2, 9], to the best of our knowledge no analytic solution for power systems with variable frequencies as well as variable voltage amplitudes is available.”
Controllability Metrics, Limitations and Algorithms for Complex Networks (2013, unknown): “Ni kij(i j); where, for the i-th generator, mi>0 and di>0 are the inertia and damping coefﬁcients, i: R !R[0;2ˇ] is the phase angle, and kij is the susceptance of the power line (i;j). As in [39] we assume that mi=di˝1, and we approximate the generator dynamics with a ﬁrst-order equation. Fina”
Foundations and Challenges of Low-Inertia Systems (Invited Paper) (2018, methodology): “Further in [172], [174], the equilibrium-based paradigm for stability is not used and allows for the continuous disturbance type analysis - see Section III-B discussion.”
Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators (2014, unknown): “[62–65]).”
A Distributed Control Strategy for Reactive Power Compensation in Smart Microgrids (2011, unknown): “See for example the stability analysis for these systems in [15] and the decentralized control synthesis in [16].”
On the Characterization of Local Nash Equilibria in Continuous Games (2014, methodology): “Coupled oscillator models are used widely for applications including power networks [15], traffic networks [16], robotics [17], biological networks [18],”
Comparative analysis of existing models for power-grid synchronization (2015, unknown): “Sufficient conditions for synchronization stability against larger perturbations have also been derived for the ENmodel, one in terms of themaximumvoltage phase difference in an approximation of the corresponding power flow solution [7] (assuming lossless transmission lines), and another in terms of”