Locating the Source of Forced Oscillations in Transmission Power Grids

A forced oscillation event in power grids refers to a state where malfunctioning or abnormally operating equipment causes persisting periodic disturbances in the system. While power grids are designed to damp most perturbations during standard operations, some of them can excite normal modes of the system and cause significant energy transfers across the system, creating large oscillations thousands of miles away from the source. Localization of the source of such disturbances remains an outstanding challenge due to a limited knowledge of the system parameters outside of the zone of responsibility of system operators. Here, we propose a new method for locating the source of forced oscillations that addresses this challenge by performing a simultaneous dynamic model identification using a principled maximum likelihood approach. We illustrate the validity of the algorithm on a variety of examples where forcing leads to resonance conditions in the system dynamics. Our results establish that an accurate knowledge of system parameters is not required for a successful inference of the source and frequency of a forced oscillation. We anticipate that our method will find a broader application in general dynamical systems that can be well described by their linearized dynamics over short periods of time.

Popular Summary

In power grids, forced oscillations pose significant risks to the critical infrastructure that lead to an undesirable transfer of energy across the system, and may cause cascading failures of equipment. Locating the source of forced oscillations remains a long-standing and open question in power grids. Here, the authors introduce a fully data-driven algorithm to localize the source of forced oscillation in power grids, even when the grid topology and system parameters are unknown and when the frequency and location of oscillations are obfuscated by the resonance with natural modes. A mixed-integer optimization problem is solved to give the correct frequency and location of oscillations in a variety of synthetic examples as well as two real-world data sets. The results yield a promising foundation for algorithms for mitigation of forced oscillations in future power grids.

Figure 1

A toy example with three nodes illustrating the challenges and solutions for locating the source of forced oscillations in a network of coupled oscillators. (a) A forced oscillation is induced on the orange node with frequency $f=0.16$ ${\mathrm{s}}^{-1}$. Time series of the network states corresponding to generalized positions ${\mathbf{x}}_t$ and momentum ${\mathbf{p}}_t$ (corresponding to phase deviations ${\mathit{\theta }}_t$ and frequency deviations ${\mathit{\omega }}_t$ in power grids; see Sec. S1 within the Supplemental Material [33]) presented in panels (b) and (c), respectively, are generated using Eq. (1) and the parameters detailed in Sec. S2 within the Supplemental Material [33]. (d) A naïve Fourier analysis does not allow one to identify either the forcing source or location: the Fourier transform (FT) of the position time series displays its largest peak at the blue node with frequency $f=0.08{\mathrm{s}}^{-1}$, and the Fourier transform of the momentum time series displays peaks at the orange node, with frequency $f=0.8$ ${\mathrm{s}}^{-1}$. (e) An eigendecomposition of the dynamic state matrix shows the natural frequencies (eigenvalues) and modes (eigenvectors, with the sizes of the nodes indicating the magnitudes of the respective components) of the system. This analysis reveals the reason for the observed behavior of the Fourier transforms: the forcing frequency is close to a natural frequency of the system, thus creating a resonance effect and exciting the natural modes of the system. (f) Our source localization algorithm confidently points to the correct source (orange node) and frequency (indicated with a dashed gray line) of the forced oscillation. The accuracy of the determination of the forcing frequency $f$ is fundamentally defined by the duration of the available time series.

Figure 2

Detection and localization of forced oscillations under the resonance conditions. (a) Synthetic test case with a topology inspired by the UK high-voltage grid, designed to reproduce features observed in real oscillatory events where oscillations interacted with the system modes. Details on the network parameters are given in Sec. S2 within the Supplemental Material [33]. The forcing at the orange node results in a highest response in Fourier spectra at the opposite side of the network, as shown for the Fourier components of the (b) generalized state and of the (c) generalized momentum. (d) SALO algorithm confidently identifies the correct forcing frequency and source without any knowledge of the system topology or parameters. The envelope of scores for nonhighlighted nodes is shown in gray.

Figure 3

Robustness of the SALO algorithm to model misspecification. (a) In the case where the source of forced oscillations is outside of the observed system (gray node), the algorithm still correctly identifies the forcing frequency and the neighbors of the hidden source inside the visible system (three overlapping peaks in likelihood at the forcing frequency correspond to orange, purple, and blue nodes). (b) In the case of several sources, both locations appear as peaks in the rescaled likelihood score at their respective forcing frequencies. (c) In the case of nonsinusoidal forcing injected into the system, the forcing location is still correctly identified, while the complex nature of the forcing shows up as likelihood peaks at different harmonics of the forcing signal. The envelope of scores for nonhighlighted nodes is shown in gray in all panels.

Figure 4

Identifying the location of forced oscillations for PMU data from a U.S. power system operator. (a) Geographical layout of PMU sensors in the system providing time series data (anonymized and modified coordinates). The node identified as the most likely source of forced oscillations is highlighted in orange. (b) According to the SALO algorithm, the identified node has a much higher likelihood for being a source for a range of frequencies in the vicinity of $f=4.6{\mathrm{s}}^{-1}$ compared to the rest of the nodes in the network (corresponding likelihoods depicted in gray). A finite range of candidate frequencies is a realistic feature that may emerge in real systems due to the finite length of the collected data. (c) The algorithm consistently points to the same candidate source in a similar frequency range even for time series collected over two time periods separated by almost a month (March 2013 versus April 2013). This consistency strongly indicates that sustained oscillations may originate from a single faulty component of the system. The envelope of scores for other nonhighlighted nodes in the system is shown in gray.