Forced oscillation source localization from generator measurements

Abstract

Malfunctioning equipment, erroneous operating conditions or periodic load variations can cause periodic disturbances that would persist over time, creating an undesirable transfer of energy across the system -- an effect referred to as forced oscillations. Wide-area oscillations may damage assets, trigger inadvertent tripping or control actions, and be the cause of equipment failure. Unfortunately, for wide-area oscillations, the location, frequency, and amplitude of these forced oscillations may be hard to determine. Recently, a data-driven maximum-likelihood-based method was proposed to perform source localization in transmission grids under wide-area response scenarios. However, this method relies on full PMU coverage and all buses having inertia and damping. Here, we extend this method to realistic scenarios which includes buses without inertia or dumping, such as passive loads and inverter-based generators. Incorporating Kron reduction directly into the maximum likelihood estimator, we are able to identify the location and frequency of forcing applied at both traditional generators and loads.

Figures (4)

• Figure 1: (a) Line network made of two generators and three load/inverter-based resource buses with its corresponding Laplacian matrix $\bf L$ . After the Kron reduction, the system reduces to two generators with a new effective coupling Laplacian matrix ${\bf L}^r$ . (b) Time-series of the angle and frequency deviations at the two generators with corresponding colors when a forcing with frequency of $0.48$Hz is applied at the load bus 2 [in green in panel (a)]. Note that on the angle time-series, we have subtracted their average value at each time step. (c) Fourier transform of the frequencies at the two generators with corresponding colors. The spectrum displays various peaks but the actual forcing frequency shown by the vertical dashed line is barely detected. (d) Correct localization of the source and identification of the frequency of the forcing showed as the largest log-likelihood peak in a challenging situation where the amplitude of the forcing is $\gamma=0.08$ and is smaller than the amplitude of the noise, $\sigma=0.2$ .. The colors correspond to those of panel (a). Generators have inertia and damping parameters, $d_1=0.5s$ , $d_2=0.8s$ , $m_1=2s^2$ , $m_2=1.5s^2$ .
• Figure 2: Detection and localization of forced oscillations when the source is at a generator. The inertia and damping parameters are first learned on the system without forcing for $10$min with measurements at $50$Hz. In both cases when the forcing of $0.48$Hz, which is close to a natural frequency of the system (see Fig. \ref{['fig1']}), is applied at the leftmost (top panel) and rightmost (bottom panel) generator, the correct source and frequency are identified by the largest log-likelihood. The dashed vertical lines give the frequency $0.48$Hz. The amplitude of the forcing and the noise are respectively $\gamma=0.3$ , $\sigma=0.2$ , and the time-series correspond to measurements at $50$Hz over 200s. Generators have inertia and damping parameters, $d_1=0.5s$ , $d_2=0.8s$ , $m_1=2s^2$ , $m_2=1.5s^2$ .
• Figure 3: Detection and localization of forced oscillations when the source is at a load/inverter-based resource bus. The inertia and damping parameters are first learned on the system without forcing for $10$min with measurements at $50$Hz. In all three cases when the forcing of $0.48$Hz, which is close to a natural frequency of the system (see Fig. \ref{['fig1']}) is applied at bus 2 (left panel), 3 (middle panel), 4 (right panel), the correct source and frequency are identified by the largest log-likelihood. The dashed vertical lines give the frequency $0.48$Hz. The amplitude of the forcing and the noise are respectively $\gamma=0.3$ , $\sigma=0.2$ , and the time-series correspond to measurements at $50$Hz over 200s. Generators have inertia and damping parameters, $d_1=0.5s$ , $d_2=0.8s$ , $m_1=2s^2$ , $m_2=1.5s^2$ .
• Figure 4: (a) Topology of the IEEE-57 bus test case, where the generators are located at the periphery of the grid. The blue and green load buses produce the same response (up to 10 decimals) at the generators when the forcing is applied at either of them. The line capacities are heterogeneous IEEE57. Detection and localization of forced oscillations when the source is at the generator bus shown in orange (b), at the load bus in pink (c), and the blue and green load buses (d). In the latter case and panel (d), both sources are indistinguishable from each other, i.e., both buses have the same likelihood as ${\bf \Gamma}_{32}\cong{\bf \Gamma}_{33}$ . The forcing frequency of $2$Hz as well as the source are correctly identified by the algorithm in all cases. The log-likelihoods for all other buses are shown in gray. The inertia and damping parameters at the generators are heterogeneous and given by $m_i=2.5s^2$ , $d_i=1s$ for $i=1,2,4,6,7$ , $m_3=4s^2$ , $m_5=1.5s^2$ , $d_3=1.6s$ , $d_5=0.6s$ . The amplitudes of the forcing and the noise are respectively, $\gamma=3$ , $\sigma=0.2$ .