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Fluctuation-induced distributed resonances in oscillatory networks

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Science Advances  31 Jul 2019:
Vol. 5, no. 7, eaav1027
DOI: 10.1126/sciadv.aav1027
  • Fig. 1 Fluctuation-induced dynamic response patterns.

    A sample network (A) driven by Brownian noise (B) at a single unit k = 1 exhibiting (C) complex dynamic response patterns that nonlinearly vary with frequency content of the input signal (D, F, and H) as well as with graph-theoretic distance between the input and the response unit (E, G, and I). Three response classes emerge: homogeneous responses at low frequencies (D and E), spatiotemporally irregular patterns at intermediate frequencies (F and G), and localized responses at high frequencies (H and I). To identify frequency regimes, we selected specific frequency bands [yellow in (D), (F), and (H)] from the spectrum of the original noise realization (B), and all others are displayed in purple. (C), (E), (G), and (I) display the time series of driving signal at unit k = 1 (top) and the response in the phase velocities of the units i ∈ {1, …,4} (bottom). In (C), (E), (G), and (I), the time series of the band-filtered signal are displayed together with the responses (yellow for unit 1 and green for units 2 to 4). Our theory (Eq. 5) (thin black lines) well predicts the system responses obtained from direct numerical simulations. (For details of further settings, see Materials and Methods.)

  • Fig. 2 Emergence of resonances and prediction of distinct network responses.

    (A) The relative response strength Ai(k) of each unit in the fluctuation-driven network illustrated in Fig. 1A versus the signal frequency ω, color coded by the graph-theoretic distance d from the driven to the responding unit. The eigenfrequencies are indicated by gray vertical lines, and the three regimes of homogeneous bulk responses, heterogeneous resonant responses, and localized responses by the gray-level gradient bar at the bottom. (B to D) The relative response strengths characteristically depend on graph-theoretic distance, shown for individual frequencies representative for each of the three regimes. (E) In the localized regime, the response amplitudes Ai(k) for units 1 to 4 (marked in Fig. 1A) are well approximated by the analytic prediction (dashed lines) (Eq. 8). (F) Relative response strengths (plotted on linear scale) at resonance peaks may be an order of magnitude (here up to 12 times) larger than in the static response limit of ω → 0.

  • Fig. 3 Generality of response patterns.

    For two exemplary networks, the response patterns for three signals of frequencies representing the three response regimes are indicated by their relative response strengths Ai(k) (color coded). Response patterns for (A) to (C), a random tree with N = 264, and (D) to (F), the topology of the British high-voltage transmission grid with N = 120, both reordered according to graph-theoretic distance from site of driving unit. For both networks, the driven unit is placed at the center with all units displayed on circles with their radii proportional to topological distance (gray concentric rings) in (D) to (F).

  • Fig. 4 Predicting response patterns of fluctuation-driven networks.

    (A) A noisy signal Fi (purple) at one unit and (B) frequency response dθi/dt at another unit in the network (Fig. 1A), with every unit driven by independent Brownian noise. (B) Response prediction (black) is based on 50 dominant Fourier modes of the signal [reconstructing the yellow time series in (A)] at each unit, which, after a transient stage, is very close to the numerical response (purple) to the original signals. (C) Prediction error E (for definition, see Materials and Methods) decreases exponentially with the number of selected Fourier modes. (D) Linear response theory well predicts responses until the maximum line load exceeds 95% (line load Lij ≔ ∣sin(θj − θi)∣), i.e., the system is almost fully loaded at the operating point.

  • Fig. 5 Frequency resonances in power grids induced by wind and photovoltaic power fluctuations.

    (A and B) Response time series of grid frequency in the network (Fig. 1A) for fluctuating power input signals (A) from wind and (B) from photovoltaic panels, with (C) and (D) as the respective power spectral density S(ω). One response time series is highlighted in purple (analytic prediction in black) and of all other units in gray. The prediction is based on 50 dominant Fourier modes of the signal, highlighted in yellow in (C) and (D). The gray vertical lines indicate the system’s eigenfrequencies. (E) The photovoltaic power fluctuations with stronger higher-frequency components induce greater nodal response inhomogeneity (for definition, see Materials and Methods) than the wind power fluctuations.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/7/eaav1027/DC1

    Section S1. Theory of dynamic response patterns in oscillatory networks

    Section S2. Frequency regimes of dynamic response patterns

    Section S3. Predicting dynamic responses to irregular and distributed noises

    Section S4. Limit of validity of linear response theory

    Fig. S1. Estimating response strength in inhomogeneous networks.

    Fig. S2. Dynamic network response patterns in three regimes.

    Fig. S3. Three frequency regimes of response patterns in networks of first-order (Kuramoto) oscillators.

    Fig. S4. An illustration of the frequency sampling method.

    Fig. S5. Grid responses to real-world power fluctuations.

    Fig. S6. Limit of validity of linear response theory under strong perturbations.

    Fig. S7. Breakdown of linear response theory at the fully loaded point.

    Fig. S8. Limit of validity of linear response theory in heavily loaded networks.

    Movie S1. Network response to Brownian noise at one node (accompanying Fig. 1).

    Movie S2. Network response to independent Brownian noise at all nodes (accompanying Fig. 4, A and B).

    Movie S3. Network response to a sinusoidal signal in bulk regime (accompanying Fig. 2B).

    Movie S4. Network response to a sinusoidal signal in resonance regime (accompanying Fig. 2C).

    Movie S5. Network response to a sinusoidal signal in localized regime (accompanying Fig. 2D).

    Movie S6. Network response to a real-world wind power fluctuation (accompanying Fig. 5A).

    Movie S7. Network response to a real-world photovoltaic power fluctuation (accompanying Fig. 5B).

    References (3945)

  • Supplementary Materials

    The PDF file includes:

    • Section S1. Theory of dynamic response patterns in oscillatory networks
    • Section S2. Frequency regimes of dynamic response patterns
    • Section S3. Predicting dynamic responses to irregular and distributed noises
    • Section S4. Limit of validity of linear response theory
    • Fig. S1. Estimating response strength in inhomogeneous networks.
    • Fig. S2. Dynamic network response patterns in three regimes.
    • Fig. S3. Three frequency regimes of response patterns in networks of first-order (Kuramoto) oscillators.
    • Fig. S4. An illustration of the frequency sampling method.
    • Fig. S5. Grid responses to real-world power fluctuations.
    • Fig. S6. Limit of validity of linear response theory under strong perturbations.
    • Fig. S7. Breakdown of linear response theory at the fully loaded point.
    • Fig. S8. Limit of validity of linear response theory in heavily loaded networks.
    • References (3945)

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    Other Supplementary Material for this manuscript includes the following:

    • Movie S1 (.mp4 format). Network response to Brownian noise at one node (accompanying Fig. 1).
    • Movie S2 (.mp4 format). Network response to independent Brownian noise at all nodes (accompanying Fig. 4, A and B).
    • Movie S3 (.mp4 format). Network response to a sinusoidal signal in bulk regime (accompanying Fig. 2B).
    • Movie S4 (.mp4 format). Network response to a sinusoidal signal in resonance regime (accompanying Fig. 2C).
    • Movie S5 (.mp4 format). Network response to a sinusoidal signal in localized regime (accompanying Fig. 2D).
    • Movie S6 (.mp4 format). Network response to a real-world wind power fluctuation (accompanying Fig. 5A).
    • Movie S7 (.mp4 format). Network response to a real-world photovoltaic power fluctuation (accompanying Fig. 5B).

    Files in this Data Supplement:

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