Research ArticleAPPLIED SCIENCES AND ENGINEERING

The key player problem in complex oscillator networks and electric power grids: Resistance centralities identify local vulnerabilities

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Science Advances  22 Nov 2019:
Vol. 5, no. 11, eaaw8359
DOI: 10.1126/sciadv.aaw8359
  • Fig. 1 Comparison between theoretical predictions and numerical results for both performance measures P1 and P2 defined in Eq. 3.

    Each point corresponds to a noisy disturbance on a single node of the European electric power grid sketched in Fig. 2A (see Materials and Methods and the Supplementary Materials) and governed by Eq. 1 with constant inertia and damping parameters. The time-dependent disturbance δPi(t) is defined by an Ornstein-Uhlenbeck noise of magnitude δP0 = 1 and correlation time γτ0 = 4 × 10−5 (red crosses), 4 × 10−4 (cyan), 4 × 10−3 (green), 4 × 10−2 (purple), 4 × 10−1 (black), and 4 (blue). Time scales are defined by the ratio of damping to inertia parameters γ = di/mi = 0.4 s−1, which is assumed constant with di = 0.02 s. The insets show 𝒫1 and 𝒫2 as a function of the resistance distance–based graph-theoretic predictions of Eq. 6 valid in both limits of very large and very short noise correlation time τ0. The limit of short τ0 for 𝒫2 gives a node-independent result (Eq. 6).

  • Fig. 2 Synchronous high-voltage power grid of continental Europe.

    (A) Topology of the European electric power grid (see Materials and Methods and the Supplementary Materials) and location of the 10 test nodes listed in Table 1. Normalized generalized resistance centralities C1(0)(i) (B) and C2(0)(i) (C) for the network Laplacian matrix of the European electric power grid.

  • Fig. 3 Comparison of the two nodal rankings WLRank1 and WLRank2 obtained from the generalized resistance centralities C1 and C2, respectively, for the 3809 nodes of the European electric power grid sketched in Fig. 2A (see Materials and Methods and the Supplementary Materials).

    Blue dots correspond to a moderate load during a standard winter weekday, and red dots correspond to a significantly heavier load corresponding to the exceptional November 2016 situation with a rather large consumption and 20 French nuclear reactors shut down.

  • Fig. 4 Comparison between LRank and WLRank corresponding to 𝒫1 for noisy disturbances with large correlation time τ0.

    (A to C) Electric power grid models for normally (blue) and more heavily loaded (red) operating states governed by Eq. 1. (A) IEEE 57 bus test case where the more loaded case has injections six times larger than the moderately loaded, tabulated case (52). (B) MATPOWER Pegase 2869 test where the more loaded case has injections 30% larger than the moderately loaded, tabulated case (53). (C) European electric power grid model sketched in Fig. 2A (see Materials and Methods and the Supplementary Materials) where the moderately loaded case corresponds to a standard winter weekday and the more heavily loaded case to the November 2016 situation with 20 French nuclear reactors offline. For both cases, the operational state is obtained from an optimal power flow including physical, technological, and economic constraints (see Materials and Methods and the Supplementary Materials). (D) Inertialess coupled oscillators governed by Eq. 1 with mi = 0, i, on a random network with 1000 nodes obtained by rewiring a cyclic graph with constant nearest and next-to-nearest neighbor coupling with a probability of 0.5 (see Materials and Methods and the Supplementary Materials) (50). Natural frequencies are randomly distributed as Pi ∈ [ − 1.8,1.63] (blue), Pi ∈ [ − 2.16,1.95] (red), and Pi ∈ [ − 2.7,2.45] (green), corresponding to maximal angle differences max(Δθ) = 31o, 70o, and 106o, respectively.

  • Fig. 5 Percentage of the nodes with highest LRank2 necessary to give the top 15% ranked nodes with WLRank2 for a random network of inertialess coupled oscillators with 1000 nodes obtained by rewiring with a probability of 0.5 of a cyclic network with constant nearest and next-to-nearest neighbor coupling (see Materials and Methods and the Supplementary Materials) (50).

    Each of the 12,000 red crosses corresponds to one of 1000 random natural frequency vector P(0) with components randomly distributed in (−0.5,0.5) and summing to zero, multiplied by a prefactor β = 0.4,0.6,…,2.4,2.6. The blue crosses correspond to running averages more than 500 red crosses with consecutive values of max(Δθ). Inset: Running averages of the Frobenius distance between the matrices L(θ(0)) and L(0). The steps in the curve reflect discrete increments of β.

  • Fig. 6 Comparison between WLRank and numerical ranking for systems with inhomogeneous inertia and damping parameters.

    (Left) Numerically obtained ranking based on the performance measure 𝒫1 plotted against the ranking WLRank2 based on the centrality C2 and (right) numerically obtained ranking based on the performance measure 𝒫2 plotted against the ranking WLRank1 based on the centrality C1. Each point is an average *over* more than 40 different noisy disturbances on a single node of the European electric power grid sketched in Fig. 2A, with independently fluctuating damping and inertia coefficients, di = d0 + δdi and mi = m0 + δmi with δmi/m0, δdi/d0 ∈ [−0.4,0.4] and γ = d0/m0 = 0.4 s−1. The noise correlation time is given by γτ0 = 4.

  • Table 1 Centrality metrics and performance measures 𝒫1,2 for the European electric power grid (see Materials and Methods and the Supplementary Materials) with noisy disturbances with large correlation time τ0 applied on the nodes shown in Fig. 2A.

    The performance measures 𝒫1 and 𝒫2 are almost perfectly correlated with the inverse resistance centralities C2−1 and C1−1, respectively, but neither with the geodesic centrality, nor the degree, nor PageRank.

    Node no.CgeoDegreePageRankC1C2𝒫1num𝒫2num2)
    17.844278231.865.180.0470.035
    26.8119922.455.680.0210.118
    35.5610380222.452.330.320.116
    44.79336221.743.790.1260.127
    57.081121721.745.340.0260.125
    64.386309121.695.650.0230.129
    75.11244519.45.890.0160.164
    84.156364819.381.830.4530.172
    95.061810.25.20.0470.449
    102.72431247.492.170.3350.64

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/11/eaaw8359/DC1

    Section S1. Calculation of the performance measures

    Section S2. Resistance distances, centralities, and Kirchhoff indices

    Section S3. Numerical models

    Section S4. Numerical comparison of LRank with WLRank.

    Fig. S1. Comparison between theoretical predictions and numerical results for both performance measures 𝒫1 and 𝒫2.

    Fig. S2. Comparison of the performance measures 𝒫1, 𝒫2 obtained numerically and in eq. S14.

    Fig. S3. Percentage of the nodes with highest LRank necessary to include the nodes with 10% and 20% highest WLRank.

    References (55, 56)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Calculation of the performance measures
    • Section S2. Resistance distances, centralities, and Kirchhoff indices
    • Section S3. Numerical models
    • Section S4. Numerical comparison of LRank with WLRank.
    • Fig. S1. Comparison between theoretical predictions and numerical results for both performance measures P1 and P2.
    • Fig. S2. Comparison of the performance measures P1, P2 obtained numerically and in eq. S14.
    • Fig. S3. Percentage of the nodes with highest LRank necessary to include the nodes with 10% and 20% highest WLRank.
    • References (55, 56)

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