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Linear stability and the Braess paradox in coupled-oscillator networks and electric power grids

Tommaso Coletta and Philippe Jacquod
Phys. Rev. E 93, 032222 – Published 31 March 2016
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  • INTRODUCTION
  • THE CHAIN MODEL
  • IMPACT OF LINE ADDITION ON POWER FLOWS
  • LINEAR STABILITY
  • LINEAR STABILITY FOR THE CHAIN MODEL
  • EXTENSION TO COMPLEX NETWORKS
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We investigate the influence that adding a new coupling has on the linear stability of the synchronous state in coupled-oscillator networks. Using a simple model, we show that, depending on its location, the new coupling can lead to enhanced or reduced stability. We extend these results to electric power grids where a new line can lead to four different scenarios corresponding to enhanced or reduced grid stability as well as increased or decreased power flows. Our analysis shows that the Braess paradox may occur in any complex coupled system, where the synchronous state may be weakened and sometimes even destroyed by additional couplings.

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    • Received 29 May 2015

    DOI:https://doi.org/10.1103/PhysRevE.93.032222

    ©2016 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Synchronization
    1. Physical Systems
    Coupled oscillatorsPower grid networks
    Nonlinear Dynamics

    Authors & Affiliations

    Tommaso Coletta and Philippe Jacquod

    • School of Engineering, University of Applied Sciences of Western Switzerland, CH-1951 Sion, Switzerland

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    Issue

    Vol. 93, Iss. 3 — March 2016

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    Images

    • Figure 1
      Figure 1

      The chain model. A single generator (square) injects a power P0>0 which is consumed by N loads (circles), each consuming a power of Pi<0 in arbitrary units. The lines have capacity Ki+1,i≥P0−∑l=1i|Pl|, except the newly added line (dotted), which has capacity δ.

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    • Figure 2
      Figure 2

      Impact of perturbative line addition on the linear stability of the power flow solution (green region, enhanced stability; yellow region, reduced stability) and on the load of the transmission lines (top quadrants, increased load; bottom quadrants, decreased loads) as a function of the value of θd,0.

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    • Figure 3
      Figure 3

      Left: UK transmission grid, 10 generators with power P=11 (squares), 110 loads with P=−1 (circles), and uniform line capacity K≡13. Power flows are represented by arrows, and their magnitude is color coded. The dashed lines (a)–(c) represent three different line additions considered, and the solid line denotes the network partition into northern and southern zones. Right: Plot of the difference in power flows between the solutions after and before the addition of line (a) of capacity δ=1.5. Arrow heads are drawn only for power flow differences larger than 0.01.

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    • Figure 4
      Figure 4

      Lyapunov exponent (dashed) and power flowing through the lines connecting the north and south areas as a function of the capacity of the additional line δ. Each of the panels refers to one of the line additions represented in Fig. 3 (labels correspond to the labels of the additional lines in Fig. 3) and illustrates one of the three Braess scenarios identified in this work. Interestingly, (c) shows the coexistence of two different stable solutions for δ≥5.98.

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    • Figure 5
      Figure 5

      Lyapunov exponent (dashed) and power flowing though the lines connecting the north and south areas as a function of the capacity of the additional line δ for the UK transmission grid in the case of line capacities uniformly distributed in the interval [9.75,16.25]. Each panel refers to one of the line additions represented in Fig. 3 and illustrates one of the three Braess scenarios identified in this work.

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