Delay master stability of inertial oscillator networks

An analytical approach to determine the stability of synchronized networks of inertial oscillators with delayed dynamics.

This repository bundles work developed in the context of my bachelor's thesis and provides supplemental material for the corresponding paper.

Click here for my presentation slides shown at the SIAM UKIE Annual Meeting 2022 on January 7. Slides from a longer talk are available here. To play around with the simulation slider, read more here.

Where to start

• What's this about? To learn more about this research, have a look at the plain language summary (0.4 pages), the peer-reviewed paper (4 pages), or the thesis (40 pages).

• Where can I find supplemental material for the paper? Providing more detail than the paper, the thesis adds physical background, derivations, an in-depth treatment of the power grid application, supplemental figures, and simulation results (read more about the thesis).
  This repo currently contains code for calculating the delay master stability function of the Decentral Smart Grid Control model on the star topology (Fig. 1c in the paper). Please contact me for additional code.

• I want to run the code. How do I get started? Read more here.

Running the code

Currently, this repository contains

• dsgc_star_dmsf.jl, a script to calculate the delay master stability function for the Decentral Smart Grid Control (DSGC) model on a 4-node star network;
• dsgc_star_slider.jl, a slider application which allows DDE simulations of the DSGC model on the star network (see chapter 6 of the thesis for details). For a given delay, the application shows the position on the delay master stability function (see figure below, left panel) and simulation results of the four nodal frequency deviations, after a random initial perturbation, as a function of time (right panel).

Prerequisites

To play around with the code, you need to have the Julia programming language installed (v1.1.0 or higher, last tested with v1.8.0).

Delay master stability function

Requires the following Julia packages: Roots, NLsolve, LinearAlgebra, Plots, LaTeXStrings, DelimitedFiles.

• Run the script dsgc_star_dmsf.jl located in the code folder.
• The script outputs a plot of the delay master stability function (dMSF) corresponding to Fig. 1c of the paper.
• For more info, please see the comments in the script.

Slider

Requires the following Julia packages: DifferentialEquations, NLsolve, Plots, LaTeXStrings, DelimitedFiles, Interact, Mux.

• Run the file dsgc_star_slider.jl located in the code folder.
• If you wish, you can customize the set of slider values (delay in seconds) by modifying tau_list in line 12.
• Once the code is executed, go to your webbrowser and enter localhost:8001/ in the search bar. The slider should appear.

Note: The precomputed datapoints of the delay master stability function (the curves in the left panel) are stored in the .txt files at the path code/data. To ensure that the Julia program can read these files, you must be in the working directory of the Julia file (i.e. code). To find out your current working directory, type the command pwd() into the REPL.

About the thesis

I conducted my bachelor's thesis in Dr. Frank Hellmann's group "Dynamics, stability and resilience of complex hybrid infrastructure networks", which is part of Research Domain 4 - Complexity Science at the Potsdam Institute for Climate Impact Research (PIK). The thesis was formally supervised by Prof. Dr. Petra Imhof from the Institute of Theoretical Physics at Freie Universität Berlin (FU).

Content

• The theoretical basics: asymptotic stability, delay differential equations, synchronization in complex networks (chapter 2)
• A step-by-step derivation of the general method, including both processing and communication delay (chapter 3)
• A detailed discussion of the application to two future power grid models, including a comparison with simulation results (chapters 5 and 6)
• Context information on power grids and their study using complex networks tools (chapter 4)
• Discussion of the results and outlook (chapters 7 and 8).

How to cite

Preferably, please cite the paper as follows when relating to this research:

R. Börner, P. Schultz, B. Ünzelmann, D. Wang, F. Hellmann, J. Kurths, Delay master stability of inertial oscillator networks, Phys. Rev. Research 2, 023409 (2020).

@article{boerner2020delay,
    title = {Delay master stability of inertial oscillator networks},
    author = {B\"orner, Reyk and Schultz, Paul and \"Unzelmann, Benjamin and Wang, Deli and Hellmann, Frank and Kurths, J\"urgen},
    journal = {Phys. Rev. Research},
    volume = {2},
    issue = {2},
    pages = {023409},
    numpages = {8},
    year = {2020},
    month = {Jun},
    publisher = {American Physical Society},
    doi = {10.1103/PhysRevResearch.2.023409},
    url = {https://link.aps.org/doi/10.1103/PhysRevResearch.2.023409}
}

To specifically cite the thesis:

R. Börner, Master Stability of Inertial Oscillator Networks with Delay, Bachelor's thesis, Freie Universität Berlin (2019).

@mastersthesis{boerner2019thesis,
    title = {Master Stability of Inertial Oscillators with Delay -- An analytical approach applied to renewable power grids},
    author = {Reyk Börner},
    year = {2019},
    type = {Bachelor's thesis},
    school = {Freie Universit\"at Berlin},
    url = {https://github.com/reykboerner/delay-networks}
}

Collaborators

Frank Hellmann, Jürgen Kurths, Anton Plietzsch, Paul Schultz, Benjamin Ünzelmann, Deli Wang

This work is part of the CoNDyNet 2 project, sponsored by the German Federal Ministry of Education and Research (BMBF).