Collective Nonlinear Dynamics of Electricity Networks (CoNDyNet) Determination of resonance frequencies of LC Determination of resonance frequencies of LC networks with binary link disorder networks with binary link disorder Daniel Jung Daniel Jung Jacobs University Bremen School of Engineering and Science Solid State Physics Group December 10, 2014 1 of 21
Table of contents Table of contents 1. Motivation LC circuit Flow equations 2. Model Basic Graph Theory The 2D grid model The Small World Model Adding Edge Attributes 3. Numerical method Resonance frequencies 4. Resonance spectra The 2D grid model The Small World Model 5. Summary 2 of 21
1 LC iωL ω 0 X 1 = Z = R + iX 1 √ = ω Z ( ω ) = ∞ lim ω → ω 0 ⇒ ω ω n + iωC Motivation Motivation A LC circuit has a resonance frequency of Frequency of AC current approaching the resonance frequency: No power transmission possible! Normal operation: should stay far below the smallest resonance. Arbitrary LC network Arbitrary LC network Coupled LC oscillators Set of resonance frequencies 3 of 21
Y ij V ij = = V ij Z ij I ij I ij − = − = I i ∑ j I ij V ij V i V j or Y = Z −1 ( I i Y Z V j j I i ∑ . ) Z ij > 0 I i < 0 = Flow equations I Flow equations I Combine Ohm's law with Kirchhoff's laws for each node and mesh An arbitrary one-phasic AC grid with line impedances . to derive the current flow equations Generator: Y ij V i Consumer: Name Symbol Impedance matrix Admittance matrix R. Huang, G. Korniss, S. Nayak, 2009 (http://dx.doi.org/10.1103 /PhysRevE.80.045101) 4 of 21
= ( L ij V j j ∑ I i I = LV or = V j ) j ∑ I i ) − = − (1 − I i < 0 I i > 0 L ij k ≠ i Y ik L G L Flow Equations II Flow Equations II Y ij V i To get a standard matrix-vector multiplication, reformulate to An arbitrary one-phasic AC grid. with Generator: Consumer: δ ij ∑ δ ij Y ij Note: is defined in analogy to the topological network Laplacian . is commonly referred to as admittance matrix as well. R. Huang, G. Korniss, S. Nayak, 2009 (http://dx.doi.org/10.1103 /PhysRevE.80.045101) 5 of 21
N N = 4 ( i , j ) i Basic Graph Theory I Basic Graph Theory I Example Example A graph is given by a set of nodes a set of edges Properties Properties Property Explanation Order Number of nodes Size Number of edges 6 of 21
1 1 0 1 0 0 1 0 1 1 0 0 ⎠ 1 0 1 1 0 ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ E = 0 0 −1 3 −1 −1 0 −1 2 −1 −1 ⎟ −1 2 ⎜ ⎜ ⎜ ⎝ ⎛ G = ⎟ ⎛ ⎟ −1 E ⎛ D = ⎞ ⎠ ⎟ ⎟ ⎟ D i ⎟ j G G = D − E = − (1 − ) G ij k ≠ i E ik ⎝ ⎜ ⎜ ⎜ ⎟ ⎠ ⎞ 1 0 0 0 0 3 0 0 0 0 2 0 0 0 0 2 0 Basic Graph Theory II Basic Graph Theory II Example Example Node Properties Node Properties Property Explanation Degree Number of incident edges Matrices Matrices Name Explanation Degree matrix Diagonal matrix containing all node degrees Adjacency Nonzero only if nodes matrix and are adjacent Laplacian matrix δ ij ∑ δ ij E ij 7 of 21
−1 −1 0 −1 2 −1 0 −1 3 2 ⎜ ⎜ ⎜ ⎝ ⎛ G = 0 −1 0 −1 1 ⎞ ⎠ ⎟ ⎟ ⎟ G G 0 0 −1 Basic Graph Theory III Basic Graph Theory III Example Example Laplacian spectrum Laplacian spectrum Certain eigenvalues (EVs) of the Laplacian matrix have special properties: All EVs are non-negative Laplacian spectrum: (as is positive semi-definite ). [0. 1. 3. 4.] At least one eigenvalue is . Number of eigenvalues equal to : Number of connected subgraphs . Second-smallest EV: Algebraic connectivity . 8 of 21
N = 9 The regular 2D grid graph The regular 2D grid graph , periodic boundary conditions 9 of 21
S = 0 p = 1.5 N = 12 p 2 S N 2 p min = = N − 3 p max = S max N ( N − 3) 2 N The Small World Model The Small World Model Example Example Closed ring with nodes, plus randomly chosen shortcuts . Shortcut density Number of shortcuts S = Np Two limiting cases: Limiting case Resulting graph Closed ring Complete graph Maximum number of shortcuts: J. Travers, S. Milgram, 1969 (http://www.jstor.org/stable/2786545) M. Newman, D. Watts, 1999 ; (http://dx.doi.org/10.1016/S0375-9601%2899%2900757-4) R. Huang, G. Korniss, S. Nayak, 2009 ; (http://dx.doi.org/10.1103/PhysRevE.80.045101) 10 of 21
iωL 1 − q Z ij Z ij q = 0.5 p = 1.5 N = 12 Adding Edge Attributes Adding Edge Attributes Example Example In order to describe LC networks , we attribute an impedance to each edge. Here, we consider random impedances, using a binary distribution : Edge type Chance Capacitance ( iωC ) −1 q Inductance R. Huang, G. Korniss, S. Nayak, 2009 (http://dx.doi.org/10.1103 /PhysRevE.80.045101) 11 of 21
h iωL −1 ⎨ ⎩ ⎧ h ij = , edge ( i , j ) carries capacitance C y 2 = ( iωL 1 − q −1 0 y 1 = iωC , h ij Y ij Z ij edge ( i , j ) carries inductance L , no edge between i and j. ⇒ E −1 1 Edge type Chance Capacitance ( iωC ) −1 q Inductance ) −1 1 Introduce matrix : Similar to , but also allowed. R. Huang, G. Korniss, S. Nayak, 2009 (http://dx.doi.org/10.1103 /PhysRevE.80.045101) 12 of 21
I = LV I i N I i = 0 L = 0 L ij V j j ∑ or ( ω ) = or LV = 0 ∑ j L ij V j ω n Resonance frequencies I Resonance frequencies I Remember: Flow equations : with "Laplacian matrix" . Consider resonance case , The system can be seen as a system of coupled LC oscillator circuits The system has resonance frequencies R. Huang, G. Korniss, S. Nayak, 2009 (http://dx.doi.org/10.1103 /PhysRevE.80.045101) 13 of 21
y 2 = y 1 , = Y ij , = 0 Y ij H − (1 − = ) , H ij − k ≠ i h ik y 2 λ = , Y ij , y 1 0 . ( H − λ G ) V = 0 ⇒ LV = 0 y 2 y 1 y 1 y 2 = iωC = ( iωL ) −1 = h ij ⎧ ⎩ ⎨ ⎪ ⎪ −1 1 + Resonance frequencies II Resonance frequencies II Remember: Define , δ ij ∑ δ ij h ij and so that the flow equations for the resonance case can be rewritten as But this is not yet a regular eigenvalue problem... R. Huang, G. Korniss, S. Nayak, 2009 (http://dx.doi.org/10.1103 /PhysRevE.80.045101) Fyodorov 1999 (http://dx.doi.org/10.1088/0305-4470 ; /32/42/314) 14 of 21
y 2 = G −1/2 ~ H H = = ( iωL ) −1 = iωC y 1 y 2 y 1 − y 2 y 1 + λ = V . V ~ G 1/2 G G 1/2 G 0 G −1 G −1/2 Resonance frequencies III Resonance frequencies III Remember: Define (real symmetric) and Notes: 1. is real and positive semi-definite , so its matrix square root is uniquely defined. 2. is positive semi-definite , i.e. it has at least one eigenvalue and hence is always singular . So its pseudo-inverse has to be considered. Fyodorov 1999 (http://dx.doi.org/10.1088/0305-4470/32/42/314) 15 of 21
~ n V . V G 1/2 ( H − λ G ) V = 0 ⇒ = H 1 − λ n ~ λ n V 1 + λ n ~ n λ n ω n = ω n 1 LC −− − = G −1/2 G −1/2 ~ √ − − − − − − λ = + y 1 y 2 − y 1 y 2 y 1 y 2 = iωC = ( iωL ) −1 = H H √ Resonance frequencies IV Resonance frequencies IV Remember: Define (real symmetric) and Then, we can rewrite ~ V So we are facing a regular eigenvalue problem , with a known relationship between the eivenvalues and the resonance frequencies : Fyodorov 1999 (http://dx.doi.org/10.1088/0305-4470/32/42/314) 16 of 21
: λ n < 1) (−1 < N R ρ ( λ ) = δ ( λ − ) 1 n =1 N R λ n Resonance spectra I: 2D grid Resonance spectra I: 2D grid Define the density of resonances (DOR): N ∑ Number of "true" resonances Obtain ensemble average (arithmetic mean) of the DOR over many disorder realizations (ADOR). R. Huang, G. Korniss, S. Nayak, 2009 (http://dx.doi.org/10.1103 /PhysRevE.80.045101) 17 of 21
p ⇒ Resonance spectra II: Small World Model Resonance spectra II: Small World Model Large- limit Confirming results by Huang et al. (http://dx.doi.org/10.1103 /PhysRevE.80.045101) R. Huang, G. Korniss, S. Nayak, 2009 (http://dx.doi.org/10.1103 /PhysRevE.80.045101) 18 of 21
p ⇒ ⇒ λ = 0 Resonance spectra III: Small World Model Resonance spectra III: Small World Model Small- limit Largely confirming results by Huang et al. (http://dx.doi.org/10.1103 /PhysRevE.80.045101) Difference: Peak at . R. Huang, G. Korniss, S. Nayak, 2009 (http://dx.doi.org/10.1103 /PhysRevE.80.045101) 19 of 21
Summary & Outlook Summary & Outlook Summary Summary Description of LC networks simple graphs binary distribution of edge impedances Calculation of resonance frequencies and the density of resonances Outlook Outlook Different topologies (triangular grid, honeycomb grid, and realistic network topologies ). Other impedance distributions (also continuous distributions ), also including ohmic resistances . Beyond the resonance case ( current and power flow calculations ). 20 of 21
Thank you for your attention! References J. Travers, S. Milgram, 1969 (http://www.jstor.org/stable/2786545) M. Newman, D. Watts, 1999 (http://dx.doi.org/10.1016 /S0375-9601%2899%2900757-4) R. Huang, G. Korniss, S. Nayak, 2009 (http://dx.doi.org/10.1103 /PhysRevE.80.045101) Fyodorov 1999 (http://dx.doi.org/10.1088/0305-4470/32/42/314) Jacobs University Bremen School of Engineering and Science Solid State Physics Group Collective Nonlinear Dynamics of Electricity Networks (CoNDyNet) 21 of 21
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