Collective Nonlinear Dynamics of
Electricity Networks (CoNDyNet)

# Determination of resonance frequencies of LC networks with binary link disorder

## Daniel Jung

Jacobs University Bremen
School of Engineering and Science
Solid State Physics Group

December 10, 2014

1. Motivation
• LC circuit
• Flow equations
2. Model
• Basic Graph Theory
• The 2D grid model
• The Small World Model
3. Numerical method
• Resonance frequencies
4. Resonance spectra
• The 2D grid model
• The Small World Model
5. Summary

## Motivation

$Z = R + i X$
$\frac{1}{X} = \frac{1}{i\omega L} + i\omega C$

A LC circuit has a resonance frequency of $\omega_0 = \frac{1}{\sqrt{LC}}$

Frequency $\omega$ of AC current approaching the resonance frequency: $\lim\limits_{\omega \rightarrow \omega_0} Z(\omega) = \infty$

$\Rightarrow$ No power transmission possible!

Normal operation: $\omega$ should stay far below the smallest resonance.

### Arbitrary LC network

• Coupled LC oscillators
• Set of resonance frequencies $\omega_n$

## Flow equations I

An arbitrary one-phasic AC grid
with line impedances $Z_{ij}$.

Generator: $I_i > 0$
Consumer: $I_i < 0$

$\mathbf{Y} = \mathbf{Z}^{-1}$

Combine Ohm's law

$V_{ij} = Z_{ij} I_{ij} \quad\text{or}\quad I_{ij} = Y_{ij} V_{ij} \qquad\qquad$

with Kirchhoff's laws for each node and mesh

$I_i = \sum_j I_{ij} \qquad V_{ij} = V_i - V_j$

to derive the current flow equations

$I_i = \sum_j Y_{ij} (V_i - V_j) \quad\rm.$

Name Symbol
Impedance matrix $\mathbf{Z}$
Admittance matrix $\mathbf{Y}$

## Flow Equations II

An arbitrary one-phasic AC grid.

Generator: $I_i > 0$
Consumer: $I_i < 0$

$I_i = \sum_j Y_{ij} (V_i - V_j)$

To get a standard matrix-vector multiplication, reformulate to

$I_i = \sum_j L_{ij} V_j \quad\mathrm{or}\quad \mathbf{I} = \mathbf{L} \mathbf{V}$

with

$L_{ij} = \delta_{ij} \sum_{k\neq i} Y_{ik} - (1-\delta_{ij}) Y_{ij}$

Note:

• $\mathbf{L}$ is defined in analogy to the topological network Laplacian $\mathbf{G}$.
• $\mathbf{L}$ is commonly referred to as admittance matrix as well.

## Basic Graph Theory I

#### Example

$N = 4$

A graph is given by

• a set of nodes $i$
• a set of edges $(i, j)$

### Properties

Property Explanation
Order $N$ Number of nodes
Size Number of edges

## Basic Graph Theory II

#### Example

$D = \begin{pmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$

$E = \begin{pmatrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{pmatrix}$

$G = \begin{pmatrix} 2 & -1 & -1 & 0 \\ -1 & 2 & -1 & 0 \\ -1 & -1 & 3 & -1 \\ 0 & 0 & -1 & 1 \\ \end{pmatrix}$

### Node Properties

Property Explanation
Degree Number of incident edges

### Matrices

Name Explanation
Degree matrix $\mathbf{D}$ Diagonal matrix containing all node degrees
Adjacency matrix $\mathbf{E}$ Nonzero only if nodes $i$ and $j$ are adjacent
Laplacian matrix $\mathbf{G}$ $\mathbf{G} = \mathbf{D} - \mathbf{E}$

$G_{ij} = \delta_{ij} \sum_{k\neq i} E_{ik} - (1-\delta_{ij}) E_{ij}$

## Basic Graph Theory III

#### Example

Laplacian spectrum:
[0.  1.  3.  4.]


$G = \begin{pmatrix} 2 & -1 & -1 & 0 \\ -1 & 2 & -1 & 0 \\ -1 & -1 & 3 & -1 \\ 0 & 0 & -1 & 1 \\ \end{pmatrix}$

### Laplacian spectrum

Certain eigenvalues (EVs) of the Laplacian matrix $\mathbf{G}$ have special properties:

• All EVs are non-negative
(as $\mathbf{G}$ is positive semi-definite).
• At least one eigenvalue is $0$.
• Number of eigenvalues equal
to $0$: Number of connected subgraphs.
• Second-smallest EV:
Algebraic connectivity.

## The regular 2D grid graph

$N=9$, periodic boundary conditions

## The Small World Model

#### Example

$N=12 \qquad p=1.5$

Closed ring with $N$ nodes, plus $S$ randomly chosen shortcuts.

Shortcut density $\frac{p}{2} = \frac{S}{N}$

Number of shortcuts $S = \frac{Np}{2}$

Two limiting cases:

Limiting case Resulting graph
$p_{\rm min} = 0$ Closed ring
$p_{\rm max} = N - 3$ Complete graph

Maximum number of shortcuts:

$S_{\rm max} = \frac{N(N-3)}{2}$

#### Example

$N=12$
$p=1.5 \qquad q=0.5$

In order to describe LC networks, we attribute an impedance $Z_{ij}$
to each edge.

Here, we consider random impedances, using a binary distribution:

Edge type $Z_{ij}$ Chance
Capacitance $(i\omega C)^{-1}$ $q$
Inductance $i\omega L$ $1-q$

Edge type $Z_{ij}$ Chance $Y_{ij}$ $h_{ij}$
Capacitance $(i\omega C)^{-1}$ $q$ $y_1 = i\omega C$ $-1$
Inductance $i\omega L$ $1-q$ $y_2 = (i\omega L)^{-1}$ $1$

Introduce matrix $\mathbf{h}$:

$h_{ij} = \begin{cases} -1 & \rm,\quad \text{edge (i,j) carries capacitance C} \\ 1 & \rm,\quad \text{edge (i,j) carries inductance L} \\ 0 & \rm,\quad \text{no edge between i and j.} \end{cases}$

$\Rightarrow$ Similar to $\mathbf{E}$, but also $-1$ allowed.

## Resonance frequencies I

Remember:

Flow equations:

$I_i = \sum_j L_{ij} V_j \quad\mathrm{or}\quad \mathbf{I} = \mathbf{L} \mathbf{V}$
with "Laplacian matrix" $\mathbf{L}$.
• Consider resonance case, $I_i = 0$
• The system can be seen as a system of coupled LC oscillator circuits
• The system has $N$ resonance frequencies $\omega_n$

$\sum_j L_{ij}(\omega)\, V_j = 0 \qquad\text{or}\qquad \mathbf{L} \mathbf{V} = \mathbf{0}$

## Resonance frequencies II

Remember: \begin{align} y_1 &= i\omega C \\ y_2 &= (i\omega L)^{-1} \end{align}
$h_{ij} = \begin{cases} -1 &\rm,\quad Y_{ij} = y_1 \\ 1 &\rm,\quad Y_{ij} = y_2 \\ 0 &\rm,\quad Y_{ij} = 0 \end{cases}$

Define $\mathbf{H}$,

$H_{ij} = \delta_{ij} \sum_{k\neq i} h_{ik} - (1-\delta_{ij}) h_{ij} \quad\rm,$

and

$\lambda = \frac{y_1 + y_2}{y_1 - y_2} \quad\rm,$

so that the flow equations for the resonance case can be rewritten as

$\mathbf{L} \mathbf{V} = \mathbf{0} \qquad\Rightarrow\qquad (\mathbf{H} - \lambda \mathbf{G}) \mathbf{V} = \mathbf{0} \quad\rm.$

But this is not yet a regular eigenvalue problem...

## Resonance frequencies III

Remember: $\lambda = \frac{y_1 + y_2}{y_1 - y_2}$
\begin{align} y_1 &= i\omega C \\ y_2 &= (i\omega L)^{-1} \end{align}

Define

$\tilde{\mathbf{H}} = \mathbf{G}^{-1/2}\, \mathbf{H}\, \mathbf{G}^{-1/2}$

(real symmetric) and

$\tilde{\mathbf{V}} = \mathbf{G}^{1/2} \mathbf{V} \quad\rm.$

Notes:

1. $\mathbf{G}$ is real and positive semi-definite, so its matrix square root $\mathbf{G}^{1/2}$
is uniquely defined.
2. $\mathbf{G}$ is positive semi-definite, i.e. it has at least one eigenvalue $0$ and hence is always singular. So its pseudo-inverse $\mathbf{G}^{-1}$
has to be considered.

## Resonance frequencies IV

Remember: $\lambda = \frac{y_1 + y_2}{y_1 - y_2}$
\begin{align} y_1 &= i\omega C \\ y_2 &= (i\omega L)^{-1} \end{align}

Define

$\tilde{\mathbf{H}} = \mathbf{G}^{-1/2}\, \mathbf{H}\, \mathbf{G}^{-1/2}$

(real symmetric) and

$\tilde{\mathbf{V}} = \mathbf{G}^{1/2} \mathbf{V} \quad\rm.$

Then, we can rewrite

$(\mathbf{H} - \lambda \mathbf{G}) \mathbf{V} = \mathbf{0} \qquad\Rightarrow\qquad \mathbf{\tilde{H}}\, \mathbf{\tilde{V}}_n = \lambda_n \mathbf{\tilde{V}}_n$

So we are facing a regular eigenvalue problem, with a known relationship between the eivenvalues $\lambda_n$
and the resonance frequencies $\omega_n$:

$\omega_n = \frac{1}{\sqrt{LC}} \sqrt{\frac{1 + \lambda_n}{1 - \lambda_n}}$

## Resonance spectra I: 2D grid

Define the density of resonances (DOR):

$\rho(\lambda) = \frac{1}{N} \sum_{n=1}^{N_{\mathrm{R}}} \delta(\lambda - \lambda_n) \qquad\qquad$

$N_{\mathrm{R}}\rm:$
Number of "true" resonances
$(-1 < \lambda_n < 1)$

Obtain ensemble average (arithmetic mean) of the DOR over many disorder realizations (ADOR).

## Resonance spectra II: Small World Model

Large-$p$ limit

$\Rightarrow$ Confirming results by Huang et al.

## Resonance spectra III: Small World Model

Small-$p$ limit

$\Rightarrow$ Largely confirming results by Huang et al.

$\Rightarrow$ Difference: Peak at $\lambda = 0$.

## Summary & Outlook

### Summary

• Description of LC networks
• simple graphs
• binary distribution of edge impedances
• Calculation of resonance frequencies
and the density of resonances

### 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).