Determination of resonance frequencies of LC Determination of - - PowerPoint PPT Presentation
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
Collective Nonlinear Dynamics of Electricity Networks (CoNDyNet)
Jacobs University Bremen School of Engineering and Science Solid State Physics Group December 10, 2014
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Motivation LC circuit Flow equations 1. Model Basic Graph Theory The 2D grid model The Small World Model Adding Edge Attributes 2. Numerical method Resonance frequencies 3. Resonance spectra The 2D grid model The Small World Model 4. Summary 5.
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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.
Coupled LC oscillators Set of resonance frequencies
1 LC √
ω→ω0
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An arbitrary one-phasic AC grid with line impedances . Generator: Consumer:
Combine Ohm's law with Kirchhoff's laws for each node and mesh to derive the current flow equations Name Symbol Impedance matrix Admittance matrix
j
j
/PhysRevE.80.045101)
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An arbitrary one-phasic AC grid. Generator: Consumer:
To get a standard matrix-vector multiplication, reformulate to with Note: is defined in analogy to the topological network Laplacian . is commonly referred to as admittance matrix as well.
j
j
k≠i
/PhysRevE.80.045101)
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Example Example
A graph is given by a set of nodes a set of edges
Property Explanation Order Number of nodes Size Number of edges
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Example Example
Property Explanation Degree Number of incident edges
Name Explanation Degree matrix Diagonal matrix containing all node degrees Adjacency matrix Nonzero only if nodes and are adjacent Laplacian matrix
D = ⎛ ⎝ ⎜ ⎜ ⎜ 2 2 3 1 ⎞ ⎠ ⎟ ⎟ ⎟ E = ⎛ ⎝ ⎜ ⎜ ⎜ 1 1 1 1 1 1 1 1 ⎞ ⎠ ⎟ ⎟ ⎟ G = ⎛ ⎝ ⎜ ⎜ ⎜ 2 −1 −1 −1 2 −1 −1 −1 3 −1 −1 1 ⎞ ⎠ ⎟ ⎟ ⎟
k≠i
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Example Example
Certain eigenvalues (EVs) of the Laplacian matrix have special properties: All EVs are non-negative (as is positive semi-definite). At least one eigenvalue is . Number of eigenvalues equal to : Number of connected subgraphs. Second-smallest EV: Algebraic connectivity.
Laplacian spectrum: [0. 1. 3. 4.]
G = ⎛ ⎝ ⎜ ⎜ ⎜ 2 −1 −1 −1 2 −1 −1 −1 3 −1 −1 1 ⎞ ⎠ ⎟ ⎟ ⎟
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, periodic boundary conditions
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Example Example
Closed ring with nodes, plus randomly chosen shortcuts. Shortcut density Number of shortcuts Two limiting cases: Limiting case Resulting graph Closed ring Complete graph Maximum number of shortcuts:
; ;
p 2 S N
2
(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)
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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 Inductance
(http://dx.doi.org/10.1103 /PhysRevE.80.045101)
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Edge type Chance Capacitance Inductance Introduce matrix : Similar to , but also allowed.
/PhysRevE.80.045101)
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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
j
j
/PhysRevE.80.045101)
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Remember:
Define , and so that the flow equations for the resonance case can be rewritten as But this is not yet a regular eigenvalue problem... ;
k≠i
/PhysRevE.80.045101) Fyodorov 1999 (http://dx.doi.org/10.1088/0305-4470 /32/42/314)
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Remember:
Define (real symmetric) and Notes: is real and positive semi-definite, so its matrix square root is uniquely defined. 1. 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. 2.
Fyodorov 1999 (http://dx.doi.org/10.1088/0305-4470/32/42/314)
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Remember:
Define (real symmetric) and Then, we can rewrite So we are facing a regular eigenvalue problem, with a known relationship between the eivenvalues and the resonance frequencies :
n
n
Fyodorov 1999 (http://dx.doi.org/10.1088/0305-4470/32/42/314)
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Define the density of resonances (DOR): Number of "true" resonances Obtain ensemble average (arithmetic mean)
realizations (ADOR).
n=1 NR
/PhysRevE.80.045101)
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Large- limit Confirming results by
Huang et al. (http://dx.doi.org/10.1103 /PhysRevE.80.045101)
/PhysRevE.80.045101)
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Small- limit Largely confirming results by Difference: Peak at .
Huang et al. (http://dx.doi.org/10.1103 /PhysRevE.80.045101)
/PhysRevE.80.045101)
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Description of LC networks simple graphs binary distribution of edge impedances Calculation of resonance frequencies and the density of resonances
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).
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References Jacobs University Bremen School of Engineering and Science Solid State Physics Group
Collective Nonlinear Dynamics of Electricity Networks (CoNDyNet)
/S0375-9601%2899%2900757-4)
/PhysRevE.80.045101) Fyodorov 1999 (http://dx.doi.org/10.1088/0305-4470/32/42/314)
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