Graph Laplacian
This section outlines the mathematical foundations of the Sparse SGWT library.
Let an undirected graph \(\mathcal{G}=\{\mathcal{V}, \mathcal{E}, \mathbf{A}, \mathbf{w}\}\) be defined by a set of vertices \(|\mathcal{V}|=N\) and a set of edges \(\mathcal{E}\) which are related by the arc-node incidence matrix \(\mathbf{A}\in\mathbb{R}^{|\mathcal{E}|\times |\mathcal{V}|}\) and the vector of branch weights \(\mathbf{w}\in\mathbb{R}^{|\mathcal{E}|}\).
A vertex domain function on the graph \(f:\mathcal{V}\to\mathbb {R}\) can be written as a vector \(\mathbf{f}\in \mathbb{R}^N\), whose \(i^{th}\) element corresponds to the evaluation of \(f\) at the \(i^{th}\) vertex.
The Graph Laplacian is denoted by \(\mathbf{L}\in\mathbb{R}^{N\times N}\), a discrete analogue of the continuous Laplace-Beltrami operator:
Alternatively, this can be viewed element-wise as \(\mathbf{L} = \mathbf{D} - \mathbf{W}\), where \(\mathbf{D}\) is the degree matrix and \(\mathbf{W}\) is the weighted adjacency matrix.
The Physics of the Weights
While any symmetric matrix can serve as a Laplacian, sgwt utilizes specific weighting schemes to ensure the graph spectral domain aligns with the physical behavior of power grids.
When working with a physical distance metric, we might want to weigh the branches by the Inverse Squared Length. For each branch of length \(\ell_{ij}\), the corresponding branch weight is assigned as:
This weighting is not arbitrary. By defining the weights as \(\ell^{-2}\), the eigenvalues of the graph Laplacian (\(\lambda\)) correspond directly to the squared wavenumber (\(k^2\)) of traveling waves on the grid.
Low Eigenvalues: Correspond to long-wavelength, inter-area oscillations that span the entire continent.
High Eigenvalues: Correspond to short-wavelength, local disturbances.
When we apply the SGWT to this graph, each scale \(a\in\mathcal{A}\) becomes physically meaningful, corresponding to a squared pseudo-wavelength, \(r\in \mathbb{R}\), where \(a=r^2\) defines this mapping. This allows for filtering based on physical spread rather than just temporal frequency.
Derivation of the Delay Weight
To derive the precise branch weights for the Delay Graph Laplacian, we must calculate the effective parameters of transformers and transmission lines, accounting for the fact that real-world grid data often aggregates these values.
Effective Branch Shunt Admittance
Shunt admittance is often stored as a nodal aggregate, the sum of the pi-model tranmission lines connected to each node. We define the effective branch shunt admittance \(Y^{sh}_{ij}\) as the average of the nodal shunt admittance at the branch terminals:
\[Y^{sh}_{ij}=\dfrac{1}{2}(Y^{sh}_{i}+Y^{sh}_{j})\]
Lossless Propagation Delay
Using these parameters, we estimate the lossless propagation delay \(\theta_{ij}\) for every branch. This represents the “electrical distance” a wave must travel:
The expression above works because the total line parameters of the line are given, rather than the per-meter values. This forms the definition of the Delay Graph Laplacian:
Which has shown to be one of the most reliable branch weights in practice.
Summary of Useful Weighting Schemes
Depending on the physical phenomenon being analyzed, different weighting schemes may be appropriate.
Reciprocal Squared Delay (Preferred):
\[w_{ij} = \frac{1}{\theta_{ij}^2}\]Used for wave propagation analysis or dynamics. The determination for the value of \(\tau\) in the previous section has proven practical for very large synthetic cases and results in a well-behaved network that outperformed other branch metrics in SGWT applications like modal analysis and FOSL.
Reciprocal Squared Distance:
\[w_{ij} = \frac{1}{d_{ij}^2}\]Used for geographic analysis when electrical parameters are unavailable but GPS coordinates are known. Aligns the graph spectrum with the squared wavenumber (\(k^2\)).
Admittance Magnitude:
\[w_{ij} = \left| \frac{1}{Z_{ij}} \right| = |Y_{ij}|\]Used where circuit analysis is directly concerned.