Local Mode Identification

Demonstrates computing the SGMA spectrum at a single bus to identify dominant oscillatory modes in the wavelength-frequency domain.

The Joint Spectrum

For a target bus \(n\) and time \(\tau\), the SGMA computes the joint wavelet transform:

\[m_{n,\tau}(\Lambda \times S) \approx L_n X R_\tau\]

The resulting spectrum \(m_{n,\tau}\) reveals the energy distribution across spatial wavelengths (related to \(\sqrt{s}\)) and temporal frequencies. Peaks in this 2D spectrum correspond to dominant oscillatory modes.

This example shows how to:

  • Initialize the SGMA engine with spatial scales and temporal frequencies.

  • Compute the spectrum for a specific bus and time using sgma.spectrum().

  • Extract modes with sgma.find_modes() to obtain frequency, damping, and wavelength.

Single Bus SGMA Analysis
from sgwt import SGMA
from sgwt import LENGTH_WECC as L

# Signals: Real or Complex Matrix (Rows: Buses, Cols: Time)
V, t = get_signal(FILEPATH, t_range=(0, 60))

# SGMA Parameters
BUS_TARGET = 36
TIME_TARGET = 2.0
ORDER = 1
TOP_N = 3

wmin = 1
wmax = 3e3
nscales = 150
spatial_scales = np.geomspace(wmin**2, wmax**2, nscales)  

temporal_freqs = np.linspace(0.05, 2.0, 100)
sgma = SGMA(L, spatial_scales, temporal_freqs, order=ORDER, w0=2*np.pi)

# Get complex spectrum (compute once)
M = sgma.spectrum(V, t, BUS_TARGET, TIME_TARGET, return_complex=True)

# Identify modes with frequency, damping ratio, wavelength, and magnitude
modes = sgma.find_modes(M, top_n=TOP_N)
SGMA Spectrum at a Single Bus

The contour plot shows the spectrum magnitude in the wavelength-frequency domain. The overlaid markers indicate the top_n most dominant oscillatory modes (peaks) identified at the target bus and time instant. Each peak provides:

  • Wavelength \(r = \sqrt{s}\): spatial extent of the mode

  • Frequency \(f_0\): oscillation rate in Hz

  • Damping \(\zeta\): decay rate estimated from phase slope

IEEE 39 Bus New England Case

The IEEE 39 bus New England ISO case demonstrates SGMA on a real-world power system topology. The signal is the complex voltage response to a balanced three-phase fault at Bus 16, occurring at \(t=0.5\) seconds and cleared at \(t=0.7\) seconds. The simulation uses a time step of 1 millisecond from \(t=0\) to \(t=10\) seconds.

SGMA Parameters

The analysis uses the following configuration:

Graph and Analysis Point:

  • Laplacian: LENGTH_NEISO (39 buses)

  • Target bus: 12

  • Target time: \(\tau = 1.5\) seconds

  • Bandpass order: \(K = 1\)

Spatial Sampling:

  • Number of scales: 150

  • Scale range: \(s \in [(0.1)^2, (100)^2]\) (geometrically spaced)

  • Wavelength range: \(r = \sqrt{s} \in [0.1, 100.0]\)

Temporal Sampling:

  • Number of frequencies: 200

  • Frequency range: \(f \in [0.01, 3.0]\) Hz (linearly spaced)

  • Wavelet parameter: \(\omega_0 = 2\pi\)

IEEE 39 Bus SGMA Analysis
from sgwt import SGMA
from sgwt import LENGTH_NEISO as L

# Signals: Real or Complex Matrix (Rows: Buses, Cols: Time)
V, t = get_signal(FILEPATH)

# SGMA Parameters
BUS_TARGET = 12
TIME_TARGET = 1.5
ORDER = 1
TOP_N = 3

wmin = 1e-1#1
wmax = 1e2#3e3
nscales = 150
spatial_scales = np.geomspace(wmin**2, wmax**2, nscales)  

temporal_freqs = np.linspace(0.01, 3, 200)
sgma = SGMA(L, spatial_scales, temporal_freqs, order=ORDER, w0=2*np.pi)

# Get complex spectrum (compute once)
M = sgma.spectrum(V, t, BUS_TARGET, TIME_TARGET, return_complex=True)

# Identify modes with frequency, damping ratio, wavelength, and magnitude
modes = sgma.find_modes(M, top_n=TOP_N)
SGMA Spectrum for IEEE 39 Bus System

Identified Modes

The find_modes function extracts the top 3 peaks from the spectrum:

------------------------------------------------------------
#   Freq (Hz)     Damping    Wavelength   Magnitude
------------------------------------------------------------
1      0.4908      0.0000          4.91      0.0209
2      1.0768      0.3920          4.48      0.0201
3      1.9182      0.1638          4.27      0.0084
------------------------------------------------------------
Each mode is characterized by:

  • Frequency (Hz): Oscillation rate extracted from the temporal frequency axis

  • Damping: Decay rate estimated from phase variation in the complex spectrum

  • Wavelength: \(r = \sqrt{s}\) at the peak location in the spatial domain

  • Magnitude: Transform magnitude at the peak location