This repository presents an abstract multi-variable power grid model interpreted through the Victoria-Nash Asymmetric Equilibrium (VNAE) framework. As we can note, the goal is not to reproduce electrical realism in detail, but to demonstrate how heterogeneity and asymmetric dissipation induce global stability in networked energy systems.
This example is intentionally structural and explanatory.
The grid is modeled as a network of interacting nodes belonging to three structural classes:
-
Generators
Nodes with higher inertia and rigidity, representing active power sources. -
Loads
Nodes with lower dissipation, representing consumption and demand variability. -
Renewables
Nodes subject to intermittent forcing, representing fluctuating power injection.
Each node is treated individually, even within the same class.
The model evolves two coupled state variables per node:
- ω(t) : frequency deviation at node i
- pᵢ(t) : power injection at node i
The core frequency dynamics are given by:
dω/dt = − L · ω − Θ · ω + p
where:
• L is the network Laplacian encoding grid connectivity
• Θ = diag(θ₁, … , θₙ) is a diagonal matrix of asymmetric dissipation parameters
• p ∈ ℝⁿ represents injected or consumed power
Power dynamics follow a dissipative evolution with intermittent forcing on renewable nodes.
- θᵢ represents structural rigidity or dissipation at node i
- Each node has its own θ value
- Generator, load, and renewable labels determine parameter ranges, not variables
Asymmetry emerges from the distribution of theta across the network, not from node type alone.
A quadratic effective metric is introduced as:
g = I + β · (Θ + A)
where:
• I is the identity matrix
• Θ = diag(θ₁, … , θₙ) is the asymmetric dissipation matrix
• A is the weighted adjacency matrix
• β > 0 controls the strength of geometric deformation
A scalar curvature proxy K is computed from heterogeneity and connectivity to interpret stability, not to drive the dynamics.
It is good to highlight that:
This is not a full Riemannian construction,
but a quadratic/canonical effective geometry derived from the VNAE framework.
Positive K corresponds to structural contraction induced by asymmetric dissipation.
This repository serves as a canonical example of how asymmetric heterogeneity in power networks can be interpreted geometrically, providing an operational illustration of stability under the VNAE framework.
Pereira, D. H. (2025). Riemannian Manifolds of Asymmetric Equilibria: The Victoria-Nash Geometry.
MIT License