This repository presents a large-scale stability validation of power grid frequency dynamics using the Victoria-Nash Asymmetric Equilibrium (VNAE) and/or Victoria-Nash Geometry framework.
The model demonstrates how global stability can emerge from asymmetric dissipation, even in massive (with 100,000 agents), heterogeneous, and sparsely connected energy networks without requiring synchronization or eigenvalue-based tuning.
Modern power grids operate under:
- high renewable penetration,
- heterogeneous inertia (turbines, inverters, storage),
- sparse and directed transmission topologies,
- persistent external disturbances (weather, demand shocks).
Classical small-signal or symmetric consensus models struggle to certify global stability at scale.
We can say that the VNAE provides a geometric alternative in which stability is governed by volume contraction on a curved manifold rather than pointwise convergence.
The grid dynamics are modeled as a dissipative flow on a networked manifold:
dω/dt = − (L + Θ) · ω + p
where:
ω ∈ Rⁿ = Vector of node frequency deviations (one component per grid node).
L = Directed graph Laplacian encoding transmission-line couplings and network topology.
Θ = diag(θ₁, θ₂, …, θₙ) Diagonal matrix of heterogeneous asymmetric dissipation parameters (e.g. generator inertia, inverter damping, load responsiveness).
p = Persistent external forcing vector (e.g. weather-induced renewable intermittency or demand shocks).
- Grid size: 100,000 nodes
- Topology: ultra-sparse directed network
- Node heterogeneity: wide inertia distribution
- Dynamics: nonlinear dissipative flow
- Numerics: sparse matrices + Monte Carlo geometry
This scale moves the framework well beyond toy models and into realistic infrastructure regimes.
The transmission network is modeled as a sparse directed graph, where each node is connected on average to approximately 15 transmission lines.
This reflects realistic large-scale power grids, in which physical, economic, and geographical constraints lead to low-degree connectivity even at national or continental scales.
The sparse structure ensures computational scalability while preserving the essential propagation pathways of frequency disturbances.
Global stability is assessed via a statistical curvature proxy:
K = E [ |θ_i − θ_j| · |A_ij| / ( 1 + β · (θ_i + θ_j) ) ]
Estimated via Monte Carlo sampling:
(i, j) ~ Uniform( {1,…,n} × {1,…,n} )
- (K > 0) → positively curved effective manifold
- Positive curvature → volume contraction
- Volume contraction → global stability (VNAE criterion)
No eigenvalue alignment or synchronization is required.
Report excerpt:
- Total Grid Nodes (n): 100,000
- Structural Curvature (K): 0.00121638
- Metric Determinant Proxy: 1.4e+120
- Status: Geometrically Stable (VNAE Certified)
- Only a random subset of nodes (n = 100) is visualized for readability.
- All trajectories rapidly contract into a narrow band.
- Late-time dispersion reflects:
- heterogeneous dissipation,
- sparse directed coupling,
- numerical integration noise.
Importantly:
Pointwise convergence is not required. Volume contraction is the stability mechanism.
- Scales to national-grid-sized systems
- Robust to heterogeneity and asymmetry
- Avoids fragile spectral assumptions
- Applicable to:
- smart grids
- inverter-dominated systems
- systemic risk networks
- large-scale cyber-physical infrastructure
Pereira, D. H. (2025). Riemannian Manifolds of Asymmetric Equilibria: The Victoria-Nash Geometry.