grid-coordination

Microgrid-inspired coordination framework: Grid-Forming (GFM) vs Grid-Following (GFL) agents with spectral gap 1/4 benchmark for direction entropy justifiability.

Core Concept

In microgrids:

• Grid-Forming (GFM) inverters: Set voltage/frequency reference (leader)
• Grid-Following (GFL) inverters: Synchronize to existing reference (follower)

In coordination:

• GFM agents: Set specification rate (drift component, deterministic)
• GFL agents: Track direction entropy (diffusion component, stochastic)
• Spectral gap 1/4: The Ramanujan benchmark for optimal mixing

Key Formula: Justifiability

Direction entropy is JUSTIFIED when:
    H(direction) ≈ specification_rate / gap_benchmark

At gap = 0.25 (Ramanujan optimal):
    H_justified = S / 0.25 = 4 × S

Justifiability Score:
    J = 1 - |H - H_justified| / max(H, H_justified)

Usage

# Run the demo
cd ~/Gay.jl && julia examples/grid_coordination_demo.jl

# In Julia REPL
include("src/grid_coordination.jl")
states = simulate_coordination(
    potential = x -> 0.5 * sum(x.^2),  # Single well
    gfm_initial = [2.0, 1.0],
    gfl_initial = [-1.0, -0.5],
    diffusion_strength = 0.4,
    gap = 0.25,  # Ramanujan benchmark
    dt = 0.01,
    T = 5.0
)
println(coordination_report(states))

Agent Types

GridFormingAgent (GFM)

mutable struct GridFormingAgent <: CoordinationAgent
    specification_rate::Float64    # Constraint generation (bits/τ)
    potential::Function            # V(x): energy landscape
    state::Vector{Float64}         # Current position
end

• Role: Creates the reference frame (like GFM inverter sets voltage)
• Dynamics: Pure drift

  dx = -∇V dt

• Metric: Specification rate S = |∇²V| / log(2)

GridFollowingAgent (GFL)

mutable struct GridFollowingAgent <: CoordinationAgent
    direction_entropy::Float64     # H(direction) in bits
    diffusion_strength::Float64    # σ² noise intensity
    state::Vector{Float64}
    reference_agent::GridFormingAgent
end

• Role: Synchronizes to reference (like GFL inverter locks to grid)
• Dynamics: Drift + diffusion

  dx = -∇V dt + σ√(gap) dW

• Metric: Direction entropy H = -Σ pᵢ log₂(pᵢ)

Spectral Gap Benchmark

Gap	Interpretation	Mixing Time	Justifiability
< 0.1	Tangled, slow mixing	> 10τ	Over-exploring
0.25	Ramanujan optimal	4τ	Aligned
> 0.5	Over-connected	< 2τ	Under-exploring

GF(3) Integration

Justifiability maps to triadic coloring:

function gf3_trit(J::Float64)
    J >= 0.67 → :PLUS     # GFM dominant (warm, 330°)
    J >= 0.33 → :ERGODIC  # Balanced (neutral, 120°)
    J <  0.33 → :MINUS    # GFL dominant (cold, 240°)
end

Conservation: Across parallel coordination streams, Σ trits ≡ 0 (mod 3)

Fokker-Planck Connection

The GFM-GFL dynamics are the agent-level realization of Fokker-Planck:

∂ρ/∂t = -∇·(μρ) + ∇·(D∇ρ)
         ↑ GFM      ↑ GFL
        drift    diffusion

• Drift μ = -∇V: GFM sets the deterministic flow
• Diffusion D = σ²: GFL provides stochastic exploration
• Stationary ρ∞ ∝ exp(-V/D): Equilibrium when GFM-GFL balanced

Practical Applications

1. Multi-Agent Coordination

# Three agents with GF(3) conservation
gfm = GridFormingAgent(...)     # :PLUS
gfl1 = GridFollowingAgent(...)  # :MINUS
gfl2 = GridFollowingAgent(...)  # :ERGODIC
# Σ = +1 - 1 + 0 = 0 ✓

2. Skill Dispersal

# Spectral gap controls how fast skills spread
gap = 0.25  # Optimal: skills reach all agents in 4τ

3. Proof Dependency Graphs

# From spectral_analyzer.jl
if gap >= 0.25
    :ramanujan  # Theorems well-connected
else
    :tangled    # Dependencies blocking mixing
end

Files

• ~/Gay.jl/src/grid_coordination.jl

  - Core framework

• ~/Gay.jl/examples/grid_coordination_demo.jl

  - Interactive demo

• ~/Gay.jl/examples/fokker_planck.jl

  - Underlying PDE

• ~/plurigrid-asi/ies/spectral_analyzer.jl

  - Gap measurement

Related Skills

• fokker-planck

  - The underlying PDE dynamics

• spectral-analyzer

  - Ramanujan gap measurement

• gf3-coloring

  - Triadic color assignment

• bisimulation-game

  - Agent equivalence via spectral properties

Quick Reference

┌─────────────────────────────────────────────────────────┐
│  GRID COORDINATION CHEAT SHEET                         │
├─────────────────────────────────────────────────────────┤
│  GFM = Grid-Forming = Drift = Specification            │
│  GFL = Grid-Following = Diffusion = Direction Entropy  │
│                                                         │
│  Spectral Gap = λ₁ - λ₂                                │
│  Benchmark = 0.25 (Ramanujan)                          │
│  Mixing Time = 1/gap ≈ 4τ at benchmark                 │
│                                                         │
│  JUSTIFIED when: H(dir) ≈ S(spec) / 0.25              │
│                                                         │
│  GF(3): +:GFM-heavy  0:Balanced  -:GFL-heavy          │
└─────────────────────────────────────────────────────────┘

GF(3) Assignment

• Trit: ERGODIC (0) - coordination/synthesis skill
• Hue: 120° (green) - balanced mixing dynamics