λ₂
Electrical · Power Systems · Spectral Graph TheoryTopological Vulnerability Map
The Fiedler value λ₂ of the network's weighted graph Laplacian is the algebraic connectivity — the quantitative measure of how hard it is to disconnect the grid. The Fiedler vector v₂ reveals the natural partition: where the network would split first under stress.
L = D − A · λ₂ = minₓ≠₀ xᵀLx / xᵀx · V(Cᵢ) = (λ₂ − λ₂(Cᵢ)) / λ₂
Baseline λ₂
5.2049
Algebraic connectivity
Cut edges
3
Bisection boundary
Islanding edges
11
Would disconnect network
Most critical
10→32 (T)
Highest vulnerability
Fiedler vector overlay(amber = partition A, blue = partition B, dashed red = removed edge, amber lines = cut edges)
● Generator busAmber/blue = Fiedler vector signYellow lines = partition cut
Baseline partition
Partition A (v₂ ≥ 0)
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 33, 34, 35, 36, 38
Partition B (v₂ < 0)
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 25, 30, 31, 32, 37, 39
Cut edges (3)
3→18, 14→15, 25→26
N-1 Contingency Ranking (46 edges)
Sort by:
| Rank | Edge | Type | Post λ₂ | λ₂ drop | Vuln % | Island? |
|---|---|---|---|---|---|---|
| 1 | 10→32 (T) | T | 0 | 5.2049 | 100.0% | YES |
| 2 | 23→36 (T) | T | 0 | 5.2049 | 100.0% | YES |
| 3 | 19→33 (T) | T | 0 | 5.2049 | 100.0% | YES |
| 4 | 2→30 (T) | T | 0 | 5.2049 | 100.0% | YES |
| 5 | 19→20 (T) | T | 0 | 5.2049 | 100.0% | YES |
| 6 | 20→34 (T) | T | 0 | 5.2049 | 100.0% | YES |
| 7 | 6→31 (T) | T | 0 | 5.2049 | 100.0% | YES |
| 8 | 22→35 (T) | T | 0 | 5.2049 | 100.0% | YES |
| 9 | 25→37 (T) | T | 0 | 5.2049 | 100.0% | YES |
| 10 | 29→38 (T) | T | 0 | 5.2049 | 100.0% | YES |
| 11 | 16→19 | L | 0 | 5.2049 | 100.0% | YES |
| 12 | 16→17 | L | 2.1311 | 3.0738 | 59.1% | — |
| 13 | 15→16 | L | 2.7486 | 2.4563 | 47.2% | — |
| 14 | 14→15 | L | 2.7726 | 2.4323 | 46.7% | — |
| 15 | 17→27 | L | 2.8407 | 2.3643 | 45.4% | — |
| 16 | 26→27 | L | 3.1273 | 2.0777 | 39.9% | — |
| 17 | 2→25 | L | 3.3089 | 1.8961 | 36.4% | — |
| 18 | 3→4 | L | 3.5833 | 1.6216 | 31.2% | — |
| 19 | 1→2 | L | 3.6299 | 1.5751 | 30.3% | — |
| 20 | 16→21 | L | 3.6940 | 1.5110 | 29.0% | — |