Loop Flow Decomposition

Every branch flow in a meshed network is the sum of a tree component (scheduled, radial) and a loop component (circulating, unscheduled). The loop component is the holonomy of the voltage angle connection around each fundamental cycle of the network graph.

dim(cycle space) = |E| − |V| + 1 = 8 · f_loop = γ / Σx · γ = ∮ dθ

Load scaling factor λ = 0.00(0% above base load)

λ = 0 (base)λ = 0.6λ = 1.2

Circulating power9741 MW26.6% of transmitted

Network — branch loop fraction(green = low loop, red = high loop, amber = selected cycle)

● Generator bus· Load/transit bus→ Flow direction

Fundamental cycles(8 co-tree chords)

Branch flow decomposition — all 46 branches

Sort by:

Branch	Total (MW)	Tree (MW)	Loop (MW)	Loop %
11→12Bus–BusXFMR	-129.0	-668.6	539.6	418.3%
12→13Bus–BusXFMR	-129.9	409.7	-539.6	415.5%
4→14Bus–Bus	-494.6	46.4	-541.0	109.4%
26→28Bus–Bus	-388.0	-796.7	408.6	105.3%
26→29Bus–Bus	-393.0	15.6	-408.6	104.0%
16→17Bus–Bus	583.7	1186.2	-602.5	103.2%
28→29Bus–Bus	-408.6	0.0	-408.6	100.0%
3→18Bus–Bus	-896.4	-1714.7	818.2	91.3%
17→18Bus–Bus	912.2	1730.5	-818.2	89.7%
17→27Bus–Bus	-328.6	-112.8	-215.7	65.7%
26→27Bus–Bus	356.7	572.4	-215.7	60.5%
25→26Bus–Bus	-410.5	-194.8	-215.7	52.6%
10→13Bus–Bus	-984.4	-1462.4	478.1	48.6%
3→4Bus–Bus	1495.2	2155.4	-660.2	44.2%
22→23Bus–Bus	-53.9	-75.9	22.0	40.9%
14→15Bus–Bus	-1608.8	-1006.3	-602.5	37.5%
15→16Bus–Bus	-1612.0	-2214.5	602.5	37.4%
10→11Bus–Bus	1634.4	1156.3	478.1	29.3%
2→3Bus–Bus	602.0	443.9	158.0	26.3%
2→25Bus–Bus	-928.1	-712.4	-215.7	23.2%
5→8Bus–Bus	-10.6	-12.6	2.1	19.4%
6→7Bus–Bus	-376.7	-432.3	55.6	14.8%
7→8Bus–Bus	-400.1	-344.5	-55.6	13.9%
9→39Bus–Slack	-462.9	-405.2	-57.7	12.5%
8→9Bus–Bus	-462.9	-405.2	-57.7	12.5%
1→39Bus–Slack	566.9	509.2	57.7	10.2%
1→2Bus–Bus	-576.1	-518.5	-57.7	10.0%
4→5Bus–Bus	1984.8	2104.0	-119.2	6.0%
5→6Bus–Bus	1995.3	2112.5	-117.1	5.9%
13→14Bus–Bus	-1114.2	-1052.7	-61.5	5.5%
16→24Bus–Bus	-450.5	-472.5	22.0	4.9%
23→24Bus–Bus	481.3	503.4	-22.0	4.6%
6→11Bus–Bus	-1763.4	-1701.8	-61.5	3.5%
16→21Bus–Bus	-676.5	-654.5	-22.0	3.3%
21→22Bus–Bus	-703.9	-681.9	-22.0	3.1%
16→19Bus–Bus	-1072.0	-1072.0	0.0	0.0%
2→30Bus–GenXFMR	-250.0	-250.0	0.0	0.0%
6→31Bus–GenXFMR	4135.4	4135.4	0.0	0.0%
10→32Bus–GenXFMR	-650.0	-650.0	0.0	0.0%
19→20Bus–BusXFMR	-440.0	-440.0	0.0	0.0%
19→33Bus–GenXFMR	-632.0	-632.0	0.0	0.0%
20→34Bus–GenXFMR	-508.0	-508.0	0.0	0.0%
22→35Bus–GenXFMR	-650.0	-650.0	0.0	0.0%
23→36Bus–GenXFMR	-560.0	-560.0	0.0	0.0%
25→37Bus–GenXFMR	-540.0	-540.0	0.0	0.0%
29→38Bus–GenXFMR	-830.0	-830.0	0.0	0.0%

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