---
title: BEP-Gravitation Optimality Proof
created: 2026-04-07
updated: 2026-04-07
status: proven
tags: [machineGroupControl, optimization, BEP, brute-force]
sources: [nodes/machineGroupControl/test/integration/distribution-power-table.integration.test.js]
---

# BEP-Gravitation vs Brute-Force Global Optimum

## Claim
The machineGroupControl BEP-Gravitation algorithm (with marginal-cost refinement) produces near-optimal flow distribution across a pump group.

## Method
Brute-force exhaustive search: 1000 steps per pump, all 2^n combinations, 0.05% flow tolerance. Station: 2x H05K-S03R + 1x C5-D03R-SHN1 @ ΔP=2000 mbar.

## Results

| Demand | Brute force | machineGroupControl | Gap |
|--------|------------|--------------------|----|
| 10% (71 m3/h) | 17.65 kW | 17.63 kW | -0.10% (MGC wins) |
| 25% (136 m3/h) | 34.33 kW | 34.33 kW | +0.01% |
| 50% (243 m3/h) | 61.62 kW | 61.62 kW | -0.00% |
| 75% (351 m3/h) | 96.01 kW | 96.10 kW | +0.08% |
| 90% (415 m3/h) | 122.17 kW | 122.26 kW | +0.07% |

Maximum deviation: **0.1%** from proven global optimum.

## Why the Refinement Matters

Before the marginal-cost refinement loop, the gap at 50% demand was **2.12%**. The BEP-Gravitation slope estimate pushed 14.6 m3/h to C5 (costing 5.0 kW) when the optimum was 6.5 m3/h (0.59 kW). The refinement loop corrects this by shifting flow from highest actual dP/dQ to lowest until no improvement is possible.

## Stability
Sweep 5-95% in 2% steps: 1 switch (rising), 1 switch (falling), same transition point. No hysteresis. See [[Pump Switching Stability]].

## Computational Cost
0.027-0.153ms median per optimization call (3 pumps, 6 combinations). Uses 0.015% of the 1000ms tick budget.