Irrigation bucket (irrigation_bucket)

Question. How does an irrigation schedule designed for a dry forecast compare with a closed loop that sees rain on the water store?

Formulation

Symbol	Role	Meaning
\(S(t)\)	state	root-zone storage (mm)
\(u(t)\)	control	irrigation (mm d\(^{-1}\))
\(P(t)\)	forcing	rain (forecast vs realised)
\(E(t)\)	forcing	evapotranspiration

Dynamics.

\[ \dot S = u + P(t) - E(t) - k S. \]

Objective. \(\min \int_0^T u\,\mathrm{d}t\).

Constraints. \(S_{\min}\le S\le S_{\max}\). Horizon. Fixed.

The OCP forecast uses \(P\equiv 0\). Closed loop can apply rain to the simulated water store while the controller still plans dry. (plant= in the API means that simulated process, not vegetation.)

What was done

The OCP is always the dry forecast: \(P\equiv 0\), sinusoidal ET, minimize irrigation, keep \(S\) between field-capacity-style bounds. That tape is the grower’s plan written on day 0.

Four integrations:

1. Open-loop NLP on the dry model (\(J^*\approx 51.7\,\mathrm{mm}\)).
2. simulate of that \(u^*\) on the same dry model — must reprint \(S^*(t)\).
3. Shrinking closed_loop on the dry model — must reprint \(u^*\) (nothing unexpected happens).
4. Unforecast rain on the water store in days 5–7: replay the dry tape vs re-solve from measured \(S\).

Rain is applied to storage \(S\), not to a vegetation canopy. The API keyword plant= is the simulated process model (control-theory name).

What we found

On the dry process, closed-loop \(u\) matches the open-loop tape (max \(|u-u^*|\) small). After rain, the tape keeps irrigating as if the soil were still dry and can pile \(S\) toward \(S_{\max}\). The closed loop sees the wetter store and cuts irrigation (more than \(5\,\mathrm{mm}\) less water in the test). Storage stays in bounds in both cases; the difference is wasted pumping, not a bound violation.

Interpretation

Open loop answers: given this forecast, what irrigation is optimal? Closed loop answers: given the store I measure now, what irrigation is optimal for the remaining season? Those coincide only if the forecast was right. In ecology that is the difference between a seasonal allocation plan and re-allocation after a rain, a litterfall pulse, or a missed mineral-N flush. The OCP does not become a different hypothesis; the state from which it is solved changes.

Public API

Full script: examples/irrigation_bucket/irrigation_bucket.py.

import eco_opt
from eco_opt import Namespace


def dry_model(t, x, u):
    S, irr = x[0], u[0]
    dS = irr - et(t) - K_DRAIN * S
    y = Namespace(S=S, u=irr)
    return dS, y


sys = eco_opt.System(states=1, controls=1, model=dry_model)
ocp = eco_opt.Ocp()
ocp.min([sys.y.u])
ocp.st(system=sys, constraints=[...])
sol = ocp.solve()

Closed loop on a rainy process: ocp.closed_loop in examples/irrigation_bucket/open_vs_closed.py.

Run and tests

cd examples/irrigation_bucket
python irrigation_bucket.py
python plot_irrigation_bucket.py
python open_vs_closed.py
python -m pytest ../../tests/test_irrigation_bucket.py -q

References

Inspired by Shang, Chen, Stroock and You (2020). Not a reproduction of their robust forecasts or water-saving number. The eco_opt trajectories are not validated against that paper’s figures.