Greenhouse climate (greenhouse)

Question. How should heating be scheduled when light and outside temperature are known functions of time, not decisions?

Formulation

Symbol	Role	Meaning
\(x_1\)	state	crop dry weight
\(x_2\)	state	indoor temperature
\(u(t)\)	control	heating
\(I(t),\,T_{\mathrm{out}}(t)\)	forcing	light and outside temperature

Dynamics.

\[ \dot x_1 = p_1 I(t)\, x_2, \qquad \dot x_2 = p_2\bigl(T_{\mathrm{out}}(t)-x_2\bigr)+p_3 u. \]

\(I\) and \(T_{\mathrm{out}}\) are interpolants of a two-day table (clipped daylight sinusoid). They are not NLP variables.

Objective.

\[ \min_u\; -p_5 x_1(T)+\int_0^{T} p_4 u\,\mathrm{d}t, \qquad T=48\,\mathrm{h}. \]

Constraints. \(x_1(0)=0\), \(x_2(0)=10\), \(0\le u\le 10\). Horizon. Fixed.

What was done

Forcing enters \(f\) as CasADi interpolants of \(t\). Heating is the only control. Collocation on a 48 h horizon.

What we found

Heating tracks the light/temperature cycle: on when the crop can use heat, off at night when \(I=0\). Objective is regression-tracked (no closed form).

Interpretation

A prescribed environment is not a control. Putting weather in \(f\) keeps the decision variable equal to what the grower actually chooses (heating), and keeps the NLP dimension from growing with the weather table.

Public API

Weather is a forcing inside \(f\), not a control (examples/greenhouse/greenhouse.py):

import eco_opt
from eco_opt import Namespace


def greenhouse_model(t, x, u):
    crop, temp, heat = x[0], x[1], u[0]
    y = Namespace(crop=crop, temp=temp, heat=heat)
    return [P1 * I_fun(t) * temp, P2 * (T_fun(t) - temp) + P3 * heat], y


sys = eco_opt.System(states=2, controls=1, model=greenhouse_model)
ocp = eco_opt.Ocp()
ocp.min([eco_opt.t_f(-P5 * sys.y.crop), P4 * sys.y.heat])
ocp.st(system=sys, constraints=[...])

Run and tests

cd examples/greenhouse
python greenhouse.py
python greenhouse_from_file.py
python plot_greenhouse.py
python -m pytest ../../tests/test_greenhouse.py -q

References

van Straten, van Ooteghem, van Willigenburg and van Henten, Optimal Control of Greenhouse Cultivation. Problem also shipped with Yop: Greenhouse. The text and code here are ours.

Not validated against the book’s figures

The eco_opt trajectories are not checked against numerical results in van Straten et al. The example follows that formulation (tabulated weather as forcing).