Optimal control of a batch reactor.

batchreactor.py

"""
Optimal control of a batch reactor.
Find the optimal temperature profile which gives maximum intermediate product
concentration in a batch reactor with two consecutive reactions. The first
reaction is of second order and the second one is of first order with known
rate constants.

Renfro J.G., Morshedi, A.M., Osbjornsen, O.A., Simultaneous optimization and
solution of systems described by differential/algebraic equations. Computer
and Chemical Engineering, vol.11, 1987, pp.503-517.
"""
import gamspy.math as gams_math
from gamspy import Alias
from gamspy import Container
from gamspy import Equation
from gamspy import Model
from gamspy import Parameter
from gamspy import Problem
from gamspy import Sense
from gamspy import Set
from gamspy import Variable


def main():
    m = Container(delayed_execution=True)

    # Set
    nh = Set(m, name="nh", records=[str(idx) for idx in range(0, 101)])
    k = Alias(m, name="k", alias_with=nh)

    # Data
    ca_0 = Parameter(m, name="ca_0", records=1.0)
    cb_0 = Parameter(m, name="cb_0", records=0.0)
    h = Parameter(m, name="h", records=1)

    # Variable
    ca = Variable(m, name="ca", domain=[nh])
    cb = Variable(m, name="cb", domain=[nh])
    t = Variable(m, name="t", domain=[nh])
    k1 = Variable(m, name="k1", domain=[nh])
    k2 = Variable(m, name="k2", domain=[nh])
    obj = Variable(m, name="obj")

    # Equation
    eobj = Equation(m, name="eobj")
    state1 = Equation(m, domain=[nh], name="state1")
    state2 = Equation(m, domain=[nh], name="state2")
    ek1 = Equation(m, domain=[nh], name="ek1")
    ek2 = Equation(m, domain=[nh], name="ek2")

    eobj[...] = obj == cb["100"]

    ek1[nh[k]] = k1[k] == 4000 * gams_math.exp(-2500 / t[k])
    ek2[nh[k]] = k2[k] == 620000 * gams_math.exp(-5000 / t[k])

    state1[nh[k.lead(1, "linear")]] = ca[k.lead(1, "linear")] == ca[k] + (
        h / 2
    ) * (
        -k1[k] * ca[k] * ca[k]
        - k1[k.lead(1, "linear")]
        * ca[k.lead(1, "linear")]
        * ca[k.lead(1, "linear")]
    )

    state2[nh[k.lead(1, "linear")]] = cb[k.lead(1, "linear")] == cb[k] + (
        h / 2
    ) * (
        k1[k] * ca[k] * ca[k]
        - k2[k] * cb[k]
        + k1[k.lead(1, "linear")]
        * ca[k.lead(1, "linear")]
        * ca[k.lead(1, "linear")]
        - k2[k.lead(1, "linear")] * cb[k.lead(1, "linear")]
    )

    ca.l[nh] = 1.0
    cb.l[nh] = 0.0

    ca.fx["0"] = ca_0
    cb.fx["0"] = cb_0

    t.lo[nh] = 110
    t.up[nh] = 280

    batchReactor = Model(
        m,
        name="batchReactor",
        equations=[eobj, state1, state2, ek1, ek2],
        problem=Problem.NLP,
        sense=Sense.MAX,
        objective=obj,
    )
    batchReactor.solve()

    import math

    assert math.isclose(batchReactor.objective_value, 0.8558, rel_tol=0.001)


if __name__ == "__main__":
    main()