Double Tank (GEKKO)

This page contains a solution of the Double Tank in GEKKO Python format. The GEKKO package is available with pip install gekko. The Python code uses orthogonal collocation and a simultaneous optimization method. The integral is converted to a differential equation through differentiation and definition of a new variable x3.

import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO
 
m = GEKKO() # create GEKKO model
 
# Add 0.01 as first step
# 0,0.01,0.1,0.2,0.3,...9.9,10.0)
m.time = np.insert(np.linspace(0,10,201),1,0.01)
 
# change solver options
m.solver_options = ['minlp_gap_tol 0.001',\
                    'minlp_maximum_iterations 10000',\
                    'minlp_max_iter_with_int_sol 100',\
                    'minlp_branch_method 1',\
                    'minlp_integer_tol 0.001',\
                    'minlp_integer_leaves 0',\
                    'minlp_maximum_iterations 200']
 
k1 = 2
k2 = 3
 
last = m.Param(np.zeros(202))
last.value[-1] = 1
 
sigma=m.MV(value=1,lb=1,ub=2,integer=True)
x1 = m.Var(value=2)
x2 = m.Var(value=2)
x3 = m.Var(value=0)
sigma.STATUS = 1
 
m.Obj(last*x3)
 
m.Equations([x1.dt() == sigma - m.sqrt(x1),\
             x2.dt() == m.sqrt(x1) - m.sqrt(x2),\
             x3.dt() == k1*(x2-k2)**2])
 
m.options.IMODE = 6
m.options.NODES = 3
m.options.SOLVER = 1
m.options.MV_TYPE = 0
m.solve()
 
plt.figure(1)
plt.step(m.time,sigma.value,'r-',label=r'$\sigma$ (1/2)')
plt.plot(m.time,x1.value,'k-',label=r'$x_1$')
plt.plot(m.time,x2.value,'g-',label=r'$x_2$')
plt.xlabel('Time')
plt.ylabel('Variables')
plt.legend(loc='best')
plt.show()

Results with APOPT (MINLP)

An MINLP solution is calculated with APOPT with an objective function value of $4.767757$.