Using the piecewise function of the IBM CPLEX python API, but the problem cannot be solved - optimization

Question

I try to use MILP (Mixed Integer Linear Programming) to calculate the unit commitment problem. (unit commitment: An optimization problem trying to find the best scheduling of generator)
Because the relationship between generator power and cost is a quadratic function, so I use piecewise function to convert power to cost.
I modify the answer on this page:
unit commitment problem using piecewise-linear approximation become MIQP
The simple program structure is like this:
from docplex.mp.model import Model
mdl = Model(name='buses')
nbbus40 = mdl.integer_var(name='nbBus40')
nbbus30 = mdl.integer_var(name='nbBus30')
mdl.add_constraint(nbbus40*40 + nbbus30*30 >= 300, 'kids')
#after 4 buses, additional buses of a given size are cheaper
f1=mdl.piecewise(0, [(0,0),(4,2000),(10,4400)], 0.8)
f2=mdl.piecewise(0, [(0,0),(4,1600),(10,3520)], 0.8)
cost1= f1(nbbus40)
cost2 = f2(nbbus30)
mdl.minimize(cost1+ cost1)
mdl.solve()
mdl.report()
for v in mdl.iter_integer_vars():
print(v," = ",v.solution_value)
which gives
* model buses solved with
objective = 3520.000
nbBus40 = 0
nbBus30 = 10.0
The answer is perfect but there is no way to apply my example.
I used a piecewise function to formulate a piecewise linear relationship between power and cost, and got a new object (cost1), and then calculated the minimum value of this object.
The following is my actual code(simply):
(min1,miny1), (pw1_1,pw1_1y),(pw1_2,pw1_2y), (max1,maxy1) are the breakpoints on the power-cost curve.
pwl_func_1phase = ucpm.piecewise(
0,
[(0,0),(min1,miny1),
(pw1_1,pw1_1y),
(pw1_2,pw1_2y),
(max1,maxy1)
],
0
)
#df_decision_vars_spinning is a dataframe store Optimization variables
df_decision_vars_spinning.at[
(units,period),
'variable_cost'
] = pwl_func_1phase(
df_decision_vars_spinning.at[
(units,period),
'production'
]
)
total_variable_cost = ucpm.sum(
(df_decision_vars_spinning.variable_cost))
ucpm.minimize(total_variable_cost )
I don’t know what causes this optimization problem can't be solve. Here is my complete code :
https://colab.research.google.com/drive/1JSKfOf0Vzo3E3FywsxcDdOz4sAwCgOHd?usp=sharing

Answer 1

With an unlimited edition of CPLEX, your model solves (though very slowly). Here are two ideas to better control what happens in solve()
use solve(log_output=True) to print the log: you'll see the gap going down
set a mip gap: setting mip gap to 5% stops the solve at 36s
ucpm.parameters.mip.tolerances.mipgap = 0.05
ucpm.solve(log_output=True)

Answer 2

Not an answer, but to illustrate my comment.
Let's say we have as the cost curve
cost = α + β⋅power^2
Furthermore, we are minimizing cost.
We can approximate using a few linear curves. Here I have drawn a few:
Let's say each linear curve has the form
cost = a(i) + b(i)⋅power
for i=1,...,n (n=number of linear curves).
It is easy to see that is we write:
min cost
cost ≥ a(i) + b(i)⋅power ∀i
we have a good approximation for the quadratic cost curve. This is exactly as I said in the comment.
No binary variables were used here.