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

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.

enter image description here

I modify the answer on this page: enter link description here

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): enter image description here (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

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.

Answer 2

With an unlimited edition of CPLEX, your model solves (though very slowly). Here are two ideas to better control what happens in solve()

1. use solve(log_output=True) to print the log: you'll see the gap going down

2. 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)