Minimum up time MINLP

I have 2 generator and 5 time periods. I wish to inclue a minimum uptime for my generators as 2 time periods. How do I go about doing this? Im not able to figure out how to represent this interms of the inequality matrix A.

Answer 1

[Shubham]

Hi Sankarshan,

To represent the minimum uptime constraint for the generators in terms of the inequality matrix A, you can follow the steps below:

1. Define your decision variables: Let x1 and x2 be the binary decision variables representing the on/off status of generator 1 and generator 2, respectively. Let y1, y2, y3, y4, and y5 be the binary decision variables representing the time periods.
2. Define the objective function: Since there is no objective function given in the problem statement, assume that the objective is to maximize the total uptime of the generators. Maximize: x1*y1 + x1*y2 + x1*y3 + x1*y4 + x1*y5 + x2*y1 + x2*y2 + x2*y3 + x2*y4 + x2*y5
3. Define the constraints:

• Generator uptime constraint: The total uptime of each generator should be at least 2 time periods.x1*y1 + x1*y2 + x1*y3 + x1*y4 + x1*y5 >= 2x2*y1 + x2*y2 + x2*y3 + x2*y4 + x2*y5 >= 2
• Time period constraint: Each time period should have only one state (either generator 1 is on or generator 2 is on).x1*y1 + x2*y1 <= 1x1*y2 + x2*y2 <= 1x1*y3 + x2*y3 <= 1x1*y4 + x2*y4 <= 1x1*y5 + x2*y5 <= 1
• Binary constraints: The decision variables should be binary.x1, x2, y1, y2, y3, y4, y5 ∈ {0, 1}.

4. Write the constraints in matrix form:

[1 1 1 1 1 0 0 0 0 0]   [x1]   [2]
[0 0 0 0 0 1 1 1 1 1] * [x2] >= [2]
[1 0 0 0 0 1 0 0 0 0]   [y1]
[0 1 0 0 0 0 1 0 0 0]   [y2]
[0 0 1 0 0 0 0 1 0 0]   [y3]
[0 0 0 1 0 0 0 0 1 0]   [y4]
[0 0 0 0 1 0 0 0 0 1]   [y5]

The first row of the matrix represents the generator 1 uptime constraint, the second row represents the generator 2 uptime constraint, and the remaining rows represent the time period constraints.

The inequality matrix A is the left-hand side of the matrix equation, and the decision variables are the column vector on the right-hand side. The right-hand side of the matrix equation is the vector of constants representing the minimum uptime constraint for the generators.