MINLP optimization using GA reaching different solutions every run

I have written a program for optimizing a set of generators. I have hourly price and cost data and need to figure out when a generator should run or just stay off. I describe the problem in more detail below. I have programed this using matlabs global optimization toolbox and the ga solver. I run the solver and each time I run the solver I get a different solution. I am thinking this is because ga is no able to properly search the solution space for an optimal solution? Is it unable to cope with the use of x_3 to turn the generator off?

The solver stops with this status:

> Optimization terminated: average change in the penalty fitness value > less than options.TolFun and constraint violation is less than > options.TolCon.

*Problem Description*

x_1 is the generator output at any one point in time and is constrained to be between a min and max capacity

x_2 is an integer variable used simply to turn the generator off completely

x_3 lastly is another integer variable used to apply a startup cost whenever the generator is switched on.

x_1, x_2 and x_3 are vectors where the index of the vector (eg. x_1(1:5) is the data for the first 5 hours)

*Objective Function (totrevenue6):*

efficiency6 = 0.2621*x_1 - 0.1229*x_1^2 + 0.2543

income6 = {-x_1*(AC_1 + FC_1 - P_1 + FC_1)}/{efficiency6}

revenue6 = - SC_1*x_3 - x_2*(income6)

**totrevenue6** = -sum(revenue6)

*Constraints:*

 - min_capacity <= generator6 power (x_1) <= max_capacity
 - 0 <= generator6 toggle (x_2)<= 1
 - The startup constraint (x_3) is x_2-x_{2-1}<=x_3 and rearranging
   -x_{2-1}+x_2-x_3<=0
 - 0 <= (x_3)<= 1

So my question(s) why does ga solver reach a new solution every time I run it? How do I get the solver to solve this type of problem?

many thanks

Answer 1

[Matt J]

The results of ga can be random, because it uses random mutation of a population of test points as part of its search.

If I take your problem statement literally, it looks like it can be simplified to a common MILP and solved with intlinprog, which should make the results more reliable. In particular, totalrevenue6 looks like it is a monotonic function of income6 which is in turn monotonic as a function of x_1. So you can optimize with respect to x1 independently, reducing the problem to an objective and constraints that are linear in the binary variables x_2 and x_3.

However, your problem description looks like it has several holes:

First, it is unclear what totalrevenue is summing over. Are you summing over hours? Over generators? Both? If x1,x2,x3 have further hidden indices over generators/hours, you should unhide them for us.

Similarly, what is x_j in the "startup constraint"? Does j=1,2,3 or is j an index for something else that's been hideen? The constraint x_2-x_1-s_1<=0 looks very strange if x_1 still refers here to generator output and x_2 still refers to the binary on/off state of the generator. Since you mention (x3), I assume you really might mean x3_j where j s an index for .... something.

Finally, it looks like you are minimizing income6, which is also strange. If I expand your expression for totalrevenue6, rearranging the minus signs, I obtain

   totalrevenue6 = sum( SC_1*x_3 + x_2*(income6) )

In other words, your objective function improves if you decrease income6, which seems counter-intuitive.

[Jesse]

thank you for the reply. Indeed I can clarify a few things

for now I am only modeling one generator

x_1, x_2 and x_3 are vectors where the index of the vector (eg. x_1(1:5) is the generator power data for the first 5 hours)

x_j is x_2. Whenever a generator switches on this constraint forces x_3 high and adds to the startup cost of the generator.

x_1, x_2 and x_3 are vectors where the index of the vector (eg. x_1(1:5) is the data for the first 5 hours)

totalrevenue is fliped negative since ga minimizes the objective function.

Could you clarify a bit more what you mean about monotonic and how I might go about simplifying?

[Matt J]

totalrevenue is fliped negative since ga minimizes the objective function.

Yes, but "revenue" as you've shown it is also flipped negative. You've double-flipped!!

Could you clarify a bit more what you mean about monotonic and how I might go about simplifying?

Your expression for income is proportional to the function,

    f=@(x_1) x_1./(0.2621*x_1 - 0.1229*x_1.^2 + 0.2543)

The plot of this function below shows that it is monotonically increasing everywhere, with a singularity at x_1=2.8569. I assume that the range of [min_capacity, max_capacity] doesn't include this singularity. Otherwise, you could have infinite income. This means, though, that the income is maximized either at min_capacity or max_capacity, depending on the sign of (AC_1 + FC_1 - P_1 + FC_1). But this means you can solve for all of the x1's in advance, leaving only the x2's and x3's. Once you do, it is a problem that intlinprog can solve.

[Jesse]

I agree the negatives are not intuitive. These are being generated from the symbolic toolbox. Ive programed

• income6 = X_1.*(P_1-CC_1-AC_1-FC_1./efficiency6);
• revenue6 = X_2.*(income6) - SC_1.*X_3;
• totrevenue6 = -sum(revenue6);

the point about the efficiency being monotonic is a great help. I see your point now about solving for x_1 first, then solving for x_2 and x_3. In the future I must add additional constraints to the generators capacity, but I believe the same will still hold then.

many thanks, Jesse

[Jesse]

I have continued to develop the program to optimize the set of generators and I now need to make sure that a minimum demand is met. In order to do so I have added the profit from two additional blocks to the objective function.

income6 = {(x_16)*(AC_1 + FC_1 - P_1 + FC_1/{el_efficiency6})}
revenue6 = - SC_16*x_36 + x_26*(income6)

income7 = {(x_17)*(AC_1 + FC_1 - P_1 + FC_1/{el_efficiency7})}
revenue7 = - SC_17*x_37 + x_27*(income6)

income8 = {(x_18)*(AC_1 + FC_1 - P_1 + FC_1/{el_efficiency8})}
revenue8 = - SC_18*x_38 + x_28*(income6)   

*totrevenue6* = -sum(revenue6+revenue7+revenue8)

In order to solve this problem before I used your suggestion to take advantage of the monotonic relationship and split the problem into two parts. Optimization for the generator output first, then a second optimization to take care of the remaining constraints. As a result however, if I want to make sure a minimum demand is met, I have to add two separate constraints, the first optimization of x_1 and the second for the optimization of x_2 and x_3:

• x_16 + x_17 + x_18 >= (thermal_demand at time t)*thermal_efficiency
• x_26*x_16 + x_27*x_17 + x_28*x_18 >= (thermal_demand at time t)*thermal_efficiency

Essentially I need to make sure that both optimizations meet the demand for each time step.

I do not believe that this breaks the monotonic relationship. But I am worried that it wont lead to an optimal result. Does this formulation make sense and can it lead to an optimal result?

Many thanks, Jesse