Solving economic dispatch problem using global optimisation methods

Hello. What are the most adapted methods/optimisation algorithms (linprog, fmincon, ...) to solve the following power flow dispatch problem (between two generating units)? Below the formulation of my problem:

- Decision variables are the power of a thermal generator (gas turbine) (P_GT), and the power of the electrical network (P_Net). These variables are regrouped in Ppl vector as follows:

-Obj function is a cost function given by:

C_Net is the purchase cost of electrical network power. This cost is linear (=0.1524 Euro/Kwh) (see Fig1). C_GT is the thermal generator power production cost. This cost function has a cubic form (see Fig2). The objectives of this optimal power flow dispatching is supplying the load (P_Load), and minimizing the total operating cost under a set of constraints.

-Equality constraint is the load balance equation:

-Inequality constraints: In this problem, I prefer using lower bound and upper bound of decision variables instead of the inequality constraints.

are respectively the nominal power of thermal generator and electrical network.

Cost functions:

Answer 1

[John D'Errico]

Your cost function has a nonlinear part. Why would you consider using linprog?

Just use fmincon.

[ET-TAOUSSI mehdi]

Dear John D'Errico,Thank you for your reply. I have tested fmincon with these different algorithms, but I m not sure that this command converges to the global optimum. During the test, I notice that the program respects the constraints, but it does not return optimal cost. For exemple, for a load demand P_Load = 20 Kw, the program returns the following results (see attached prog):

>> */Optimal power of gas turbine: 20 Kw

>>*/Optimal power of electrical network: 1.7324e-007 Kw

>>*/Optimal total cost: 3.5129 Euro

Contrariwise, if one gets P_Load just from the electrical network, one will have a lower cost (C_Net*20 = 2 Euro).

I think the prob is due to the form of the objective function. Indeed, this function is composed of the gas turbine cost (cubic form) and the linear cost of the electrical network. In order that fmincon works well, the obj function must have several directions (hollows and bumps).This is not the case in this economic dispatching problem.

Please let me know if there is an error in prog? Is there the cost functions of the two sources are unadapted? And if possible can I solve this economic dispatching problem by other methods of the global optimisation (methods more adapted to my global cost function)?