30 ECONOMIC DISPATCH OF THERMAL UNITS

F , -

FIG. 3.1 N thermal units committed to serve a load of P,oad.

This is a constrained optimization problem that may be attacked formally using advanced calculus methods that involve the Lagrange function.

In order to establish the necessary conditions for an extreme value of the objective function, add the constraint function to the objective function after the constraint function has been multiplied by an undetermined multiplier. This is known as the Lagrange function and is shown in Eq. 3.3.

2’ = FT + Lip (3.3)

The necessary conditions for an extreme value of the objective function result when we take the first derivative of the Lagrange function with respect to each of the independent variables and set the derivatives equal to zero. In this case, there are N + 1 variables, the N values of power output, pi, plus the undetermined Lagrange multiplier, 2. The derivative of the Lagrange function with respect to the undetermined multiplier merely gives back the constraint equation. On the other hand, the N equations that result when we take the partial derivative of the Lagrange function with respect to the power output values one at a time give the set of equations shown as Eq. 3.4.

or (3.4)

That is, the necessary condition for the existence of a minimum cost- operating condition for the thermal power system is that the incremental cost rates of all the units be equal to some undetermined value, L. Of course, to this

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