Two generators have cost functions F₁ and F₂ . Their incremental-cost characteristics are

$\frac{d{F}_1}{d{P}_1}=40+0.2P_1$ and $\frac{d{F}_2}{d{P}_2}=32+0.4P_2$

They need to deliver a combined load of 260 MW. Ignoring the network losses, for economic operation, the generations P₁ and P₂ (in MW) are

The correct option is: P₁=160 ,P₂=100

To achieve the most economic operation of a power system consisting of multiple generating units, the incremental fuel costs of all operating generators must be equal, provided network losses are ignored and no generator limits are violated.
This is known as the equal incremental cost criterion:

$\frac{d{F}_1}{d{P}_1}=\frac{d{F}_2}{d{P}_2}=\lambda$

where λ is the incremental cost of the system.

We are given the incremental-cost characteristics of the two generators:
Incremental cost of Generator 1:

$\frac{d{F}_1}{d{P}_1}=40+0.2P_1$

Incremental cost of Generator 2:

$\frac{d{F}_2}{d{P}_2}=32+0.4P_2$

Applying the equal incremental cost criterion:

$40+0.2P_1=32+0.4P_2$

Rearranging the equation to relate P₁ and P₂:

$0.2P_1-0.4P_2=32-40$

$0.2P_1-0.4P_2=-8$

Multiplying the entire equation by 5 to simplify:

$P_1-2P_2=-40$ (Equation 1)

We also know that the combined load to be delivered is 260 MW:

$P_1+P_2=260$ (Equation 2)

Now, we can solve this system of linear equations. Subtracting Equation 1 from Equation 2:

$(P_1+P_2)-(P_1-2P_2)=260-(-40)$

$3P_2=300$

$P_2=100\text{MW}$

Substitute the value of P₂ back into Equation 2:

$P_1+100=260$

$P_1=160\text{MW}$

Thus, the generations for the most economic operation are P₁ = 160 MW and P₂ = 100 MW.