Question

Rework Problem 6.62, except assume the $1000MW$value also includes losses, and the penalty factor for the first unit is 1.0 and for the second unit 0.95 .

Short answer

The optimal dispatch value for generator $P_1$and $P_2$are 423.36MW and 576.64MW. The incremental operating cost for generator 1 and 2 are $51.87\frac{\$}{MWh}$and $54.6\frac{\$}{MWh}$ respectively. The total operating cost is $37598.03\frac{\$}{hr}$.

Step-by-step solution

Write the given data from the question.

Write the fuel cost curve for two generators.

$$\begin{gathered} C_1\left(P_1\right)=600+18.P_1-0.04P_1^2 \\ {C}_2\left(P_2\right)=700+20.P_2-0.03P_2^2 \end{gathered}$$

Write the total load for $P_1$and $P_2$,

$P_1+P_2=P_T$

$P_T=1000MW$

The losses in the system $P_{loss}=1000MW$.

The penalty factor for unit 1 $L_1=1$.

The penalty factor for unit 2 $L_2=0.95$.

Determine the formulas to calculate the optimal dispatch values, incremental cost and operating cost.

The equation to calculate the incremental operating cost is given as follows.

${\left(\frac{dC}{dP}\right)}_p$ …... (1)

The equation to calculate the marginal cost is given as follows.

$\lambda =\frac{dC}{dP}\left(L\right)$ …... (2)

The total operation cost is given as follows.

$$\begin{gathered} C_T=C_1\left(P_1\right)+C_2\left(P_2\right) \\ {C}_T=600+18.P_1-0.04P_1^2+700+20.P_2-0.03P_2^2 \end{gathered}$$ …... (3)

Calculate the optimal dispatch values, incremental cost and operating cost.

The incremental operating cost can be calculated by differentiate the fuel cost curve of the generator.

Calculate the incremental operating cost for generator 1,

$$\begin{gathered} \frac{d{C}_1}{d{P}_1}=\frac{d}{d{P}_1}\left(600+18.P_1+0.04P_1^2\right) \\ \frac{d{C}_1}{d{P}_1}=18+0.08P_1\frac{\$}{MWh} \end{gathered}$$ ……. (4)

Calculate the incremental operating cost for generator 2,

$$\begin{gathered} \frac{d{C}_2}{d{P}_2}\frac{d}{d{P}_2}\left(700+20.P_1+0.03P_2^2\right) \\ \frac{d{C}_2}{d{P}_2}=20+0.06P_2\frac{\$}{MWh} \end{gathered}$$ ……. (5)

Calculate the marginal cost for the unit 1,

Substitute $18+0.08P_1$for $\frac{d{C}_1}{d{P}_1}$and 1 for $L_1$into equation (2).

$$\begin{gathered} {\lambda }_1=\left(18+0.08P_1\right)\left(1\right) \\ {\lambda }_1=18+0.08P_1 \end{gathered}$$

Calculate the marginal cost for the unit 2,

Substitute $20+0.06P_2$for $\frac{d{C}_2}{d{P}_2}$and 0.95 for $L_2$into equation (2).

$$\begin{gathered} {\lambda }_2=\left(20+0.06P_2\right)\left(0.95\right) \\ {\lambda }_2=19+0.057P_2 \end{gathered}$$

The marginal cost of both the unit should be equal for the economic dispatch.

${\lambda }_1={\lambda }_2$

Substitute $18+0.08P_1$for ${\lambda }_1$and $19+0.058P_2$for ${\lambda }_2$into above equation.

$$\begin{gathered} 18+0.08P_1=19+0.057P_2 \\ 0.08P_1-0.057P_2=19-1 \\ 0.08P_1-0.057P_2=1 \end{gathered}$$ ……. (6)

The total for $P_1$and $P_2$is the 1000MW.

$$\begin{gathered} P_1+P_2=1000MW \\ {P}_1=1000-P_2 \end{gathered}$$ …… (7)

Substitute the value of $P_1$from the equation (7) into equation (6).

$$\begin{gathered} 0.08\left(1000-P_2\right)-0.057P_2=1 \\ 80-0.08P_2-0.057P_2=1 \\ 0.137P_2=79 \\ {P}_2=576.64MW \end{gathered}$$

Substitute value of $P_2$into equation (7).

$$\begin{gathered} P_1=1000-576.64 \\ {P}_1=423.36MW \end{gathered}$$

Calculate the incremental cost for generator 1.

Substitute $423.36MW$for $P_1$into equation (4).

$$\begin{gathered} \frac{d{C}_1}{d{P}_1}=18+0.08\times 423.36 \\ \frac{d{C}_1}{d{P}_1}=51.87\frac{\$}{MWh} \end{gathered}$$

Calculate the incremental cost for generator 2.

Substitute $576.64MW$for $P_2$into equation (5).

$$\begin{gathered} \frac{d{C}_2}{d{P}_2}=20+0.06\times 576.64 \\ \frac{d{C}_2}{d{P}_2}=54.6\frac{\$}{MWh} \end{gathered}$$

Calculate the total operating cost.

Substitute 423.36MW for $P_1$and 576.64MW for $P_2$into equation (2).

$C_T=600+18\times 423.36+0.04\times 423.{36}^2+700+20\times 576.64+0.03\times 576.{64}^2$

$$\begin{gathered} C_T=15389.83+22208.21 \\ {C}_T=37598.03\frac{\$}{hr} \end{gathered}$$

Hence, the optimal dispatch value for generator $P_1$and $P_2$are 423.36MW and 576.64MW. The incremental operating cost for generator 1 and 2 are $51.87\frac{\$}{MWh}$and $54.6\frac{\$}{MWh}$respectively. The total operating cost is $37598.03\frac{\$}{hr}$.