Question

The fuel-cost curves for two generators are given as follows:

$\begin{array}{l}{\mathbf{C}}_{\mathbf{1}}\mathbf{\left(}{\mathbf{P}}_{\mathbf{1}}\mathbf{\right)}\mathbf{=}\mathbf{600}\mathbf{+}\mathbf{18}\mathbf{.}{\mathbf{P}}_{\mathbf{1}}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{04}{\mathbf{P}}_{\mathbf{1}}^{\mathbf{2}}\\ {\mathbf{C}}_{\mathbf{2}}\mathbf{\left(}{\mathbf{P}}_{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{700}\mathbf{+}\mathbf{20}\mathbf{.}{\mathbf{P}}_{\mathbf{2}}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{03}{\mathbf{P}}_{\mathbf{2}}^{\mathbf{2}}\end{array}$

Assuming the system is lossless, calculate the optimal dispatch values of${\mathbf{P}}_{\mathbf{1}}$ and${\mathbf{P}}_{\mathbf{2}}$ for a total load of $\mathbf{1000}\mathbf{MW}$, the incremental operating cost, and the total operating cost.

Short answer

The optimal dispatch value for generator$P_1$ and$P_2$ are$442.86\text{\hspace{0.17em}MW}$ and$553.14\text{\hspace{0.17em}MW}$ . The incremental operating cost is$53.43\frac{\text{\$}}{\text{MWh}}$ and the total operating cost is $37571.43\frac{\text{\$}}{\text{hr}}$.

Step-by-step solution

Write the given data from the question.

Write the fuel cost curve for two generators.

$\begin{array}{l}{\mathrm{C}}_1\left({\mathrm{P}}_1\right)=600+18.{\mathrm{P}}_1-0.04{\mathrm{P}}_1^2\\ {\mathrm{C}}_2\left({\mathrm{P}}_2\right)=700+20.{\mathrm{P}}_2-0.03{\mathrm{P}}_2^2\end{array}$

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

$\begin{array}{c}{P}_1+P_2=P_T\\ P_T=1000\text{\hspace{0.17em}MW}\end{array}$

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.

${\left(\frac{\mathrm{dC}}{\mathrm{dP}}\right)}_{\mathbf{p}}$ …… (1)

The total operation cost is given as follows.

$\begin{array}{l}{\mathbf{C}}_{\mathbf{T}}\mathbf{=}{\mathbf{C}}_{\mathbf{1}}\mathbf{\left(}{\mathbf{P}}_{\mathbf{1}}\mathbf{\right)}\mathbf{+}{\mathbf{C}}_{\mathbf{2}}\mathbf{\left(}{\mathbf{P}}_{\mathbf{2}}\mathbf{\right)}\\ {\mathbf{C}}_{\mathbf{T}}\mathbf{=}\mathbf{600}\mathbf{+}\mathbf{18}\mathbf{.}{\mathbf{P}}_{\mathbf{1}}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{04}{\mathbf{P}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}\mathbf{700}\mathbf{+}\mathbf{20}\mathbf{.}{\mathbf{P}}_{\mathbf{2}}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{03}{\mathbf{P}}_{\mathbf{2}}^{\mathbf{2}}\end{array}$…… (2)

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

The incremental operating cost is calculated by differentiating the fuel cost curve of the generator.

Calculate the incremental operating cost for generator 1.

$\begin{array}{c}\frac{d{C}_1}{d{P}_1}=\frac{d}{d{P}_1}(600+18.P_1+0.04P_1^2)\\ \frac{d{C}_1}{d{P}_1}=18+0.08P_1\frac{\text{\$}}{\text{MWh}}\end{array}$ ……. (3)

Calculate the incremental operating cost for generator 2,

$\begin{array}{l}\frac{d{C}_2}{d{P}_2}=\frac{d}{d{P}_2}(700+20.P_1+0.03P_2^2)\\ \frac{d{C}_2}{d{P}_2}=20+0.06P_2\frac{\text{\$}}{\text{MWh}}\end{array}$ ……. (4)

The minimum cost occurs when the incremental cost of both generators becomes equal to each other.

$\frac{d{C}_1}{d{P}_1}=\frac{d{C}_2}{d{P}_2}$

Substitute $18+0.08P_1\frac{\text{\$}}{\text{MWh}}$for $\frac{d{C}_1}{d{P}_1}$and $20+0.06P_2\frac{\text{\$}}{\text{MWh}}$for $\frac{d{C}_2}{d{P}_2}$into above equation.

$\begin{array}{c}18+0.08P_1=20+0.06P_2\\ 0.08P_1-0.06P_2=20-18\\ 0.08P_1-0.06P_2=2\end{array}$ …… (5)

The total for$P_1$and$P_2$isrole="math" localid="1656144282228" $1000MV$.

$\begin{array}{c}{P}_1+P_2=1000\text{\hspace{0.17em}MW}\\ P_1=1000-P_2\end{array}$ …… (6)

Substitute the value of$P_1$from the equation (6) into equation (5).

$\begin{array}{l}0.08(1000-P_2)-0.06P_2=2\\ 80-0.08P_2-0.06P_2=2\\ 0.14P_2=78\\ P_2=557.14\text{\hspace{0.17em}MW}\end{array}$

Substitute value of $P_2$into equation (6).

$\begin{array}{l}{P}_1=1000-557.14\\ P_1=442.86\text{\hspace{0.17em}MW}\end{array}$

Calculate the incremental cost.

Substitute$442.86\text{\hspace{0.17em}MW}$for$P_1$into equation (3).

$\begin{array}{l}\frac{d{C}_1}{d{P}_1}=\frac{d{C}_2}{d{P}_2}\\ \frac{d{C}_1}{d{P}_1}=18+0.08\times 442.86\\ \frac{d{C}_1}{d{P}_1}=18+35.4288\\ \frac{d{C}_1}{d{P}_1}=53.43\frac{\text{\$}}{\text{MWh}}\end{array}$

Calculate the total operating cost.

Substitute $442.86\text{\hspace{0.17em}MW}$for $P_1$and $557.14\text{\hspace{0.17em}MW}$for $P_2$into equation (2).

$\begin{array}{l}{C}_T=600+18\times 442.86+0.04\times {442.86}^2+700+20\times 557.14+0.03\times {557.14}^2\\ C_T=600+7971.48+7845+700+11142.8+9312.14\\ C_T=37571.43\frac{\text{\$}}{\text{hr}}\end{array}$

Hence, the optimal dispatch value for generator$P_1$ and$P_2$ are$442.86\text{\hspace{0.17em}MW}$ and $553.14\text{\hspace{0.17em}MW}$. The incremental operating cost is$53.43\frac{\text{\$}}{\text{MWh}}$ and the total operating cost is $37571.43\frac{\text{\$}}{\text{hr}}$.