Question

Rework Problem 6.62, except assume that the limit outputs are subject to the following inequality constraints:

$\begin{array}{l}\mathbf{200}\mathbf{\le }{\mathbf{P}}_{\mathbf{1}}\mathbf{\le }\mathbf{800}\mathbf{MW}\\ \mathbf{100}\mathbf{\le }{\mathbf{P}}_{\mathbf{2}}\mathbf{\le }\mathbf{400}\mathbf{MW}\end{array}$

Short answer

The optimal dispatch value for generator$P_1$ and $P_2$are$600\text{\hspace{0.17em}MW}$ and$400\text{\hspace{0.17em}MW}$ . The incremental operating cost for generator 1 and 2 are$66\frac{\text{\$}}{\text{MWh}}$ and $66\frac{\text{\$}}{\text{MWh}}$respectively. The total operating cost is $39300\frac{\text{\$}}{\text{hr}}$.

Step-by-step solution

Write the given data from the question:

Write the fuel cost curve for two generators.

$\begin{array}{l}{C}_1\left(P_1\right)=600+18.P_1-0.04P_1^2\\ C_2\left(P_2\right)=700+20.P_2-0.03P_2^2\end{array}$

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.

${\left(\frac{dC}{dP}\right)}_{\mathbf{p}}$ …... (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 can be calculated by differentiate the fuel cost curve of the generator.

Calculate the incremental operating cost for generator 1,

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

role="math" localid="1656146842235" $\begin{array}{c}18+0.08P_1=20+0.06P_2\\ 0.08P_1-0.06P_2=20-18\\ 0.08P_1-0.06P_2=2\end{array}$ …… (5)

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

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

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

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

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

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

From the above discussion, it is clear that generator 2 is operating higher than its maximum limits that is$400MW$.To obtain the optimum operation the generator 2 will operate at its maximum limit.

Therefore, optimal dispatch value of generator 1 and 2 are $600MW$and $400MW$respectively.

Calculate the incremental cost for generator 1.

Substitute$600\text{\hspace{0.17em}MW}$for$P_1$into equation (3).

$\begin{array}{l}\frac{d{C}_1}{d{P}_1}=18+0.08\times 600\\ \frac{d{C}_1}{d{P}_1}=66\frac{\text{\$}}{\text{MWh}}\end{array}$

Calculate the incremental cost for generator 2.

Substitute$400\text{\hspace{0.17em}MW}$ for$P_2$into equation (4).

$\begin{array}{l}\frac{d{C}_2}{d{P}_2}=20+0.06\times 400\\ \frac{d{C}_2}{d{P}_2}=44\frac{\text{\$}}{\text{MWh}}\end{array}$

Calculate the total operating cost.

Substitute $600MW$for $P_1$ and $400MW$for $P_2$into equation (2).

$\begin{array}{l}{C}_T=600+18\times 600+0.04\times {600}^2+700+20\times 400+0.03\times {4000}^2\\ C_T=24800+13500\\ C_T=39300\frac{\$}{\text{hr}}\end{array}$

Hence the optimal dispatch value for generator$P_1$ and$P_2$ are$600MW$ and$400MW$ . The incremental operating cost for generator 1 and 2 are$66\frac{\text{\$}}{\text{MWh}}$ and$44\frac{\text{\$}}{\text{MWh}}$ respectively. The total operating cost is$39300\frac{\text{\$}}{\text{hr}}$ .