Question

Given two generating units with their respective variable operating costs as

$$\begin{gathered} {\mathit{C}}_{\mathbf{1}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{01}{\mathbf{p}}_{\mathbf{G}\mathbf{1}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathbf{P}}_{\mathbf{G}\mathbf{1}}\mathbf{+}\mathbf{100}\mathbf{\backslash (}\mathbf{/}\mathbf{hr}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{for}\mathbf{}\mathbf{25}\mathbf{\le }{\mathbf{P}}_{\mathbf{g}\mathbf{1}}\mathbf{\le }\mathbf{150}\mathbf{MW} \\ {\mathbf{C}}_{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{004}{\mathbf{p}}_{\mathbf{G}\mathbf{2}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{6}{\mathbf{P}}_{\mathbf{G}\mathbf{2}}\mathbf{+}\mathbf{80}\mathbf{\backslash )}\mathbf{/}\mathbf{hr}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{for}\mathbf{}\mathbf{30}\mathbf{\le }{\mathbf{p}}_{\mathbf{2}}\mathbf{\le }\mathbf{200}\mathbf{}\mathbf{MW} \end{gathered}$$

Determine the economically optimum division of generation for $\mathbf{55}\mathbf{\le }{\mathbf{P}}_{\mathbf{L}}\mathbf{\le }\mathbf{350}\mathbf{}\mathbf{MW}$. In particular, for ${\mathbf{P}}_{\mathbf{L}}\mathbf{=}\mathbf{282}\mathbf{}\mathbf{MW}\mathbf{,}\mathbf{compute}\mathbf{}{\mathbf{P}}_{\mathbf{G}\mathbf{1}}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{P}}_{\mathbf{G}\mathbf{2}}$. Neglect transmission losses.

Short answer

For ${\mathrm{P}}_{\mathrm{L}}=55\mathrm{MW}$the optimum division value of generator 1 and generator 2 is$25\mathrm{MW}\mathrm{and}30\mathrm{MW}$.

For ${\mathrm{P}}_{\mathrm{L}}=100\mathrm{MW}$the optimum division value of generator 1 and generator 2 is$50\mathrm{MW}\mathrm{and}50\mathrm{MW}$.

For${\mathrm{P}}_{\mathrm{L}}=150\mathrm{MW}$the optimum division value of generator 1 and generator 2 is$64.29\mathrm{MW}\mathrm{and}85.71\mathrm{MW}$.

For${\mathrm{P}}_{\mathrm{L}}=200\mathrm{MW}$the optimum division value of generator 1 and generator 2 is$78.57\mathrm{MW}\mathrm{and}121.43\mathrm{MW}.$

For ${\mathrm{P}}_{\mathrm{L}}=250\mathrm{MW}$the optimum division value of generator 1 and generator 2 is $92.86\mathrm{MW}\mathrm{and}157.14\mathrm{MW}$.

For${\mathrm{P}}_{\mathrm{L}}=300\mathrm{MW}$the optimum division value of generator 1 and generator 2 is $107.14\mathrm{MW}\mathrm{and}192.86\mathrm{MW}.$

For${\mathrm{P}}_{\mathrm{L}}=350\mathrm{MW}$the optimum division value of generator 1 and generator 2 is $150\mathrm{MW}\mathrm{and}200\mathrm{MW}.$

For ${\mathrm{P}}_{\mathrm{L}}=282\mathrm{MW}$the optimum division value of generator 1 and generator 2 is$102\mathrm{MW}\mathrm{and}180\mathrm{MW}.$

Step-by-step solution

Write the given data.

The variable operating costs for two generators are,

$$\begin{gathered} C_1=0.01{\left(P_{G1}\right)}^2+2P_{G1}+100\$/\mathrm{hr} \\ {C}_1=0.04{\left(P_{G2}\right)}^2+2.6P_{G1}+80\$/\mathrm{hr} \end{gathered}$$

Consider the range for power.

role="math" localid="1655290278704" $$\begin{gathered} \mathrm{for}25\le P_{G\mathit{1}}\le 150\mathrm{MW} \\ \mathrm{for}30\le {\mathrm{P}}_{\mathit{1}}\mathit{\le }200\mathrm{MW} \\ \mathrm{Total}\mathrm{load}\mathrm{limits}\mathrm{are}55\le P_L\le 350\mathrm{MW}. \end{gathered}$$

Determine the required formula.

Total load of Generator

${\mathbf{P}}_{\mathbf{L}}\mathbf{=}{\mathbf{P}}_{\mathbf{G}\mathbf{1}}\mathbf{+}{\mathbf{P}}_{\mathbf{G}\mathbf{2}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

Here,${\mathit{P}}_{\mathbf{G}\mathbf{1}}$ is generator one${\mathbf{P}}_{\mathbf{G}\mathbf{2}}$ is generator two.

Step 3: Determine the variable operating costs for two generators PG1 and PG2.

Determine the incremental operating costs of the units are,

$\frac{d{C}_1}{d{P}_{g1}}=0.02P_{g1}+2\$/\mathrm{MWh}$ …… (2)

Here, is $P_{G1}$generator one.

$\frac{d{C}_2}{d{P}_{G2}}=0.008P_{g2}+2.6\$/\mathrm{MWh}$ …… (3)

The minimum operating cost occurs when:

$\frac{d{C}_1}{d{P}_1}=\frac{d{C}_2}{d{P}_2}=0$ …… (4)

Substitute the equations (2) and (3) into equation (4).

$$\begin{gathered} 0.02P_{\mathit{G}\mathit{1}}\mathit{+}2=0.008{\mathrm{P}}_{\mathrm{G}2}+2.6 \\ \mathit{20}{P}_{\mathit{G}\mathit{1}}+2000=8P_{\mathit{G}\mathit{1}}\mathit{+}2600 \\ \mathit{20}{P}_{\mathit{G}\mathit{1}}\mathit{-}\mathit{8}{P}_{\mathit{G}\mathit{2}}\mathit{=}600 \end{gathered}$$ …… (5)

Here, ${\mathrm{P}}_{\mathrm{G}1}$and $P_{G2}$is generator one and generator two.

Determine the total load of generator.

As further solve of equation (1).

$P_{\mathit{G}\mathit{2}}\mathit{=}{P}_{\mathit{L}}\mathit{-}{P}_{\mathit{G}\mathit{1}}$ …… (6)

Here,$P_{\mathit{G}\mathit{1}}$is generator one,localid="1655291987505" $P_{\mathit{G}\mathit{2}}$is generator two and localid="1655292013153" $P_{\mathit{L}}$is total load of generator.

Substitute equation (6) into equation (5).

$$\begin{gathered} 20P_{G1}-8\left(P_L-P_{G1}\right)=600 \\ 20P_{G1}-8P_L-8P_{G1}=600 \\ 28P_{G1}-8P_L=600 \end{gathered}$$ …… (7)

For various values of the load, calculate the optimum division of generation.

Substitute$P_L=55\mathrm{MW}$into equation (7).

$$\begin{gathered} 28P_{G1}-8\left(55\right)=600 \\ {P}_{G1}=25\mathrm{MW} \end{gathered}$$

Here, $P_{G1}$is the optimal dispatch value of generator 1.

Substitute localid="1655292905416" $P_{G1}=25\mathrm{MW}$into equation (6).

$$\begin{gathered} P_{G2}=55-25 \\ =30\mathrm{MW} \end{gathered}$$

Here,${\mathrm{P}}_{\mathrm{G}2}$is the optimal dispatch value of generator 2.

Substitute$P_L=100\mathrm{MW}$into equation (7).

localid="1655293210945" $$\begin{gathered} 28P_{G1}-8\left(100\right)=600 \\ {P}_{G1}=50\mathrm{MW} \end{gathered}$$

Here,$P_{G1}$is the optimal dispatch value of generator 1.

$$\begin{gathered} \mathrm{Substitute}{P}_{G1}=100\mathrm{MW}\mathrm{into}\mathrm{equation}\left(6\right). \\ {P}_{G2}=100-50 \\ =50\mathrm{MW} \end{gathered}$$

Here, $P_{G2}$is the optimal dispatch value of generator 2.

role="math" localid="1655294576252" $$\begin{gathered} \mathrm{Substitute}{P}_L=150\mathrm{MW}\mathrm{into}\mathrm{equation}\left(7\right). \\ 28P_{\mathrm{G}2}\mathit{-}\mathit{8}\left(150\right)\mathit{=}\mathit{600} \\ {P}_{G\mathit{1}}\mathit{=}64.29\mathrm{MW} \end{gathered}$$

Here,$P_{G1}$is the optimal dispatch value of generator 1.

$$\begin{gathered} Substitute64.29\mathrm{MW}for{P}_{g1}\mathrm{into}\mathrm{equation}\left(6\right). \\ {P}_{\mathit{G}\mathit{2}}\mathit{=}150-64.29 \\ =85.71\mathrm{MW} \end{gathered}$$

Here,$P_{G2}$is the optimal dispatch value of generator 2.

Substitute$200\mathrm{MW}$for $P_L$into equation (7).

$$\begin{gathered} 28P_{G1}-8\left(200\right)=600 \\ {P}_{G1}=78.57\mathrm{MW} \end{gathered}$$

Here,$P_{G1}$is the optimal dispatch value of generator 1.

Substitute $78.57\mathrm{MW}$for$P_{G1}$into equation (6).

$$\begin{gathered} P_{G2}=200-78.57 \\ =121.43\mathrm{MW} \end{gathered}$$

Here,$P_{G2}$is the optimal dispatch value of generator 2.

Substitute $250\mathrm{MW}$for $P_L$into equation (7).

$$\begin{gathered} 28P_{G1}-8\left(250\right)=600 \\ {P}_{G1}=92.86\mathrm{MW} \end{gathered}$$

Here,$P_{G1}$is the optimal dispatch value of generator 1.

$$\begin{gathered} \mathrm{Substitute}92.86\mathrm{MW}\mathrm{for}{P}_{G1}\mathrm{into}\mathrm{equation}\left(6\right). \\ {P}_{G2}=250-92.86 \\ =157.14\mathrm{MW} \end{gathered}$$

Here,$P_{G2}$is the optimal dispatch value of generator 2.

$$\begin{gathered} \mathrm{Substitute}300\mathrm{MW}for{P}_L\mathrm{into}\mathrm{equation}\left(7\right). \\ 28{\mathrm{P}}_{\mathrm{G}1}-8\left(300\right)=600 \\ {\mathrm{P}}_{\mathrm{G}1}=107.14\mathrm{MW} \\ \mathrm{Here},P_{\mathit{G}\mathit{1}}\mathrm{is}\mathrm{the}\mathrm{optimal}\mathrm{dispatch}\mathrm{value}\mathrm{of}\mathrm{generator}1. \\ \mathrm{Substitute}107.14\mathrm{MW}\mathrm{for}{P}_{\mathit{G}\mathit{1}}\mathrm{into}\mathrm{equation}\left(6\right). \\ {P}_{\mathrm{G}2}=300-107.14 \\ =192.86\mathrm{MW} \\ \mathrm{Here},P_{\mathit{G}\mathit{2}}\mathrm{is}\mathrm{the}\mathrm{optimal}\mathrm{dispatch}\mathrm{value}\mathrm{of}\mathrm{generator}2. \\ \mathrm{Substitute}350\mathrm{MW}\mathrm{for}{\mathrm{P}}_{\mathrm{L}}\mathrm{into}\mathrm{equation}\left(7\right). \\ 28P_{\mathit{G}\mathit{1}}\mathit{-}8\left(350\right)=600 \\ {\mathrm{P}}_{\mathrm{G}\mathit{1}}=150\mathrm{MW} \\ \mathrm{Here},{\mathrm{P}}_{\mathrm{G}1}\mathrm{is}\mathrm{the}\mathrm{optimal}\mathrm{dispatch}\mathrm{value}\mathrm{of}\mathrm{generator}1. \\ \mathrm{Substitute}150\mathrm{MW}\mathrm{for}{P}_{\mathit{G}\mathit{1}}\mathrm{into}\mathrm{equation}\left(6\right). \\ \mathit{}{P}_{\mathit{G}\mathit{2}}\mathit{=}\mathit{350}\mathit{-}\mathit{150} \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}=150\mathrm{MW} \\ \mathrm{Here},\mathit{}{P}_{\mathit{G}\mathit{2}}\mathit{}\mathrm{is}\mathrm{the}\mathrm{optimal}\mathrm{dispatch}\mathrm{value}\mathrm{of}\mathrm{generator}2. \\ \mathrm{Substitute}282\mathrm{MW}\mathrm{for}{P}_{\mathit{L}}\mathit{}\mathrm{into}\mathrm{equation}\left(7\right). \\ 28{\mathrm{P}}_{\mathrm{G}\mathit{1}}\mathit{-}8\left(282\right)=600 \\ {\mathrm{P}}_{\mathrm{G}\mathit{1}}=102\mathrm{MW} \\ \mathrm{Here},P_{G1}\mathrm{is}\mathrm{the}\mathrm{optimal}\mathrm{dispatch}\mathrm{value}\mathrm{of}\mathrm{generator}1. \\ \mathrm{Substitute}102\mathrm{MW}\mathrm{for}{\mathrm{P}}_{\mathrm{G}1}\mathrm{into}\mathrm{equation}\left(6\right). \\ {\mathrm{P}}_{\mathrm{G}1}=282-102 \\ =180\mathrm{MW} \\ \mathrm{Here},P_{\mathit{G}\mathit{1}}\mathit{}\mathrm{is}\mathrm{the}\mathrm{optimal}\mathrm{dispatch}\mathrm{value}\mathrm{of}\mathrm{generator}2. \end{gathered}$$