### correction tut5

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 ... ... @@ -133,7 +133,7 @@ Assume that the company is maximising its net surplus for a given electricity pr \end{enumerate} %=============== ====================================================== \paragraph{Problem VI.2 \normalsize (Economic dispatch in a single bidding zone).}~\\ \paragraph{Problem IV.2 \normalsize (Economic dispatch in a single bidding zone).}~\\ %===================================================================== Consider an electricity market with two generator types, one with the cost function $C_1(G_1)=c_1G_1$ with variable cost $c_1 = 20\emwh$, capacity $K_1 = 300\mw$ and a dispatch rate of $G_1$~[MW], and another with the cost function $C_2(G_2)=c_2G_2$ with variable cost $c_2=50\emwh$, capacity $K_2=400\mw$ and a dispatch rate of $G_2$~[MW]. The demand has utility function $U(D) = 8000D - 5D^2$~[\euro/h] for a consumption rate of $D$~[MW]. ... ... @@ -158,8 +158,8 @@ Consider an electricity market with two generator types, one with the cost funct Consider the two-bus power system shown in Figure \ref{test}, where the two nodes represent two markets, each with different total demand, and one generator at each node. At node A the demand is $D_A = 2000 \si{\mega\watt}$, whereas at node B the demand is $D_B = 1000 \si{\mega\watt}$. Furthermore, there is a transmission line with a capacity denoted by $F_{AB}$. The marginal cost of production of the generators connected to buses A and B are given respectively by the following expressions: \begin{align*} MC_A & = 20 + 0.03 G_A \hspace{1cm}\eur/\si{\mega\watt\hour} \\ MC_B & = 15 + 0.02 G_B \hspace{1cm} \eur/\si{\mega\watt\hour} MC_A & = 20 + 0.02 G_A \hspace{1cm}\eur/\si{\mega\watt\hour} \\ MC_B & = 15 + 0.03 G_B \hspace{1cm} \eur/\si{\mega\watt\hour} \end{align*} Assume that the demands $D_A$ and $D_B$ are constant and insensitive to price, that energy is sold at its marginal cost of production and that there are no limits on the output of the generators. ... ...
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 ... ... @@ -230,8 +230,8 @@ Consider an electricity market with two generator types, one with the cost funct Consider the two-bus power system shown in Figure \ref{twobus}, where the two nodes represent two markets, each with different total demand, and one generator at each node. At node A the demand is $D_A = 2000 \si{\mega\watt}$, whereas at node B the demand is $D_B = 1000 \si{\mega\watt}$. Furthermore, there is a transmission line with a capacity denoted by $F_{AB}$. The marginal cost of production of the generators connected to buses A and B are given respectively by the following expressions: \begin{align*} MC_A & = 20 + 0.03 G_A \hspace{1cm}\eur/\si{\mega\watt\hour} \\ MC_B & = 15 + 0.02 G_B \hspace{1cm} \eur/\si{\mega\watt\hour} MC_A & = 20 + 0.02 G_A \hspace{1cm}\eur/\si{\mega\watt\hour} \\ MC_B & = 15 + 0.03 G_B \hspace{1cm} \eur/\si{\mega\watt\hour} \end{align*} Assume that the demands $D_A$ and $D_B$ are constant and insensitive to price, that energy is sold at its marginal cost of production and that there are no limits on the output of the generators. ... ...
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 ... ... @@ -101,7 +101,7 @@ %===================================================================== Two generators are connected to the grid by a single transmission line (with no electrical demand on their side of the transmission line). A single company owns both the generators and the transmission line. Generator 1 has a linear cost curve $C(g) = 5 g$ [\euro/h] and a capacity of 300~MW and Generator 2 has a linear cost curve $C(g) = 10 g$ [\euro/h] and a capacity of 900~MW. The transmission line has a capacity of 1000~MW. Suppose the demand in the grid is always high enough to absorb the line (with no electrical demand on their side of the transmission line). A single company owns both the generators and the transmission line. Generator 1 has a linear cost curve $C_1(g_1) = 5 g_1$ [\euro/h] and a capacity of 300~MW and Generator 2 has a linear cost curve $C_2(g_2) = 10 g_2$ [\euro/h] and a capacity of 900~MW. The transmission line has a capacity $K$ of 1000~MW. Suppose the demand in the grid is always high enough to absorb the generation from the two generators and that the market price of electricity $\pi$ is never below 15 \euro/MWh and averages 20 \euro/MWh. ... ...
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 ... ... @@ -118,7 +118,7 @@ %===================================================================== Two generators are connected to the grid by a single transmission line (with no electrical demand on their side of the transmission line). A single company owns both the generators and the transmission line. Generator 1 has a linear cost curve $C(g) = 5 g$ [\euro/h] and a capacity of 300~MW and Generator 2 has a linear cost curve $C(g) = 10 g$ [\euro/h] and a capacity of 900~MW. The transmission line has a capacity of 1000~MW. Suppose the demand in the grid is always high enough to absorb the line (with no electrical demand on their side of the transmission line). A single company owns both the generators and the transmission line. Generator 1 has a linear cost curve $C_1(g_1) = 5 g_1$ [\euro/h] and a capacity of 300~MW and Generator 2 has a linear cost curve $C_2(g_2) = 10 g_2$ [\euro/h] and a capacity of 900~MW. The transmission line has a capacity $K$ of 1000~MW. Suppose the demand in the grid is always high enough to absorb the generation from the two generators and that the market price of electricity $\pi$ is never below 15 \euro/MWh and averages 20 \euro/MWh. ... ... @@ -159,12 +159,12 @@ electricity $\pi$ is never below 15 \euro/MWh and averages 20 From stationarity we have for $g_1$ the non-zero terms: \begin{align*} 0 & = \frac{\d}{\d g_1} \left( \pi (g_1+g_2) - 5g_1 - 10g_2\right) - \bar{\m}_1 \frac{\d}{\d g_1} (g_1-\hat g_1)- \ubar{m}_1 \frac{\d}{\d g_1} (-g_1) -\m_T \frac{\d}{\d g_1} (g_1+g_2-K) \nn \\ 0 & = \frac{\d}{\d g_1} \left( \pi (g_1+g_2) - 5g_1 - 10g_2\right) - \bar{\m}_1 \frac{\d}{\d g_1} (g_1-\hat g_1)- \ubar{\m}_1 \frac{\d}{\d g_1} (-g_1) -\m_T \frac{\d}{\d g_1} (g_1+g_2-K) \nn \\ & = \pi -5 - \bar{\m}_1 + \ubar{\m}_1 - \m_T \end{align*} For $g_2$ we have \begin{align*} 0 & = \frac{\d}{\d g_2} \left( \pi (g_1+g_2) - 5g_1 - 10g_2\right) - \bar{\m}_2 \frac{\d}{\d g_2} (g_2-\hat g_2)- \ubar{m}_2 \frac{\d}{\d g_2} (-g_2) -\m_T \frac{\d}{\d g_2} (g_1+g_2-K) \nn \\ 0 & = \frac{\d}{\d g_2} \left( \pi (g_1+g_2) - 5g_1 - 10g_2\right) - \bar{\m}_2 \frac{\d}{\d g_2} (g_2-\hat g_2)- \ubar{\m}_2 \frac{\d}{\d g_2} (-g_2) -\m_T \frac{\d}{\d g_2} (g_1+g_2-K) \nn \\ & = \pi - 10- \bar{\m}_2 + \ubar{\m}_2 - \m_T \end{align*} ... ...
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