correction tut5

tutorial-4/sheet04.tex

@@ -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.

tutorial-4/solution04.tex

@@ -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.

tutorial-5/sheet05.tex

@@ -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.

tutorial-5/solution05.tex

@@ -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*}