Commit 4dab58dd authored by sp2668's avatar sp2668

correction tut4

parent 1c842aa2
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......@@ -117,7 +117,7 @@
Suppose that the utility for the electricity consumption of an industrial company is given by
\[
U(d) = 70d - 3d^2 [\textrm{\euro}/h] \quad , \quad d_{min}=2\leq d \leq d_{max}=10,
U(d) = 70d - 3d^2 ~[\textrm{\euro}/\si{\mega\watt\hour}] \quad , \quad d_{min}=2\leq d \leq d_{max}=10,
\]
where $d$ is the demand in MW and $d_{min}, d_{max}$ are the minimum and maximum demand. \\
[1em]
......@@ -136,7 +136,7 @@ Assume that the company is maximising its net surplus for a given electricity pr
\paragraph{Problem VI.2 \normalsize (Economic dispatch in a single bidding zone).}~\\
%=====================================================================
Consider an electricity market with two generator types, one with variable cost $c = 20\emwh$, capacity $K = 300\mw$ and a dispatch rate of $D_1$~[MW] and another with variable cost $c=50\emwh$, capacity $K=400\mw$ and a dispatch rate of $D_2$~[MW]. The demand has utility function $U(D) = 8000D - 5D^2$~[\euro/h] for a consumption rate of $D$~[MW].
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].
\begin{enumerate}[(a)]
\item What are the objective function and constraints required for an optimisation problem to maximise short-run social welfare in this market?
\item Write down the Karush-Kuhn-Tucker (KKT) conditions for this problem.
......@@ -158,11 +158,11 @@ Consider an electricity market with two generator types, one with variable cost
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 P_A \hspace{1cm}\eur/\si{\mega\watt\hour} \\
MC_B & = 15 + 0.02 P_B \hspace{1cm} \eur/\si{\mega\watt\hour}
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}
\end{align*}
Assume that the demand $D_*$ is 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.
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.
\begin{enumerate}[(a)]
\item Calculate the price of electricity at each bus, the production
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