... ... @@ -164,13 +164,14 @@ Consider an electricity market with two generator types, one with the cost funct \includegraphics[width=14cm]{two-bus} \caption{A simple two-bus power system.} \label{test} \label{twobus} \end{figure} 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: Consider the two-bus power system shown in Figure \ref{twobus}, where the two nodes represent two markets, each with different total demand $D_i$, and one generator at each node producing $P_i$. 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 + 0.03 P_A \hspace{1cm}\eur/\si{\mega\watt\hour} \\ MC_B & = 15 + 0.02 P_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. ... ...
 ... ... @@ -247,10 +247,10 @@ Consider an electricity market with two generator types, one with the cost funct \label{twobus} \end{figure} 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: Consider the two-bus power system shown in Figure \ref{twobus}, where the two nodes represent two markets, each with different total demand $D_i$, and one generator at each node producing $G_i$. 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 + 0.02 G_A \hspace{1cm}\eur/\si{\mega\watt\hour} \\ MC_B & = 15 + 0.03 G_B \hspace{1cm} \eur/\si{\mega\watt\hour} MC_A & = 20 + 0.02 P_A \hspace{1cm}\eur/\si{\mega\watt\hour} \\ MC_B & = 15 + 0.03 P_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. ... ... @@ -264,32 +264,57 @@ The price of electricity is the value of the dual variable at the nodal balance Use the following nomenclature: price $\lambda_{A/B}$, generation $G_{A/B}$, flow $F_{AB}$. \begin{enumerate}[(i)] \begin{shaded}\item The line between buses A and B is disconnected.\end{shaded} Where to start: $P_A=D_A$ and $P_B=D_B$ and substitute into $MC_i$. $\l_A= 80\emwh$, $\l_B=35\emwh$, $G_{A}=2000$ MW, $G_B=1000$ MW, $F_{AB}=0$ $G_{A}=2000$ MW, $P_B=1000$ MW, $F_{AB}=0$ \begin{shaded}\item The line between buses A and B is in service and has an unlimited capacity.\end{shaded} $\l_A= 53\emwh$, $\l_B=53\emwh$, Where to start: No restriction in transmission, so prices must be the same for the two nodes: $\l_A = \l_B$, therefore, $MC_A = MC_B$. Also: $P_A+P_B = D_A+D_B$. $G_{A}=1100\mw$, $G_B=1900$ MW, $F_{AB}=-900\mw$ $\l_A= 53\emwh$, $\l_B=53\emwh$, $G_{A}=1100\mw$, $P_B=1900$ MW, $F_{AB}=-900\mw$ \begin{shaded}\item The line between buses A and B is in service and has an unlimited capacity, but the maximum output of Generator B is 1500~MW.\end{shaded} $\l_A= 65\emwh$, $\l_B=65\emwh$, Where to start: $P_B=1500$ MW since it is now constrained but would have been higher in the unconstrained case (ii). Also, since there are no transmission constraints: $\l_A = \l_B$. Also: $P_A+P_B = D_A+D_B$. $G_{A}=1500\mw$, $G_B=1500$ MW, $F_{AB}=-500\mw$ $\l_A= 65\emwh$, $\l_B=65\emwh$, $G_{A}=1500\mw$, $P_B=1500$ MW, $F_{AB}=-500\mw$ \begin{shaded}\item The line between buses A and B is in service and has an unlimited capacity, but the maximum output of Generator A is 900~MW. The output of Generator B is unlimited.\end{shaded} $\l_A= 57\emwh$, $\l_B=57\emwh$, Where to start: $P_A=900$ MW since it is now constrained but would have been higher in the unconstrained case (ii). Also, since there are no transmission constraints: $\l_A = \l_B$. Also: $P_A+P_B = D_A+D_B$. $\l_A= 57\emwh$, $\l_B=57\emwh$, $G_{A}=900\mw$, $G_B=2100$ MW, $F_{AB}=-1100\mw$ $G_{A}=900\mw$, $P_B=2100$ MW, $F_{AB}=-1100\mw$ \begin{shaded}\item The line between buses A and B is in service but its capacity is limited to 600~MW. The output of the generators is unlimited.\end{shaded} $\l_A= 62\emwh$, $\l_B=47\emwh$, Where to start: $F_{AB} = - 600$ MW since we would want even more transmission, if it were not constrained. Also, $P_A+F_{AB}=2000$ MW. $\l_A= 62\emwh$, $\l_B=47\emwh$, $G_{A}=1400\mw$, $G_B=1600$ MW, $F_{AB}=-600\mw$ $G_{A}=1400\mw$, $P_B=1600$ MW, $F_{AB}=-600\mw$ \end{enumerate} \begin{shaded}\item Calculate the generator revenues, generator profits, consumer payments and consumer net surplus for all the cases considered in the above problem. Who benefits from the line connecting these two buses?\end{shaded} Generator revenues $R_{i}$, generator costs $C_{i}$, generator profits $P_{i}$, consumer payments $E_{i}$. Find the generator profits by subtracting the costs from the revenue. Costs are given by integrating the marginal cost, i.e. $C_A = 20G_A + 0.015G_A^2$ and $C_B = 15G_B + 0.01G_B^2$. The generator at $B$ and the consumers at $A$ benefit from the line (price increases at $B$, decreases at $B$). Generator revenues $R_{i}$, generator costs $C_{i}$, generator profits $P_{i}$, consumer payments $E_{i}$. Find the generator profits by subtracting the costs from the revenue. Costs are given by integrating the marginal cost, i.e. $C_A = 20P_A + 0.015P_A^2$ and $C_B = 15P_B + 0.01P_B^2$. The generator at $B$ and the consumers at $A$ benefit from the line (price increases at $B$, decreases at $B$). \begin{table}[!h] \centering \begin{tabular}{lrrrrr} ... ...