Commit 39fe4c23 authored by Fabian Neumann's avatar Fabian Neumann
Browse files

tutorial-4: clarifications and improvement of nomenclature

parent a0b011e3
...@@ -177,8 +177,7 @@ Assume that the demands $D_A$ and $D_B$ are constant and insensitive to price, t ...@@ -177,8 +177,7 @@ Assume that the demands $D_A$ and $D_B$ are constant and insensitive to price, t
\begin{enumerate}[(a)] \begin{enumerate}[(a)]
\item Calculate the price of electricity at each bus, the production \item Calculate the price of electricity at each bus, the production
of each generator, the flow on the line, and the value of any KKT of each generator, and the flow on the line for the following cases. You may also calculate the values of any KKT multiplier as a bonus.
multipliers for the following cases:
\begin{enumerate}[(i)] \begin{enumerate}[(i)]
\item The line between buses A and B is disconnected. \item The line between buses A and B is disconnected.
\item The line between buses A and B is in service and has an unlimited capacity. \item The line between buses A and B is in service and has an unlimited capacity.
......
...@@ -158,76 +158,82 @@ Assume that the company is maximising its net surplus for a given electricity pr ...@@ -158,76 +158,82 @@ Assume that the company is maximising its net surplus for a given electricity pr
& = U'(d) - \pi - \m_{max} + \m_{min} \label{eq:2stat} & = U'(d) - \pi - \m_{max} + \m_{min} \label{eq:2stat}
\end{align} \end{align}
Note that it does not matter whether you pull in the constant of the right-hand side of the respective constraint.
The marginal utility curve is $U'(d) = 70 - 6d$ [\euro/MWh]. At The marginal utility curve is $U'(d) = 70 - 6d$ [\euro/MWh]. At
$\pi = 5$, the demand would be determined by $5=70-6d$, i.e. $d = $\pi = 5$ and if the demand were unconstrained, the demand would be determined by $5=70-6d$, i.e. $d =
65/6 = 10.8333$, which is above the consumption limit 65/6 = 10.8333$, which is above the consumption limit
$d_{max} = 10$. Therefore the optimal demand is $d^* = 10$, the upper limit is binding $\mu_{max} $d_{max} = 10$. Therefore the optimal demand is $d^* = 10$, the upper limit is binding such that $\mu_{max}
\geq 0$ and the lower limit is non-binding $\mu_{min} = 0$. \geq 0$ and the lower limit is non-binding such that $\mu_{min} = 0$.
To determine the value of $\mu_{max}$ we use \eqref{eq:2stat} to get To determine the value of $\mu_{max}$ we use \eqref{eq:2stat} to get
$\m_{max} = U'(d^*) - \pi = U'(10) - 5 = 5$. $\m_{max} = U'(d^*) - \pi + \mu_{min} = U'(10) - 5 + 0= 5$.
\begin{shaded} \begin{shaded}
\item Suppose now the electricity price is $\pi = 60$~\euro/MWh. What are \item Suppose now the electricity price is $\pi = 60$~\euro/MWh. What are
the optimal demand $d^*$, $\mu_{max}$ and $\mu_{min}$ now? the optimal demand $d^*$, $\mu_{max}$ and $\mu_{min}$ now?
\end{shaded} \end{shaded}
At $\pi = 60$, the demand would be determined by $60=70-6d$, i.e. $d = 10/6 = 1.667$, which is below the consumption limit $d_{min} = 2$. Therefore the optimal demand is $d^* = 2$, the upper limit is non-binding $\mu_{max} At $\pi = 60$, the demand would be determined by $60=70-6d$, i.e. $d = 10/6 = 1.667$, which is below the consumption limit $d_{min} = 2$. Therefore the optimal demand is $d^* = 2$, the upper limit is non-binding such that $\mu_{max}
= 0$ and the lower limit is binding $\mu_{min} \geq 0$. = 0$ and the lower limit is binding such that $\mu_{min} \geq 0$.
To determine the value of $\mu_{min}$ we use \eqref{eq:2stat} to get To determine the value of $\mu_{min}$ we use \eqref{eq:2stat} to get
$\m_{min} = \pi - U'(d^*) = 60 - U'(2) = 2$. $\m_{min} = \pi - U'(d^*) + \mu_{max} = 60 - U'(2) + 0 = 2$.
\end{enumerate} \end{enumerate}
%=============== ====================================================== %=============== ======================================================
\paragraph{Solution IV.2 \normalsize (Economic dispatch in a single bidding zone).}~\\ \paragraph{Solution 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]. 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 $G_1 = 300\mw$ 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 $G_2=400\mw$ 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)] \begin{enumerate}[(a)]
\begin{shaded}\item What are the objective function and constraints required for an optimisation problem to maximise short-run social welfare in this market?\end{shaded} \begin{shaded}\item What are the objective function and constraints required for an optimisation problem to maximise short-run social welfare in this market?\end{shaded}
The optimisation problem has objective function: The optimisation problem has the objective function:
\begin{equation*} \begin{equation*}
\max_{D,G_1,G_2}\left[ U(D) - C_1(G_1) - C_2(G_2) \right] = \max_{D,G_1,G_2} \left[8000D-5D^2 - c_1G_1 - c_2G_2 \right] \max_{d,g_1,g_2}\left[ U(d) - C_1(g_1) - C_2(g_2) \right] = \max_{d,g_1,g_2} \left[8000d-5d^2 - c_1g_1 - c_2g_2 \right]
\end{equation*} \end{equation*}
with constraints: with constraints:
\begin{align*} \begin{align*}
D - G_1 - G_2 & = 0 \leftrightarrow \l \\ d - g_1 - g_2 & = 0 \leftrightarrow \l \\
G_1 & \leq K_1 \leftrightarrow \bar{\m}_1 \\ g_1 & \leq G_1 \leftrightarrow \bar{\m}_1 \\
G_2 & \leq K_2 \leftrightarrow \bar{\m}_2 \\ g_2 & \leq G_2 \leftrightarrow \bar{\m}_2 \\
-G_1 & \leq 0 \leftrightarrow \ubar{\m}_1 \\ -g_1 & \leq 0 \leftrightarrow \ubar{\m}_1 \\
-G_2 & \leq 0 \leftrightarrow \ubar{\m}_2 -g_2 & \leq 0 \leftrightarrow \ubar{\m}_2
\end{align*} \end{align*}
\begin{shaded}\item Write down the Karush-Kuhn-Tucker (KKT) conditions for this problem.\end{shaded} \begin{shaded}\item Write down the Karush-Kuhn-Tucker (KKT) conditions for this problem.\end{shaded}
Stationarity gives for $D$: Stationarity gives for $d$:
\begin{equation*} \begin{equation*}
\frac{\d U}{\d D} - \l = 8000 - 10D - \l = 0 \frac{\d U}{\d d} - \l = 8000 - 10d - \l = 0
\end{equation*} \end{equation*}
Stationarity for $G_1$ gives: Stationarity for $g_1$ gives:
\begin{equation*} \begin{equation*}
-\frac{\d C_1}{\d G_1} + \l - \m_1 = -c_1+ \l - \bar{\m}_1 + \ubar{\m_1} = 0 -\frac{\d C_1}{\d g_1} + \l - \m_1 = -c_1+ \l - \bar{\m}_1 + \ubar{\m_1} = 0
\end{equation*} \end{equation*}
Stationarity for $G_2$ gives: Stationarity for $g_2$ gives:
\begin{equation*} \begin{equation*}
-\frac{\d C_2}{\d G_2} + \l - \m_2 = -c_2+ \l - \bar{\m}_2 + \ubar{\m_2} = 0 -\frac{\d C_2}{\d g_2} + \l - \m_2 = -c_2+ \l - \bar{\m}_2 + \ubar{\m_2} = 0
\end{equation*} \end{equation*}
Primal feasibility is just the constraints above. Dual feasibility is $\bar{\m}_i,\ubar{\m}_i \geq 0$ and complementary slackness is $\bar{\m}_i(G_i-K) = 0$ and $\ubar{\m}_i G_i = 0$ for $i=1,2$. Primal feasibility is just the generator limits above in (a). Dual feasibility is $\bar{\m}_i,\ubar{\m}_i \geq 0$ and complementary slackness is $\bar{\m}_i(G_i-K) = 0$ and $\ubar{\m}_i G_i = 0$ for $i=1,2$.
\begin{shaded}\item Determine the optimal rate of production of the generators and the value of all KKT multipliers. What is the interpretation of the respective KKT multipliers?\end{shaded} \begin{shaded}\item Determine the optimal rate of production of the generators and the value of all KKT multipliers. What is the interpretation of the respective KKT multipliers?\end{shaded}
The marginal utility at the full output of the generators, $K_1 The marginal utility at the full output of the generators, $G_1
+ K_2 = $ 700~MW is $U'(700) = 8000 - 10\cdot700 = 1000$ \euro/MWh, + G_2 = $ 700~MW is $U'(700) = 8000 - 10\cdot700 = 1000$ \euro/MWh,
which is higher than the costs $c_i$, so we'll find optimal rates which is higher than the costs $c_i$, so we'll find optimal rates
$G_1^* = K_1$ and $G_2^* = K_2$ and $D^* = K_1+K_2$. This means $\l $g_1^* = G_1$ and $g_2^* = G_2$ and $d^* = G_1+G_2$. This means $\l
= U'(K_1+K_2) = 1000$ \euro/MWh, which is the market price. Because = U'(G_1+G_2) = 1000$ \euro/MWh, which is the market price. Because
the lower constraints on the generator output are not binding, from the lower constraints on the generator output are not binding, from
complementary slackness we have $\ubar{\m}_i = 0$. The upper complementary slackness we have $\ubar{\m}_i = 0$. The upper
constraints are binding, so $\bar{\m}_i \geq 0$. constraints are binding, so $\bar{\m}_i \geq 0$.
From stationarity $\bar{\m}_i = From stationarity $\bar{\m}_i =
\l - c_i$, which is the increase in social welfare if Generator $i$ \l - c_i + \ubar{\m}_i$, which is the increase in social welfare if Generator $i$
could increase its capacity by a marginal amount. could increase its capacity by a marginal amount.
$$\bar{\m}_1=1000-20=980 \text{\euro/MWh}$$
$$\bar{\m}_2=1000-50=950 \text{\euro/MWh}$$
\end{enumerate} \end{enumerate}
%=============== ====================================================== %=============== ======================================================
...@@ -251,8 +257,10 @@ Assume that the demands $D_A$ and $D_B$ are constant and insensitive to price, t ...@@ -251,8 +257,10 @@ Assume that the demands $D_A$ and $D_B$ are constant and insensitive to price, t
\begin{enumerate}[(a)] \begin{enumerate}[(a)]
\begin{shaded}\item Calculate the price of electricity at each bus, the production \begin{shaded}\item Calculate the price of electricity at each bus, the production
of each generator, the flow on the line, and the value of any KKT of each generator, and the flow on the line for the following cases. You may also calculate the values of any KKT multiplier as a bonus.\end{shaded}
multipliers for the following cases:\end{shaded}
The price of electricity is the value of the dual variable at the nodal balance equation.
Use the following nomenclature: price $\lambda_{A/B}$, generation $G_{A/B}$, flow $F_{AB}$. Use the following nomenclature: price $\lambda_{A/B}$, generation $G_{A/B}$, flow $F_{AB}$.
\begin{enumerate}[(i)] \begin{enumerate}[(i)]
\begin{shaded}\item The line between buses A and B is disconnected.\end{shaded} \begin{shaded}\item The line between buses A and B is disconnected.\end{shaded}
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment