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

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
\begin{enumerate}[(a)]
\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
multipliers for the following cases:
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.
\begin{enumerate}[(i)]
\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.
......
......@@ -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}
\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
$\pi = 5$, 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
$d_{max} = 10$. Therefore the optimal demand is $d^* = 10$, the upper limit is binding $\mu_{max}
\geq 0$ and the lower limit is non-binding $\mu_{min} = 0$.
$d_{max} = 10$. 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 such that $\mu_{min} = 0$.
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}
\item Suppose now the electricity price is $\pi = 60$~\euro/MWh. What are
the optimal demand $d^*$, $\mu_{max}$ and $\mu_{min}$ now?
\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 $\mu_{max}
= 0$ and the lower limit is binding $\mu_{min} \geq 0$.
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 such that $\mu_{min} \geq 0$.
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}
%=============== ======================================================
\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{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*}
\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*}
with constraints:
\begin{align*}
D - G_1 - G_2 & = 0 \leftrightarrow \l \\
G_1 & \leq K_1 \leftrightarrow \bar{\m}_1 \\
G_2 & \leq K_2 \leftrightarrow \bar{\m}_2 \\
-G_1 & \leq 0 \leftrightarrow \ubar{\m}_1 \\
-G_2 & \leq 0 \leftrightarrow \ubar{\m}_2
d - g_1 - g_2 & = 0 \leftrightarrow \l \\
g_1 & \leq G_1 \leftrightarrow \bar{\m}_1 \\
g_2 & \leq G_2 \leftrightarrow \bar{\m}_2 \\
-g_1 & \leq 0 \leftrightarrow \ubar{\m}_1 \\
-g_2 & \leq 0 \leftrightarrow \ubar{\m}_2
\end{align*}
\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*}
\frac{\d U}{\d D} - \l = 8000 - 10D - \l = 0
\frac{\d U}{\d d} - \l = 8000 - 10d - \l = 0
\end{equation*}
Stationarity for $G_1$ gives:
Stationarity for $g_1$ gives:
\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*}
Stationarity for $G_2$ gives:
Stationarity for $g_2$ gives:
\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*}
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}
The marginal utility at the full output of the generators, $K_1
+ K_2 = $ 700~MW is $U'(700) = 8000 - 10\cdot700 = 1000$ \euro/MWh,
The marginal utility at the full output of the generators, $G_1
+ 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
$G_1^* = K_1$ and $G_2^* = K_2$ and $D^* = K_1+K_2$. This means $\l
= U'(K_1+K_2) = 1000$ \euro/MWh, which is the market price. Because
$g_1^* = G_1$ and $g_2^* = G_2$ and $d^* = G_1+G_2$. This means $\l
= U'(G_1+G_2) = 1000$ \euro/MWh, which is the market price. Because
the lower constraints on the generator output are not binding, from
complementary slackness we have $\ubar{\m}_i = 0$. The upper
constraints are binding, so $\bar{\m}_i \geq 0$.
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.
$$\bar{\m}_1=1000-20=980 \text{\euro/MWh}$$
$$\bar{\m}_2=1000-50=950 \text{\euro/MWh}$$
\end{enumerate}
%=============== ======================================================
......@@ -251,8 +257,10 @@ Assume that the demands $D_A$ and $D_B$ are constant and insensitive to price, t
\begin{enumerate}[(a)]
\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
multipliers for the following cases:\end{shaded}
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}
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 $F_{AB}$.
\begin{enumerate}[(i)]
\begin{shaded}\item The line between buses A and B is disconnected.\end{shaded}
......
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