Commit 4992e412 authored by sp2668's avatar sp2668

Change nomenclature q/Q -> d/D for demand + g/G for generation

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Suppose that the utility for the electricity consumption of an industrial company is given by Suppose that the utility for the electricity consumption of an industrial company is given by
\[ \[
U(q) = 70q - 3q^2 [\textrm{\euro}/h] \quad , \quad q_{min}=2\leq q \leq q_{max}=10, U(d) = 70d - 3d^2 [\textrm{\euro}/h] \quad , \quad d_{min}=2\leq d \leq d_{max}=10,
\] \]
where $q$ is the demand in MW and $q_{min}, q_{max}$ are the minimum and maximum demand. \\ where $d$ is the demand in MW and $d_{min}, d_{max}$ are the minimum and maximum demand. \\
[1em] [1em]
Assume that the company is maximising its net surplus for a given electricity price $\pi$, i.e. it maximises $\max_{q} \left[U(q) - Assume that the company is maximising its net surplus for a given electricity price $\pi$, i.e. it maximises $\max_{d} \left[U(d) -
\pi q\right]$. \pi d\right]$.
\begin{enumerate}[(a)] \begin{enumerate}[(a)]
\item If the price is $\pi = 5$~\euro/MWh, what is the optimal \item If the price is $\pi = 5$~\euro/MWh, what is the optimal
demand $q^*$? What is the value of the KKT multiplier $\mu_{max}$ demand $d^*$? What is the value of the KKT multiplier $\mu_{max}$
for the constraint $q \leq q_{max}=10$ at this optimal solution? for the constraint $d \leq d_{max}=10$ at this optimal solution?
What is the value of $\mu_{min}$ for $q \geq q_{min} = 2$? What is the value of $\mu_{min}$ for $d \geq d_{min} = 2$?
\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 $q^*$, $\mu_{max}$ and $\mu_{min}$ now? the optimal demand $d^*$, $\mu_{max}$ and $\mu_{min}$ now?
\end{enumerate} \end{enumerate}
%=============== ====================================================== %=============== ======================================================
\paragraph{Problem VI.2 \normalsize (Economic dispatch in a single bidding zone).}~\\ \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 $Q_1$~[MW] and another with variable cost $c=50\emwh$, capacity $K=400\mw$ and a dispatch rate of $Q_2$~[MW]. The demand has utility function $U(Q) = 8000Q - 5Q^2$~[\euro/h] for a consumption rate of $Q$~[MW]. 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].
\begin{enumerate}[(a)] \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 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. \item Write down the Karush-Kuhn-Tucker (KKT) conditions for this problem.
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%===================================================================== %=====================================================================
Two generators are connected to the grid by a single transmission 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 $C(q) = 5 q$ [\euro/h] and a capacity of 300~MW and Generator 2 has a linear cost curve $C(q) = 10 q$ [\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 $C(g) = 5 g$ [\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
generation from the two generators and that the market price of generation from the two generators and that the market price of
electricity $\pi$ is never below 15 \euro/MWh and averages 20 electricity $\pi$ is never below 15 \euro/MWh and averages 20
\euro/MWh. \euro/MWh.
...@@ -120,7 +120,7 @@ electricity $\pi$ is never below 15 \euro/MWh and averages 20 ...@@ -120,7 +120,7 @@ electricity $\pi$ is never below 15 \euro/MWh and averages 20
\paragraph{Problem V.2 \normalsize (duration curves and generation investment).}~\\ \paragraph{Problem V.2 \normalsize (duration curves and generation investment).}~\\
%===================================================================== %=====================================================================
Let us suppose that demand is inelastic. The demand-duration curve is given by $Q=1000-1000z$. Suppose that there is a choice between coal and gas generation plants with a variable cost of 2 and 12~\euro/MWh, together with load-shedding at 1012\euro/MWh. The fixed costs of coal and gas generation are 15 and 10~\euro/MWh, respectively. Let us suppose that demand is inelastic. The demand-duration curve is given by $D=1000-1000z$. Suppose that there is a choice between coal and gas generation plants with a variable cost of 2 and 12~\euro/MWh, together with load-shedding at 1012\euro/MWh. The fixed costs of coal and gas generation are 15 and 10~\euro/MWh, respectively.
\begin{enumerate}[(a)] \begin{enumerate}[(a)]
\item Describe the concept of a screening curve and how it helps to determine generation investment, given a demand-duration curve. \item Describe the concept of a screening curve and how it helps to determine generation investment, given a demand-duration curve.
......
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"\n", "\n",
"**Classic screening curve analysis for generation investment**\n", "**Classic screening curve analysis for generation investment**\n",
"\n", "\n",
"Let us suppose that demand is inelastic. The demand-duration curve is given by $Q=1000-1000z$. Suppose that there is a choice between coal and gas generation plants with a variable cost of 2 and 12 €/MWh, together with load-shedding at 1012 €/MWh. The fixed costs of coal and gas generation are 15 and 10 €/MWh, respectively." "Let us suppose that demand is inelastic. The demand-duration curve is given by $D=1000-1000z$. Suppose that there is a choice between coal and gas generation plants with a variable cost of 2 and 12 €/MWh, together with load-shedding at 1012 €/MWh. The fixed costs of coal and gas generation are 15 and 10 €/MWh, respectively."
] ]
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%===================================================================== %=====================================================================
Two generators are connected to the grid by a single transmission 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 $C(q) = 5 q$ [\euro/h] and a capacity of 300~MW and Generator 2 has a linear cost curve $C(q) = 10 q$ [\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 $C(g) = 5 g$ [\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
generation from the two generators and that the market price of generation from the two generators and that the market price of
electricity $\pi$ is never below 15 \euro/MWh and averages 20 electricity $\pi$ is never below 15 \euro/MWh and averages 20
\euro/MWh. \euro/MWh.
...@@ -133,38 +133,38 @@ electricity $\pi$ is never below 15 \euro/MWh and averages 20 ...@@ -133,38 +133,38 @@ electricity $\pi$ is never below 15 \euro/MWh and averages 20
owned the transmission line, he could take the congestion revenue for owned the transmission line, he could take the congestion revenue for
himself. himself.
If we label the dispatch of Generator 1 by $q_1$ and of Generator 2 by $q_2$, then the objective function is to maximise total profit If we label the dispatch of Generator 1 by $g_1$ and of Generator 2 by $g_2$, then the objective function is to maximise total profit
\begin{equation*} \begin{equation*}
\max_{q_1,q_2} \left[ \pi (q_1+q_2) - C_1(q_1) - C_2(q_2) \right] = \max_{q_1,q_2} \left[ \pi (q_1+q_2) - 5q_1 - 10q_2 \right] \max_{g_1,g_2} \left[ \pi (g_1+g_2) - C_1(g_1) - C_2(g_2) \right] = \max_{g_1,g_2} \left[ \pi (g_1+g_2) - 5g_1 - 10g_2 \right]
\end{equation*} \end{equation*}
The constraints are The constraints are
\begin{align*} \begin{align*}
q_1 & \leq \hat q_1 & \leftrightarrow & \bar{\m}_1 \\ g_1 & \leq \hat g_1 & \leftrightarrow & \bar{\m}_1 \\
-q_1 & \leq 0 & \leftrightarrow & \ubar{\m}_1 \\ -g_1 & \leq 0 & \leftrightarrow & \ubar{\m}_1 \\
q_2 & \leq \hat q_2 & \leftrightarrow & \bar{\m}_2 \\ g_2 & \leq \hat g_2 & \leftrightarrow & \bar{\m}_2 \\
-q_2 & \leq 0 & \leftrightarrow & \ubar{\m}_2 \\ -g_2 & \leq 0 & \leftrightarrow & \ubar{\m}_2 \\
q_1+q_2 & \leq K & \leftrightarrow & \m_T g_1+g_2 & \leq K & \leftrightarrow & \m_T
\end{align*} \end{align*}
Where the first four constraints come from generation, where $\hat q_1 = $ 300 MW and $\hat q_1 = $ 900 MW and the final constraint comes from the transmission, where $K = $ 1000~MW is the capacity of the export transmission line. Where the first four constraints come from generation, where $\hat g_1 = $ 300 MW and $\hat g_1 = $ 900 MW and the final constraint comes from the transmission, where $K = $ 1000~MW is the capacity of the export transmission line.
\begin{shaded}\item What is the optimal dispatch?\end{shaded} \begin{shaded}\item What is the optimal dispatch?\end{shaded}
Since the market price is always higher than the marginal price Since the market price is always higher than the marginal price
of the generators, they will both run as high as possible given the of the generators, they will both run as high as possible given the
constraints. Since Generator 1 is cheaper than Generator 2, it will constraints. Since Generator 1 is cheaper than Generator 2, it will
max-out its capacity first, so that $q_1^* = \hat q_1 =$ 300~MW. Generator 2 will output as much as it can given the transmission constraint, so that $q_2^* =$ 700~MW. max-out its capacity first, so that $g_1^* = \hat g_1 =$ 300~MW. Generator 2 will output as much as it can given the transmission constraint, so that $g_2^* =$ 700~MW.
\begin{shaded}\item What are the values of the KKT multipliers for all the constraints in terms of $\pi$?\end{shaded} \begin{shaded}\item What are the values of the KKT multipliers for all the constraints in terms of $\pi$?\end{shaded}
From stationarity we have for $q_1$ the non-zero terms: From stationarity we have for $g_1$ the non-zero terms:
\begin{align*} \begin{align*}
0 & = \frac{\d}{\d q_1} \left( \pi (q_1+q_2) - 5q_1 - 10q_2\right) - \bar{\m}_1 \frac{\d}{\d q_1} (q_1-\hat q_1)- \ubar{m}_1 \frac{\d}{\d q_1} (-q_1) -\m_T \frac{\d}{\d q_1} (q_1+q_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 & = \pi -5 - \bar{\m}_1 + \ubar{\m}_1 - \m_T
\end{align*} \end{align*}
For $q_2$ we have For $g_2$ we have
\begin{align*} \begin{align*}
0 & = \frac{\d}{\d q_2} \left( \pi (q_1+q_2) - 5q_1 - 10q_2\right) - \bar{\m}_2 \frac{\d}{\d q_2} (q_2-\hat q_2)- \ubar{m}_2 \frac{\d}{\d q_2} (-q_2) -\m_T \frac{\d}{\d q_2} (q_1+q_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 & = \pi - 10- \bar{\m}_2 + \ubar{\m}_2 - \m_T
\end{align*} \end{align*}
...@@ -181,11 +181,11 @@ electricity $\pi$ is never below 15 \euro/MWh and averages 20 ...@@ -181,11 +181,11 @@ electricity $\pi$ is never below 15 \euro/MWh and averages 20
\begin{shaded}\item A new turbo-boosting technology can increase the capacity of Generator 1 from 300~MW to 350~MW. At what annualised capital cost would it be efficient to invest in this new technology?\end{shaded} \begin{shaded}\item A new turbo-boosting technology can increase the capacity of Generator 1 from 300~MW to 350~MW. At what annualised capital cost would it be efficient to invest in this new technology?\end{shaded}
The value of $\bar{\m}_1$ gives us the increase in profit for a small increase in $\hat{q}_1$. We want to understand a large increase in $\hat{q}_1$ of 50 MW, therefore we have to integrate over $\bar{\m}_1$ as a function of $\hat{q}_1$, since the value of $\bar{\m}_1$ may change as $\hat{q}_1$ changes. The total increase in profitability for expanding $\hat{q}_1$ from 300~MW to 350~MW is then The value of $\bar{\m}_1$ gives us the increase in profit for a small increase in $\hat{g}_1$. We want to understand a large increase in $\hat{g}_1$ of 50 MW, therefore we have to integrate over $\bar{\m}_1$ as a function of $\hat{g}_1$, since the value of $\bar{\m}_1$ may change as $\hat{g}_1$ changes. The total increase in profitability for expanding $\hat{g}_1$ from 300~MW to 350~MW is then
\begin{equation*} \begin{equation*}
\int_{300}^{350} \bar{\m}_1(\hat{q}_1) d\hat{q}_1 \int_{300}^{350} \bar{\m}_1(\hat{g}_1) d\hat{g}_1
\end{equation*} \end{equation*}
Because of the linearity of the problem, $\bar{\m}_1$ is actually constant as we expand $\hat{q}_1$ in the region from 300~MW to 350~MW. The extra profit would be per year: 5 \euro/MWh * 50 MW * 8760h/a = \euro 2.19 million/a. Because of the linearity of the problem, $\bar{\m}_1$ is actually constant as we expand $\hat{g}_1$ in the region from 300~MW to 350~MW. The extra profit would be per year: 5 \euro/MWh * 50 MW * 8760h/a = \euro 2.19 million/a.
At or below this annualised capital cost, it would be worth investing. At or below this annualised capital cost, it would be worth investing.
\begin{shaded}\item A new high temperature conductor technology can increase the capacity of the transmission line by 200~MW. At what annualised capital cost would it be efficient to invest in this new technology?\end{shaded} \begin{shaded}\item A new high temperature conductor technology can increase the capacity of the transmission line by 200~MW. At what annualised capital cost would it be efficient to invest in this new technology?\end{shaded}
...@@ -201,7 +201,7 @@ electricity $\pi$ is never below 15 \euro/MWh and averages 20 ...@@ -201,7 +201,7 @@ electricity $\pi$ is never below 15 \euro/MWh and averages 20
\paragraph{Solution V.2 \normalsize (duration curves and generation investment).}~\\ \paragraph{Solution V.2 \normalsize (duration curves and generation investment).}~\\
%===================================================================== %=====================================================================
Let us suppose that demand is inelastic. The demand-duration curve is given by $Q=1000-1000z$. Suppose that there is a choice between coal and gas generation plants with a variable cost of 2 and 12~\euro/MWh, together with load-shedding at 1012\euro/MWh. The fixed costs of coal and gas generation are 15 and 10~\euro/MWh, respectively. Let us suppose that demand is inelastic. The demand-duration curve is given by $D=1000-1000z$. Suppose that there is a choice between coal and gas generation plants with a variable cost of 2 and 12~\euro/MWh, together with load-shedding at 1012\euro/MWh. The fixed costs of coal and gas generation are 15 and 10~\euro/MWh, respectively.
\begin{enumerate}[(a)] \begin{enumerate}[(a)]
\begin{shaded} \begin{shaded}
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"\n", "\n",
"**Classic screening curve analysis for generation investment**\n", "**Classic screening curve analysis for generation investment**\n",
"\n", "\n",
"Let us suppose that demand is inelastic. The demand-duration curve is given by $Q=1000-1000z$. Suppose that there is a choice between coal and gas generation plants with a variable cost of 2 and 12 €/MWh, together with load-shedding at 1012 €/MWh. The fixed costs of coal and gas generation are 15 and 10 €/MWh, respectively." "Let us suppose that demand is inelastic. The demand-duration curve is given by $D=1000-1000z$. Suppose that there is a choice between coal and gas generation plants with a variable cost of 2 and 12 €/MWh, together with load-shedding at 1012 €/MWh. The fixed costs of coal and gas generation are 15 and 10 €/MWh, respectively."
] ]
}, },
{ {
......
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