Commit 4992e412 authored by sp2668's avatar sp2668

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

parent 4a0e0cb9
This diff is collapsed.
......@@ -24,7 +24,7 @@
},
{
"cell_type": "code",
"execution_count": 1,
"execution_count": 3,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -52,7 +52,7 @@
},
{
"cell_type": "code",
"execution_count": 2,
"execution_count": 4,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -80,7 +80,7 @@
},
{
"cell_type": "code",
"execution_count": 3,
"execution_count": 5,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -94,7 +94,7 @@
},
{
"cell_type": "code",
"execution_count": 4,
"execution_count": 6,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -108,7 +108,7 @@
},
{
"cell_type": "code",
"execution_count": 5,
"execution_count": 7,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -122,7 +122,7 @@
},
{
"cell_type": "code",
"execution_count": 6,
"execution_count": 8,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -136,7 +136,7 @@
},
{
"cell_type": "code",
"execution_count": 7,
"execution_count": 9,
"metadata": {
"slideshow": {
"slide_type": "subslide"
......@@ -150,7 +150,7 @@
},
{
"cell_type": "code",
"execution_count": 8,
"execution_count": 10,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -164,7 +164,7 @@
},
{
"cell_type": "code",
"execution_count": 9,
"execution_count": 11,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -178,7 +178,7 @@
},
{
"cell_type": "code",
"execution_count": 10,
"execution_count": 12,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -203,7 +203,7 @@
},
{
"cell_type": "code",
"execution_count": 11,
"execution_count": 13,
"metadata": {
"slideshow": {
"slide_type": "skip"
......@@ -227,7 +227,7 @@
},
{
"cell_type": "code",
"execution_count": 12,
"execution_count": 14,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -266,7 +266,7 @@
},
{
"cell_type": "code",
"execution_count": 13,
"execution_count": 15,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -318,7 +318,7 @@
},
{
"cell_type": "code",
"execution_count": 14,
"execution_count": 16,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -341,7 +341,7 @@
"0.73913043212890694"
]
},
"execution_count": 14,
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
......@@ -365,7 +365,7 @@
},
{
"cell_type": "code",
"execution_count": 15,
"execution_count": 17,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -378,7 +378,7 @@
},
{
"cell_type": "code",
"execution_count": 16,
"execution_count": 18,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -401,7 +401,7 @@
"0.56521739196777387"
]
},
"execution_count": 16,
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
......@@ -413,7 +413,7 @@
},
{
"cell_type": "code",
"execution_count": 17,
"execution_count": 19,
"metadata": {
"slideshow": {
"slide_type": "fragment"
......@@ -440,7 +440,7 @@
},
{
"cell_type": "code",
"execution_count": 18,
"execution_count": 20,
"metadata": {
"slideshow": {
"slide_type": "subslide"
......@@ -450,10 +450,10 @@
{
"data": {
"text/plain": [
"[<matplotlib.lines.Line2D at 0x7f2c3ef2d3c8>]"
"[<matplotlib.lines.Line2D at 0x7fb9fc9f0710>]"
]
},
"execution_count": 18,
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
},
......@@ -496,7 +496,7 @@
},
{
"cell_type": "code",
"execution_count": 19,
"execution_count": 21,
"metadata": {
"slideshow": {
"slide_type": "subslide"
......@@ -506,10 +506,10 @@
{
"data": {
"text/plain": [
"[<matplotlib.lines.Line2D at 0x7f2c3ee857f0>]"
"[<matplotlib.lines.Line2D at 0x7fb9fc948b38>]"
]
},
"execution_count": 19,
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
},
......
No preview for this file type
......@@ -117,26 +117,26 @@
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]
Assume that the company is maximising its net surplus for a given electricity price $\pi$, i.e. it maximises $\max_{q} \left[U(q) -
\pi q\right]$.
Assume that the company is maximising its net surplus for a given electricity price $\pi$, i.e. it maximises $\max_{d} \left[U(d) -
\pi d\right]$.
\begin{enumerate}[(a)]
\item If the price is $\pi = 5$~\euro/MWh, what is the optimal
demand $q^*$? What is the value of the KKT multiplier $\mu_{max}$
for the constraint $q \leq q_{max}=10$ at this optimal solution?
What is the value of $\mu_{min}$ for $q \geq q_{min} = 2$?
demand $d^*$? What is the value of the KKT multiplier $\mu_{max}$
for the constraint $d \leq d_{max}=10$ at this optimal solution?
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
the optimal demand $q^*$, $\mu_{max}$ and $\mu_{min}$ now?
the optimal demand $d^*$, $\mu_{max}$ and $\mu_{min}$ now?
\end{enumerate}
%=============== ======================================================
\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)]
\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.
......
This diff is collapsed.
No preview for this file type
......@@ -101,7 +101,7 @@
%=====================================================================
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
electricity $\pi$ is never below 15 \euro/MWh and averages 20
\euro/MWh.
......@@ -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).}~\\
%=====================================================================
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)]
\item Describe the concept of a screening curve and how it helps to determine generation investment, given a demand-duration curve.
......
......@@ -12,7 +12,7 @@
"\n",
"**Classic screening curve analysis for generation investment**\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."
]
},
{
......@@ -639,9 +639,8 @@
}
],
"metadata": {
"celltoolbar": "Slideshow",
"kernelspec": {
"display_name": "Python [default]",
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
......@@ -655,7 +654,7 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.5"
"version": "3.6.4"
},
"varInspector": {
"cols": {
......
......@@ -118,7 +118,7 @@
%=====================================================================
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
electricity $\pi$ is never below 15 \euro/MWh and averages 20
\euro/MWh.
......@@ -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
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*}
\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*}
The constraints are
\begin{align*}
q_1 & \leq \hat q_1 & \leftrightarrow & \bar{\m}_1 \\
-q_1 & \leq 0 & \leftrightarrow & \ubar{\m}_1 \\
q_2 & \leq \hat q_2 & \leftrightarrow & \bar{\m}_2 \\
-q_2 & \leq 0 & \leftrightarrow & \ubar{\m}_2 \\
q_1+q_2 & \leq K & \leftrightarrow & \m_T
g_1 & \leq \hat g_1 & \leftrightarrow & \bar{\m}_1 \\
-g_1 & \leq 0 & \leftrightarrow & \ubar{\m}_1 \\
g_2 & \leq \hat g_2 & \leftrightarrow & \bar{\m}_2 \\
-g_2 & \leq 0 & \leftrightarrow & \ubar{\m}_2 \\
g_1+g_2 & \leq K & \leftrightarrow & \m_T
\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}
Since the market price is always higher than the marginal price
of the generators, they will both run as high as possible given the
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}
From stationarity we have for $q_1$ the non-zero terms:
From stationarity we have for $g_1$ the non-zero terms:
\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
\end{align*}
For $q_2$ we have
For $g_2$ we have
\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
\end{align*}
......@@ -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}
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*}
\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*}
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.
\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
\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{shaded}
......
......@@ -8,7 +8,7 @@
"\n",
"**Classic screening curve analysis for generation investment**\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."
]
},
{
......
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