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.
The marginal utility curve is $U'(q)=70-6q$ [\euro/MWh]. At
$\pi=5$, the demand would be determined by $5=70-6q$, i.e. $q=
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. $d=
65/6=10.8333$, which is above the consumption limit
$q_{max}=10$. Therefore the optimal demand is $q^*=10$, the upper limit is binding $\mu_{max}
$d_{max}=10$. Therefore the optimal demand is $d^*=10$, the upper limit is binding $\mu_{max}
\geq0$ and the lower limit is non-binding $\mu_{min}=0$.
To determine the value of $\mu_{max}$ we use \eqref{eq:2stat} to get
$\m_{max}= U'(q^*)-\pi= U'(10)-5=5$.
$\m_{max}= U'(d^*)-\pi= U'(10)-5=5$.
\begin{shaded}
\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{shaded}
At $\pi=60$, the demand would be determined by $60=70-6q$, i.e. $q=10/6=1.667$, which is below the consumption limit $q_{min}=2$. Therefore the optimal demand is $q^*=2$, 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 $\mu_{max}
=0$ and the lower limit is binding $\mu_{min}\geq0$.
To determine the value of $\mu_{min}$ we use \eqref{eq:2stat} to get
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{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}
Primal feasibility is just the constraints above. Dual feasibility is $\bar{\m}_i,\ubar{\m}_i \geq0$ and complementary slackness is $\bar{\m}_i(Q_i-K)=0$ and $\ubar{\m}_i Q_i =0$ for $i=1,2$.
Primal feasibility is just the constraints above. Dual feasibility is $\bar{\m}_i,\ubar{\m}_i \geq0$ and complementary slackness is $\bar{\m}_i(D_i-K)=0$ and $\ubar{\m}_i D_i =0$ 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,
which is higher than the costs $c_i$, so we'll find optimal rates
$Q_1^*= K_1$ and $Q_2^*= K_2$ and $Q^*= K_1+K_2$. This means $\l
$D_1^*= K_1$ and $D_2^*= K_2$ and $D^*= K_1+K_2$. This means $\l
= U'(K_1+K_2)=1000$\euro/MWh, which is the market price. Because
the lower constraints on the generator output are not binding, from
complementary slackness we have $\ubar{\m}_i =0$. The upper
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@@ -240,35 +240,35 @@ Assume that the demand $D_*$ is constant and insensitive to price, that energy i
\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}
Use the following nomenclature: price $\lambda_i$, production $Q^{S}_i$, flow $F$.
Use the following nomenclature: price $\lambda_i$, generation $G^{S}_i$, flow $F$.
\begin{enumerate}[(i)]
\begin{shaded}\item The line between buses A and B is disconnected.\end{shaded}
$\l_A=80\emwh$, $\l_B=35\emwh$,
$Q_{A}^S=2000$ MW, $Q_B^S=1000$ MW, $F=0$
$G_{A}^S=2000$ MW, $G_B^S=1000$ MW, $F=0$
\begin{shaded}\item The line between buses A and B is in service and has an unlimited capacity.\end{shaded}
$\l_A=53\emwh$, $\l_B=53\emwh$,
$Q_{A}^S=1100\mw$, $Q_B^S=1900$ MW, $F=-900\mw$
$G_{A}^S=1100\mw$, $G_B^S=1900$ MW, $F=-900\mw$
\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}
$\l_A=65\emwh$, $\l_B=65\emwh$,
$Q_{A}^S=1500\mw$, $Q_B^S=1500$ MW, $F=-500\mw$
$G_{A}^S=1500\mw$, $G_B^S=1500$ MW, $F=-500\mw$
\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}
$\l_A=57\emwh$, $\l_B=57\emwh$,
$Q_{A}^S=900\mw$, $Q_B^S=2100$ MW, $F=-1100\mw$
$G_{A}^S=900\mw$, $G_B^S=2100$ MW, $F=-1100\mw$
\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}
$\l_A=62\emwh$, $\l_B=47\emwh$,
$Q_{A}^S=1400\mw$, $Q_B^S=1600$ MW, $F=-600\mw$
$G_{A}^S=1400\mw$, $G_B^S=1600$ MW, $F=-600\mw$
\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 substracting the costs from the revenue. Costs are given by integrating the marginal cost, i.e. $C_A =20Q+0.015Q^2$ and $C_B =15Q+0.01Q^2$. 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 substracting the costs from the revenue. Costs are given by integrating the marginal cost, i.e. $C_A =20G+0.015G^2$ and $C_B =15G+0.01G^2$. The generator at $B$ and the consumers at $A$ benefit from the line (price increases at $B$, decreases at $B$).
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 $C(q)=5q$ [\euro/h] and a capacity of 300~MW and Generator 2 has a linear cost curve $C(q)=10q$ [\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 $C(g)=5g$ [\euro/h] and a capacity of 300~MW and Generator 2 has a linear cost curve $C(g)=10g$ [\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.
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@@ -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.
**Classic screening curve analysis for 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 €/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.
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 $C(q)=5q$ [\euro/h] and a capacity of 300~MW and Generator 2 has a linear cost curve $C(q)=10q$ [\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 $C(g)=5g$ [\euro/h] and a capacity of 300~MW and Generator 2 has a linear cost curve $C(g)=10g$ [\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
Where the first four constraints come from generation, where $\hatq_1=$ 300 MW and $\hatq_1=$ 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 $\hatg_1=$ 300 MW and $\hatg_1=$ 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 $q_1^*=\hatq_1=$ 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 $g_1^*=\hatg_1=$ 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:
@@ -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}
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@@ -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.
**Classic screening curve analysis for 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 €/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.
