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

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 ... ... @@ -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. ... ...
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 ... ... @@ -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": { ... ...
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 ... ... @@ -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." ] }, { ... ...
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