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

tutorial-1/solution01-2.ipynb

@@ -24,7 +24,7 @@
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@@ -52,7 +52,7 @@
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@@ -80,7 +80,7 @@
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@@ -94,7 +94,7 @@
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@@ -108,7 +108,7 @@
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@@ -122,7 +122,7 @@
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@@ -136,7 +136,7 @@
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@@ -150,7 +150,7 @@
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@@ -164,7 +164,7 @@
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@@ -178,7 +178,7 @@
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@@ -203,7 +203,7 @@
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@@ -227,7 +227,7 @@
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@@ -266,7 +266,7 @@
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@@ -318,7 +318,7 @@
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@@ -341,7 +341,7 @@
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@@ -365,7 +365,7 @@
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@@ -378,7 +378,7 @@
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@@ -401,7 +401,7 @@
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@@ -413,7 +413,7 @@
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@@ -440,7 +440,7 @@
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@@ -450,10 +450,10 @@
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@@ -496,7 +496,7 @@
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tutorial-4/sheet04.tex

@@ -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.

tutorial-5/sheet05.tex

@@ -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.

tutorial-5/solution05-1.ipynb

@@ -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."
    ]
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   {
@@ -639,9 +639,8 @@
   }
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+   "display_name": "Python 3",
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@@ -655,7 +654,7 @@
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    "pygments_lexer": "ipython3",
-   "version": "3.6.5"
+   "version": "3.6.4"
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tutorial-5/solution05.tex

@@ -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}

tutorial-5/tutorial05-1.ipynb

@@ -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."
    ]
   },
   {