tutorial-3: clarifications and nomenclature

tutorial-3/worksheet/sheet03.tex

@@ -156,23 +156,23 @@ For now, assume no power exchange between the regions and that the stores are lo
  \item How much wind capacity $G^{N}_{w}$ must be installed in the North and solar capacity $G_s^S$ in the South so that on average generation matches demand?
 
        % (b)
- \item For a system to work, generation must match demand not on average but at each location and each point in time. Storages are one way of ensuring this constraint with variable renewable generation. What is the amount of store and dispatch power capacity $G_{st,store}=\max(-\Delta(t))$ and $G_{st,dispatch} = \max \Delta(t)$ the storage units must have in the North and in the South to account for the mismatch $\Delta(t)=L(t)-G_{w/s}\cdot g_{w/s}(t)$?
+ \item For a system to work, generation must match demand not on average but at each location and each point in time. Storages are one way of ensuring this constraint with variable renewable generation. What is the amount of store and dispatch power capacity $G_{st,store}=\max(-\Delta(t))$ and $G_{st,dispatch} = \max \Delta(t)$ the storage units must have in the North and in the South to account for the mismatch $\Delta(t)=L(t)-G\cdot g(t)$?
 \newpage
        % (c)
- \item What is the amount of energy capacity one needs in the North and in the South?
+ \item What is the amount of energy capacity $E_{st}$ one needs for either storage in the North and in the South? The energy capacity is given by
 
        \begin{equation*}
         E_{st} = \max_t e_{st}(t) = \max_t \int_{0}^{t} -\Delta(t') \;\mathrm{d}t'
        \end{equation*}
 
        % (d)
- \item Should they choose hydrogen or battery storage? And how much would it cost them with the prices in Table 1? Is the South or the North paying more for their energy?
+ \item Should they choose hydrogen or battery storage? And how much would it cost them with the prices in Table 1? Is the South or the North paying more for their energy? Disregard losses!
 
 % (e)
  \item What do you imagine would change if you considered the storage losses given in Table 1 in your results (a)-(d)?
 
 \end{enumerate}
-Now we lift the restriction against transmission and allow them to bridge their 500 km separation with a transmission line.
+Now we lift the restriction against transmission and allow the two regions to bridge their 500 km separation with a transmission line.
 
 \begin{enumerate}[(f)]
  % (f)

tutorial-3/worksheet/solution03.tex

@@ -149,38 +149,37 @@ For now, assume no power exchange between the regions and that the stores are lo
 
 \begin{enumerate}[(a)]
  % (a)
- \begin{shaded}\item How much wind capacity $G^{N}_{w}$ must be installed in the North and solar capacity $G_s^S$ in the South so that on average generation matches demand?
-?\end{shaded}
+ \begin{shaded}\item How much wind capacity $G^{N}_{w}$ must be installed in the North and solar capacity $G_s^S$ in the South so that on average generation matches demand?\end{shaded}
 
  In the North:
 
- $$\expect{L^N} = \expect{G^N_w \cdot G^N_w(t)}$$
+ $$\expect{L^N} = \expect{G^N_w \cdot g^N_w(t)}$$
 
- $$\Rightarrow \quad A^N_l = G^N_w\cdot Cf_w $$
+ $$\Rightarrow \quad A^N_l = G^N_w\cdot c_w $$
 
  \DIVIDE{20}{0.3}\res
 
- $$\Rightarrow \quad G^N_w = \frac{A^N_l}{Cf_w} = \frac{20\si{\giga\watt}}{0.3} = \rd{\res}\si{\giga\watt}$$
+ $$\Rightarrow \quad G^N_w = \frac{A^N_l}{c_w} = \frac{20\si{\giga\watt}}{0.3} = \rd{\res}\si{\giga\watt}$$
 
  In the South:
 
- $$\expect{L^S} = \expect{G^S_s \cdot G^S_s(t)}$$
+ $$\expect{L^S} = \expect{G^S_s \cdot g^S_s(t)}$$
 
- $$\Rightarrow \quad A^S_l = G_{S,w}\cdot Cf_w $$
+ $$\Rightarrow \quad A^S_l = G^S_s\cdot c_s $$
 
  \DIVIDE{30}{0.12}\res
 
- $$\Rightarrow \quad G^S_s = \frac{A^S_l}{Cf_s} = \frac{30\si{\giga\watt}}{0.12} = \rd{\res}\si{\giga\watt}$$
+ $$\Rightarrow \quad G^S_s = \frac{A^S_l}{c_s} = \frac{30\si{\giga\watt}}{0.12} = \rd{\res}\si{\giga\watt}$$
 
  % (b)
- \begin{shaded}\item For a system to work, generation must match demand not on average but at each location and each point in time. Storages are one way of ensuring this constraint with variable renewable generation. What is the amount of store and dispatch power capacity $G_{st,store}=\max(-\Delta(t))$ and $G_{st,dispatch} = \max \Delta(t)$ the storage units must have in the North and in the South to account for the mismatch $\Delta(t)=L(t)-G_{w/s}\cdot g_{w/s}(t)$?\end{shaded}
+ \begin{shaded}\item For a system to work, generation must match demand not on average but at each location and each point in time. Storages are one way of ensuring this constraint with variable renewable generation. What is the amount of store and dispatch power capacity $G_{st,store}=\max(-\Delta(t))$ and $G_{st,dispatch} = \max \Delta(t)$ the storage units must have in the North and in the South to account for the mismatch $\Delta(t)=L(t)-G\cdot g(t)$?\end{shaded}
 
  In the North:
 
  \begin{align*}
   G_{s,storage,dispatch}^N & = \max ( \pm \Delta^N(t))                                                              \\
-                           & = \max (\pm [L^N(t) - G^N_w \cdot G^N_w(t)])                                    \\
-                           & = \max (\pm [L^N(t) - \frac{A^N_l}{Cf_w}\cdot Cf_w\cdot (1+A_w \sin \omega_w t)]) \\
+                           & = \max (\pm [L^N(t) - G^N_w \cdot g^N_w(t)])                                    \\
+                           & = \max (\pm [L^N(t) - \frac{A^N_l}{c_w}\cdot c_w\cdot (1+A_w \sin \omega_w t)]) \\
                            & = \max (\pm [L^N(t) - A^N_l + A^N_l A_w \sin \omega_w t)])                      \\
                            & = \max (\pm [A^N_l A_w \sin \omega_w t)])                                         \\
                            & = A^N_l A_w = 0.9 \cdot 20 \si{\giga\watt} = 18 \si{\giga\watt}
@@ -190,15 +189,15 @@ For now, assume no power exchange between the regions and that the stores are lo
 
  \begin{align*}
   G_{s,storage,dispatch}^S & = \max ( \pm g_s^S(t))                                                              \\
-                           & = \max (\pm [L^S(t) - G^S_s \cdot G^S_s(t)])                             \\
-                           & = \max (\pm [L^S(t) - \frac{A^S_l}{Cf_s}\cdot Cf_s\cdot (1+A_s \sin \omega_s t)]) \\
+                           & = \max (\pm [L^S(t) - G^S_s \cdot g^S_s(t)])                             \\
+                           & = \max (\pm [L^S(t) - \frac{A^S_l}{c_s}\cdot c_s\cdot (1+A_s \sin \omega_s t)]) \\
                            & = \max (\pm [L^S(t) - A^S_l + A^S_l A_s \sin \omega_s t)])                      \\
                            & = \max (\pm [A^S_l A_s \sin \omega_s t)])                                         \\
                            & = A^S_l A_s = 1.0 \cdot 30 \si{\giga\watt} = 30 \si{\giga\watt}
  \end{align*}
 
  % (c)
- \begin{shaded}\item What is the amount of energy capacity one needs in the North and in the South?
+ \begin{shaded}\item What is the amount of energy capacity $E_{st}$ one needs for either storage in the North and in the South? The energy capacity is given by
  	
  	\begin{equation*}
  	E_{st} = \max_t e_{st}(t) = \max_t \int_{0}^{t} -\Delta(t') \;\mathrm{d}t'
@@ -229,7 +228,7 @@ For now, assume no power exchange between the regions and that the stores are lo
  \end{equation*}
 
  % (d)
- \begin{shaded}\item Should they choose hydrogen or battery storages? And how much would it cost them? Is the South or the North paying more for their energy?\end{shaded}
+ \begin{shaded}\item Should they choose hydrogen or battery storages? And how much would it cost them? Is the South or the North paying more for their energy? Disregard losses!\end{shaded}
 
  \textbf{In the North:}
 
@@ -240,7 +239,7 @@ For now, assume no power exchange between the regions and that the stores are lo
  The minimal (lossless) corresponding cost to supply constant demand by using hydrogen as storage technology are
 
  \begin{align*}
-  P_h^N & = 750 \text{\EUR{}}\si{\per\kilo\watt} \cdot G_{storage,dispatch}^N + 10 \text{\EUR{}}\si{\per\kilo\watt\hour} \cdot E_s^N \\
+  P_h^N & = 750 \text{\EUR{}}\si{\per\kilo\watt} \cdot G_{storage,dispatch}^N + 10 \text{\EUR{}}/\text{kWh} \cdot E_{st}^N \\
         & = 13.5 \cdot 10^9 \eur + 10 \cdot 10^9 \eur                                                                                  \\
         & = 23.5 \cdot 10^9 \eur
  \end{align*}
@@ -248,18 +247,18 @@ For now, assume no power exchange between the regions and that the stores are lo
  The minimal (lossless) corresponding cost to supply constant demand by using batteries as storage technology are:
 
  \begin{align*}
-  P_b^N & = 300 \text{\EUR{}}\si{\per\kilo\watt} \cdot G_{storage,dispatch}^N + 200 \text{\EUR{}}\si{\per\kilo\watt\hour} \cdot E_s^N \\
+  P_b^N & = 300 \text{\EUR{}}\si{\per\kilo\watt} \cdot G_{storage,dispatch}^N + 200 \text{\EUR{}/kWh} \cdot E_{st}^N \\
         & = 5.4 \cdot 10^9 \eur + 200 \cdot 10^9 \eur                                                                                   \\
         & = 205.4 \cdot 10^9 \eur
  \end{align*}
 
  The minimal (lossless) total system cost using hydrogen storages accumulates to
 
- $$P_w^N + P_h^N = 104 \cdot 10^9\eur \leq P_{w+h}^N$$
+ $$P_{w+h}^N = P_w^N + P_h^N = 104 \cdot 10^9\eur$$
 
  whereas the system cost using batteries are
 
- $$P_w^N + P_b^N = 285 \cdot 10^9\eur \leq P_{w+b}^N.$$
+ $$P_{w+b}^N = P_w^N + P_b^N = 285 \cdot 10^9\eur$$
 
  Thus, the North should choose hydrogen storages.\\~\\
 
@@ -272,7 +271,7 @@ For now, assume no power exchange between the regions and that the stores are lo
  The minimal (lossless) corresponding cost to supply constant demand by using hydrogen as storage technology are
 
  \begin{align*}
-  P_h^S & = 1200 \text{\EUR{}}\si{\per\kilo\watt} \cdot G_{storage,dispatch}^S + 10 \text{\EUR{}}\si{\per\kilo\watt\hour} \cdot E_{st}^S \\
+  P_h^S & = 1200 \text{\EUR{}}\si{\per\kilo\watt} \cdot G_{storage,dispatch}^S + 10 \text{\EUR{}/kWh} \cdot E_{st}^S \\
         & = 22.5 \cdot 10^9 \eur + 2.3 \cdot 10^9 \eur                                                                                  \\
         & = 24.8 \cdot 10^9 \eur
  \end{align*}
@@ -280,18 +279,18 @@ For now, assume no power exchange between the regions and that the stores are lo
  The minimal (lossless) corresponding cost to supply constant demand by using batteries as storage technology are:
 
  \begin{align*}
-  P_b^S & = 300 \text{\EUR{}}\si{\per\kilo\watt} \cdot G_{storage,dispatch}^S + 200 \text{\EUR{}}\si{\per\kilo\watt\hour} \cdot E_{st}^S \\
+  P_b^S & = 300 \text{\EUR{}}\si{\per\kilo\watt} \cdot G_{storage,dispatch}^S + 200 \text{\EUR{}/kWh} \cdot E_{st}^S \\
         & = 9 \cdot 10^9 \eur + 46 \cdot 10^9 \eur                                                                                      \\
         & = 54 \cdot 10^9 \eur
  \end{align*}
 
  The minimal (lossless) total system cost using hydrogen storages accumulates to
 
- $$P_s^S + P_h^S = 175 \cdot 10^9\eur \leq P_{s+h}^S$$
+ $$P_{s+h}^S = P_s^S + P_h^S = 175 \cdot 10^9\eur$$
 
  whereas the system cost using batteries are
 
- $$P_s^S + P_b^S = 204 \cdot 10^9\eur \leq P_{s+b}^S$$
+ $$P_{s+b}^S = P_s^S + P_b^S = 204 \cdot 10^9\eur$$
 
  Thus, the South should also choose hydrogen storages.\\~\\
 
@@ -299,19 +298,21 @@ For now, assume no power exchange between the regions and that the stores are lo
 
  Without taking losses into account, both regions should choose hydrogen storages. Overall, the North can provide electricity at a lower rate than the South:
 
- $$P_{w+h}^N \geq \frac{104 \cdot 10^9 \eur}{20 \si{\giga\watt}} = 5 \cdot 10^9 \eur \si{\per\giga\watt}$$
+ $$P^N = \frac{104 \cdot 10^9 \eur}{20 \si{\giga\watt}} = 5 \cdot 10^9 \eur \si{\per\giga\watt}$$
 
- $$P_{s+h}^S \geq \frac{175 \cdot 10^9 \eur}{30 \si{\giga\watt}} = 6 \cdot 10^9 \eur \si{\per\giga\watt}$$
+ $$P^S = \frac{175 \cdot 10^9 \eur}{30 \si{\giga\watt}} = 6 \cdot 10^9 \eur \si{\per\giga\watt}$$
+
+ Note that we can consider energy as $\si{\giga\watt}$ as we have a constant load! Otherwise, we would have to pay more attention!
 
  % (e)
  \begin{shaded}
-  \item What do you imagine would change if you considered the storage losses given in Table 1 in your results (a)-(d)? Support your statement with a graphical illustration.
+  \item What do you imagine would change if you considered the storage losses given in Table 1 in your results (a)-(d)?
  \end{shaded}
 
- To compensate for the energy losses wind and solar capacity $G_{w/s}$, store and dispatch power capacities $G_{storage,dispatch}$ and storage energy capacities $E_{st}$ have to increase.
+ To compensate for the energy losses wind and solar capacity $G_{w/s}$, store and dispatch power capacities $G_{storage,dispatch}$ and storage energy capacities $E_{st}$ have to increase. Since hydrogen storage has lower efficiencies than batteries, the optimal choice (particularly in the South) might change. You can prove that by taking into account the losses in your preceding calculations.
 
  % (f)
- \begin{shaded}\item Now we lift the restriction against transmission and allow them to bridge their 500 km separation with a transmission line. Estimate the cost-optimal technology mix by assuming wind energy in the North is only stored in the North and solar energy in the South is likewise only stored in the South! What would happen if you dropped that assumption?\end{shaded}
+ \begin{shaded}\item Now we lift the restriction against transmission and allow the two regions to bridge their 500 km separation with a transmission line. Estimate the cost-optimal technology mix by assuming wind energy in the North is only stored in the North and solar energy in the South is likewise only stored in the South! What would happen if you dropped that assumption?\end{shaded}
 
  Because $P_{w+h}^N < P_{w+h}^S$ there will be energy exports from North to South:
 
@@ -327,7 +328,7 @@ For now, assume no power exchange between the regions and that the stores are lo
           & = \frac{E^N (P_{w+h}^N - P_{s+h}^S + 400 \eur \si{\per\kilo\watt}) + 50 \si{\giga\watt} (P_{s+h}^S - 200 \eur \si{\per\kilo\watt})}{E^N + E^S} \\
  \end{align*}
 
- Now, minimising the term for a choice of $E^N$ will yield
+ Now, minimising the term for a choice of $E^N$ and filling in the costs calculated in the preceding tasks will yield
 
  \begin{align*}
   E^N & = 50 \si{\giga\watt}