
\newcommand{\thistitle}{Lecture 11: Cost Recovery \& Renewables}

\input{title.tex}


\section{Cost Recovery in Long-Term Equilibrium}


\begin{frame}{Cost recovery in the long-term equilibrium}

  We will demonstrate that all players in the power network
  (generators, storage and network operators) \alert{recover their
    costs} in theory with perfect markets in long-term equilibrium and
  linear (actually convex) costs.

  If they didn't cover their costs, they would leave the market.

  If they made a profit, others would join the market and competition would reduce the profit.

  This is a direct consequence of the investment equations we considered in Lecture 11.

  We will discuss at the end why this \alert{does not work in real life}, i.e. the consequences of imperfect markets, frictions  and non-convexities.
\end{frame}


\begin{frame}{Single node with optimised capacities and dispatch}

  Suppose we have generators labelled by $s$ at a single node with \alert{marginal costs} $o_s$ arising from each unit of
  production $g_{s,t}$ and \alert{capital costs} $c_s$ that arise from fixed costs
  regardless of the rate of production (such as the investment in building
  capacity $G_s$).   For a variety of demand values $d_t$ in representative situation $t$ we optimise the total annual system costs
  \begin{equation*}
    \min_{\{g_{s,t}\},\{G_s\}}  \left[\sum_{s}c_s G_s +  \sum_{s,t} o_{s} g_{s,t} \right]
  \end{equation*}
  such that (NB: we now include availability hourly capacity factor $G_{s,t} \in [0,1]$ for wind/solar)
  \begin{align*}
    \sum_s g_{s,t} & = d_t  \hspace{1cm}\leftrightarrow\hspace{1cm} \l_t \\
    - g_{s,t}  & \leq  0  \hspace{1cm}\leftrightarrow\hspace{1cm} \ubar{\m}_{s,t} \\
    g_{s,t} - G_{s,t} G_s  & \leq 0  \hspace{1cm}\leftrightarrow\hspace{1cm} \bar{\m}_{s,t}
  \end{align*}

  We will now show using KKT that every generator exactly recovers their costs if the market price is set by $\l_t^*$, the \alert{no/zero profit rule}.
\end{frame}


\begin{frame}{Single node with optimised capacities and dispatch}

  Take the costs of generator $s$ at the optimal point:
  \begin{align*}
    c_s G_s^* +  \sum_{t}  o_{s} g_{s,t}^*
  \end{align*}

  Use stationarity for $g_{s,t}^*$
  \begin{align*}
        0 = \frac{\d \cL}{\d g_{s,t}}  = o_s - \l_t^* - \bar{\m}^*_{s,t} + \ubar{\m}^*_{s,t}
  \end{align*}
  to substitute for $o_s$ in the costs:
  \begin{align*}
    c_s G_s^* +  o_{s} \sum_{t}  g_{s,t}^* & =  c_s G_s^* + \sum_{t}( \l_t^* + \bar{\m}^*_{s,t} - \ubar{\m}^*_{s,t}) g_{s,t}^*
  \end{align*}
\end{frame}



\begin{frame}{Single node with optimised capacities and dispatch}

  Next use complementarity
  \begin{align*}
    \bar{\m}^*_{s,t}(g_{s,t}^* -  G_{s,t} G_s^*) & = 0 \\
    \ubar{\m}^*_{s,t}g_{s,t}^* & = 0
  \end{align*}
  to substitute for the terms $\m^*g_{s,t}^*$
  \begin{align*}
    c_s G_s^* +  o_{s} \sum_{t}  g_{s,t}^* & =  c_s G_s^* + \sum_{t}( \l_t^* + \bar{\m}^*_{s,t} - \ubar{\m}^*_{s,t}) g_{s,t}^* \\
    & = c_s G_s^* +  \sum_{t} \l_t^* g_{s,t}^* + \sum_t  \bar{\m}^*_{s,t}  G_{s,t} G_s^*
  \end{align*}
  Finally use stationarity for the capacity $G_s^*$
    \begin{align*}
        0 = \frac{\d \cL}{\d G_{s}}  = c_s + \sum_t \bar{\m}^*_{s,t}  G_{s,t}
  \end{align*}
  to get \alert{full cost recovery} from the market price:
  \begin{align*}
    c_s G_s^* +  o_{s} \sum_{t}  g_{s,t}^* =  \sum_{t} \l_t^* g_{s,t}^*
  \end{align*}

\end{frame}


\begin{frame}{Network of nodes with optimised capacities and dispatch}

  Suppose now we have a network of nodes $i$ connected by lines $\ell$.

  Our investment problem is now:
    \begin{equation*}
    \min_{\{g_{i,s,t}\},\{G_{i,s}\}, f_{\ell,t}, F_\ell}  \left[\sum_{i,s}c_s G_{i,s} +  \sum_{i,s,t} o_{s} g_{i,s,t}  + \sum_\ell c_\ell F_\ell \right]
  \end{equation*}
  such that
  \begin{align*}
    \sum_s g_{i,s,t} - \sum_\ell K_{i\ell}f_{\ell,t} & = d_{i,t}  \hspace{1cm}\leftrightarrow\hspace{1cm} \l_{i,t} \\
    - g_{i,s,t}  & \leq  0  \hspace{1cm}\leftrightarrow\hspace{1cm} \ubar{\m}_{i,s,t} \\
    g_{i,s,t} - G_{i,s,t}G_{i,s}  & \leq 0  \hspace{1cm}\leftrightarrow\hspace{1cm} \bar{\m}_{i,s,t} \\
    f_{\ell,t} - F_\ell & \leq 0  \hspace{1cm}\leftrightarrow\hspace{1cm} \bar{\m}_{\ell,t} \\
        - f_{\ell,t} - F_\ell & \leq 0  \hspace{1cm}\leftrightarrow\hspace{1cm} \ubar{\m}_{\ell,t}
  \end{align*}


\end{frame}

\begin{frame}{Network of nodes with optimised capacities and dispatch}

  The cost recovery of the generators follows through exactly as before.

  What about the costs $c_\ell F_\ell^*$ of each transmission line?


  Use stationarity for the capacity $F_{\ell}^*$:
    \begin{align*}
        0 = \frac{\d \cL}{\d F_{\ell}}  = c_\ell   +\sum_t \bar{\m}^*_{\ell,t}  + \sum_t \ubar{\m}^*_{\ell,t}
    \end{align*}
    to get
    \begin{align*}
      c_\ell F_\ell^* = - F_\ell^* \sum_t \left[ \ubar{\m}^*_{\ell,t} +  \bar{\m}^*_{\ell,t} \right]
    \end{align*}
   `At the optimal point, fixed costs equal the sum of marginal
    benefits of expanding the line at each time.'


  Next use complementarity for the flows $\bar{\m}_{\ell,t}^* ( f_{\ell,t}^* - F_\ell^*) = 0$ and $\ubar{\m}_{\ell,t}^* ( - f_{\ell,t}^* - F_\ell^*) = 0$ to get
    \begin{align*}
      c_\ell F_\ell^* =  -\sum_t \left[ \bar{\m}^*_{\ell,t} -  \ubar{\m}^*_{\ell,t} \right] f_{\ell,t}^*
    \end{align*}

\end{frame}


\begin{frame}{Network of nodes with optimised capacities and dispatch}

  Finally use stationarity for each $f_{\ell,t}^*$:
    \begin{align*}
        0 = \frac{\d \cL}{\d f_{\ell,t}}  = \sum_i \l_{i,t}^*K_{i\ell}  - \bar{\m}^*_{\ell,t} + \ubar{\m}^*_{\ell,t}
  \end{align*}
    to substitute for the $\m^*$:
    \begin{align*}
      c_\ell F_\ell^*  &=  -\sum_t \left[ \bar{\m}^*_{\ell,t} -  \ubar{\m}^*_{\ell,t}   \right] f_{\ell,t}^* \\
      & =-\sum_t \sum_i \l_{i,t}^*K_{i\ell} f_{\ell,t}^*
    \end{align*}
    $ -\sum_i \l_{i,t}^*K_{i\ell} f_{\ell,t}^*$ is nothing other than
    the \alert{congestion rent} on line $\ell$ at time $t$, i.e. the flow $f_{\ell,t}^*$ multiplied by the price difference across the line $\sum_i \l_{i,t}^*K_{i\ell}$.

    At the long-term equilibrium, the network operator covers the costs of the line exactly with the congestion rent. The optimum requires congestion at least some of the time!
\end{frame}

\begin{frame}{Storage cost recovery}

  The proof for storage is a bit grizzly, but you can find it in \hrefc{https://arxiv.org/abs/2002.05209}{this paper}.

  The result is for storage unit $r$ at node $i$:
    \begin{align*}
      & c_{r,\textrm{discharge}} G_{i,r,\textrm{discharge}}^* +        c_{r,\textrm{charge}} G_{i,r,\textrm{charge}}^* +  c_{r,\textrm{energy}}  E^*_{i,r} \\
      & = \sum_t \lambda_t^* g_{i,r,t,\textrm{discharge}}^* - \sum_t \lambda_t^* g_{i,r,t,\textrm{charge}}^*
    \end{align*}

    All the costs, including the costs of the electricity to charge the storage, are recovered when the storage discharges, thereby selling its electricity to the market.

    At the equilibrium, the profits from arbitrage in the market (`buy low, sell high') exactly cover the investment costs.

    From KKT we can deduce the optimal levels at which storage should bid into the market as demand or offer as supply (more later maybe).


\end{frame}


\begin{frame}{Adding a CO2 constraint for a single node}

If we add a constraint on the total \co2 emissions
\begin{equation*}
  \sum_{s,t} \frac{\varepsilon_s}{\eta_{s}} g_{s,t} \leq \textrm{CAP}  \leftrightarrow \m_{CO2}
\end{equation*}
where $\varepsilon_s$ are the specific \co2 emissions of technology $s$ per fuel
thermal energy and $\eta_s$ is the efficiency of the generator
(i.e. the ratio between thermal energy and electrical energy). CAP
could correspond to e.g. political targets for \co2 reduction.

All that changes is stationarity for the generator
\begin{align*}
  0 =     \frac{\d \cL}{\d g_{s,t}} =   o_{s}-\l_t^* + \ubar{\m}^*_{s,t} - \bar{\m}^*_{s,t} - \m_{CO2}^* \frac{\varepsilon_s}{\eta_{s}}
\end{align*}
and now for each generator cost recovery becomes
\begin{align*}
  c_s G_s^* +  o_{s}\sum_{t} g_{s,t}^*
  & = \sum_{t} \l_t^* g_{s,t} +\m_{CO2}^* \sum_{t}  \frac{\varepsilon_s}{\eta_{s}} g_{s,t}^*
\end{align*}

This shows nicely the duality for exchanging the CO2 constraint for a CO2 price $o_{s} \to o_{s} - \m^*_{CO2} \frac{\varepsilon_s}{\eta_{s}} $ (remember $\m^*_{CO2} \leq 0$ for minimisation problems).

\end{frame}



\begin{frame}{Introduction to Lagrangian Relaxation}

  This switching between costs and constraints is a special case of \alert{Lagrangian relaxation}.

  Consider the optimisation problem:
  \begin{equation*}
  \max_{x} f(x)
\end{equation*}
[$x = (x_1, \dots x_k)$] subject to some \alert{constraints} within $\R^k$:
\begin{align*}
  g_i(x) & = c_i \hspace{1cm}\leftrightarrow\hspace{1cm} \l_i \hspace{1cm} i = 1,\dots n \\
  h_0(x) & \leq d_0  \hspace{1cm}\leftrightarrow\hspace{1cm} \m_0 \\
  h_j(x) & \leq d_j \hspace{1cm}\leftrightarrow\hspace{1cm} \m_j \hspace{1cm} j = 1,\dots m
\end{align*}

\end{frame}



\begin{frame}{Introduction to Lagrangian Relaxation}

  Now consider the related problem where $\tilde\mu_0$ is fixed to a constant:
  \begin{equation*}
  \max_{x} f(x) - \tilde\mu_0 (h_0(x) - d_0)
\end{equation*}
[$x = (x_1, \dots x_k)$] subject to some \alert{constraints} within $\R^k$:
\begin{align*}
  g_i(x) & = c_i \hspace{1cm}\leftrightarrow\hspace{1cm} \l_i \hspace{1cm} i = 1,\dots n \\
  h_j(x) & \leq d_j \hspace{1cm}\leftrightarrow\hspace{1cm} \m_j \hspace{1cm} j = 1,\dots m
\end{align*}

We have \alert{relaxed} the problem by removing one of the constraints.

You can show that the new problem has the same solution as the old $(x^*,\l^*,\m^*)$ if we fix the constant $\tilde\mu_0 = \mu_0^*$ by comparing the KKT stationarity constraints of the two problems.

We have lifted the constraint into the objective function, where it penalises solutions with $h_0(x) > d_0$.

In general, if we don't know $\mu_0^*$ beforehand, we can iteratively solve to find it.

Often it can be easier to solve the relaxed problem.


\end{frame}


\begin{frame}{Fundamental Welfare Theorem is Lagrangian Relaxation}

  Consider the maximisation of total welfare:
  \begin{align*}
    \max_{\{d_b\}, \{g_s\}} f(\{d_b\}, \{g_s\}) = \left[ \sum_b U_b (d_b)  -  \sum_s C_s (g_s) \right]
  \end{align*}
  subject to the balance constraint:
  \begin{align*}
    g(\{d_b\}, \{g_s\}) = \sum_b d_b -  \sum_s g_s  = 0 \hspace{1cm} \leftrightarrow \hspace{1cm} \l
  \end{align*}
  Now let's relax the constraint:
  \begin{align*}
    \max_{\{d_b\}, \{g_s\}} f(\{d_b\}, \{g_s\}) = \left[ \sum_b U_b (d_b)  -  \sum_s C_s (g_s) - \tilde\lambda ( \sum_b d_b -  \sum_s g_s) \right]
  \end{align*}
  This problem is \alert{separable} and \alert{decomposes} into separate problems for each $d_b$:
  \begin{equation*}
    \max_{d_b} \left[ U_b(d_b) - \tilde\lambda d_b\right]
  \end{equation*}
  and for each $g_s$:
  \begin{equation*}
    \max_{g_s} \left[ \tilde\lambda g_s  -  C_s(g_s)\right]
  \end{equation*}

\end{frame}

\begin{frame}{Grit in the machine for generation 1/2}

  Several factors make this theoretical picture quite different in reality:
  \begin{itemize}
  \item Generation investment is \alert{lumpy} i.e. you can often only
    build power stations in e.g. 500~MW blocks, not at any size
  \item Some older generators have \alert{sunk costs}, i.e. costs which have been incurred once and investments that cannot be reversed $\Rightarrow$ they have no incentive to withdraw from the market if they are no longer cost-optimal in the long-term
  \item \alert{Returns on scale} in building plant are not taken into account (specific capital costs [\euro/kW] going down would be a non-convexity; we did everything linear)
  \item Site-specific concerns ignored (e.g. lignite might need to be near a mine and have limited capacity)
  \item Variability of production for wind/solar ignored
    \item There is considerable  uncertainty given load/weather conditions during a year, which makes investment risky; economic downturns reduce electricity demand
  \end{itemize}

\end{frame}



\begin{frame}{Grit in the machine for generation 2/2}

  Several factors make this theoretical picture quite different in reality:
  \begin{itemize}
    \item Fuel cost fluctuations, building delays which cost money
  \item Risks from third-parties:  Changing costs of other generators, political risks (\co2 taxes,  Atomausstieg, subsidies for renewables, price caps)
  \item Political or administrative constraints on wholesale energy
    prices may prevent prices from rising high enough for long enough
    to justify generation investment (``Missing Money Problem'')
  \item  Lead-in time for planning and building, behaviour of others, boom-and-bust investment cycles resulting from periods of under- and over-investment in capacity
    \item Exercise of \alert{market power} - single companies can dominate the market and alter the price by changing their supply bids - they are no longer price takers
  \end{itemize}

\end{frame}


\begin{frame}{Episodes of High Prices are an Essential Part of an Energy-Only Market}

  In an energy-only market (in which generators are only compensated
  for the energy they produce), the wholesale spot price must at times
  be higher than the variable cost of the highest-variable-cost
  generating unit in the market. Episodes of high prices and/ or price
  spikes are not in themselves evidence of market power or evidence of
  market failure.

  However, there may be political or administrative restrictions on
  prices going to very high levels (i.e. consumer protection, concerns about market abuse).




\end{frame}


\begin{frame}{Today's market does not (usually) have enough times of high prices}

  This makes it hard for e.g. gas generators to make back their
  costs. Day ahead spot market prices in 2016 in Germany-Austria bidding zone:

  \centering
  \includegraphics[width=8cm]{germany-2016-price-duration}

  \raggedright
  Gas generators can bid into other markets, such as the intra-day or
  reserve power markets, or provide redispatch services.



\end{frame}


\begin{frame}{Market prices from highly renewable simulations}

  In our simulations for high renewable penetrations (taken from \hrefc{https://arxiv.org/abs/1704.05492}{this paper}), the theory does
  however work:

  \centering
  \includegraphics[width=8.5cm]{benefits-lmp-lvopt.pdf}
  %data from https://zenodo.org/record/804338
  %supplementary_data_benefits_of_cooperation/results/
  %diw2030-CO0-T1_8761-wWsgrpHb-LVNone_c0_base_diw2030_solar1_7_angles_N1_5-2017-02-22-17-08-45/buses-marginal_price.csv

  \raggedright
  Prices are zero around a quarter of the time, but spike above 10,000 \euro/MWh in some hours.

\end{frame}

\begin{frame}{Price cap}

  Some markets implement a maximum market price cap (MPC), which may be below the Value of Lost Load (VoLL)
  ($V$ for the inelastic case).

  In the Eastern Australian National Electricity Market (NEM), a MPC
  of A\$15,000/MWh (\euro~9,300/MWh) for the 2020-2021 financial year
  is set, corresponding to the price automatically triggered when AEMO
  directs network service providers to interrupt customer supply in
  order to keep supply and demand in the system in balance.

  \centering

\begin{columns}[T]
\begin{column}{6.5cm}



  \includegraphics[width=6.5cm]{price-cap}
\end{column}
\begin{column}{5cm}

  The Electric Reliability Council of Texas (ERCOT) has an energy only market with an MPC of \$9000/MWh.

  \vspace{.5cm}

  MPC can introduce distortions which make it difficult for some generators to recover costs.

\end{column}

\end{columns}


  \source{Biggar and Hesamzadeh, 2014}
\end{frame}




\begin{frame}{Capacity Remuneration Mechanisms vary widely}

  Capacity Remuneration Mechanisms (CRM) in 2019 in Europe and the US:

  \centering
  \includegraphics[height=6.8cm]{cpm-bublitz-2019.jpg}


  \source{\href{https://doi.org/10.1016/j.eneco.2019.01.030}{Bublitz et al, 2019}}

\end{frame}

\begin{frame}{Features of Transmission Investment 1/2}


  \begin{enumerate}
  \item \alert{Rationale for transmission}: Load and generation do not coincide in location at all times, so electricity must be transported for some of the time.
  \item \alert{Transmission is a natural monopoly}: Like railways or water provision, it is unlikely that a parallel electricity network would be built, given cost and limits on installing infrastructure due to space and public acceptance. Natural monopolies require \alert{regulation}.
  \item \alert{Transmission is a capital-intensive business}:
    Transmitting electric power securely and efficiently over long
    distances requires large amounts of equipment (lines,
    transformers, etc.) which dominate costs compared to the operating
    costs of the grid. Making good investment decisions is thus the
    most important aspect of running a transmission company.
  \end{enumerate}



\end{frame}


\begin{frame}{Features of Transmission Investment 2/2}


  \begin{enumerate}
  \item \alert{Transmission assets have a long life}: Most
    transmission equipment is designed for an expected life ranging
    from 20 to 40 years or even longer (up to 60-80 years). A lot can
    change over this time, such as load behaviour and generation costs
    and composition.
  \item \alert{Transmission investments are irreversible}: Once a transmission line has been built, it
  cannot be redeployed in another location where it could be used more profitably.
  \item \alert{Transmission investments are lumpy}: Manufacturers sell transmission equipment in
    only a small number of standardized voltage and MVA ratings. It is therefore often not
    possible to build a transmission facility whose rating exactly matches the need.
    \item \alert{Economies of scale}: Transmission investment more
      proportional to length (costs of rights of way, terrain, towers,
      which dominate costs) than to power rating (which depends only
      on conductoring, which is cheap).

  \end{enumerate}


\end{frame}



\section{Integrating Renewables in Power Markets}


\begin{frame}{Characteristics of Renewables}

  \begin{itemize}
  \item \alert{Variability}: Their production depends on weather (wind speeds for wind, insolation for solar and precipitation for hydroelectricity)
  \item \alert{No Upwards Controllability}: Variable Renewable Energy (VRE) like wind and solar can only reduce their output; raising is hard
  \item \alert{No Long-Term Forecastability}: Although short-term forecasting is improving steadily
  \item \alert{Low Marginal Cost} (no fuel costs)
  \item \alert{High Capital Cost}
  \item \alert{No Direct Carbon Dioxide Emissions} (but some indirect ones from manufacturing)
  \item \alert{Small unit size} (onshore wind turbine is 3-5~MW; coal/nuclear is 1000~MW)
  \item \alert{Somewhat Decentralised Distribution} for some VRE (e.g. solar panels on household rooves); offshore is however very centralised
    \item \alert{Provision of system services}: Increasing
  \end{itemize}

\end{frame}

\begin{frame}{RE Levelised Cost in 2019 USD/kWh  (already at/below fossil fuels)}

  \centering
      % left bottom right top
  \includegraphics[trim=2.6cm 2cm 0 0cm,width=11cm,clip=true]{irena-2020.jpg}

  \source{\hrefc{https://www.irena.org/publications/2020/Jun/Renewable-Power-Costs-in-2019}{IRENA Renewable Generation Costs in 2019}}
\end{frame}


\begin{frame}{RE Forecasting}

  Just like the weather on which it depends, Variable RE (wind and
  solar) production can be forecast in advance.   (Shaded area is the uncertainty.)

  \centering
  \includegraphics[width=9.5cm]{wind_farm_forecast_2}
\end{frame}



\begin{frame}{RE Forecasting}

  Like the weather, the forecast in the short-term (e.g. day ahead) is
  fairly reliable, particularly for wind, but for several days ahead
  it is less useful. In addition, it is subject to more uncertainty
  than the load. For example, fog and mist is very local, hard to
  predict, and has a big impact on solar power production.


  This makes scheduling more challenging and has led to the introduction of
  more regular auctions in the intraday market.

  Forecasting has also become a big business.

\end{frame}


\begin{frame}{Effect on effective `residual' load curve}

  Since RE often have priority feed-in (i.e. network operators are obliged to take their power), we often subtract the RE production from the load to get the \alert{residual load}, plotted here as a demand-duration-curve.

  \centering
  \includegraphics[width=10cm]{residual-load}

  \source{Biggar and Hesamzadeh, 2014}
\end{frame}


\begin{frame}{Residual load curve and screening curve}



\begin{columns}[T]
\begin{column}{6.5cm}



  \centering
    \begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[width=5.4cm]{residual-screening_curve-clean}};
\draw (0.3,4.9) node{$c_2$};
\draw (0.3,6.5) node{$c_1$};
  \end{tikzpicture}

\end{column}
\begin{column}{6cm}

  The residual load must be met by conventional generators.

  The changed duration curve interacts differently with the screening curve, so that we may require less baseload generation and peaking plant and more load shedding, depending on the shape of the curve.

  In some markets, there is increased demand for medium-peaking plant.

\end{column}



\end{columns}


  \centering

  \source{Biggar and Hesamzadeh, 2014}
\end{frame}


\begin{frame}
  \frametitle{Effect of varying renewables: fixed demand, no wind}

  \centering
  \includegraphics[width=9cm]{demand-supply-no_wind}

\end{frame}



\begin{frame}
  \frametitle{Effect of varying renewables: fixed demand, 35~GW wind}

  \centering
  \includegraphics[width=9cm]{demand-supply-with_wind}

\end{frame}


\begin{frame}
  \frametitle{Spot market price development}

  As a result of so much zero-marginal-cost renewable feed-in, spot
  market prices steadily decreased until 2016. This is called the
  \alert{Merit Order Effect}.  Since then prices have been rising due
  to rising gas and CO$_2$ prices.


  \centering
  \includegraphics[width=10cm]{strompreis-2019.png}

  \source{\hrefc{https://www.agora-energiewende.de/fileadmin2/Projekte/2019/Jahresauswertung_2019/171_A-EW_Jahresauswertung_2019_WEB.pdf}{Agora Energiewende Jahresauswertung 2019}}
\end{frame}




\begin{frame}
  \frametitle{Merit Order Effect}

  To summarise:
  \begin{itemize}
  \item Renewables have zero marginal cost
  \item As a result they enter at the bottom of the merit order, reducing the price at which the market clears
  \item This pushes non-CHP gas and hard coal out of the market
  \item This is unfortunate, because among the fossil fuels, gas is the most flexible and produces lower \co2
    per MWh\el~than e.g. lignite
  \item It also reduces the profits that nuclear and lignite make
  \item Will there be enough backup power plants for times with no wind/solar?
  \end{itemize}

  This has led to lots of political tension, but has been counteracted in recent years by the rising CO$_2$ price.

\end{frame}


\begin{frame}{Market value}

  VRE have the property that they cannibalise their own market, by
  pushing down prices when lots of other VRE are producing.

  We define the \alert{market value} of a technology by the average
  market price it receives when it produces, i.e.

  \begin{equation*}
    MV_s = \frac{\sum_{t} \l^*_t g_{s,t}}{\sum_{t} g_{s,t}}
  \end{equation*}

  We can compare this to the average market price,
    defined either as the simple average $\frac{1}{T} \sum_t \l^*_t$ or the demand-weighted average $ \frac{\sum_{t} \l^*_td_t}{\sum_{t}d_t}$.

\end{frame}



\begin{frame}{Historic market values in Germany}

  \centering

  \includegraphics[width=11cm]{market_value-historical}



  \source{\hrefc{https://doi.org/10.1016/j.eneco.2013.02.004}{Lion Hirth, 2013}}
\end{frame}

\begin{frame}{Market value at higher shares}


  At low shares of VRE the market value may be higher than the average market price (because for example, PV produces a midday when prices are higher than average), but as VRE share increases the market value goes down.

\begin{columns}[T]
\begin{column}{6.5cm}



  \centering

  \includegraphics[width=6cm]{market_value_us}


\end{column}
\begin{column}{4cm}

  The effect is particularly severe for PV, since the production is highly correlated; for wind smoothing prevents a steeper drop off. The bigger the catchment area, the longer wind preserves its market value.

  \source{Mills \& Wiser, 2014}
\end{column}



\end{columns}


\end{frame}



\begin{frame}{Market value mitigation}

To halt the drop in market value (and hence revenue for wind and
solar) we can use networks to do price arbitrage in space, storage to
do arbitrage in time, or introduce CO2 prices that push up the prices in times when fossil fuel
plants are running.

\begin{columns}[T]
\begin{column}{5.5cm}



  \centering

  \includegraphics[width=5cm]{market_value-networks}


\end{column}
\begin{column}{5.5cm}


  \centering

  \includegraphics[width=5cm]{market_value-storage}
\end{column}

\end{columns}


  \source{\hrefc{https://doi.org/10.1016/j.eneco.2013.02.004}{Lion Hirth, 2013}}

\end{frame}


\begin{frame}
  \frametitle{Market value from our 95\% renewable simulations}
  \begin{columns}[T]
    \begin{column}{8.3cm}
      \vspace{1cm}
\centering
  \includegraphics[width=8.3cm]{pre-7-market_value}
  %\includegraphics[width=6cm]{pre-2-costs_de_storage}
\end{column}
\begin{column}{4cm}
  \begin{itemize}
%\item   %Big reduction in curtailment
  \item Storage charges at low market prices and dispatches at high
    prices.
  \item Dispatchable power sources take advantage of high prices.
  \item Variable renewables get lower prices, but saved by storage, networks and high CO2 price.
  \end{itemize}
\end{column}
\end{columns}


\end{frame}

\begin{frame}
  \frametitle{Relation of LCOE to market value}

  From the first section we had for a perfect market in long-term equilibrium that all costs are recovered from market revenue:
  \begin{align*}
    c_s G_s^* +  o_{s} \sum_{t}  g_{s,t}^* =  \sum_{t} \l_t^* g_{s,t}^*
  \end{align*}
  If we divide both sides by the total yearly generation $\sum_{t} g_{s,t}^*$ then we get:
  \begin{align*}
    \frac{c_s G_s^* +  o_{s} \sum_{t}  g_{s,t}^*}{\sum_{t} g_{s,t}^*} =  \frac{\sum_{t} \l_t^* g_{s,t}^*}{\sum_{t} g_{s,t}^*}
  \end{align*}
  This is none other than the identity between the LCOE and market value:
  \begin{align*}
    LCOE = MV
  \end{align*}
  This \emph{only} holds in a perfect equilibrium. I.e. the
  equilibrium is found by increasing the penetration until the market
  value equals the LCOE.

  In reality the market is far from equilibrium: subsidies support
  technologies (with a longer-term view of pushing them down the
  learning curve), there are sunk costs for existing plants, excess
  capacity supported outside the energy-only market, etc.
\end{frame}

\begin{frame}{Market value: more details}

  For more details on market value, the zero profit rule and how market value is affected by CO$_2$ prices and VRE subsidies, see the paper

  \hrefc{https://arxiv.org/abs/2002.05209}{Decreasing market value of variable renewables can be avoided by policy action} (2020)

  This paper examines how to maintain market value even at high shares of wind and solar.


\end{frame}


\section{Value of hydrogen storage}

\begin{frame}
  \frametitle{Value of hydrogen}

  We can now use the machinery of shadow prices to explore the \alert{value of
    hydrogen} and how it relates to the value of electricity in a
  system with hydrogen storage.

  Suppose we have electrolysis with efficiency $\eta_e$ and a fuel cell or hydrogen turbine with efficiency $\eta_f$. They have respective market values in the electricity market of $MV_e$ and $MV_f$.

  The value of hydrogen is given by the KKT multiplier of the storage constraint
    \begin{align*}
    e_{t} = e_{t-1} + \eta_eg_{e,t} -  \eta_f^{-1} g_{f,t}   \hspace{1cm}\leftrightarrow\hspace{1cm}  \tilde \lambda_{t}
    \end{align*}
    where $e_t$ is the state of charge (amount of hydrogen at time $t$), $g_{e,t}$ is the power consumption of the electrolyser and $g_{f,t}$ is the electricity dispatch of the fuel cell or hydrogen turbine.

    $\tilde \lambda_{t}$ is the \alert{price/value of hydrogen} since
    it tells us the change in objective function if we increase
    hydrogen use in this hour.
\end{frame}



\begin{frame}
  \frametitle{Value of hydrogen turbine}

  The KKT stationarity for the discharge variable of the fuel cell or hydrogen turbine $g_{f,t}$ is
  \begin{equation*}
      0 =     \frac{\d \cL}{\d g_{f,t}} = \eta_f^{-1} \tilde \lambda_{t}^*  -\l_t^* + \ubar{\m}^*_{f,t} - \bar{\m}^*_{f,t} \quad \forall t
  \end{equation*}
  Note that this has exactly the same structure as a conventional generator with marginal cost $o_f =  \eta_f^{-1} \tilde \lambda_{t}^*$ based on a fuel cost $\tilde \lambda_t^*$. This sets how the storage bids into the electricity market.

  Now multiply this equation by $g_{f,t}^*$, sum over $t$, then divide by $\sum_t g_{f,t}^*$.
  \begin{equation*}
   0 =  \eta_f^{-1}\frac{ \sum_t \tilde \lambda_{t}^*g_{f,t}^*}{\sum_t g_{f,t}^*} - \frac{\sum_t\l_t^* g_{f,t}^*}{\sum_t g_{f,t}^*} + \frac{ \sum_t\ubar{\m}^*_{f,t}g_{f,t}^*}{\sum_t g_{f,t}^*} - \frac{ \sum_t\bar{\m}^*_{f,t}g_{f,t}^*}{\sum_t g_{f,t}^*}
  \end{equation*}
  The 1st term is the turbine-demand-averaged hydrogen price  $\langle \tilde \lambda_{t}^* \rangle_f$; the 2nd term is the market value of the turbine; the 3rd term vanishes by complementarity and by complementarity plus stationarity for $G_{f}$ we have $\sum_t\bar{\m}^*_{f,t}g_{f,t}^* = \sum_t\bar{\m}^*_{f,t}G_f  = - c_f G_f$. Rearranging
    \begin{equation*}
      MV_f =   \frac{c_f G_f}{\sum_t g_{f,t}} + \frac{\langle \tilde \lambda_t^* \rangle_f}{\eta_f} = LCOE_f
    \end{equation*}
    This is the LCOE of the hydrogen turbine (average capital cost plus the marginal cost).
\end{frame}
\begin{frame}
    \frametitle{Value of hydrogen electrolyser}

    Doing the same for the power consumption of the electrolyser $g_{e,t}$ we get from stationarity
  \begin{equation*}
      0 =     \frac{\d \cL}{\d g_{e,t}} = - \eta_e \tilde \lambda_{t}^*   + \l_t^* + \ubar{\m}^*_{e,t} - \bar{\m}^*_{e,t} \quad \forall t
  \end{equation*}
  Note that this has exactly the same structure as a flexible demand bidding with a willingness to pay of $\eta_e \tilde \lambda_{t}^*$.
  The electrolyser is willing to pay up to $\eta_e \tilde \lambda_{t}^*$ for electricity because if it wants to produce 1~MWh of hydrogen, it needs $1/\eta_e$~MWh of electricity. If it pays $\eta_e \tilde \lambda_{t}^*$~\euro/MWh or less for $1/\eta_e$~MWh it will pay up to  $\lambda_{t}^*$ and still can make a profit in the hydrogen market.

  If we do the same tricks by multiplying by $g_{e,t}^*$, summing over $t$ and dividing by $\sum_t g_{e,t}^*$ we get
  \begin{equation*}
      0 = - \eta_e  \langle \tilde \lambda_{t}^* \rangle_e   + MV_e + \frac{c_e G_e}{\sum_t g_{e,t}}
  \end{equation*}
  Rearranging and dividing by $\eta_e$ we have
  \begin{equation*}
    \langle \tilde \lambda_t^* \rangle_e =  \frac{c_e G_e}{\eta_e\sum_t g_{e,t}} + \frac{MV_e}{\eta_e} = LCOH
  \end{equation*}
   The average value of hydrogen is the levelised cost of hydrogen, LCOH, at equilibrium, i.e. the averaged capital cost of the electrolyser plus the average cost of electricity used.

\end{frame}


\begin{frame}
  \frametitle{Example calculation}

  Consider the example in this \hrefc{https://nworbmot.org/courses/es-23/hydrogen_storage.ipynb}{Jupyter notebook}. This runs over a year of weather for Germany with a flat demand met by wind, solar, hydrogen storage and load-shedding for 1000~\euro/MWh.

  The hydrogen price is give by $\tilde \lambda^*_t = 67.3$~\euro/MWh. It is constant in time because we set the storage cost to zero $c_s=0$ so that hydrogen can be moved between hours with no cost.

  If we look at the electricity price duration curve for $\lambda^*_t$

  \centering
\includegraphics[width=8.3cm]{price_duration-storage.pdf}

\end{frame}

\begin{frame}
  \frametitle{Example calculation}

What causes these price steps?
\begin{itemize}
  \item The high price 1\% of hours of 1000~\euro/MWh are set by the load-shedding.
  \item The next highest step for 38\% of the time is set by the hydrogen turbine bidding as a supplier with marginal cost of $ \eta_f^{-1} \tilde \lambda_{t}^* = (1/0.58) * 67.3$~\euro/MWh $=116.0$~\euro/MWh.
  \item The next step for 43\% of the time is set by the hydrogen electrolyser bidding as a demand with willingness to pay of $\eta_e  \tilde \lambda_{t}^* = 0.62 * 67.3$~\euro/MWh $=41.7$~\euro/MWh.
  \item For 18\% of the time the price is set by wind and solar with zero marginal cost.
\end{itemize}

The hydrogen turbine has market value $MV_f = 162.2$~\euro/MWh and recovers its capital costs in the hours of load-shedding.

The electrolyser has market value $MV_e = 22.3$~\euro/MWh and recovers its capital costs by selling into the hydrogen market when the electricity price is zero.

Wind and solar have market values of 46.1 and 28.7~\euro/MWh respectively.

\end{frame}

\begin{frame}
  \frametitle{Hydrogen storage lessons}

  \begin{itemize}
  \item First, this is only a \alert{simulation of a zero-emission system} - we don't know exactly how the world will turn out (we won't have perfect foresight or equilibrium)
  \item Electricity prices are \alert{intimately tied to hydrogen prices} (sector-coupling, just as prices are tied to fossil gas today)
  \item Unlike today, prices in many hours are \alert{set by the demand side} (1\% by load-shedding, 43\% by electrolysers in our example; in a more complex world, prices could be set by flexible electric vehicle charging, stationary battery charging, heat pumps, flexible industrial loads)
  \item As a result of demand flexibility, prices are set by wind and solar to zero \alert{only for a small fraction of hours} (here 18\%)
  \end{itemize}

\end{frame}

\section{Networks Versus Storage for Highly-Renewable European Electricity System}



\begin{frame}
  \frametitle{Warm-up: Determine optimal electricity system}


\begin{columns}[T]
\begin{column}{7cm}
  \begin{itemize}
  \item Meet all electricity demand.
  \item Reduce \co2{} by 95\% compared to 1990.
  \item \alert{Generation} (where potentials allow): onshore and offshore
    wind, solar, hydroelectricity, backup from natural gas.
  \item \alert{Storage}: batteries for short term, electrolyse hydrogen gas for long term.
  \item \alert{Grid expansion}: simulate everything from no grid expansion (like a \alert{decentralised solution}) to optimal grid expansion (with significant \alert{cross-border trade}).
  \end{itemize}
\end{column}
\begin{column}{6.5cm}


  \vspace{0.3cm}
\centering

\includegraphics[width=7cm]{europe_map}

\end{column}
\end{columns}


\source{PyPSA-Eur, based on ENTSO-E map}
\end{frame}



\begin{frame}
  \frametitle{Linear optimisation problem}


  Objective is the minimisation of \alert{total annual system costs}, composed of \alert{capital costs} $c_*$ (investment costs) and \alert{operating costs} $o_*$ (fuel ,etc.):
  \begin{equation*}
\min    f(F_\ell, f_{\ell,t}, G_{i,s},  g_{i,s,t}) =  \sum_{\ell} c_l F_\ell + \sum_{i,s} c_{i,s} G_{i,s} + \sum_{i,s,t} w_t o_{i,s} g_{i,s,t}
  \end{equation*}
  We optimise for $i$ nodes, representative times $t$ and transmission lines $l$:
  \begin{itemize}
  \item the transmission capacity $F_\ell$ of all the lines $\ell$
  \item the flows $f_{\ell,t}$ on each line $\ell$ at each time $t$
  \item the generation and storage capacities $G_{i,s}$ of all technologies (wind/solar/gas etc.) $s$ at each node $i$
  \item the dispatch $g_{i,s,t}$ of each generator and storage unit at each point in time $t$
  \end{itemize}

  Representative time points are weighted $w_t$ such that $\sum_t w_t = 365*24$ and the capital costs $c_*$ are annualised, so that the objective function represents the annual system cost.

\end{frame}



\begin{frame}
  \frametitle{Constraints 1/6: Nodal energy balance}

  Demand $d_{i,t}$ at each node $i$ and time $t$ is always met by
  generation/storage units $g_{i,s,t}$ at the node or from transmission
  flows $f_{\ell,t}$ on lines attached at the node (Kirchhoff's Current Law):
    \begin{equation*}
      \sum_{s} g_{i,s,t} - d_{i,t} =  \sum_{\ell } K_{i\ell} f_{\ell,t}  \hspace{1cm}\leftrightarrow\hspace{1cm} \l_{i,t}
    \end{equation*}

    Nodes are shown as thick busbars connected by transmission lines (thin lines):

\centering
\begin{circuitikz}
  \draw (1.5,14.5) to [short,i^=$f_1$] (1.5,13);
  \draw [ultra thick] (0,13) node[anchor=south]{i} -- (4,13);
  \draw(2.5,13) -- +(0,0.5) to [short,i^=$f_2$] +(5,0.5) -- +(5,0);
  \draw [ultra thick] (6,13) node[anchor=south]{j} -- +(4,0);
  \draw (8.5,14.5) to [short,i^=$f_3$] +(0,-1.5);
  \draw (0,-0.5) ;
  \draw (0.5,13) -- +(0,-0.5) node[sground]{};
  \draw (2,12) node[vsourcesinshape, rotate=270](V2){}
  (V2.left) -- +(0,0.6);
  \draw (3.5,12) node[vsourcesinshape, rotate=270](V2){}
  (V2.left) -- +(0,0.6);
  \draw (0.5,11) node{$d_i$};
  \draw (2,11) node{$g_{i,w}$};
  \draw (3.5,11) node{$g_{i,s}$};
  \draw (2,10.3) node{$ g_{i,w} + g_{i,s} - d_i = f_2 -f_1$};
  \draw (6.5,13) -- +(0,-0.5) node[sground]{};
  \draw (8,12) node[vsourcesinshape, rotate=270](V2){}
  (V2.left) -- +(0,0.6);
  \draw (9.5,12) node[vsourcesinshape, rotate=270](V2){}
  (V2.left) -- +(0,0.6);
  \draw (6.5,11) node{$d_j$};
  \draw (8,11) node{$g_{j,w}$};
  \draw (9.5,11) node{$g_{j,s}$};
  \draw (8,10.3) node{$ g_{j,w} + g_{j,s} - d_{j} = -f_2 - f_3$};

\end{circuitikz}


\end{frame}



\begin{frame}
  \frametitle{Constraints 2/6: Generation availability}
  Generator/storage dispatch $g_{i,s,t}$ cannot exceed availability $G_{i,s,t}\cdot G_{i,s}$, made up of per unit availability $0 \leq G_{i,s,t} \leq 1$ multiplied by the capacity $G_{i,s}$. The capacity is bounded by the installable potential $\hat{G}_{i,s}$.
    \begin{equation*}
      0 \leq g_{i,s,t} \leq G_{i,s,t}\cdot G_{i,s} \leq  G_{i,s} \leq  \hat{G}_{i,s}
    \end{equation*}

    \centering
        \begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[width=10.5cm]{scigrid-curtailment}};
\draw (3,2.85) node[text=blue]{$g_{i,s,t}$};
\draw (3,3.6) node[text=red]{$G_{i,s,t}\cdot G_{i,s}$};
\draw (3,4.95) node[text=cyan]{$G_{i,s}$};
  \end{tikzpicture}

\end{frame}




\begin{frame}
  \frametitle{Installation potentials limited by geography}

  Expansion potentials are limited by \alert{land usage} and
  \alert{conservation areas}; potential yearly energy yield at each site
  limited by \alert{weather conditions}:

  \centering
  \includegraphics[width=14.5cm]{average_power_density_potential-eps-converted-to.pdf}
\end{frame}




\begin{frame}
  \frametitle{Constraints 3/6: Storage consistency}

  Storage units such as batteries or hydrogen storage can work in both
  storage and dispatch mode. This has to be consistent with the state
  of charge $e_{i,s,t}$:
  \begin{align*}
    e_{i,s,t} = \eta_0e_{i,s,t-1} + \eta_1g_{i,s,t,\textrm{store}} -  \eta_2^{-1} g_{i,s,t,\textrm{dispatch}}
  \end{align*}

  The state of charge is limited by the energy capacity $E_{i,s}$:
  \begin{align*}
    0 \leq e_{i,s,t} \leq E_{i,s} \quad \forall i,s,t
  \end{align*}


  There are efficiency losses $\eta$; hydroelectric dams can also have a river inflow.

\end{frame}




\begin{frame}
  \frametitle{Constraints 4/6: Kirchoff's Laws for Physical Flow}

  The linearised \alert{power flows} $f_\ell$ for each line $\ell \in
  \{1,\dots L\}$ in an AC network are determined by the
  \alert{reactances} $x_\ell$ of the transmission lines and the
  \alert{net power injection} at each node $p_i$ for $i\in\{1,\dots
  N\}$.


  We have to satisfy Kirchoff's Laws, which can be compactly expressed
  using the \alert{incidence matrix} $K \in \R^{N\times L}$ (boundary
  operator in homology theory) of the graph and the \alert{cycle
    basis} $C \in \R^{L\times(L-N+1)}$ (kernel of $K$)

  \begin{itemize}
  \item  Kirchoff's Current Law: $p_i = \sum_{\ell} K_{i\ell} f_\ell$
  \item  Kirchoff's Voltage Law: $\sum_\ell C_{\ell c} x_\ell f_\ell = 0$
  \end{itemize}

\end{frame}



\begin{frame}
  \frametitle{Constraints 5/6: Transmission Line Thermal Limits}

    Transmission flows cannot exceed the thermal capacities of the transmission lines (otherwise they sag and hit buildings/trees):
    \begin{equation*}
      | f_{\ell,t} | \leq  F_\ell
    \end{equation*}

\end{frame}



\begin{frame}
  \frametitle{Constraints 6/6: Global constraints on \co2 and transmission volumes}
  CO${}_2$ limits are respected, given emissions $\varepsilon_{i,s}$ for each fuel source $s$:
    \begin{equation*}
      \sum_{i,s,t}  g_{i,s,t} \frac{\varepsilon_{i,s}}{\eta_{s}} \leq \textrm{CAP}_{\textrm{CO}_2} \hspace{1cm} \leftrightarrow \hspace{1cm} \m_{\textrm{CO}_2}
    \end{equation*}
    We enforce a reduction of \co2 emissions by 95\% compared to 1990
    levels, in line with German and EU targets for 2050.

    Transmission volume limits are respected, given length $d_\ell$ and capacity $F_\ell$ of each line:
    \begin{equation*}
      \sum_{\ell} d_\ell   F_\ell \leq \textrm{CAP}_{\textrm{trans}} \hspace{1cm} \leftrightarrow \hspace{1cm} \m_{\textrm{trans}}
    \end{equation*}
    We  successively change the transmission limit, to assess the costs of balancing power in time (i.e. storage) versus space (i.e. transmission networks).

\end{frame}



\begin{frame}
\frametitle{Model Inputs and Outputs}

\begin{columns}
  \begin{column}{6cm}


    \begin{table}[!t]
	\centering
	\begin{tabular}{@{}p{1.15cm}p{4.54cm}@{}}
\toprule
\alert{Inputs} & Description \\
\midrule
$d_{i,t}$ & Demand (inelastic) \\
$G_{i,s,t}$ & Per unit availability for wind and solar \\
$\hat{G}_{i,s}$ & Generator installable potentials \\
various & Existing hydro data \\
various & Grid topology \\
$\eta_*$ & Storage efficiencies \\
$c_{i,s}$ & Generator capital costs \\
$o_{i,s,t}$ & Generator marginal costs \\
$c_\ell$ & Line costs \\
\bottomrule
	\end{tabular}
\end{table}
  \end{column}
    \begin{column}{1cm}


      \vspace{1cm}

      $\to$

    \end{column}
  \begin{column}{6cm}


    \begin{table}[!t]
	\centering
	\begin{tabular}{@{}p{1.15cm}p{4.54cm}@{}}
\toprule
\alert{Outputs} & Description \\
\midrule
$G_{i,s}$ &  Generator capacities \\
$g_{i,s,t}$ & Generator dispatch \\
$F_\ell$ & Line capacities \\
$f_{\ell,t}$ & Line flows \\
$\lambda_*,\mu_*$ & Lagrange/KKT multipliers of all constraints \\
f & Total system costs \\
\bottomrule
	\end{tabular}
\end{table}
  \end{column}

\end{columns}
\end{frame}




\begin{frame}
  \frametitle{Costs and assumptions for the electricity sector (projections for 2030)}

  \begin{table}
\centering
\begin{tabular}{@{}lrlrr@{}}
\toprule
Quantity                &Overnight  Cost [\euro]  &Unit & FOM [\%/a] & Lifetime [a] \\
\midrule
Wind onshore    &1182   &kW\el &3 & 20   \\
Wind offshore  &2506   &kW\el  &3& 20  \\
Solar PV           &600   &kW\el &4 & 20   \\
Gas             &400    &kW\el  &4& 30  \\
%Gas marginal            &75     &\euro{}/MWh$_{\textrm{e}}$    \\
Battery storage         &1275   &kW\el  & 3 & 20 \\
Hydrogen storage        &2070   &kW\el  & 1.7 &20 \\
Transmission line       &400    &MWkm & 2 & 40\\
\bottomrule
\end{tabular}
\end{table}
  Interest rate of 7\%, storage efficiency losses, only gas has \co2 emissions, gas marginal costs.

  Batteries can store for 6 hours at maximal rating (efficiency $0.9\times 0.9$), hydrogen storage for 168 hours (efficiency $0.75\times 0.58$).
\end{frame}






\begin{frame}
  \frametitle{Costs: No interconnecting transmission allowed}



\begin{columns}[T]
  \begin{column}{4.9cm}

  Technology~by~energy:
  % left bottom right top
  \begin{tabular}{cc}
    \includegraphics[trim=0 0cm 0 0cm,width=3.2cm,clip=true]{total-pie-0-0} &
  \end{tabular}


  Average~cost~\alert{\euro 86/MWh}:

  \includegraphics[width=3.6cm]{total-costs-joint-1}

  \end{column}

  \begin{column}{9cm}
    % left bottom right top
    \centering
    \includegraphics[trim=0 4cm 0 4cm,width=8cm,clip=true]{euro-pie-0-0}

    \raggedright
    Countries must be self-sufficient at all times; lots of storage and some gas to deal with fluctuations of wind and solar.
  \end{column}

\end{columns}
\end{frame}


\begin{frame}
  \frametitle{Dispatch with no interconnecting transmission}

  For Great Britain with no interconnecting transmission, excess wind is either stored as hydrogen or curtailed:

  \centering
  \includegraphics[width=13cm]{GB-0}
\end{frame}


\begin{frame}
  \frametitle{Costs: Cost-optimal expansion of interconnecting transmission}


\begin{columns}[T]
  \begin{column}{5.3cm}

  Technology~by~energy:
  % left bottom right top
    \begin{tabular}{cc}
    \includegraphics[trim=0 0cm 0 0cm,width=3.2cm,clip=true]{total-pie-0-5} &
  \end{tabular}

  Average~cost~\alert{\euro 64/MWh}:

  \includegraphics[width=4.7cm]{total-costs-joint-2}

  \end{column}

  \begin{column}{9cm}
    % left bottom right top
    \centering
    \includegraphics[trim=0 4cm 0 4cm,width=8cm,clip=true]{euro-pie-0-5}

    \raggedright
    Large transmission expansion; onshore wind dominates. This optimal solution may run into public acceptance problems.
  \end{column}


\end{columns}

\end{frame}


\begin{frame}
  \frametitle{Dispatch with cost-optimal interconnecting transmission}

  Almost all excess wind can be now be exported:

  \centering
  \includegraphics[width=13cm]{GB-1}
\end{frame}



\begin{frame}
  \frametitle{Electricity Only Costs Comparison}

\begin{columns}[T]
  \begin{column}{9cm}

    \vspace{0.5cm}
  \includegraphics[width=9cm]{opteu_paper2-elec_only-costs-curve}

  \end{column}

  \begin{column}{5cm}
    \begin{itemize}
    \item Average total system costs can be as low as \euro~64/MWh
     \item Energy is dominated by wind (64\% for the cost-optimal system), followed
       by hydro (15\%) and solar (17\%)
     \item Restricting transmission results in more storage to deal with variability, driving up the costs by up to 34\%
     \item Many benefits already locked in at a few multiples of today's grid
    \end{itemize}

  \end{column}

\end{columns}
\end{frame}


\begin{frame}
  \frametitle{Grid expansion CAP shadow price $\m_{\textrm{trans}}$ as CAP relaxed}


\begin{columns}[T]
\begin{column}{9cm}
\centering
  \includegraphics[width=9cm]{shadow_price_LV_N1_5_diw2030_solar1_7-eps-converted-to}
  %\includegraphics[width=6cm]{pre-2-costs_de_storage}
\end{column}
\begin{column}{4cm}
  \begin{itemize}
%\item   %Big reduction in curtailment
  \item With overhead lines the optimal system has around 7 times today's transmission volume
  \item With underground cables (5-8 times more expensive) the optimal system has around 3 times today's transmission volume
  \end{itemize}
\end{column}
\end{columns}

\end{frame}





\begin{frame}
  \frametitle{Distribution of costs}

  As transmission volumes increase, costs become more unequally distributed...

  \centering
  \includegraphics[width=13cm]{costs_ct_N1_5_HFOM-diw2030_solar1_7-LVs_0-0_0-25_Opt-eps-converted-to.pdf}

\end{frame}


\begin{frame}
  \frametitle{Distribution of prices}

  ...while market prices converge.

  \centering

  \includegraphics[width=13cm]{local_marginal_price_ct_N1_5-diw2030_solar1_7_angles-LVs_0-0_0-25_Opt-eps-converted-to.pdf}

\end{frame}



\begin{frame}
  \frametitle{Different flexibility options have difference temporal scales}

\begin{columns}[T]
  \begin{column}{10.5cm}

    \vspace{0.5cm}
  \includegraphics[width=11cm]{soc_series_LV0-25_H2_hydro_diw2030_solar1_7-eps-converted-to.pdf}

  \end{column}

  \begin{column}{3cm}
    \vspace{1cm}
    \begin{itemize}
    \item Hydro reservoirs are \alert{seasonal}
    \item Hydrogen storage is \alert{synoptic} (i.e. weekly)
    \end{itemize}

  \end{column}

\end{columns}
\end{frame}



\begin{frame}
  \frametitle{Different flexibility options have difference temporal scales}

\begin{columns}[T]
  \begin{column}{10.5cm}

    \vspace{0.5cm}
  \includegraphics[width=11cm]{soc_series_LV0-25_all_2011-08_diw2030_solar1_7-eps-converted-to.pdf}

  \end{column}

  \begin{column}{3cm}
        \vspace{2cm}
    \begin{itemize}
    \item Pumped hydro and battery storage are \alert{daily}
    \end{itemize}

  \end{column}

\end{columns}
\end{frame}


\begin{frame}
  \frametitle{You Try}

  For more details, see the paper \hrefc{https://arxiv.org/abs/1704.05492}{The Benefits of Cooperation} (2017).


  The basic result (benefit of European interconnection versus national balancing) can also be seen using the online toy model:

  \vspace{.5cm}
  \centering
  \urlc{https://model.energy/}
  \vspace{.5cm}

  \raggedright
  Look at the differences of wind and solar feed-in and optimal storage solutions for:
  \begin{itemize}
  \item \alert{City: Karlsruhe}
  \item \alert{Country: Germany}
    \item \alert{Continent: Europe}
  \end{itemize}

\end{frame}


\end{document}
