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\let\olditem\item
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\title{Electricity Markets: Summer Semester 2016, Lecture 1}
%\subtitle{---}
\author{
  Tom Brown, Mirko Schäfer
}

\institute{
  Frankfurt Institute of Advanced Studies (FIAS), Goethe-Universit\"at Frankfurt \\
  FIAS Renewable Energy System and Network Analysis (FRESNA)\\
  \\
  \url{{brown, schaefer}@fias.uni-frankfurt.de}
}


\date{11th April 2016}


\titlegraphic{%
  \vspace{7cm}
    \includegraphics[width=4cm]{FIAS-logo}
}

\begin{document}

\maketitle


\begin{frame}

  \frametitle{Table of Contents}
  \setbeamertemplate{section in toc}[sections numbered]
  \tableofcontents[hideallsubsections]
\end{frame}


\section{Administration}



\begin{frame}
  \frametitle{Who we are}

\begin{itemize}
\item Tom Brown, Postdoctoral Researcher, FIAS (Riedberg), brown@fias.uni-frankfurt.de
\item Mirko Schäfer, Postdoctoral Researcher, FIAS (Riedberg), schaefer@fias.uni-frankfurt.de
\end{itemize}

We are both physicists who have specialised in the optimisation of energy
systems and the interactions of complex networks.

\end{frame}


\begin{frame}
  \frametitle{Lectures and Exercise Classes}

  Lectures:
  \begin{itemize}
    \item Weekly, Mondays 16:00 - 18:00 Seminarhaus SH 0.109
    \item 11.04.2016 until 11.07.2016 (with exception of Pfingsten 16.05.2016)
  \end{itemize}

  Exercise Classes:
  \begin{itemize}
    \item Biweekly, Mondays 18:00 - 20:00 Seminarhaus SH 0.109
    \item 18.04.2016 until 11.07.2016
    \item Exercise Sheets: Given out the week before, starting today
  \end{itemize}

\end{frame}




\begin{frame}
  \frametitle{Exam and Exercise Sheets}

  Only the final exam is graded.

  Every two weeks there will be an \alert{Exercise Sheet}, which we will go
  through in the Exercise Class in the following week.

  The exam will be in the style and at the difficulty level of non-starred question in the Exercise Sheet.

  Therefore if you want to do well in the exam, you need to do the Exercise Sheets.

  Exercise classes are biweekly, starting 18.04 (next week).

  There are also starred questions which are non-compulsory but should
  be interesting - you'll learn better by doing these exercises.

\end{frame}




\begin{frame}
  \frametitle{Literature}

  There are lots of textbooks covering the material of this course and
  {\bf lots} of supporting material on the internet. We will roughly follow:
  \begin{itemize}
  \item Daniel Kirschen and Goran Strbac, ``Fundamentals of Power System
    Economics,'' Wiley, 2004
  \end{itemize}

 The following books are also useful:
  \begin{itemize}
  \item D.R.~Biggar, M.R.~Hesamzadeh, ``The Economics of Electricity
  Markets,'' Wiley, 2014
  \item Steven Stoft, ``Power System Economics: Designing Markets for
  Electricity,'' Wiley, IEEE Press, 2002
  \item Joshua Adam Taylor, ``Convex Optimization of Power Systems,'' CUP,
  2015
  \item J.M. Morales et al., ``Integrating Renewables in Electricity
  Markets,'' Springer, 2014
  \end{itemize}

\end{frame}




\begin{frame}
  \frametitle{Kirschen and Strbac}
 Daniel Kirschen and Goran Strbac, ``Fundamentals of Power System
    Economics,'' Wiley, 2004


\begin{columns}[T]
\begin{column}{6.5cm}
\includegraphics[width=5.5cm]{kirschen-strbac}
\end{column}
\begin{column}{4cm}
  \begin{itemize}
   \item Focus on examples, concrete calculations
   \item Less mathematical theory
   \item We will broadly follow this book
  \end{itemize}

\end{column}
\end{columns}

\end{frame}




\begin{frame}
  \frametitle{Biggar and Hesamzadeh}
  D.R.~Biggar, M.R.~Hesamzadeh, ``The Economics of Electricity
  Markets,'' Wiley, 2014


\begin{columns}[T]
\begin{column}{6.5cm}
\includegraphics[width=5.5cm]{biggar-hesamzadeh}
\end{column}
\begin{column}{4cm}
  \begin{itemize}
   \item More detail on the optimisation
  theory, engineering background and coupling electricity markets with transmission networks
  \end{itemize}

\end{column}
\end{columns}

\end{frame}




\begin{frame}
  \frametitle{Stoft}
  Steven Stoft, ``Power System Economics: Designing Markets for
  Electricity,'' Wiley, IEEE Press, 2002


\begin{columns}[T]
\begin{column}{6.5cm}
\includegraphics[width=5.5cm]{stoft}
\end{column}
\begin{column}{4cm}
  \begin{itemize}
  \item Economics focus
  \item Looks at real-world implementations of
  electricity markets
  \end{itemize}

\end{column}
\end{columns}

\end{frame}


\begin{frame}
  \frametitle{Course Website}

  On the course website:

  \url{http://fias.uni-frankfurt.de/~brown/courses/electricity_markets/}

  we will publish links to material on the internet and to the course
  notes and exercise sheets.
\end{frame}



\section{Introduction: Why Electricity Markets are Important}





\begin{frame}
  \frametitle{Why is electricity useful?}

  Electricity is a versatile form of energy carried by electrical
  charge which can be consumed in a wide variety of ways (with selected examples):
  \begin{itemize}
  \item Lighting (lightbulbs, halogen lamps, televisions)
  \item Mechanical work (hoovers, washing machines, electric vehicles)
  \item Heating (cooking, resistive room heating, heat pumps)
  \item Cooling (refrigerators, air conditioning)
  \item Electronics (computation, data storage, control systems)
  \item Industry (electrochemical processes)
  \end{itemize}

  Compare the convenience and versatility of electricity with another
  energy carrier: the chemical energy stored in natural gas (methane),
  which can only be accessed by burning it.

\end{frame}




\begin{frame}
  \frametitle{How is electricity generated?}

  {\bf Conservation of Energy}: Energy cannot be created or destroyed:
  it can only be converted from one form to another.

  There are several `primary' sources of energy which are converted
  into electrical energy in modern power systems:
  \begin{itemize}
  \item Chemical energy, accessed by combustion (coal, gas, oil, biomass)
  \item Nuclear energy, accessed by fission reactions
  \item Hydroelectric energy, allowing water to flow downhill
  \item Wind energy
  \item Solar energy (accessed with photovoltaic (PV) panels or
    concentrating solar thermal power (CSP))
  \item Geothermal energy
  \end{itemize}
  NB: The definition of `primary' is somewhat arbitrary.

\end{frame}


\begin{frame}
  \frametitle{Generators}

  With the exception of solar photovoltaic panels (and electrochemical
  energy and a few other minor exceptions), all generators convert to
  electrical energy via rotational kinetic energy and electromagnetic
  induction in an \emph{alternating current generator}.

  %https://en.wikipedia.org/wiki/Fossil-fuel_power_station#/media/File:Coal_fired_power_plant_diagram.svg

  \includegraphics[width=10cm]{Coal_fired_power_plant_diagram}

  \source{Wikipedia}
%  https://de.wikipedia.org/wiki/Drehstrom-Synchronmaschine#/media/File:Walchenseekraftwerk-1_Turbinenhalle.jpg
% Walchenseekraftwerk-1_Turbinenhalle.jpg



\end{frame}


\begin{frame}
  \frametitle{Example of electricity generation across major EU countries in 2013}


  \includegraphics[width=11cm]{european_countries-energy}


\end{frame}



\begin{frame}
  \frametitle{Electricity generation in Germany per year}

  In 15 years Germany has gone from a system dominated by nuclear and
  fossil fuels, to one with 33\% renewables in electricity consumption.

  \includegraphics[width=11cm]{germany_yearly_electricity}

  \source{Author's own representation based on \url{https://www.energy-charts.de/energy_en.htm}}
\end{frame}



\begin{frame}
  \frametitle{The Economic Operation of the Electricity Sector}


  Given the many different ways of consuming and generating
  electricity, the questions naturally arise:

  \begin{itemize}
    \item   What is the most efficient way to deploy consuming and generating
  assets in the short-run?
    \item How should we invest in assets in the long-run to maximise economic welfare?
  \end{itemize}


  In the past and still in many countries today, this was done centrally
  by `vertically-integrated' monopoly utilities that owned generating
  assets, the electricity networks and retailing.

  Given that these utilities owned all the infrastructure, it was hard
  for third-party generators to compete, even if they were allowed to.

  From the 1980s onwards, countries began to liberalise their
  electricity sectors, separating generation from transmission, and
  allowing regulated competition in the generation sector.
\end{frame}


\begin{frame}
  \frametitle{Electricity Markets}

  Electricity markets have several important differences compared to
  other commodity markets.

  At every instant in time, consumption must be balanced with
  generation.

  If you throw a switch to turn on a light, somewhere a generator will
  be increasing its output to compensate.

  If the power is not balanced in the grid, the power supply will
  collapse and there will be blackouts.

  It is not possible to run an electricity market for every single
  second, for practical reasons (the network must be checked for
  stability, etc.).

  So electricity is traded in blocks of time, e.g. hourly,
  14:00-15:00, or quarter-hourly, 14:00-14:15, well in advance of the
  time when it is actually consumed (based on forecasts).

  Further markets trade in backup balancing power, which step in if the
  forecasts are wrong.


\end{frame}



\begin{frame}
  \frametitle{Discrete Consumers Aggregation}

  The discrete actions of individual consumers smooth out
  statistically if we aggregate over many consumers.

  \centering
  \includegraphics[width=10cm]{demand_smoothing}

\end{frame}



%picture of negative prices (NB: prices negative also because of
%priority dispatch: this removes zero-marginal REs from market and
%makes them part of load shifting left - if they were in the market,
%the price would be zero, not negative as high-priced generators are
%shifted right)


\begin{frame}
  \frametitle{Baseload versus Peaking Plant}

  \alert{Load} (= Electrical Demand) is low during night; in Northern Europe in the winter, the peak is in the evening.

  To meet this load profile, cheap \alert{baseload} generation runs the whole time; more expensive \alert{peaking plant} covers the difference.

  \centering
  \includegraphics[width=9cm]{baseload-peaking}

  \source{Tom Brown}

\end{frame}


\begin{frame}
  \frametitle{Effect of varying demand for fixed generation}

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

  \source{Tom Brown}

\end{frame}




\begin{frame}
  \frametitle{Example market 1/3}

  \centering

  \includegraphics[width=9cm]{germany_generation-price-2015-01-06-2015-01-12}

\end{frame}

\begin{frame}
  \frametitle{Example market 2/3}

  \centering
  \includegraphics[width=9cm]{germany_generation-price-2015-03-15-2015-03-31}

\end{frame}

\begin{frame}
  \frametitle{Example market 3/3}

  \centering
  \includegraphics[width=9cm]{germany_generation-price-2015-04-07-2015-04-14}

\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}

  \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 have been steadily decreasing:

  \includegraphics[width=11cm]{price_development}

  \source{Agora Energiewende}
\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 and
    hard coal are the most flexible and produce the \emph{lowest} \co2
    per MWh
  \item It also massively reduces the profits that nuclear and brown coal 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...

\end{frame}

\begin{frame}
  \frametitle{Consumption metering}

\begin{columns}[T]
\begin{column}{6.5cm}
\includegraphics[width=7.5cm,angle=270]{stromzaehler}
\end{column}
\begin{column}{4cm}
  \begin{itemize}
  \item Look for your electricity meter at home
  \item Mine showed last Friday 42470.3 kWh
  \item Check what the value is a week later
  \end{itemize}

\end{column}
\end{columns}


\end{frame}



\begin{frame}
  \frametitle{Electricity bill}

  My bill: 1900 kWh for a year, at a cost of \euro 570, which
  corresponds to 0.3 \euro/kWh or 300 \euro/MWh. But the spot market price is 30 \euro/MWh, so what's going on??

  \includegraphics[width=10cm,angle=180]{bill}

\end{frame}



\begin{frame}
  \frametitle{Household price breakdown}

  Although the wholesale price is going down, other taxes, grid
  charges and renewables subsidy (EEG surcharge) have kept the price
  high.

  \includegraphics[width=11cm]{household_price}

  HOWEVER the EEG is only high because it is paying for solar panels
  bought at a time when they were still comparatively expensive; but
  through the German subsidy, production volumes were high and the
  learning curve has brought the costs down exponentially.

  \source{Agora Energiewende}

\end{frame}


\begin{frame}
  \frametitle{Learning curve for renewables}


  Onshore wind has already been through the learning process and has
  been for some years highly competitive with conventional power
  sources.
  In some parts of the world wind is being installed without
  subsidies. Solar has also reaped the benefits of mass production,
  including the 40~GW of PV installed in Germany:

  \centering
  \includegraphics[width=8cm]{solar-learning-curve}



\end{frame}



\begin{frame}
  \frametitle{Electricity Transport from Generators to Consumers}

%https://upload.wikimedia.org/wikipedia/commons/thumb/4/41/Electricity_grid_simple-_North_America.svg/640px-Electricity_grid_simple-_North_America.svg.png

  Electricity is also easy to transport over long distances using the
  high voltage transmission grid:

  \includegraphics[width=10cm]{Electricity_grid_simple-_North_America}

  Usually in houses the voltage is $230~V$, but in the transmission
  grid it is hundreds of thousands of Volts.


  \source{Wikipedia}
\end{frame}



\begin{frame}[fragile]
  \frametitle{European transmission network}

  Flows in the European transmission network must respect both
  Kirchoff's laws for physical flow and the thermal limits of the power lines.

  Taking account of network flows and constraints in the electricity market is a major and exciting topic at the moment.

\centering
\includegraphics[width=8cm]{europe-transmission}
\source{ENTSO-E}


\end{frame}



\begin{frame}
  \frametitle{Network Bottlenecks and Loop Flows}


  Electricity is traded in large market zones. Power trades between
  zones (``scheduled flows'') do not always correspond to what flows
  according to the network physics (``physical flows''). This
  leads to political tension as wind from Northern Germany flows to Southern Germany via Poland and the Czech Republic.

  \centering
  \includegraphics[width=9cm]{germany_loop_flows}

  \source{THEMA Consulting Group}
  %https://ec.europa.eu/energy/sites/ener/files/documents/201310_loop-flows_study.pdf
\end{frame}



\begin{frame}
  \frametitle{CO2 emissions from electricity sector}

  Despite the increase in renewables in the electricity sector, \co2
  emission have not been reduced substantially in Germany. This is
  partly because German exports have also increased.

  \includegraphics[width=11cm]{co2}

  \source{Agora Energiewende}
\end{frame}





\begin{frame}
  \frametitle{Other sectors}

  The \co2 emissions from electricity generation contribute only a
  fraction to total global greenhouse gas emissions.
  However electricity generation is one of the easiest places to
  reduce emissions, aside from directly reducing energy consumption.

  \centering
  \includegraphics[width=6cm]{global_emissions_sector_2015}

  \source{EPA, IPCC (2014)}
  %https://www3.epa.gov/climatechange/ghgemissions/global.html

\end{frame}




\begin{frame}
  \frametitle{Sector integration}

  Replacing heating and transport fossil fuels with renewable sources and renewable electricity increases efficiency and reduces emissions of GHG.

  \includegraphics[width=11cm]{weissbuch-sector_coupling}

  \source{BMWi Weißbuch}
\end{frame}



\begin{frame}
  \frametitle{Several changes happening simultaneously}

  \emph{Energiewende}: The Energy Transition

  \begin{itemize}
  \item Renewables replace conventional generation
  \item Increasing integration of international electricity markets
  \item Better integration of transmission constraints in electricity markets
  \item Sector coupling: heating and transport electrify
  \item {\bf Reform of electricity market to allow these transactions and investments to take place efficiently}
  \end{itemize}

  In summer 2016 both the German Electricity Market Law and the
  Renewable Energy Law will be reformed. Fun times!
\end{frame}



\begin{frame}
  \frametitle{Newspaper Articles: Lots going on}

  \includegraphics[width=10cm]{ft-fossils_stranded}

\end{frame}


\begin{frame}
  \frametitle{Newspaper Articles: Lots going on}

  \includegraphics[width=10cm]{ft-national_grid_coal}

\end{frame}


\begin{frame}
  \frametitle{Newspaper Articles: Lots going on}

  \includegraphics[width=10cm]{hb-kampf_co2}

\end{frame}



\begin{frame}
  \frametitle{Newspaper Articles: Lots going on}

  \includegraphics[width=10cm]{hb-kleine_motoren}

\end{frame}



\begin{frame}
  \frametitle{Newspaper Articles: Lots going on}

  \includegraphics[width=10cm]{hb-stromspeicher}

\end{frame}


\section{Units: Watts, Kilowatts, Megawatts, WTF?}


%power versus energy: Germany example

% peak != average != capacity

% converting TWh/a to GW to MJ


\begin{frame}
  \frametitle{Power: Flow of energy}

  \alert{Power} is the rate of consumption of energy.

  It is measured in \alert{Watts}:
  \begin{equation*}
     1 \textrm{ Watt } = 1 \textrm{ Joule per second }
  \end{equation*}
  The symbol for Watt is W, 1 W = 1 J/s.

  \centering
  1 kilo-Watt = 1 kW = 1,000 W

  1 mega-Watt = 1 MW = 1,000,000 W

  1 giga-Watt = 1 GW = 1,000,000,000 W

  1 tera-Watt = 1 TW = 1,000,000,000,000 W


\end{frame}


\begin{frame}
  \frametitle{Power: Examples of consumption and generation}

  At full power, the following items consume/generate:

  \ra{1.1}
  \begin{table}[!t]
    \begin{tabular}{lr}
      \toprule
      Item & Power\\
      \midrule
      New efficient lightbulb & 10 W \\
      Old-fashioned lightbulb & 70 W \\
      Single room air-conditioning & 1.5 kW \\
      Kettle & 2 kW \\
      Wind turbine & 3 MW \\
      Coal power station & 1 GW \\
      Germany total demand & 50-80 GW \\
      \bottomrule
    \end{tabular}
  \end{table}

\end{frame}



\begin{frame}
  \frametitle{Energy}

  In the electricity sector, energy is usually measured in
  `Watt-hours', Wh.

  1 kWh = power consumption of 1 kW for one hour

  E.g. a 10 W lightbulb left on for two hours will consume

  10 W * 2 h = 20 Wh

  It is easy to convert this back to the SI unit for energy, Joules:

  1 kWh = (1000 W) * (1 h) = (1000 J/s)*(3600 s) = 3.6 MJ
\end{frame}


\begin{frame}
  \frametitle{Yearly energy to power}

  Germany consumes around 600 TWh per year, written 600 TWh/a.

  What is the \emph{average} power consumption?

  \begin{align*}
    600\textrm{ TWh/a} & = \frac{(600\textrm{ TW}) * (1\textrm{ h})}{(365*24 \textrm{ h})} \\
    & =  \frac{600}{8760} \textrm{ TW} \\
    & = 68.5 \textrm{ GW}
  \end{align*}

\end{frame}



\begin{frame}
  \frametitle{Efficiency}

  When fuel is consumed, much/most of the energy of the fuel is lost
  as waste heat rather than being converted to electricity.

  The thermal energy, or calorific value, of the fuel is given in
  terms of MWh${}_{\textrm{th}}$, to distinguish it from the
  electrical energy MWh${}_{\textrm{el}}$.

  The ratio of input thermal energy to output electrical energy is the
  \alert{efficiency}.

  \ra{1.1}
  \begin{table}[!t]
    \begin{tabular}{lrrrrr}
      \toprule
      Fuel & Calorific energy & Per unit efficiency & Electrical energy \\
       & MWh\(_{\text{th}}\)/tonne & MWh\(_{\text{el}}\)/MWh\(_{\text{th}}\) & MWh\(_{\text{el}}\)/tonne \\
\midrule
Lignite & 2.5 & 0.4 & 1.0 \\
Hard Coal & 6.7 & 0.45 & 2.7\\
Gas & 15.4 & 0.4 & 6.16\\
Uranium & 150000 & 0.33 & 50000 \\
      \bottomrule
    \end{tabular}
  \end{table}




\end{frame}


\begin{frame}
  \frametitle{Fuel costs to marginal costs}


  The cost of a fuel is often given in \euro/kg or \euro/MWh\(_{\text{th}}\).

  Using the efficiency, we can convert this to \euro/MWh\(_{\text{el}}\).


  \ra{1.1}
  \begin{table}[!t]
    \begin{tabular}{lrrrrr}
      \toprule
      Fuel &  Per unit efficiency & Cost per thermal & Cost per elec. \\
      &  MWh\(_{\text{el}}\)/MWh\(_{\text{th}}\)  & \euro/MWh\(_{\text{th}}\) & \euro/MWh\(_{\text{el}}\)\\
\midrule
Lignite  & 0.4  & 4.5 & 11\\
Hard Coal  & 0.45   & 10 & 22\\
Gas & 0.4  & 23 & 58\\
Uranium & 0.33 & 3.3 & 10\\
      \bottomrule
    \end{tabular}
  \end{table}



\end{frame}





\begin{frame}
  \frametitle{Real marginal costs}

  Some actual data on marginal costs for different fuels; these don't
  agree fully with the previous table due to differing assumptions on
  efficiencies, etc.

  \includegraphics[width=11cm]{marginal_prices}

  \source{Agora Energiewende}

\end{frame}



\begin{frame}
  \frametitle{CO2 emissions per MWh}

  The \co2 emissions of the fuel.

  \ra{1.1}
  \begin{table}[!t]
    \begin{tabular}{lrrr}
      \toprule
Fuel & t\(_{\text{CO2}}\)/t & t\(_{\text{C02}}\)/MWh\(_{\text{th}}\) & t\(_{\text{CO2}}\)/MWh\(_{\text{el}}\)\\
\midrule
Lignite &  0.9 & 0.36 & 0.9 \\
Hard Coal  & 2.4 & 0.36 & 0.8 \\
Gas & 3.1 & 0.2 & 0.5\\
Uranium & 0 & 0 & 0 \\
      \bottomrule
    \end{tabular}
  \end{table}

  Current \co2 price in EU Emissions Trading Scheme (ETS) \euro 5.27/t${}_{\textrm{CO2}}$

\end{frame}


\section{Electricity Markets from the Consumer Perspective}


\begin{frame}{Consumer behaviour: Theory}

  Suppose for some given period a consumer consumes electricity at a rate of
  $Q$~MW.


  Their \alert{utility or value function} $U(Q)$ in \euro/h is a
  measure of their benefit for a given consumption rate $Q$.

  For a firm this could be the profit related to this electricity
  consumption from manufacturing goods.


  Typical the consumer has a higher utility for higher $Q$, i.e. the
  first derivative is positive $U'(Q) > 0$. By assumption, the rate of
  value increase with consumption decreases the higher the rate of
  consumption, i.e. $U''(Q) < 0$.

\end{frame}


\begin{frame}{Utility: Example}


  A widget manufacturer has a utility function which depends on the
  rate of electricity consumption $Q$ [\euro/h] as
  \begin{equation*}
    U(Q) = 0.0667~Q^3  - 8~Q^2 + 300~Q
  \end{equation*}


  \centering
  \includegraphics[width=6cm]{utility}


  Note that the slope is always positive, but becomes less positive
  for increasing $Q$.
\end{frame}


\begin{frame}{Optimal consumer behaviour}



  We assume to begin with that the consumer is a price-taker,
  i.e. they cannot influence the price by changing the amount they
  consume.

  Suppose the market price is $\lambda$ \euro/MWh. The consumer should
  adjust their consumption rate $Q$ to maximise their \alert{net surplus}
  \begin{equation*}
    \max_Q U(Q) - \l Q
  \end{equation*}

  This optimisation problem is optimised for $Q=Q^*$ where
  \begin{equation*}
    U'(Q^*) \equiv \frac{dU}{dQ} (Q^*) =  \l
  \end{equation*}

  [Check units: $\frac{dU}{dQ}$ has units $\frac{\textrm{\euro/h}}{\textrm{MW}} = $ \euro/MWh.]

  I.e. the consumer increases their consumption until they make a net loss for any increase of consumption.

  $U'(Q)$ is known as the \alert{inverse demand curve}, which shows,
  for each rate of consumption $Q$ what price $\l$ the consumer is
  willing to pay.


\end{frame}



\begin{frame}{Inverse demand function: Example}
  For our example the inverse demand function is given by
  \begin{equation*}
    U'(Q) = 0.2~Q^2 - 16~Q + 300
  \end{equation*}

  \centering
  \includegraphics[width=6cm]{inverse_demand_function}

  [It's called the \emph{inverse} demand function, because the demand function is the function you get from reversing the axes. The \alert{demand function} $Q(\l)$ gives the demand is as a function of the price.]
\end{frame}





\begin{frame}{Gross consumer surplus}

  The area under the curve is the \alert{gross consumer surplus},
  which as the integral of a derivative, just gives the utility
  function $U(Q)$ again, up to a constant.

  \centering
  \includegraphics[width=6cm]{gross_consumer_surplus}

\end{frame}



\begin{frame}{Net consumer surplus}

  The more relevant \alert{net consumer surplus}, or just
  \alert{consumer surplus} is the `net profit' the consumer makes by
  having utility above the electricity price.

  \centering
  \includegraphics[width=6cm]{net_consumer_surplus}

\end{frame}


\begin{frame}{Limits to consumption}


  Note that it is quite common for consumption to be limited by other
  factors before the electricity price becomes too expensive, e.g. due
  to the size of electrical machinery. This gives an upper bound
  \begin{equation*}
    Q \leq Q^{\textrm{max}}
  \end{equation*}
  In the following case the optimal consumption is at
  $Q^{\textrm{max}} = 10$ MW.

  \centering
  \includegraphics[width=6cm]{net_consumer_surplus-limited}

\end{frame}



\begin{frame}
  \frametitle{Consumers can delay their consumption}

  Besides changing the amount of electricity consumption, consumers
  can also shift their consumption in time.

  For example electric storage heaters use cheap electricity at night
  to generate heat and then store it for daytime.

  The LHC particle accelerator does not run in the winter,
  when prices are higher (see
  \url{http://home.cern/about/engineering/powering-cern}).  Summer
  demand: 200~MW, corresponds to a third of Geneva, equal to peak
  demand of Rwanda (!); winter only 80~MW.


  \centering
  \includegraphics[width=6cm]{csm_1998_LHCUeberblick_CERN_02_ed178e98ea}


  \source{CERN}




\end{frame}


\begin{frame}
  \frametitle{Consumers can also move location}

  Aluminium smelting is an electricity-intensive process. Aluminium
  smelters will often move to locations with cheap and stable
  electricity supplies, such as countries with lots of hydroelectric
  power.  For example, 73\% of Iceland's total power consumption in
  2010 came from aluminium smelting.

  Aluminium costs around US\$ 1500/ tonne to produce. %http://www.lme.com/metals/non-ferrous/aluminium/

  Electricity consumption: 15 MWh/tonne.

  At Germany consumer price of \euro 300 / MWh, this is \euro 4500 / tonne. Uh-oh!!!

  If electricity is 50\% of cost, then need \$750/tonne to go on electricity $\Rightarrow$ 750/15 \$/MWh = 50 \$/MWh.

  %https://arcticecon.wordpress.com/2012/02/15/aluminium-smelting-in-iceland-alcoa-rio-tinto-alcan-century-aluminum-corp/
  %15.7 kWh of electricity to produce 1 kg of aluminium.  Al203 (alumina) -> Al

  % The Hall-Héroult electrolysis process is the major production route for primary aluminium.

  %Due to the nature of the process, power outages have the potential
  %to cause damage to production cells as the molten liquids could
  %solidify in absence of adequate current.  For this reason,
  %production facilities need to be near secure and reliable sources
  %of energy.  step before smelting is bauxite -> alumina. Bauxite is
  %gibbsite Al(OH)3, boehmite γ-AlO(OH), and diaspore α-AlO(OH).


%What is notable is that the aluminium industry’s total power usage
  %amounted to roughly 73% of Iceland’s total power consumption in 2010.

  %https://en.wikipedia.org/wiki/Economy_of_Iceland#Aluminium

  %https://agmetalminer.com/2015/11/24/power-costs-the-production-primary-aluminum/

  %making domestic electricity costs some 44% of the sales price of
  %ingot and may explain why Chinese smelters combined with low
  %capacity utilization due to reduced demand are widely reported

  %http://www.crugroup.com/about-cru/cruinsight/Aluminium_Smelter_Power_Tariffs_Winners_and_Losers

%Power typically represents more than a third of an aluminium
%smelter's operating costs and is the single most important cost
%variant between smelting assets. As a result access to cheap power is
%one of the key sources of competitive advantage in primary aluminium
%smelting.

  % https://en.wikipedia.org/wiki/List_of_aluminium_smelters

  % THere are some in Germany

  \end{frame}

\begin{frame}{Copyright}


  Unless otherwise stated the graphics and text is Copyright
  \copyright Tom Brown, 2016.

  We hope the graphics borrowed from others have been attributed
  correctly; if not, drop a line to the authors and we will correct
  this.

  The source \LaTeX, self-made graphics and Python code used to
  generate the self-made graphics are available on the course website:

  \url{http://fias.uni-frankfurt.de/~brown/courses/electricity_markets/}

  The graphics and text for which no other attribution are given are licensed under a
  \href{http://creativecommons.org/licenses/by-sa/4.0/}{Creative Commons
  Attribution-ShareAlike 4.0 International License}.

  \begin{center}\ccbysa\end{center}

\end{frame}


\end{document}
