% NB: use pdflatex to compile NOT pdftex.  Also make sure youngtab is
% there...

% converting eps graphics to pdf with ps2pdf generates way too much
% whitespace in the resulting pdf, so crop with pdfcrop
% cf. http://www.cora.nwra.com/~stockwel/rgspages/pdftips/pdftips.shtml




\documentclass[10pt]{beamer}
%\usetheme[everytitleformat=uppercase]{m}
\usetheme[everytitleformat=regular]{m}
\usepackage[absolute,overlay]{textpos}
\usepackage{booktabs}


\usepackage[official]{eurosym}


%use this to add space between rows
\newcommand{\ra}[1]{\renewcommand{\arraystretch}{#1}}

% for sources http://tex.stackexchange.com/questions/48473/best-way-to-give-sources-of-images-used-in-a-beamer-presentation

\setbeamercolor{framesource}{fg=gray}
\setbeamerfont{framesource}{size=\tiny}


\newcommand{\source}[1]{\begin{textblock*}{4cm}(8.0cm,8.0cm)
    \begin{beamercolorbox}[ht=0.5cm,right]{framesource}
        \usebeamerfont{framesource}\usebeamercolor[fg]{framesource} Source: {#1}
    \end{beamercolorbox}
\end{textblock*}}

\usepackage{hyperref}


%\usepackage[pdftex]{graphicx}


\graphicspath{{graphics/}}

\DeclareGraphicsExtensions{.pdf,.jpeg,.png}



\def\goat#1{{\scriptsize\color{green}{[#1]}}}



\let\olditem\item
\renewcommand{\item}{%
\olditem\vspace{5pt}}

\title{Reducing Complexity in RE System Optimisations}
%\subtitle{---}
\author{
  Tom Brown
}

\institute{
  Frankfurt Institute of Advanced Studies (FIAS), Goethe-Universit\"at Frankfurt \\
  FIAS Renewable Energy System and Network Analysis (FRESNA)
}

\date{NEXT ENERGY 2. Themenfeldseminar Systemmodellierung und -analyse, 22nd February 2016}


\titlegraphic{%
  \vspace{6.5cm}
    \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{Introduction}


\begin{frame}[fragile]
  \frametitle{Goal}

Given a desired CO${}_2$ reduction, what is the most cost-effective system?

\begin{equation*}
  \textrm{Min} \left(\parbox{6em}{Yearly system costs}\right) = \sum_n \left(\parbox{6em}{Annualised capital costs}\right) + \sum_{n,t} \left(\parbox{6em}{Marginal costs}\right)
\end{equation*}
subject to
\begin{itemize}
\item meeting energy demand at each node $n$ and time $t$
\item wind, solar, hydro (VRE) availability $\forall n,t$
\item electricity/gas transmission constraints between nodes
\item (installed capacity) $\leq$ (geographical potential)
\item CO${}_2$ constraint / RE share covering demand
\end{itemize}


\end{frame}


\begin{frame}[fragile]
  \frametitle{Interesting Questions}
  How do system characteristics and costs change as we \dots
  \begin{itemize}
  \item \dots restrict transmission expansion?
  \item \dots impose import/export balance constraints?
  \item \dots relax CO${}_2$ constraint?
  \item \dots include non-pumped-hydro storage?
  \item \dots include meshed overlay direct current network and/or more phase-shifting transformers?
  \item \dots couple electricity sector to heating/cooling and transport?
  \end{itemize}

Some of these questions are driven by politics/social factors: compromises that take us away from economic optimum.

\end{frame}



\begin{frame}[fragile]
  \frametitle{Problem 1: Spatial resolution}

Need high spatial resolution to represent VRE resource variation and
transmission constraints in electricity and gas networks.


\centering
\includegraphics[width=8cm]{ENTSO-E_map_2012-low}
\source{ENTSO-E}


\end{frame}




\begin{frame}[fragile]
  \frametitle{Problem 2: Temporal resolution}

Need high temporal resolution to represent load and VRE resource
variability and correlations. Wind generation in Europe in July 2013:


\centering
\includegraphics[width=11cm]{various_wind}


\end{frame}



\begin{frame}[fragile]
  \frametitle{Problem 3: Model accuracy}


Modelling must respect physics.



  \begin{itemize}
  \item How much detail in the input data do we need?
\item Optimise transmission simultaneously with generation capacity?
\item How bad are linear approximations?
\item Can we make the algorithms faster, to add detail in other areas?
\item By looking at static situations, do we miss dynamic effects?
  \end{itemize}



\end{frame}




\begin{frame}
  \frametitle{Examples from literature of electricity optimisation}


\ra{1.05}
\begin{table}[!t]
	\centering
	\begin{tabular}{@{}lllll@{}}
\toprule
Study &  Spatial      &  Temporal & What? & Flow \\
& resolution & resolution &  & physics \\
\midrule
Czisch (2005) & low & high & g and t & transport \\
Hagspiel et al. (2014) & medium & low & g and t & linear \\
Egerer et al. (2014) & high & low &  t only & linear \\
\bottomrule
	\end{tabular}
\end{table}

\vspace{0.5cm}

\begin{columns}[T]
\begin{column}{3.5cm}

\centering

Czisch

\includegraphics[width=3cm]{czisch-cropped}
\end{column}
\begin{column}{3.5cm}


\centering


Hagspiel et al.

\includegraphics[width=3cm]{ena-ewi-cropped}
\end{column}
\begin{column}{3.5cm}

\centering

Egerer et al.

\includegraphics[width=3cm]{egerer-cropped}
\end{column}

\end{columns}




\end{frame}




\begin{frame}[fragile]
  \frametitle{Overarching goal}


  Find the ``sweet spot'' where:
  \begin{itemize}
  \item Computation time is finite (i.e. a week)
  \item Temporal resolution is ``good enough''
  \item Spatial resolution is ``good enough''
  \item Model detail is ``good enough''
  \end{itemize}

  AND quantify the error we make by only being ``good enough''

  (e.g. are important metrics $\pm 10\%$ or $\pm 50\%$ correct?)


\end{frame}





\section{Example: Electricity system optimisation}



\begin{frame}
  \frametitle{Objective function}


  Objective function is total annual system cost, whose minimisation represents maximisation of ``social welfare'' (for fixed inelastic demand):
  \begin{equation*}
    f(\bar{P}_l,  \bar{g}_{n,s},  g_{n,s,t}) =  \sum_{l} (c_l + o_l) \bar{P}_l + \sum_{n,s} c_{n,s} \bar{g}_{n,s} + \sum_{n,s,t} w_t o_{n,s} g_{n,s,t}
  \end{equation*}
  We optimise:
  \begin{itemize}
  \item the transmission capacity of all the lines $l$ ($\bar{P}_l$)
  \item the generation capacities of all technologies (wind/solar/gas etc.) $s$ at each node $n$
  \item the dispatch of each generator 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{Main constraints}

  \begin{itemize}
  \item Demand $d_{n,t}$ is always met by generation or transmission
    \begin{equation*}
      d_{n,t} = \sum_{s} g_{n,s,t} + \sum_{l \in n} f_{l,t}
    \end{equation*}
  \item Dispatch $g_{n,s,t}$ cannot exceed availability $\bar{g}_{n,s,t}$
    \begin{equation*}
      0 \leq g_{n,s,t} \leq \bar{g}_{n,s,t} \leq  \bar{g}_{n,s}
    \end{equation*}
  \item Rated capacity cannot exceed the installable potential $\hat{g}_{n,s}$
    \begin{equation*}
       \bar{g}_{n,s}  \leq  \hat{g}_{n,s}
    \end{equation*}
  \item Transmission flows cannot exceed capacities
    \begin{equation*}
      | f_{l,t} | \leq  \bar{P}_l
    \end{equation*}
  \item CO${}_2$ constraint is respected
    \begin{equation*}
      \sum_{n,s,t} \frac{1}{\eta_{n,s}} g_{n,s,t} e_{n,s} \leq CAP
    \end{equation*}
  \end{itemize}

\end{frame}


\begin{frame}
  \frametitle{Nota bene}

  \begin{itemize}
  \item Renewables can be arbitrarily curtailed if that's cost-effective
  \item No assumptions made about which technology is preferred
  \item No assumptions made about spatial distribution of generation
  \item No assumptions made about transmission capacity
  \item Transmission capacity is optimized jointly with generation,
    e.g.  sites far from loads requiring lots of transmission are
    balanced against sites nearer loads; also balancing made in time,
    e.g. transmission line not necessarily built out for extreme
    events
  \end{itemize}
\end{frame}



\begin{frame}
  \frametitle{Example application: 30-node model of Europe}


  `Toy' model: 30 European countries optimised over all 8760 hours of a sample year, only wind, solar and gas generation (hydro is fiddly)

\begin{columns}[T]
\begin{column}{6.5cm}
\includegraphics[width=7cm]{europe-map}
\end{column}
\begin{column}{4.5cm}
  \begin{itemize}
  \item Minimise electricity system costs (generation \& transmission)
  assuming CO${}_2$ reduction of 90\%
  \item Energy consumed:
86\% wind, 10\% gas, 4\% solar
\item 16\% curtailment of available wind energy
\item Total system cost of \euro~131 billion / year $\sim$ \euro~41~/MWh
  \end{itemize}

\end{column}
\end{columns}



\end{frame}



\begin{frame}
  \frametitle{Cost and other assumptions}

\begin{table}[!t]
	\centering
	\begin{tabular}{@{}lrl@{}}
\toprule
Quantity &  Cost     &  Unit \\
\midrule
Wind onshore capital & 1000  & \euro/kW \\
Solar capital & 1000  & \euro/kW \\
Gas capital & 700  & \euro/kW \\
Gas marginal & 50 & \euro/MWh\\
Transmission line & 400 & \euro/MW/km \\
Gas CO${}_2$ emissions & 0.2 & t/MWh${}_{\textrm{thermal}}$ \\
EEA electricity CO${}_2$ emissions 2015 & $1.5$ & Gt/a  \\
Gas plant efficiency & 40 & \% \\
Interest rate &  5  & \% \\
Line lifetime &  40  & years \\
Generators lifetime &  20 & years \\
\bottomrule
	\end{tabular}
\end{table}

\end{frame}





\begin{frame}
  \frametitle{Projects at FIAS over next 18 months}

  \begin{itemize}
  \item Look at coupling electricity with heating and transport,
    probably at country-scale aggregation (most studies on sector
    coupling have either no spatial resolution or no temporal resolution, missing
    Europe-wide smoothing benefits)
  \item Look at electricity sector optimisation with high spatial resolution to see how far it deviates from country-aggregated solutions
  \end{itemize}
\end{frame}




\section{Network reduction}

\begin{frame}
  \frametitle{Reducing spatial resolution}

We need spatial resolution to:
\begin{itemize}
\item capture the geographical variation of renewables resources (Niedersachsen versus Bayern) and the load
\item capture spatio-temporal effects (e.g. size of wind correlations across the continent)
\item represent important transmission constraints
\end{itemize}


BUT we do not want to have to model all 10,000 network nodes of the European system.

We want to aggregate nodes where load/RE resources are highly correlated
and where there are no network bottlenecks inside the clusters (very
similar to the concept of the electricity market bidding zone).


\end{frame}




\begin{frame}
  \frametitle{Many algorithms in the literature}

  There are lots of algorithms, particularly in the engineering literature:
  \begin{itemize}
  \item $k$-means clustering on (electrical) distance
    \item $k$-means on load distribution
    \item Community clustering (e.g. Louvain)
    \item Spectral analysis of Laplacian matrix
      \item Clustering of Locational Marginal Prices with nodal pricing (sees congestion and RE generation)
      \item PTDF clustering
        \item Cluster nodes with correlated RE time series
  \end{itemize}


  The algorithms all serve different purposes (e.g. reducing part of
  the network on the boundary, to focus on another part).

  Not always tested on real network data.

\end{frame}




\begin{frame}
  \frametitle{Our metrics of interest}

  What we want from a network aggregation algorithm:

  \begin{enumerate}
  \item Preservation of major flows within original network
  \item Preservation of overall volume of flows
  \item For capacity optimisation: representative capacity extensions with aggregated network
  \item Preservation of spatial distribution of generation capacity
  \end{enumerate}

\end{frame}


\begin{frame}
  \frametitle{Best method so far: load custering}

  Cluster nodes based on load using $k$-means.

  I.e. find $k$ centroids and the corresponding $k$-partition of the
  original nodes that minimises the sum of squared distances from each
  centroid to its nodal members:

  \begin{equation}
    \min_{\{x_c\}} \sum_{c=1}^k \sum_{n \in N_c} w_n || x_c - x_n ||^2
  \end{equation}

  where each node is weighted $w_n$ by the average load there.


  NB: Totally ignores grid topology. It works because network is
  principally laid out to serve the load (with exception of large
  conventional power plants situated near e.g. mines/rivers).
\end{frame}



\begin{frame}
  \frametitle{$k$-means clustering}

\centering
\includegraphics[width=11cm]{k_means_clustering-six}
\source{SciGRID (2015), OpenStreetMap}


\end{frame}



\begin{frame}
  \frametitle{Compare flows}

  Compare the aggregated `microscopic' flows in the original network to the
  `macroscopic' flows in the aggregated network.

  We have two networks, our original one $1$ and our aggregated one $2$.

  We have some $N_2 \times N_1$ matrix map $B$ from the buses in $1$ to $2$ with ones wherever a bus in $1$ is aggregated to a bus in $2$.

  Similarly we have an $L_2 \times L_1$ matrix map $L$ from the lines in $1$ to the lines in $2$.

  Aggregated microscopic flows:
  \begin{equation}
    L \cdot PTDF_1 \cdot P
  \end{equation}
  where $P$ is an $N_1$ vector of power imbalances at the buses of $1$ and $PTDF$ is the linear Power Transfer Distribution Factor.

  Macroscopic flows in aggregated network:
  \begin{equation}
    PTDF_2 \cdot B \cdot P
  \end{equation}


\end{frame}



\begin{frame}
  \frametitle{$k$-means clustering correlation}

\centering
\includegraphics[width=11cm]{k_means-correlation-six}


\end{frame}



\begin{frame}
  \frametitle{$k$-means clustering Pearson}

\centering
\includegraphics[width=11cm]{k_means-pearson}


\end{frame}


\begin{frame}
  \frametitle{$k$-means clustering total loading}

\centering
\includegraphics[width=11cm]{k_means-total_loading}


\end{frame}



\section{Time series reduction}



\begin{frame}
  \frametitle{Basic idea}

It
is often desirable to take historical data, for example electrical
load, wind and solar generation for many locations sampled hourly
over several years, and then reduce the snapshots
to a representative sample.

This is often done fairly manually, e.g. summer/winter + high/low wind/load days.


Can we do this more systematically and algorithmically?

The sample should be representative in the following senses:
\begin{enumerate}
\item It should reproduce statistics of the full historical data, such
  as the mean, (co-)variance and higher moments of the time series (so
  that, for example, capacity factors and spatial correlations for
  wind and solar are preserved).
\item It should preserve extreme events, such as high electrical demand
  and low VRE feed-in, which determine backup generation capacities.
\end{enumerate}

\end{frame}


\begin{frame}
  \frametitle{More mathsy version}


Suppose we have a set $I$ of time series over times $T$, so that the
value of each time series $i \in I$ at time $t\in T$ is given by
$x_i(t)$. We treat this like an $|I|$-dimensional random vector ${\bf X} = \{X_i\}$ over the event space of
time (with uniform probability measure).

The goal is to find a smaller sample $|S| \leq |T|$ and a weighting
$w:S \to [0,1]$ such that common statistics are preserved
\begin{equation*}
 \mathbb{E} \left[f(\mathbf{X})\right] \sim \sum_{t\in T} \frac{1}{|T|} f(\mathbf{x}(t)) \sim    \sum_{s\in S} w(s) f(\mathbf{x}(s))
\end{equation*}
(depending on the statistic, corrections can be made to get unbiased estimators)
and that extremes are also preserved
\begin{equation*}
  \max_T/\min_T  x_i(t) \sim   \max_S/\min_S  x_i(s)
\end{equation*}

Note that the choice of $S$ and $w$ depends on all the samples
$x_i(t)$.  The weight imitates a probability measure and satisfies
$\sum_{s\in S}w(s) = 1$.


\end{frame}



\begin{frame}
  \frametitle{2d example: wind production in Germany and Denmark}

\begin{columns}[T]
\begin{column}{6.5cm}
\includegraphics[width=7.5cm]{DE-wind-DK-wind-fourplot.png}
\end{column}
\begin{column}{4cm}
  \begin{itemize}
  \item $k$-means clustering creates $k$ clusters
\item Centroids chosen  to minimise the sum of
  squared distances between the centroids and the original points
\item $w(s) \propto$  number of points assigned to each cluster
\item Envelope $\sim$ convex hull not so well captured
  \end{itemize}

\end{column}
\end{columns}

\end{frame}




\begin{frame}
  \frametitle{Statistics over time for clustering of $DE-DK$ wind}

\begin{columns}[T]
\begin{column}{8cm}
\includegraphics[width=8.5cm]{DE-DK-stats}
\end{column}
\begin{column}{4cm}
  \begin{itemize}
  \item Mean and (co-)variance well-preserved as number of clusters
    decreases
  \item Below 10 clusters, the extremes (min and max) are not so well
    captured
  \end{itemize}

\end{column}
\end{columns}


\end{frame}




\begin{frame}
  \frametitle{European example}

  For a bigger European example, enlarge the dimensionality of the random vector from 2 to
90 $=$ (30 countries) $\times$ (wind, solar, load)


\begin{columns}[T]
\begin{column}{8cm}
\includegraphics[width=8.5cm]{EU-stats}
\end{column}
\begin{column}{4cm}
  \begin{itemize}
  \item Means always preserved, thanks to algorithm
  \item With higher dimensionality, more clusters are required to
    preserve other statistics
  \end{itemize}

\end{column}
\end{columns}



\end{frame}



\begin{frame}
  \frametitle{European example without logarithmic scale}

\centering

  Removing the logarithmic scale, it's not so bad...

\includegraphics[trim=0cm 0.7cm 0cm 0cm,width=9cm,clip=True]{EU-stats-non_log}


\end{frame}



\begin{frame}
  \frametitle{Example application: 30-node model of Europe}

Demonstrate effects on optimisation with example we can solve for all 8760 hours of a sample year: the 30-node European model

\begin{columns}[T]
\begin{column}{6.5cm}
\includegraphics[width=7cm]{europe-map}
\end{column}
\begin{column}{4.5cm}
  \begin{itemize}
  \item Minimise electricity system costs (generation \& transmission)
  assuming CO${}_2$ reduction of 90\%
  \item Energy consumed:
86\% wind, 10\% gas, 4\% solar
\item 16\% curtailment of available wind energy
\item Total system cost of \euro~131 billion / year $\sim$ \euro~41~/MWh
  \end{itemize}

\end{column}
\end{columns}



\end{frame}


\begin{frame}
  \frametitle{Energy progression}



The European electrical load of $\sim 3200$ TWh/a is mostly met by wind.


\begin{columns}[T]
\begin{column}{8cm}
\includegraphics[width=8.5cm]{nc_energy}

\end{column}
\begin{column}{4cm}
  \begin{itemize}
    \item Main feature is preserved: domination of wind
  \item For low number of clusters, optimisation doesn't see hours with
    high load and low VRE, so removes backup energy
  \end{itemize}

\end{column}
\end{columns}


\end{frame}




\begin{frame}

  \frametitle{Cost progression}



\begin{columns}[T]
\begin{column}{8cm}
\includegraphics[width=8.5cm]{nc_cost}

\end{column}
\begin{column}{4cm}
  \begin{itemize}
    \item This story is repeated in the cost summation
    \item Costs with just one cluster are 34\% lower than with all snapshots
    \item Balance between generation and transmission is well preserved
  \end{itemize}

\end{column}
\end{columns}


\end{frame}


\begin{frame}

  \frametitle{Installed capacity / peak load progression}



\begin{columns}[T]
\begin{column}{8cm}
\includegraphics[width=8.5cm]{nc_power}

\end{column}
\begin{column}{4cm}
  \begin{itemize}
    \item Total wind and solar capacities remain relatively consistent
      \item Backup gas capacity determined by maximum residual load (load -
        VRE), which drops from around 260~GW to 0~GW as number of clusters decreases
  \end{itemize}

\end{column}
\end{columns}


\end{frame}





\begin{frame}

  \frametitle{Installed capacity distribution}



\begin{columns}[T]
\begin{column}{8cm}
\includegraphics[width=8.5cm]{nc_power_distribution}

\end{column}
\begin{column}{4cm}
  \begin{itemize}
    \item Distribution of capacity around Europe shows some variation
      in countries which are not at the upper capacity potential limit (France,
      Germany for wind and both Italy and Greece for solar)
  \end{itemize}

\end{column}
\end{columns}


\end{frame}







\begin{frame}

  \frametitle{Computation time progression}

  The pay-off is big in terms of computational time:

\begin{columns}[T]
\begin{column}{8cm}
\includegraphics[width=8.5cm]{nc_time-log}

\end{column}
\begin{column}{4cm}
  \begin{itemize}
    \item Reducing number of snapshots by factor 8 reduces computation
      time from 9.4 hours to 11 minutes.
      \item This gain comes with little loss of modelling accuracy.
  \end{itemize}

\end{column}
\end{columns}


\end{frame}




\begin{frame}
  \frametitle{Forcing extreme events}

  By analysing tail of high residual load (load - VRE), can force in
  extreme events with appropriate weighting/measure (i.e. include
  once-in-ten-years event) to capture backup capacity correctly.

\centering
\includegraphics[trim=0cm 0cm 0cm 0cm,width=9cm]{residual_load_plot}


\end{frame}








\begin{frame}
  \frametitle{Distribution of extreme events: 5 year sample QQ}


\begin{columns}[T]
\begin{column}{5cm}
\includegraphics[width=5.5cm]{loads_43822_quantiles.png}

\end{column}
\begin{column}{4cm}
  \begin{itemize}
    \item Both load and residual exhibit shorter tails than the normal
      distribution
    \item This is GOOD - it means extreme events are less likely than
      normal
  \end{itemize}

\end{column}
\end{columns}


\end{frame}





\begin{frame}
  \frametitle{Fitting the tails}


\begin{columns}[T]
\begin{column}{5cm}
\includegraphics[width=5.5cm]{load_selection_43822_lambda_quantiles.png}

\end{column}
\begin{column}{4cm}
  \begin{itemize}
  \item Can fit them to Tukey-Lambda with value $\lambda = 0.239$ (normal distribution is $\lambda = 0.14$; fatter tails are $\lambda < 0.14$, like logistic or t-distribution; $\lambda=1$ is the uniform distribution)
  \end{itemize}

\end{column}
\end{columns}


\end{frame}


\begin{frame}
  \frametitle{Results: extreme event maintain total cost better}

  Inserting the once-in-five-years extreme event of high residual load into the data with the correct weight means the total cost is better kept as you reduce the number of nodes
\includegraphics[width=10cm]{cost_extreme_comparison.pdf}

\end{frame}



\begin{frame}

  \frametitle{Line cost experiment 1/2}

  Can do simple sensitivity analysis very quickly, e.g. on changing line cost.

  See effect of restricting transmission: Far left is Europe as copper-plate (as much transmission as necessary), far right is no transmission.

\begin{columns}[T]
\begin{column}{8cm}
\includegraphics[width=8.5cm]{line_cost-high_res-example}

\end{column}
\begin{column}{4cm}
  \begin{itemize}
    \item Model results {\bf very} sensitive to change in line costs
      \item If line costs increase by factor 10 ($\sim$ using underground cables instead of overhead lines), only 32\% of lines get built and system costs increase by over 40\%
  \end{itemize}

\end{column}
\end{columns}


\end{frame}





\begin{frame}

  \frametitle{Line cost experiment 2/2}

\begin{columns}[T]
\begin{column}{8cm}
\includegraphics[width=8.5cm]{line_cost-high_res-energy}

\end{column}
\begin{column}{4cm}
  \begin{itemize}
  \item Curtailment of wind jumps from 14\% to 59\% as transmission restricted
    \item Increasing shares of solar as transmission is restricted, with even higher curtailment than for wind (68\% curtailment)
  \end{itemize}

\end{column}
\end{columns}


\end{frame}






\begin{frame}

  \frametitle{CO${}_2$ restriction experiment 1/3}

  See effect of reducing backup energy, which in this model is $\propto$ CO${}_2$ (although in reality we have carbon-free hydroelectricity)

\begin{columns}[T]
\begin{column}{8cm}
\includegraphics[width=8.5cm]{vary_co2-energy}

\end{column}
\begin{column}{4cm}
  \begin{itemize}
  \item From 20\% and higher, CO${}_2$ restriction is no longer
    binding  $\implies$ low-CO${}_2$ is cheaper!
  \item Cost-optimal backup around 15-20\% by energy, which is exactly
    level of hydroelectricity
  \item Lower than 1\%, curtailment sky-rockets
  \end{itemize}

\end{column}
\end{columns}


\end{frame}







\begin{frame}

  \frametitle{CO${}_2$ restriction experiment 2/3}

\begin{columns}[T]
\begin{column}{8cm}
\includegraphics[width=8.5cm]{vary_co2-cost}

\end{column}
\begin{column}{4cm}
  \begin{itemize}
  \item Cost sky-rockets below 1\% but quite flat about 5\%, so cost-optimum does not require much backup energy
  \end{itemize}

\end{column}
\end{columns}


\end{frame}




\begin{frame}

  \frametitle{CO${}_2$ restriction experiment 3/3}

\begin{columns}[T]
\begin{column}{8cm}
\includegraphics[width=8.5cm]{vary_co2-power-gas}

\end{column}
\begin{column}{4cm}
  \begin{itemize}
  \item But backup capacity still remains high
  \item Ideally still require 50\% of peak load
    \item Political/security considerations might still require 100-110\% backup capacity to peak load ratio
  \end{itemize}

\end{column}
\end{columns}


\end{frame}



\section{Improving optimisation formulation}

\begin{frame}
  \frametitle{Attacking model formulation}

Reformulate and benchmark linear optimal power flow formulation:
\begin{columns}[T]
\begin{column}{7cm}
\begin{itemize}
\item Power Transfer Distribution Factor (PTDF) (dense but no auxilliary variables)
  \begin{equation*}
    F_\ell = \sum_i PTDF_{\ell i} P_i
  \end{equation*}
\item Voltage angles (sparse but $N$ auxilliary variables)
  \begin{equation*}
    F_\ell = \sum_{k,i} B_{\ell k} K^T_{ki} \theta_i
  \end{equation*}
\end{itemize}
\end{column}
\begin{column}{4cm}
\vspace{1cm}
\includegraphics[width=4.5cm]{cycles-cropped}
\end{column}
\end{columns}

\begin{itemize}
\item Kirchoff's voltage law around closed cycles (cf. \emph{Ronellenfitsch, Timme,
  Witthaut (2015)} - semi-sparse and only $L-N+1$ auxilliary variables)
  \begin{equation*}
    F_\ell = T_{\ell} + \sum_{c } C_{\ell c} L_c
  \end{equation*}
\end{itemize}


\end{frame}



\section{Conclusions}



\begin{frame}
  \frametitle{Conclusions}

  \begin{itemize}
    \item There is lots of room for improvement in overall system optimisation.
    \item Data reduction in time and space can improve optimisation speed, allowing more interesting modelling complexity.
    \item Moreoever data reduction is essential as the modelling
      becomes more detailed (integrating other sectors like transport
      and heating)
    \item Learning where data reduction does not affect the results tells us important information about the system sensitivities.
    \item Extreme events must be balanced carefully against ``typical'' system behaviour, depending on their effect on the results.
    \item New algorithms can also speed up the optimisation.
  \end{itemize}

\end{frame}

\begin{frame}
  \frametitle{Open Modelling Adverts}

Small advert 1: We'd like our modelling and results to be as
transparent and repeatable as possible. Therefore we aim to make all
our code and data available. All the software for the optimisation is
available here:

https://github.com/FRESNA/PyPSA

``Python for Power System Analysis'' also does non-linear power flow for those interested in voltage
and reactive power.

\vspace{.7cm}

Small advert 2: Open Energy Modelling Initiative
\url{http://openmod-initiative.org/} - join the mailing list! Let's
put an end to pointlessly-repeated data munging and increase transparency.
Workshop at KTH, Stockholm, 28-29 April, 2016.

\end{frame}



\end{document}
