esm-lecture-13.tex

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\title{Energy System Modelling\\ Summer Semester 2020, Lecture 13}
%\subtitle{---}
\author{
  {\bf Dr. Tom Brown}, \href{mailto:tom.brown@kit.edu}{tom.brown@kit.edu}, \url{https://nworbmot.org/}\\
  \emph{Karlsruhe Institute of Technology (KIT), Institute for Automation and Applied Informatics (IAI)}
}

\date{}


\titlegraphic{
  \vspace{0cm}
  \hspace{10cm}
    \includegraphics[trim=0 0cm 0 0cm,height=1.8cm,clip=true]{kit.png}

\vspace{5.1cm}

  {\footnotesize

  Unless otherwise stated, graphics and text are Copyright \copyright Tom Brown, 2020.
  Graphics and text for which no other attribution are given are licensed under a
  \href{https://creativecommons.org/licenses/by/4.0/}{Creative Commons
  Attribution 4.0 International Licence}. \ccby}
}

\begin{document}

\maketitle


\begin{frame}

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


\section{Workflow management for complex models}


\begin{frame}{Motivation: More detailed data-driven models, more scenarios}


When we upgrade from the 30-node model to 5000+ nodes for a detailed grid model, and calculate 100 scenarios with different settings, \alert{data management becomes important}.

\begin{columns}[T]
  \begin{column}{6cm}

    % left bottom right top
    \includegraphics[trim=0 0.7cm 0 3.5cm,width=6.4cm]{europe_map}
  \end{column}

  \begin{column}{6cm}

    \includegraphics[width=6.5cm]{pypsa-eur-grid.pdf}
  \end{column}
\end{columns}



\end{frame}




\begin{frame}
  \frametitle{Data-Driven Modelling}

  Lots of different types of data come together for the modelling...


    \includegraphics[width=14cm]{data_driven.png}

\end{frame}



\begin{frame}
  \frametitle{Problems}

  \begin{itemize}
  \item Many different data sources
  \item Many data sources need cleaning and processing before they can be used
  \item Many intermediate scripts and datasets
  \item Many colleagues hack something together in a folder - hard to reproduce later
  \item Often dependencies are not clear (both data and software)
  \item Data and code change over time
  \item Want to run many parameteric scenarios for same model
  \end{itemize}

  \vspace{.5cm}
  \pause
  What we need is a \alert{workflow management tool}.

\end{frame}

\begin{frame}
  \frametitle{Workflow Management Tool: Snakemake}

  ``The \href{https://snakemake.readthedocs.io/en/stable/}{Snakemake} workflow management system is a tool to create \alert{reproducible and scalable data analyses}. Workflows are described via a human readable, Python based language. They can be seamlessly scaled to server, cluster, grid and cloud environments, without the need to modify the workflow definition. Finally, Snakemake workflows can entail a description of required software, which will be automatically deployed to any execution environment.''

 \vspace{1cm}

  See \href{https://slides.com/johanneskoester/snakemake-short}{\bf\color{blue}\underline{Snakemake presentation}}.

\end{frame}


\begin{frame}
  \frametitle{Snakemake for PyPSA-Eur: Building Model}

    The \href{https://snakemake.readthedocs.io/en/stable/}{\bf\color{blue}\underline{Snakemake}} workflow management system is a tool to create \alert{reproducible and scalable data analyses}. Dependency graph: \alert{nodes} for scripts, \alert{directed edges} map outputs to inputs.


    \vspace{.3cm}

  \includegraphics[width=14.8cm]{workflow.png}
\end{frame}


\begin{frame}
  \frametitle{Snakemake for PyPSA-Eur: Managing Scenarios}

  \centering
  \includegraphics[width=14cm]{scenarios.png}
\end{frame}



\begin{frame}
  \frametitle{PyPSA-Eur}

\begin{columns}[T]
  \begin{column}{4cm}

    \vspace{1.5cm}

    See \href{https://docs.google.com/presentation/d/1mzj4X9uuO58gUvkhVMRCFWOJUWbs6NR9SNZe-RIkkNo/edit?usp=sharing}{\bf\color{blue}\underline{PyPSA-Eur slidedeck}} for breakdown of PyPSA-Eur snakemake rules.

    \vspace{.5cm}

    Checkout also the \href{https://github.com/PyPSA/pypsa-eur}{\bf\color{blue}\underline{PyPSA-Eur GitHub repository}}.
  \end{column}

  \begin{column}{8cm}

    \vspace{.2cm}


\includegraphics[width=8.5cm]{pypsa-eur-grid}
  \end{column}
\end{columns}


\end{frame}

\begin{frame}
  \frametitle{PyPSA-Eur-Sec builds on PyPSA-Eur}

  \begin{columns}[T]

  \begin{column}{7cm}

    \vspace{.2cm}

    \begin{itemize}
      \item \href{https://github.com/PyPSA/pypsa-eur-sec}{\bf\color{blue}\underline{PyPSA-Eur-Sec}} includes PyPSA-Eur as a \alert{snakemake subworkflow}.
      \item PyPSA-Eur builds the clustered power grid and renewable generators.
      \item Then PyPSA-Eur-Sec adds other sectors.
    \end{itemize}

    \vspace{.2cm}
    \centering
    \includegraphics[trim={1.5cm 1.5cm 1.5cm 1.5cm},clip,width=4.5cm]{today.pdf}
  \end{column}

    \begin{column}{5.5cm}

    \vspace{1cm}

  \centering
  \includegraphics[width=6cm]{20200223_multisector_figure.pdf}
  \end{column}
\end{columns}

\end{frame}



\section{Effect of spatial scale on results of energy system optimisations}




\begin{frame}{Motivation: Transmission bottlenecks}

  Many of the results we've examined so far have aggregated countries
  to a single node. However, there are also transmission network
  bottlenecks \alert{within} countries (e.g. North to South Germany).


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


\end{frame}




\begin{frame}{Motivation: Wind and solar resource variation}

  There is also considerable variation in wind and solar resources...

    \begin{columns}[T]
\begin{column}{5cm}
  \includegraphics[trim=0 0cm 0 0cm,width=4.7cm,clip=true]{SolarGIS-Solar-map-Germany-de}
\end{column}
\begin{column}{6cm}
  \vspace{.3cm}
  \includegraphics[trim=0 0cm 0 0cm,width=6cm,clip=true]{43_Mittlere_Windgeschwindigkeit_100_m_Deutschland}
\end{column}
    \end{columns}

\end{frame}





\begin{frame}
  \frametitle{Spatial resolution}

\begin{columns}
\begin{column}{7cm}
We need spatial resolution to:
\begin{itemize}
\item capture the \alert{geographical variation} of renewables resources and the load
\item capture \alert{spatio-temporal effects} (e.g. size of wind correlations across the continent)
\item represent important \alert{transmission constraints}
\end{itemize}

BUT we do not want to have to model all 5,000 network nodes of the European system.
\end{column}
\begin{column}{6cm}
\centering
% No, we really do not not want to optimise a model of 4653 substation, 5613 AC lines and 26 DC lines of the European HV electricity network.
\includegraphics[width=6.5cm]{pre-2-network-full.pdf}
\source{Own representation of Bart Wiegman's GridKit extract of the
  online ENTSO-E map, \url{https://doi.org/10.5281/zenodo.55853}}
\end{column}
\end{columns}
\end{frame}

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

  There are lots of algorithms for clustering networks, 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{$k$-means clustering on load \& conventional generation}

  Our \alert{goal}: maintain main transmission corridors of today to
  investigate highly renewable scenarios with no grid expansion. Since generation fleet is totally rebuilt, do not want to rely on current generation dispatch (like e.g. LMP algorithm).

  Today's grid was laid out to connect big generators and load centres.

  \begin{columns}

    \begin{column}{5cm}

      \vspace{0.5cm}
  \alert{Solution}:
  Cluster nodes based on spatial distribution using $k$-means, with a weighting to sites with higher average load and conventional generation capacity.
    \end{column}

    \begin{column}{7cm}
      \vspace{.3cm}
            \includegraphics[width=6.5cm]{clustering.pdf}
    \end{column}
\end{columns}
\end{frame}

\begin{frame}
  \frametitle{$k$-means clustering on load \& conventional generation}

  Suppose the $N$ nodes $i$ have spatial coordinates $(x_i,y_i)$. The $k$-means algorithm works by partitioning them into $k\leq N$ sets $N_c$ for $c = 1, \dots k$ such that the sum of squared distance to the centroid $(x_c,y_c)$ (mean point inside each set) is minimised:
  \begin{equation*}
    \min_{\{(x_c,y_c)\}} \sum_{c=1}^k \sum_{i \in N_c} w_i \left|\left| \left(\begin{matrix}
      x_c \\
      y_c
\end{matrix}\right) -  \left(\begin{matrix}
      x_i \\
      y_i
\end{matrix}\right)\right|\right|^2
  \end{equation*}
  Each node $i$ is weighted $w_i$ by the average load and the
  average conventional generation there.

  Use the centroid as the location of the new clustered node.
\end{frame}



\begin{frame}
  \frametitle{Reconstitution of network}

  Once the partition of nodes is determined:
  \begin{itemize}
  \item A new node is created to represent each set of clustered nodes
  \item Hydro capacities and load is aggregated at the node; VRE (wind and solar) time series are aggregated, weighted by capacity factor; potentials for VRE aggregated
  \item Lines between clusters replaced by single line with length 1.25 $\times$ crow-flies-distance, capacity and impedance according to replaced lines
    \item $n-1$ blanket safety margin factor grows from 0.3 with $\geq 200$ nodes to 0.5 with 37 nodes (to account for aggregation)
  \end{itemize}

\end{frame}

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

\centering
 \includegraphics[width=13cm]{networks-clustered-crop}
  % $\vcenter{\hbox{\includegraphics[width=4cm]{pre-2-network-full}}}$
  % $\vcenter{\hbox{\includegraphics[width=4cm]{pre-2-network-362-LV-1}}}$
  % $\vcenter{\hbox{\includegraphics[width=4cm]{pre-2-network-181-LV-1}}}$ \\
  % $\vcenter{\hbox{\includegraphics[width=4cm]{pre-2-network-128-LV-1}}}$
  % $\vcenter{\hbox{\includegraphics[width=4cm]{pre-2-network-64-LV-1}}}$
  % $\vcenter{\hbox{\includegraphics[width=4cm]{pre-2-network-37-LV-1}}}$



\end{frame}



\begin{frame}
  \frametitle{Question of spatial resolution}

  How is the overall minimum of the cost objective (building and
  running the electricity system) affected by an increase of spatial
  resolution in each country?

  We expect
  \begin{itemize}
  \item A better representation of existing internal bottlenecks will prevent the transport of e.g. offshore wind to the South of Germany.
  \item Localised areas of e.g. good wind can be better exploited by the optimisation.
  \end{itemize}

  Which effect will win?

  First we only optimize the gas, wind and solar generation
  capacities, the long-term and short-term storage capacities and
  their economic dispatch including the available hydro facilities
  \alert{without grid expansion}.
\end{frame}



\begin{frame}
  \frametitle{Nodal energy shares per technology (w/o grid expansion)}

\includegraphics[width=15cm]{legend-flat}
\begin{columns}[T]
\begin{column}{8cm}


  \includegraphics[trim=0 0cm 0 2cm,width=8cm]{euro-pie-pre-7-branch_limit-1-37.png}

\end{column}
\begin{column}{8cm}

  \raisebox{-6.6cm}{

  \includegraphics[trim=0 0cm 0 2cm,width=8cm]{euro-pie-pre-7-branch_limit-1-256.png}
  }

\end{column}

\end{columns}


\end{frame}

\begin{frame}
   \frametitle{Costs: System cost w/o grid expansion}


\begin{columns}[T]
\begin{column}{8.5cm}
  \includegraphics[width=9.5cm]{cost_eu_1}
\end{column}
\begin{column}{6cm}

  \vspace{2cm}

  \begin{itemize}
  \item Steady total system cost at \euro~260 billion per year
   \item This translates to \euro~82/MWh (compared to today of \euro~50/MWh to \euro~60/MWh)
  \end{itemize}
\end{column}
\end{columns}

\end{frame}


\begin{frame}
  \frametitle{Costs: System cost and break-down into technologies {\large (w/o grid expansion)}}
\begin{columns}[T]
\begin{column}{8.5cm}
  \includegraphics[width=8.5cm]{cost_eu_breakdown_by_cap-1}
\end{column}
\begin{column}{6cm}

  If we break this down into technologies:
  \begin{itemize}
  \item 37 clusters captures around half of total network volume
  \item Redistribution of capacities from offshore wind to solar
  \item Increasing solar share is accompanied by an increase of
    battery storage
  \item Single countries do not stay so stable
  \end{itemize}
\end{column}
\end{columns}
\end{frame}

\begin{frame}
  \frametitle{Costs: Focus on Germany (w/o grid expansion)}

\begin{columns}[T]
\begin{column}{8.5cm}
\centering
  \includegraphics[width=8.5cm]{cost_de_breakdown-1-0}
  %\includegraphics[width=6cm]{pre-2-costs_de_storage}
\end{column}
\begin{column}{6cm}
  \begin{itemize}
  \item Offshore wind replaced by onshore wind at better sites and solar (plus batteries), since
  the represented transmission bottlenecks make it impossible to
  transport the wind energy away from the coast
  \item the effective onshore wind capacity factors increase from $26\%$ to up to $42\%$
  \item Investments stable at 181 clusters and above
  \end{itemize}
\end{column}
\end{columns}


\end{frame}




\begin{frame}
  \frametitle{Interaction between network expansion and spatial scale}

  6 different scenarios of network expansion by constraining the
  overall transmission line volume in relation to today's line volume $\mathrm{CAP}_{\mathrm{trans}}^{\mathrm{today}}$, given length $d_\ell$ and capacity
  $F_\ell$ of each line $\ell$:
  \begin{align}
    F_\ell & \geq F_\ell^{\mathrm{today}} \\
    \sum_\ell d_\ell F_\ell & \leq \mathrm{CAP}_{\mathrm{trans}}
  \end{align}
  where
  \begin{equation}
    \mathrm{CAP}_{\mathrm{trans}} = x \, \mathrm{CAP}_{\mathrm{trans}}^{\mathrm{today}}
  \end{equation}

  for $x = 1$ (today's grid) $x = 1.125,1.25,1.5,2$, $x=3$ (optimal for overhead line at high number of cluster).
\end{frame}





\begin{frame}
  \frametitle{With expansion}

  \includegraphics[width=15cm]{legend-flat}
  \begin{columns}[T]

\begin{column}{8cm}

  \includegraphics[trim=0 0cm 0 2cm,width=8cm]{euro-pie-pre-7-branch_limit-1-5-256.png}

\end{column}
\begin{column}{8cm}

  \raisebox{-6.6cm}{

  \includegraphics[trim=0 0cm 0 2cm,width=8cm]{euro-pie-pre-7-branch_limit-3-256.png}
  }

\end{column}

\end{columns}


\end{frame}



\begin{frame}
  \frametitle{Costs: Total system cost}

\centering

\begin{columns}[T]
\begin{column}{8.5cm}
  \includegraphics[width=9.5cm]{cost_eu}

\end{column}
\begin{column}{7cm}

  \begin{itemize}
  \item Steady cost for No Expansion (1)
  \item For expansion scenarios, as clusters increase, the better expoitation of good sites decreases costs faster than transmission bottlenecks increase them
  \item Decrease in cost is v. non-linear as grid expanded (25\% grid expansion gives 50\% of optimal cost reduction)
  \item Only a moderate $20-25\%$ increase in costs from the Optimal Expansion
    scenario (3) to the No Expansion scenario (1).
  \end{itemize}

\end{column}

\end{columns}

\end{frame}


\begin{frame}
  \frametitle{Costs: Break-down into technologies}

\centering

  \includegraphics[width=12cm]{cost_eu_breakdown_by_cap}
\end{frame}



\begin{frame}
  \frametitle{Costs: Focus on Germany (CAP = 3)}

\begin{columns}[T]
\begin{column}{8.5cm}
\centering
  \includegraphics[width=8.5cm]{cost_de_breakdown-3-0}
  %\includegraphics[width=6cm]{pre-2-costs_de_storage}
\end{column}
\begin{column}{6cm}
  \begin{itemize}
  \item Investment reasonably stable at 128 clusters and above
  \item System consistently dominated by wind
  \item No solar or battery for any number of clusters
  \end{itemize}
\end{column}
\end{columns}


\end{frame}




\begin{frame}
  \frametitle{Behaviour as CAP is changed}


\begin{columns}[T]
\begin{column}{8.5cm}
\centering
  \includegraphics[width=8.5cm]{costs_per_cluster_181}
  %\includegraphics[width=6cm]{pre-2-costs_de_storage}
\end{column}
\begin{column}{6cm}
  \begin{itemize}
%\item   %Big reduction in curtailment
    \item Same non-linear development with high number of nodes that we saw with one node per country
      \item Most of cost reduction happens with small expansion; cost
        rather flat once capacity has doubled, reaching minimum (for
        overhead lines) at 3 times today's capacities
      \item Solar and batteries decrease significantly as grid expanded
  \item  Reduction in storage losses too
  \end{itemize}
\end{column}
\end{columns}
\end{frame}



\begin{frame}
  \frametitle{Locational Marginal Prices CAP=1 versus CAP=3}



  \begin{columns}[T]

\begin{column}{8cm}

  \hspace{1cm}  With today's capacities:

  \includegraphics[width=8cm]{lmp-1-256.pdf}

\end{column}
\begin{column}{8cm}

  \hspace{1cm}With three times today's grid:

  \includegraphics[width=8cm]{lmp-3-256.pdf}

\end{column}

\end{columns}


\end{frame}




\begin{frame}
  \frametitle{Grid expansion CAP shadow price for 181 nodes as CAP relaxed}


\begin{columns}[T]
\begin{column}{10cm}
\centering
  \includegraphics[width=11cm]{shadows-branch_limit}
  %\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 3 times today's transmission volume
  \item With underground cables (5-8 times more expensive) the optimal system has around 1.3 to 1.6 times today's transmission volume
  \end{itemize}
\end{column}
\end{columns}

\end{frame}


\begin{frame}
  \frametitle{CO2 prices versus line cap for 181 clusters}



\begin{columns}[T]
\begin{column}{10cm}
\centering
  \includegraphics[width=11cm]{shadows-co2}
  %\includegraphics[width=6cm]{pre-2-costs_de_storage}
\end{column}
\begin{column}{4cm}
  \begin{itemize}
    \vspace{2cm}
%\item   %Big reduction in curtailment
  \item CO2 price of between 150 and 250 \euro/tCO2 required to reach these solutions, depending on line volume cap
  \end{itemize}
\end{column}
\end{columns}

\end{frame}




% \begin{frame}
%   \frametitle{Comparison of results: energy}

% \centering

%   \includegraphics[width=4.5cm]{pre-2-clusters-yearly_energy-LV-inf}
%   \includegraphics[width=4.5cm]{pre-2-clusters-yearly_energy-LV-4}
%   \includegraphics[width=4.5cm]{pre-2-clusters-yearly_energy-LV-1}

% \end{frame}


% \section{Conclusions}



\begin{frame}
  \frametitle{More Details in Paper}

  For more details, see the following paper:
  \begin{itemize}
  \item J. Hörsch, T. Brown, ``The role of spatial scale in joint optimisations of generation and transmission for European highly renewable scenarios,'' EEM 2017, \href{https://arxiv.org/abs/1705.07617}{\bf\color{blue}\underline{link}}.
  \end{itemize}

  In an upcoming paper with Martha Frysztacki and the same authors, we disentangle the effects of the network resolution from the renewable resource resolution.

\end{frame}


\begin{frame}
  \frametitle{Conclusions}

  \begin{itemize}
    \item Generation costs always dominate grid costs, but the grid can cause higher generation costs if expansion is restricted
    \item Systems with no grid extension beyond today are up to 25\% more expensive, but small grid extensions (e.g. 25\% more capacity than today) can lock in big savings
    \item Need at least around 200 clusters for Europe to see grid bottlenecks if no expansion
    \item Can get away with $\sim 120$ clusters for Europe if grid expansion is allowed
    \item This is \alert{no single solution} for highly renewable systems, but a \alert{family of solutions}  with different costs and compromises
    \item Much of the stationary storage needs can be eliminated by sector-coupling: DSM with electric vehicles, thermal storage; this makes grid expansion less beneficial
    \item Understanding the need for \alert{flexibility at different temporal and spatial scales} is key to mastering the complex interactions in the energy system
  \end{itemize}
\end{frame}

\section{Cycle formulations of optimal power flow}

\begin{frame}
  \frametitle{Angle-based formulation of linear optimal power flow is slow}

  The most common way of implementing optimization models with linear power flow is to use the \alert{angle formulation}.
  We start with energy conservation (KCL)
  \begin{equation*}
    p_i = \sum_{\ell} K_{i\ell}f_\ell
  \end{equation*}
  This puts $N-1$ constraints on the $L$ flows $f_\ell$ (since $\sum_i K_{i\ell} = 0$). For KVL we add $N$ \alert{auxiliary variables} for the voltage angles $\theta_i$ with $L+1$ additional constraints on $f_\ell$ and $\theta_i$:
  \begin{align*}
    f_\ell  & = \frac{1}{x_\ell}\sum_{i} K_{i\ell} \theta_i \\
    \theta_0 & = 0
  \end{align*}

  Check totals: $N+L$ variables $\theta_i,f_\ell$ with $N-1 + L+1 = L+N$ independent, sparse constraints $\Rightarrow$ $\theta_i,f_\ell$ fully determined by the $p_i$, which is what we want.

  But we don't really care about the angles $\theta_i$ and they introduce more variables and constraints.

  \alert{Is there a better way?}

\end{frame}

\begin{frame}
  \frametitle{Cycle-based ``Kirchhoff'' formulation is faster}

  The cycle-based \alert{Kirchhoff formulation} avoids the auxiliary variables $\theta_i$ altogether by implementing the Kirchhoff Voltage Law (KVL) directly on the flows $f_\ell$ themselves.

  We start again with our $N-1$ KCL constraints:
  \begin{equation*}
    p_i = \sum_{\ell} K_{i\ell}f_\ell
  \end{equation*}
  and add the $L-N+1$ cycle constraints of KVL from Lecture 4:
  \begin{equation*}
    \sum_\ell C_{\ell c} x_\ell f_\ell = 0
  \end{equation*}

  Check totals: $L$ variables $f_\ell$ with $N-1 + L-N+1 = L$  independent, sparse constraints $\Rightarrow$ $f_\ell$ fully determined by the $p_i$, which is what we want.

  This has fewer variables and fewer constraints than the angle-based formulation, so we can expect it to perform better.
\end{frame}


\begin{frame}
  \frametitle{Cycle formulation of linear power flow}

  A third \alert{cycle formulation} decomposes the flows in the network into two parts:
  \begin{enumerate}
  \item A flow on a spanning tree of the network, uniquely determined by nodal $\mathbf{p}$ (ensuring KCL)
  \item Cycle flows, which don't affect KCL; their strength is fixed by enforcing KVL
  \end{enumerate}

\begin{circuitikz}
  \draw
  (0,3)
  to [short,i^=$f_1$,*-*] (3,3)
  to [short,i>=$f_2$,*-*] (3,0)
  to [short,i>=$f_3$,*-*] (0,0)
  to [short,i>=$f_4$,*-*] (0,3);
  \draw (3,3)   to [short,i^=$f_5$,*-*] (0,0);

  \draw (1.5,-1.2) node{$f_\ell$};


  \draw (4,1.5) node{$=$};

  \draw (4,-1.2) node{$=$};


  \draw[red]
  (5,3)
  to [short,i^=$t_1$,*-*] (8,3)
  to [short,i>=$t_2$,*-*] (8,0)
  to [short,i>=$t_3$,*-*] (5,0);

  \draw (6.5,-1.2) node{$t_\ell$};


  \draw (9,1.5) node{$+$};

  \draw (9,-1.2) node{$+$};



  \draw[blue]
  (10,3)
  to [short,i>=$$,*-*] (12.8,3)
  to [short,i>=$$,*-*] (10,0.2)
  to [short,i>=$c_1$,*-*] (10,3);

  \draw[blue]
  (13,0)
  to [short,i>=$$,*-*] (10.2,0)
  to [short,i>=$$,*-*] (13,2.8)
  to [short,i>=$c_2$,*-*] (13,0);


  \draw (11.5,-1.2) node{$\sum_k C_{\ell,k} c_k$};


\end{circuitikz}


\end{frame}


\begin{frame}
  \frametitle{Cycle formulation of linear power flow}

  The $N-1$ tree flows $\mathbf{t}$ are determined directly from the
  $N$ nodal powers $p_n$ and the network power balance constraint
  $\sum_n p_n = 0$.

  We solve for the $L-N+1$ cycle flows $c_k$ by enforcing  the $L-N+1$ KVL equations:
  \begin{equation*}
    C^t X \mathbf{f} = C^t X (\mathbf{t} + C \mathbf{c}) = 0
  \end{equation*}

  The matrix $C$ is the incidence matrix of the \alert{weak dual
    graph}, $C^t X C$ is the weighted Laplacian of the dual graph and
  the above equation becomes a discrete Poisson equation:
  \begin{equation*}
    C^t X C \mathbf{c} =  - C^t X \mathbf{t}
  \end{equation*}

  Now we have only $L-N+1$ variables $c_k$ with $L-N+1$ independent, \alert{semi-dense} constraints.
\end{frame}



\begin{frame}
  \frametitle{LOPF speedup with cycle flows}
\begin{columns}[T]
  \begin{column}{7cm}
  \includegraphics[width=7.5cm]{lopf-ne-speedup-per-nodes-mean-with-ci-99-regression}
  \end{column}
  \begin{column}{6cm}
    \vspace{0.5cm}
    Using \alert{cycle flows instead of voltage angles} we found for generation expansion optimisation (fixed grid):
    \begin{itemize}
    \item A speed-up of up to \alert{200 times}
    \item Average speed-up of \alert{factor 12}
    \item Speed-up is highest for \alert{large networks with lots of renewables}
    \end{itemize}
  \end{column}

\end{columns}

\vspace{.3cm}

  \footnotesize
H.~Ronellenfitsch, D.~Manik, J.~Hörsch, {\bf T.~Brown, D.~Witthaut}, \emph{``Dual theory of transmission line outages,''} 2017, IEEE Transactions on Power Systems, \href{https://arxiv.org/abs/1606.07276}{\bf\color{blue}\underline{link}}.

J.~Hörsch, H.~Ronellenfitsch, {\bf D.~Witthaut, T.~Brown}, \emph{``Linear Optimal Power Flow Using Cycle Flows,''} 2017, Electric Power Systems Research,
\href{https://arxiv.org/abs/1704.01881}{\bf\color{blue}\underline{link}}.
\end{frame}



\end{document}