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\title{Energy System Modelling\\ Summer Semester 2019, Lecture 12}
%\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{\vspace{.3cm}14th June 2019}


\titlegraphic{%
  \vspace{6cm}
  \hspace{6.7cm}
    \includegraphics[trim=0 0cm 0 0cm,height=1.8cm,clip=true]{kit.png}
}

\begin{document}

\maketitle


\begin{frame}

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





\section{Oral examination sample questions}


\begin{frame}{Oral exam}

  The following question are \emph{not} identical to the oral exam,
  but give an indication of the difficulty level of the questions.

\end{frame}

\begin{frame}{General Questions}

\begin{itemize}
  \item Describe variations of wind and solar in space and time (lecture 1)
  \item Describe what you would expect from the Fourier transform of wind and solar time series (lecture 1)
  \item What are the options for balancing variable renewables?
  \item What happens to feed-in when integrated over larger areas? (lecture 2)
  \item What is the correlation length of the wind? (lecture 2)
  \item When balancing with only storage, how does storage volume depend on frequency of variations for a sinusoidal generation pattern? (lecture 4, sheet 2)
 \item When balancing only with networks, how does network distance depend on wavelength of variations?
\end{itemize}

\end{frame}


\begin{frame}{Network physics}

\begin{itemize}
\item What is the incidence matrix? Write down the incidence matrix of a given network. (lecture 2)
\item What is the Laplacian? (lecture 2)
\item How are power, voltage angles and the Laplacian related for a linear power flow calculation? (lecture 3)
  \item How is the PTDF derived? (lecture 3)
\end{itemize}
\end{frame}


\begin{frame}{Optimisation}


\begin{itemize}
\item Write down the typical form of an optimisation problem. (lecture 6)
\item What are the KKT conditions? (lecture 6)
\item Describe each KKT condition. (lecture 6)
\item How does the maximisation of economic welfare relate to optimisation and KKT?
\item Write down KKT for a given two-node problem. (lecture 8)
\item Screening curves, demand duration curve: derive generation fleet for given parameters (lecture 8, sheet 4)
\item Explain the merit order effect (lecture 10, 11)
\item  What is the `no profit rule' (lecture 10)
  \item Are extreme prices in markets a problem? How can we incentivise capacity? (lecture 10)
\end{itemize}

\end{frame}




\begin{frame}{Other topics}

  \begin{itemize}
  \item Write down the relation between storage dispatch/storing and energy. (lecture 4)
  \item How is Demand Side Management different from Storage? (lecture 5)
  \item Describe the opportunities of coupling to other energy sectors (lecture 9)
  \item What techniques can we use to reduce the complexity of energy system modelling calculations? (spatial clustering in lecture 11, PCA)
    \item What algorithms can we use to trace power flow from sources to sinks? (flow tracing)
  \item How is the frequency affected by power imbalances? (lecture 12)
\end{itemize}
\end{frame}

\section{Grid Dynamics and Synchronization}


\begin{frame}{From Static to Dynamic}

  So far we have considered the power system only in its static
  \alert{steady state} (or hourly snapshots of the steady state),
  i.e. where the time derivatives of all system quantities are zero.

  However, the power system is changing all the time, both in slow,
  predictable ways (the load changing in a statistically-smoothed way)
  and in sudden, unpredictable ways, e.g. if a generator fails and
  500~MW is lost, or if a power line or transformer fails while
  heavily loaded (e.g. due to lightning, trees falling, diggers or
  human operator error). The continental European is designed to
  survive the sudden loss of 3~GW of generation.

  In the next few slides we will look at the \alert{dynamics} of the
  power system.

\end{frame}


\begin{frame}{Time scales}

  Dynamic phenomena in power systems can be classified by the time scales on which they occur, for example:
  \begin{enumerate}
  \item  Electro-magnetic transients (e.g. switching, lightning strikes) (100 Hz – MHz)
  \item Electro-mechanical swings (e.g. rotor swings in synchronous machines) (0.1 –
  3 Hz)
  \item Non-electric dynamics (e.g. mechanical phenomena and thermodynamics) (up to tens of Hz)
  \end{enumerate}

  We will focus on number 2, i.e. frequency dynamics due to
  interactions between changing loads and rotating machines
  (generators and motors).

\end{frame}


\begin{frame}{Balancing Power via Steady Frequency}



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

    \vspace{2cm}
\includegraphics[width=5.5cm]{balance-crop}
  \end{column}

  \begin{column}{6cm}
    \vspace{0.5cm}

    Generation and demand have to be kept in balance at all times
    (since we cannot create or destroy energy).

    This is reflected in the alternating current grid frequency (50~Hz
    in Europe and most of Africa/Asia; 60~Hz in much of Americas).

    If there is more generation than load, the extra energy is
    converted to rotational energy of the synchronous rotating
    machines and the frequency goes up.

    If there is more load than generation, the deficit energy is
    substituted by rotational energy of the synchronous rotating
    machines and the frequency goes down.


  \end{column}

\end{columns}


\end{frame}



\begin{frame}{Example: Grid split and partial blackout 4.11.2006}

  On 4th November 2006 a cascading line outage caused the continental
  European network to split into three parts. The individual parts
  were not perfectly balanced: excess generation in North-East, too little in West.


  \includegraphics[width=10cm]{grid_split}


  \source{\href{https://www.entsoe.eu/fileadmin/user_upload/_library/publications/ce/otherreports/Final-Report-20070130.pdf}{UCTE 2006 Report}}
\end{frame}



\begin{frame}{Example: Grid split and partial blackout 4.11.2006}

  The frequency was only stabilised by load shedding (blackouts) once the frequency hit 49~Hz in the West.

  \includegraphics[width=10cm]{frequency_after_grid_split}


  \source{\href{https://www.entsoe.eu/fileadmin/user_upload/_library/publications/ce/otherreports/Final-Report-20070130.pdf}{UCTE 2006 Report}}
\end{frame}


\begin{frame}{Synchronous machines}

For a single synchronous machine (left in generation mode; right in motor
mode) we will now examine the relation between the \alert{mechanical
  power} $P_m$, the \alert{electrical power} $P_e$, the
\alert{frequency} $\omega$, the \alert{angle} $\theta$ and the \alert{mechanical and electrical
  torques} $T_m,T_e$.

\includegraphics[width=11cm]{machine}

For the rotor dynamics we have for the moment of inertia $J$
\begin{equation*}
  J \frac{d^2 \theta}{dt^2} = T_m - T_e
\end{equation*}

\source{\href{https://pdfs.semanticscholar.org/0ecf/220a7d41a404e755d472a80c4ed0357b26ed.pdf}{Göran Andersson lecture notes}}

\end{frame}



\begin{frame}{Swing equation}

  Multiplying by the frequency $\omega = \dot{\theta}$ we get the \alert{swing equation}
\begin{equation*}
  \omega J \frac{d^2 \theta}{dt^2} = P_m - P_e
\end{equation*}
The total inertia is given by $H = \frac{1}{2} \omega^2 J$ so we get
\begin{equation*}
  \frac{2H}{\omega} \frac{d^2 \theta}{dt^2} = P_m - P_e
\end{equation*}

It is now clear what will happen if $P_m \neq P_e$.
\end{frame}


\begin{frame}{Networks of synchronous machines}
  Networks of oscillators, i.e. an oscillator with angle $\theta_i$ at
  every node, can be studied. We get using a simplification of the load flow equations for $P_e$
  \begin{equation*}
  \frac{2H_i}{\omega_i} \frac{d^2 \theta_i}{dt^2} = P_m - \sum_j \frac{1}{x_{ij}} \sin(\theta_i-\theta_j)
  \end{equation*}


\begin{columns}[T]
  \begin{column}{7.5cm}

\includegraphics[width=6.8cm]{kuramoto_stability}
  \end{column}

  \begin{column}{4.5cm}
    \vspace{0.5cm}

  The solutions have several interesting properties: basins of stability, limit cycles, etc.
    \vspace{0.5cm}
    We can study them in a simpler physics model called the \alert{Kuramoto model}.

  \end{column}

\end{columns}
\source{Paul Schultz}
\end{frame}


\begin{frame}{Kuramoto model}

  This very similar model for synchronization was introduced in 1975 by Kuramoto:
  \begin{equation*}
    \frac{d\theta_i}{dt} = \omega_i + \frac{K}{N} \sum_{j=1}^N \sin(\theta_j- \theta_i)
  \end{equation*}
  where $\theta_i$ is a phase, $K$ is the coupling strength,
  multiplied by the sine of the phase differences, and $\omega_i$ is
  the natural frequency of the oscillator.

  Suppose the natural frequencies are distributed with pdf $g(\omega)$.

  To understand synchronisation in this model, Kuramoto introduced \alert{order parameters} $R$ and $\psi$ defined by
  \begin{equation*}
    Re^{i\psi(t)} = \frac{1}{N}  \sum_{j=1}^N e^{i\theta_j(t)}
  \end{equation*}

\end{frame}


\begin{frame}{Kuramoto model}

  This allows us to re-write the equations as
  \begin{equation*}
    \frac{d\theta_i}{dt} = \omega_i + \frac{KR}{N}\sin(\psi- \theta_i)
  \end{equation*}
  Now the interaction with the other $\theta_i$ is concealed inside the $\psi$.

  The coupling between the $\theta_i$ grows with $R$; if $R=0$ there
  is no synchrony and the phases are smeared around the circle. If
  $R\sim 1$ then the phases are synchronized.

  We will show that the oscillators with $|\omega_i| \leq KR$ are
  locked at $\theta_i = \arcsin(\frac{\omega_i}{KR})$, whereas those
  with $|\omega_i| > KR$ are drifting.

  See also:  \url{https://en.wikipedia.org/wiki/Kuramoto_model},
\url{http://www.stevenstrogatz.com/articles/from-kuramoto-to-crawford-exploring-the-onset-of-synchronization-in-populations-of-coupled-oscillators-pdf},
  \url{https://arxiv.org/abs/1511.07139},
  \url{http://scala.uc3m.es/publications_MANS/PDF/finalKura.pdf}

  Videos: \url{https://www.youtube.com/watch?v=sROKYelaWbo},
  \url{https://www.youtube.com/watch?v=eAXVa__XWZ8},
  \url{https://www.youtube.com/watch?v=W1TMZASCR-I}


\end{frame}




\section{Cycles Flows and Grid Outages}


\begin{frame}
  \frametitle{Line Outages}


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

    \vspace{1cm}

  \includegraphics[width=6.5cm]{2006_blackout}

  \end{column}

  \begin{column}{5.5cm}

  When there are faults in the network, transmission lines can
  disconnect. This forces power flows onto the remaining network. If
  there is insufficient capacity, there can be a cascading line outage.

  \vspace{0.5cm}
  This happened in November 2006, where a cascading line outage caused
  the European network to split into three isolated parts, causing a
  major blackout.

  \vspace{0.5cm}
  Such outages may be exacerbated by high levels of variable renewable energy.
  \end{column}

\end{columns}



\end{frame}



\begin{frame}
  \frametitle{Recall angle formulation of linear power flow}

  Recall that for net power injections $p_n$ at nodes $n$, flows $f_\ell$
  on lines $\ell$, network incidence matrix $K$ and cycle basis $C$ (kernel of $K$, $KC = 0$) we
  can express Kirchhoff's Circuit Laws as:
  \begin{itemize}
  \item  Kirchhoff's Current Law (KCL): $\mathbf{p} = K \mathbf{f}$
  \item  Kirchhoff's Voltage Law (KVL): $C^t X \mathbf{f} = 0$
  \end{itemize}

  Using voltage angle $\theta_n$ for each node and then using $f_l = \frac{\theta_i - \theta_j}{x_l}$:
    \begin{align*}
    \mathbf{f} & = X^{-1} K^t \mathbf{\theta} \hspace{2cm} (\sim E = -\nabla \phi)\\
    \mathbf{p} & = K X^{-1} K^t \mathbf{\theta} \hspace{2cm} (\sim \rho = \Delta \phi)
  \end{align*}
  NB: $K X^{-1} K^t$ is a weighted Laplacian matrix for the graph, so
  the final equation has the form of a discrete Poisson equation sourced by $\mathbf{p}$.
\end{frame}



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

  We can use dual graph theory to decompose 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{figure}[!t]
\begin{adjustbox}{scale=0.75}


\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{adjustbox}
  \end{figure}

\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*}

\end{frame}

\begin{frame}
  \frametitle{Line outages from cycles}


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

    \includegraphics[trim = 3.5cm 2.5cm 6.5cm 1.5cm,clip,width=6cm]{5bus2}

  \end{column}

  \begin{column}{5.5cm}

    \vspace{0.5cm}

    The outage of a line \alert{only affects the cycle flows}.

    \vspace{0.5cm}

    (a) shows the flows before the outage;

    (b) shows
    the flows after the outage of the diagonal line;

    (c) shows the
    change in flows, decomposed into cycle flows.


    \vspace{0.5cm}

    Therefore the effect of the line outage on the other flows in the network can be entirely understood in terms of the cycle flows, corresponding to the nodes of the weak dual graph.
  \end{column}

\end{columns}
\end{frame}




\begin{frame}
  \frametitle{Line outages from cycles}


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

\includegraphics[trim=2cm 2cm 6cm 1cm, clip, width=6.5cm]{scheme2}
  \end{column}

  \begin{column}{5.5cm}
    \vspace{1cm} If we consider the outage of a line, the change in
    flows can be expressed very compactly in terms of the
    change in the cycles flows, $\Delta \mathbf{c}$, which are determined by a Poisson equation sourced by
    a  term $\mathbf{q}$ that only has non-zero entries for cycles bordering the failed line
    \begin{equation*}
      C^t X C \Delta \mathbf{c} = \mathbf{q}
    \end{equation*}
    For a plane network where the line fails away from the boundary, $\mathbf{q}$ is a dipole source for the two cycles that border the failed line.

  \end{column}

\end{columns}
\end{frame}



\begin{frame}
  \frametitle{Line outages from cycles}
\begin{columns}[T]
  \begin{column}{6.5cm}
  \includegraphics[width=7cm]{speedup-per-nodes-mean-with-ci-99-regression}
  \end{column}
  \begin{column}{5cm}
    \vspace{0.5cm}
    If we reformulate the linear optimal power flow using \alert{cycle flows instead of voltage angles} we find:
    \begin{itemize}
    \item A speed-up of up to \alert{20 times}
    \item Average speed-up of \alert{factor 3}
      \item Speed-up is highest for \alert{large networks with lots of decentralised generators}
    \end{itemize}

  \end{column}
\end{columns}


\end{frame}



\section{Effect of Climate Change on Renewable Energy Systems}

\begin{frame}
  \frametitle{Climate change}

  Ongoing MA thesis by Markus Schlott:
  \begin{itemize}
  \item   What are the consequences of climate change for highly renewable energy systems?
  \item How will generation patterns for wind and solar change?
    \item What will be the effects on the dimensioning of wind,
      solar, storage, networks and backup generation?
  \end{itemize}


  If we can answer these questions, we know our energy system will be
  robust against expected changes to the climate.


\end{frame}

\begin{frame}
  \frametitle{Climate change scenarios: RCP 8.5}

  Markus took a simulated dataset of how the weather would look
  between today and the year 2100 with a scenario of high
  concentrations of greenhouse house gases.

  The scenario is called Representative Concentration Pathways 8.5
  (RCP~8.5), since it estimates a radiative forcing of $\Delta P = 8.5$~
  $\mathrm{W}/ \mathrm{m}^2$ (difference between insolation and energy
  radiated into space) at the end of the century. It is a `business as
  usual' scenario and extrapolates the current greenhouse gas emission
  without reduction efforts. This corresponds to a
  CO$_2$-equivalent-concentration (including all forcing agents) of
  approximately 1250 ppm (today around 410~pmm for CO$_2$) and an
  average temperature increase of $\Delta T = 3.7 \pm 1.1$~C at the
  end of the century, dependent on the model used.

  Compare historical values (his) to begin/middle/end of the century (b/m/eoc).

\end{frame}

\begin{frame}
  \frametitle{Changes to wind speeds}

  \begin{columns}[T]
  \begin{column}{7.5cm}
    \includegraphics[width=7.5cm]{mean_ensemble_sfcWind_abs.eps}
  \end{column}
  \begin{column}{4.5cm}
    \vspace{0.5cm}
    \begin{itemize}
    \item Very small changes to mean wind speeds (also very local)
    \item Small ($\sim 5\%$) average increase in Northern Europe
    \item Small ($\sim 5\%$) average decrease in Southern Europe
    \end{itemize}

  \end{column}
\end{columns}

      %corr_ensemble_sfcWind_abs.eps

  \source{Markus Schlott}

\end{frame}


\begin{frame}
  \frametitle{Changes to mean short-wave radiation}

  \begin{columns}[T]
  \begin{column}{7.5cm}
    \includegraphics[width=7.5cm]{mean_ensemble_rsds_abs.eps}
  \end{column}
  \begin{column}{4.5cm}
    \vspace{0.5cm}
    \begin{itemize}
    \item Small ($\sim 5\%$) increase in downward short-wave radiation
      in Southern Europe around Mediterranean
    \item Smallish ($\sim 10\%$) decrease in Northern Europe (due to
      increased cloud cover)
    \item Solar results known to be a little unreliable because of
      cloud modelling etc.
    \end{itemize}

  \end{column}
\end{columns}


    %mean_ensemble_sfcWind_abs.eps
      %corr_ensemble_sfcWind_abs.eps


  \source{Markus Schlott}

\end{frame}

\begin{frame}
  \frametitle{Correlation Length}


  \begin{columns}[T]
  \begin{column}{5.5cm}
    \includegraphics[height=8cm]{simeon-correlation}
  \end{column}
  \begin{column}{4.5cm}
    \vspace{0.5cm}
    \begin{itemize}
      \item The correlation of wind time series with a point in
        northern Germany decays with distance.
      \item Determine the \alert{correlation length} $L$ by fitting the
        function:
        \begin{equation*}
          \rho \sim e^{-\frac{x}{L}}
        \end{equation*}
        to the radial decay with distance $x$.
    \end{itemize}

  \end{column}
\end{columns}


  \source{\href{https://doi.org/10.1016/j.apenergy.2011.10.039}{Hagspiel et al, 2012}}
\end{frame}


\begin{frame}
  \frametitle{Changes to wind speed correlation lengths}

  \begin{columns}[T]
  \begin{column}{6.8cm}
    \includegraphics[width=7.5cm]{corr_ensemble_sfcWind_abs.eps}
  \end{column}
  \begin{column}{5.3cm}
    \begin{itemize}
      \item Correlation lengths are typically longer in the North than
        the South because of the big weather systems that roll in from
        the Atlantic to the North (in the South they get dissipated).
      \item With global warming, correlation lengths grow marginally
        longer in the North and marginally shorter in the South.
      \item This is because weather systems have more energy and are
        bigger in the North.
    \end{itemize}

  \end{column}
\end{columns}


  \source{Markus Schlott}

\end{frame}



\begin{frame}
  \frametitle{Effects of climate change on power system}

  What kind of changes do we expect to the optimal power system?



    \begin{itemize}
      \item Most effects are small ($\sim 5-10\%$) so we don't expect
        massive changes.
      \item More wind capacity built in the North to take advantage of improving wind speeds?
       \item Somewhat counteracted by higher correlation lengths in North -
         need bigger networks or more long-term storage?
    \end{itemize}

    For results, see `The Impact of Climate Change on a Cost-Optimal Highly Renewable European Electricity Network,' \url{https://arxiv.org/abs/1805.11673}

\end{frame}


\section{Sensitivity of Optimisation to Cost and Weather Data}


\begin{frame}
  \frametitle{Sensitivity of Optimisation to Cost and Weather Data}

  See `Cost optimal scenarios of a future highly renewable European
  electricity system: Exploring the influence of weather data, cost
  parameters and policy constraints,'
  \url{https://arxiv.org/abs/1803.09711}

\end{frame}

\end{document}
