sheet01.tex 9.32 KB
Newer Older
sp2668's avatar
sp2668 committed
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
\documentclass[11pt,a4paper,fleqn]{scrartcl}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[colorlinks=true, citecolor=blue, linkcolor=blue, filecolor=blue,urlcolor=blue]{hyperref}
\hypersetup{
     colorlinks   = true,
     citecolor    = gray
}
\usepackage{wrapfig}

\usepackage{caption}
\captionsetup{format=plain, indent=5pt, font=footnotesize, labelfont=bf}


\setkomafont{disposition}{\scshape\bfseries}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{bbm}
\usepackage{mathtools}
% \usepackage{epsfig}
% \usepackage{grffile}
%\usepackage{times}
\usepackage{palatino}
\usepackage{mathpazo}
\setlength\parindent{0pt}

%\usepackage{babel}
\usepackage{tikz}
\usepackage{paralist}
\usepackage{color}
\usepackage[top=3cm, bottom=2.5cm, left=2.5cm, right=3cm]{geometry}
%\setlength{\mathindent}{1ex}

% PGF
\usepackage{pgfplots}
\pgfplotsset{
  compat=newest,
  every axis/.append style={small, minor tick num=3}
}

%\usepackage[backend=biber,style=alphabetic,url=false,doi=false]{biblatex}
%\addbibresource{sheet01_biber.bib}
% \addbibresource{/home/coroa/papers/refs.bib}

\newcommand{\id}{\mathbbm{1}}
\newcommand{\NN}{{\mathbbm{N}}}
\newcommand{\ZZ}{{\mathbbm{Z}}}
\newcommand{\RR}{{\mathbbm{R}}}
\newcommand{\CC}{{\mathbbm{C}}}
\renewcommand{\vec}[1]{{\boldsymbol{#1}}}

\renewcommand{\i}{\mathrm{i}}

\newcommand{\expect}[1]{\langle\,#1\,\rangle}
\newcommand{\e}[1]{\ensuremath{\,\mathrm{#1}}}

\renewcommand{\O}{\mc{O}}
\newcommand{\veps}{\varepsilon}
\newcommand{\ud}[1]{\textup{d}#1\,}

\newcommand{\unclear}[1]{\color{green}#1}
\newcommand{\problem}[1]{\color{red}#1}

%=====================================================================
%=====================================================================
\begin{document}

\begin{center}
sp2668's avatar
sp2668 committed
72 73 74 75 76
 \textbf{\Large Energy System Modelling }\\
 {SS 2018, Karlsruhe Institute of Technology}\\
 {Institute of Automation and Applied Informatics}\\ [1em]
 \textbf{\textsc{\Large Tutorial I: Time Series Analysis}}\\
 \small Will be worked on in the exercise session on Wednesday, 11 July 2018.\\[1.5em]
sp2668's avatar
sp2668 committed
77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92
\end{center}

\vspace{1em}

%=============== ======================================================
\paragraph{Problem I.1 \normalsize (data analysis).}~\\
%=====================================================================

The following data are made available to you on the course home
page\footnote{\url{https://nworbmot.org/courses/complex_renewable_energy_networks/}}:
\begin{verbatim}
  de_data.csv, gb_data.csv, eu_data.csv, (wind.csv, solar.csv, load.csv).
\end{verbatim}
They describe (quasi-real) time series for wind power generation \(W(t)\), solar power generation \(S(t)\) and load \(L(t)\) in Great Britain (GB), Germany (DE) and Europe (EU). The time step is \(1\e{h}\) and the time series are several years long.

\begin{enumerate}[(a)]
sp2668's avatar
sp2668 committed
93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116
 \item Check that the wind and solar time series are normalized to 'per-unit of installed \mbox{capacity}', and that the load time series is normalized to MW.
 \item For all three regions, calculate the maximum, mean, and variance of the time series.
 \item For all three regions, plot the time series \(W(t)\),
       \(S(t)\),
       \(L(t)\) for a winter month (January) and a summer month (July).
 \item For all three regions, plot the duration curve for \(W(t)\), \(S(t)\), \(L(t)\).
 \item For all three regions, plot the probability density function of \(W(t)\), \(S(t)\), \(L(t)\).
 \item Apply a (Fast) Fourier Transform to the the three time series $X \in W(t), S(t), L(t)$:
       \begin{equation*}
        \tilde{X}(\omega) = \int_0^T X(t) e^{\i \omega t} \,\ud t \, .
       \end{equation*}
       For all three regions, plot the energy spectrum
       $\left| \tilde{\Delta}(\omega) \right|^2$ as a function of
       $\omega$. Discuss the relationship of these results with the
       findings obtained in (b)-(e).
 \item Normalize the time series to one, so that \(\expect{W} = \expect{S} = \expect{L} = 1\).
       Now, for all three regions, plot the mismatch time series
       \begin{equation*}
        \Delta(t) = \gamma \alpha W(t) + \gamma (1 - \alpha) S(t) - L(t)
       \end{equation*}
       for the same winter and summer months as in (c). Choose
       \(\alpha \in \{0.0, 0.5, 0.75, 1.0\}\) with \(\gamma = 1\),
       and $\gamma \in \{0.5, 0.75, 1.0, 1.25, 1.5\} $ with $\alpha = 0.75$.
 \item For all three regions, repeat (b)-(f) for the mismatch time series.
sp2668's avatar
sp2668 committed
117 118 119 120 121 122 123 124
\end{enumerate}

\pagebreak
%=====================================================================
\paragraph{Problem I.2 \normalsize (analytical).}~\\
%=====================================================================

\begin{wrapfigure}[11]{r}{0pt}
sp2668's avatar
sp2668 committed
125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141
 \begin{tikzpicture}
  \begin{axis}[
    domain=0:1, no markers, samples=200
    % xlabel = $x$, ylabel = $f(x)$
   ]
   \addplot+[dashed] {1 + 0.4 * cos(deg(2*pi*x))}; \label{figref:w}
   \addplot+[densely dotted] {1 - 0.75 * cos(deg(2*pi*x))}; \label{figref:s}
   \addplot+[solid] {1 + 0.1 * cos(deg(2*pi*x))}; \label{figref:l}
   \addplot+[dotted] {1}; \label{figref:1}
  \end{axis}
 \end{tikzpicture}
 \caption{Seasonal variations of wind and solar power generation
  \(W(t)\)
  \autoref{figref:w} and \(S(t)\)
  \autoref{figref:s}, and load \(L(t)\)
  \autoref{figref:l} around the mean \(1\) \ref{figref:1}.}
 \label{fig:seasonalvariations}
sp2668's avatar
sp2668 committed
142 143 144 145 146 147
\end{wrapfigure}

Figure \ref{fig:seasonalvariations} shows approximations to the
seasonal variations of wind and solar power generation \(W(t)\)
and \(S(t)\) and load \(L(t)\):
\begin{align*}
sp2668's avatar
sp2668 committed
148 149 150
 W(t) & = 1 + A_W \cos \omega t \\
 S(t) & = 1 - A_S \cos \omega t \\
 L(t) & = 1 + A_L \cos \omega t
sp2668's avatar
sp2668 committed
151 152 153 154 155 156
\end{align*}

The time series are normalized to
\(\expect{W} = \expect{S} = \expect{L} := \frac{1}{T} \int_0^T L(t)
\ud t = 1\), and the constants have the values
\begin{align*}
sp2668's avatar
sp2668 committed
157 158
 \omega & = \frac{2\pi}{T} & T   & = 1 \e{year}               \\
 A_W    & = 0.4            & A_S & = 0.75       & A_L & = 0.1
sp2668's avatar
sp2668 committed
159 160 161 162
\end{align*}

~\\
\begin{enumerate}[(a)]
sp2668's avatar
sp2668 committed
163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
 \item What is the seasonal optimal mix \(\alpha\), which minimizes
       \begin{equation*}
        \expect{\left[ \alpha W(\cdot) + (1-\alpha) S(\cdot) - L(\cdot) \right]^2} = \frac1T \int_0^T \left[ \alpha W(t) + (1-\alpha) S(t) - L(t) \right]^2 \,\mathrm d t
        ,
       \end{equation*}
 \item How does the optimal mix change if we replace \(A_L \to -A_L\)?
 \item Now assume that there is a seasonal shift in the wind signal
       \begin{equation*}
        W(t) = 1 + A_W \cos \left( \omega t - \phi \right)
        .
       \end{equation*}
       Express the optimal mix \(\alpha\) as a function of \(\phi\).
 \item A constant conventional power source \(C(t) = 1 - \gamma\) is now introduced. The mismatch then becomes
       \begin{equation}
        \Delta(t) = \gamma \left[ \alpha W(t) + (1-\alpha) S(t) \right] + C(t) - L(t)
        .
       \end{equation}
       Analogously to (a), find the optimal mix \(\alpha\) as a function of \(0 \leq \gamma \leq 1\), which minimizes \(\expect{\Delta^2}\).
sp2668's avatar
sp2668 committed
181 182 183 184 185 186 187 188 189 190
\end{enumerate}

\pagebreak
%=============== ======================================================
\paragraph{Remarks (Python pointers or where to start).}~\\
%=====================================================================

I found the python notebook based notes of Robert Johansson to be a
comprehensive kick starter\footnote{\url{http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/tree/master/}}.
\begin{itemize}
sp2668's avatar
sp2668 committed
191 192 193 194 195 196 197 198 199
 \item
       \href{http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-0-Scientific-Computing-with-Python.ipynb}{Lecture~0} covers installation and getting ready.
 \item
       \href{http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-1-Introduction-to-Python-Programming.ipynb}{Lecture~1}
       zooms through most basic general python control structures (only
       brush over it and stop reading early, i.e. if you read the word
       \verb+classes+ you already know too much).
 \item \href{http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-2-Numpy.ipynb}{Lecture~2} is the most important and closely connected to the exercises.
 \item You might as well stop now, but if you \emph{are} hooked, I recommend \href{http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-3-Scipy.ipynb}{Lecture~3} for more physics and \href{http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-4-Matplotlib.ipynb}{Lecture~4} for prettier graphs.
sp2668's avatar
sp2668 committed
200 201 202 203
\end{itemize}

Further reference material of help is:
\begin{itemize}
sp2668's avatar
sp2668 committed
204 205
 \item The website-books \url{http://python-course.eu/} (english), \url{http://python-kurs.eu/} (german); especially if you only \emph{very} quickly skim over the \href{http://www.python-course.eu/course.php}{python2 tutorial} and switch over to the \href{http://www.python-course.eu/numerical_programming.php}{numerical python} stuff early; especially of interest might be the \href{http://www.python-course.eu/pandas.php}{pandas} bit in the end, which will make the exercises a breeze at the expense of yet another package to learn.
 \item the exhaustive (overly so) official python tutorial\footnote{\url{https://docs.python.org/2/tutorial/}} available in \href{https://docs.python.org/2/tutorial/}{english} and \href{https://py-tutorial-de.readthedocs.org/de/python-3.3/index.html}{german}; which will NOT introduce you to numpy or scipy.
sp2668's avatar
sp2668 committed
206 207 208 209
\end{itemize}


\end{document}