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\begin{document}

\newpage
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\ Spread of Plug-in-Electric Vehicles}


\markboth{Spread of Plug-in-Electric Vehicles}{Spread of Plug-in-Electric Vehicles}

\author{Brian Winkel,
Director SIMIODE\\ 
Chardon OH 44024 USA\\
BrianWinkel@simiode.org
}

\makeStitlePDFLaTex


\begin{abstract}
We present data on world sales data of plug-in electric vehicles (PEVs) and request a model on the rate of change in sales over time, leading to prediction as to number of PEVs in the future.
\end{abstract}


\section*{SCENARIO DESCRIPTION}

We quote from the source\cite{EVTC2015}: 

``The object of this research is to present the current market status of plug-in-electric vehicles (PEVs) and to predict their future penetration within the world and U.S.markets. The sales values for 2015 show that China leads in yearly sales at 214,283 (triple increase over 2014) followed by Western Europe at 184,500 vehicles sold. The U.S. is third at 115,262 followed by Japan at 46,339 vehicles.These four countries comprise 95\% of the global sales market. The world total of EV sales for 2015 is estimated to be 565,668 up from 315,519 in 2014. The data also shows that the overall world growth is 79\% and that Western Europe is now ahead of the U.S.in total cumulative vehicles.

``Within the U.S., the PEV sales results for 2015 show that 115,262 vehicles were sold as compared to 118,882 vehicles in 2014 which is lower by 3\%.However, sales for the first 6 months of 2016 are 19\% greater than the sales for the same period in 2015.  On a state basis, California is the largest market with about 55\% of the sales for 2015 and a growth of 4.6\% in 2015. The total cumulative number of EVs sold in the U.S. over the six year lifetime is now at more than 400,000 vehicles. Looking at the PEV sales data the future world market is extremely bright and is rapidly growing due to the various country policies and to the development of lithium - ion batteries from both a technological and manufacturing standpoint.

``The current work also evaluated the types of barriers to EV usage and the actions, incentives and research to overcome the barriers.  The barriers to large scale EV usage are costs, range, availability of charging stations, charging time, battery life and infrastructure, standard and permitting. Results are presented for overcoming each of the barriers that includes incentives and technical progress.''


\subsection*{Assignment}
Using the data from the Research Results \cite{EVTC2015} as displayed in Figure~\ref{fig:1-122-DataTable} create a mathematical model for the rate of change in  the number of PEVs over time and then of the predicted number of PEVs over time.

State your assumptions and discuss model limitations.


\begin{figure}[h]
\begin{center}
\includegraphics[width=14.9cm]{1-122-DataTable.jpg}



\parbox{12.2 true cm}{\caption{\label{fig:1-122-DataTable} Data from Prediction of Electric Vehicle Penetration\cite{EVTC2015}. }}
\end{center}

\vspace*{-1.3 true cm}
\end{figure}


\begin{thebibliography}{98}


 \bibitem{EVTC2015} Electric Vehicle Transportation Center. 2015.  University of Central Florida, Cocoa FL USA. {\it Project 5: Prediction of Electric Vehicle Penetration\/}.  \url{http://evtc.fsec.ucf.edu/research/project5.html}.

\end{thebibliography}



\teacher{
\section*{NOTES FOR TEACHERS}

We basically considered two versions of the logistic differential equation (\ref{eq:SOT1}) and (\ref{eq:SOT2}) for modeling the spread of technology. 

\begin{equation}\label{eq:SOT1}
y(t) = r  y(t) \frac{K - y(t)}{K} \, , 
\end{equation}
\noindent where $y(t)$ is the number   of PEV sales in year $t$, $r$ is the intrinsic growth rate, and $K$ is the "carrying capacity" or the market cap, say, due to the number of charging stations.
\noindent and 
\begin{equation}\label{eq:SOT2}
y(t) = r  y(t) \frac{K(t) - y(t)}{K(t)} \,  , 
\end{equation}
\noindent where $K(t) = a + b t$ is the time dependent carrying capacity as a function of time $t$.  Here $a$ is the initial cap and $b$ is the rate at which that cap increases per year.

In a Mathematica notebook, 1-122-T-Mma-SPreadPEV-TeacherVersion.nb, and a pdf version of the notebook for those who do not use Mathematica we offer analyses of these two models and use the Akaike Information Criterion (AIC)\cite{Ledder2013, Wiki2017} to compare the results. Model (\ref{eq:SOT2}) is found to be the better model.


\begin{thebibliography}{98}

\bibitem{Ledder2013} Ledder, G. 2013. {\it Mathematics for the Life Sciences\/}. New York: Springer.

\bibitem{Wiki2017} Wikipedia. 2017. Akaike information criterion. Wikipedia, The Free Encyclopedia, \href{https://en.wikipedia.org/w/index.php?title=Akaike\_informatio\_criterion\&oldid=758283417}{https://en.wikipedia.org/w/index.php?title=Akaike\_informatio\_criterion\&oldid=758283417}.  

\end{thebibliography}


}

\end{document}
