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Super-linear Scaling Behavior for Electric Vehicle Chargers and Road Map to Addressing the Infrastructure Gap
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\dropcap{C}onsumer interest in Electric Vehicles (EV) has been rising as EVs approach price parity with internal combustion engines (ICE).
However, the lack of sufficient Electric Vehicle Supply Equipment (EVSE), popularly called charging stations, could slow future adoption\cite{shareefReviewStageoftheartCharging2016,ahmedEnablingFastCharging2017}.
Thus identifying the placement of EVSE to meet this growing demand is essential and has been an area of extensive research~\cite{shareefReviewStageoftheartCharging2016,guttenbergINCEPTSSoftwareHighfidelity2021,ahmedEnablingFastCharging2017,weiPersonalVehicleElectrification2021}.
Prior work in the area has relied predominantly on mathematical optimization to maximize captured vehicle flow, minimize economic costs, or account for power grid-related issues~\cite{shareefReviewStageoftheartCharging2016}.
Bottom-up optimization based-methods can precisely place and size EVSE based on the local demand and peculiarities.
However, they can become challenging and computationally prohibitive for larger-scale analysis, as a complex system of trade-offs emerges, which brings into question which factors are feasible to include and what form the loss function should take~\cite{shareefReviewStageoftheartCharging2016}.
On the other hand, a coarse-grained scaling analysis approach provides a tractable way to assess the infrastructure needs for large regions.
In particular, scaling relationships demonstrate how features change with system size and often illustrate that a single set of mechanisms is governing a system~\cite{bettencourtOriginsScalingCities2013,kempes2019scales,taylor2019scalability,westScaleUniversalLaws2017,marquetScalingPowerlawsEcological2005}.
Even when the specific mechanisms are unidentified, the coarse-grained regularities of a system, as captured by the scaling exponents, often provide a powerful foundation for building models, making forecasts, and interpreting dominant trade-offs~\cite{bettencourtOriginsScalingCities2013,bettencourtGrowthInnovationScaling2007,taylor2019scalability,kempes2019scales,leitaoThisScalingNonlinear2016,westScaleUniversalLaws2017,marquetScalingPowerlawsEcological2005}.
Here we formulate the EVSE infrastructure problem in a scaling analysis framework \(\left( Y = Y_0 N^{\beta} \right)\), which connects the EVSE infrastructure needs for a particular region to its population size.
We find that the current EVSE infrastructure follows scaling similar to gas stations; however, the scaling exponent is super-linear (\(\beta > 1\)) while that of gas stations is sub-linear (\(\beta < 1\)).
Sub-linear scaling behavior is well established for infrastructure assets and is motivated by theory~\cite{bettencourtOriginsScalingCities2013,bettencourtGrowthInnovationScaling2007}.
The super-linear exponent of the current EVSE build-out indicates interesting differences.
The appearance of super-linear exponents in human systems has been observed for quantities that rely on social interactions, such as patents or gross domestic product~\cite{bettencourtGrowthInnovationScaling2007}.
The argument here is that as cities increase in size, they densify, leading to increased social interactions per person in settings with less infrastructure requirements.
This paradigm implies that all new infrastructures should be sub-linear.
However, this is not necessarily true for an emerging infrastructure, which may be initially driven by social phenomena and later by standard infrastructural geometry constraints.
Thus, we argue that the emergent behavior of the current EVSE market should initially follow super-linear scaling due to social adoption dynamics but will transition to sub-linear behavior as the infrastructure matures.
Finally, we estimate the EVSE infrastructure gap as the number of additional stations required to match the power delivery of existing gasoline stations.
To our knowledge, scaling analysis has not been used to forecast future infrastructure requirements of emerging technology.
Our application here to EVSEs opens up a variety of future work in scaling theory.
\subsection*{Results}
\begin{figure}[t]
\centering
\parbox{0pt}{\phantomsubcaption\label{fig:charger_scale_stations}}
\parbox{0pt}{\phantomsubcaption\label{fig:charger_scale_power}}
\includegraphics[width=\linewidth]{img/charger_scale.pdf}
\caption{\subref*{fig:charger_scale_stations})~Power Law Scaling for Gas Station and Electric Vehicle Supply Equipment for United State Counties (n = 3143), showing novel super-linear behavior for EVSE stations and expected sub-linear behavior for gas stations. Super-linear behavior suggests EVSE infrastructure has been consolidated in larger population centers. 95\% confidence intervals for the scaling exponent \(\beta\) are shown in the legend. \subref*{fig:charger_scale_power})~Comparison of the power delivery of existing gas stations (Assuming 12 pumps per station and improved efficiency of EVs) to existing EVSE infrastructure. While the EVSE infrastructure of some counties has reached parity, in quantity with gasoline stations, no counties have reached parity in terms of power delivery.\label{fig:charger_scale}
}
\end{figure}
We curated a dataset of county-level EVSE charger, gasoline station counts, and the corresponding county population.
We fit several Generalized Linear Models (GLM) to the data using maximum likelihood estimation (MLE) to predict the number of EVSE (\(Y_{EVSE}\)) and gasoline (\(Y_{GS}\)) stations for all counties (\(n = 3143\)) in the United States from their population (Figure~\ref{fig:charger_scale_stations}).
Using a likelihood ratio test, we compared all models to their null counterparts and have tabulated the test statistic \(\lambda_{LR}\) for each model in Table~\ref{tab:model_zoo}.
All models were found to be highly significant with a p-value of less than \(10^{-99}\), as such, we have only reported the test statistic.
For the power-law scaling model, we initially fit models assuming a Poisson distribution, \(Pois(\mu)\), but found the assumption that the variance is equal to the mean to be a poor fit for our data.
We relaxed this by fitting a Negative Binomial (NB) distribution, \(NB(r, \mu)\) parameterized by a dispersion parameter \(r\) and its expected value \(\mu \).
Using this model, our fitted values for \(\beta \), with a 95\% confidence interval, was \(\beta = 1.17 \pm 0.051\) for the EVSE stations and \(\beta = 0.77 \pm 0.0092\) for the gasoline stations.
We do see a reduction in McFadden's \(R^2_{McF}\) for the NB models~(Table~\ref{tab:model_zoo}) driven by a higher likelihood for their null counterpart, and not a reduction in the model's fit.
We performed a Wald test to determine if \(\beta \neq 1\) was statistically significant for the NB power law models;
and found \(\beta \) to be significantly different from 1 for both the EVSE (\(\mbox{SE} = 0.026\mbox{, W} = 6.4\mbox{, p} < 10^{-9}\)) and gas stations (\(\mbox{SE} = 0.0047\mbox{, W} = -50\mbox{, p} < 10^{-99}\)).
As all models achieved similar Root-Mean-Squared-Deviations (RMSD) and are statically significant (\(\lambda_{LR}\)), we used the Bayesian Information Criterion (BIC) to compare models.
Using the criteria that \(\Delta BIC > 6\) indicates a significant difference in model performance~\cite{leitaoThisScalingNonlinear2016}, we found strong evidence for the NB power scaling models over both linear and quadratic models.
An in-depth discussion of the above statistical tests can be found in the SI appendix.
We performed an ordinary least squares (OLS) regression on the log-log transformed data in line with prior scaling works~\cite{bettencourtOriginsScalingCities2013,bettencourtGrowthInnovationScaling2007,taylor2019scalability,kempes2019scales,leitaoThisScalingNonlinear2016,marquetScalingPowerlawsEcological2005}.
However, as this method does not support zero-count data, it excludes a significant portion of our dataset (43.8\% for EVSE and 1\% for gasoline stations).
We did not find the OLS models compelling and excluded the fits from this analysis; see the SI Appendix for additional details.
\subsubsection*{Gas Station to EVSE Scaling}
Driven by an analogous utility, as EV adoption increases, EVSE infrastructure should tend towards the same coarse-grained regularities as gasoline stations.
With the number of stations in an area to be proportional to the vehicle miles travel \(M\), average efficiency \(E\), and power delivery of a station \(P\): \(Y \propto \eta M / P\).
Assuming no change in consumer driving behavior, the number of EVSE stations to replace a single gasoline station is:
\begin{equation}\label{eq:ev_power_parity}
\frac{Y_{EVSE}}{Y_{GS}} = \frac{\eta_{EV} P_{GS}}{\eta_{ICE} P_{EVSE}}
\end{equation}
Using the regulated gasoline flow rate of 10 \si{gpm} for a consumer pump in the United States~(40 CFR \S 1090.1550) and the EPA's 33.705\si{kWh/gallon} of gasoline equivalency (40 CFR \S 600.002), the max power delivery of a consumer gasoline pump is \(P_{EVSE} = 20.2 \si{MW}\).
We assume \( \eta_{EV} / \eta_{ICE} \approx 3 \), or that on average EV's consume 1/3 the energy per mile traveled compared to ICE vehicles.
Assuming \(P_{EVSE} = 400\si{kW}\) or Extreme Fast Charging~\cite{ahmedEnablingFastCharging2017}, gives \( Y_{EVSE} / Y_{GS} = 17 \), or to replace one gasoline pump, 17 ports are required to reach power parity.
Reducing \(P_{EVSE}\) to 11.5kW, or the upper end of available home chargers, we find \( Y_{EVSE} / Y_{GS} \approx 586 \).
From this, holding the average number of pumps/ports constant, the number of EVSE stations needed in an area is given by Eq.~\ref{eq:ev_charger_prediction}.
\begin{equation}\label{eq:ev_charger_prediction}
\hat{Y}_{EVSE} = 3 \frac{P_{GS}}{P_{EV}} \times Y_{0,GS} N^{\beta_{GS}}
\end{equation}
\begin{table}[h]
\caption{Model Statistics for Fitted Generalized Linear Models\hspace*{\fill}}\label{tab:model_zoo}
\centering
\begin{tabular}{llccccc}
& Model & RMSD & $R^2_{McF}$ & \shortstack{$\lambda_{LR}$ \\ $\times 10^{3}$} & \shortstack{BIC \\ $\times 10^{3}$} \\
\midrule
\multirow{4}{*}{\rotatebox[origin=c]{90}{EVSE}} & $NB \left (r, \mu = Y_0 N ^\beta \right)$ & $49.5$ & $0.195$ & $3.24$ & $13.4$ \\
& $\mathcal{N}(a N^2 + bN + c, \sigma^2)$ & $45.3$ & $0.104$ & $3.81$ & $32.9$ \\
& $\mathcal{N}(a N + b, \sigma^2)$ & $49.6$ & $0.088$ & $3.25$ & $33.5$ \\
& $Pois(Y_0 N ^\beta)$ & $46.4$ & $0.804$ & $167$ & $40.7$ \\
\midrule
\multirow{4}{*}{\rotatebox[origin=c]{90}{Gasoline}} & $NB \left (r, \mu = Y_0 N ^\beta \right)$ & $29.2$ & $0.283$ & $8.02$ & $20.3$ \\
& $Pois(Y_0 N ^\beta)$ & $28.8$ & $0.862$ & $176$ & $28.2$ \\
& $\mathcal{N}(a N^2 + bN + c, \sigma^2)$ & $28.6$ & $0.168$ & $6.08$ & $30.0$ \\
& $\mathcal{N}(a N + b, \sigma^2)$ & $30.6$ & $0.156$ & $5.63$ & $30.5$ \\
\bottomrule
\end{tabular}
\end{table}
\subsubsection*{Mean-field Model Relating Charging Power and Vehicle Speed}
In this section, we develop a mean-field model to estimate the maximum average speed of an EV for a given charger power.
We assume that the EV charges at \(P_{EVSE}\) for \(\alpha \) percent of the time, then drives at a speed of \(v\) for the remaining \(1 - \alpha \) percent of the time.
We further make the simplifying assumption that aerodynamic forces dominate the overall power draw of the EV.\@
For a continuous drive cycle, the energy delivered during charging must match the energy demand of the drive cycle:
\begin{equation}
P_D (1 - \alpha) = \alpha P_{EVSE} \to
\frac{1}{2} \rho C_d A v^3 = \frac{\alpha}{1 - \alpha} P_{EVSE}
\end{equation}
From this, we can relate the power of the charger \(P_{EVSE}\) to the vehicle's speed while driving \(v\):
\begin{equation}\label{eq:home_charging_speed}
v = \sqrt[3]{ \frac{\alpha}{1-\alpha} \frac{2}{\rho C_d A} P_{EVSE} }
\end{equation}
As the vehicle is stationary while charging, the average vehicle speed is \(\bar{v} = v (1-\alpha) \propto \alpha^{1/3} {\left( 1-\alpha \right)}^{2/3} \).
The maximum average speed occurs when the vehicle spends \(\alpha = 1/3\) of its time charging at 2x the power it consumes while driving~(Figure~\ref{fig:home_chargers_avg_speed}).
For our analysis we assume \( \rho = 1.225 \si{kg/m^3}\) and \(C_d A = 0.75 \si{m^2} \); or a boxy sedan at International Standard Metric Conditions.
In the case of \(P_{EVSE} = 1.92 \si{kW}\), this translates to a driving speed of \(v = 28.6 \si{mph}\), and an average speed of \( \bar{v} = 19 \si{mph} \), speeds well suited for residential driving.
Higher driving speeds are possible with 1.92\si{kW}, but only at the expense of increased charging times, which may be tenable for sort trips if charging can occur before the next trip.
However, operating above \(\alpha > 1/3\) will increase the overall trip duration for long-range trips.
At the other extreme, 400 \si{kW} delivers a max driving speed (\textasciitilde170mph) far beyond highway speed limits.
Instead, the higher charging rates enable shorter charging times for a given driving speed (Figure~\ref{fig:home_chargers_driving_speed}).
Our assumption that aerodynamic forces dominate the overall power consumption breaks down at low speeds.
More complex models are possible\cite{guttenbergINCEPTSSoftwareHighfidelity2021,weiPersonalVehicleElectrification2021}, but do not provide a closed-form solution for velocity as a function of charger power.
Additionally, as a lower bound on the vehicle's power consumption, Eq.~\ref{eq:home_charging_speed} provides an upper bound on the vehicle's speed for a given charger power.
Thus the qualitative conclusion that charger power limits the types of trips that are feasible remains valid.
\subsubsection*{Home Charging}
At present, consumer-grade chargers range from 1.92kW (120V@12A) to 11.5kW (240V@48A), significantly slower than commercial or a ``gasoline-equivalent'' charger.
As shown in Figure~\ref{fig:home_chargers_avg_speed}, home chargers are insufficient for a continuous operating cycle at highway speeds.
The mean-field analysis provides a framework to understand the scaling relationship between EVSE power and vehicle speed.
A more detailed analysis is needed to account for real-world factors such as mixed power level charging stations and idle time for the vehicle.
Home charging may be sufficient for many consumers' local commutes~\cite{weiPersonalVehicleElectrification2021}; however, it will be insufficient for predominantly long-distance highway travel.
\begin{figure}[t]
\centering
\parbox{0pt}{\phantomsubcaption\label{fig:home_chargers_avg_speed}}
\parbox{0pt}{\phantomsubcaption\label{fig:home_chargers_driving_speed}}
\includegraphics[width=\linewidth]{img/home_chargers.pdf}
\caption{\subref*{fig:home_chargers_avg_speed})~Average vehicle speed vs.~time charging for varying charger power outputs. The solid black lines of constant driving speed (\(v\)) reflect the impact of increasing charge time, when the vehicle is stationary, on the average speed (\(\bar{v}\)). The maximum \(\bar{v}\) for a given charger power occurs at \(\alpha = 1/3\), indicated by the dashed vertical bar. \subref*{fig:home_chargers_driving_speed})~Trade-off between driving speed \(v\) and charger power for various charging times, with vertical lines at notable charger power levels. For example, a vehicle driving at 30mph would spend \textasciitilde10\% of it's time charging at 11.5kW vs.\ \textasciitilde0.2\% at 400kW.\label{fig:home_chargers}}
\end{figure}
\subsection*{EVSE Infrastructure Gap}
Using Eq.~\ref{eq:ev_charger_prediction}, we estimated the EVSE Station Gap: \(\Delta Y_{EVSE} = \hat{Y}_{EVSE} - Y_{EVSE}\), or the number of additional stations needed for each county, assuming all current and future chargers are 400kW~(Figure~\ref{fig:us_charger_gap}).
The gap between the existing EVSE infrastructure and the existing ICE infrastructure is large (Figure~\ref{fig:charger_scale_power}).
Our model assumes a fixed average pumps/ports ratio, ignores consumer behavior regarding longer charging times, the role of home chargers, and variations in consumer behavior over time.
Our treatment of existing gasoline stations neglect factors, such as gasoline purchases outside of on-highway transportation, that may inflate the number of stations in a region.
Finally, scaling laws can not provide precise EVSE placement or sizing information as a coarse-grained model.
Further refinement is left to optimization-based methods~\cite{shareefReviewStageoftheartCharging2016} or complex systems analysis~\cite{guttenbergINCEPTSSoftwareHighfidelity2021}.
\subsection*{Temporal Evolution of Scaling Relationships}
Beyond the planning and policy implications of the scaling relationships presented here, these findings also provide new insights into scaling phenomena in general.
The central question in scaling theory is how to connect scaling exponents with fundamental mechanisms.
For many biological systems and human infrastructure, the optimal solution to dominant physical constraints directs the scaling exponent~\cite{kempes2019scales,bettencourtOriginsScalingCities2013,westScaleUniversalLaws2017,marquetScalingPowerlawsEcological2005}.
At the same time, recent studies show that human institutions can adjust their exponent values based on their distinct goals and missions~\cite{taylor2019scalability}.
However, in biology, we have not observed the evolution of scaling exponent values in time towards the equilibrium optimum.
Such temporal changes would have occurred during much deeper histories than we can observe.
EVSE infrastructure is an example of an out-of-equilibrium scaling relationship, where predicting the equilibrium scaling exponent provides unique planning forecasts.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{img/us_charger_gap.pdf}
\caption{The EVSE Station Gap between the number of existing EVSE stations and the number predicted with Eq.~\ref{eq:ev_charger_prediction}, assuming all current and future chargers are capable of 400kW.
At present, no counties have sufficient EVSE stations to meet power parity, even when assuming all existing stations have been upgraded to 400 kW.
}\label{fig:us_charger_gap}
\end{figure}
\matmethods{
County-level population estimates were obtained from the United States Census Bureau's Population Estimates Program using the 2020 Vintage estimates of the 2010 Census; at the time of writing, this was the most up-to-date estimate available~\cite{CountyPopulationTotalsC2010V2020}.
We used the Forth Quarter 2020 counts for ``Gasoline Stations'' from the United States Bureau of Labor Statistics Quarterly Census of Employment and Wages Program~\cite{QuarterlyCensusEmployment2021}.
EVSE charger locations, and other metadata, were obtained from the National Renewable Laboratory's Alternative Fuel Stations Application Programming Interface~\cite{nrelAlternativeFuelStations2021} and then geocoded to counties using shapefiles obtained from the United States Census~\cite{census2020TIGERLine2021}.
We then fit various GLMs to the data to predict station counts, EVSE and Gasoline, for each county using maximum likelihood estimation as implemented in the Julia Package GLM.jl.
Due to difficulty fitting some parameters, we increased the maximum iterations from 30 to 60 but otherwise used the default settings.
For an extended discussion of model fitting, see the GLM section of the SI Appendix.
Total EVSC Power for each station was computed by multiplying the counts of Level 1 (1.4\si{kW}), Level 2 (7.2\si{kW}), and DC Fast Chargers (50\si{kW}) by their respective power outputs shown in parenthesis~\cite{ahmedEnablingFastCharging2017}.
Individual EVSE station power estimates were summed to produce the county-level EVSE power estimates shown in Figure~\ref{fig:charger_scale_power}.
}
\showmatmethods{}
\acknow{
The authors from CMU acknowledge the support from Mobility21, A United States Department of Transportation National University Transportation Center.
CPK thanks CAF Canada and Toby Shannan for generously supporting this work.
}
\showacknow{}
\bibliography{references}
\clearpage
\includepdf[pages=-]{SI.pdf}