# Placement of EV Charging Stations—Balancing Benefits among Multiple Entities

Chao Luo, Yih-Fang Huang, and Vijay Gupta C. Luo, Y.-F. Huang, and V. Gupta
are with the Department of Electrical Engineering, University of Notre Dame,
Notre Dame, IN, 46556 USA e-mail: {cluo1, huang<EMAIL_ADDRESS>This paper
was presented, in part, at the 2015 IEEE 81st Vehicular Technology Conference.
This work has been partially supported by the National Science Foundation
under grants CNS-1239224 and ECCS-0846631.

###### Abstract

This paper studies the problem of multi-stage placement of electric vehicle
(EV) charging stations with incremental EV penetration rates. A nested logit
model is employed to analyze the charging preference of the individual
consumer (EV owner), and predict the aggregated charging demand at the
charging stations. The EV charging industry is modeled as an oligopoly where
the entire market is dominated by a few charging service providers
(oligopolists). At the beginning of each planning stage, an optimal placement
policy for each service provider is obtained through analyzing strategic
interactions in a Bayesian game. To derive the optimal placement policy, we
consider both the transportation network graph and the electric power network
graph. A simulation software—The EV Virtual City 1.0—is developed using Java
to investigate the interactions among the consumers (EV owner), the
transportation network graph, the electric power network graph, and the
charging stations. Through a series of experiments using the geographic and
demographic data from the city of San Pedro District of Los Angeles, we show
that the charging station placement is highly consistent with the heatmap of
the traffic flow. In addition, we observe a spatial economic phenomenon that
service providers prefer clustering instead of separation in the EV charging
market.

###### Index Terms:

Electric vehicle, charging station placement, consumer behavior, nested logit
model, Bayesian game, oligopoly.

##  I Introduction

The continued technological innovations in battery and electric drivetrain
have made electric vehicles (EVs) a viable solution for a sustainable
transportation system. Currently, most EV charging is done either at
residences, or for free at some public charging infrastructure provided by
municipalities, office buildings, etc. As the EV industry continues to grow,
commercial charging stations will need to be strategically added and placed.
Development of effective management and regulation of EV charging
infrastructure needs to consider the benefits of multiple
constituencies—consumers, charging station owners, power grid operators, local
government, etc. In this paper, we concentrate on striking a balance among the
profits of charging station owners, consumer satisfaction, and power grid’s
reliability.

Our work is motivated by the desire of service providers to make a forward-
looking decision on charging station placement to obtain a good return on
their investment. We use the most up-to-date information (i.e., travel
pattern, traffic flow, road network, power grid, etc.) to make the best-effort
decisions on charging station placement, hoping that service providers will
have a good chance to profit over the next few years. In this paper, we do not
consider factors such as uncertainties in fuel prices, climate change,
population migration etc., which are random and unpredictable. Instead, we
assume that some revenue management techniques (i.e., real-time pricing) may
be applied to deal with the potential effects of these factors.

We assume that the service providers aim to strike a balance between the
competing goals of maximizing their profits and minimizing the disturbance to
the electric power network due to large-scale EV charging. Accordingly, we
construct a utility function that incorporates both of these aims. Each
charging service provider attempts to maximize his/her own expected utility
function while satisfying the Quality-of-Service (QoS) constraints through
choosing the optimal locations of charging stations that s/he owns. The nested
logit model is used to analyze and predict the charging preference of EV
owners. At the beginning of each stage, the service providers predict the
charging demand of each charging station candidate using the nested logit
model. The optimal placement strategy is obtained through a Bayesian game. As
the EV penetration rate increases, the existing charging stations may no
longer satisfy the QoS constraints and a new stage shall be initiated to place
more charging stations.

There is a growing literature addressing the issues relevant to EV charging
station placement. [2]-[5] formulated charging station placement as an
optimization problem. However, they did not take into account the overall
consumer satisfaction and the impact of EV charging on the electric power
network in their works. Besides, their optimization models were formulated
from the perspective of a central urban planner rather than that of service
providers in a free competitive market. In [6], the authors presented a
strategy to deploy charging stations by analyzing the patterns of residential
EV ownership and driving activities. In their work, they deploy the new
charging stations either randomly with no weight or only based on the weights
of population. They did not consider the mobility of EVs and the overall
consumer experience. In [7], Bernardo _et al._ proposed a discrete choice
model (DCM) based framework to study the optimal locations for fast charging
stations. They treat each charging station as a player in a noncooperative
game. However, the underlying assumption in their work is that each player has
complete information about other players, which may be overly restrictive and
infeasible in a practical competitive market. In this paper, we propose a
Bayesian game framework that does not require the complete information of
other players.

The main contributions of our work are: (1) A multi-stage charging station
placement strategy with incremental EV penetration rates is first formulated,
which takes into account the interactions among EVs, road network, and the
electric power grid; (2) The nested logit model is then employed to
characterize the overall consumer satisfaction and predict the aggregated
charging demand, which provides insights into the preferences and decision-
making processes of EV owners; (3) An oligopolistic market model of EV
charging service providers is studied and a Bayesian game framework is applied
to analyze the strategic interactions among service providers. (4) A
simulation software has been developed to analyze the interplay among EV
owners, road network, power grid, urban infrastructure and charging stations
111The EV Virtual City 1.0 simulator can be downloaded from
https://github.com/chaoluond/EVVirtualCity.

The paper is organized as follows: Section ii@ presents the problem
formulation. Section iii@ discusses the nested logit model and how charging
demand is calculated. In Section iv@, we describe the impact of EV charging on
the power grid. In Section v@, a Bayesian game is used to characterize the
competition among service providers, and the optimal station placement policy
is obtained. Section vi@ shows the architecture and applications of the
simulation software and discusses a case study in San Pedro District.
Conclusions are given in Section<EMAIL_ADDRESS>Table i@ provides a full description of
parameters and symbols used in the paper.

##  II Problem Formulation

Table I: Parameter Description Parameter | Description | Unit
---|---|---
$L$ | Total Candidate Locations | -
$N$ | Total EVs | -
$\psi_{j,k}^{n}$ | Charging Demand | kWh
$s_{j,k}$ | Placement Indicator | -
$F_{j,k}$ | Setup Cost | $
$R_{k}$ | Total Revenue | $
$c_{j,k}$ | Locational Marginal Price | $
$\Pi_{k}$ | Total Profit | $
$U_{k}$ | Overall utility | $
w | Coef. of EV Charging Penalty | -
$\Upsilon_{k}$ | Average Service Probability | -
$\Xi_{k}$ | Average Service Coverage | -
$t_{k}$ | Average Charging Time | min
$p_{k}$ | Retail Charging Price | $/kWh
$i_{n}$ | Income of the $n$th EV owner | $
$d_{j,k}^{n}$ | Deviating distance | km
$z_{j,k}^{n}$ | Indicator of Destination | -
$d_{th}$ | Distance threshold | km
$r_{j,k}$ | Indicator of Restaurant | -
$g_{j,k}$ | Indicator of Shopcenter | -
$m_{j,k}$ | Indicator of Supermarket | -
$\mathbf{P}_{\textrm{g}}^{\textrm{base}}$ | Active power vector without EV | -
$\mathbf{P}_{\textrm{g}}^{\textrm{EV}}$ | Active power vector with EV | -
$\mathbf{Q}_{\textrm{g}}^{\textrm{base}}$ | Reactive power vector without EV | -
$\mathbf{Q}_{\textrm{g}}^{\textrm{EV}}$ | Reactive power vector with EV | -
$v_{i}$ | Voltage at bus $i$ | Volts
$\phi_{ik}$ | Voltage angle between bus $i$, $k$ | Radian
$\alpha$ | Coef. of $t_{k}$ | -
$\beta$ | Coef. of $p_{k}/i_{n}$ | -
$\mu_{k}$ | Coef. of $d_{j,k}$ | -
$\eta_{k}$ | Coef. of $z_{j,k}$ | -
$\gamma_{k}$ | Coef. of $r_{j,k}$ | -
$\lambda_{k}$ | Coef. of $g_{j,k}$ | -
$\delta_{k}$ | Coef. of $m_{j,k}$ | -

In this paper, we postulate the problem of EV charging with an oligopolistic
market structure that has multiple charging service providers (oligopolists).
The service providers aim to maximize their expected utility while satisfying
the QoS constraints by selecting optimal station placements.

In particular, we consider the case of three service providers that offer
three EV charging services [8], namely, Level 1, Level 2, and Level 3 (see
Table II for details). Level 1 and Level 2 are AC charging. Level 3 charging
is DC Fast charging. Let $\mathcal{O}=\\{1,2,3\\}$ denote the set of charging
service providers. Moreover, we assume that service provider 1 offers Level 1
charging, service provider 2 offers Level 2 charging, and service provider 3
offers Level 3 charging. The three charging levels represent three charging
services, which have different charging voltages and currents, charging times,
and charging experiences. In economic terms, they are imperfect substitutes to
each other. In our model, we are interested in investigating how the different
charging services compete with each other in choosing locations and prices.
Each service provider can run multiple charging stations. At each planning
stage, service providers select some charging stations from a given set of
candidates, denoted as $\mathcal{I}=\\{1,2,3\cdots,L\\}$. The set of EVs is
denoted as $\mathcal{E}=\\{1,2,3,\cdots,N\\}$.

###  II-A The Profit of EV Charging

Table II: Charge Method Electrical Ratings Charging Method | Nominal Supply Voltage(Volts) | Max Current (Amps) | Time from fully depleted to fully charged
---|---|---|---
Level 1 | 120 vac, 1-phase | 12 A | 16-18 hours
Level 2 | 208 to 240 vac, 1-phase | 32 A | 3-8 hours
Level 3 | 600 vdc maximum | 400 A maximum | less than 30 minutes

We assume that service providers run the charging stations like “chain
stores”, so that charging stations affiliated with the same service provider
have the same retail charging price. The charging stations purchase the
electricity from the wholesale market at the locational marginal price (LMP).
In a deregulated electricity market (like PJM, NYISO, NEISO, MISO, ERCOT,
California ISO in USA, the New Zealand and Singapore markets), LMP is computed
at every node (bus) by the market coordinator. LMP primarily consists of three
components: system energy price, transmission congestion cost, and cost of
marginal losses [9]. Let $p_{k}$ represent the retail charging price of
provider $k\;(k=1,2,3)$, and $p_{-k}$ be the retail charging prices of the
other two service providers except $k$. Let $c_{j,k}$ be the LMP of the $j$th
charging station candidate of service provider $k$, and $\psi_{j,k}$ be the
predicted charging demand at the $j$th charging station candidate of service
provider $k$. The vector $S_{k}=[s_{1,k},s_{2,k},\cdots,s_{L,k}]^{\textrm{T}}$
represents the placement policy of service provider $k$, where
$s_{j,k}\in\\{0,1\\}$ is an indicator with $s_{j,k}=1$ implying that service
provider $k$ will place the $j$th charging station. Let $S_{-k}$ represent the
placement policies of the other two service providers, $\theta_{j,k}$ be the
placement cost of the $j$th charging station. The total profit of service
provider $k$ is given by

$\Pi_{k}=p_{k}\Psi_{k}^{\textrm{T}}S_{k}-\textrm{diag}[C_{k}]\Psi_{k}^{\textrm{T}}S_{k}-\Theta_{k}^{\textrm{T}}S_{k},$
(1)

and the total revenue of service provider $k$ is,

$R_{k}=p_{k}\Psi_{k}^{\textrm{T}}S_{k}-\textrm{diag}[C_{k}]\Psi_{k}^{\textrm{T}}S_{k},$
(2)

where
$\Psi_{k}=[\psi_{1,k},\psi_{2,k},\psi_{3,k},\cdots,\psi_{L,k}]^{\textrm{T}}$,
$C_{k}=[c_{1,k},c_{2,k},\cdots,c_{L,k}]^{\textrm{T}}$ and
$\Theta_{k}=[\theta_{1,k},\theta_{2,k},\cdots,\theta_{L,k}]^{\textrm{T}}$. The
notation $\textrm{diag}[.]$ is an operator to create a diagonal matrix using
the underlying vector, and $[.]^{\textrm{T}}$ is the transpose operation. In
Equation (2), the total revenue from the sales is
$p_{k}\Psi_{k}^{\textrm{T}}S_{k}$, the cost of purchasing electricity is
$\textrm{diag}[C_{k}]\Psi_{k}^{\textrm{T}}S_{k}$, and the placement cost is
$\Theta_{k}^{\textrm{T}}S_{k}$.

###  II-B The Disturbance on Power Grid Due to EV Charging

It is conceivable that the simultaneous large-scale EV charging can disrupt
the normal operation of the power grid in terms of frequency variation,
voltage imbalance, voltage variation, power loss, etc [10]-[12]. In a
conventional power grid, the generators will be incentivized to cooperatively
control the output of real power and reactive power to maintain system
stability, perform frequency regulation and voltage regulation. The charging
service providers typically cannot participate in such market. Instead, we
assume that they are fined in proportion to the “disturbance” they impose on
the grid. Thus, the providers must optimally place the charging stations to
mitigate the “disturbance” to the power grid. Accordingly, the overall utility
function of service provider can be defined as:

$U_{k}=\Pi_{k}-wB_{k},$ (3)

where $\Pi_{k}$ is the total profits from EV charging. $B_{k}$ characterizes
the penalty arising from large-scale EV charging. The variable $w$ is a
weighting coefficient, reflecting the tolerance to the penalty. In the
following section, we will further discuss how to develop a proper metric to
evaluate the penalty $B_{k}$. We should note that the weighting factor $w$ in
Equation (3) offers a mechanism for the charging service providers to strike a
balance between their own profit and the “stress” their charging adds to the
power system. If $w=0$, the impact of EV charging on the grid is not
considered at all, and a non-zero value of $w$ implies some impact on the
grid—the larger $w$ is, the larger the impact is. Generally, if the focus is
on the charging provider’s profit, a small $w$ is used. In practice, the value
of $w$ needs to be determined, e.g., with the help of heuristic and empirical
data.

###  II-C Quality-of-Service Constraints

We use two quality-of-service (QoS) metrics for the service provider: (1) the
average service delay probability $\Upsilon_{k}$, and (2) the average service
coverage $\Xi_{k},\;(k=1,2,3)$.

$\Upsilon_{k}=\frac{1}{N}\sum_{i=1}^{N}\upsilon_{i,k},$ (4)

$\Xi_{k}=\frac{1}{N}\sum_{i=1}^{N}\xi_{i,k},$ (5)

where $\upsilon_{i,k}$ is the average service delay probability for the $i$-th
EV owner getting the EV charged at service provider $k$. For the $i$th EV
owner, $\upsilon_{i,k}$ is defined as the ratio of the number of delayed
charging to the total number of charging attempts; $\xi_{i,k}$ is the average
number of accessible Level $k$ charging stations along the route from origin
to destination. Notice that $\upsilon_{i,k}$ and $\xi_{i,k}$ are random
variables that depend on the travel patterns of all EVs, the urban road
network and the charging stations. It is, thus, difficult to use a simple
formula to compute them. Instead, we employ Mento Carlo method to estimate
those two values.

###  II-D Multi-stage Charging Station Planning Scheme

At each planning stage, the service providers aim at solving the following
fundamental problem to obtain the optimal placement policy subjected to the
QoS constraints.

$\displaystyle[S_{k}^{T}|S_{1}^{T-1},S_{2}^{T-1},S_{3}^{T-1}]=$ (6)
$\displaystyle\operatorname*{argmax}_{\begin{subarray}{c}{s_{1,k},\cdots,s_{L,k}}\\\
{s_{j,k}\in\\{0,1\\}}\end{subarray}}\left\\{\mathbb{E}_{S_{-k}}[U_{k}]|S_{1}^{T-1},S_{2}^{T-1},S_{3}^{T-1}\right\\},$

subject to

$\Upsilon_{k}\leq\Upsilon^{0},$ (7) $\Xi_{k}\leq\Xi^{0},$ (8)

where $\mathbb{E}_{S_{-k}}[.]$ denotes the expectation over $S_{-k}$, and
$S_{k}^{T}$ is the placement policy at stage $T$. The variables $\Upsilon^{0}$
and $\Xi^{0}$ are the predetermined QoS constraints.

To solve this problem, we are confronted with three principal questions: (1)
How to predict the aggregated charging demand $\psi_{j,k}$ at each charging
station candidate? (2) How to find an appropriate metric to characterize the
impacts of EV charging on the power grid? (3) How to derive the optimal
placement policy in a tractable way? For the first question, we assume that
the service providers estimate the charging demand using a nested logit model.
In Section iii@, we will describe the nested logit model and elaborate on how
to use this model to analyze consumer behavior and estimate the charging
demand. In Section iv@, we will discuss how the EV charging may impact the
power grid, and propose a metric to assess the impacts of EV charging. For the
last question, notice that the optimization problem formulated by Equation (6)
is intractable since the optimal placement decision for every service provider
also depends on the decisions taken by other service providers. To this end,
we employ a Bayesian game model to characterize the strategic interaction and
price competition among the service providers. We will calculate the optimal
placement strategies and prices in Section<EMAIL_ADDRESS>
##  III Charging Demand of EV Charging Station

In this paper, the aggregated charging demand at a charging station candidate
is defined as the sum of the product of the probability that EV owners choose
that particular charging station and the electricity required to charge the
EVs. The charging behaviors of the EV owners may be influenced by many factors
that include the charging price, travel cost, amenities at or near the
charging station, the travel purpose, EV owner’s income, and so on. We use the
nested logit model to characterize the attractiveness of a charging station.

###  III-A Nested Logit Model And Probability of Choice

The nested logit model is widely used in the analysis and prediction of a
consumer’s choice from a finite set of choice alternatives [13]. The main idea
of nested logit model is that a consumer is a utility maximizer. The consumer
will choose the product which brings him/her the maximum utility.

In our problem, the utility that the $n$-th EV owner can obtain from choosing
charging station $j\;(j=1,2,\cdots,L)$ of service provider $k\;(k=1,2,3)$ is
denoted as $U_{j,k}^{n}=\overline{U}_{j,k}^{n}+\epsilon_{j,k}^{n}$, where
$\overline{U}_{j,k}^{n}$ is the observable utility and $\epsilon_{j,k}^{n}$ is
the unobservable utility. The vector of unobservable utility
$\epsilon^{n}=[\epsilon_{1,1}^{n},\cdots,\epsilon_{L,1}^{n},\epsilon_{1,2}^{n},\cdots,\epsilon_{L,2}^{n},\epsilon_{1,3}^{n},\cdots,\epsilon_{L,3}^{n}]^{\textrm{T}}$
is assumed to have a generalized extreme value (GEV) distribution with
cumulative distribution function (CDF) [13]

$F(\epsilon^{n})=\exp\left(-\sum_{k=1}^{3}\left(\sum_{l=1}^{L}e^{-\epsilon_{l,k}^{n}/\sigma_{k}}\right)^{\sigma_{k}}\right),$
(9)

where $\sigma_{k}$ is a measure of the degree of independence in the
unobservable utility among the charging stations owned by service provider
$k$. For the nested logit model, $\epsilon_{j,k}$ is correlated within each
charging level, and uncorrelated across different charging levels.

We can decompose the observable utility $\overline{U}_{j,k}^{n}$ into two
components—the utility of choosing service provider $k$ and the utility of
choosing a charging station $j$. In addition, we assume home charging is the
“outside good” in this market [14]-[15]. Thus, $\overline{U}_{j,k}^{n}$ for EV
owner $n$ can be expressed as

$\overline{U}_{j,k}^{n}=\overline{W}_{k}^{n}+\overline{V}_{j,k}^{n},$ (10)

where $\overline{W}_{k}^{n}$ is the observable utility of choosing service
provider $k$ (choosing nest $k$), and $\overline{V}_{j,k}^{n}$ is the
observable utility of choosing charging station $j$ given that service
provider $k$ has been chosen; $\overline{W}_{k}^{n}$ and $\overline{V}_{j,k}$
are linear weighted combinations of attributes of the charging stations and
the EV owner.

Note that the retail charging price and the charging time are two factors
differentiating the three charging services. In addition, we assume the income
of EV owners will also play a role in choosing charging services. In contrast
to our previous work [16], we use a different formula to calculate
$\overline{W}_{k}^{n}$ here.

$\overline{W}_{k}^{n}=\alpha\frac{1}{t_{k}}+\beta\frac{p_{k}}{i_{n}},$ (11)

where $t_{k}$, $p_{k}$ and $i_{n}$ represent, respectively, the average
charging time, the retail charging price, and the income of the $n$-th EV
owner; and $\alpha,\beta$ are the corresponding weighting coefficients. This
model is similar to Ben-Akiva and Lerman’s utility model in their study of
public transportation mode [17]. The value of $\alpha$ is positive because
shorter charging time implies a better charging service experience; therefore,
leading to higher utility. The value of $\beta$ is negative because a higher
retail charging price results in less utility. However, the retail charging
price is divided by the income, which reflects that the retail charging price
for the EV owners becomes less important as their income increases. As an
“outside good”, the utility of home charging is normalized, i.e.
$\overline{W}_{0}^{n}=0$.

Furthermore, we define $\overline{V}_{j,k}^{n}$ as follows,

$\displaystyle\overline{V}_{j,k}^{n}=$
$\displaystyle\mu_{k}d_{j,k}^{n}+\eta_{k}z_{j,k}^{n}+\gamma_{k}r_{j,k}+\lambda_{k}g_{j,k}+\delta_{k}m_{j,k},$
(12)

where $z_{j,k}^{n}$ is the destination indicator. If the $j$th charging
station is near the EV owner’s travel destination (within a threshold distance
$d_{th}$), $z_{j,k}^{n}=1$, otherwise, $z_{j,k}^{n}=0$. The term $d_{j,k}^{n}$
is the deviating distance due to EV charging. We use Dijkstra’s shortest path
algorithm [18] to calculate the travel route for each EV owner from his/her
origin to destination. If an EV owner needs to go to charging station $j$
halfway, we define the deviating distance $d_{j,k}^{n}$ as the route length of
this new route minus the route length of the original route. Additionally,
each candidate charging station has a vector of characteristics
$[r_{j,k},g_{j,k},m_{j,k}]^{\mathrm{T}}$, which characterizes the
attractiveness of this charging station in terms of those amenities. For
instance, if there exists a restaurant near location $j$, we may set
$r_{j,k}=1$, otherwise we may set $r_{j,k}=0$. Similarly, $g_{j,k}$ and
$m_{j,k}$ are the indicators for shopping center and supermarket,
respectively. The corresponding weighting coefficients are
$\mu_{k},\eta_{k},\gamma_{k},\lambda_{k},\delta_{k}$.

The EV owner’s choice is not deterministic due to the random unobservable
utility. However, we can derive the average probability that s/he will choose
a certain charging station by taking the expectation over the unobservable
utilities defined in Equation (9). The probability that the $n$-th EV owner
will choose the $j$-th charging station of service provider $k$ is [13]

$\displaystyle\Phi_{j,k}^{n}$
$\displaystyle=\mathbf{Prob}\left(\overline{U}_{j,k}^{n}+\epsilon_{j,k}^{n}>\overline{U}_{i,l}^{n}+\epsilon_{i,l}^{n},\forall
i\neq j,\textrm{or }l\neq k\right)$ (13)
$\displaystyle=\int_{-\infty}^{+\infty}F_{j,k}(\overline{U}_{j,k}^{n}-\overline{U}_{1,1}^{n}+\epsilon_{j,k}^{n},\overline{U}_{j,k}^{n}-\overline{U}_{2,1}^{n}+\epsilon_{j,k}^{n},$
$\displaystyle\cdots,\epsilon_{j,k}^{n},\cdots,\overline{U}_{j,k}^{n}-\overline{U}_{L,3}^{n}+\epsilon_{j,k}^{n})d\epsilon_{j,k}^{n},$

where $F_{j,k}$ denotes the derivative of $F$ with respect to
$\epsilon_{j,k}^{n}$, i.e. $F_{j,k}=\partial F/\partial\epsilon_{j,k}^{n}$.
Evaluating this integral with the above assumptions, we obtain

$\Phi_{j,k}^{n}=\frac{e^{\overline{U}_{j,k}^{n}/\sigma_{k}}\left(\sum_{l=1}^{L}e^{\overline{U}_{l,k}^{n}/\sigma_{k}}\right)^{\sigma_{k}-1}}{\sum_{t=1}^{3}\left(\sum_{l=1}^{L}e^{\overline{U}_{l,t}^{n}/\sigma_{t}}\right)^{\sigma_{t}}}.$
(14)

###  III-B Charging Demand Estimation

Once the EV owners’ choice probability is computed, we can predict the
charging demand at any charging station. Let $q_{n}\;(n=1,2,\cdots,N)$ denote
the total electricity (measured in kWh) that the $n$-th EV owner purchases
from the charging station. Further, let $q_{n}$ be a random variable uniformly
distributed in the range $[Q_{a},Q_{b}]$, where $Q_{a}$ and $Q_{b}$ are,
respectively, the lower and upper limit of charging demand for all EVs. The
total predicted charging demand of charging station $j$ of service provider
$k$ is modeled as

$\psi_{j,k}=\sum_{n=1}^{N}q_{n}\Phi_{j,k}^{n}.$ (15)

All coefficients in the nested logit model can be estimated and calibrated
from preference survey data. The nested logit model enables us to compute the
probability that an EV owner will go to a certain charging station, even
though, an EV owner’s decision may not always comply with the calculated
probabilities. An individual EV owner may go to a fixed charging station at
his/her discretion. However, employing the nested logit model provides a
statistically meaningful prediction for the charging demand based on ensemble
averages.

##  IV The Impact of EV Integration on Power Grid

The main function of the power grid is to deliver electricity to users
reliably and economically. However, large-scale EV integration can potentially
disrupt the normal operation of power grid in terms of system stability,
severe power loss, frequency variation, voltage imbalance, etc. Generally, the
variations in voltage and frequency of electricity are considered as the major
factors to characterize the power quality.

Assume that the power system has $M$ generators and $D$ buses (substations),
and the power flow study approach [19] is applied to solve the voltage, real
power, and reactive power in the power system. Consider the node power
equations, which can be written as real and reactive power for each bus.

$0=-P_{i}+\sum_{k=1}^{N}|v_{i}||v_{k}|(G_{ik}\cos\phi_{ik}+B_{ik}\sin\phi_{ik}),$
(16)
$0=-Q_{i}+\sum_{k=1}^{N}|v_{i}||v_{k}|(G_{ik}\sin\phi_{ik}-B_{ik}\cos\phi_{ik}),$
(17)

where $P_{i}$ and $Q_{i}$ are, respectively, the injected real power and
reactive power at bus $i$. The variable $G_{ik}$ is the real part of the
element in the bus admittance matrix corresponding to the $i$-th row and
$k$-th column, and $B_{ik}$ is the imaginary part of the element. In Equations
(16) and (17), $\phi_{ik}$ is the voltage angle between the $i$-th bus and the
$k$-th bus, while $|v_{i}|$ and $|v_{k}|$ are the voltage magnitudes at bus
$i$ and bus $k$, respectively. The stress on the voltage and frequency imposed
by concentrated charging at EV charging station is expected to be significant.
However, good models to calculate and penalize this stress are not yet
considered.

In [20]-[24], the authors have proposed different frameworks to coordinate EV
charging to ensure stable and economical operation of power grid. In this
work, we will consider how to alleviate the “stress” added to the power grid
by EV charging when determining the optimal charging station deployment.

Note that the voltage and system frequency are the two important factors of
power quality in the power grid. The imbalance of the active power will lead
to frequency drift, while the imbalance of the reactive power will cause the
voltage variation [25]. Specifically, if the active power needed by the loads
exceeds the generation, the extra active power is supplied by decreasing the
generator’s rotation speed, which results in a downward drift in frequency
[26]. On the other hand, the balance of the reactive power can influence the
voltage stability. The excess of reactive power will lead to voltage increase,
while the insufficiency of reactive power will lead to voltage decrease [27].
Therefore, the demand and supply of both active power and reactive power
should always be balanced.

Generally, the system operator schedules the power plants by estimating the
load. If the load fluctuates significantly around the predefined value, the
power plants are incentivized to ramp up and ramp down. However, this results
in low efficiency and high cost from committing spinning reserves [28].
Therefore, the fluctuation of the active power and reactive power at the
generators with and without EV charging can be used as a metric to evaluate
the stress that the charging stations impose on the grid. In particular, we
use the 2-norm deviation of generating power (real power and reactive power)
of all generators in the power system to calculate the impacts of EV charging.

$B=\left\lVert\mathbf{P}_{\textrm{g}}^{\textrm{base}}-\mathbf{P}_{\textrm{g}}^{\textrm{EV}}\right\rVert^{2}_{2}+\left\lVert\mathbf{Q}_{\textrm{g}}^{\textrm{base}}-\mathbf{Q}_{\textrm{g}}^{\textrm{EV}}\right\rVert^{2}_{2},$
(18)

where
$\mathbf{P}_{\textrm{g}}^{\textrm{base}}=[P_{1}^{\textrm{base}},P_{2}^{\textrm{base}},\cdots,P_{M}^{\textrm{base}}]$
is a vector representing the active power generated by the $M$ generators
under the base power load scenario (without EV charging), and
$\mathbf{P}_{\textrm{g}}^{\textrm{EV}}=[P_{1}^{\textrm{EV}},P_{2}^{\textrm{EV}},\cdots,P_{M}^{\textrm{EV}}]$
is the vector of active power with EV charging (i.e., base power load
superposed by EV charging load). Similarly,
$\mathbf{Q}_{\textrm{g}}^{\textrm{base}}=[Q_{1}^{\textrm{base}},Q_{2}^{\textrm{base}},\cdots,Q_{M}^{\textrm{base}}]$
is the vector of reactive power of the base load, and
$\mathbf{Q}_{\textrm{g}}^{\textrm{EV}}=[Q_{1}^{\textrm{EV}},Q_{2}^{\textrm{EV}},\cdots,Q_{M}^{\textrm{EV}}]$
is the vector of reactive power with EV charging. For a specific power system,
$\mathbf{P}_{\textrm{g}}^{\textrm{base}},\mathbf{P}_{\textrm{g}}^{\textrm{EV}},\mathbf{Q}_{\textrm{g}}^{\textrm{base}}$,
and $\mathbf{Q}_{\textrm{g}}^{\textrm{EV}}$ can be calculated through solving
the global power flow equations.

##  V Spatial Competition and Optimal Placement in A Bayesian Game

It is pivotal for firms to choose the right location and product to compete
with rivals in the same industry. Business locations will affect business
competition, and conversely intensive competition will affect how firms choose
the appropriate locations. One question arises naturally is whether or not
firms from the same industry like to cluster their stores. There are some
classical literature on spatial competition, e.g. Hotelling’s location model
[29] and Salop’s circle model [15]. Firms have incentives for both clustering
and separation. On one hand, firms prefer clustering so that they can learn
from each other on how to improve manufacturing and research productivity
[30]-[31], and learn demand from each other to reduce the cost of searching
for the optimal location. Firms also cluster for the labor pool and supplies.
In addition, firms can benefit from the spinoffs that are located near parent
firms. On the other hand, the fear of intensive price competition due to
clustering may motivate the firms to separate locations from each other.

The EV charging service providers face the similar dilemma. Therefore, we need
to investigate how the service providers will interact with each other in
choosing their charging station locations and setting the retail charging
prices. In practice, the exact placement costs and utility functions of the
competing service providers may not be known to the service provider _a
priori_. We thus formulate the problem as a Bayesian game [32] among the
service providers at each planning stage.

For notational simplicity, we omit the stage superscript in the following
definitions since the Bayesian game has the same mechanism at each stage. The
main components of a Bayesian game include the set of players $\mathcal{I}$,
the strategy space $S_{k}$, the type space $\Theta_{k}$, the payoff function
$u_{k}$, and the joint probability of the types
$f(\Theta_{1},\Theta_{2},\Theta_{3})$.
$S_{k}=[s_{1,k},s_{2,k},\cdots,s_{L,k}]^{\textrm{T}}$ accounts for all
possible placement policies for player $k$. $S_{-k}$ is the placement policies
of the competing players. Denote $f(S_{-k})$ as probability mass function
(PMF) of the placement policies of the other players. The type space
$\Theta_{k}=[\theta_{1,k},\theta_{2,k},\cdots,\theta_{L,k}]^{\textrm{T}}$
corresponds to the placement costs of all charging station candidates of
service provider $k$. In this paper, we assume that a service provider knows
its own type, but not the types of the other two competing players.

Let $\theta_{j,k}$ denote the placement cost for $j$-th charging station
candidate of service provider $k$, which includes the charging equipment cost,
installation fee, construction cost, land rental cost, etc. $\theta_{j,k}$ has
i.i.d. uniform distribution.

Before proceeding to analyze the Bayesian game, we need the following
assumptions.

$\mathbf{Assumption\;1}$: $f(S_{-k})$ is binomially distributed with parameter
0.5, _i.e._ $S_{-k}\backsim\textrm{Binomial}(2L,0.5)$.

$\mathbf{Remark\;1}$: The distribution of $S_{-k}$ reflects player $k$’s
conjecture on how other players will act during the game. Each player can form
their conjectures about other players according to their beliefs about the
competitors. For instance, a player may be risk neutral, risk aversion or risk
seeking. For simplicity, we assume that $S_{-k}$ has a binomial distribution
with parameter 0.5. However, the theoretical analysis can be applied to any
other distribution of $S_{-k}$.

$\mathbf{Assumption\;2}$: All service providers in the market are Bertrand
competitors.

$\mathbf{Remark\;2}$: Bertrand competitors are players that do not cooperate
with each other. Their goal is to maximize their own utility. They will not
form any type of “coalition” to manipulate the market.

For each player, the Bayesian Nash Equilibirum (BNE) of placement policy can
be derived from Equation (6). To solve Equation (6), we need to know the
retail charging prices of all the service providers. In a Bertrand
competition, the retail prices for every combination of the charging station
placement policies are determined by the first order of conditions (FOC):

$\frac{\partial\Pi_{1}}{\partial
p_{1}}=\sum_{n=1}^{N}\sum_{j=1}^{L}q_{n}s_{j,1}\left[\Phi_{j,1}^{n}+(p_{1}-c_{j,1})\frac{\partial\Phi_{j,1}^{n}}{\partial
p_{1}}\right]=0$ (19) $\frac{\partial\Pi_{2}}{\partial
p_{2}}=\sum_{n=1}^{N}\sum_{j=1}^{L}q_{n}s_{j,2}\left[\Phi_{j,2}^{n}+(p_{2}-c_{j,2})\frac{\partial\Phi_{j,2}^{n}}{\partial
p_{2}}\right]=0$ (20) $\frac{\partial\Pi_{3}}{\partial
p_{3}}=\sum_{n=1}^{N}\sum_{j=1}^{L}q_{n}s_{j,3}\left[\Phi_{j,3}^{n}+(p_{3}-c_{j,3})\frac{\partial\Phi_{j,3}^{n}}{\partial
p_{3}}\right]=0$ (21)

where $c_{j,1},c_{j,2},$ and $c_{j,3}$ represent the LMP at each charging
station candidate.

$\mathbf{Remark\;3}$: For simulation simplicity, we assume the charging
stations affiliated to the same service provider have the same retail charging
prices ($p_{1},p_{2},\textrm{ and }p_{3}$). However, our analysis can be
easily generalized to the case where each charging station sets its own retail
price. Note that those retail prices obtained from Equations (19)-(21) may not
be the real-time prices used in practice. They are only the equilibrium prices
in this market under the assumption of Bertrand competition. They can be
interpreted as the averaged charging prices of the service providers over a
long period of time. In practice, the service providers take turns to set the
retail price in response to the prices of the competitors. Additionally, if
some of the other factors change (i.e. consumer’s preference, crude oil price
soaring, etc.), the existing equilibrium breaks and a new equilibrium must be
computed using the same procedure.

$\mathbf{Theorem\;1}$ [Strategy Decision Condition]: Service provider $k$ will
choose placement policy $l(l=1,2,3,\cdots,2^{L})$ if the type space
$\Theta_{k}$ falls into the hypervolume specified by

$\displaystyle\mathcal{H}(l)=$ (22)
$\displaystyle\\{\Theta_{k}\in\mathbb{R}_{+}^{L}:\Theta_{k}^{\textrm{T}}\left(S_{k,j}-S_{k,l}\right)-(\mathbb{E}R_{k,j}-\mathbb{E}R_{k,l})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+w(B_{k,j}-B_{k,l})>0;\forall
j\neq l\\},$

where $S_{k,j}$ and $S_{k,l}$ denote the placement strategy $j$ and $l$,
respectively. $\mathbb{E}R_{k,j}$ and $\mathbb{E}R_{k,l}$ denote the expected
total revenue with deployment strategy $j$ and $l$, respectively.

###### Proof.

Each service provider has $L$ location candidates, so there are $2^{L}$
different placement policies. The type space can be seen as an $L$-dimensional
space, and
$\Theta_{k}=[\theta_{1,k},\theta_{2,k},\cdots,\theta_{L,k}]^{\textrm{T}}$
represents a point in this space.

By Equation (6), strategy $l$ is optimal if

$\displaystyle\mathbb{E}[R_{k,l}]-\Theta_{k}^{\textrm{T}}S_{k,l}-wB_{k,l}>\mathbb{E}[R_{k,j}]-$
(23)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\Theta_{k}^{\textrm{T}}S_{k,j}-wB_{k,j};(j=1,2,\cdots,2^{L},j\neq
l).$

Rearranging the terms, we get

$\displaystyle\Theta_{k}^{\textrm{T}}\left(S_{k,j}-S_{k,l}\right)-(\mathbb{E}R_{k,j}-\mathbb{E}R_{k,l})+$
(24)
$\displaystyle\;\;\;\;\;\;\;w(B_{k,j}-B_{k,l})>0;(j=1,2,\cdots,2^{L},j\neq
l),$

where each inequality represents a hyperplane and the intersection of all the
inequalities defines a hypervolume in the type space. ∎

##  VI Simulation Platform and Case Study

We have developed a general-purpose simulation software—The EV Virtual City
1.0 using Repast [1]. Our simulation software is designed to construct a
virtual digital city by integrating a variety of data and information, such as
geographic information, demographic information, spatial infrastructure data,
urban road network graph, electric power network graph, travel pattern,
diurnal variation in traffic flow, seasonal fluctuation of driving activities,
social interaction, etc. The platform is flexible in that one can include or
exclude many modules to satisfy different simulation needs. See Fig. 1 for the
architecture of the simulation software.

Figure 1: The Architecture of The EV Virtual City 1.0

In this paper, we conduct a case study using the data of San Pedro District of
Los Angeles. We import the shapefiles of zip code tabulation area (ZCTA) and
road network data from the U.S. Census Bureau into our simulation software. In
addition, we calculate the centroids of locations of residence, restaurants,
supermarkets, shopping centers and workplaces using Google Maps, see Fig. 2.

Figure 2: Roads and Buildings of San Pedro District

From the California Energy Commission website, we obtained the maps of
transmission line and substations of San Pedro District. This area has 107
substations in total. Thus we use the IEEE 118-bus power system test case in
our simulation. For each charging station placement policy, we used MATPOWER
[33] to calculate the LMP of each bus and the output power of each generator
with and without EV charging.

The service providers must also satisfy the QoS constraints when planning the
charging stations at each new stage. In the simulations, we consider four
stages with 5000 EVs, 10000 EVs, 15000 EVs, and 20000 EVs. Since travel
pattern also plays a significant role in analysing the charging behavior of EV
owners, it is necessary to have a thorough study on the statistics of travel
pattern. From the 2009 National Household Travel Survey (2009 NHTS) [34], we
obtained the travel pattern statistics. See Fig. 3.

The Simulation Algorithm

  * •

Initialization: Initialize road network, spatial infrastructure data, travel
pattern statistics.

  * •

EV Movement: For $1\leq j\leq N$ (total number of EVs), randomly assign a
destination $Des(j)$ to EV $j$; calculate a route $Route(j)$ from $Home(j)$ to
$Des(j)$ using Dijkstra’s shortest path algorithm.

  * •

Bayesian Game Solution:

    1. 1.

Prices and Revenue Calculation: For $1\leq i\leq M$ (total number of
deployment strategies), calculate retail charging prices
$p_{1},p_{2},\textrm{and }p_{3}$. Calculate
$R_{1,i},R_{2,i},\textrm{and},R_{3,i}$.

    2. 2.

Optimal Deployment Strategy Search: Find the strategy for each service
provider $k(k=1,2,3)$

$S_{k}^{*}=l^{*}=\operatorname*{argmax}_{l\in\\{1,2,\cdots,M\\}}\left\\{R_{k,l}-\Theta_{k}^{\textrm{T}}S_{k,l}\right\\}$

  * •

Report: $S_{1}^{*},S_{2}^{*},S_{3}^{*},p_{1}^{*},p_{2}^{*},p_{3}^{*}$

Figure 3: The Statistics of Travel Patterns

Figure 4: A Snapshot of EVs Movement

A snapshot of the moving EVs is shown in Fig. 4. Each red star represents an
EV owner. The traffic flow heatmap of EV owners is also plotted in this
figure. The simulation results are summarized in Table III. Figs. 10 to 10
correspond to the charging station placement for stages 1 to 4, respectively.
Fig. 10 is an overview of charging station placement by superposing Figs. 10
to 10. Fig. 10 shows how the number of charging station increases as the EV
penetration rate increases.

Figure 5: Stage 1 Charging Station Placement

Figure 6: Stage 2 Charging Station Placement

Figure 7: Stage 3 Charging Station Placement

Figure 8: Stage 4 Charging Station Placement

Figure 9: Superimposing all stages

Figure 10: Charging Station verses EV Penetration Rate

Table III: Charging Station Placement Strategy Stage | Level | Delay prob. | Coverage | Newly Built Stations | Total # of Stations
---|---|---|---|---|---
Stage 1 | 1 | 0.29 | 1.7 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14 | 13
(Penetration Rate 0.32%, | 2 | 0.17 | 1.6 | 2, 3, 4, 7, 9, 10, 12, 14, 15 | 9
5000 EVs) | 3 | 0.04 | 0.8 | 2, 4, 7, 12 | 4
Stage 2 | 1 | 0.28 | 2.5 | 17, 20, 22, 23, 24, 25 | 20
(Penetration Rate 0.64%, | 2 | 0.17 | 2.3 | 18, 20, 22, 23, 24, 25 | 15
10000 EVs) | 3 | 0.10 | 1.15 | 18, 22, 23, 25 | 8
Stage 3 | 1 | 0.29 | 3.6 | 26, 27, 29, 30, 31, 32 | 26
(Penetration Rate 0.96%, | 2 | 0.18 | 3.1 | 26, 27, 28, 31, 32 | 20
15000 EVs) | 3 | 0.08 | 1.9 | 26, 27, 28, 29, 31 | 13
Stage 4 | 1 | 0.22 | 4.6 | 33, 34, 35, 36, 37 | 31
(Penetration Rate 1.28%, | 2 | 0.16 | 3.9 | 33, 37, 38, 39, 44 | 25
20000 EVs) | 3 | 0.10 | 2.9 | 36, 38, 40, 41, 44 | 18

From the simulation results, we can make the following observations:

  * •

The optimal charging station deployment is consistent with the EV traffic flow
heatmap. This suggests that our model can adequately capture the mobility of
EVs and provide EV owners with convenient charging services.

  * •

As for the number of charging stations, Level 1 charging station is
predominant over Level 2 and Level 3. Level 3 has the least number of charging
stations. Notice that it takes a much longer time to finish charging for Level
1, so Level 1 service provider must place more charging stations to meet the
average delay probability constraint. The difference in quantity also reveals
that the service providers have different marketing strategies. Service
provider 1 tries to place the charging stations evenly across the entire area,
while service provider 3 is more likely to place the charging stations at some
“hot” locations.

  * •

The number of charging stations grows almost linearly with the number of EVs
except for the initial stage. At the initial stage, Level 1 and Level 2
service providers tend to place more charging stations than the next stages.
This is because service providers must place more charging stations to meet
the average service coverage constraints. As the number of charging stations
increases, however, the service coverage constraint is less of a concern for
the service providers.

  * •

Service providers prefer clustering instead of spatial separation. The three
service providers have segmented the EV charging market by providing three
different products (different charging level services) in terms of voltage,
current, charging speed and charging price. Due to product differentiation,
they significantly soften the price competition so that they do not need to
spatially separate from each other to further relax competition. This
observation supports the opinions in [35]-[36] that firms do not have to
maximize differentiation in every characteristic of the product. In general,
differentiation in one dominant characteristic is sufficient to soften price
competition.

##  VII Conclusion

In this paper, we have proposed a solution to the placement of EV charging
stations which balances the benefits of EV owner, charging station owner, and
power grid operator. We formulate the competition among charging stations as a
Bayesian game. Solving the game renders the optimal placement policies for the
service providers. In addition, we develop a simulation software—The EV
Virtual City 1.0 on Repast, and conduct a case study of San Pedro District of
Los Angeles. The simulation illustrates that charging station placement is
highly consistent with the traffic flow of EVs, and the service providers
prefer clustering to separating the charging stations.

## References

  * [1] Michael J North, Nicholson T Collier, Jonathan Ozik, Eric R Tatara, Charles M Macal, Mark Bragen, and Pam Sydelko, “Complex adaptive systems modeling with Repast Simphony,” Complex Adaptive Systems Modeling, Springer, Heidelberg, FRG (2013).
  * [2] Shaoyun Ge, Liang Feng, and Hong Liu, “The Planning of Electric Vehicle Charging Station Based on Grid Partition Method,” in _Proc. 2011 International Conference on Electrical and Control Engineering (ICECE)_ , 2011, pp. 2726-2730.
  * [3] Sara Mehar, and Sidi Mohammed Senouci, “An Optimization Location Scheme for Electric Charging Stations,” in _Proc. 2013 International Conference on Smart Communications in Network Technologies (SaCoNeT)_ , 2013, pp. 1-5.
  * [4] Ines Frade, Anabela Ribeiro, Goncalo Goncalves, and Antonio Pais Antunes, “Optimal location of charging stations for electric vehicles in a neighborhood in lisbon, portugal,” In _Transportation Research Record: Journal of the Transportation Research Board_ , vol. 2252, no.12, pp. 91-98, Feb. 2011.
  * [5] Zonggen Yi , and Peter H. Bauer, “Energy Consumption Model and Charging Station Placement for Electric Vehicles,” In _Proc. 3rd International Conference on Smart Grids and Green IT Systems (SMARTGREENS)_ , 2014\.
  * [6] Timothy M. Sweda, and Diego Klabjan, “An Agent-Based Decision Support System for Electric Vehicle Charging Infrastructure Deployment,” in _Proc. 2011 IEEE Vehicle Power and Propulsion Conference (VPPC)_ , 2011, pp. 1-5.
  * [7] Valeria Bernardo, Joan-Ramon Borrell, and Jordi Perdiguero, “Fast Charging Stations: Network Planning versus Free Entry,” _Available online: http: //www.cemfi.es/ftp/pdf/papers/wshop/version_3.pdf_ , (accessed on 27 March, 2014)
  * [8] Craig B. Toepfer, _SAE Electric Vehicle Conductive Charge Coupler, SAE J1772, REV. MONTH01 (DOC)_ , California Air Resources Board, 2009.
  * [9] Ezra Hausman, Robert Fagan, David White, Kenji Takahashi, and Alice Napoleon, “LMP Electricity Markets: Market Operations, Market Power, and Value for Consumers” (online), _available at: https://www.publicpower.org/files/PDFs/SynapseLMPElectricityMarkets 013107.pdf_ , (accessed on February 5, 2006)
  * [10] Lopes, J.A.P. , Soares, F.J., and Almeida, P.M.R., “Integration of Electric Vehicles in the Electric Power System”, _Proceedings of the IEEE_ , vol. 99, no. 1, pp. 168 - 183, Jan. 2011.
  * [11] M. Kinter-Meyer, K. Schneider, and R. Pratt, “Impacts assessment of plug-in hybrid electric vehicles on electric utilities and regional U.S. power grids. Part I–Technical analysis”, Pacific North West National Lab., Richland, WA, PNNL-SA-61669, Jan. 2007.
  * [12] M. J. Scott, M. Kintner-Meyer, D. Elliott, and W. Warwick, “Economic assessment and impacts assessment of plug-in hybrid vehicles on electric utilities and regional U.S. power grids. Part II”, Pacific North West National Lab., Richland, WA, PNNL-SA-61687, Jan. 2007.
  * [13] Kenneth Train, _Discrete Choice Methods with Simulation_ , Cambridge University Press, Cambridge UK, 2003.
  * [14] Steven T. Berry, “Estimating Discrete-Choice Models of Product Differentiation”, _Rand Journal Of Economics_ , vol. 25(2), pp.242-262, 1994.
  * [15] Steven C. Salop, “Monopolistic Competition with Outside Goods”, _The Bell Journal of Economics_ , vol. 10(1), pp.141-156, 1979.
  * [16] Chao Luo, Yih-Fang Huang, and Vijay Gupta, “A Consumer Behavior Based Approach to Multi-stage EV Charging Station Placement”, accepted by _IEEE 81st Vehicular Technology Conference_ , 2015\.
  * [17] Moshe E. Ben-Akiva and Steven R. Lerman, _Discrete choice analysis : theory and application to travel demand_ , MIT Press, Cambridge, Massahusetts, 1985.
  * [18] Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, _Introduction to Algorithms_ Third Edition, The MIT Press, 2009.
  * [19] John Grainger, and William Stevenson, _Power system analysis_ , New York : McGraw-Hill, 1994.
  * [20] Shiyao Chen, and Lang Tong, “iEMS for Large Scale Charging of Electric Vehicles: Architecture and Optimal Online Scheduling”, _2012 SmartGridComm_ , Nov. 2012.
  * [21] Michael Caramanis, and Justin M. Foster, “Management of Electric Vehicle Charging to Mitigate Renewable Generation Intermittency and Distribution Network Congestion”, in _Proceedings of the 48th IEEE Conference on Decision and Control_ , pp. 4717 C4722, Dec. 2009.
  * [22] Lingwen Gan, Ufuk Topcu, and Steven H. Low, “Optimal Decentralized Protocol for Electric Vehicle Charging”, _IEEE Trans. on Power System_ , vol. 28, no. 2, May 2013.
  * [23] Somayeh Sojoudi, and Steven H. Low, “Optimal Charging of Plug-in Hybrid Electric Vehicles in Smart Grids”, in _Proc. IEEE PES General Meeting_ , 2011\.
  * [24] Eric Sortomme, Mohammad M. Hindi, S. D. James MacPherson, and S. S. Venkata, “Coordinated Charging of Plug-In Hybrid Electric Vehicles to Minimize Distribution System Losses”, _IEEE Trans. on Smartt Grid_ vol. 2, no. 1, March 2011.
  * [25] Prabha Kundur, _Power System Stability and Control (1st edition)_ , McGraw-Hill Education, January 22, 1994.
  * [26] RENAC Online Academy, “ReGrid: Frequency and voltage regulation in electrical grids” (online), _Available at: http://docplayer.net/6038884-Regrid-frequency-and-voltage-regulation-in-electrical-grids.html_ , (accessed on October, 2015).
  * [27] Tore Petersson, “Reactive Power Compensation”, ABB Power Systems AB, Sweden, 1993.
  * [28] Y. Rebours, and D. Kirschen, “What is spinning reserve?”, Available at: _http:http://www.ee.washington.edu/research/real/Library/Reports/What_is _spinning_reserve.pdf_ , (accessed on September, 2005).
  * [29] Harold Hotelling, “Stability in Competition”, _The Economic Journal_ , vol. 39, no. 153, pp. 41-57, March 1929.
  * [30] Edward L. Glaeser, Heddi D. Kallal, Jose A. Scheinkman, and Andrei Shleifer, “Growth in cities”, _Journal of Political Economy_ , vol. 100(6), pp.1126-1152, 1992.
  * [31] J. Myles Shaver, and Fredrick Flyer, “Agglomeration economies, firm heterogeneity, and foreign direct investment in the United States”, _Strategic Management Journal_ , vol. 21(12), pp.1175-1193, , 2000.
  * [32] John Harsanyi, “Games with incomplete information played by Bayesian players, I-III”, _Management Science_ , vol. 14, no. 3, Nov. 1967.
  * [33] R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas, “MATPOWER: Steady-State Operations, Planning, and Analysis Tools for Power Systems Research and Education”, _Power Systems, IEEE Transactions on_ , vol. 26, no. 1, pp. 12-19, Feb. 2011.
  * [34] U.S. Department of Transportation, Federal Highway Administration, “2009 National Household Travel Survey,” (online), _Available at: http://nhts.ornl.gov_ , (accessed on October, 2009).
  * [35] Andreas Irmen, and Jacques-Francois Thisse, “Competition in Multi-characteristics Spaces: Hotelling Was Almost Right,” _Journal of economic theory_ , vol.78 iss.1 pp. 76 -102, 1998.
  * [36] Joel A. C. Baum and Heather A. Haveman, “Love Thy Neighbor? Differentiation and Agglomeration in the Manhattan Hotel Industry”, _Administrative Science Quarterly_ , vol. 42, no. 2, pp. 304-338, 1997\.

|  Chao Luo received the B.Eng degree with distinction in Communication
Engineering from Harbin Institute of Technology (HIT), China, in 2012.
Currently, he is pursuing his Ph.D. in Electrical Engineering in the
University of Notre Dame, USA. Chao’s research interests include electric
vehicle (EV) integration into power grid, network protocol design of smart
grid, and electricity market.
---|---
|  Yih-Fang Huang is Professor of Department of Electrical Engineering and
Senior Associate Dean of College of Engineering at University of Notre Dame.
Dr. Huang received his BSEE degree from National Taiwan University, MSEE
degree from University of Notre Dame and Ph.D. from Princeton University. He
served as chair of the Electrical Engineering Department at the University of
Notre Dame from 1998 to 2006. His research interests focus on theory and
applications of statistical signal detection and estimation, and adaptive
signal processing. In Spring 1993, Dr. Huang received the Toshiba Fellowship
and was Toshiba Visiting Professor at Waseda University, Tokyo, Japan. From
April to July 2007, he was a visiting professor at the Munich University of
Technology, Germany. In Fall, 2007, Dr. Huang was awarded the Fulbright-Nokia
scholarship for lectures/research at Helsinki University of Technology in
Finland (which is now Aalto University). Dr. Huang received the Golden Jubilee
Medal of the IEEE Circuits and Systems Society in 1999, served as Vice
President in 1997-98 and was a Distinguished Lecturer for the same society in
2000-2001. At the University of Notre Dame, he received Presidential Award in
2003, the Electrical Engineering department s Outstanding Teacher Award in
1994 and in 2011, the Rev. Edmund P. Joyce, CSC Award for Excellence in
Undergraduate Teaching in 2011, and the College of Engineering s Teacher of
the Year Award in 2013. Dr. Huang is a Fellow of the IEEE.
---|---
|  Vijay Gupta is an Associate Professor in the Department of Electrical
Engineering at the University of Notre Dame. He received his B. Tech degree
from the Indian Institute of Technology, Delhi and the M.S. and Ph.D. degrees
from the California Institute of Technology, all in Electrical Engineering.
Prior to joining Notre Dame, he also served as a research associate in the
Institute for Systems Research at the University of Maryland, College Park. He
received the NSF CAREER award in 2009, and the Donald P. Eckman Award from the
American Automatic Control Council in 2013. His research interests include
cyber-physical systems, distributed estimation, detection and control, and, in
general, the interaction of communication, computation and control.
---|---