# The Promise of EV-Aware Multi-Period OPF Problem: Cost and Emission Benefits

Sezen Ece Kayacık, Burak Kocuk, Tuğçe Yüksel  S. E. Kayacık is with the
Faculty of Economics and Business, University of Groningen, Groningen,
Netherlands (e-mail: s.e.kayacik@rug.nl). B. Kocuk and T. Yüksel are with the
Faculty of Engineering and Natural Sciences, Sabancı University, Istanbul,
Turkey (e-mail: burak.kocuk, tugce.yuksel@sabanciuniv.edu).

###### Abstract

In this paper, we study the Multi-Period Optimal Power Flow problem (MOPF)
with electric vehicles (EV) under emission considerations. We integrate three
different real-world datasets: household electricity consumption, marginal
emission factors, and EV driving profiles. We present a systematic solution
approach based on second-order cone programming to find globally optimal
solutions for the resulting nonconvex optimization problem. To the best of our
knowledge, our paper is the first to propose such a comprehensive model
integrating multiple real datasets and a promising solution method for the EV-
aware MOPF problem. Our computational experiments on various instances with up
to 2000 buses demonstrate that our solution approach leads to high-quality
feasible solutions with provably small optimality gaps. In addition, we show
the importance of coordinated EV charging to achieve significant emission
savings and reductions in cost. In turn, our findings can provide insights to
decision-makers on how to incentivize EV drivers depending on the trade-off
between cost and emission.

###### Index Terms:

Multi-Period Optimal Power Flow Problem, Second-Order Cone Programming,
Integration of Electric Vehicles to Power Grid

##  I Introduction

Many countries, including the United States, grant tax credits and other
incentives to increase electric vehicle (EV) usage aiming to reduce emissions
[1]. Since EV charging demand is expected to comprise a major part of the
electricity consumption in the future, charging strategies will become more
critical from different perspectives such as network security, financial
parties, environmental policy, and EV drivers.

EVs create a considerable opportunity to reduce greenhouse gas emissions from
transportation compared to conventional vehicles. However, the additional
electricity generation to charge EVs may cause greenhouse gas emissions
depending on the source of electricity generation [2]. Therefore, it is also
important to consider the additional amount of emission generated by the power
plants to correctly judge the environmental benefits of EVs.

Integrating EV charging stations to power systems changes the dynamics of
conventional power distribution and poses considerable challenges to network
operations under an uncoordinated charging scenario. These challenges include
increased network losses, overloaded transmission lines, unbalanced voltage,
and increased peak load, resulting in a significant increase in electricity
generation cost and the risk of damage to power system security. Recently,
various coordinated charging strategies of EVs have been developed to overcome
economic and security issues by flattening the load profiles [3], filling the
load valley [4] and minimizing the stress on transformers [5].

With the increased integration of electric vehicles, Optimal Power Flow (OPF)
studies that consider large-scale integration of EVs have become increasingly
important to protect network security. The classical OPF problem considers
single-period planning and optimizes operations of the power grid with respect
to a certain objective and network constraints. The interconnection of the
energy storage of EVs with the power network makes it necessary to solve a
multi-period version of the OPF problem (in short, MOPF).

In recent years, several studies have focused on the OPF problem that
considers the integration of EVs with limited scope. Some papers study single-
period OPF and analyze the effects of EV penetration by resolving the problem
with the inclusion of EVs [6]. Some studies do not consider the effects of EV
charging on power system security and ignore operational limits such as limits
of line flow, generation and, voltage [7]. The authors in [8] do not consider
the physical limitations of batteries and assume that EVs can be fully charged
during a single period. Many existing studies incorporate only a small number
of EVs [9] or charging stations [10]. The work in [11] considers only economic
aspects without environmental impacts. The authors of [12] consider both
operational cost and emission objectives; however, they do not guarantee to
produce globally optimal solutions and carry out tests only on small-size
networks. Study by [10] do not consider the behavior of EV users and randomly
generate arrival times.

Systematic treatments to solving large-scale MOPF problems with EVs are
limited in the literature. Since Alternating Current (AC) OPF is a nonlinear,
nonconvex, and NP-hard problem, a globally optimal solution cannot be
guaranteed in general. The integration of EVs requires a multi-period setting
and additional constraints, increasing the complexity and computational effort
even more. Primarily, simulation-oriented [13, 14] and heuristic approaches
[15, 16, 17] are developed, however, these methods fail to provide a quality
measure for global optimality. A comprehensive overview of heuristic
approaches can be found in [18].

Convex relaxations of the single-period OPF problem have been widely studied
[19], however very few studies focus on convex relaxations of the MOPF
problem. The authors in [20] model a demand response problem in radial
distribution networks as an AC OPF problem ignoring reactive power. This
optimization problem is convexified and extended to an AC MOPF problem.
Reference [21] solves the AC MOPF problem using a Semidefinite Programming
relaxation, and reference [22] proposes a Second-Order Cone Programming (SOCP)
relaxation of the AC MOPF problem for distribution networks with
photovoltaics. The study by [23] formulates the AC MOPF with the planning of
EV charging as an SOCP problem.

In this paper, we study an EV-aware AC MOPF problem with an integrated
perspective to preserve system security, capture the trade-off between cost
and emission, and satisfy the charging needs of EV owners simultaneously. We
develop a convex programming framework to find globally optimal solutions for
large-scale networks in a reasonable time. We consider both Grid to Vehicle
(G2V) and Vehicle to Grid (V2G) concepts, where the latter allows EVs to
supply power back to the grid. To implement V2G, we integrate an aggregator
concept in order to control a group of EVs to function as a source of
electricity. We analyze the impacts of this bidirectional flow on the grid
considering several other aspects, including penetration level, driving
profiles, and battery characteristics. The main contributions of this paper
can be summarized as follows:

  * •

We introduce a new EV-aware MOPF formulation for the joint problem of OPF and
EV charging by bringing together test cases and real data for electricity
consumption, marginal emission factors, and EV driving profiles.

  * •

We develop a convex optimization framework to solve the resulting challenging
problem. Specifically, we introduce an SOCP model in which an emission
constraint is imposed to restrict emissions caused due to EV charging from
power plants. EV charging constraints and their effects on the power system
are also considered.

  * •

We demonstrate the effectiveness of the proposed approach on three test cases
with up to 2000 buses from PGLIB-OPF. Our algorithm can solve large-scale test
cases in reasonable computation time and offer guarantees of global optimality
with very small optimality gaps.

  * •

Our extensive computational experiments suggest that through coordinated
charging of EVs, marginal emission can be decreased while keeping the cost
virtually constant, and the integration of the V2G concept leads to cost
savings, despite assuming hourly electricity prices are constant.

The remainder of this paper is organized as follows. Section II describes the
problem, provides an alternative formulation and its SOCP relaxation. Section
III introduces our approach to solve this optimization problem. Section IV
introduces the input data used in the formulation. Section V presents the
experimental setup and the computational results. The concluding remarks are
made in Section VI.

##  II Mathematical Model

###  II-A Formulation

In this section, we present our mathematical programming formulation and its
SOCP relaxation. We formulate an MOPF problem that coordinates the charging of
EVs. The model minimizes the total cost of generation over a finite planning
horizon while determining the optimal schedule of power generation and EV
charging under emission limits. We take into account both network constraints
and EV charging constraints.

We consider power station emissions associated with EV charging only.
Therefore, we utilize marginal emissions, which occur to satisfy the
additional demand due to EVs. The EV charging constraints are formulated based
on actual driving profiles that provide arrival and departure times of each
trip and energy requirements for each period. We assume that only the real
power is consumed and fed back to the grid by EVs. All EVs connected to the
same bus are aggregated as an entity. In the formulation, we will use the term
EV to refer to a group of EVs charged at the same bus. Note that our
formulation considers aggregate charging load and requires perfect knowledge
of the user’s driving behavior and electricity consumption.

Consider a power network $\mathcal{N}=(\mathcal{B},\mathcal{L})$, where
$\mathcal{B}$ denotes the node set, i.e., the set of buses, and $\mathcal{L}$
denotes the edge set, i.e., the set of transmission lines. Let
$\mathcal{G}\subseteq\mathcal{B}$ denote the set of generators,
$\mathcal{D}\subseteq\mathcal{B}$ be the set of buses with real power load and
$\mathcal{T}=\\{1,...,T\\}$ represent the set of time periods.

Rest of the parameters are given as follows:

  * •

For each bus $i\in\mathcal{B}$ and time $t\in\mathcal{T}$; $p_{it}^{d}$ and
$q_{it}^{d}$ are the real and reactive power load, $\underline{V}_{i}$ and
$\overline{V}_{i}$ are the bounds on the voltage magnitude, $\delta(i)$ is the
set of its neighbors, $b_{ii}$ is the shunt susceptance,
$\overline{a}_{it},\overline{b}_{it}$ are the maximum allowable charging and
discharging rates of EV, $\underline{s}_{it},\overline{s}_{it}$ are the
minimum energy requirement and maximum capacity of EV battery, $c_{it}$ is the
energy requirement of EV, $\tilde{p}_{it}^{g}$ is the reference power
generation without EVs, $\eta_{i}$ the charging efficiency of EV and $I_{i}$
is the initial battery state of charge of EV for bus $i$.

  * •

For each generator located at bus $i\in\mathcal{G}$ and time
$t\in\mathcal{T}$; active and reactive outputs must be in the intervals
$[\underline{p}_{i},\overline{p}_{i}]$ and
$[\underline{q}_{i},\overline{q}_{i}]$.

  * •

For each line $(i,j)\in\mathcal{L}$ and time $t\in\mathcal{T}$;
$G_{ij},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}G_{ji}}$
and
$B_{ij},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}B_{ji}}$
are the elements of the conductance and susceptance matrices,
$\overline{S}_{ij}$ is the apparent power flow limit and
$\overline{\theta}_{ij}$ is the bound on the phase angle.

  * •

For each time $t\in\mathcal{T}$; $e_{t}$ is the marginal emission parameter
and $\overline{E}$ is the upper bound on the total emission over the planning
horizon.

We define the following decision variables:

  * •

For each bus $i\in\mathcal{B}$ and time $t\in\mathcal{T}$, $|V_{it}|$ and
$\theta_{it}$ are the voltage magnitude and phase angle, $a_{it},b_{it}$ are
the charging and discharging power of EV, $s_{it}$ is the stock variable of
EV.

  * •

For each generator located at bus $i\in\mathcal{G}$ and time
$t\in\mathcal{T}$, $p_{it}^{g}$ and $q_{it}^{g}$ are the real and reactive
power output.

  * •

For each line $(i,j)\in\mathcal{L}$ and time $t\in\mathcal{T}$, $p_{ijt}$ and
$q_{ijt}$ are the real and reactive power flow.

We present the following formulation of the EV-aware MOPF problem:

$\displaystyle\min$ $\displaystyle\hskip
2.5pt\sum_{i\in\mathcal{G}}\sum_{t\in\mathcal{T}}f(p_{it}^{g})$ (1a)
$\displaystyle\mathrm{s.t.\ }$ $\displaystyle\text{For each bus
}i\in\mathcal{B}\text{ and time t}\in\mathcal{T},$ $\displaystyle\hskip
2.5ptp_{it}^{g}-p_{it}^{d}-a_{it}+\eta_{i}b_{it}=\
g_{ii}|V_{it}|^{2}+\sum_{j\in\delta(i)}p_{ijt}$ (1b) $\displaystyle\hskip
2.5ptq_{it}^{g}-q_{it}^{d}=-b_{ii}|V_{it}|^{2}+\sum_{j\in\delta(i)}q_{ijt}$
(1c) $\displaystyle\text{For each line }(i,j)\in\mathcal{L}\text{ and time
t}\in\mathcal{T},$ $\displaystyle\hskip
2.5ptp_{ijt}=G_{ij}|V_{it}|^{2}+|V_{it}||V_{jt}|[G_{ij}\cos(\theta_{it}-\theta_{jt})$
(1d) $\displaystyle\hskip 122.34692pt-B_{ij}\sin(\theta_{it}-\theta_{jt})]$
$\displaystyle\hskip
2.5ptq_{ijt}=-B_{ij}|V_{it}|^{2}-|V_{it}||V_{jt}|[B_{ij}\cos(\theta_{it}-\theta_{jt})$
(1e) $\displaystyle\hskip 122.34692pt+G_{ij}\sin(\theta_{it}-\theta_{jt})]$
$\displaystyle\hskip
2.5pt\underline{V}_{i}\leq|V_{it}|\leq\overline{V}_{i}\hskip
85.35826pti\in\mathcal{B},t\in\mathcal{T}$ (1f) $\displaystyle\hskip
2.5pt\underline{p}_{i}\leq p_{it}^{g}\leq\overline{p}_{i}\hskip
99.58464pti\in\mathcal{G},t\in\mathcal{T}$ (1g) $\displaystyle\hskip
2.5pt\underline{q}_{i}\leq q_{it}^{g}\leq\overline{q}_{i}\hskip
99.58464pti\in\mathcal{G},t\in\mathcal{T}$ (1h) $\displaystyle\hskip
2.5ptp_{ijt}^{2}+q_{ijt}^{2}\leq\overline{S}_{ij}^{2}\hskip
68.28644pt(i,j)\in\mathcal{L},t\in\mathcal{T}$ (1i) $\displaystyle\hskip
2.5pt|\theta_{it}-\theta_{jt}|\leq\overline{\theta}_{ij}\hskip
71.13188pt(i,j)\in\mathcal{L},t\in\mathcal{T}$ (1j) $\displaystyle\hskip
2.5pts_{it}+\eta_{i}a_{it}-b_{it}-c_{it}=s_{i(t+1)}\hskip
19.91684pti\in\mathcal{B},t\in\mathcal{T}\ $ (1k)
$\displaystyle\underline{s}_{it}\leq s_{it}\leq\overline{s}_{it}\hskip
93.89418pti\in\mathcal{B},t\in\mathcal{T}$ (1l) $\displaystyle
s_{i0}=s_{iT}=I_{i}\hskip 122.34692pti\in\mathcal{B}$ (1m) $\displaystyle
0\leq a_{it}\leq\overline{a}_{it}\hskip
98.16191pti\in\mathcal{B},t\in\mathcal{T}$ (1n) $\displaystyle 0\leq
b_{it}\leq\overline{b}_{it}\hskip 99.58464pti\in\mathcal{B},t\in\mathcal{T}$
(1o)
$\displaystyle\sum_{i\in\mathcal{B}}\sum_{t\in\mathcal{T}}e_{t}(p^{g}_{it}-\tilde{p}_{it}^{g})\leq\overline{E}.$
(1p)

Here, the objective function (1a) aims to minimize the total cost of power
generation. Constraint (1b) ensures real power flow balance at bus $i$, while
considering EV charging and discharging power. Constraint (1c) ensures
reactive power flow balance at bus $i$. Constraints (1d) and (1e) represent
the real and reactive power flow, respectively. Constraint (1f) enforce bus
voltage magnitude to maintain a level under acceptable limitations.
Constraints (1g) and (1h) limit real and reactive power outputs of generator
$i$ (we set
$\underline{p}_{i}=\overline{p}_{i}=\underline{q}_{i}=\overline{q}_{i}=0$ for
$i\in\mathcal{B}\setminus\mathcal{G}$). Constraint (1i) satisfy transmission
capacity limitations of line $(i,j)$. Constraint (1j) sets restrictions on
phase angle. Constraint (1k) ensures balance between power supply and EV
demand. Constraint (1l) controls the load of EV battery in a specified range
between the minimum charging requirement and the maximum battery capacity in
order to satisfy the charging demand of the EV. Constraint (1m) set the
battery state of charge at the beginning and end of the planning horizon.
Constraints (1n) and (1o) limit the charging and discharging rate of EV. If an
EV is connected to the grid at time $t$, its charging/discharging rate should
be between zero and the maximum limit. Otherwise, maximum allowable charging
and discharging limits $(\overline{a}_{it},\overline{b}_{it})$ are set to
zero. Constraint (1p) sets an upper limit on the total amount of marginal
emission allowed, according to the difference between generation with and
without EVs.

###  II-B Alternative Formulation

We now present an alternative formulation for the mathematical model (1)
motivated by [24]. Let us first define the new decision variables:

$\displaystyle c_{iit}:=|V_{it}|^{2}$ $\displaystyle i$
$\displaystyle\in\mathcal{B},t\in\mathcal{T}$ $\displaystyle c_{ijt}:=\
|V_{it}||V_{jt}|\cos(\theta_{ij}-\theta_{jt})$ $\displaystyle($ $\displaystyle
i,j)\in\mathcal{L},t\in\mathcal{T}$ $\displaystyle
s_{ijt}:=-|V_{it}||V_{jt}|\sin(\theta_{it}-\theta_{jt})$ $\displaystyle($
$\displaystyle i,j)\in\mathcal{L},t\in\mathcal{T}.$

To eliminate the nonlinearities, we rewrite the constraints (1b)–(1f) for each
$t\in\mathcal{T}$, replacing the newly defined variables as follows:

$\displaystyle p_{it}^{g}-p_{it}^{d}-a_{it}+\eta_{i}b_{it}=\
g_{ii}c_{iit}+\sum_{j\in\delta(i)}p_{ijt}\hskip 8.5359pti\in\mathcal{B}\ \ \ $
(3a) $\displaystyle
q_{it}^{g}-q_{it}^{d}=-{b_{ii}}{c_{iit}}+\sum_{j\in\delta(i)}q_{ijt}\hskip
65.44142pti\in\mathcal{B}$ (3b) $\displaystyle
p_{ijt}=G_{ij}c_{iit}+G_{ij}c_{ijt}-B_{ij}s_{ijt}\hskip
42.67912pt(i,j)\in\mathcal{L}$ (3c) $\displaystyle q_{ijt}=\
-B_{ij}c_{iit}-B_{ij}c_{ijt}-G_{ij}s_{ijt}\hskip
34.14322pt(i,j)\in\mathcal{L}$ (3d) $\displaystyle\underline{V}_{i}^{2}\leq
c_{iit}\leq\overline{V}_{i}^{2}\hskip 130.88284pti\in\mathcal{B}.$ (3e)

We define the following consistency constraints which preserves the
trigonometric relation between the variables $c_{iit},c_{ijt}$ and $s_{ijt}$:

$\displaystyle c_{ijt}^{2}+s_{ijt}^{2}=c_{iit}c_{jjt}$ $\displaystyle($
$\displaystyle i,j)\in\mathcal{L},t\in\mathcal{T}$ (4a)
$\displaystyle\theta_{jt}-\theta_{it}=\operatorname{atan2}(s_{ijt},c_{ijt})$
$\displaystyle i$ $\displaystyle\in\mathcal{B},t\in\mathcal{T}.$ (4b)

The alternative formulation minimizes the objective function (1a) subject to
constraints (1g)–(1p), (3) and (4).

###  II-C SOCP Relaxations

The feasible region of the alternative NLP formulation is nonconvex due to
constraints (4). We eliminate the constraint (4b) and relax the constraint
(4a) as follows:

$c_{ijt}^{2}+s_{ijt}^{2}\leq c_{iit}c_{jjt}\ \ \
(i,j)\in\mathcal{L},t\in\mathcal{T}.$ (5)

Then, an SOCP relaxation of the EV-aware MOPF problem is obtained as
minimizing (1a) subject to (1g)–(1p), (3) and (5).

##  III Solution Approach

Due to its nonconvexity, obtaining a globally optimal solution of the original
problem (1) is quite challenging. In this section, we propose a solution
approach based on its SOCP relaxation. The exactness of this convex relaxation
would guarantee that the resulting solution is also globally optimal for the
original problem. Although the SOCP relaxation is rarely exact in practice, we
still make use of it in two aspects: Firstly, it provides a lower bound for
the optimal value of the original problem (1). Secondly, we utilize the
optimal solution of the relaxation to guide the local solver to obtain a
feasible solution, hence, an upper bound, for the original nonconvex problem.
This outlined procedure provides both lower and upper bounds for the optimal
value of problem (1), from which we can compute a quality measure for global
optimality.

We introduce Algorithm 1 in order to explore the trade-off between cost and
marginal emission. We aim to approximate the Pareto frontier following a
multi-objective optimization approach. Since we are interested in marginal
emissions related to the additional generation to satisfy EV demand, we first
find generation values without EVs, denoted by $\tilde{p}_{it}$. To find a
lower bound on marginal emission (LBE), we solve the problem under the
emission minimization objective. To find an upper bound on marginal emission
(UBE), we first solve the problem under the cost minimization objective and
then calculate the marginal emission. We create a list of emission bounds
varying between LBE and UBE. For each fixed bound $\overline{E}$, we
repeatedly perform the optimization algorithm to obtain non-dominated
solutions, which make up the Pareto frontier.

We start the optimization algorithm by solving the SOCP relaxation from
Section II-C. Since the original problem is challenging to solve, we fix the
charging and discharging power variables $a_{it}$ and $b_{it}$ to their
optimal values obtained from the SOCP relaxation in the MOPF model.
Consequently, the EV charging constraints (1k)–(1p) are no longer needed.
These simplifications enable us to decompose the remaining problem into $T$
subproblems with respect to time periods. We solve these subproblems, denoted
by $SP_{t}$, in parallel and sum up their objective function values to obtain
an upper bound (UB). We also calculate corresponding marginal emission values.
Since we eliminate emission constraint (1p) from the original problem,
marginal emission values of original and SOCP problems may not match.
Therefore, we solve SOCP relaxation one more time fixing $\overline{E}$ to
marginal emission from the original problem to obtain a lower bound (LB). The
optimality gap is computed as $\%\text{Gap}=100\times(1-\text{LB}/\text{UB})$.

Algorithm 1

1: Solve $\min${(1a) : (1b)–(1j)} where $(a_{it},b_{it})$ equal to zero to
obtain $p_{it}^{g}{{}^{*}}$

2: Set $\tilde{p}_{it}^{g}=p_{it}^{g}{{}^{*}}$

3:
LBE=$\min\\{\sum_{i\in\mathcal{B}}\sum_{t\in\mathcal{T}}e_{t}(p^{g}_{it}-\tilde{p}_{it}^{g})$
: (1g)–(1o), (3), (5)}

4: Solve $\min$ { (1a) : (1g)–(1o), (3), (5) } to obtain
$(p^{g}_{it}{{}^{*}})$

5: Compute
UBE=$\sum_{i\in\mathcal{B}}\sum_{t\in\mathcal{T}}e_{t}(p^{g}_{it}{{}^{*}}-\tilde{p}_{it}^{g})$

6: Generate a list of size $n$, which consists of numbers spaced between LBE
and UBE as $\\{\rho^{1},\rho^{2},...,\rho^{n}\\}$

7: for $k=1\ to\ n$ do

8: Set $\overline{E}=\rho^{k}$

9: Solve $\min${ (1a) :(1g)–(1p), (3), (5) } to obtain
$(a_{it}^{*},b_{it}^{*})$

10: Fix $(a_{it}^{*},b_{it}^{*})$ and decompose $\min${ (1a) : (1b)–(1j) }
into $\mathcal{T}$ sub-problems

11: for all $t\in\mathcal{T}\ (\text{in parallel})$ do

12: Solve $SP_{t}$ to obtain UBkt

13: Calculate
Emission${}_{kt}=\sum_{i\in\mathcal{B}}\sum_{t\in\mathcal{T}}e_{t}(p^{g}_{it}-\tilde{p}_{it}^{g})$

14: UB${}_{k}=\sum_{t\in\mathcal{T}}\text{UB}_{kt}$

15: Emission${}_{k}=\sum_{t\in\mathcal{T}}\text{Emission}_{kt}$

16: Set $\overline{E}=\text{Emission}_{k}$

17: LB${}_{k}=\min${ (1a) :(1g)–(1p), (3), (5) }

18: Compute $\%\text{Gap}=100\times(1-\text{LB}_{k}/\text{UB}_{k})$

19: Plot coordinates $\\{(\text{UB}_{k},\text{Emission}_{k}):k=1\ ,\dots,\
n\\}$

##  IV Input

In this section, we provide information on input datasets utilized for
constructing the optimization model. We aim to use realistic datasets in order
to suggest practical solutions to cope with possible grid challenges. Most of
the researchers use synthetic grid data in OPF studies since access to
confidential information on power grids is restricted, and actual grid data is
not publicly available. In our study, we use a synthetic but realistically
created test case and integrate real datasets to increase the applicability of
the optimization model. In particular, our model draws on three real datasets:
hourly electricity consumption, hourly marginal emission factors, and EV
driving profiles. We select the regions where a realistic OPF test case and
real datasets are available. In the following sections, we explain these
datasets and describe their variations across different regions. In Section
V-A, we will discuss in detail how to integrate these datasets into our model
formulation.

###  IV-A OPF Test Case

The first important piece of information is the OPF test case. We test our
algorithm on 200-bus, 500-bus, and 2000-bus Texas A&M University (TAMU)
instances from the Power Grid Library (PGLIB-OPF) [25], geographically
situated in Illinois (IL), South Carolina (SC), and Texas (TX), respectively.
These synthetic test cases do not match any actual grid; they are constructed
based on the statistical characteristics of the actual grid. The creators of
these test cases initially situate substations in a specified region and
determine loads and generators of these substations in such a way that
generation and load profiles are similar to real profiles. Then, they link
these substations with an automated line placement process based on realistic
choices. They also consider additional complexities such as voltage control
and transient stability to make the test cases more realistic. For detailed
information on how to generate these test cases, see [25]. The main reason we
prefer to use TAMU instances is their similarity in size, complexity, and
characteristics to real networks and the availability of real datasets given
below for the same regions.

In the test cases used in this study, substations are situated in the central
part of the IL, northwestern part of the SC, and the whole state of TX.

###  IV-B Electricity Demand

We retrieve the hourly electricity demand data available from Energy
Information Administration [26]. We match each test case with the demand data
in the corresponding region. The hourly demand data for Illinois is not
available; instead, we consider the Midcontinent Independent System Operator’s
demand. For SC and TX, we use demand data for the SC Public Service Authority
and regional demand of Texas, respectively.

Seasonal variations in electricity demand may result in differences in the
operation scheduling of the network. Therefore, the numerical experiments are
conducted using data both for a summer’s day and a winter’s day to demonstrate
the influences of the variations on the model results. To generate an average
day, we first calculate the hourly averages of electricity consumption for a
certain month. Then, these hourly averages are normalized by the maximum
consumption and plotted in Fig. 1 for August 2018 and December 2018. We will
explain how to process the normalized data to determine the real and reactive
power demand of the grid in more detail in Section V-A.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour)Electricity demand (normalized) (a) Illinois 200-bus

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour) (b) South Carolina 500-bus

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour)August 2018December 2018 (c) Texas 2000-bus

Figure 1: Hourly electricity demand.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$400$$600$$800$$1{,}000$Time
(hour)Emission factor (kg/MWh) (a) Illinois 200-bus

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$400$$600$$800$$1{,}000$Time
(hour) (b) South Carolina 500-bus

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$400$$600$$800$$1{,}000$Time
(hour)Summer 2018Winter 2018 (c) Texas 2000-bus

Figure 2: Marginal $\text{CO}_{2}$ emission factors.

Hourly electricity consumption follows a similar trend across all regions. The
total consumption in summer is higher than in winter. In winter, hourly
electricity consumption varies slightly and peaks in the morning and evening,
while it peaks in the afternoon in summer.

###  IV-C Emission Factors

We use the marginal emissions that occur from power stations to satisfy the
additional demand for EV charging. We retrieve marginal emission factors
estimates, available from the Climate and Energy Decision Making Center [27].
We only take into account $\text{CO}_{2}$ emissions. For the winter and
summer, the hourly marginal $\text{CO}_{2}$ emission factors in kg/MWh are
shown in Fig. 2.

The marginal emission factors, as shown in Fig. 2, vary widely across regions,
seasons, and time of the day.

###  IV-D Driving Profiles

Our model also considers driving profiles to determine charging profiles. We
use the National Household Travel Survey (NHTS) [28] dataset, a household
survey that tracks trends in personal travel and includes all transportation
modes. The NHTS dataset contains household, person, vehicle, and travel day
trip files. We use vehicle and travel day trip files. The vehicle file
includes records for each household vehicle, while the travel day trip file
includes records for each personal trip. Regional data is available for IL,
SC, and TX. We select trips made by passenger cars and eliminate nonhousehold
vehicles. We electrify the gasoline vehicles in the NHTS dataset to
investigate how their electricity consumption will affect the OPF problem.
Then we obtain driving profiles over 24-hour that include energy requirement
at time $t$, which is explained in detail in Algorithm 2.

Algorithm 2 Driving Profile

Input: Trip index $r\in[0,R]$, vehicle index $v\in[0,V]$
From vehicle and travel day trip file import $D_{r}$: Total distance travelled
during trip $r$, $S_{r}$: Start time of trip $r$, $E_{r}$: End time of trip
$r$, $\mathcal{S}_{v}$: Set of trips for vehicle $v$, $c^{avg}$: Average
energy consumption per mile, $\overline{B}$: Battery capacity
Output: Driving Profile

1: Select passenger cars (Car, SUV, Van, Pickup Truck)

2: Delete nonhousehold vehicles

3: if $D_{r}\geq\overline{B}/c^{avg}$ then

4: Delete trip $r$

5: Create a drive duration matrix $\Delta$ of size $T\times R$

6: for all $t\in[0,T],r\in[0,R]$ do

7: if $S_{r}\leq t-1$ and $E_{r}\geq t$ then

8: $\Delta_{rt}=60$

9: if $S_{r}\geq t$ or $E_{r}\leq t-1$ then

10: $\Delta_{rt}=0$

11: else

12: $\Delta_{rt}=\min(t,E_{r})-\max(t-1,S_{r})$

13: Calculate average speed $S_{r}^{avg}=D_{r}/(E_{r}-S_{r})$

14: Create an EV energy consumption matrix $\Omega$ of size $T\times V$

15: for all $t\in[0,T],v\in[0,V]$ do

16: $\Omega_{vt}=\sum_{r\in\mathcal{S}_{v}}S_{r}^{avg}\Delta_{tr}c^{avg}$

17: Driving profile of vehicle $v$ $\Omega_{v}=\\{\Omega_{vt}:t\in[0,T]\\}$

To obtain the energy requirement of vehicles at time $t$, we make some
assumptions. Nissan Leaf has the highest sales in the moderate cost electric
car segment in US [29]. Therefore, we assume all of the EVs are similar to
Nissan Leaf. The standard battery of Nissan Leaf has a usable capacity of 32
kWh, that is 80% of its 40 kWh capacity [30]; we set the battery capacity
$\overline{B}$ to 32. We assume the energy consumption is constant and set the
corresponding parameter $c^{avg}$ to average energy consumption value of 0.3
kWh/mile [31]. Lastly, we also eliminate trips that require more than a full
battery. After filtering the data, we calculate energy requirements at each
time $t$ for vehicle $v$ as outlined in Algorithm 2.

Since the data is only available for one typical day, we use the same EV
driving profiles for both summer and winter, ignoring possible seasonal
variations in EV energy consumption. Fig. 3 provides an overview of these
driving profiles. We will explain how to integrate these driving profiles into
our model in detail in Section V-A.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$2$$4$$6$$8$$10$Time
(hour)Percentage of energy requirement (%)IllinoisSouth CarolinaTexas Figure
3: Percentage of hourly energy requirement of EVs.

##  V Computational Experiments

This section presents the results of our computational experiments, which have
been conducted to investigate the effectiveness of our approach and assess the
aggregate impact of EV charging load on optimal network operations under
different scenarios. We first discuss the experimental setup to explain how we
integrate input datasets in Section IV into the MOPF model in Section II-A.
Then, we graphically illustrate the results of the computational experiments.

###  V-A Experimental Setup

This section presents how we process input datasets so as to be compatible
with the multi-period formulation. For the EV-aware MOPF model, we consider a
24-hour period, in 1-hour slots, from 00:00 to 00:00 the following day.

The TAMU instances include network parameters for only a single period. Most
of the parameters remain constant throughout the day, but some parameters vary
on an hourly basis, such as power load and electricity price. To extend to the
multi-period version, we multiply the real and reactive power load of the grid
with the corresponding normalized values as follows:

$p_{it}^{d}=p_{i}^{d}\times\overline{u}_{t},\
q_{it}^{d}=q_{i}^{d}\times\overline{u}_{t}\quad
i\in\mathcal{B},t\in\mathcal{T},$
$p_{it}^{d}=p_{i}^{d}\times\overline{w}_{t},\
q_{it}^{d}=q_{i}^{d}\times\overline{w}_{t}\quad
i\in\mathcal{B},t\in\mathcal{T}.$

Here $\overline{u}_{t}$ and $\overline{w}_{t}$ are the normalized values for
summer and winter seasons (see Fig. 1). Since hourly electricity price data is
not publicly available, we assume that the price is constant over the 24-hour
period [26]. We adjust emission parameters $e_{t}$ to corresponding marginal
emission factors in Section IV-C.

We utilize EV driving profiles to adjust parameter settings related to EV
charging. We assume that EVs can only be charged at a bus with a nonzero real
power load. If there is no real load at bus $i$, we set
$c_{it}=\overline{a}_{it}=\overline{b}_{it}=\underline{s}_{it}=\overline{s}_{it}=0$
for all $t\in\mathcal{T}$.

Instead of individual EVs, we consider EV groups, each containing EVs with the
same driving profile. These EV groups are assigned to buses according to their
weighted energy demand, $\sum_{t\in\mathcal{T}}w_{v}\Omega_{vt}$, in such a
way that each bus $i\in\mathcal{D}$ has one EV group. Here, $w_{v}$ represents
a weighting factor obtained from the NHTS dataset. Assuming that if the bus
load is high, the EV demand around the bus is also high, we assign an EV group
with a higher demand to a bus with a higher load.

Now, each bus with a real power load has one group of EV, and thus we can use
bus indices $i$ instead of vehicle indices $v$ for brevity. We adjust
parameter settings for EV groups instead of individual EVs. We first set
energy requirements $c_{it}$ to $w_{i}\Omega_{it}$ for
$i\in\mathcal{D},t\in\mathcal{T}$. According to the energy requirements, we
determine stock parameter $\underline{s}_{it}$ for each period. If an EV has
demand during successive periods $(t,t+1,\dots,t+n)$, the amount of battery
charge at the beginning of the period $t$ should be sufficient to satisfy the
demand in these successive periods.

As previously mentioned in Section IV-D, we use the Nissan Leaf as a baseline
vehicle. The Nissan Leaf has a usable battery capacity of 32kWh with a 6.6kW
onboard charger [30]. We assume that charging rates are constant over a
period. For each bus $i\in\mathcal{D}$, the maximum limit of charging and
discharging powers, $\overline{a}_{it}$ and $\overline{b}_{it}$, is set to
$w_{i}\times 6.6$. We set the stock parameter $\overline{s}_{it}$ to usable
battery capacity, that is $w_{i}\times 32$, and the charging efficiency to 90%
[32]. We also set the initial battery SOC $I_{i}$ for EV at bus
$i\in\mathcal{D}$ to zero.

To make the ratio of the electricity consumption and EV charging demand
consistent with the actual data, we calculate a parameter called weight as the
ratio between the total power demand of the grid and daily electricity
consumption as follows:

$\texttt{weight}=\frac{\sum_{i\in\mathcal{B}}\sum_{t\in\mathcal{T}}p_{it}^{d}}{\max(\sum_{t\in\mathcal{T}}u_{t},\sum_{t\in\mathcal{T}}w_{t})}.$
(6)

Here, $u_{t}$ and $w_{t}$ are the electricity consumption for summer and
winter obtained by normalization. We multiply EV charging requirement $c_{it}$
and thereby all charging parameters
$\overline{a}_{it},\overline{b}_{it},\underline{s}_{it},\overline{s}_{it},I_{i}$
with weight. Note that we use the same weight for different seasons. Table I
summaries the daily demand data at a macro level.

TABLE I: Daily demands in kWh. | Region
---|---
| IL | SC | TX
August demand ($\sum_{t\in\mathcal{T}}u_{t}$) | 149690 | 82575 | 1256288
December demand ($\sum_{t\in\mathcal{T}}w_{t}$) | 129297 | 69901 | 923604
Grid demand ($\sum_{i\in\mathcal{B}}p_{i}^{d}$) | 1476 | 7751 | 67109
Total grid demand ($\sum_{i\in\mathcal{B}}\sum_{t\in\mathcal{T}}p_{it}^{d}$) | 29276 | 151214 | 1313994
EV demand ($\sum_{t\in\mathcal{T}}\sum_{v\in\mathcal{V}}\Omega_{vt}w_{v}$) | 9632 | 3312 | 22646
weight | 0.19 | 1.83 | 1.05

We assume that an EV cannot be driven and connected to the grid in the same
period. If the EV is in driving mode at time $t$, we set $a_{it}$ and $b_{it}$
to be zero. If EV has no energy requirement at time $t$, we assume that it is
connected to the grid, and optimization decides exactly one mode among the
following three based on the driving profile: charging, discharging, or rest.
It can only be in one of these modes during time $t$. In addition, if the
operating cost of a generator is zero, then we set the lower bound on the
power generation as $\underline{p}_{i}=0$.

The proposed MOPF model allows bidirectional flow, G2V, and V2G. In the case
where only G2V is allowed, the maximum allowable discharging rate
$\overline{b}_{it}$ is set to zero.

###  V-B Results

In this section, we report the result of our computational experiments on
three instances from PGLIB-OPF. A 64-bit desktop with Intel Xeon CPU E5-2630
v4 with a 2.20GHz processor and 32 GB RAM is used for all experiments. Our
code is written in Python programming language using Spyder environment. The
solvers GUROBI and IPOPT are used to solve the SOCP relaxation and the NLP
models, respectively.

We perform Algorithm 1 for each region, considering four different cases
depending on the season of the year and direction of flow, namely G2V in
summer, G2V-V2G in summer, G2V in winter, G2V-V2G in winter. We also create a
benchmark strategy for comparison purposes, where we assume all EV drivers
start charging their EVs at midnight.

We illustrate how $\%$ changes in total generation cost and marginal emission
move in relation to each other in Fig. 4. This figure shows the G2V case (in
solid blue line) and G2V-V2G case (in red dashed line) computed via NLP
models. The cyan cross represents the corresponding result for the benchmark
case. We calculate the change in cost as
$\%\text{Change}=100\times(\text{generation cost with EVs}/\text{generation
cost without EVs}-1)$. For $\%$ change in emission, we first assume all
vehicles in our model belong to the average gasoline car class [33] and
calculate their resulting emission. We then provide the changes in emission as
$\%\text{Change}=100\times(\text{marginal emission by EVs}/\text{emission by
gasoline cars}-1)$. In Figs. 5 and 6, we plot the hourly electricity
generation excluding EVs (in dashed blue line), hourly electricity generated
for charging EVs (in solid red line), and hourly V2G power profile (in orange
dotted line) for cost minimization, intermediate, and emission minimization
cases.

$-1$$0$$1$$2$$3$$4$$-40$$-35$$-30$$-25$% Change in emission (a) Illinois
summer

$0$$0.5$$1$$1.5$$2$$2.5$$-35$$-30$$-25$$-20$$-15$ (b) Illinois winter

$0$$2$$4$$6$$8$$-50$$-45$$-40$$-35$% Change in cost% Change in emission (c)
South Carolina summer

$5$$6$$7$$8$$9$$-42$$-41$$-40$$-39$$-38$$-37$% Change in
costG2VG2V-V2GBenchmarkIntermediate (d) South Carolina winter

Figure 4: Pareto frontier of % changes in costs and emissions.

####  V-B1 Cost-Emission Trade-off

From Fig. 4, one can observe that changes in cost and emission are inversely
correlated. When we set tight restrictions on emission, the cost tends to
rise; while we aim to minimize cost, emission increases. For example, as
depicted in Fig. 4(a), emission decreases by approximately 3% in Illinois
during summer for the G2V case with no change in the generation cost. Similar
interpretations can also be made for the results observed in Figs. 4(c) and
4(d). When we integrate the V2G concept, we can decrease the generation cost
much more, albeit with an increase in emission (see Figs. 4(a) and 4(c)).

In some cases, despite the tighter restrictions on emission, the cost remains
the same up to a point. At this point, the reduction of emission is free. This
can be clearly observed in Fig. 4(b). In this case, we do not observe
significant changes in cost since network demand is lower in winter than
summer (see Table I), and generator lower bounds almost match the demand in
winter. However, by coordinating the charging time of EVs, we can achieve
emission savings.

For the 2000-bus Texas case, since the added EV demand compromise a smaller
part (1.8%) of the total network demand, different charging patterns do not
significantly affect the solution in terms of cost and emission. In
particular, the solution that provides the least cost is almost identical to
the solution that yields the least emission. Hence, we only provide optimal
and benchmark values for the Texas case in Table II rather than the Pareto
frontier. We note that when we artificially increase EV penetration level in
the Texas case, we obtain Pareto Frontiers similar to Illinois and South
Carolina test cases.

TABLE II: Results for Texas. | Summer | Winter
---|---|---
|  G2V-V2G |  Benchmark |  G2V-V2G |  Benchmark
Cost |  2.49 |  2.57 |  3.65 |  3.68
Emission |  -39.88 |  -37.13 |  -42.15 |  -41.86

Finally, we would like to also note that, in each case, there exist multiple
points on the Pareto frontier that are strictly better than the benchmark
strategy. This shows the promise of coordinated charging in terms of both cost
and emission.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour)Energy (kWh) (normalized) (a) Cost minimization G2V

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour) (b) Intermediate case G2V

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour) (c) Emission minimization G2V

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour)Energy (kWh) (normalized) (d) Cost minimization G2V and V2G

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour) (e) Intermediate case G2V and V2G

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour)Electricity generation excluding EVsElectricity generated for charging
EVsV2G power profile (f) Emission minimization G2V and V2G

Figure 5: Hourly load variations for Illinois Summer (Intermediate cases are
indicated in Fig. 4(a)).

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour)Energy (kWh) (normalized) (a) Cost minimization G2V

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour) (b) Intermediate case G2V

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour) (c) Emission minimization G2V

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour)Energy (kWh) (normalized) (d) Cost minimization G2V and V2G

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour) (e) Intermediate case G2V and V2G

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$0$$0.2$$0.4$$0.6$$0.8$$1$Time
(hour)Electricity generation excluding EVsElectricity generated for charging
EVsV2G power profile (f) Emission minimization G2V and V2G

Figure 6: Hourly load variations for South Carolina Summer (Intermediate cases
are indicated in Fig. 4(c)).

####  V-B2 Hourly Load Variations

We now analyze the effects of hourly electricity consumption and marginal
emission factors with respect to the trade-off between the cost and emission
objectives. Under the cost minimization objective, we observe that the peak of
electricity generation for EV charging occurs when the generation excluding
EVs is the lowest. For example, in Figs. 5(a) and 6(a), when the conventional
load is relatively low early in the morning, the electricity generated for
charging EVs is high, while at the peak of the conventional load, the
electricity generated for charging EVs is minimal. If the problem aims to
minimize emission, the electricity is generated when the emission factor is
relatively low (see Figs. 5(c) and 6(c)). This results in higher peaks of the
grid; therefore, grid losses and, consequently, total generation cost
increases. It is also important to consider the impacts of driving profiles
and the initial battery state of charge. For example, in Fig. 5(c), although
emission factors are higher in the early morning, EVs have to be charged in
the early morning to satisfy their demand (see Fig. 3). As expected, EV
charging curves of intermediate cases are in between cost and emission
minimization curves (see Figs. 5(b) and 6(b)).

Our findings can help decision-makers to design incentivization mechanisms for
EV users depending on the preferred trade-off between cost and emission. For
example, if cost-efficiency is the primary objective, then charging in the
morning and feeding back to the grid in the evening can be promoted for peak
shaving (see Figs. 5(d) and 6(d)).

####  V-B3 Effect of V2G

We investigate how $\%$ changes in cost and emission are affected by the V2G
concept. The results show that the bidirectional flow can decrease cost (e.g.,
Fig. 4(c)). In the G2V-V2G case, EVs are charged during off-peak hours of
electricity consumption and give energy back to the grid during peak hours to
reduce the peak load of the power grid (see Fig. 6(d)). This flattens the load
profile, resulting in a reduction of power losses, thereby decreasing the
total generation cost. For the G2V-V2G case in Fig. 4(a), we even observe a
cost reduction compared to the no-EV case through coordinated charging of EVs.
According to Fig. 4(d), the integration of V2G does not significantly affect
cost-emission trade-off since the demand profile is relatively straight in
winter (see Fig. 1(b)).

####  V-B4 Computational Performance

To demonstrate the computational accuracy of our solution approach, we provide
optimality gaps. For the 200-bus system in IL and the 2000-bus system in TX,
we can provide approximately globally optimal solutions with an optimality gap
below $0.1\%$ and $0.6\%$, respectively. For the 500-bus system in SC, the
optimality gaps are no more than 0.6% in winter; however, in summer, the
optimality gaps vary between $1.8\%$ and $6.3\%$. The authors in [34] also
note that the 500-bus case exhibits a significant SOCP optimality gap.
Overall, our solution approach achieves relatively small percentage optimality
gaps for all the instances we consider. We calculate computational times of
obtaining Pareto frontiers of Illinois 200-bus, South Carolina 500-bus, and
Texas 2000-bus cases as approximately 5 minutes, 21 minutes, and 42 minutes,
respectively.

##  VI Conclusions

This paper proposes a new mathematical formulation for the EV-aware MOPF
Problem, and a corresponding solution approach based on SOCP. We consider a
realistic OPF test case with three real datasets: hourly electricity
consumption, hourly marginal emission factors, and EV driving profiles. We
develop a convex programming framework based on SOCP to approximate globally
optimal solutions. We conduct computational experiments to investigate the
aggregate effect of the EV charging load on the grid. Computational
experiments on different sizes of OPF instances from the PGLIB-OPF library
show that our solution approach provides high quality solutions with small
optimality gaps.

We conduct our experiments at two different seasons and three different
locations. Our results show that coordinated charging of EVs can decrease
marginal $\text{CO}_{2}$ emissions significantly with either no change or
minimal increase in cost. In addition, the integration of the V2G concept
leads to cost savings.

Our approach can have implications for real world management of EV charging
and OPF. As an example, it is possible to provide incentives to EV owners to
charge their vehicles at times that are cost and/or emission effective. Our
methodology can provide insights to decision makers on when and how to
incentivize EV charging.

Further investigations can improve our results. In this study, we consider
constant electricity prices. V2G might have greater cost reduction potential
with variable electricity prices. In addition, we consider only one type of EV
and all vehicles have the same constant charging rate. Including a mix of
different electric vehicle models and charging rates would make a more
realistic case, however this would complicate the optimization problem.
Finally, our model requires hourly electricity consumption, EV charging
demands, and the charging location of each EV to be known a priori.
Considering the variability and stochasticity of these settings reveals itself
as a challenging problem to be studied.

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