# Voltage Constrained Heavy Duty Vehicle Electrification:
Formulation and Case Study

Apurv Shukla, Rayan El Helou, , and Le Xie AS, REL, LX are with the Department
of Electrical and computer Engineering at Texas A&M University, College
Station, TX<EMAIL_ADDRESS>This work is
supported in part by NSF Grant ECCS-2038963 and U.S. Department of Energy’s
Office of Energy Efficiency and Renewable Energy (EERE) under the Solar Energy
Technologies Office (SETO) Award Number DE-EE0009031.

###### Abstract

The electrification of heavy-duty vehicles (HDEVs) is a rapidly emerging
avenue for decarbonization of energy and transportation sectors. Compared to
light duty vehicles, HDEVs exhibit unique travel and charging patterns over
long distances. In this paper, we formulate an analytically tractable model
that considers the routing decisions for the HDEVs and their charging
implications on the power grid. Our model captures the impacts of increased
vehicle electrification on the transmission grid, with particular focus on
HDEVs. We jointly model transportation and power networks coupling them
through the demand generated for charging requirements of HDEVs. In
particular, the voltage constraint violation is explicitly accounted for in
the proposed model given the signifcant amount of charging power imposed by
HDEVs. We obtain optimal routing schedules and generator dispatch satisfying
mobility constraints of HDEVs while minimizing voltage violations in electric
transmission network. Case study based on an IEEE 24-bus system is presented
using realistic data of transit data of HDEVs. The numerical results suggest
that the proposed model and algorithm effectively mitigate the voltage
violation when a significant amount of HDEVs are integrated to the power
transmission network. Such mitigation includes reduction in the voltage
magnitude, geographical dispersion of voltage violations and worst-case
voltage violations at critical nodes.

##  I INTRODUCTION

Recent surge of market penetration of electric vehicles (EVs), calls for
development of scalable design and operation of charging infrastructure in
support of the decarbonization of the mobility sector. Of 30% U.S. total
green-house gas emissions [1], $25\%$ can be attributed to medium-duty and
heavy-duty vehicles (HDVs) [2], i.e., 7.5% of the total emissions are due to
HDVC. Of particular interest is the potential impact of the electrification of
HDVs, specifically Class 7, 8 and 9 vehicles exceeding $26,000$ pounds [3].
The state of Texas has $7\%$ of the nationwide total of $3.91$ million Class 8
heavy vehicles [4] of the approximately $275,000$ HDVs registered. Each HDV
travels on average over $62,000$ miles annually - nearly $5.5$ times the
distance traveled by a typical passenger car [3]. The significance of
electrification is further amplified by noting the average heavy-duty electric
vehicle (HDEV) is expected to draw anywhere between 75 kW and 600 kW while
charging [5]. Such fundamental shifts in electrification require a substantial
restructuring and shift from fuel-based to electricity-based charging
infrastructure. This would lead to significant disruptions for both electric
grid operators and external market participants. This makes it urgent and
imperative for policy makers to adopt a data-driven, scientific approach to
reconcile the market adoption of HDEVs with the realities of existing grid
infrastructure There has been a burgeoning interest in studying recent the
impact of HDEV on the power grid. [6] demonstrate the impact of HDEV charging
decisions on the power grid.[7] study how different types of fuels effect
decarbonization of heavy-duty vehicles.

From the perspective of an electric or transportation network operator two
critical concerns arise given the surge in EV penetration: (i) mismatch of
demand allocation and infrastructure capacity, leading to inefficient use of
charging stations and (ii) mismatch of demand and supply in the power grid,
leading to voltage instability, and line capacity violations [8]. Existing
literature on these issues can be divided into three categories:

  1. (a)

The first category aims to manage the effects of EV charging load on the power
grid, at both the distribution and transmission levels, with most previous
work capturing plug in decisions at specific charging stations as exogenous
random processes. Some recent works model the effect of EV mobility patterns
on the spatio-temporal distribution of charging load [9, 10]. Recent work [11]
focuses on designing mechanisms for fleet deployment and charging at optimal
times and locations.

  2. (b)

Other work focuses on the scheduling problem of individual EV fleets
considering the time needed to charge, limited energy stored in the battery,
and range anxiety [12, 13, 14] assuming that the electricity prices are given.
The potential of an EV fleet for vehicle-to-grid (V2G) services considering
their mobility patterns is studied in [15].

  3. (c)

The third category considers the interaction between power and transportation
networks. For EVs that are parked while charging, [16] consider a case where
the operator tracks the mobility of large shared fleets of EVs and their
energy consumption and designs optimal multi-period vehicle-to-grid (V2G)
strategies. For individual EVs (i.e., those not belonging to a fleet) [17]
consider mobility-aware EV demand management problem to calculate wholesale
market clearing prices. [18] study the optimal deployment of EV public
charging stations by formulating it as a mathematical program with equilibrium
constraints.

On top of the above challenges, HDEVs have several specific challenges
unaddressed in antecedent literature: i) the threat of grid instability due to
line fluctuations and voltage deviation from nominal levels ii) sub-optimal
design of routing and scheduling policies that do not account for vehicle
charging levels especially since HDEVs are on road 80% of time. The purpose of
this paper is to manage the effects of HDEV movement and charging on the power
grid, at transmission levels.

Figure 1: Time-Energy Expanded Graph: Illustrations of movements along time
and energy dimensions

In this paper, we consider a dynamic energy and transportation network to
capture variations in travel times, routing requests, and energy requirements
from the perspective of an Independent System Operator (ISO) who is interested
in suggesting a optimal route and energy consumption of a fleet of HDEVs. To
do so, we consider the interaction of ISO with individual EV drivers and
propose a model that simultaneously prescribes the optimal route (subject to
the driver’s constraints) as well as the optimal power dispatch schedule for
the ISO. We do not focus on pricing strategies and equilibrium considerations
that arise from those. Our first primary contribution is to propose a
tractable and computationally efficient model of the transportation and energy
networks associated with scheduling HDEVs. Existing models are either
intractable due to combinatorial explosion in the number of variables or the
associated optimization problem are non-convex. Further, existing models
completely eschew aspects of HDEV scheduling pertinent for power and
transportation network operation. In contrast, our model uses significantly
smaller number of variables and the associated optimization problem is convex.
Further, in line with our main goal of analyzing the impact of HDEV
integration on the reliability of transmission and distribution grids the
model minimizes grid voltage violations while determining optimal routes for
HDEV scheduling. Our second contribution is the development of a Python-based
simulation tool, to study the consequences of HDEV electrification, exposing
the limitations imposed by existing grid infrastructure. The parameters used
to realistically model different components of the simulation (such as wind
[19], solar [20], vehicle data [21] and electric grid simulation [22]). We
utilize an IEEE-24 bus system that realistically represents multiple
distribution grids in Travis county, Texas [23], and transit data pertaining
to HDEVs. Our simulation studies demonstrate the advantages of co-optimizing
our power flow and transportation objectives as compared to a decoupled
approach on grid reliability, congestion on the transportation network and
charging decisions of the HDEV. To the best of our knowledge, this is the
first study on the effects of HDEVs on a combined energy and transportation
network.

##  II MODEL

Our main goal in this section is to propose a dynamic energy-transportation
network model that captures scheduling and routing requirements for a fleet of
HDEVs. A transportation network can be modeled as a graph
$\mathcal{G}\left(\mathcal{V},\mathcal{E}\right)$, where $\mathcal{V}$ denote
the set of vertices and $\mathcal{E}$ denote the set of edges. To model the
movement of vehicles within the state of Texas, we use such a graph where
nodes represents the charging station locations and edges represent the roads
along which any fleet of HDEVs must travel to relocate from one node to
another. Edges are weighted by the average distance required to traverse from
either node to the other (where each edge connects two nodes). Typically,
vehicles would use a path on the transportation network that can take them
from source to destination, obtaining such a network using a shortest path
problem, where each arc has a weight equal to the travelling time on that arc.
We now extend this transportation network to incorporate energy levels and
time.

A heavy-duty electric vehicle runs on electric storage, where each vehicle is
associated with a state of charge (SOC), which decreases only when the vehicle
moves, and increases only when the vehicle is charging. In contrast to smaller
electric vehicles, HDEVs not only require larger amounts of energy (kWh) to
fully charge, but also demand more charging power (kW) to reasonably serve the
vehicles. To model these energy requirements on top of a transportation
network, we expand a vertex in the transportation graph to include (discrete)
energy levels $e\in[e_{\min},\ldots,e_{\max}]:=\mathcal{E}$. Finally, in order
to consider discrete time steps and the planning horizon $T$, we add a third
dimension representing time to each vertex. This leads us to define an
extended energy-transportation network in Definition 1.

######  Definition 1 (Extended Energy-Transportation Network)

Let $\mathcal{G}^{e}=(\mathcal{N},\mathcal{A})$ denote a directed graph where
each vertex $v\in\mathcal{V}$ is a $3$-tuple
$(i,e,t)\in\mathcal{N}\subseteq[|\mathcal{V}|]\times[|\mathcal{E}|]\times[T]$
with the first component representing the physical location, the second
representing the energy levels and the third time. An arc $a\in\mathcal{A}$
connects tail node $(i,e,t)$ and head node
$(i^{\prime},e^{\prime},t^{\prime})$ iff:

  1. 1.

There is no movement backward in time, i.e., $t^{\prime}>t$

  2. 2.

Charging Arc: Increase in energy level and same physical location implying
$e<e_{\max}$ and $e^{\prime}=e+1,\ i^{\prime}=i$.

  3. 3.

V2G Discharging Arc: Decrease in energy level and same physical location,
i.e., V2G discharging implying $e>e_{\min}$ and $e^{\prime}=e-1$ and
$i^{\prime}=i$.

  4. 4.

Driving arc: $e>e_{min},\ e^{\prime}=e-1,\ i^{\prime}=\text{neighbor}(i)$,
where $\text{neighbor}(i)$ represents

  5. 5.

Resting arc: $e^{\prime}=e$ and $i^{\prime}=i$

We consider the problem of managing a given fleet of HDEVs. The purpose of
fleet management is to ensure that there is no congestion in the
transportation graph $\mathcal{G}$ and the expanded energy-transportation
$\mathcal{G}^{e}$. In the subsequent section, we will suppress the dependence
of the constraint and equation on $t$ which implies that these relationships
need to hold for all $t\in[T]$.

###  II-A Fleet Management System

We begin by considering a fleet $\mathcal{H}$ of HDEVs. The fleet management
system also needs to prescribe routes based on the fleet’s requirements. Let
$\mathcal{O}_{a}$ denote the set of arcs orginating at node $v$ and
$\mathcal{I}_{v}$ denote the incoming arcs at node $v$.Further,
$\psi^{h,\text{in}}_{v}$ is the exogenous injection of traffic of fleet $h$ at
node $v$ and $\psi^{h,\text{out}}_{v}$ denotes the exogeneous disappearence of
the HDEV from the network. This process is subject to the following
constraints:

$\displaystyle\overline{\psi}^{h,\texttt{in}},\overline{\psi}^{h,\texttt{out}}\sim\text{Travel
Criteria Distribution}$
$\displaystyle\text{s.t.}~{}\sum_{v\in\mathcal{V}}\overline{\psi}^{h,\texttt{in}}_{v}=\sum_{v\in\mathcal{V}}\overline{\psi}^{h,\texttt{out}}_{v}$
$\displaystyle\text{s.t.}~{}~{}\overline{\psi}^{h,\texttt{in}}_{v},\overline{\psi}^{h,\texttt{out}}_{v}\geq
0$
$\displaystyle\sum_{u\in\mathcal{U}_{v}}\psi^{h,\texttt{out}}_{u}=\overline{\psi}^{h,\texttt{out}}_{v}$
$\displaystyle\psi^{h,\texttt{out}}_{v}\geq 0$

where, $\bar{\psi}^{h,\text{in}}_{v}$ and $\bar{\psi}^{h,\text{out}}_{v}$ is
the system operator specified injection/rejection from the system. Let
$\lambda_{a}(t)$ denote the flow of traffic on arc $a\in\mathcal{A}$ in the
extended energy-transportation network. The flow of HDEV $h\in\mathcal{H}$
satsifes the following flow-balance constraints at all times $t\in[T]$:

$\displaystyle\sum_{a\in\mathcal{O}_{v}}\lambda_{a}^{h}-\sum_{a\in\mathcal{I}_{v}}\lambda_{a}^{h}=\overline{\psi}^{h,\texttt{in}}_{v}-\psi^{h,\texttt{out}}_{v}$

###  II-B Power Dispatch Problem

Each HDEV has direct impact on the electric power grid only when it is
charging at an electric vehicle charging station (EVCS). While there may be
several EVCS’s connected to a single distribution grid, we assume that no HDEV
charges at more than one EVCS within the same distribution grid in the short
run, since HDVs travel long dis tances after each charge. Thus, when modelling
the impact of HDEVs on distribution grids, it suffices to capture only their
local power consumption, without their movement within the neighborhood
itself. Conversely, since EVCS’s are geographically dispersed across multiple
locations in the transmission grid (state-wide or wider), we believe that it
is necessary to consider the movement of a fleet of vehicles within each
transmission grid’s encompassing geographic region to capture heir spatio-
temporal impact on critical grid parameters such as grid voltages. Given the
demand generated by considering the mobility in the transportation network, we
determine the optimal power flow in the power network using the AC optimal
power flow (AC-OPF) [24, 25]. The AC-OPF problem is a nonlinear, nonconvex
problem that models the power flow in electrical networks, determining nodal
voltages levels, angles and line flows [26, 27, 28, 29]. From the
transportation graph $\mathcal{G}$, we assume that every physical location
$\mathcal{V}$ also represents a bus and the edges $\mathcal{E}$ represent a
power line. We assume that there is one conventional generator with active and
reactive power outputs ${p}_{G,i},q_{G,i}$, one demand $p_{D,i},q_{D,i}$ per
node that is influenced due to the presence of HDEVs. Nodes without generation
or load can be handled by setting the respective entries to zero, and nodes
with multiple entries can be handled through a summation. To model generation
and control at different types of buses, we add subscripts $PQ,PV$, and
$\theta V$ to distinguish between $PQ,PV$ and $\theta V$ (reference) buses. We
model the problem of determining optimal power flow in the energy network,
using AC-OPF. Therefore, every node has the following associated constraints:

  1. 1.

Nodal Constraints: At every node $i\in\mathcal{V}$, there are three
constraints:

    1. (a)

Nodal demand balance which ensures both active, $P_{i}$ and reactive power
$Q_{i}$ are balanced.

$\displaystyle P_{i}^{d}$ $\displaystyle=$
$\displaystyle\overline{P}_{i}^{d}+x_{i}$ $\displaystyle Q_{i}^{d}$
$\displaystyle=$ $\displaystyle\overline{Q}_{i}^{d}+\alpha^{PQ}x_{i}$

In this formulation, $x_{i}$ denotes the excess demand due to the presence of
HDEVs at node $i$. A fraction of this demand $\alpha^{PQ}$ of the real demand
is incurred as reactive power.

    2. (b)

Nodal flow balance: Constraints representing that the total throughput at a
node is equal to the net generation and demand at that node.

    3. (c)

Generation Limits: The generation limits reflect the minimum and maximum
capacity of generation at individual nodes.

$\displaystyle\underline{P}_{i}^{g}\leq P_{i}^{g}\leq\overline{P}_{i}^{g}$
$\displaystyle\underline{Q}_{i}^{g}\leq Q_{i}^{g}\leq\overline{Q}_{i}^{g}$

  2. 2.

Line Flow: For every line $ij$, let $\theta_{ij}$ denote the difference in
phase angles between nodes $i$ and $j$ and $y_{ij}$ denote its admittance. We
add the following thermal limit to line constraints:

$|y_{ij}|^{2}\left[|V_{i}|^{2}+|V_{j}|^{2}-2|V_{i}||V_{j}|\cos{(\theta_{ij})}\right]\leq\overline{I}_{i,j}^{2}$

The AC-OPF formulation mentioned before is non-linear and non-convex and is
computationally inefficient without further assumptions. In order to make our
formulation tractable, we propose an analytical reformulation of the AC-OPF
problem by utilizing a first-order Taylor approximation of the branch-flow and
thermal-limit constraints about an operating point. For this purpose, let
$(\mathbf{\theta}_{0},\mathbf{V}_{0},\mathbf{P}_{0},\mathbf{Q}_{0})$ denote
the vector of operating point and let $\mathbf{J}^{P}_{V}$ denote the Jacobian
of $\mathbf{P}$ wrt $\mathbf{V}$. For example, for the branch flow constraint,
this becomes:

$\displaystyle\mathbf{P^{g}}-\mathbf{P^{d}}=\mathbf{P^{g}}_{0}-\mathbf{P^{d}}_{0}+\mathbf{J}^{P}_{V}(\mathbf{V}-\mathbf{V}_{0})+\mathbf{J}^{P}_{\theta}(\mathbf{\theta}-\mathbf{\theta}_{0})$
$\displaystyle\mathbf{Q^{g}}-\mathbf{Q^{d}}=\mathbf{Q^{g}}_{0}-\mathbf{Q^{d}}_{0}+\mathbf{J}^{Q}_{V}(\mathbf{V}-\mathbf{V}_{0})+\mathbf{J}^{Q}_{\theta}(\mathbf{\theta}-\mathbf{\theta}_{0})$

Other constraints can be linearized similarly. The final ingredient in our
formulation is the objective function that penalizes the deviation of the
nodal voltage from nominal operating point. To this end, we propose the
following objective:

$\textbf{cost}(\mathbf{P}^{g}(t))=\sum_{i\in\mathcal{V}}c_{1,g}P^{2}_{i,g}+c_{1,i}P_{i,g}+c_{0,i}+\Phi(V_{i})$

where, $\Phi(\cdot)$ denotes a penalty term on the voltage magnitude. Of
particular interest to us are the following penalties:

  1. 1.

$\ell_{2}$ co-optimization:
$\Phi(V_{i}):=\sum_{i\in\mathcal{V}}\left(V_{i}-\bar{V}_{i}\right)^{2}$: This
penalty term ensures that deviation from operation point $\bar{V}_{i}$ will be
minimized.

  2. 2.

$\ell_{1}$ co-optimization:
$=\Phi(V_{i}):=\sum_{i\in\mathcal{V}}|V_{i}-\bar{V}_{i}|$: In addition to
ensuring that deviations are minimized, this penalty ensures a sparse
geographical distribution of voltage deviations enhancing grid reliability.

  3. 3.

$\ell_{\infty}$ co-optimization:
$\Phi(V_{i}):=\max_{i\in\mathcal{L}}|V_{i}-\bar{V}_{i}|$: This penalty focuses
on minimizing the largest $|\mathcal{L}|$ nodal voltages where the set
$\mathcal{L}$ is the set of top-$L$ largest nodal voltages $\bar{V}_{i}$.

Several remarks are in order regarding our modeling choices. First, it is
widely established that DC-OPF is the standard for computing power flows in
practice. However, given our motivation to study the impact of placement of
HDEVs on voltage deviation in the grid, DC-OPF doesn’t suit our purposes since
it doesn’t take into account nodal voltages. Secondly, among the several
possible linearization schemes, our proposed reformulation includes the
complete set of non-convex AC power flow equations for the specified operating
point. Crucial for our purposes, is to ensure feasibility of the power flow
problem about an operating point which is not guaranteed by approximations,
partial linearization or relaxations of the problem. Finally, the choice of
the penalty function is critical in ensuring reliability of the network. Our
combined model (transportation with energy network) is presented in Appendix
IV-A.

##  III CASE STUDY

In this section, we present a set of experiments investigating the impact of
realistic HDEV transportation patterns on a transmission grid under a set of
alternative fleet dispatch mechanims. We focus comparison on two types of
scenarios: (a) one where we adopt a grid-agnostic HDEV spatiotemporal
scheduling mechanism to determine where and when charging occurs, and (b) one
where we adopt the solution to the proposed joint ACOPF-HDEV co-optimization
presented in (P1). Our results indicate that under the co-optimization, there
is a clear improvement in the voltage profile while maintaining required
business travel requirements on heavy-duty vehicles.

The simulation is performed as follows. First, we consider a simplified model
of the Texas grid by using an IEEE 24-bus [30] transmission grid model scaled
to meet realistic Texas-based generation and demand criteria. Second, we
consider between 30,000 to 50,000 HDEVs present in the network, which a
suitable baseline for HDEV impact in Texas according to reference [6], and
they are each constrained by having to meet some business travel criteria.
Travel criteria is defined in the context of this work as a set of required
arrival and departure times, initial and final levels of charge, and initial
and final physical locations.

For the remainder of this case study, we reference the following example for
illustration purposes. Consider some fleet $h\in\mathcal{H}$ with the
following travel criteria:

$\displaystyle\overline{\psi}^{h,\texttt{in}}_{v}\leftarrow N^{h}$
$\displaystyle\quad\text{for}\quad v=(\text{Bus 1, 50\% Charge, Hour 1})$
$\displaystyle\overline{\psi}^{h,\texttt{out}}_{v}\leftarrow N^{h}$
$\displaystyle\quad\text{for}\quad v=(\text{Bus 22, 25\% Charge, Hour 7})$

That is, this fleet of vehicles begins its business trip at hour 1 and the
location corresponding to bus 1, with 50% of battery charged. It is required
to arrive at no later than hour 7 at the location corresponding to bus 22 with
a minimum battery charge level of 25%.

With this set of business requirements on the fleet of vehicles, we compare
two spatiotemporal HDEV dispatch approaches to study the impact on the system-
wide transmission-level voltage:

  1. 1.

The vehicles follow the shortest path and stop to charge when they run out of
energy. This yields a path of buses ($1\to 4\to 14\to 20\to 22$). See Fig. (5)
for the numbers of buses in the IEEE 24-bus case.

  2. 2.

Solve the joint ACOPF-HDEV co-optimization, which yields a spatiotemporal
redistribution where vehicles strategically time their charging to avoid
congestion at key stations and key hours.

The results for voltage violations are shown in Fig. (2, 3, 4), illustrating
that the first approach yields many voltage violations across time unlike the
proposed approach which only yields much less voltage deviations. To
illustrate acceptable voltage ranges, we shade the region between 0.95 and
1.05 per unit voltage in either figure, which is typically acceptable in a
power systems context. While Fig. (3, 4) are both based on the proposed
approach, the former is based on the $\ell_{\infty}$ norm objective function
for voltage violations and the latter is based on the $\ell_{1}$ norm.

Figure 2: Voltage distribution without co-optimization. Figure 3: Voltage
distribution with $\ell_{\infty}$ co-optimization. Figure 4: Voltage
distribution with $\ell_{1}$ co-optimization.

As shown in Fig. (2), under the grid-agnostic HDEV spatiotemporal distribution
of vehicles, the voltage violations significantly exceed the acceptable band
of voltages. In contrast, under both $\ell_{\infty}$ and $\ell_{1}$ norms
using the proposed co-optimization, voltage is kept within the acceptable
band. The results in Fig. (3, 4) suggest that perhaps the $\ell_{1}$ norm is
better for voltage regulation purposes. However, this could also yield an
increase in usage of physical resources to provide this regulation. A deeper
exploration of the optimal use of such resources is beyond the scope of this
work. In this paper, we simply assume that all resources are constrained by
upper and lower limits on active and reactive power and a cost function, as
shown in formulation (P1).

To further illustrate the difference between the grid-agnostic approach and
the proposed approach, we present a visual representation of both the electric
transmission grid and the HDEV transportation network at a certain snapshot in
time, specifically showing a spatial depiction of the condition of the system
in the simulation for any of the cases. The results are shown in Fig. (5, 6,
7). From these results, we can observe that the proposed approach manages to
spatially dilute the voltage deviations across the electric transmission grid
(figures on left side), mainly by strategically changing the following on the
HDEV transportation side (figures on right side):

  * •

location to assign most charging congestion

  * •

time to wait before charging

Note that we do not explicitly state how to spatiotemporal redistribute the
charging, but the proposed co-optimization automatically yields such results.
We now direct the readers attention to Fig (6, 7) specifically. Indeed, both
results clearly out-perform the grid-agnostic approach (Fig. (5)) voltage-
wise. However, when using the $\ell_{\infty}$ norm as opposed to the
$\ell_{1}$ norm for voltage deviations in the objective function, i.e. Fig.
(6) as opposed to , Fig. (7) respectively, more number of buses incur voltage
violations, albeit small in magnitude. In contrast, the $\ell_{1}$ norm yields
voltage violations which are more at the worst buses yet less number of buses.
This kind of distinction between the two norms is expected as described in the
formulation section, and we leave it to a power systems operator to judge
which is more suitable for their grid management application.

Nonetheless, for either norm, as shown on the right sides of Fig. (6, 7),
there is clearly less congestion per charging station than in the case without
co-optimization. These results suggest that with co-optimization of HDEV
spatiotemporal scheduling and optimal power flow, business travel criteria can
still be met while yielding significantly improved voltage profiles in the
electric transmission grid.

##  IV CONCLUSION AND FUTURE WORK

In this paper, we propose a first step towards modeling the routing and
scheduling of heavy duty vehicles with particular focus on their impact on the
power grid. Our simulation results corroborate that adding penalty functions
to the joint transportation and power flow problem does indeed result in more
even geographic distribution of HDEVs and helps us in minimizing voltage
violations from nominal levels. As expected, different penalty measures serve
different purpose but all of them perform better than separately optimizing
transportation and energy networks.

In the future we would like to pursue several directions pertaining to model
and algorithm development. First, our model can be expanded to take into
account distribution network. On the algorithmic side we would like to develop
efficient algorithms for solving the co-optimization problem involving both
energy and transportation networks. Finally, reformulations and efficient
practical deployment of the proposed model in real-life systems remains a
challenge.

Figure 5: Voltage distribution without co-optimization. Figure 6: Voltage
distribution with $\ell_{\infty}$ co-optimization. Figure 7: Voltage
distribution with $\ell_{1}$ co-optimization.

## References

  * [1] USEPA, “Sources of greenhouse gas emissions.”
  * [2] U.S.TSE, “Center for climate and energy solutions.”
  * [3] USDOE, “Average annual vehicle miles traveled by major vehicle category.”
  * [4] Truck and T.-T. Registrations, “Federal highway administration,” 2019.
  * [5] F. Unterlohner, “Comparison of hydrogen and battery electric trucks,” 2020.
  * [6] R. El Helou, S. Sivaranjani, D. Kalathil, A. Schaper, and L. Xie, “The impact of heavy-duty vehicle electrification on large power grids: A synthetic texas case study,” _Advances in Applied Energy_ , vol. 6, p. 100093, 2022.
  * [7] M. Prussi, L. Laveneziana, L. Testa, and D. Chiaramonti, “Comparing e-fuels and electrification for decarbonization of heavy-duty transports,” _Energies_ , vol. 15, no. 21, p. 8075, 2022.
  * [8] W. Tushar, C. Yuen, S. Huang, D. B. Smith, and H. V. Poor, “Cost minimization of charging stations with photovoltaics: An approach with ev classification,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 17, no. 1, pp. 156–169, 2015.
  * [9] Y. Mu, J. Wu, N. Jenkins, H. Jia, and C. Wang, “A spatial–temporal model for grid impact analysis of plug-in electric vehicles,” _Applied Energy_ , vol. 114, pp. 456–465, 2014.
  * [10] M. De Gennaro, E. Paffumi, H. Scholz, and G. Martini, “Gis-driven analysis of e-mobility in urban areas: An evaluation of the impact on the electric energy grid,” _Applied Energy_ , vol. 124, pp. 94–116, 2014.
  * [11] M. Alizadeh, H.-T. Wai, A. Goldsmith, and A. Scaglione, “Retail and wholesale electricity pricing considering electric vehicle mobility,” _IEEE Transactions on Control of Network Systems_ , vol. 6, no. 1, pp. 249–260, 2018\.
  * [12] D. Goeke and M. Schneider, “Routing a mixed fleet of electric and conventional vehicles,” _European Journal of Operational Research_ , vol. 245, no. 1, pp. 81–99, 2015.
  * [13] T. Wang, C. G. Cassandras, and S. Pourazarm, “Optimal motion control for energy-aware electric vehicles,” _Control Engineering Practice_ , vol. 38, pp. 37–45, 2015.
  * [14] A. Artmeier, J. Haselmayr, M. Leucker, and M. Sachenbacher, “The shortest path problem revisited: Optimal routing for electric vehicles.” _KI_ , vol. 6359, pp. 309–316, 2010.
  * [15] R. Yu, W. Zhong, S. Xie, C. Yuen, S. Gjessing, and Y. Zhang, “Balancing power demand through ev mobility in vehicle-to-grid mobile energy networks,” _IEEE Transactions on Industrial Informatics_ , vol. 12, no. 1, pp. 79–90, 2015.
  * [16] M. E. Khodayar, L. Wu, and Z. Li, “Electric vehicle mobility in transmission-constrained hourly power generation scheduling,” _IEEE Transactions on Smart Grid_ , vol. 4, no. 2, pp. 779–788, 2013.
  * [17] M. Alizadeh, H.-T. Wai, M. Chowdhury, A. Goldsmith, A. Scaglione, and T. Javidi, “Optimal pricing to manage electric vehicles in coupled power and transportation networks,” _IEEE Transactions on control of network systems_ , vol. 4, no. 4, pp. 863–875, 2016.
  * [18] F. He, D. Wu, Y. Yin, and Y. Guan, “Optimal deployment of public charging stations for plug-in hybrid electric vehicles,” _Transportation Research Part B: Methodological_ , vol. 47, pp. 87–101, 2013.
  * [19] J. M. Freeman, N. A. DiOrio, N. J. Blair, T. W. Neises, M. J. Wagner, P. Gilman, and S. Janzou, “System advisor model (sam) general description (version 2017.9. 5),” National Renewable Energy Lab.(NREL), Golden, CO (United States), Tech. Rep., 2018.
  * [20] M. Sengupta, Y. Xie, A. Lopez, A. Habte, G. Maclaurin, and J. Shelby, “The national solar radiation data base (nsrdb),” _Renewable and sustainable energy reviews_ , vol. 89, pp. 51–60, 2018.
  * [21] K. Walkowicz, A. Duran, and E. Burton, “Fleet dna project data summary report,” _National Renewable Energy Laboratory, Tech. Rep_ , 2014.
  * [22] B. L. Thayer, Z. Mao, Y. Liu, K. Davis, and T. Overbye, “Easy simauto (esa): A python package that simplifies interacting with powerworld simulator,” _Journal of Open Source Software_ , vol. 5, no. 50, p. 2289, 2020.
  * [23] A. Birchfield, “Combined transmission and distribution synthetic dataset on the footprint of austin, texas.”
  * [24] D. Bienstock and A. Verma, “Strong np-hardness of ac power flows feasibility,” _Operations Research Letters_ , vol. 47, no. 6, pp. 494–501, 2019.
  * [25] L. Roald and G. Andersson, “Chance-constrained ac optimal power flow: Reformulations and efficient algorithms,” _IEEE Transactions on Power Systems_ , vol. 33, no. 3, pp. 2906–2918, 2017.
  * [26] J. Lavaei and S. H. Low, “Zero duality gap in optimal power flow problem,” _IEEE Transactions on Power Systems_ , vol. 27, no. 1, pp. 92–107, 2011.
  * [27] D. K. Molzahn, I. A. Hiskens _et al._ , “A survey of relaxations and approximations of the power flow equations,” _Foundations and Trends® in Electric Energy Systems_ , vol. 4, no. 1-2, pp. 1–221, 2019.
  * [28] B. Kocuk, S. S. Dey, and X. A. Sun, “Matrix minor reformulation and socp-based spatial branch-and-cut method for the ac optimal power flow problem,” _Mathematical Programming Computation_ , vol. 10, pp. 557–596, 2018.
  * [29] C. Chen, A. Atamtürk, and S. S. Oren, “A spatial branch-and-cut method for nonconvex qcqp with bounded complex variables,” _Mathematical Programming_ , vol. 165, pp. 549–577, 2017.
  * [30] C. Ordoudis, P. Pinson, J. M. Morales, and M. Zugno, “An updated version of the ieee rts 24-bus system for electricity market and power system operation studies,” _Technical University of Denmark_ , vol. 13, 2016.

## Appendices

###  IV-A Combined Transportation and Power Problem

The complete formulation is given in the next page.

$\displaystyle\underset{\mathbf{P^{g}},\mathbf{\lambda}}{\text{minimize}}\quad$
$\displaystyle\sum_{t=1}^{T}\textbf{cost}(\mathbf{P^{g}}(t))+\Phi(V(t))$ (P1a)
s.t. $\displaystyle\forall t\in[1,T]:$ $\displaystyle\textbf{Power Grid
Constraints}:\quad$ $\displaystyle\theta_{\text{slack}}=0$ (reference angle)
(P1b) $\displaystyle V_{i}=\overline{V_{i}}\quad\forall
i\in\mathcal{N}^{\text{AVR}}$ $\displaystyle{\begin{aligned}
&(\text{automatic}\\\ &~{}~{}~{}\text{voltage regulation})\end{aligned}}$
(P1c) $\displaystyle\forall i\in\mathcal{N}$ $\displaystyle:$ $\displaystyle
P_{i}^{d}=\overline{P}_{i}^{d}+x_{i}\quad$ (bus demand) (P1d) $\displaystyle
Q_{i}^{d}=\overline{Q}_{i}^{d}+\alpha^{PQ}x_{i}\quad$ (bus demand) (P1e)
$\displaystyle
P_{i}^{g}-P_{i}^{d}=\sum_{j\in\mathcal{N}}V_{i}V_{j}\left[G_{ij}\cos(\theta_{i}-\theta_{j})+B_{ij}\sin(\theta_{i}-\theta_{j})\right]$
(flow balance) (P1f) $\displaystyle
Q_{i}^{g}-Q_{i}^{d}=\sum_{j\in\mathcal{N}}V_{i}V_{j}\left[G_{ij}\sin(\theta_{i}-\theta_{j})-B_{ij}\cos(\theta_{i}-\theta_{j})\right]$
(flow balance) (P1g) $\displaystyle\underline{P}_{i}^{g}\leq
P_{i}^{g}\leq\overline{P}_{i}^{g}$ (gen. limits) (P1h)
$\displaystyle\underline{Q}_{i}^{g}\leq Q_{i}^{g}\leq\overline{Q}_{i}^{g}$
(gen. limits) (P1i) $\displaystyle\forall(i,j)\in\mathcal{L}$ $\displaystyle:$
$\displaystyle\theta_{ij}:=\theta_{i}-\theta_{j}$ (angle deviation) (P1j)
$\displaystyle
P_{i,j}=V_{i}V_{j}\left[G_{ij}\cos(\theta_{ij})+B_{ij}\sin(\theta_{ij})\right]$
(branch flow) (P1k) $\displaystyle
Q_{i,j}=V_{i}V_{j}\left[G_{ij}\sin(\theta_{ij})-B_{ij}\cos(\theta_{ij})\right]$
(branch flow) (P1l) $\displaystyle
P_{i,j}^{\text{loss}}+P_{j,i}^{\text{loss}}=P_{i,j}+P_{j,i}$ (branch losses)
(P1m) $\displaystyle
Q_{i,j}^{\text{loss}}+Q_{j,i}^{\text{loss}}=Q_{i,j}+Q_{j,i}$ (branch losses)
(P1n)
$\displaystyle|y_{ij}|^{2}\left[V_{i}^{2}+V_{j}^{2}-2V_{i}V_{j}\cos(\theta_{i}-\theta_{j})\right]\leq\overline{I}_{i,j}^{2}$
(thermal limit) (P1o) $\displaystyle\textbf{HDEV Constraints}:\quad$
$\displaystyle\forall i\in\mathcal{N}$ $\displaystyle:$ $\displaystyle
x_{i}(t)=\sum_{a\in\mathcal{X}_{i}(t)}\delta_{a}\sum_{h\in\mathcal{H}}\lambda_{a}^{h}$
(bus demand) (P1p) $\displaystyle\underline{x}_{i}\leq
x_{i}(t)\leq\overline{x}_{i}$ (station limit) (P1q) $\displaystyle\forall
h\in\mathcal{H},a\in\mathcal{A}$ $\displaystyle:$
$\displaystyle\lambda_{a}^{h}\geq 0$ (flow direction) (P1r)
$\displaystyle\forall h\in\mathcal{H},v\in\mathcal{V}$ $\displaystyle:$
$\displaystyle\overline{\psi}^{h,\texttt{in}},\overline{\psi}^{h,\texttt{out}}\sim\textbf{Travel
Criteria Distribution}$ $\displaystyle{\begin{aligned} &(\text{user-
specified}\\\ &~{}~{}~{}\text{arrival \& departure})\end{aligned}}$ (P1s)
$\displaystyle\qquad\qquad\qquad\qquad\text{s.t.}~{}\sum_{v\in\mathcal{V}}\overline{\psi}^{h,\texttt{in}}_{v}=\sum_{v\in\mathcal{V}}\overline{\psi}^{h,\texttt{out}}_{v}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\overline{\psi}^{h,\texttt{in}}_{v},\overline{\psi}^{h,\texttt{out}}_{v}\geq
0$
$\displaystyle\sum_{a\in\mathcal{O}_{v}}\lambda_{a}^{h}-\sum_{a\in\mathcal{I}_{v}}\lambda_{a}^{h}=\overline{\psi}^{h,\texttt{in}}_{v}-\psi^{h,\texttt{out}}_{v}$
(flow balance) (P1t) $\displaystyle\psi^{h,\texttt{out}}_{v}\geq 0$ (departure
flexibility) (P1u)
$\displaystyle\sum_{u\in\mathcal{U}_{v}}\psi^{h,\texttt{out}}_{u}=\overline{\psi}^{h,\texttt{out}}_{v}$
(departure flexibility) (P1v)