# A Reliability-aware Distributed Framework to Schedule Residential Charging
of Electric Vehicles

Rounak Meyur1111Contact Author Swapna Thorve1 Madhav Marathe1 Anil Vullikanti1
Samarth Swarup1 Henning Mortveit1
1UVA Biocomplexity Institute Charlottesville, Virginia, USA
{rm5nz, st6ua, marathe, vsakumar, swarup<EMAIL_ADDRESS>
###### Abstract

Residential consumers have become active participants in the power
distribution network after being equipped with residential EV charging
provisions. This creates a challenge for the network operator tasked with
dispatching electric power to the residential consumers through the existing
distribution network infrastructure in a reliable manner. In this paper, we
address the problem of scheduling residential EV charging for multiple
consumers while maintaining network reliability. An additional challenge is
the restricted exchange of information: where the consumers do not have access
to network information and the network operator does not have access to
consumer load parameters. We propose a distributed framework which generates
an optimal EV charging schedule for individual residential consumers based on
their preferences and iteratively updates it until the network reliability
constraints set by the operator are satisfied. We validate the proposed
approach for different EV adoption levels in a synthetically created digital
twin of an actual power distribution network. The results demonstrate that the
new approach can achieve a higher level of network reliability compared to the
case where residential consumers charge EVs based solely on their individual
preferences, thus providing a solution for the existing grid to keep up with
increased adoption rates without significant investments in increasing grid
capacity.

##  1 Introduction

Studies have shown that home-charging units are pivotal infrastructure for
promoting EV adoption Wei et al. (2021). As EV adoption increases over the
next few years, the power drawn from the grid will increase and may cause
disturbances in the distribution power network. It is quite possible that the
network operation may undergo modifications in order to accommodate these
unconventional loads in the future without affecting the reliability of the
grid.

From the standpoint of power engineering, a _reliable_ power grid is one which
has adequate generation to support the consumer load demand and can be
operated without violating standard power engineering constraints Billinton
and Li (1994). In this paper, we consider an operational problem and therefore
are not concerned about the aspect of adequate generation. Thus, we use the
term _reliable distribution network_ to mean a network that can satisfy the
loads without violating the node voltages and line flow (edge flow)
capacities. While flows above line rating causes overheating of conductors and
subsequent physical damage, node voltages represent the quality of power
delivered at the node. Undervoltage and overvoltage at residence nodes lead to
eventual failure of household appliances. Relevant basic concepts of power
distribution networks are described in the Appendix.

Traditionally, the distribution network is designed to sustain the peak load
demand of consumers Heidari et al. (2015). The predictable growths in consumer
peak demands and energy consumption enables the network operator to plan and
operate the network reliably. However, adoption of EVs in residential
communities leads to a significant deviation in the predictability of consumer
loads Shao et al. (2009). Residential EV charging constitutes a significant
percentage of net household demand leading to large power consumed from the
grid. The problem is more dominant when residential consumers opt to charge
EVs according to their personal convenience Putrus et al. (2009). The excess
load consumed by the EV charging units adversely affects distribution grid by
causing transformer overloading or high voltage drop at feeder ends Farkas et
al. (2011). As a result, it is desirable to develop a framework which aids
residential consumers with their goal of scheduling EV charging based on their
individual preferences, and simultaneously taking into account the grid
reliability requirements of the network operator.

Contributions. The contributions of the paper are summarized here: (i) A novel
‘reliability-aware distributed EV charging scheduling framework’ is proposed.
It uses information such as the hourly electricity rate, household energy
demand profiles & their preferences as inputs for consumers, and power
engineering constraints of distribution network as inputs for operator to aid
residential EV adopters in scheduling their EV charging units in an optimal
manner without affecting the reliability of the power grid. (ii) The
distributed framework uses ADMM based iterative methodology which guarantees
an optimal solution for our problem. Each iteration involves solving a mixed
integer quadratic program (MIQP) for each residential consumer and a quadratic
program (QP) for the operator. The optimal solutions are exchanged and used in
succeeding iterations until a consensus is reached. This minimum exchange of
information between the consumers/residences/households and the network
operator can be executed using present smart grid infrastructure and avoids
sharing of private and proprietary data. (iii) We use digital duplicates of
residential consumer load demand profile and power distribution networks
resembling the actual physical counterparts for our case studies. This
facilitates conducting real-world test scenarios to explore the impact of
using the proposed framework while considering multiple levels of EV adoption.
Our experiment results demonstrate that the proposed distributed framework
helps maintain network reliability compared to the case where EV adopters
charge their vehicles based on their personal (individualized) preferences.

##  2 Related Work

Several works have been presented in the literature for scheduling EV
charging. In general, optimization techniques are a popular choice for solving
the problem of scheduling EV charging at household level Cao et al. (2012);
Zhao et al. (2018); Lee and Choi (2020); Blonsky et al. (2021); Wi et al.
(2013); Tang et al. (2016); Gonçalves et al. (2018); Khonji et al. (2018);
Clement-Nyns et al. (2010); Dall’Anese et al. (2014). Recently, ML techniques
such as neural networks Shuvo and Yilmaz (2021) and reinforcement learning
frameworks Cao et al. (2022) have been proposed to study the problem of
scheduling EV charging loads in smart homes.

Most of these works implement a centralized approach to schedule EV charging.
A centralized optimization algorithm/framework evaluates the optimal power
consumption patterns which are beneficial to only one of the entities –
consumers or network operators Liu et al. (2015). This approach may not be
realistic since details of the individual consumer load is usually not
accessible to the network operator. At the same time, the network topology and
parameters are unknown to the consumers. Under such circumstances, a de-
centralized/distributed framework is useful since it can help network
operators and consumers communicate essential information that can respect
both – network reliability and consumer preferences. The current smart grid
infrastructure supports the development of such a framework due to the
availability of two-way communication.

Dall’Anese _et al._ Dall’Anese et al. (2014) use an alternating direction
method of multipliers (ADMM) based approach to evaluate inverter set points at
different locations in a network while maintaining network reliability. The
results show that this method provides superior convergence guarantees in
comparison with other methods while dealing with mixed integer linear programs
(MILP). In this work, we propose a distributed optimization framework based on
the ADMM method for scheduling EV charging in a power distribution network.
Our framework satisfies two goals – maintaining grid reliability for the
network operator while respecting consumer preferences.

##  3 Problem Formulation

In this paper, we are interested in power consumption trajectory over a finite
horizon time window of $T$ intervals from time instant $k=0$ to $k=T$. We
define an interval $t$ as the duration between time instants $k=t-1$ and
$k=t$. Table 1 summarizes the index variables and sets used in the paper.

Table 1: Summary of index variables and sets used Symbol | Description
---|---
$T$ | Number of intervals in time window
$N$ | Number of non-substation nodes in network
$i$ | Index of node in network
$t$ | Index of time interval
$k$ | Index of time instant
$\alpha,\beta$ | Limits of squared voltage magnitude
$\mathscr{V}$ | Set of all nodes in network
$\mathscr{H}$ | Set of all residence nodes in network
$\mathscr{N}$ | Set of all non-substation nodes in network

###  3.1 Distribution Network Model

The power distribution network is a _tree_ comprising of $N+1$ nodes collected
in the set
$\mathscr{V}:=\mathscr{N}\cup\\{0\\},\mathscr{N}:=\\{1,2,\cdots,N\\}$. The
tree is rooted at substation node $\\{0\\}$ and consists of primary and
secondary distribution lines collected in the edge set. The set $\mathscr{N}$
includes residences, local transformers and auxiliary nodes required to
connect the transformers and residences Meyur et al. (2020); Kersting (2012);
Bolognani and Dörfler (2015). Here, we are interested in the variables
associated with the set of residence nodes $\mathscr{H}\subset\mathscr{N}$.
The power consumption and squared voltage magnitude at node $i$ and time $t$
are denoted respectively by $p_{i}^{t}$ and $v_{i}^{t}$. We respectively stack
these variables for $i\in\mathscr{N}$ to corresponding vectors
$\mathbf{p}^{t}$ and $\mathbf{v}^{t}$ for every time interval $t$. The non-
linear relation between power injections and squared voltages in the network
can be simplified to linear expression using the Linearized Distribution Flow
(LDF) model Bolognani and Dörfler (2015).

$\displaystyle\mathbf{v}^{t}=-2\mathbf{Rp}^{t}+\mathbf{1}$ (1a)
$\displaystyle\alpha\mathbf{1}\leq\mathbf{v}^{t}\leq\beta\mathbf{1}$ (1b)

Here, $\mathbf{1}$ is a vector of all $1$’s. The matrix $\mathbf{R}$ is a
function of network topology and edge parameters which is only accessible by
the network operator. The primary objective of the operator is to maintain the
network reliability where the node voltages are within acceptable ANSI C.84
Range A limits ANSI (2020). In most practical distribution networks, these
limits are 0.95 pu and 1.05 pu. The operator ensures that the power
consumption at different nodes in the network satisfy (1) where $\alpha,\beta$
denote the squared voltage limits.

###  3.2 Residence Load Models

This section describes load demand of a residence node $i\in\mathscr{H}$. The
aggregate power consumption for the time interval $t$ is given by
$p_{i}^{t}\in\mathbb{R}$. This load comprises of an uncontrollable base load
demand denoted by $p_{i,0}^{t}$ and the controllable counterpart. In this
paper, we use the synthetically generated residential load demand data
described in Thorve et al. (2018) for the uncontrollable base load demand. We
consider residence owned EV charging stations as the only controllable load
which is denoted by $p_{i,\textrm{EV}}^{t}$. The power consumption at node $i$
and over time interval $t$ can be expressed as

$p_{i}^{t}=p_{i,0}^{t}+p_{i,\textrm{EV}}^{t}\quad\forall t=1,2,\cdots,T$ (2)

EV Charging Model. We assume that the EV charging unit is responsible to
charge only a single EV owned by the customer. Let the charge capacity of the
EV be $Q_{i,\textrm{EV}}$ and the power rating of the EV charging unit be
$P_{i,\textrm{EV}}$. The state of charge (SOC) evolves over the time interval
$t$ from $s_{i,\textrm{EV}}^{t-1}$ to $s_{i,\textrm{EV}}^{t}$ following (3b).
Further, the constraint (3c) limits the SOC to suitable lower and upper
bounds. The scheduling problem aims to find out the optimal time intervals
when the EV can be charged while maintaining a secure power grid. Let
$z_{i,\textrm{EV}}^{t}\in\\{0,1\\}$ be a binary variable which takes the value
$1$ if the EV is charged at time interval $t$ and $0$ otherwise.

Further, the EV is available for charging only at particular time interval
denoted by the closed interval
$\mathscr{T}_{i,\textrm{EV}}:=\left[t_{\textrm{start}},t_{\textrm{end}}\right]$.
Let the SOC at time $t=t_{\textrm{start}}$ be $s_{i,\textrm{init}}$ and it is
expected that by the end of the interval, the SOC of the battery needs to be
at least $s_{i,\textrm{final}}$. Note that we consider a very simple case
where the EV is available to be charged within a single continuous time
interval $\mathscr{T}_{i,\textrm{EV}}$ in the time window of $T$ intervals. We
also assume that the EV is not used in this interval. In realistic scenarios,
these intervals of charging are discontinuous and usage of EV would result in
different SOC at different time intervals.

$\displaystyle p_{i,\textrm{EV}}^{t}=z_{i,\textrm{EV}}^{t}P_{i,\textrm{EV}}$
$\displaystyle\forall t=1,2,\cdots,T$ (3a) $\displaystyle
s_{i,\textrm{EV}}^{t}=s_{i,\textrm{EV}}^{t-1}+\frac{p_{i,\textrm{EV}}^{t}}{Q_{i,\textrm{EV}}}$
$\displaystyle\forall t=1,2,\cdots,T$ (3b) $\displaystyle 0\leq
s_{i,\textrm{EV}}^{k}\leq 1$ $\displaystyle\forall k=0,1,\cdots,T$ (3c)
$\displaystyle z_{i,\textrm{EV}}^{t}=0$ $\displaystyle\forall
t\notin\mathscr{T}_{i,\textrm{EV}}$ (3d) $\displaystyle
s_{i,\textrm{EV}}^{t_{\textrm{start}}}=s_{i,\textrm{init}},\quad
s_{i,\textrm{EV}}^{t_{\textrm{end}}}\geq s_{i,\textrm{final}}$ (3e)

###  3.3 Optimization problem

Each residence aims to compute the optimal power usage trajectory of its EV
charging unit over a finite horizon time window of length $T$ denoted by
$\\{p_{i,\textrm{EV}}^{t}\\}_{t=1}^{T}$. Given the hourly rate of electricity
$c^{t}$ for each time interval in the time window, the optimization problem
for each residence involves minimizing the total cost of consumption given the
EV constraints (3). This results in the MILP (4).

$\displaystyle\min\quad$ $\displaystyle\sum_{t=1}^{T}c^{t}p_{i}^{t}$ (4a) over
$\displaystyle p_{i,\textrm{EV}}^{t}\quad\quad\quad\quad~{}~{}~{}\forall t$
(4b) s.t. $\displaystyle(\ref{eq:load-bal}),(\ref{eq:res-
ev})~{}~{}~{}\quad\quad\quad\forall t$ (4c)

At the same time, the network operator needs to ensure node voltages are
within acceptable limits to maintain a reliable system. Additionally the
operator might aim to optimize other aspects such as minimize losses or reduce
voltage deviation Dall’Anese et al. (2014). This can be expressed by
$C\left(\mathbf{p}^{t}\right)$ which is a function of power usage of all
residences at interval $t$. In this paper, we do not consider any particular
objective of the network operator and treat $C\left(\mathbf{p}^{t}\right)=0$.
We define the _Reliability-aware EV Charge Scheduling_(REVS) problem which
satisfies consumer preferences as well as ensures network reliability by (5).

$\displaystyle\min\quad$
$\displaystyle\sum_{t=1}^{T}C\left(\mathbf{p}^{t}\right)+\sum_{i\in\mathscr{H}}\sum_{t=1}^{T}c^{t}p_{i}^{t}$
(5a) over $\displaystyle
p_{i,\textrm{EV}}^{t}\quad\quad\quad\quad~{}~{}~{}\forall t~{}\forall
i\in\mathscr{H}$ (5b) s.t. $\displaystyle(\ref{eq:volt-
constraint})\quad\quad\quad\quad\quad~{}~{}\forall t$ (5c)
$\displaystyle(\ref{eq:load-bal}),(\ref{eq:res-
ev})~{}~{}~{}\quad\quad\quad\forall t~{}~{}\forall i\in\mathscr{H}$ (5d)
$\displaystyle p_{i}^{t}=0\quad\quad\quad\quad\forall t~{}~{}\forall
i\notin\mathscr{H}$ (5e)

##  4 Proposed Methodology

The REVS problem in (5) is an MILP with binary variables arising from the
on/off status of the EV charging unit if we consider . This problem can be
solved from a central location (such as the operator) if the load information
of residences are known. However, this is not always the case due to privacy
concern associated with sharing personal data of consumers. Similarly, the
network topology and parameters are considered proprietary information and
cannot be shared with the consumers. However, limited information exchange
such as total power consumption can be done without violating privacy concerns
using the current smart grid infrastructure. In this section, we propose an
iterative method based on the ADMM technique to reach the optimal solution for
the REVS problem.

To this end, we separate the problem for the network operator and individual
residences. Each residence $i$ aims to compute the optimal power usage
trajectory $\left\\{p_{i}^{t}\right\\}_{t=1}^{T}$ over the time window given
the EV charging constraints. The network operator computes consumption
trajectories $\left\\{\tilde{p}_{i}^{t}\right\\}$ for all nodes (in the vector
form $\tilde{\mathbf{p}}^{t}$) such that the network reliability constraints
are satisfied. Additionally, we add constraint (6f) to force these
trajectories to match each other. Therefore, we get (6) as the alternate
version of the REVS problem.

$\displaystyle\min\quad$
$\displaystyle\sum_{i\in\mathscr{H}}\sum_{t=1}^{T}c^{t}p_{i}^{t}$ (6a) over
$\displaystyle p_{i}^{t},\tilde{p}_{i}^{t}~{}~{}\quad\quad\quad\quad~{}\forall
t~{}~{}\forall i$ (6b) s.t. $\displaystyle(\ref{eq:volt-
constraint})~{}\quad\quad\quad\quad\quad~{}~{}\forall t$ (6c)
$\displaystyle(\ref{eq:load-bal}),(\ref{eq:res-ev})\quad\quad\quad\quad\forall
t~{}~{}\forall i\in\mathscr{H}$ (6d) $\displaystyle
p_{i}^{t}=0=\tilde{p}_{i}^{t}~{}\quad\quad\forall t~{}~{}\forall
i\notin\mathscr{H}$ (6e)
$\displaystyle\tilde{p}_{i}^{t}=p_{i}^{t}~{}\quad\quad\quad~{}~{}~{}\forall
t~{}~{}\forall i\in\mathscr{H}$ (6f)

Table 2: Summary of variables in optimization problem Symbol | Description
---|---
$v_{i}^{t}$ | Voltage at node $i$ for interval $t$
$p_{i}^{t}$ | Power consumed at node $i$ for interval $t$
$\tilde{p}_{i}^{t}$ | Power consumption computed by operator
$p_{i,0}^{t}$ | Power consumed by fixed load
$p_{i,\textrm{EV}}^{t}$ | Power consumed by EV charging unit
$z_{i,\textrm{EV}}^{t}$ | On/Off status of EV charging unit
$s_{i,\textrm{EV}}^{k}$ | SOC of EV at time instant $k$
$\gamma_{i}^{t}$ | Dual variable corresponding to (6f)
$\mathbf{v}^{t}$ | Vector of $v_{i}^{t}$ for all nodes $i\in\mathscr{N}$
$\mathbf{p}^{t}$ | Vector of $p_{i}^{t}$ for all nodes $i\in\mathscr{N}$
$\tilde{\mathbf{p}}^{t}$ | Vector of $\tilde{p}_{i}^{t}$ for all nodes $i\in\mathscr{N}$

Now, we use the conventional ADMM steps to iteratively update the optimization
variables for the operator and residences. Let
$\tilde{\mathcal{P}}[l]:=\left\\{\tilde{\mathbf{p}}^{t}[l]\right\\}_{t=1}^{T}$
denote the optimal trajectories for all nodes computed by the operator for
iteration $l$. Similarly, let
$\mathcal{P}_{i}[l]:=\left\\{{p}_{i}^{t}[l]\right\\}_{t=1}^{T}$ denote the
optimal power usage trajectory computed by residence $i$. We abuse the
notation $\left\\{\mathcal{P}_{i}[l]\right\\}$ to denote the optimal
trajectories $\left\\{{p}_{i}^{t}[l]\right\\}_{t=1}^{T}$ computed by all
residences $i\in\mathscr{H}$ individually. The two steps of iteration are
listed below. Note that the first step is carried out simultaneously for all
residences and network operator. Fig. 1 illustrates the proposed message
passing based distributed framework.

  1. S1a.

At the operator side, we update the operator estimated power consumption
$\tilde{p}_{i}^{t}$ for all residences using (7).

$\displaystyle\tilde{\mathcal{P}}[l+1]:=$
$\displaystyle\textrm{arg}~{}\textrm{min}\quad
F(\tilde{\mathcal{P}}[l],\\{\mathcal{P}_{i}[l]\\})$ (7a)
$\displaystyle\textrm{s.to.}\quad\alpha\leq
1-2\sum_{j=1}^{N}R_{ij}\tilde{p}_{j}^{t}\leq\beta\quad\forall t~{}~{}\forall
i$ (7b)

where the function $F(\tilde{\mathcal{P}}[l],\\{\mathcal{P}_{i}[l]\\})$ is
defined as

$\displaystyle F(\tilde{\mathcal{P}}[l],$
$\displaystyle\\{\mathcal{P}_{i}[l]\\}):=\sum_{i\in\mathscr{H}}\sum_{t=1}^{T}\frac{\kappa}{2}\left(\tilde{p}_{i}^{t}\right)^{2}$
(8) $\displaystyle+$
$\displaystyle\sum_{i\in\mathscr{H}}\sum_{t=1}^{T}\tilde{p}_{i}^{t}\left(\gamma_{i}^{t}[l]-\frac{\kappa}{2}\tilde{p}_{i}^{t}[l]-\frac{\kappa}{2}p_{i}^{t}[l]\right)$

  2. S1b.

For residence $i$, we update using (9).

$\displaystyle\mathcal{P}_{i}[l+1]:=\textrm{arg}~{}\textrm{min}\quad$
$\displaystyle\sum_{t=1}^{T}c^{t}p_{i}^{t}+F_{i}(\tilde{p}_{i}^{t}[l],p_{i}^{t}[l])$
(9a) s.to. $\displaystyle(\ref{eq:load-bal})-(\ref{eq:res-ev})$ (9b)

where the function $F_{i}\left(\tilde{p}_{i}^{t}[l],p_{i}^{t}[l]\right)$ is
defined as

$\displaystyle F_{i}\left(\tilde{p}_{i}^{t}[l],p_{i}^{t}[l]\right)=$
$\displaystyle\sum_{t=1}^{T}\frac{\kappa}{2}\left({p}_{i}^{t}\right)^{2}$ (10)
$\displaystyle-$
$\displaystyle\sum_{t=1}^{T}{p}_{i}^{t}\left(\gamma_{i}^{t}[l]+\frac{\kappa}{2}\tilde{p}_{i}^{t}[l]+\frac{\kappa}{2}p_{i}^{t}[l]\right)$

  3. S2.

At the operator and residence sides, the dual variable update is updated.

$\gamma_{i}^{t}[l+1]=\gamma_{i}^{t}[l]+\frac{\kappa}{2}\left(\tilde{p}_{i}^{t}[l+1]-p_{i}^{t}[l+1]\right)$
(11)

The resulting decentralized procedure involves a two-way message exchange of
the iterates $\left\\{\tilde{\mathbf{p}}^{t}[l]\right\\}_{t=1}^{T}$ and
$\left\\{\mathbf{p}^{t}[l]\right\\}_{t=1}^{T}$ between the network operator
and residential consumers. At an iteration $l>0$, the network operator updates
the power trajectories based on (7) whose objective includes a regularization
term $F(\tilde{\mathcal{P}}[l],\\{\mathcal{P}_{i}[l]\\})$. This term enforces
consensus with the power usage trajectories computed at the residences. The
constraints ensure the reliability aspects in of the network. Note that (7) is
a QP because of the quadratic regularization term. The operator relays to each
residential consumer $i$ a copy of the iterate value
$\left\\{\tilde{p}_{i}^{t}[l+1]\right\\}_{t=1}^{T}$. At the same time, the
consumer optimal trajectories are updated using (9) and copy of the iterate
value $\left\\{{p}_{i}^{t}[l+1]\right\\}_{t=1}^{T}$ is sent to the operator.
We note that (9) is a MIQP because of the quadratic regularization term
ensuring consensus with the operator objective and binary constraints for the
EV charging unit. We note that the REVS problem which was originally an MILP
is converted to a QP for the operator and MIQPs for individual residences
using the proposed ADMM based framework. Once the updated local iterates are
exchanged, the operator and residential updates the local dual variables using
(11).

The centralized approach to solve the MILP guarantees convergence to the
global optimum solution. However, the concern of sharing private consumer
information with the network operator hinders the approach. The proposed ADMM
based distributed framework avoids sharing of private and proprietary
information and only uses exchange of power consumption data. The approach
converts the problem into a QP for the operator and MIQPs for each residence.
However, the size of each problem is significantly smaller than the original
MILP. The convergence of the algorithm to the optimal solution of (5) is
formally stated next.

Figure 1: Message exchange aided proposed distributed framework to schedule
residential EV charging.

######  Proposition 1.

The iterates $\left\\{\tilde{\mathbf{p}}^{t}[l]\right\\}_{t=1}^{T}$ and
$\left\\{\mathbf{p}^{t}[l]\right\\}_{t=1}^{T}$ produced by
$[\mathbf{S1}]-[\mathbf{S2}]$ are convergent, for any $\kappa>0$. Further,

$\lim_{l\rightarrow\infty}\left\\{\tilde{\mathbf{p}}^{t}[l]\right\\}_{t=1}^{T}=\lim_{l\rightarrow\infty}\left\\{\mathbf{p}^{t}[l]\right\\}_{t=1}^{T}=\left\\{\mathbf{p}^{t}_{\textrm{opt}}\right\\}_{t=1}^{T}$

where $\mathbf{p}^{t}_{\textrm{opt}}$ denotes the optimal power usage
trajectory.

ADMM has been proved to converge to the optimal solution for convex problems
Boyd et al. (2011). However, in our paper the problem involves binary
constraints for the EV charging unit which renders the problem to be non-
convex. Interestingly, ADMM can converge to optimality in the exact sense for
non-convex problems with binary variables. Therefore, we can guarantee that
the proposed framework converges to the optimal solution.

##  5 Experiments

The experiments are conducted in order to study the effects of EV adoptions at
different levels (30%, 60%, 90%). We compare effects of two optimization
scenarios (individual vs. distributed) on EV scheduling behavior in different
communities. Under the _individual optimization_ scenario, customers charge
their EVs based on individual preferences without thinking about the impact on
the network. The optimal schedule is obtained by solving (4) for each EV
adopter. With _distributed optimization_ , the customers agree to an optimal
EV charging schedule where they are benefited as well as the network
reliability is maintained. This allows the network operator to maintain
acceptable node voltage level throughout the network. The optimal schedule is
obtained by iteratively solving Equations (7), (9), and (11) until
convergence.

Particularly, we aim to compare the reliability of the network when these two
methods are used to schedule residential EV charging for varied levels of EV
adoption. Note that network reliability is the ability to operate with edge
power flows within the line capacities and node voltage within the bandwidth
($0.95-1.05$ p.u) ANSI (2020). Hence, these two measures – node voltage and
edge power flow are used to quantify the impact of network reliability at
different levels (30%, 60%, 90%) of EV adoptions in multiple communities in
the distribution network.

Table 3: Hourly electricity rate for the experiments | Time
---
interval
(HH:MM)
| 00:00-
---
-05:00
| 05:00-
---
-15:00
| 15:00-
---
-18:00
| 18:00-
---
-00:00
| Cost
---
($/kWhr)
0.07866 | 0.09511 | 0.21436 | 0.09511

Figure 2: Impact of residential EV charging is analyzed for two different
residential communities within the same network: ‘Com-A’(top) and
‘Com-B’(bottom). The orange nodes shown in the network denote the residences
in the two communities. The individual optimization leads to undervoltage
(less than 0.95 p.u.) at a significant number of residences. This can be
avoided by using the proposed distributed optimization method even for higher
levels of EV adoption.

A small area of Montgomery county in Virginia is considered as the region of
interest for our study. Household level synthetic hourly electricity
consumption profiles are used. These timeseries are created using several
population surveys, statistical models, and physics based models of household
devices and validated using real data Thorve et al. (2018). The data also has
household level demographics and spatial attributes. Synthetically generated
distribution networks created using electrical engineering concepts and
resembling actual networks are used for the purpose of our analysis Meyur et
al. (2020). Hourly electricity rate (in $/kWhr) is known to the residential
customers (Table 3). Time of use (TOU) hourly electricity rate provided by the
off-peak plan of a utility company serving the particular geographical region
Dominion Energy (2021) are used.

_Assumptions._ All EVs have a uniform charge capacity of $20$kWhr and are
available to be charged between 4:00p.m. and 5:00a.m. The initial state of
charge is assumed to be $20\%$ and the EVs are required to be charged to at
least $90\%$. A uniform rating of $4.8kW$ of the residential EV charger.
Households are randomly selected as EV adopters in the network. All adopters
have necessary provisions to charge their EVs at their residential premises.

##  6 Results

Figure 3 describes edge power flow as a percentage of line capacities and node
voltages in ‘Com-A’ community in the network when EV adoption has reached 90%.
Figure 3 (top) shows that the edge flow (line rating capacity) levels in the
network are well within limits even when 90% of residences have adopted EV.
However, the same cannot be observed for node voltages in the network. Figure
3 (bottom) shows that the node voltages at several residences are outside the
acceptable limits. We notice that maximum number of node voltage violations
occur in the period where the hourly electricity rate (in $/kWhr) is minimum.

Figure 3: Comparison of line loading level (edge flows) and node voltages for
residential EV adoption of $90\%$ in ‘Com-A’ of network. The high EV adoption
does not significantly affect the line loading level. However, node voltages
at multiple residences in the network are outside the acceptable voltage
limits of $0.95-1.05$ p.u.

We further explore effects of node voltage violation, at different adoption
levels in two communities (‘Com-A’, ‘Com-B’) of residences in the network.
Figure 2 shows the selected communities in the network and node voltage
violation at different adoption levels. The results are obtained after
performing optimizations under two scenarios : individual and distributed. We
focus our attention only on the time intervals where the hourly electricity
rate (in $/kWhr) is minimum (i.e. time windows of maximum node violations).
The node voltage violation is divided into 3 ranges : less than $0.92$ p.u.,
between $0.92-0.95$ p.u. and between $0.95-0.98$ p.u. Though the last voltage
range is not considered as voltage violation by the practised ANSI standard
ANSI (2020), it can be considered as a sign of reduced network reliability.
The clustered bar chart shows significantly higher number of residences with
voltage violation under individual optimization scenario as compared to the
distributed optimization scenario. The number of residences with voltage less
than $0.95$ p.u. is close to zero at all considered time intervals. The
observations are similar for both communities in the network. This shows that,
if the proposed distributed framework is used to schedule EV charging units,
the network operator is able to dispatch the power without compromising on
system reliability.

Figure 2 shows that under the individual optimization approach the number of
residences with undervoltage during the cheap electricity hours increases with
an increase in the level of adoption. However, this trend is not consistent in
the distributed optimization approach. This is because the later approach also
ensures system reliability along with consuming electricity during cheap
hourly rates. The distributed framework does this by allocating small amount
of EV charging during time intervals where the hourly electricity rates are
relatively higher.

We also notice that the number of residences experiencing undervoltage issues
for the same level of adoption differs significantly when we consider
different communities for EV adoption. These differences can be attributed to
location and energy usage of adopter households in the network and the
resulting voltages at different nodes. The error bars on the bar chart (Figure
2) show variation in number of residences violating node voltage in each
category after multiple runs of the adoption level.

##  7 Conclusion

In this paper we propose an ADMM method based distributed framework for
scheduling EV charging for consumers at residential premises while maintaining
reliability of the network. We observe that individual consumer loads, spatial
layout of the network, and electricity pricing all affect network reliability.
We provide a set of results on a small distribution network in Montgomery
county of Virginia, USA for EV adoption at different levels in multiple
residential communities. We inspect the network reliability with two measures
: node voltage and edge flows (line rating/line loading). Results show that
the proposed distributed approach for scheduling residential EV charging
enables the network operator to maintain system reliability by respecting line
limits and having fewer undervoltage households and almost no households with
node violation even for higher levels of EV adoption as opposed to an
individual approach.

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## Appendix

### Preliminaries of power networks

Figure 4: Schematic of power distribution network with few residential EV
adopters.

Fig. 4 shows a typical power distribution network with few EV adopters. The
network is a tree rooted at the substation node and connects residences
through local pole top transformers. A branch originating from the substation
is also termed as a ‘feeder’. The power flowing along each edge results in
voltage drop between the nodes. Therefore, the node at the extreme end of the
feeder experiences the minimum voltage.

### Convergence of ADMM methodology

In the paper, we have discussed the need of a distributed methodology to
respect the privacy of consumer information. However, this alters the original
MILP problem into one QP and multiple MIQPs. Here, we discuss the deviation of
the optimal solution obtained by the distributed framework from that obtained
by a centralized approach. Note that the centralized approach involves solving
the MILP in (5).

Figure 5: Percentage deviation of the optimal solution obtained using the
proposed framework from the solution obtained by centrally solving (5)

Fig. 5 shows the percentage deviation in the optimal cost incurred by the EV
adopting residences. This is done for an adoption level of $50\%$ in ‘Com-A’
of the network. The solution obtained by the proposed distributed approach
converges with the solution obtained by the centralized approach for majority
of residences. For the other residences the deviation in the optimal cost lies
mostly below $5\%$ with one residence having a large deviation of $20\%$.
Therefore, we can conclude that the optimal solution converges for most of the
residences. The deviation is the price we pay to respect the privacy of
consumers while dealing with this problem.

### Additional Results

We provide the plots for line loading levels and node voltages for multiple
adoption levels in the two communities Fig. 6 and Fig. 7. In all the cases, we
note that the edges in the network are not loaded to their full capacities.
However, the voltages of multiple residences fall outside the acceptable
limits of $0.95-1.05$p.u.

Figure 6: Comparison of line loading level (edge flows) for residential EV
adoption of $30\%$ and $60\%$ in ‘Com-A’ of network. The EV adoption does not
significantly affect the line loading level for either of the optimization
methods.

Figure 7: Comparison of residence node voltages for residential EV adoption of
$30\%$ and $60\%$ in ‘Com-A’ of network. The EV adoption adversely affects the
node voltages during cheap electricity hours for the individual optimization
scenario.

Figure 8: Impact of residential EV charging is analyzed for three more
residential communities within the same network: ‘Com-C’(top), ‘Com-D’(middle)
and ‘Com-E’(bottom). The orange nodes shown in the network denote the
residences in the two communities. The individual optimization leads to
undervoltage (less than 0.95 p.u.) at a significant number of residences. This
can be avoided by using the proposed distributed optimization method even for
higher levels of EV adoption.

We further compare the distributed and individual optimization frameworks for
three more residential communities in the network. These are shown in Fig. 2.
We note that the impact on reliability of the network in terms of number of
residences experience undervoltage is different for each community in the
network. This proves the spatial dependency of network reliability on EV
adoption location.