Simulation Study for the Comparison of Power Flow Models for a Line

Distribution Network with Stochastic Load Demands

Mark Christianen

1 a

, Maria Vlasiou

1,2 b

and Bert Zwart

1,3

Eindhoven University of Technology, Eindhoven, The Netherlands

University of Twente, Enschede, The Netherlands

Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

Keywords:

Electric Vehicle Charging, Power Flow Models, Bandwidth-Sharing Networks.

Abstract:

We use simulation to compare different power ﬂow models in the process of charging electric vehicles (EVs)

by considering their random arrivals, their stochastic demand for energy at charging stations, and the char-

acteristics of the electricity distribution network. We assume the distribution network is a line with charging

stations located on it. We consider the Distﬂow and the Linearized Distﬂow power ﬂow models and we assume

that EVs arrive at the network with an exponential rate, have an exponential charging requirement, and that

voltage drops on the distribution network stay under control. We provide extensive numerical results investi-

gating the effect of using different power ﬂow models on the performance of the network.

1 INTRODUCTION

In recent years, the growing electricity consumption,

the active adoption of renewable generation, and the

energy transition result in congestion in the electric-

ity network. On one side, more companies use elec-

tricity for their production, more houses are heated

with heat pumps and more people drive electric cars.

On the other side, companies and citizens are gener-

ating more and more electricity from wind and sun,

which they mostly feed back to the electricity net-

work. This causes network capacity problems, or in

other words, congestion. This is illustrated in (Hoog-

steen et al., 2017), where the impact of the energy

transition on a real electricity network is evaluated.

Here, the authors showed that charging a small num-

ber of EVs is enough to cause a blackout in a neigh-

borhood. Therefore, it is imperative to study the per-

formance of the network under different power ﬂow

models, since these models are used for the design

and control of the network.

Irrespective of the power ﬂow model, it is impor-

tant to respect the physical laws of the network and

its constraints. In an electricity network, an impor-

tant constraint is the requirement of keeping voltage

losses, or in other words, the voltage drop, on a cable

https://orcid.org/0000-0002-9611-500X

https://orcid.org/0000-0002-0457-2925

in the network under control. These voltage losses

are caused by the physical properties of the cables in

the network. Keeping the voltage losses under con-

trol ensures that every user in the network receives

safe and reliable power at a voltage that is within

some standard range, which varies from one country

to another (Kersting, 2018). For example, according

to Dutch law, the voltage drop in a distribution net-

work, a small part of the electricity network, is not al-

lowed to be more than 4.5% (van Westering and Hel-

lendoorn, 2020).

In this paper, we consider the stochastic process

of charging EVs in a neighborhood such that the volt-

age drop in the distribution network stays under con-

trol. In this process, the stochasticity comes from ran-

dom arrivals (at parking lots with charging stations)

and charging requirements of EVs, while the power

allocated to each EV in the network depends on the

number of EVs and the corresponding location of EVs

that are currently charging in the network. We model

this process as a queue, with EVs representing jobs,

charging stations classiﬁed as servers, and the ser-

vice being delivered as the power supplied to EVs,

constrained by physical laws and network constraints.

The particular queuing model that we employ can be

seen as falling under a general class of queuing net-

works called bandwidth-sharing networks. To model

the physical laws and network constraints, we use two

Christianen, M., Vlasiou, M. and Zwart, B.

Simulation Study for the Comparison of Power Flow Models for a Line Distribution Network with Stochastic Load Demands.

DOI: 10.5220/0011670600003396

In Proceedings of the 12th International Conference on Operations Research and Enterprise Systems (ICORES 2023), pages 167-174

ISBN: 978-989-758-627-9; ISSN: 2184-4372

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

167

approximations of the alternating current (AC) power

ﬂow equations (Molzahn and Hiskens, 2019), i.e.; we

study the Distﬂow and a linearized version of the Dis-

tﬂow model, by ignoring some power losses, that is

called the Linearized Distﬂow (Baran and Wu, 1989b;

Baran and Wu, 1989a).

In this simulation study, we assess the accuracy

and effectiveness of the Linearized Distﬂow model

compared to the Distﬂow model. Our goal is to com-

pare the different power ﬂow models on the perfor-

mance of the stochastic process of EV charging, by

the mean number of EVs and the mean charging time

of each EV in the network. Furthermore, we gain in-

sights into the behavior of the network by including

variability in the distribution of the arrival rates to dif-

ferent parking lots.

The contributions we obtained from analyzing

such a network are summarized as follows. We ob-

serve that the performance of the Linearized Distﬂow

model is comparable to the Distﬂow model, i.e. the

mean number of EVs and the mean charging time of

an EV under both power ﬂow models are similar and

the relative difference between critical arrival rates

(the speciﬁc arrival rates under both power ﬂow mod-

els for which the mean number of EVs and the mean

charging time of an EV grow to its maximum capac-

ity) is below 5%. Thus, our ﬁrst contribution is that

we obtain more evidence that using the much simpler

Linearized Distﬂow model is a valid and accurate ap-

proximation of the Distﬂow model, even if the system

is highly heterogeneous. Namely, in our numerical

examples, we consider cases where one station has

almost all the load of the system. Even for such het-

erogeneous cases, the performance of the network is

the same under the Linearized Distﬂow and the Dis-

tﬂow model. In other words, we do not lose much

accuracy from a performance perspective by ignoring

some power losses. The second contribution is rather

surprising. It is very well known from queuing the-

ory that variability in the network causes worse per-

formance. However, what is surprising in this case is

that the network does not perform symmetrically un-

der the same loads. If the load of an individual park-

ing lot is way larger than the other loads of the other

parking lots, the performance of the network is differ-

ent from the performance of the network if the (same)

largest load is put on another parking lot.

The structure of the paper is as follows. In Sec-

tion 2, we provide a literature review on work that has

been done on stability for EV-charging and speciﬁ-

cally in the context of a queuing network, and a com-

parison between the Linearized Distﬂow model and

the Distﬂow power ﬂow model. In Section 3, we pro-

vide a detailed model description. In particular, we

introduce the queuing network, the constraints and as-

sumptions of the electrical distribution network, and

the power ﬂow models we consider. In Section 4, we

present several numerical experiments showing the

accuracy and effectiveness of the Linearized Distﬂow

model and the effect of including variability in the dis-

tribution of the arrival rates to the performance of the

network.

2 LITERATURE REVIEW

First, we discuss the literature on stability for EV-

charging and stability in queuing networks for EV-

charging which relate to the network capacity prob-

lems that we mentioned earlier. Second, we provide

literature that compares the Distﬂow model and the

Linearized Distﬂow model.

The term stability, from a queuing perspective,

means the stability of the queuing model and is de-

ﬁned as the positive recurrence of a Markov process.

Informally speaking, stability means the ability of all

queues to complete the service of all jobs, without the

number of outstanding building up inﬁnitely. In our

simulations, we consider a ﬁnite number of parking

spaces at all parking lots, which implies that the queu-

ing model is always stable. Therefore, we investigate

the performance of the network in terms of the mean

number of EVs and the mean charging time of an EV

in the network but still discuss the literature on stabil-

ity results.

Stability Results for EV-Charging. The literature

on stability results for EV charging is limited to nu-

merical experiments. An early paper on stability anal-

ysis in EV-charging is by Huang et al (Huang et al.,

2013). The authors present a new quasi-Monte Carlo

stability analysis method to assess the dynamic ef-

fects of plug-in electric vehicles in power systems.

They conclude that improvements in stability control

are worth further study since the number of EVs is

growing. Other simulation studies are conducted to

obtain stability conditions. In (de Hoog et al., 2014),

the authors explore the constraint of requiring a mini-

mum voltage to charge EVs throughout a network and

demonstrate that the physical locations of individual

demand and generation of energy play a signiﬁcant

role in determining whether voltages throughout the

network remain within required limits or not. Simi-

larly, in (Zhang et al., 2016), the impact of charging

EVs on the voltage stability of the distribution net-

work is simulated and analyzed. The simulation re-

sults show that the voltage stability is related to the

individual loads, total load in the network, and physi-

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

168

cal properties of the network. In (Ul-Haq et al., 2015),

simulations are performed on another test network.

For this network, different charging strategies are im-

plemented and it is shown that for some scenarios this

can cause signiﬁcant voltage instability. Last, in (Deb

et al., 2018), the authors perform a numerical study

on a speciﬁc test system, where they investigate the

impact of a single EV charging station on the volt-

age stability, power losses, and economic losses of the

distribution network. Here, it is also observed that the

location of the EV charging station is important in the

smooth operation of the grid.

Stability Conditions for Queuing Networks. The

literature of the stability of queuing networks for EV

charging is very limited. It has been studied in (Car-

valho et al., 2015). Here, the authors ﬁnd by simu-

lation that there is a threshold on the arrival rates of

EVs, such that if the actual arrival rate is greater than

this threshold, some cars have to wait for increasingly

long times to fully charge. The ﬁrst analytical study

is (Christianen et al., 2022), where the authors com-

pare these thresholds on the arrival rates under dif-

ferent power ﬂow models and compute the difference

between these rates explicitly as the number of park-

ing lots grows to inﬁnity.

Comparison of the Linearized Distﬂow and Dis-

tﬂow Model. The practical use of the Linearized

Distﬂow compared to the Distﬂow model is based on

the assumption that power losses on cables are typi-

cally small. It has been shown experimentally that this

only introduces a small relative error, on the order of

1% (Farivar et al., 2013). However, these small rela-

tive errors may be exaggerated when used in a com-

plex stochastic process and this is what we examine

in this paper. Multiple other numerical studies have

been conducted to verify the accuracy and effective-

ness of the Linearized Distﬂow model (Baran and Wu,

1989b; Wang et al., 2014a; Chen et al., 2016; Tan

et al., 2013; Yuan et al., 2016; Wang et al., 2014b;

Yeh et al., 2012; Li et al., 2019; Cao et al., 2019).

3 MODEL DESCRIPTION AND

FORMULATION

This section describes the main components of the

EV-charging model, i.e.; we describe the characteris-

tics of the queuing, the distribution network, and the

power ﬂow models.

3.1 Queuing Model of EV-Charging

We use a queuing model to study the process of charg-

ing EVs in a distribution network. EVs referred to

as jobs require service. This service is delivered by

charging stations, referred to as servers and the ser-

vice being delivered is the power supplied to EVs.

At all parking lots, there is one charging station with

K > 0 parking spaces, and each parking space has its

own EV charger.

Thus, in the queuing system, we consider N

single-server queues, each having its own arrival

stream of jobs. Denote by X(t) = (X

(t), . . . , X

(t))

the vector giving the number of jobs at each queue

at time t. At all parking lots, all EVs arrive inde-

pendently according to Poisson processes with rate

, i = 1, . . . , N and have independent service require-

ments which are Exp(1) random variables. If all park-

ing spaces are occupied, a newly arriving EV does not

enter the system but is assumed to leave immediately.

At each queue, all jobs are served simultaneously

and start service immediately (there is no queuing).

Furthermore, each job receives an equal fraction of

the service capacity, i.e. the charging rate, allocated

to a queue. Denote by

p(t) = ( ˜p

(t), . . . , ˜p

(t)) the

vector of service capacities allocated to each queue

at time t. From now on, for simplicity, we drop the

dependence on time t from the notation. For example,

we write X

and ˜p

instead of X

(t) and ˜p

(t).

Service capacities are state-dependent and subject

to changes to the current vector X = (X

, . . . , X

) of

number of jobs. For each state of the system, i.e. a

given number of EVs charging at each parking lot,

we assume that the charging rates

p are the unique

solution of the optimization problem

max

∑

log



˜p



, (1)

which are called proportional fair allocations. For

the optimization problem, the feasible region can take

many forms and depends heavily on the power ﬂow

model that is used. In Section 3.3, we discuss the

feasible regions for both power ﬂow models in more

detail.

We can then represent the number of electric ve-

hicles charging at every station as an N-dimensional

continuous-time Markov process. The evolution of

the queue at node j is given by

(t) → X

(t) + 1 at rate λ

and

(t) → X

(t) − 1 at rate ˜p

Simulation Study for the Comparison of Power Flow Models for a Line Distribution Network with Stochastic Load Demands

169

3.2 Distribution Network Model

As distribution networks are typically radial (Avek-

louris et al., 2019), we model the distribution network

as a directed graph G = (I , E), where we denote by

I = {0, . . . , N} the set of nodes and by E its set of

directed edges, assuming that node 0 is the root node.

We assume that G has a line topology. Each edge

j−1, j

∈ E represents a line connecting nodes j − 1

and j where node j is further away from the root node

than node j −1. Each edge ε

j−1, j

∈ E is characterized

by the impedance z = r + ix, where r, x ≥ 0 denote the

resistance and reactance along the lines, respectively.

We make the following natural assumption, given that

r >> x in distribution networks (Khatod et al., 2006;

Tonso et al., 2005).

Assumption 3.1. All edges have the same resistance

value r > 0 and reactance value x = 0.

Furthermore, let ˜s

= ˜p

+ i ˜q

be the complex

power consumption at node j. Here, ˜p

and ˜q

denote

the active and reactive power consumption at node

j, respectively. By convention, a positive active (re-

active) power term corresponds to consuming active

(reactive) power. Since EVs can only consume active

power (Carvalho et al., 2015), it is natural to make the

following assumption.

Assumption 3.2. The active power ˜p

is non-

negative and the reactive power ˜q

is zero at all charg-

ing stations j ∈ I .

Let

denote the voltage at node j. Given As-

sumptions 3.1 and 3.2, the voltages at each node j,

, can be chosen to have zero imaginary components

(Carvalho et al., 2015; Aveklouris et al., 2019). For

each ε

j−1, j

∈ E , let I

j−1, j

be the complex current and

j−1, j

+ i

j−1, j

be the complex power ﬂow-

ing from node j − 1 to j. Here,

j−1, j

and

j−1, j

de-

note the active and reactive power ﬂowing from node

j − 1 to j. The model is illustrated in Figure 1.

Voltage Drop Constraint. The distribution net-

work constraints, that is in our case only the voltage

drop constraint, represent the feasible region of (1)

and are described by a set C. The set C is contained

in an N-dimensional vector space and represents fea-

sible power allocations. In our setting, a power al-

location is feasible if the maximal voltage drop; i.e.,

the relative difference between the base voltage

and the minimal voltage in all buses between the root

node and any other node is bounded by some value

∆ ∈ (0,

]. Thus, the distribution network constraints

can be described as

C :=



p :

− min

1≤ j≤N

≤ ∆



, (2)

where 0 < ∆ ≤

. In Section 3.3, we give more con-

crete deﬁnitions of the constraint set C for each power

ﬂow model.

3.3 Power Flow Models

We introduce two commonly used models to repre-

sent the power ﬂow that is valid for radial systems;

i.e., systems where all charging stations have only one

(and the same) source of supply. They are called the

Distﬂow and Linearized Distﬂow model (Low, 2014;

Baran and Wu, 1989b). Both models are valid when

the underlying network topology is a tree, which is the

case in this paper (as we consider a line topology).

Given the impedance r, the voltage at the root

node

and the power consumptions ˜p

, j = 1, . . . , N,

both power ﬂow models satisfy three relations. First,

we have power balance at each node:

j−1, j

− r



j−1, j



= ˜s

j, j+1

, j ∈ I \{0}. (3)

Here, on the one hand, the quantity r|I

j−1, j

rep-

resents line loss so that

j−1, j

− r|I

j−1, j

is the

receiving-end complex power at node j from node

j − 1. On the other hand, the delivering-end complex

power is the sum of the consumed power at node j

and the complex power ﬂowing from node j to node

j + 1. Second, by Ohm’s law, we have for each edge,

j−1

−

= rI

j−1, j

, ε

j−1, j

∈ E , (4)

and third, due to the deﬁnition of complex power, we

have for each edge,

j−1, j

j−1

j−1, j

, ε

j−1, j

∈ E . (5)

However, for a line topology, the distribution net-

work constraints in (2), using (3)–(5), can be rewrit-

ten for both power ﬂow models (Christianen et al.,

2022). Following the approach in (Christianen et al.,

2022), the distribution network constraints under the

Distﬂow model C

reduce to



p :

≤

1 − ∆



, (6)

where

can be found recursively by

N−1

= 1 + r ˜p

, (7)

j−1

= 2

−

j+1

r ˜p

, j = 1, . . . , N − 1. (8)

In the Linearized Distﬂow model, it is assumed that

the active power losses r|I

j−1, j

are much smaller

than the active power ﬂows

j−1, j

. In other words,

the Linearized Distﬂow model neglects the loss

terms associated with the squared current magnitudes

j−1, j

. In that case, the distribution network con-

straints under the Linearized Distﬂow C

reduce to

(

p : 2r

∑

j=1

∑

k= j

˜p

≤

∆(2 − ∆)

(1 − ∆)

)

. (9)

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

170

Root node

(

i j

, I

i j

, r)

Figure 1: Line network with N charging stations and arriving vehicles at rate λ

, i ∈ {1, . . . , N}.

4 NUMERICAL STUDY

In the previous section, we discussed our model for

the EV-charging process. Here, we obtain general in-

sights into the performance of the model by simula-

tion on a large range of parameter settings. Moreover,

this allows us to compare the behavior of the model

under the Distﬂow model and the Linearized Distﬂow

model. We vary the total arrival rate to the network

and the distribution of the total arrival rate to differ-

ent parking lots. We focus on the effect of the mean

number of EVs in the network and the mean charging

time of an EV, possibly at each parking lot.

4.1 Critical Arrival Rate

To control the network, we observe that there is a crit-

ical arrival rate λ

(≈ 0.18) if the arrival rate to each

parking lot is assumed to be the same. At every park-

ing lot, the mean number of EVs and the mean charg-

ing time of an EV grow to maximum capacity as soon

as the actual arrival rate is greater than the critical ar-

rival rate. See Figure 2, where the mean number of

EVs and the mean charging time of an EV are plotted

versus the individual arrival rate to each parking lot.

We observe the mean number of EVs and the mean

charging time of an EV for both power ﬂow models

at each parking lot. We ﬁx the number of parking lots

at N = 2, the resistance for each cable at r = 0.1, the

maximal capacity at each parking lot at K = 100 and

the parameter to control the voltage drop at ∆ = 0.05

for the Distﬂow model (dashed) and the Linearized

Distﬂow model (solid) at parking lot 1 (blue) and at

parking lot 2 (red). Although the total arrival rate is

varied, the solid curves are close to the dashed lines

for the two parking lots. Moreover, the solid curves

are below the dashed curves. This is to be expected;

in (Christianen et al., 2022), the authors observed that

the Linearized Distﬂow power ﬂow model allows for

too optimistic arrival rates since the Linearized Dist-

ﬂow model overestimates the voltages in comparison

with the voltages given by the Distﬂow model. From

this, we observe that the allocated power to each park-

ing lot is higher under the Linearized Distﬂow than

the allocated power under the Distﬂow model. Higher

allocated power means faster charging. Hence, EVs

leave the parking lots faster and the mean number of

EVs charging is lower.

This statement is reinforced by Figure 3, where

the relative difference in the mean number of EVs

in the network in percentages between the Distﬂow

model and the Linearized Distﬂow model is plotted

versus the total arrival rate to the parking lots. It is still

assumed that N = 2, r = 0.1, K = 100 and ∆ = 0.05

for all of these lines. For the blue curve, we have

equal arrival rates for all parking lots. From Figure 3,

it is apparent that the relative difference in the mean

number of EVs between both power ﬂow models is

below 5% for almost all total arrival rates to the net-

work. However, when the total arrival rate is close to

two times the critical arrival rate λ

(≈ 0.36), we have

already seen in Figure 2 that the mean number of cars

for both power ﬂow models seem to reach their max-

imum capacity and that this happens slightly earlier

for the Distﬂow model than for the Linearized Dist-

ﬂow model. This causes the high relative difference

in the mean number of EVs for both power ﬂow mod-

els.

4.2 Variability of the Distribution of the

Total Arrival Rate

The previous section brings us naturally to the discus-

sion of adding variability to the distribution of the to-

tal arrival rate. Instead of assuming equal arrival rates

for all parking lots, we vary the fraction of EVs that

arrive at each parking lot for a wide range of the total

arrival rate to the network.

For all combinations of total arrival rates and frac-

tions of EVs that arrive at each parking lot, the heat

map of the mean number of cars has an interesting

structure. In Figure 4, we show the total mean num-

ber of EVs in the network as a function of the total

arrival rate and the fractions of EVs that arrive at each

parking lot. As we see in Figure 4, the mean num-

ber of EVs has a non-symmetric structure. When we

increase the fraction of EVs that arrive at parking lot

1 (and thus decrease the fraction of EVs that arrive

at parking lot 2) from the situation of equal arrival

Simulation Study for the Comparison of Power Flow Models for a Line Distribution Network with Stochastic Load Demands

171

(a) Mean number of cars.

(b) Mean charging time.

Figure 2: Performance measures vs. the individual arrival rate per parking lot for the Distﬂow model (dashed) and the

Linearized Distﬂow model (solid) at parking lot 1 (blue) and at parking lot 2 (red).

Figure 3: Relative difference in mean number of EVs be-

tween both power ﬂow models.

Figure 4: Mean number of cars.

rates, the total mean number of EVs in the network

decreases faster than the total number of EVs in the

network decreases when we increase the fraction of

EVs that arrive at parking lot 2. This is natural given

the total available power that can be allocated to each

parking lot. Due to the power loss on the cables, the

available power that can be allocated to parking lot 1

is approximate twice the available power that can be

allocated to parking lot 2. We compare the following

two situations; one where we have a given number of

EVs charging at parking lot 1 and no EVs charging at

parking lot 2 (which corresponds to a situation where

the fraction of EVs that arrive at parking lot 1 is high),

and one where we have the same given number of EVs

charging at parking lot 2 and no EVs charging at park-

ing lot 1 (which corresponds to a situation where the

fraction of EVs that arrive to parking lot 2 is high).

Since the allocated power to parking lot 1 in the ﬁrst

situation is higher than the allocated power to parking

lot 2 in the second situation, the mean number of EVs

at parking lot 1 tends to be smaller than the number

of EVs at parking lot 2. Moreover, if we consider a

ﬁxed total arrival rate, e.g., λ

+ λ

= 0.8, the vari-

ability of the distribution of the total arrival rate has

a small inﬂuence on the mean number of EVs for a

wide range of the ratio of arrival rates. In Figure 5a,

where the relative difference between the total mean

number of EVs given any distribution of the total ar-

rival over both parking lots and the total mean number

of EVs given equal arrival rates to both parking lots is

plotted, we observe that this relative difference is be-

low 5% if the fraction of EVs that arrive at parking lot

1 range between 20% and 60%.

Another observation on the heat map of the mean

number of cars is that there is a clear distinction be-

tween networks that have reached their capacity or

not. In the blue region, the mean number of EVs

is relatively low. However, in the green and yellow

regions, the mean number of EVs is relatively high

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

172

(a) Relative difference for λ

+ λ

= 0.8.

(b) Approximation of critical arrival rates.

Figure 5: Observations based on the heat map of the mean number of cars.

and close to its maximum capacity, namely 200 EVs.

Moreover, the green region indicates a network where

the number of EVs charging at either one of the park-

ing lots reaches its capacity. The yellow region in-

dicates a network where the number of EVs at both

parking lots reaches its capacity.

In Figure 5b, we also plot a rough approximation

of the evolution of the critical arrival rates under both

power ﬂow models as we vary the fraction of EVs that

arrive at parking lot 1 that we obtained as follows. As

can be seen in Figure 2, there is a steep increase in

the mean number of EVs and the mean charging time

of an EV at a certain point. Using a ﬁne grid search,

we ﬁnd the arrival rate for which the absolute differ-

ence between subsequent measurements in the mean

number of cars is the largest. In Figure 5b, we ob-

serve that as the fraction of EVs that arrive at park-

ing lot 1 increases, the critical arrival rates for both

power ﬂow models decrease. Furthermore, the crit-

ical arrival rates for the Distﬂow model are always

below the critical arrival rates for the Linearized Dist-

ﬂow model; a behavior that we observed in the setting

of equal arrival rates for both parking lots already.

Adding variability to the distribution of the total

arrival also inﬂuences the relative difference of the to-

tal mean number of EVs in the network between both

power ﬂow models. This effect is observed in Figure

3, where we see that for a fraction of EVs that arrive

at parking lot 1 that deviates from the nominal situ-

ation of equal arrival rates, the maximal relative dif-

ference between the two power ﬂow models is lower.

This has to do with parking lots reaching their maxi-

mum capacity. In the situation that we have different

arrival rates to the parking lots, the mean number of

EVs grows to maximum capacity only at one park-

ing lot while in the situation of equal arrival rates, the

mean number of EVs grows to maximum capacity at

both parking lots. The relative difference in the mean

number of EVs for both power ﬂow models is highest

in these cases and therefore the maximal relative dif-

ference between the Linearized Distﬂow and the Dis-

tﬂow model is highest in the situation of equal arrival

rates. Furthermore, also for other distributions of the

total arrival rate than equal arrival rates to both park-

ing lots (red and yellow curves), we observe that the

relative difference is below 5% for almost all total ar-

rival rates to the network, except around those values

for the total arrival rate where we turn from a network

with a relatively low mean number of EVs to a net-

work with a relatively high number of EVs. For such

values, we could compute the critical arrival rate for

the Linearized Distﬂow model and adjust by a small

percentage to compensate for ignoring power losses,

since the critical arrival rates under both power ﬂow

models are close to each other, see Figure 5b.

In summary, the performance of both power ﬂow

models is approximately the same. Simulation results

show that for a wide range of total arrival rates to the

network the relative difference in the mean number of

EVs between the Distﬂow model and the Linearized

Distﬂow model is below 5%. Furthermore, the critical

arrival rates under both power ﬂow models are close

to each other. Moreover, we can say that the variabil-

ity in the distribution of the total arrival to parking

lots for heavily loaded instances, as long as the frac-

tion of EVs that arrive at parking lot 1 range between

20% and 60%, does not inﬂuence the performance of

the network much, in the sense that the mean num-

ber of EVs and the mean charging time of an EV are

comparable to the mean number of EVs and the mean

charging time of an EV in the case where the arrival

rates to all parking lots are the same, respectively.

Simulation Study for the Comparison of Power Flow Models for a Line Distribution Network with Stochastic Load Demands

173

REFERENCES

Aveklouris, A., Vlasiou, M., and Zwart, B. (2019). A

stochastic resource-sharing network for electric vehi-

cle charging. IEEE Transactions on Control of Net-

work Systems, 6(3):1050–1061.

Baran, M. E. and Wu, F. F. (1989a). Optimal capacitor

placement on radial distribution systems. IEEE Trans-

actions on Power Delivery, 4(1):725–734.

Baran, M. E. and Wu, F. F. (1989b). Optimal Sizing of

Capacitors Placed On a Radial Distribution System.

IEEE Transactions on Control of Network Systems,

4(1):735–743.

Cao, Y., Wei, W., Wang, J., Mei, S., Shaﬁe-Khah, M., and

Catalao, J. P. (2019). Capacity Planning of Energy

Hub in Multi-carrier Energy Networks: A Data-driven

Robust Stochastic Programming Approach. IEEE

Power and Energy Society General Meeting, (1):3–14.

Carvalho, R., Buzna, L., Gibbens, R., and Kelly, F. (2015).

Critical behaviour in charging of electric vehicles.

New Journal of Physics, 17(9):95001.

Chen, C., Wang, J., Qiu, F., and Zhao, D. (2016). Re-

silient Distribution System by Microgrids Formation

after Natural Disasters. IEEE Transactions on Smart

Grid, 7(2):958–966.

Christianen, M., Cruise, J., Janssen, A., Shneer, S., Vla-

siou, M., and Zwart, B. (2022). Comparison of

stability regions for a line distribution network with

stochastic load demands. Preprint available at:

https://arxiv.org/abs/2201.06405.

de Hoog, J., Muenzel, V., Jayasuriya, D. C., Alpcan, T.,

Brazil, M., Thomas, D. A., Mareels, I., Dahlenburg,

G., and Jegatheesan, R. (2014). The importance of

spatial distribution when analysing the impact of elec-

tric vehicles on voltage stability in distribution net-

works. Energy Systems, 6(1):63–84.

Deb, S., Tammi, K., Kalita, K., and Mahanta, P. (2018). Im-

pact of electric vehicle charging station load on distri-

bution network. Energies, 11(1):1–25.

Farivar, M., Chen, L., and Low, S. (2013). Equilibrium

and dynamics of local voltage control in distribution

systems. Proceedings of the IEEE Conference on De-

cision and Control, pages 4329–4334.

Hoogsteen, G., Molderink, A., Hurink, J. L., Smit, G. J.,

Kootstra, B., and Schuring, F. (2017). Charging elec-

tric vehicles, baking pizzas, and melting a fuse in

Lochem. CIRED - Open Access Proceedings Journal,

2017(1):1629–1633.

Huang, H., Chung, C. Y., Chan, K. W., and Chen, H. (2013).

Quasi-Monte Carlo based probabilistic small signal

stability analysis for power systems with plug-in elec-

tric vehicle and wind power integration. IEEE Trans-

actions on Power Systems, 28(3):3335–3343.

Kersting, W. (2018). Distribution System Modeling and

Analysis. CRC Press, fourth edition.

Khatod, D. K., Pant, V., and Sharma, J. (2006). A novel ap-

proach for sensitivity calculations in the radial distri-

bution system. IEEE Transactions on Power Delivery,

21(4):2048–2057.

Li, R., Wei, W., Mei, S., Hu, Q., and Wu, Q. (2019). Partic-

ipation of an Energy Hub in Electricity and Heat Dis-

tribution Markets: An MPEC Approach. IEEE Trans-

actions on Smart Grid, 10(4):3641–3653.

Low, S. H. (2014). Convex relaxation of optimal power

ﬂow - Part {I}: Formulations and equivalence. IEEE

Transactions on Control of Network Systems, 1:15–27.

Molzahn, D. and Hiskens, I. (2019). A Survey of Relax-

ations and Approximations of the Power Flow Equa-

tions. A Survey of Relaxations and Approximations of

the Power Flow Equations, 4(1):1–221.

Tan, S., Xu, J. X., and Panda, S. K. (2013). Optimization of

distribution network incorporating distributed genera-

tors: An integrated approach. IEEE Transactions on

Power Systems, 28(3):2421–2432.

Tonso, M., Morren, J., De Haan, S. W., and Ferreira, J. A.

(2005). Variable inductor for voltage control in dis-

tribution networks. 2005 European Conference on

Power Electronics and Applications, 2005.

Ul-Haq, A., Cecati, C., Strunz, K., and Abbasi, E. (2015).

Impact of electric hehicle charging on voltage unbal-

ance in an urban distribution network. Intelligent In-

dustrial Systems, 1(1):51–60.

van Westering, W. and Hellendoorn, H. (2020). Low voltage

power grid congestion reduction using a community

battery: Design principles, control and experimental

validation. International Journal of Electrical Power

and Energy Systems, 114:105349.

Wang, Z., Chen, B., Wang, J., Kim, J., and Begovic,

M. M. (2014a). Robust optimization based optimal

DG placement in microgrids. IEEE Transactions on

Smart Grid, 5(5):2173–2182.

Wang, Z., Chen, H., Wang, J., and Begovic, M. (2014b).

Inverter-less hybrid voltage/var control for distribu-

tion circuits with photovoltaic generators. IEEE

Transactions on Smart Grid, 5(6):2718–2728.

Yeh, H. G., Gayme, D. F., and Low, S. H. (2012). Adap-

tive VAR control for distribution circuits with photo-

voltaic generators. IEEE Transactions on Power Sys-

tems, 27(3):1656–1663.

Yuan, W., Wang, J., Qiu, F., Chen, C., Kang, C., and Zeng,

B. (2016). Robust Optimization-Based Resilient Dis-

tribution Network Planning Against Natural Disasters.

IEEE Transactions on Smart Grid, 7(6):2817–2826.

Zhang, Y., Song, X., Gao, F., and Li, J. (2016). Re-

search of voltage stability analysis method in distribu-

tion power system with plug-in electric vehicle. Asia-

Paciﬁc Power and Energy Engineering Conference,

APPEEC, 2016-Decem(51177152):1501–1507.

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

174