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Enhancing the efficiency of charging & parking processes for Autonomous Mobile Robot fleets: A simulative evaluation

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  • Tesla Manufacutring Brandenburg SE

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The allocation of tasks to Autonomous Mobile Robots in a production setting in combination with the most efficient parking and charging processes are the focus of this paper. This study presents a simulative evaluation of the theoretical allocation methods developed in Selmair and Maurer (2020) combined with either hard or dynamic availability rules to ascertain the most efficient parameters of an Autonomous Mobile Robot System. In order to quantify this efficiency, the following Key Performance Indicator (KPI) were considered: number of delayed orders, driven fleet metres and the percentage of available Autonomous Mobile Robot as determined by their state of charge. Additionally, as an alternative energy source, a fast-charging battery developed by Battery Streak Inc. was included in this study. The results show that, in comparison to a conventional and commonly used trivial strategy, our developed strategies provide superior results in terms of the relevant KPI.
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Journal of Power Sources 521 (2022) 230894
Available online 8 January 2022
0378-7753/© 2022 Elsevier B.V. All rights reserved.
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Journal of Power Sources
journal homepage: www.elsevier.com/locate/jpowsour
Enhancing the efficiency of charging & parking processes for Autonomous
Mobile Robot fleets: A simulative evaluation
Maximilian Selmair a,, Tobias Maurer b, Chun-Han Lai c, David Grant c
aTesla Manufacturing Brandenburg SE, Berlin, Germany
bRegensburg University of Applied Sciences, Regensburg, Germany
cBattery Streak, Inc., Newbury Park, CA 91320, United States
HIGHLIGHTS
Material flow and efficient allocation of parking and charging stations of AMRs.
Novel strategies for charging & parking AMRs are introduced and evaluated.
Strategies are compared by means of a single controlled agent-based simulation.
New strategies are superior based on availability, delays, and traffic density.
Exploration of suitability of fast-charging alternative energy source.
ARTICLE INFO
Keywords:
AMR
Autonomous Mobile Robots
Battery charging
Vehicle Fleet Charging
ABSTRACT
The allocation of tasks to Autonomous Mobile Robots in a production setting in combination with the most
efficient parking and charging processes are the focus of this paper. This study presents a simulative evaluation
of the theoretical allocation methods developed in Selmair and Maurer (2020) combined with either hard or
dynamic availability rules to ascertain the most efficient parameters of an Autonomous Mobile Robot System.
In order to quantify this efficiency, the following Key Performance Indicator (KPI) were considered: number of
delayed orders, driven fleet metres and the percentage of available Autonomous Mobile Robot as determined
by their state of charge. Additionally, as an alternative energy source, a fast-charging battery developed by
Battery Streak Inc. was included in this study. The results show that, in comparison to a conventional and
commonly used trivial strategy, our developed strategies provide superior results in terms of the relevant KPI.
1. Introduction
To meet the increasing demand for individualised goods and tighter
production schedules, production-related logistics must become more
and more flexible [1]. It follows that internal logistics have to adapt
efficiently to the increasing dynamics and complexity of production
processes. The visionary concept of Logistics 4.0 is proposed to offer a
wide range of solutions for this purpose [2,3]. The use of Autonomous
Mobile Robots (AMRs), in particular, is gaining attention and has
become the focal point of researchers across different industries [47].
AMRs are readily used in a variety of application environments
due to easy installation and high mobility without any need for phys-
ical guiding systems [8]. With increasing environmental flexibility,
AMRs can be used for scenarios of a greater range [5]. Taking into
consideration the progressing flexibility and technological advances,
Autonomous Mobile Robots have the potential to play an important role
Corresponding author.
E-mail address: mselmair@tesla.com (M. Selmair).
URL: https://maximilian.selmair.de (M. Selmair).
in factory logistics [3]. Yet, it is proposed that the organisational com-
plexity increases alongside with greater flexibility of application [9].
Besides the apparent problem of efficient task processing, other
issues relevant to the resource efficient operation of the entire system
are still to be resolved, and more specifically, focus on scheduling the
charging and parking tasks of the AMRs. Although this issue may seem
to be trivial for a limited number of robots, it does become increasingly
complex for large-scale and highly utilised systems. By choosing an
optimal strategy, the AMR fleet can be reduced to a minimum number
of AMRs and thus fewer resources are necessary. The lower number
of AMRs results in less traffic density, which in turn leads to fewer
traffic jams and blocked pathways [5]. As such, the charging strategy is
proposed to be an important aspect and has the potential to improve the
https://doi.org/10.1016/j.jpowsour.2021.230894
Received 24 August 2021; Received in revised form 15 November 2021; Accepted 8 December 2021
Journal of Power Sources 521 (2022) 230894
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M. Selmair et al.
Fig. 1. BMW’s iw.hub in its natural habitat.
total performance of such a material flow system [10]. Unfortunately,
this topic has not been explored in great detail so far [11].
Therefore, this paper evaluates existing methods discussed in the
relevant literature, and proposes concepts to improve operation effi-
ciency among the tasks of the AMRs. A simulation study was per-
formed to compare all previously developed allocation methods, as
described in [12], in order to generate recommendations for future
implementation.
More specifically, four different Allocation Methods in combination
with two Availability Rules are applied to the same production scenario
with set parameters regarding this transport of materials. The strategies
that result from combining the allocation methods and availability rules
are evaluated by means of the following KPIs: delays of tasks, the
average SOC of the AMR fleet, traffic density, availability rate of the
AMR fleet and the utilisation of charging stations. Furthermore, we
have added a further variable, a newly developed fast charging battery
to provide a promising outlook for future developments.
This paper is structured as follows: in Section 2, we outline the
requirements that provide the theoretical framework of this study.
Section 3provides a review of the literature relevant to this study.
The strategies that we have developed are introduced in Section 4.
Sections 5–7 constitute the core of this paper, as these sections describe
the experimental parameters, the simulation itself as well as the results.
The final Section 8discuss the study’s results and present a conclusion.
2. General requirements
This section describes the parameters of our study, stipulated by
the conditions of a research project undertaken by the BMW Group,
to which we are contributing. For the automatisation of the internal
material flow, the BMW Group has developed the so-called iw.hub (see
Fig. 1), designed to substitute common tucker trains which currently
play a central role in the automotive material flow. The industrial
use-case of BMW Group specifies the following requirements:
1. Every AMR can carry one load carrier at a time.
2. All transportation requests are issued randomly and therefore
unknown prior to their receipt.
3. AMRs without a task are required to either recharge or park.
2.1. Optimisation objective and KPIs
The overall goal of an automotive transport AMR system is to
complete all tasks on time. This requirement is considered to be a hard
constraint, which is non-negotiable to the detriment of any other KPIs.
That is, in order to meet this requirement, other resource intensive
variables are minimised to save resources. An important resource in
an automotive production environment are the pathways. These are
not only utilised by AMRs, but also by forklifts, tow trains, bicycles
and pedestrians. In order to maximise their availability for purposes
other than AMRs, the subsequent charging and parking strategy was
developed.
Additional KPIs applied to evaluate the quality of loading strategies
in terms of their flexibility, efficiency and scalability will be introduced
below. In this context, flexibility is defined as the availability of as
many AMRs as possible at any time, allowing for an immediate reaction
to short notice events, such as highly-prioritised urgent tasks.
In order to assess the efficiency of a charging strategy, it is necessary
to take into account the non-linear quality of the charging process. That
is, the amount of time required to recharge a battery to a predetermined
level depends on the initial SOC. This KPI can be evaluated by accessing
the utilisation records of the charging stations.
2.2. Energy source
The iw.hub is supplied with energy from a lithium-ion battery mod-
ule from BMW’s electric car i3, which consists of eight such modules.
Each module contains twelve battery cells and has 48 V as well as a
capacity of 120 Ah. The time required to fully charge one module by
means of a common method like Constant-Current-Constant-Voltage
(CCCV) is 3.2 h. However, fast charge methods are likely to reduce this
duration substantially. The charging current for this method decreases
from 38 A to 36 A. CCCV and also other fast charge methods have in
common that they are able to charge faster in a relatively small SOC
than in a high one [1315].
Lithium-ion batteries have a number of advantageous properties
relevant to the practical application of AMRs. Especially the minimal
negative memory effects of short charging periods over a few minutes
should be emphasised. The memory effect refers to a loss of capacity
which results from frequent partial discharges. This feature allows for
efficient, short charging processes and continuous use over 24 h [16,
17]. More advantageous properties are, e. g. the long life-cycle, low
levels of self-discharge, their price and their relatively small size and
weight [16].
A variety of fast charging methods for lithium-ion batteries are in
application to date. These include the pulse charging process, Multi-
stage Constant Current Charging (MSCC) and
Constant-Current-Constant-Voltage (CCCV). The CCCV charging method
is currently the most common method to recharge lithium-ion batteries
in electric AMRs [1315].
Another viable approach to decrease the time span at which a
battery is charged is to use batteries made of different materials.
Battery Streak Inc. (BSI) offers fast charging batteries made of meso-
porous nanostructured materials which were originally developed at
the University of California Los Angeles [18]. The benefit of the bat-
tery’s porosity is the ability to house the electrolyte like a sponge,
which provides additional surface area per unit volume for high-current
charging. Compared to common lithium-ion cells, BSI cells simply op-
erate as capacitors without the necessity for a chemical phase change.
Consequently, this new technology minimises excess heat generated
by the conventional charging process, keeping the cells cool to the
touch which allows for safely charging by means of 6C charge. The
C-rate is defined by the chargeable current (A) divided by the battery’s
capacity (Ah), and the charge time can be determined by 1
𝐶per hour.
Each BSI battery designed for the AMR-application comprises
24 cells, resulting in two modules of 24 V and a total capacity of
18.8 Ah. It is suggested that one of the KPIs for an Autonomous Mo-
bile Robot System (AMRS), operational efficiency, could be enhanced
substantially by utilising BSI batteries, on account of their fast charging
ability. Even when a conventional charger, the BSI batteries are fully
recharged in 1 hour with a charging current of 18.8 A. Additionally, the
module offers a 10-minute-to 80% SOC quick charge, which requires
Journal of Power Sources 521 (2022) 230894
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M. Selmair et al.
Fig. 2. BSI Pack Charge/Discharge Profile. (For interpretation of the references to
colour in this figure legend, the reader is referred to the web version of this article.)
a CCCV protocol with a charge current of 110 A (6C). Both mentioned
charge protocols can be realised by means of a charging station with a
conventional European 230 V / 16 A-Socket.
Fig. 2 exhibits the charge/discharge profile of the novel BSI battery
pack (24 V, 18.8 Ah). The red line represents the charge profile and
depicts the first Constant-Current (CC) stage of charging up to 60%
SOC after which it levels off at Constant-Voltage (CV) (34.8 V), and
ends at 80% SOC which equates to 15 Ah. In contrast, the blue line
that represents the discharge profile, visualises the constant-current
discharge to the lower voltage limit (6 V). This profile is calculated
from a typical BSI 1 Ah cell capable of 10-minute charge to 80% SOC,
which followed the same CCCV charging protocol and a CC discharging
protocol.
2.3. Task allocation and prioritisation
The tasks are allocated by a central operating optimisation algo-
rithm in this experimental setting. The algorithm ensures that all tasks
are performed on time, whilst minimising the driving effort of the
AMR fleet measured in metres. For this purpose, all driving efforts are
transferred to a matrix that constitutes the associated Assignment Prob-
lem (AP) (for more details see Section 4.1.4). The solution to the AP,
referred to as Vogels Approximation Method for non-quadratic Matrices
(VAM-nq), was developed by Selmair et al. [3], and is an extension
of the original Vogels Approximation Method (VAM) introduced by
Reinfeld et al. [19]. Unlike exact methods like the Hungarian Method,
proposed by Kuhn [20], and Integer Linear Programming, VAM-nq ap-
proximates a solution for the AP. As a result, the quality of the solution
is sufficient for most cases, but the algorithm computes solutions sub-
stantially faster than exact methods. For a detailed review, we refer the
reader to Selmair et al. [3].
To ensure that all tasks are performed on time, the system uses a
prioritisation method to schedule all tasks. This method ensures that the
system uses the temporal flexibility of each task to minimise the driving
effort of the AMRs. Extensive research has been conducted about this
prioritisation method by Selmair et al. [21].
3. Relevant literature
This section presents a description of the state-of-the-art planning
strategies summarised in Table 1 and explains their findings, identifies
possible weaknesses and defines potential for consolidation. Based on
a review of the relevant literature, the following key components of an
Allocation Method are identified:
Timing of the charging process
Choice of a charging station
Duration of the charging process
Table 1 serves as the basis for a comparison and summary of plan-
ning strategies provided by studies on battery charging. On the basis
of these studies, the findings are evaluated and examples of specific
research are selected.
3.1. Timing of the charging process
The trigger that prompts the charging process is usually a predefined
low SOC and it coincides with a point in time when an AMR is not
performing a task [3,10,22,24,25,27]. In an effort to determine the op-
timal timing, Kawakami et al., for instance, focus on minimising battery
deterioration [23]. However, as previously established, this aspect is
deemed to be negligible when it comes to lithium-ion batteries.
Interestingly, Colling et al. distributed AMRs equally, by using
charging cycles and thus focused the attention on the overall sys-
tem [26]. However, the mentioned study is carried out on a system
with a total of six AMRs, it remains uncertain whether this method can
be implemented for a larger scale.
Charging idle AMRs, even if their SOCs are above the triggering
SOC, also referred to as intermediate charging, is also suggested to be a
feasible tool to enhance a charging strategy [3,22,24]. Possibilities for
intermediate charging and the consideration of the SOC in comparison
with other AMRs in the field or in charging stations could lead to an
increase in flexibility, efficiency and scalability of the system. These
considerations should also be taken into account and tested when
developing the charging strategy.
3.2. Selecting a charging station
Heuristics can be particularly helpful in selecting a suitable charging
station. Commonly, this involves selecting charging stations based on
either their distance to an agent or that can be used with the least
overall delay [10,22,27]. The total delay consists of the travel time
to a station and the waiting time required by an AMR to use the
charging station. In contrast, predetermining charging stations without
monitoring the distance to and availability of charging stations, as
implemented by Colling et al., does not appear to be feasible for a
flexible and scalable system [26].
In fact, using heuristics can increase efficiency and flexibility. Fur-
thermore, if the parameters allowed for the displacement of AMRs cur-
rently at a charging station by AMRs with a higher need for recharging,
it is assumed that efficiency and flexibility can be increased.
3.3. Duration of the charging process
The duration of a charging process can either be defined as the
time required to physically replace a battery [10,22,23] or to reach
its planned SOC [24,26]. Kabir et al. and Zhan et al. suggest that
batteries should only be charged to a level of 90 or 95% to shorten the
inefficient phase of a battery charge (see Section 2.2) [25,27]. All the
mentioned studies have in common that only the condition of one AMR
is considered in any decision. Selmair et al. provides a first method
that take into account the condition of more than a single AMR [3].
In their paper, two AMRs are compared with each other by using an
objective function that includes the distance of an AMR to an occupied
station as well as the SOC of both. The following conclusions can be
drawn from Selmair et al.: the possibility to interrupt a charging process
due to tasks or displacements can increase flexibility. For this purpose,
accurate parameters are necessary in order to achieve robust processes
to attain a high level of flexibility.
Journal of Power Sources 521 (2022) 230894
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Table 1
Categorisation of previous Literature.
Timing Selection Criteria Duration
Ebben 2001 [22] Intermediate/Charge Range Heuristics Duration of Battery Exchange
Kawakami & Takata 2012 [23] Min. Deterioration Duration of Battery Exchange
Zou et al. 2018 [24] Intermediate/Below Threshold Handling Time/Fully Loaded
Kabir & Suzuki 2018a [25] Below Threshold Heuristics Duration of Battery Exchange
Kabir & Suzuki 2018b [10] Below Threshold Nearest Station Fully Loaded/Above Threshold
Colling et al. 2019 [26] Scheduled Permanent Assignment Fully Loaded
Selmair et al. 2019 [3] Intermediate/Below Threshold Minimal Costs Fully Loaded/Displaced
Zhan et al. 2019 [27] Below Threshold Heuristics Fully Loaded/Above Threshold
3.4. Fast charging demand and charging methods for lithium-ion batteries
The fast charging ability of lithium-ion batteries has become one of
the major milestones in achieving large-scale deployment of electrified
mobility [28,29]. Nowadays, warehouse robots powered by conven-
tional lithium-ion batteries still require at least an hour to recharge be-
tween their tasks [24,30]. This duration of downtime can be substantial
when robots handle heavy-duty tasks, and further prevents operators
from using the automated system in larger industrial areas due to
limited travel distance per minute charge [31,32]. Therefore, several
charging strategies have been developed to improve the charging
processes and the efficiency of operation [27]. Constant-Current (CC),
Constant-Current-Constant-Voltage (CCCV), pulse-current-charging, and
boost-charging by means of increasing the charging current are the
common fast charging strategies, because their protocols can be simply
applied with predefined charging variables such as fixed currents
and voltages [14,33,34]. However, due to the low battery charging
efficiency at certain State of Charge (SOC), applying a high charging
current can also overheat the batteries, which often shortens the battery
lifespan [35,36]. To resolve these degradation issues, adaptive fast
charging was developed to maintain battery health as well as to ensure
user safety. By assigning different charging steps according to real-
time battery properties, including SOC and State of Health (SOH), pack
temperature, and internal resistance, etc., [3740] these protocols not
only improve the battery operation efficiency but also meet the safety
requirement under excessive usage.
In addition to the charging protocols, the selection of power sources
can also play a crucial role in realising the economical fast charging
AMR model [27,41,42]. Lithium-iron-phosphate and lithium-titanium-
oxide batteries have recently become popular choices, not only for their
fast charging ability, but also for their excellent chemical stability [43
46]. Consequently, such stability leads to a markedly extended lifespan
of the battery, better safety during the charging process, a substantial
decrease in the cost of ownership, and has been widely adapted in
electrical bus systems for frequent and short-distance commutes. On
top of the rapid charge, niobium-based batteries (in replace of graphite
anode) have also demonstrated reduced heat generation due to unique
pseudocapacitive storage [47,48].
4. Developed strategies
This section describes potential strategies for charging and parking
AMRs, which maintain the availability of the AMR fleet on the one
hand, and minimise the distance travelled on the other. Every strategy
can be divided into two aspects: the first is the Allocation Method
itself (Subsection A) and the second pertains to the availability of an
AMR, referred to as Availability Rule, for tasks or charging and parking
processes (Subsection B).
Table 2 illustrates the two aspects of the strategies in the form of a
morphological box. Every combination of the left and the right column
represents a strategy
Table 2
Two Components of a Charging & Parking Strategy.
Allocation Method Availability Rule
Trivial Hard Condition (HC)
Trivial+ Dynamic Condition (DC)
Pearl Chain
AP
4.1. Allocation Methods
The four Allocation Methods are presented in the order of their scope,
in terms of numbers of AMRs considered by each method. Therefore,
the first Allocation Method – namely Trivial – only takes into account the
AMR that requires either charging or parking. This Allocation Method is
common in today’s industrial transport AMR applications (see Table 1).
The Trivial Allocation Method is followed by three new Alloca-
tion Methods, which consider other factors than merely the AMR that
requires a charging or parking station. These are in namely Trivial+,
Pearl Chain and AP. Their scope is illustrated in Fig. 3.
4.1.1. Trivial Allocation Method
The Trivial Allocation Method combines the methods most frequently
encountered in the literature review. They are mostly utilised in thresh-
old rules and nearest-agent-first allocations. Table 3 shows an overview
structured by the defined key components, followed by their specifica-
tion.
The triggering moment, in which the charging process is prompted,
is predefined as the moment when the SOC falls below a certain value.
In the included study, this value equals 25%. As soon as the current
transport tasks are completed, the AMR drives to the nearest vacant
charging station and the charging process continues until the battery
has reached a specific SOC, in this study above 99%). This process can
only be disrupted by the Availability Rules of AMRs (see Section 4.2). If
an AMR is not performing a task and all charging stations are occupied,
it is assigned to the nearest parking station. AMRs in a parking station
can be pulled out by getting elected for a task or if their SOC falls below
a specific threshold, in this study below 25%.
4.1.2. Trivial+ Allocation Method
The Trivial+ Allocation Method is designed to be more flexible, with
the help of the insights gained from the literature review. The key
difference to the Trivial Allocation Method is that intermediate charging
are now feasible and AMRs can be displaced from their charging
stations. This requires that the AMR which has to be allocated must
be compared to an AMR within a occupied station. A description of
this method is included in Table 4 and text below.
With Trivial+, AMRs can now be assigned to charging stations
regardless of their SOC. This introduces the concept of intermediate
charging. AMRs with a SOC of 25% will still be allocated exclusively
and therefore with priority to charging stations.
Additionally, occupied charging stations will also be considered for
allocation if the occupying AMR has a 20% higher SOC than the AMR
seeking a charging station. This ensures that charging stations are not
Journal of Power Sources 521 (2022) 230894
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M. Selmair et al.
Fig. 3. The Four Allocation Methods and their Illustrated Scope: Trivial, Trivial+, Pearl Chain and Assignment Problem.
Table 3
Scheduling Rules for the Trivial Charging & Parking Allocation Method.
Timing Selection Criteria Duration
Charging Below Threshold Nearest Station Above Threshold/Call for Task
Parking No Task/Above Threshold/No vacant Charging Station Nearest Station Above Threshold/Call for Task
Table 4
Scheduling Rules for the Trivial+ Charging & Parking Allocation Method.
Timing Selection Criteria Duration
Charging Intermediate/Below Threshold Minimal Costs Above Threshold/Call for Task/Displaced
Parking No task/Above Threshold/No vacant Charging Station Nearest Station Above Threshold/Call for Task
occupied by AMRs that have a significantly higher SOC than those seek-
ing to recharge. Within the parameters of Trivial+, the driving distance
to the stations remains the most important criterion for selecting a
charging station. The relevant station with the lowest costs (sum of
driving distance and optional additional costs) will be assigned. For
occupied stations, additional costs are imposed on the AMR that has
to vacate its current station and travel to a new one. These extra costs
ensure that AMRs in charging stations are not immediately displaced
at will if there is a more efficient alternative for the seeking AMR.
The charging process is carried out until the AMR has either reached
a sufficient SOC of 99%, gets displaced or receives a task as soon as
the SOC has reached a minimum level in subject to the later expressed
Availability Rules (see Section 4.2).
The allocation of parking stations is unaffected by these modifi-
cations to the methods and works the same way as described for the
Trivial Allocation Method.
4.1.3. Pearl Chain Allocation Method
The Pearl Chain Allocation Method is based on the basic principle of
the Trivial+ Allocation Method. However, this Allocation Method not only
compares the SOC of both AMRs, like the Trivial+, but also examines
whether displacements are efficient considering the total distance in
metres that the AMRs would have to travel. Furthermore, a distinction
is made between the following situations:
If the seeking AMR has a 40% lower SOC than the occupying
AMR, displacement is permitted without further conditions.
If the seeking AMR’s SOC is between 20 and 40% lower than
that of the occupying AMR, displacement is permitted if the total
distance travelled by both AMRs is shorter than the distance to
other charging station options. The travelling effort of the AMR
that needs to charge for a potential displace is calculated as
follows:
𝐶=𝐷𝑉𝑆+𝐷𝑉𝐷+𝑀𝐷,
where 𝐷𝑉𝑆represents the travelling effort of the seeking AMR
to the occupied charging station, whereas 𝐷𝑉𝐷stands for ef-
fort involved for the displaced AMR to travel to another station
as described below. Finally, 𝑀𝐷represents the potential effort
required for the displaced AMR to manoeuvre out of its station.
The displaced AMR is permitted to choose between stations listed
below this paragraph. This principle prevents performance issues,
by a reasonable restriction of the solution space. Therefore, the
‘‘pearl chain’’ has a maximum length of three AMRs (see Fig. 3).
1. Vacant charging stations
2. Vacant parking stations
3. Occupied charging stations which fulfil the case described
in Situation 1
If the seeking AMR’s SOC is not at least 20% lower than that of
the occupied AMR, a displacement is not permitted.
4.1.4. Allocation Method based on the Assignment Problem
The AP finds its roots in the research field of applied mathemat-
ics [49]. This problem, in its original form, consists of a number of
agents and a number of tasks. Any task can be assigned to one of the
agents. The total costs arising from a solution may vary, depending
on each chosen agent-task assignment [50]. The optimal solution is
defined as the one minimising the associated costs.
Our idea for charging &parking AMRs by using the AP, is to transfer
the efforts of AMRs for driving to a station into an Assignment Problem
(in the mathematical context the costs). By using this method, a number
of AMRs can be assigned to a number of stations by minimising the
driving effort and meeting defined requirements at the same time.
Therefore, the costs can be manipulated to ensure that AMRs are being
assigned to the closest stations on the basis of their SOC, even if this
means bypassing some in order to allow prioritised AMRs access to
these. This mechanism ensures that AMRs of a fleet with a relatively
low SOC will have lower costs for charging stations than others, and
will have greater priority to access charging stations. The conditions
are defined as extra costs added to the driving distance to a station
measured in metres. For the underlying use case, these extra costs are
defined as follows:
1. An occupied charging station is only accessible to AMRs whose
SOC is 10 or more percentage points lower than that of the
occupying AMR. The theoretical allocation of an AMR that does
not fulfil this requirement is sanctioned with a penalty cost of
10,000 units and should not be accepted by a balanced system
for any reason.
Journal of Power Sources 521 (2022) 230894
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M. Selmair et al.
2. An occupied parking station is not accessible for any seeking
AMRs with an SOC lower than the level of the AMR currently
occupying the station. The theoretical allocation of an AMR that
does not fulfil this requirement is sanctioned with a penalty cost
of 10,000 units and should not be accepted by a balanced system
for any reason.
3. The theoretical allocation of AMRs with an SOC below 30% to
a parking station will be sanctioned with a value of 1000 units.
This ensures that such AMRs will only be assigned to a parking
station if no option for charging is available.
4. If an AMR is occupying a charging or parking station, additional
manoeuvre costs of 7 cost units are added to the agent’s accrued
costs. One cost unit is the equivalent of 1 metre.
The above formulated constraints increase the likelihood of assign-
ing the AMRs efficiently to charging and parking stations. Constraint 1
permits the assignment to an occupied charging station if the occupying
AMR can be assigned to any other station and the seeking AMR’s SOC is
at least 10% below the SOC of the AMR currently occupying the charg-
ing station. Due to Constraint 4, manoeuvre costs are also factored into
every effort undertaken by an AMR to vacate a station. Simultaneously,
Constraint 2 allows AMRs in parking stations to become displaced by
others if it is beneficial to the overall system in terms of distance
travelled. Constraint 3 ensures that AMRs with a SOC lower than 30%
are more likely to be assigned to charging stations than to parking
stations. This was determined to ensure that the AMRs’ batteries are
not depleted entirely — provided that the system contains a sufficient
number of charging stations.
4.2. Availability Rules
Availability Rules for an AMRS constitute an interface between the
task allocation and the charging- & parking strategy. These rules regulate
the proportion of AMRs available to perform tasks within the system
and can be applied within the following two modelling paradigms:
either hard or dynamic conditions (HC and DC) can be formulated.
A Hard Condition cannot be disregarded for any reason, which may
be disadvantageous in some instances, however, their simple structure
and comprehensibility can be useful. For this study, such a condition
was formulated as follows:
An AMR in the field or in a parking station is available for tasks
when its SOC is above 25%
If an AMR is charging, it will become available for tasks as soon
as its SOC exceeds 90%
DC can be modelled, for example, by manipulating costs in some
manner in order to perform a desired behaviour. In the presented
case, where the system assigns AMRs to tasks by taking into account
their distance to the task’s source and their SOC, such a dynamic
condition can be formulated to sanction AMRs with extra costs if they
are currently charging or their SOC is too low. Such a condition can be
defined as follows:
An AMR incurs extra costs in addition to the travelling distance
in metres to a new task if it is located at a charging station. We
initially proposed to calculate these extra costs with the following
formula:
(−1 ∗ SOC) + 70
These extra costs will be negative for AMRs with a SOC above 70%.
As such, the system will be rewarded if such AMRs receive tasks. They
are more likely to be chosen than other AMRs in the field or parking,
even if the distance to the task is the same. On the other hand, the
extra costs will be positive if a charging AMR’s SOC is below 70%.
Then, this AMR will receive penalty costs and is therefore less likely
to be considered in comparison to other AMRs with the same distance
to the task’s source. The lower an AMR’s SOC, the higher the penalty
costs and the less probable their chance of receiving a new task. The
70% threshold parameter applied here was chosen on account of the
underlying charging curve: once SOC exceeds 70%, the charging speed
begins to decrease significantly. Therefore, AMR with a SOC below this
parameter are prioritised for the charging process.
While developing the simulation study, we devised a more complex
Dynamic Condition (DC) which allows for more flexibility and also
takes the number of open tasks and the occupied charging stations (CS)
into account.
Number of AMRs in system Number of occ. CS
Number of open Tasks
For example, if we assume a system of 40 AMRs with three occupied
charging stations and 70 unassigned tasks, the resulting SOC threshold
would be 52%. This means that all AMRs in charging stations with an
SOC of this or above this value, would become available for performing
tasks. The availability of AMRs in the field remains at a SOC above 25%
similar to the Hard Condition (HC) described above.
To ensure the long-term and sustainable operation of such a system,
we chose to restrict this formula and defined a minimum and maximum
value of the allowed threshold. This range was set between 40% and
80% SOC. That is, if the formula yielded results within this range, they
were applied directly. However, if the results fell below or above the
predetermined range, the minimum or maximum value was set as the
threshold.
5. Experimental environment
The layout used in this study represents a typical automotive
continuous-flow production system (see Fig. 4). There are three types
of warehouses embedded in this layout:
1. a full goods warehouse,
2. a mixed storage facility that serves as a full and empties ware-
house and
3. a storage for empties.
The full goods warehouses (1 and 2 in Fig. 4) are the starting points
of the intracompany supply chain. The empty dollies are collected to
be prepared for reuse in Warehouse 2 and 3. In the centre of the
layout there are three production lines that represent a flow production.
Along the production lines are work stations, also called fitpoints. Every
work station is supplied by a predefined warehouse at certain intervals.
These intervals are determined by the number of parts stored in a
dolly, the frequency at which a part is required and the cycle-time
of the production line. The replenishment is carried out by using a
two-bin system. There is only space for one AMR at each station at
a time. Thus, if a second AMR was required to enter a station that
is currently occupied, it has to wait in front of it and will block the
pathway for other AMRs, potentially causing delays. One way to satisfy
this restriction will be explained later in Section 6. Parking stations are
established between some fitpoints along the production line (see blue
house-symbols in Fig. 4). The charging stations are visualised by green
dots and are placed at the end of each production line.
Table 5 illustrates the parametrisation of the simulation study. The
first four parameters describe the travelling behaviour of each AMR
including the maximum speed, an applied speed factor for the loaded
AMRs as well as the acceleration and deceleration. The subsequent
four parameters show the battery settings for the i3 and BSI with the
corresponding battery capacity and charging current.
Journal of Power Sources 521 (2022) 230894
7
M. Selmair et al.
Fig. 4. Simulation environment consisting of four production lanes, two warehouses (1+2) and one area for empty dollies (3).
Table 5
Parameter Settings.
Parameter Value Unit of Measurement
Max. AMR speed 1.5 m/s
Speedfactor when loaded 0.65
Acceleration 1 m/s2
Deceleration 5 m/s2
i3 Battery capacity 120 A h
i3 Charge in 792 mAh/min
BSI Battery capacity 19 A h
BSI Charge in 1900 mAh/min
Load period 25 s
Unload period 20 s
Supply period 9 h
Simulation period until all tasks are completed
Fig. 5. Released orders per 10-minute time periods during each simulation run with
highlighted stress periods.
6. Simulation study
The previous section described the experimental setting in which
we utilised a total of 40 AMRs to deliver tasks from one place to
another. Initially, each AMR starts with an uniformly distributed SOC
between 30 and 75%. In order to ‘‘stress’’ the system temporarily, we
decided to define two periods with a greater number of tasks, referred
to as a ‘‘stress test’’. All in all, within a period of 9 hours (540 min),
a total of 6582 tasks are released by 125 fitpoints with a uniformly
distributed call-off rate between 10 and 60 min (referring to the ‘‘stan-
dard utilisation’’ in Fig. 5). The same predefined release pattern was
adhered to in each simulation run. Stress tests were performed in
the period between minute 60 and 120 where the cycle time of all
fitpoints was shortened by 25%, which temporarily lead to a higher
call-off rate. Between minute 300 and 400, the call-off time span was
shortened by 30% which correlates to the definition of ‘‘periods of high
utilisation’’. Fig. 5 illustrates the total number of released tasks in 10-
minute time windows with the respectively highlighted periods. We
selected the Trivial Allocation Method in combination with the Dynamic
Availability Rule in order to ascertain the most promising AMR to task
ratio.
In line with the previously described two-bin principle, each order
call-off creates two different tasks in the system: one transport task
for the full bin from the warehouse to the respective fitpoint and one
transport task of the empty bin from the fitpoint to the empties storage
or, if the situation requires it, to the mixed storage facility. To ensure
that the system satisfies the previously described restrictions of the
empty container having been removed before the full container arrives,
we utilised a binary prioritisation flag. Therefore, tasks involving an
empty container were prioritised when the corresponding full container
was being picked up at the warehouse by an AMR. This rule ensured
that the fitpoint is free to receive the full container upon arrival.
This simulation study investigated the behaviour pattern and the
efficiency of the previously introduced charging strategies (Table 2)
developed in [12]. The combination of four Allocation Methods with two
Availability Rules lead to a total of eight combinations of strategies for
the conventional i3-lithium-ion battery. For the BSI battery, we added
one more simulation run using only the Trivial Allocation Method and
the Hard Availability Rule. Due to the fact that this battery can be fully
charged in less than 10 min and the Trivial Allocation Method condition
requires the AMRs to be fully charged, it was necessary to perform a full
charge, also due to the relatively long setup time for an AMR to drive
into and out of a charging station. Interrupting this comparatively short
charging process would lead to an even higher number of charging
events and be counterproductive regarding the optimisation goal of
keeping traffic density as low as possible.
For the parking process, a total of 64 parking stations were available
(marked by blue houses in Fig. 4). The charging process can be per-
formed at one of four charging stations on the right hand side of each
production line (marked by green dots in Fig. 4). In order to assess the
performance of the different charging strategies, the number of charg-
ing stations was restricted to four, on the basis of some preliminary
findings. These demonstrated that when AMRs were only able to access
three charging stations, the systems itself came to a standstill.
In order to evaluate the efficiency of the strategies, the combination
of allocation methods and availability rules, described above, we ran a
total of nine scenarios. Their results will be introduced in the following
section.
Journal of Power Sources 521 (2022) 230894
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M. Selmair et al.
Table 6
Order Delays with the i3-battery.
Allocation Method Availability Rule Total Number of Delays Percentage of Delays
Trivial *HC 1,454 22.09%
Trivial DC 340 5.17%
Trivial+ HC 4 0.06%
Trivial+ DC 23 0.35%
Pearl Chain HC 10 0.15%
Pearl Chain DC 48 0.73%
AP HC 7 0.11%
AP DC 5 0.08%
*most commonly applied solution in industry
Fig. 6. Illustrated SOC and resulting availability of the AMR fleet.
7. Results and discussion
For the purpose of evaluating and comparing all strategies, we have
adopted the approach proposed by [12], who analysed the results on
the basis of five KPIs:
Delays of tasks
Average SOC of the AMR fleet
Density of traffic
Availability rate of the AMR fleet
Utilisation of charging stations
Fig. 7. Traffic caused by the AMR fleet.
Table 6 presents both the total number and percentage of delayed
tasks of each strategy with the i3-battery. The Trivial Allocation Method
in combination with the Hard Availability Rule can be described as in-
flexible and thus unsuitable for the tested order volume in combination
with 40 AMRs in the fleet. Therefore, this strategy will be disregarded
and not be considered in all other investigated KPI. As mentioned previ-
ously, the simulation run with the BSI battery was only performed with
the Trivial Allocation Method and the Hard Availability Rule resulting in
5 delays (0.08%).
The average SOC across time can be seen in Fig. 6 a. Firstly, it is
particularly noticeable that using the BSI battery results in a entirely
different average SOC pattern. This can be easily explained by the
different parameters this battery has in comparison to the i3 battery
(see Table 5). The lines of the other data sets show that the three
advanced methods (Trivial+, Pearl Chain and AP) combined with the
hard availability rule follow a similar trajectory. Applying the Trivial Al-
location Method with the Dynamic Availability Rule results in the lowest
average SOC and consequently the worst outcome beside the Triv-
ial / HC combination. The application of the DC Rule in combination
with the three advanced Allocation Methods also results in similar SOC
Journal of Power Sources 521 (2022) 230894
9
M. Selmair et al.
Fig. 8. Average utilisation of the AMR fleet, charging stations and parking stations.
averages across time, which are a little lower than when combining
the advanced strategies with the HC. It is proposed that the dynamic
request of AMRs when having a large number of transport tasks to
complete lead to this outcome. This means that AMRs are called out
of charging stations and allocated to transport tasks even when they
are not at full battery capacity. The box plot diagram (Fig. 6 b) depicts
a summary of the resulting SOC pattern.
In order to evaluate the traffic density, we evaluated the driving
effort of the AMR fleet. With our focus being the optimisation of the
charging and parking processes, we initially investigated the driving
effort for both of these processes. Fig. 7 a illustrates this effort mea-
sured in metres driven for each strategy. In comparison to the Trivial
Allocation Method, all advanced Allocation Methods reduced the driving
effort of both the charging and parking processes by two to 19%.
If we consider the total driving effort of all processes (see Fig. 7 b),
the effort required by charging/parking processes is relatively low
(2%). Consequently, the optimisation potential of the driving effort
to charging and parking stations is also deemed to be low.
Fig. 6 c summarises the number of AMRs available during each
simulation run. An AMR is rated as ‘‘available’’ when its applied Avail-
ability Rule is fulfilled. For the Hard Condition (HC) Rule, this means
an AMR is available at a SOC above 90% when charging and above
25% when idle. For the Dynamic Condition (DC) Rule, the availability
is defined as already described in Section 4.2.Fig. 6 c shows that all
advanced Allocation Methods, regardless of the applied Availability Rule,
result in a relatively high rate of availability. The lowest level of AMR
availability is 17 AMR, which corresponds to a proportion of 42%. The
availability of AMRs in Trivial Allocation Method scenarios is close to
zero in many cases, which is insufficient to meet the required targets,
as far too many tasks are inevitably delayed.
Fig. 8 a, b and c illustrate the utilisation of the AMRs, charging sta-
tions and parking stations and show that all advanced Allocation Meth-
ods improve the utilisation of AMRs and charging stations. In regards
to the Availability Rules, the results presented in Fig. 8 b: show that the
HC Rule results in a higher utilisation of the charging stations, whereas
the DC Rule leads to a relatively lower utilisation. This data can be
explained by the larger number of AMR call-offs from a charging station
when applying the DC Rule. This, in turn, leads to more and longer idle
times of the charging stations.
Finally, the results associated with our alternative energy storage,
the BSI battery, are discussed. As previously mentioned, the accelerated
charging process makes any Advanced Allocation Method like Trivial+,
Pearl Chain and AP obsolete. The average SOC of the BSI fleet was
higher than all average SOC for the i3-battery scenarios, as the charging
process is 2.4×faster than that of the i3-battery. With the BSI’s ability
for fast charging, nearly all other resulting KPIs for the BSI-scenario
can compete with the i3-battery and the Advanced Allocation Methods,
where AMRs started with a randomly distributed SOC between 30 and
75%, leading to the randomly distributed necessity for charging regard-
less of the charging speeds. However, without the random distribution
of the SOC, all 40 AMRs could potentially commence with a low SOC
and require initial charging at nearly the same time. This circumstance
of high charging demand could cause significant slow down to the sys-
tem’s performance due to the limited number of charging stations, but
the utilisation of the BSI battery is likely to mitigate such a problematic
scenario. On the other hand, the relatively low capacity of the BSI bat-
tery (see Table 5) correlates with the higher number of metres driven
to charging stations, as the demand for charging processes occurs more
frequently. Specifically, the BSI-scenario requires more than twice as
many charging processes than e. g. the AP / DC-scenario (182 vs. 66).
This clearly indicates that increasing the BSI-battery’s capacity is also
critical to the overall performance. A possible solution is to increase
the battery capacity with compatible charging station that facilitates
the fast charging safely [51].
8. Conclusion
With this study, we contributed our ideas and expertise to improve
systems consisting of AMRs in terms of charging and parking processes.
A total of three advanced Allocation Methods and two expressions of
Availability Rules are combined into seven charging and parking strate-
gies, which sustain AMRs in an even more efficient manner than the
widely used trivial strategies. The overall optimisation objective was
to satisfy the order deadlines and to reduce the number of delays to a
minimum. The secondary objective was to minimise the resulting traffic
density of the entire AMR fleet. The results of our simulation study
showed that all of our strategies met both objectives — minimising the
metres driven by the fleet and maximising the availability of this fleet.
Using different KPIs to evaluate the system’s behaviour also demon-
strated that the proposed strategies even lead to a higher availability
level of the AMR fleet, a higher average SOC and to a better utilisation
of available charging and parking stations.
Beside investigating different strategies with AMRs running on con-
ventional lithium-ion batteries, we also evaluated the performance
of a fleet utilising niobium-based oxide batteries developed by BSI.
Such batteries have a lower capacity, but can be charged much faster
than conventional lithium-ion batteries. With the fast charging ability,
using the BSI-battery not only leads to only minimal delays, but also
to a higher average SOC. In comparison to conventional lithium-ion
batteries, the only slight disadvantage is the more frequent demand for
Journal of Power Sources 521 (2022) 230894
10
M. Selmair et al.
charging and the subsequent greater driving effort to charging stations.
These disadvantages can be resolved when the batteries with greater
capacity become available. All in all, the application of our proposed
Advanced Strategies leads to better results in many ways: fewer delays,
less traffic density, higher fleet availability and also a higher average
SOC.
CRediT authorship contribution statement
Maximilian Selmair: Conceptualization, Methodology, Software,
Data curation, Writing – original draft. Tobias Maurer: Conceptual-
ization, Writing- review & editing, Validation. Chun-Han Lai: Writing
– review & editing. David Grant: Writing – review & editing.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
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