Automated Design of Routing Policies for the Dynamic Electric Vehicle

Routing Problem with Genetic Programming

Marko Ðurasevi

1 a

and Francisco Javier Gil Gala

2 b

University of Zagreb Faculty of Electrical Engineering and Computing, Unska 3, Zagreb, Croatia

University of Oviedo, Gijon, Spain

Keywords:

Genetic Programming, Electric Vehicle Routing Problem, Hyper-Heuristics, Routing Policy.

Abstract:

The dynamic electric vehicle routing problem (EVRP) with time windows (DEVRPTW) is an important com-

binatorial optimisation problem gaining on importance in today’s world due to environmental concern and the

requirement of dealing with dynamic and uncertain environments. This represents a problem when solving

such problems, as standard improvement based heuristics cannot be used to solve them, since not all infor-

mation about the problem is known beforehand. This provides a motivation for applying improvement based

heuristics, most notably routing policies (RPs), which determine only the next decision that needs to be per-

formed an execute it. Since these RPs do not construct the schedule in advance, they can easily react to any

changes in the problem. However, RPs are difﬁcult to design, which motivated the use of genetic programming

(GP) in automatically designing such heuristics for various problems. Unfortunately, in the context of EVRP

only static problems were considered. This study investigates the application of GP to automatically design

new RPs for DEVRPTW under different levels of dynamism. The results demonstrate that GP performs well

for certain levels of dynamism, although as it increases it is more difﬁcult to perform good decisions.

1 INTRODUCTION

The vehicle routing problem (VRP) represents an im-

portant combinatorial optimisation problem often en-

countered in the real world in various areas of logistic

and transportation (Erdeli

c and Cari

c, 2019). When

solving the VRP the main goal is to construct a set of

routes for a number of vehicles that need to visit and

serve a given number of customers, in a way that one

or more user deﬁned criteria are optimised. Due to the

growing environmental concern and the desire to re-

duce the negative inﬂuence of humans on the environ-

ment, the research is shifted towards the green VRP

problem variants (Moghdani et al., 2021), most com-

monly the electric VRP (EVRP) (Qin et al., 2021).

VRP, and by extension EVRP, are both NP-hard

problems, which in most cases leads to the application

of various heuristic and metaheuristic based meth-

ods for solving such problems (Erdeli

c and Cari

2019). Although these methods can solve various

EVRP problems efﬁciently, they are usually not ap-

propriate when solving dynamic, uncertain, or large

https://orcid.org/0000-0001-8732-4769

https://orcid.org/0000-0002-0606-7009

scale problem variants (Mardeši

c et al., 2023). The

reason for this is that they search the entire search

space for the best possible solution, however, the en-

tire information about the problem is either unavail-

able (as in dynamic problems) or certain information

is susceptible to change (such as in uncertain prob-

lems) (Mardeši

c et al., 2023). Therefore, the solutions

that these methods would construct would likely be-

come infeasible, and would have to be modiﬁed ac-

cording to the changing conditions in the problem.

Under such conditions, simple constructive based

heuristics, usually denoted as routing policies (RPs),

represent a viable alternative to standard improve-

ment based metaheuristics (Jacobsen-Grocott et al.,

2017). Instead of searching the space of all solutions,

RPs construct a solution in a step-wise manner using

a certain strategy. This means, each time that a certain

decision needs to be made, they determine the "best"

decision using some kind of a heuristic rule, for ex-

ample that the vehicle that becomes free should visit

the closest customer to it. By constructing the solu-

tion in such a way RPs can quickly react to changes

that occur in the system, such as the arrival of new

customers, or changes in their orders, since they do

not construct the entire solution until the end.

346

Durasevic, M. and Gil Gala, F.

Automated Design of Routing Policies for the Dynamic Electric Vehicle Routing Problem with Genetic Programming.

DOI: 10.5220/0013058900003837

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 16th International Joint Conference on Computational Intelligence (IJCCI 2024), pages 346-353

ISBN: 978-989-758-721-4; ISSN: 2184-3236

However, such methods also have certain limita-

tions, with one of the most important being that it is

difﬁcult to manually design high quality RPs. Due to

this reason researchers have resorted to using various

hyper-heuristic methods (Burke et al., 2013), most

notably genetic programming (GP) (Poli et al., 2008)

to aid them in the design of new RPs for various VRP

problem variants that they were faced with. For exam-

ple, GP was used to design RPs for the VRP variant

with time windows for both static and dynamic en-

vironments (Jacobsen-Grocott et al., 2017; Jakobovi

et al., 2023). Furthermore, GP was also successfully

used to generate RPs for VRP with zone based pric-

ing (Afsar et al., 2021), a variant in which not all cus-

tomers need to be served and in which it is also re-

quired to determine the prices for different customer

zones (Gil-Gala et al., 2023). Regarding the EVRP

variant, it was investigated in a short study where dif-

ferent scenarios were considered, although all were

static (Gil-Gala et al., 2022).

As we can see, the literature dealing with auto-

mated design of RPs with VRPs and especially on the

EVRP variant is still scarce, although the problem is

of practical relevance. Therefore, in this paper we in-

vestigate the ability of GP to generate RPs for the dy-

namic EVRP with time windows (DEVRPTW) prob-

lem variant in which the requests of the customers are

not known in advance, but rather become available

over time. We consider problems with several lev-

els of dynamism and different properties in order to

analyse how the RPs generated with GP perform un-

der different conditions. The obtained results demon-

strate that automatically generated RPs can perform

well also on dynamic problems, although on certain

problem types their performance signiﬁcantly deteri-

orates as the level of dynamism increases.

2 DYNAMIC EVRP

The DEVRP problem is usually modelled as a fully

connected graph in which the nodes represent differ-

ent types of locations, with edges representing the

connections (for example roads) between those loca-

tions (Schneider et al., 2014; Lin et al., 2016). Each

node has 2 coordinates that denote its position in a

2-dimensional Euclidean space, whereas each edge

has a certain weight that denotes the Euclidean dis-

tance between those nodes. In DEVRP, these nodes

represent either a customer, charging station, or de-

pot. Each customer i has a certain demand dem

that

needs to be served by a vehicle, the service time s

that

speciﬁes the time required to serve the customer, and

time window during which the service of the customer

needs to begin, which is deﬁned with the release time

and deadline d

of the customer. This variant of

the DEVRP is denoted as DEVRP with time windows

(EVRPTW) in the literature (Schneider et al., 2014).

In order to serve the demands of the customers, a

ﬂeet of electric vehicles is available, all with a given

cargo capacity, battery level, and speed. At the start,

all vehicles are located at the depot and they leave the

depot with full cargo and battery capacity. In the con-

sidered problem it is presumed that there is only a sin-

gle depot. Each vehicle has a cargo capacity, which

is reduced by the demand of the customer once the

vehicle ﬁnishes servicing him. Furthermore, each ve-

hicle also has a battery level, which is reduced when

traversing an edge. The reduction of the battery level

depends linearly on the distance that is traversed by

the vehicle from the source to the destination. When

the vehicle does not have enough cargo capacity to

serve a customer, it returns to the depot and ﬁnishes

the route, since in the considered problem the cargo

cannot be reloaded. The energy level of the vehi-

cle can be recharged by visiting one of the available

charging stations. When a vehicle reaches a charg-

ing station its battery level is fully recharged, with the

recharging time depending on the amount of energy

being recharged. The charging station have no limit

on the number of vehicles or the capacity that can be

recharged. In the problem considered in this study it

is presumed that the ﬂeet of vehicles is homogeneous,

meaning all vehicles have the same properties.

There are various objectives that can be consid-

ered when solving the DEVRPTW. However, the

most important objectives is the minimisation of the

number of vehicles used to service all the customers

(Lin et al., 2016). The reason why this objective is the

most important one is since it usually has the largest

inﬂuence on the cost (due to vehicle maintenance and

driver cost). Therefore, this objective is usually op-

timised as the primary one, after which a secondary

objective is optimised. Since in the considered prob-

lem it is required to service customers during their

time windows, the secondary objective will be de-

ﬁned as the total lateness L, which determines the

total time that all vehicles were late when servicing

customers on their route. The total objective is ﬁnally

deﬁned as a weighted sum of these two objectives:

Ob j(x) = c ∗ V(x) + L(x ), where Ob j(x) denotes the

total objective value for a solution x, V denotes the

total number of vehicles used in solution x, c denotes

a scaling constant, and L denotes the total lateness of

vehicles in solution x. In this study, c is set to 10

that the algorithm primarily focuses on the optimisa-

tion of the number of vehicles.

In this study we investigate the dynamic variant

Automated Design of Routing Policies for the Dynamic Electric Vehicle Routing Problem with Genetic Programming

347

of the aforementioned problem. This dynamic na-

ture of the problem is manifested in the way that

the information about the customers becomes avail-

able as the problem is being solved. Namely, not

all customer information is available from the start,

rather only a portion of the problem information is

available, which is deﬁned as the look-ahead win-

dow. The look-ahead window is deﬁned as an interval

[0, ct + (1 − dl) ∗ max

], where ct represents the cur-

rent time of the system, max

the maximum release

time of a customer, and dl the level of dynamism.

For all the customers whose release times fall into

this interval, their information is available to the rout-

ing algorithm and they can be considered for routing.

The information about all other customers will still

be unknown until their release time does not fall into

the look-ahead window. We see that as the current

time of the system advances, the upper interval will

be increasing, thus more customers will fall within

the look-ahead window and their information will be-

come known, until ﬁnally the information about all

the customers becomes known. The dynamic level pa-

rameter dl is used to determine the level of dynamism

in the system. If it is equal to 0, this means that the

look-ahead window will cover the entire horizon and

the information about all the jobs will be known from

the start, which amounts to the static variant of the

problem. On the other hand, if the parameter value

is equal to 0, only the customers with a release time

until the current moment in time will be known.

3 DESIGNING RPs WITH GP

Genetic programming (GP) is an evolutionary com-

putation method with a similar structure as genetic al-

gorithms (GAs) (Poli et al., 2008). This means that

the algorithm starts with a random set of solutions

which are iteratively improved using various genetic

operators such as selection, crossover, and mutation.

The main difference between GP and GAs is the solu-

tion representation, since in GP the solutions are rep-

resented as expression trees. This representation en-

ables GP to be used as hyper-heuristic, i.e., a method

to develop new heuristics for solving a combinatorial

optimisation problem (Burke et al., 2013). Until now,

GP has been successfully utilised as a hyper-heuristic

method for various combinatorial problems, such as

scheduling (Branke et al., 2016), container relocation

(Ðurasevi

c et al., 2024), bin packing (Burke et al.,

2012), and travelling salesman (Duﬂo et al., 2019).

Although there are other methods that can be used

as hyper-heuristics, such as neural networks (Branke

et al., 2014) or methods similar to GP (Planinic et al.,

2022), GP still remained dominant.

The RPs designed with GP can be divided into two

components, the routing scheme (RS) and the priority

function (PF) (Gil-Gala et al., 2022). The RS deﬁnes

the outline of the RP, which determines when a rout-

ing decision needs to be performed and ensures that

the constructed solution is feasible. The RS used in

this study to construct the solution to the problem is

outlined in Algorithm 1. The RS starts with a set of

vehicles, and then it iteratively selects the ﬁrst avail-

able vehicle (at the start all vehicles are available at

the same time) and determines the next location that

it should visit. This location is determined by the PF,

which is used to rank all the remaining available and

unserved customers. The best ranked customer is then

selected for service by the current vehicle. If the vehi-

cle does not have enough cargo capacity to serve the

demand of the selected customer, then it returns to the

depot. Otherwise, it visits the selected customer and

serves him. If the vehicle arrives before the release

time of customer, then it needs to wait until the cus-

tomer becomes available. If the vehicle arrives during

the time window of the customer, then the customer

can immediately be served. On the other hand, if the

vehicle arrives after the due date of the customer, the

customer can still be served, but a certain penalty in

the form of lateness is incurred. During the service of

customers it is possible that the vehicle does not have

enough energy to visit the selected customer. In this

case the vehicle will ﬁrst visit one or more charging

stations in order to reﬁll its energy. The charging sta-

tions it will visit will be selected so that the path be-

tween the source and destination is the shortest possi-

ble. This is possible since the number of charging sta-

tions is not large and therefore the best path between

the source and destination can be determined.

Algorithm 1: Outline of the routing scheme.

1: while not all customers are served do

2: v ← ﬁrst available vehicle.

3: for customer c

out of all the considered cus-

tomers C do

4: Calculate the priority π

5: Determine the customer k with the smallest pri-

ority value π

6: if Vehicle v has not enough capacity to serve

customer c

, or no customer k was selected

then

7: Return vehicle v to the depot.

8: else

9: Visit customer c

by vehicle v.

As previously outlined, the PF is used to deter-

mine the customer that will be scheduled next. The PF

ECTA 2024 - 16th International Conference on Evolutionary Computation Theory and Applications

348

represents a mathematical expression in which vari-

ables represent different information about the current

state of the problem. Since the PF is just a mathemat-

ical expression that can easily be represented as an

expression tree, this makes GP suitable for develop-

ing new PFs. However, in order to be able to do so, it

is required to deﬁne the building blocks used by GP to

construct such a PF. These building blocks represent

the functions and variables that will be used in the

PF. The functions used by GP are the addition, sub-

traction, multiplication, protected division (returns 1

in case division by 0 occurs), maximum, and mini-

mum operators, all of which are binary. The variables,

also denoted as terminal nodes since they represent

the leafs of the expression tree, used to construct the

PFs are outlined in Table 1. These terminals represent

various kind of information about the system, ranging

from basic system information, such as the demand of

the customer, or the remaining cargo capacity of the

vehicle. However, this set also includes certain nodes

that represent some extended information about the

problem, such as the average distance of a customer

to other customers, or the remaining time until the ve-

hicle would become late for serving a customer.

With the given set of function and terminal nodes,

GP can construct the PF required to solve the problem

under consideration. It should be stressed out that the

generation process of a new RP is a computationally

expensive task. The reason for this is that the PFs

need to be evaluated on a set of problem instances

to ensure that GP has obtained a PF that performs

well across different problems, and not just on a sin-

gle problem. However, once the PF has been evolved,

it can be used in conjuncture with the RS as a RP that

construct schedules for new and unseen DEVRPTW

problems in time that is negligible compared to im-

provement based heuristics. This characteristic of the

generated routing policies represents an important ad-

vantage over improvement based methods.

4 EXPERIMENTAL ANALYSIS

4.1 Experimental Setup

To generate the PFs that will be used by RPs we use

a steady state tournament GP algorithm. This algo-

rithm starts with a randomly initialised population of

200 individuals, where each individual is initialised

using the ramped half-and-half method. The size of

the individuals is limited to a maximum depth of 5.

In each iteration of the algorithm, the 3-tournament

selection procedure is performed, meaning that 3 in-

dividuals are randomly selected from the population,

Table 1: The set of terminal nodes used by GP to construct

PFs.

Parameter Description

Distance from the vehicle’s current

location to customer j

Ready time of customer j

Due date of customer j

dem

Demand of customer j

t Current time

Slack (time remaining until being

late) of vehicle for customer j

ddc Distance of customer j to the depot

dc Distance of customer j to the near-

est charging station

du Distance of customer i to the near-

est unserved customer

vc Remaining vehicle cargo capacity

bc Remaining vehicle battery capacity

avg

Average distance of customer j to

all other locations

avg

Average distance to 10 closest des-

tinations.

nn Number of locations within a given

radius (10% of the maximum dis-

tance between any customers)

with the better two being used in crossover to generate

a new individual. The new individual is then mutated

with a probability of 0.3 and inserted into the popu-

lation by replacing the worst individual in the tourna-

ment. For crossover, the uniform, size fair, context

preserving, subtree and one point operators are used,

whereas for mutation the hoist, subtree, permutation,

node complement, node replacement, and shrink mu-

tation operators (Poli et al., 2008). Since several op-

erators are deﬁned for crossover and mutation one of

them is randomly selected each time either crossover

or mutation need to be performed. The algorithm ter-

minates when 50 000 iterations were executed. All

the parameter values were selected based on a prelim-

inary experimental analysis.

To evaluate the ﬁtness of each individual, or rather

PF, it is coupled with the RS outlined in Algorithm

1 and used to solve a set of predeﬁned problem in-

stances. The problem instances that are used for

this purpose are those proposed in (Schneider et al.,

2014). This dataset consists of 56 instances that con-

tain 100 customers that need to be served, 21 charg-

ing stations, and a single depot. However, regarding

the way in which they were generated, the instances

can be divided into several types. Regarding the way

in which the customers are distributed, the instances

can be divided into random (R), cluster (C), or ran-

Automated Design of Routing Policies for the Dynamic Electric Vehicle Routing Problem with Genetic Programming

349

dom cluster (RC). In the random instances the posi-

tion of customers are generated completely randomly

within a given interval. On the other hand, in the clus-

ter instances the position of customers are generated

randomly around certain clusters, meaning that cus-

tomers are organised in certain groups. Finally, the

random cluster instances represent a combination of

both previous types. The second distinction between

the problem instances comes from the way in which

the time windows of customers were generated. In

this case the problem instances can be grouped into

instances with tight (T) time windows or loose time

windows (L). In instances with tight time windows

the size of the time window will be small, meaning

it will be more difﬁcult for vehicles to arrive in a

timely manner at the customer. On the other hand,

in problem instances with loose time windows there

is more time available to serve each customer, thus

making it easier to timely serve more customers. The

original dataset deﬁned in (Schneider et al., 2014) is

used as the test set, meaning it will be used to evalu-

ate the generalisation ability of the generated PFs af-

ter they have been evolved by GP. However, GP also

requires a dataset that is used to evaluate the quality

of solutions during the evolution process. Therefore,

another dataset has been generated based on the de-

scription from (Schneider et al., 2014) and used as a

training set, i.e., a set used by GP during the gener-

ation process of PFs. This set can be obtained from

https://github.com/nfridFER/EVRP-data.

The aforementioned instances were designed with

static scheduling conditions in mind. However, they

can easily be considered as dynamic instances in a

way described in Section 2. This means that a dy-

namic level is deﬁned that speciﬁes how many cus-

tomers in the future are known at the current moment

in time. To test the performance of the algorithm in

different situations, scenarios with different levels of

dynamism were used, which include levels of 0, 0.1,

0.3, 0.5, and 0.7, where 0 represents the static case.

Finally, to obtain an objective measure of the per-

formance of the algorithm on the different scenarios,

each experiment was repeated 30 times, and the best

individual on the training set from the ﬁnal population

was stored. This resulted in 30 PFs being obtained for

each scenario, out of which the average value was cal-

culated and is outlined in the results table.

4.2 Experimental Results

Table 2 outlines the results obtained by the automati-

cally designed RPs. They are evolved by GP for dif-

ferent dynamic levels and problem sets, with the best

result for each dynamic level being outlined in bold.

As expected, the best results are obtained in scenarios

with the lowest levels of dynamism. For two prob-

lem types we can see that the best result was obtained

in the case of 0.1. Although surprising, it seems that

in these scenarios the knowledge about the customers

that are available at the end does not inﬂuence the al-

gorithm much. The worst results are obtained when

the level of dynamism reaches the largest levels, since

in those cases the least amount of information is avail-

able.

The results are also illustrated in Figures 1, 2 and

3 as box plots, to gain a better notion on the distribu-

tion of the results across the 30 executions of the al-

gorithm. Figure 1 represents the results for problems

in which the customers are randomly distributed. It

it interesting to note how for the tight due dates for

the dynamism levels of 0.5 and 0.7 the results greatly

deteriorate. And although sometimes the algorithm

can ﬁnd a good rule, in most cases the evolved rules

achieve a poor result. The situation is better for the

case with loose time windows, although for level of

0.5 the algorithm again performed poorly. However,

this might simply be a unlucky case, as for a larger

level of dynamism this behaviour is not observed.

The results for the case when the customers are

distributed randomly are shown in Figure 2. Here we

can observe a completely different situation, in which

for the instances with tight time windows we have

similar results for all levels of dynamic levels, except

for the static case. Therefore, it seems that here not

knowing the complete problem from the start can be

quite penalising for the algorithm. On the other hand,

in the case of loose time windows, we see that there

are almost no difference between the rules that were

generated for the different levels of dynamism.

Results for random clustered instances are out-

lined in Figure 3. In this case the tight time windows

the algorithm are more sensitive to the level of dy-

namism, although the case for 0.1 could again indi-

cate a special case, as for larger levels we see that the

algorithm performs again better. For the case of loose

time windows there is very little difference between

the RPs generated for different level of dynamism.

Based on the previously described results, we can

draw certain conclusions on the performance of GP

when generating RPs for dynamic DEVRPTW prob-

lems. Regarding the level of dynamism, we see that

almost consistently the generated RPs perform well

for smaller levels of dynamism, like 0.1 and 0.3, with

no difference between these results and those ob-

tained by the RPs that were generated for the static

case. This suggests that knowing the information

about customers that have their time windows far in

the future does not play a vital role in most of the

ECTA 2024 - 16th International Conference on Evolutionary Computation Theory and Applications

350

Table 2: Results obtained by automatically designed routing rules for different dynamic levels and problem types.

0.0 0.1 0.3 0.5 0.7

Cluster 1 4474 3139 8278 206258 183120

Cluster 2 99865 105233 107392 235837 105710

Random 1 93246 337253 286211 360598 272148

Random 2 875653 745569 793060 844117 760951

Random cluster 1 107385 169637 109494 370116 188407

Random cluster 2 863325 868767 870696 852218 876266

Lateness

(a) Results for instances with tight time windows.

Lateness

(b) Results for instances with loose time windows.

Figure 1: Results for the problem instances with clustered customer distribution.

tested scenarios and that the RPs can perform as well

as in the case then all customer information is avail-

able from the start. However, for the larger levels of

dynamism we see that the behaviour of the generated

RPs depends on the problem instances that are consid-

ered. In the case of problem instances with loose due

dates the increase in the level of dynamism does not

lead to a signiﬁcant deterioration of the performance

of the generated RPs. On the other hand, in the cases

where the due dates are tight we observe that the per-

formance of the automatically generated PRs deterio-

rates signiﬁcantly as the level of dynamism increases

to 0.5 and 0.7. The reason for this is that in the case of

problem instances with loose time windows the vehi-

cles have a lot of more ﬂexibility to serve a customer.

For example, even if the vehicle is quite far away it

would be possible to reach the customer in time, as

the time windows are wide. On the other hand, when

customers have tight time windows they can be served

only during a very limited time, thus if all the vehicles

are far away it can be difﬁcult to serve the customer

timely. In this case it is important to plan the route

while considering future requests, however, not all are

available, and therefore the RP has a myopic view on

the problem and cannot follow a long term strategy.

The approach proposed in this study has certain

limitations. Although it is applicable in dynamic envi-

ronments, it suffers from the problem that without the

information about future requests the solutions will be

of poor quality as it cannot develop a long term strat-

egy. The reason for this is the way in which the dy-

namic nature of the problem was modelled, since no

information about future customers is available. Such

a model can be considered pessimistic, as it provides

the least possible information about the problem. By

relaxing this model it would be possible to obtain bet-

ter solutions, as more information could be integrated

into terminal nodes, and used by GP. For example, the

positions of the customers could be known, only their

demands and time windows could be modelled with

uncertainty.

Automated Design of Routing Policies for the Dynamic Electric Vehicle Routing Problem with Genetic Programming

351

Lateness

(a) Results for instances with tight time windows.

Lateness

(b) Results for instances with loose time windows.

Figure 2: Results for the problem instances with random customer distribution.

(a) Results for instances with tight time windows.

Lateness

(b) Results for instances with loose time windows.

Figure 3: Results for the problem instances with random clustered customer distribution.

5 CONCLUSION

This study investigated the application of GP to au-

tomatically generate RPs for DEVRPTW. The goal

was to analyse how GP performs for various levels

of dynamism. The algorithm was examined on sev-

eral scenarios and results demonstrate that GP can

generate efﬁcient RPs for lower values of dynamism.

As the level of dynamism increases to larger levels,

GP struggles to generate RPs that perform well. This

shows the limitation of the approach as the RPs do not

have enough information to perform good decisions

ECTA 2024 - 16th International Conference on Evolutionary Computation Theory and Applications

352

and motivates the investigation of problems in which

at least some information about customers is known.

This study dealt only with one form of dynamic

behaviour, but other forms of dynamic changes like

the weights of the edges, or various uncertainties

about the travel times or customer demands, could

also be considered. Furthermore, in this paper we

considered what we could call a pessimistic variant

of the problem, in which nothing was known about

future customers. However, it is very likely that

some information is known and could be used by the

heuristics to better perform their decisions. Such in-

formation could be modelled through various termi-

nal nodes that would be used by GP when designing

new RPs. Finally, subsequent studies should focus

on problems that model more real world characteris-

tics, such as nonlinear recharging functions or partial

recharging.

ACKNOWLEDGEMENTS

This research has been supported by the Euro-

pean Union - NextGenerationEU under the grant

NPOO.C3.2.R2-I1.06.0110. and the Spanish Gov-

ernment under projects MCINN-23-PID2022 and

TED2021-131938B-I00.

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