A (max - plus)-based approach for charging

management of electric vehicles

Azizbek Ruzmetov

Laboratory of OPERA-

IRTES

University of Technology of

Belfort-Montbéliard

Belfort, France

azizbek.ruzmetov@utbm.fr

Ahmed Nait-Sidi-Moh*

Laboratory of Innovative

Technologies

University of Picardie Jules

Verne-INSSET,

Saint - Quentin, France

ahmed.nait@u-picardie.fr *Corresponding author

Mohamed Bakhouya

Computer Science

Department

International University of

Rabat

Technopolis Sala el Jadida,

Morocco

bakhouya@gmail.com

Jaafar Gaber

Laboratory of OPERA-

IRTES

University of Technology of

Belfort-Montbéliard

Belfort, France

gaber@utbm.fr

Abstract – The management of electric vehicles charging and

adequate assignment to charging stations are the major

challenges facing managers of EV fleet. These issues required

then to be optimized considering all functioning constraints and

real situation of the charging process. Therefore development of

a faithful model to the context integrating all entities of the

process while considering their interaction and collaboration is

required. In this paper we propose a (max-plus) based approach

to represent a sequence of events and states of the charging

process involving the principal entities. Three main components

of the system will be represented: EV as a discrete entity,

charging stations as energy providers and an integrated platform

that ensures the synergy between the two mentioned entities.

Each entity is modeled and the occurrence date of each event will

be evaluated. A predictive charging approach is proposed to

anticipate and improve the provided service to drivers and to

suggest the adequate charging station according to their

characteristics.

Keywords- Electric vehicles, Charging process, Modeling and evaluation, Prediction, (Max – plus) algebra, Timed Event Graphs.

I. INTRODUCTION

In the literature, many research efforts have been made to further develop and promote electric vehicles (EV). However, little attention has been paid so far to the fact that charging process for electric vehicles is completely different from refueling process of vehicles that are powered by conventional power. Indeed, for charging management of EVs many parameters should be taken into account in order to adequately satisfy users and optimize the quality of provided services.

Nowadays, EVs play a significant role in the road traffic. The main features of EVs are limited cruising range, long recharge times, and the ability to regain energy during deceleration. This requires novel predictive methods, since the task is to suggest the adequate charging station rather than just the nearest one.

In our recent research work subject to the management of electric vehicles charging [13], we have addressed one of the most major issues related to the uncertainty of the drivers to get a suitable and vacant place at a charging station. In this work, we continue these efforts and try to propose a formal approach aiming to anticipate, plan and propose adequate charging solutions for EVs. These solutions should take into account several parameters such as the location of the EV, the remaining energy in the battery, traffic condition, the length of queuing in each charging station, etc.

The charging management of EVs within a charging station with performance metrics such as arriving of vehicles to a charging point, number of vehicles to serve, required charging time, etc. can be seen as discrete events. In this point of view, many appropriate tools have been developed in the literature to model and analyze such systems using discrete event systems theory (DES). In this work, we are interested in the use of Timed Event Graphs (TEG) which is a subclass of Petri nets (PN), combined with (max-plus) algebra for charging management of EVs. These tools have been proposed in the literature as powerful tools for modeling and performance analysis issues [9, 12, 14]. More precisely, the goal of the proposed models in this contribution is to act appropriately on the service time of each involved service in the process in order to serve a maximum of charging demands while satisfying all EV constraints. The proposed model allows defining a predictive functioning of the charging process by providing useful information and suggesting adequate charging station for each EV when it is necessary.

The reminder of this paper is organized as follows. The section II represents a survey of related work. The problem statement and system description are given in section III. The section IV is devoted to the modeling and evaluation issues. In the section V, we present a predictive charging approach. The last section concludes the paper and gives the future work.

978-1-4799-4647-1/14/$31.00 ©2014 IEEE

II. RELATED WORK

A charging and discharging were the main problems after beginning to produce the EVs. In other word, the charging process will take more time. Many researchers are working to solve this problem and several approaches were proposed in the literature. For example, in [5], a multi-agent system has been used to model and control the charging and discharging of PHEVs. Furthermore, authors compared the reducing imbalance costs by reactive scheduling and proactive scheduling. Simulations show that reactive scheduling is able to reduce imbalance costs by 14%, while proactive scheduling yields the highest imbalance cost reduction of 44%.

The authors examine in [6] the problem of optimizing the charge pattern of a PHEV. The optimization goal is to simultaneously minimize the total cost of fuel and electricity, and the total battery health degradation over a 24 h naturalistic drive cycle. The first objective is calculated using stochastic optimization for power management, whereas the second objective is evaluated through an electrochemistry-based model of anode-side resistive film formation in lithium-ion batteries.

In [7], a genetic optimization algorithm is applied to optimize the charging behavior of a PHEV connected to the grid with respect to maximizing energy trading profits in a vehicle-to-grid (V2G) context and minimizing battery aging costs at the same time. The study proposes a method to use the vehicle batteries in an optimized way under the consideration of battery aging costs and variable electricity prices. In [15], a charging/discharging process has been formulated as a scheduling optimization problem, in which powers of charging are considered to minimize the total cost of EVs. For fast charging and increasing cycle of battery life, charging algorithms are developed in [1, 2]. In [3, 4] authors have proposed some approaches for effective planning charging times.

The research work presented in [13] introduces an integrated platform for increasing the synergy between electric vehicles and charging stations. The interaction and communication between the EVs and the platform is ensured by the use of strengths of information and communication technologies, Web services and geo-positioning techniques. Based on information provided by the intermediate platform mainly the status of charging stations and the status of the EVs as well as their positions and the remaining power in their batteries, the scheduling and guidance optimization problem of EVs to charging stations are addressed. The problem was formulated and solved with linear programming.

III. SYSTEM DESCRIPTION

A. Charging process architecture

As depicted in Fig.1, the architecture of charging process mainly consists of three components: electric vehicles, a collaborative platform (CPL) and charging stations (CSs). The CPL is an intermediate layer between EVs and CSs. EVs connect to the platform and an in-vehicle application sends requests and receives responses from the platform. This last is too connected to many CSs. All exchanges between involved entities are ensured using communication technologies as

detailed hereafter. Information about the status of CSs are regularly updated in the platform database. These information relate, for example, number of free/occupied charging points within CSs, charging power of charging points, the location of charging stations, etc... Fig.1 represents all required information as input of the CPL. Based on these input parameters, and internal interactions between the CPL and other involved services such as calculation algorithms for shortest path and road traffic information, drivers of EVs will be assisted and guided to the appropriate charging stations corresponding to the characteristics of EVs and meeting desired points of interest (POI) of drivers.

B. Communication architecture

The interaction and communication of the three components of the system are based on the communication architecture as detailed in [13]. In this paper, we just recall the principle of this architecture. After geo-positioning the EV that needs to be charged with GNSS (Global Navigation Satellite Systems - GPS/EGNOS) by determining its GPS coordinates, its request is communicated, manually or automatically, to the platform using GPRS/3G networks (V2I communication). The CPL consults its own database and/or communicates directly with CSs and other Web services to find and provide a satisfactory response to each request. The communication between the platform and each CS is mainly done using the Internet (I2I communication) [8, 10].

Figure 1. Charging process architecture

IV. MODELLING AND EVALUATION

We start this section while recalling basic definitions of the proposed graphical tool used to describe the behavior of the system. We introduce TEG as a high level subclass of PN already used efficiently for modeling, evaluation and analysis for discrete event systems such as transportation, manufacturing and telecommunication systems [11, 14].

A. Graphical modeling

1) Introduction to TEG

A PN is a graph with of two kinds of nodes: places and transitions. The oriented arcs connect some places to some

Battery

level

GPS

coordin

ates

Other

POI

Charging

stations

Adequate charging

station

Collaborative Platform

Road traffic

status DB Calculation

algorithm

Electric Vehicles

transitions, or conversely. To each arc, we associate a weight (non negative integer). The dynamic of the graph is governed by a set of tokens that participate to the firing of transitions and change the system states. A Timed Event Graph is a subclass of Petri nets where each place has exactly one upstream transition and one downstream transition. TEG is well known to be rather adapted to problems with synchronization and parallelism phenomena, and then which suppose the absence of conflicts and resources sharing. More details about this formalism and its properties can be found in [11].

We denote Pij, representing a system state or a task to accomplish, the output (resp. input) place of the transition xi to xj. These transitions represent respectively the beginning and the ending of an event. The required time to accomplish the task that related to the state Pij is denoted τij and associated with the place Pij. The temporization τij corresponds to the minimal sojourn time of tokens in the place Pij. In the context of charging process of EVs, these temporizations correspond to the required times to accomplish assigned tasks to each involved service in charging process. It is worth noting that the main objective of these temporizations is to act appropriately on the service time (temporizations of the TEG model) of each involved service in the process in order to serve a maximum of requests in the case of several instant demands.

2) Global TEG model of the system.

TEG representing the behavior of the three components of the system is given in Fig.2. Various elements and parameters of the TEG are defined in the tables I, II and III.

Figure 2. Charging process TEG model

TABLE I. SIGNIFICANCE OF TEG ELEMENTS OF THE EV COMPONENT

Places Transitions/Temporizations

P1: Warning for charging x1: Sending a request to the CPL

P2: Waiting for a response from CPL x2: Receiving a response from CPL

P3: Number of EV to support a the same time

x3: End of charging

P4: Charging operation x4: Sending warning to driver

P5: Driving after charging operation t1: Required time for EV charging

P6: Sending request from EV to CPL t2:Driving time after charging (or battery autonomy) P7: Sending response from CPL to EV

TABLE II. SIGNIFICANCE OF TEG ELEMENTS OF THE CPL

Places Transitions/temporizations

P8: CPL availability to handle a request

x5: Receiving request from an EV

P9: Searching free/adequate CS x6: Asking for booking a charging point

P10: Updating CPL data base (DB) x7: Sending response to EV

P12: Sending request from CPL to CS for reserving a free charging point

t3: Spent time for searching free/adequate CS

P13: Confirmation from CS.

TABLE III. SIGNIFICANCE OF TEG ELEMENTS OF THE CS COMPONENT

Places Transitions/temporizations

P14: Availability of a free CS x8: Confirmation of booking

P11: Waiting for an EV P15: Waiting for charging operation

x9: Start of charging operation

t4: Time from sending response till starting charging operation

P16: Charging operation

x10: End of charging and releasing of charging point for next use

t5 (t1): Charging time

B. Mathematical modeling

1) Basics of (max-plus) algebra The (max-plus) algebra is defined with two main operators:

maximization and addition, which are denoted respectively by ⊕ and ⊗. The set ℝ𝜀 ≝ ℝ ∪ −∞ endowed these two operators is called a dioid (i.e. (ℝ𝜀 , ⊕, ⊗)). These operators are defined as follows:

𝑥 ⊕ 𝑦 = max 𝑥, 𝑦 𝑎𝑛𝑑 𝑥 ⊗ 𝑦 = 𝑥 + 𝑦

For all 𝑥, 𝑦 ∈ ℝ𝜀 . Usually we call ⊕ the (max-plus) addition, and ⊗ the

(max-plus) multiplication. The zero element for ⊕ is 𝜀 ≝ −∞: we have ∀𝑎 ∈ ℝ𝜀 , a⊕ = a = ⊕a. The neutral element of ⊗

is 𝑒 ≝ 0: we have, ∀𝑎 ∈ ℝ𝜀 , a⊗e = a= e⊗a. The element is

called absorbing element for ⊗, ∀𝑎 ∈ ℝ𝜀 , a⊗ = = ⊗ a.

Let 𝑟 ∈ ℝ the rth (max-plus) algebraic power of 𝑥 ∈ ℝ𝜀 is

denoted by 𝑥⨂𝑟 and corresponds to 𝑟. 𝑥 (with “.” is the

multiplication in conventional algebra). For 𝑥 ∈ ℝ𝜀 then

𝑥⨂0= 𝑒 and the inverse element of 𝑥 w.r.t. ⊗ is 𝑥⨂−1

= −𝑥. There is no inverse element for ε since ε is absorbing for ⊗. If

r > 0 then ε⨂r= ε. If 𝑟 < 0 then ε⨂r

is not defined [9].

Like for the conventional algebra, matrix calculation in the (max-plus) algebra is also possible and enables to solve infinity of problems. The basic (max-plus) algebraic operations are

extended to matrices as follows. ∀𝐴, 𝐵 ∈ ℝ𝜀𝑚×𝑛 and 𝐶 ∈ ℝ𝜀

𝑛×𝑝

(where ℝ𝜀𝑚×𝑛 is the dioid of matrices with m lines and n

columns. The elements of these matrices are scalars in ℝ𝜀 ), then for all i, j:

𝐴⨁𝐵 𝑖𝑗 = 𝑎𝑖𝑗 ⨁𝑏𝑖𝑗 = max 𝑎𝑖𝑗 , 𝑏𝑖𝑗

A⨂C ij =⊕k=1n aik⨂ckj = max

k(aik + ckj )

The matrix 𝜀𝑚×𝑛 is the m×n (max-plus) algebraic zero matrix: (𝜀𝑚×𝑛 )𝑖𝑗 = 𝜀 for all i, j; and the matrix 𝐸𝑛 is the n×n

(max-plus) algebraic identity matrix: 𝐸𝑛 𝑖𝑖 = 𝑒 for all i and 𝐸𝑛 𝑖 ,𝑗 = 𝜀 for all i, j with i ≠ j. The (max-plus) algebraic

matrix power of 𝐴 ∈ ℝ𝜀𝑛×𝑛 is defined as follows: 𝐴⊗0

= 𝐸𝑛

and 𝐴⊗𝑘= 𝐴 ⊗ 𝐴⊗𝑘−1

for k =1, 2....

The Kleene star of a matrix A is given by 𝐴∗ =

𝐸𝑛⨁ 𝐴 ⨁𝐴⊗2⨁ …. We show later, how this matrix will be

used to evaluate different states of the process whose behavior is expressed with a (max-plus) implicit equation (see (2)).

It is well known that the dynamical behavior of a TEG can be expressed by a system of linear inequalities in the (max- plus) algebra [12]. To do so, we associate to each transition xi of the TEG a dater xi(k) which corresponds to the date of the k

th firing of the transition xi. This parameter will play a major

role in the evaluation of the accomplishment time of each task of the charging process. A complete definition and all properties of these operators are detailed in [9, 12].

2) (Max-plus) linear model

The behavior of the TEG model of Fig.2 is translated into

the (max-plus) linear equation as follows. For all k >1,

kxtkx

kxtkxkx

kxkxkx

kxkxkx

kxtkx

kxkxkx

kxtkx

kxtkx

kxkxkxkx

kxkx

9510

8429

1068

867

536

715

324

213

7312

41

1

11

1

1

This system can be written in a matrix form as follows: for

all k >1,

𝑋 𝑘 = 𝐴0𝑋 𝑘 ⨁𝐴1𝑋 𝑘 − 1 (2)

Where:

X(k) = [x1(k), x2(k),…,x10(k)]t

regroups all daters of the

model. It called also the state vector.

A0 and A1 are the characteristic matrices of the system

whose components represent needed times to accomplish

various tasks of the process.

When we consider the system input, a third member may

be added to (2) expressing the impact of this input on the

system evolution. As we will see in the section V, the new

expression of (2) with the system input becomes: For all k >1,

𝑋 𝑘 = 𝐴0𝑋 𝑘 ⨁𝐴1𝑋 𝑘 − 1 ⨁ 𝐵𝑈(𝑘) (3)

Where:

U(k) = [u1(k), u2(k), …]t regroups all inputs of the system.

These inputs are known, a priori, for each k ≥ 1.

B is the characteristic matrix of the system representing the

impact of the input system on its evolution.

3) Evaluation and analysis

In order to calculate the starting and ending times of each

process task for each kth

charging request, we determine the

values of the state vector X(k), for k = 1, 2, ….

The solution of (2) is given by replacing X(k), iteratively,

by its expression as follows: ∀𝑘 > 1,

)X(k AA

kXAAkXA

) X(kAEAA X(k)A

)X(kAEA X(k)A

) X(kA)- X(kAA X(k)A

) X(kA)]- X(kA X(k) [AA

)X(kA X(k)X(k)=A

*

ii

i

ε

1

1

41

1

11

11

1

10

10

9

0

10

0

1100

2

0

3

0

1100

2

0

110

2

0

1100

10

Where ∀𝑘 > 1, 𝐴010𝑋 𝑘 = 𝜀, 𝑠𝑖𝑛𝑐𝑒 𝐴0

⊗10 = 𝜀

𝐴0∗ = 𝐸10⨁𝐴0⨁𝐴0

⊗2 … ⨁𝐴0⊗9 and 𝐸10ℝ𝜀

10×10

The state vector will be then calculated as follows: ∀𝑘 > 1,

)X(AAX(k)

) X(AA)X( AA)X(

)X( A)=AX(

k

1

5

;123

;12

1

2

1

*

0

1

*

01

*

0

1

*

0

Where X(1) is the initial condition of the system

corresponding to the first firing of the model transitions. In

other words, the components of this initial vector correspond

to the starting and ending times of each process task for the

first charging request. Formally, and while considering the

immediate response of each process component and the

behavior of the associated TEG, the initial vector is given by

𝑋 1 = 0, 0, 𝑡1, 𝑡1 + 𝑡2, 0, 𝑡3, 𝑡3, 𝑡3, 𝑡3 + 𝑡4, 𝑡3 + 𝑡4 + 𝑡5 𝑡 (6)

Considering this vector, and the characteristic matrices of

the model, response times of all received charging requests

can be calculated iteratively as given by (5).

V. PREDICTIVE CHARGING

A. Adopted approach

Based on the (max-plus) model describing the analytical

behavior of the process, we show how to act on the charging

time, or also charging rate, for each charging demand in order

to serve a maximum request of EVs while avoiding a long

waiting of vehicles. To prove the feasibility of the proposed

approach, we will consider in this study only one charging

station with many charging requests. In this case, all EV

requests are addressed to the unique CS successively with

random time slots.

For the rest of the paper, let consider these parameters:

- CTmax and CTmin: Required times for respectively a full

charging (100% by convention in this study) and

minimum acceptable charging rate within the CS.

Concretely with fast charging, the state of charging

(SoC) of a battery can reach 80% within 30 to 40

minutes. In order to reach full charging (100%) the

required charging time is about 4 to 5 hours for a given

power of charging. More details about this charging issue

are addressed in [16].

- : Authorized charging rate of each EV within the CS.

- DTi: Driving Time (under normal conditions) with the

remaining battery power of the ith

EV.

In this study we show how to regulate the charging process

according to the number of charging requests. Through the

predictive charging approach we propose in this paper, we try

to find a compromise between the number of charging

requests and allocated time to the charging point for each

charging operation. More precisely, with the aim to maintain

the waiting times of each EV less than an acceptable

threshold, we reduce the charging rate of certain EV in order

to satisfy a maximum number of charging demands.

The new TEG representing the charging process while

considering the number of charging requests as system input is

given in by following graph.

Figure 3. Predictive TEG model of the charging process

For this step of modeling and simulation we assume what

follows:

- H1: spent time for searching free/adequate CS by the

CPL is supposed to be fixed; t3 equals to a constant.

- H2: the CS should ensure at least 50% of charging rate

for each request (50 % ≤ ≤ 100 %).

For all k >1,

maxmin59510

8429

1068

867

536

715

maxmin2324

maxmin1213

7312

11

,,

1

71

,,

,,

1

CTCTktwithkxktkx

kxtkxkx

kxkxkx

kxkxkx

kxtkx

kxkxkx

DTDTktwithkxktkx

CTCTktwithkxktkx

kxkxkxkx

kukx

Figure 4. Predictive charging times and rates for EV

The analytical behavior, of the new TEG is represented by

the system (7). The matrix form of the system (7) can be

expressed like the function (3) with a light modification of

characteristic matrices which become matrices with variable

coefficients (depend on the parameter k). The (max-plus) state

equation obtained from the system (7) can be expressed as

given in (8).

𝑋 𝑘 = 𝐴0 𝑘 𝑋 𝑘 ⨁𝐴1 𝑘 𝑋 𝑘 − 1 ⨁𝐵 𝑘 𝑈 𝑘 (8)

In our case only the matrix A0 depends on the parameter k.

Other characteristic matrices are expressed with constant

components. For all k, 𝐴1 𝑘 = 𝐴1 and 𝐵 𝑘 = 𝐵.

P1, 0

P2, 0

P4,[CTmin, CTmax]

P5, [DTmin, DTmax]

P10, 0

P9, t3

P8, 0

P6, 0

P14, 0

P15, t4

P12, 0

P13, 0

P3, 0

P11, 0

P7, 0

P16, [CTmin, CTmax]

x1

0

x2

0

x31

x43

x7

0

x61

x50 x80

x90

x10

1

U1

t1(k-1) = CTmax

Yes

Yes

No

No

CTmax: Full charging time (=100 %)

CTmin: minimum charging time (with =50%) k=1... N, t1(k) charging time of kth EV

u1(1) ... u1(N): known arrival dates of EV demands

k> 1, (k) = |u1(k) - u'1(k-1)|

(k) ≥ CTmax

(k-1)= 100 %

wt(k) = 0

t1(k-1) = (k)

(k-1)= 100 ∙(k)/CTmax

wt(k) = 0

(k-1)= 100 ∙CTmin/CTmax

wt(k) = CTmin - (k)

(k) ≥ CTmin

t1(k-1) = CTmin

k =k+1

Solving (8) leads us based on (4) to the solution (9).

𝑋 𝑘 = 𝐴0∗ 𝑘 𝐴1𝑋 𝑘 − 1 ⨁𝐴0

∗ 𝑘 𝐵𝑈 𝑘

= Ψ⊗(𝑘−1)𝑋 1 ⨁⨁𝑖=2𝑖=𝑘Ψ⊗(𝑘−𝑖)Φ 𝑈(𝑖) (9)

With Ψ = 𝐴0∗ 𝑘 𝐴1 and Φ = 𝐴0

∗ 𝑘 𝐵.

As said previously, we predict the charging time, as well as

charging rate, of each EV asking to be charged according to

the frequency of arrival dates of all EV demands. These arrival

dates, u1(k), for all k >1, are known a priori and registered

within the database of the CPL. Charging times t1(k) and

charging rates are defined according to the flowchart of Fig.4.

For all k > 1, we define (k) that represents the time slot

between two consecutive arrivals of charging demands k and

(k-1) including the waiting time of the kth

demand before

freeing the charging point by the (k-1)th

EV. Let us denote

wt(k) this waiting time. The parameter (k) is defined by:

k> 1, (k) = u1(k) - u'1(k-1) (10)

with u'1(k-1) = u1(k-1) + wt(k-1) and wt(1) = 0

B. Numerical example: analysis and discussions

The obtained results in this section are based on random

numerical values of arrival dates of charging demands. The

maximal and minimal charging times are given by CTmax=40,

CTmin=20. Constrained by the limit number of pages, we limit

ourselves in this version to a few obtained results using the

flowchart of Fig.4. The extended version of this work will

expose more details about the obtained results while using the

obtained predictive charging times using (max-plus) model.

As given in Fig.5, we can see the evolution of the

charging time and waiting time versus the arrival dates of

charging demands. If the waiting time of the kth

EV is high,

then the (k-1)th

EV cannot be charged fully. In this case, the

charging time decreases toward CTmin. Otherwise, if there is

no waiting time of the kth

, it can be fully charged and the

charging time reaches the maximum value.

Figure 5. Charging times and waiting times versus arrivals of requests

The Fig.6 shows the evolution of the charging time and

inter-arrival (or time slot) of charging demands versus the

number of demands k. We remark that when the inter-arrival

of two consecutive charging demands is widely large, the

charging time reach the maximum value CTmax. When the time

slot is small, the charging time decreases toward CTmin.

Figure 6. Comparaison of the evolution of charging time and time slot

VI. CONCLUSION AND FUTURE WORK

In this paper, we propose a (max-plus) approach for

charging management of electric vehicles. The behavior of the

charging process including EVs, a charging station and an

intermediate collaborative platform is represented and

evaluated with a (max-plus) state model. The process is

studied in the point of view of discrete event systems for

which (max-plus) combined with TEG were already

efficiently used in the literature. The developed (max-plus)

model is used to evaluate the occurrence date of each charging

process task.

Furthermore, we have proposed a predictive charging

policy in order to anticipate the assignment and the guidance

of EV to charging stations. A numerical example is worked

out and simulation results are reported and show the added

value of the proposed predictive charging approach.

In our future work, we will extend this study by developing

further the proposed model while combining (max-plus)

algebra with queuing theory to predict the charging process for

each EV taking into account the random evolution of the

system. Furthermore, real time issue will be taken into account

and integrated in the developed model.

REFERENCES

[1] H.L.N. Wiegman, "A review of battery charging algorithms and

methods", University of Wisconsin – Madison, Nov 28, 1999.

[2] Abd El-Shafy A. Nafeh, "An effective and safe charging algorithm for

lead-acid batteries in PV systems", International Journal of Energy

Research. Volume 35, Issue 8, pages 733–740, 25 June 2011.

[3] O.Sundstrӧm and C.Binding, "Planning electric-drive vehicle charging

under constrained grid conditions", International Conference on Power

System Technology, IEEE 2010.

[4] H. –J. Kim, J. Lee, G. –L. Park, M. –J. Kang and M. Kang, "An efficient

scheduling scheme on charging stations for smart transportation",

Springer, 2010.

[5] S. Vandael, N. Boucke, T. Holvoet, K. D. Craemer and G. Deconinck,

"Decentralized coordination of plug-in hybrid vehicles for imbalance

reduction in a Smart Grid", AAMAS, 2011.

[6] S. Bashash, S. J. Moura, J. C. Forman, H. K. Fathy, "Plug-in hybrid

electric vehicle charge pattern optimization for energy cost and battery

longevity", Journal of Power Sources, vol.196, no.1, pp. 541-549, 2011.

[7] B. Lunz, H. Walz, D. U. Sauer, "Optimizing vehicle-to-grid charging

strategies using genetic algorithms under the consideration of battery

aging", IEEE Conf. of Vehicle Power and Propulsion - VPPC, 2011.

[8] A Nait-Sidi-Moh, M Bakhouya, J Gaber, M Wack, "Geopositioning and

Mobility" Wiley-ISTE, Networks and Telecommunications series; 272

pages. isbn:978-1-84821-567-2. 2013.

[9] B. De Schutter and T. van den Boom, "Max – plus algebra and max –

plus linear discrete event systems: An introduction", Proceedings of the

9th International Workshop on Discrete Event Systems (WODES’08),

Gӧteborg, Sweden, pp.36–42, May 2008.

[10] Kashif Dar, Mohamed Bakhouya, Jaafar Gaber, Maxime Wack, Pascal

Lorenz: Wireless communication technologies for ITS applications.

IEEE Communications Magazine 48(5): 156-162, 2010.

[11] T. Murata, "Petri nets, Properties, Analysis and applications" Proceeding

of the IEEE, Vol. 77(4) p.p. 541-579, 1989.

[12] F. L. Baccelli, G. Cohen, G. –J. Olsder and J. –P. Quadrat,

"Synchronization and linearity". John Wiley, New York, 1992.

[13] A. Ruzmetov, A. Nait-Sidi-Moh, M. Bakhouya, J. Gaber. "Towards an optimal assignment and scheduling for charging electric vehicles”. Renewable and Sustainable Energy Conference (IRSEC)-2013 International, Ouarzazate - Morocco 7-9 March 2013.

[14] A. Nait-Sidi-Moh, M-A. Manier, A. El. Moudni "Spectral analysis for performance evaluation in a bus network" European Journal of Operational Research Volume 193: Issue 1. Pages 289-302. 2009.

[15] Y. He, B. Venkatesh, L. Guan, "Optimal scheduling for charging and discharging of electric vehicles", IEEE transactions on smart grid, vol. 3, no. 3, September 2012.

[16] Rania Mkahl, Ahmed Nait-Sidi-Moh, Maxime Wack. “Modeling and Simulation of Standalone Photovoltaic Charging Stations for Electric Vehicles”. Accepted for publication In proc. of ICCCISE 2015 : XIII International Conference on Computer, Communication and Information Sciences, and Engineering. January 26 - 27, 2015, Jeddah, Saudi Arabia.