# Comparing lithium-ion battery architecture performances with Colored Petri Net

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## Abstract

Conventionally, the basic cells constituting the multicellular energy storage systems are modeled by electrical schemes based on Thevenin’s model. Other, more complex models incorporate the aging phenomena, resulting in a decrease in the State of Health of each cell. All these models remain analytical models and not discrete event systems. In this article, a discrete model is proposed, by detailing how each physical parameter is modeled. It is based on a no-timed Colored High-Level Petri Net. An example of a battery is simulated to validate this theoretical model. Its structure (number of cells) is declined under different architectures (connections between the cells) and is subjected to different resource dynamic allocation strategies. This Petri Net (PN) model makes easy, by adding a sub-network, to simulate different control laws and different resource management algorithms, whether or not allow commutations by forbiding all configurations that do not meet the specification or that will lead to accelerated cell aging. PN is used as a tool for comparing hardware architectures and cell control logic for a battery. Various conventional and innovative architecture are simulate. Different control laws can be compared in terms of performance, as lifespan and use of resources.

## Keywords

Battery Battery aging Electric power storage system Discrete event system Colored Petri Net## List of symbols

*T*Continuous time

- \(\varUpsilon\)
Dummy discrete time variable

- \(T_{cycle}\)
Cycle time (sum of charge, discharge and rest times)

*i*,*j*Dummy variable for a row, a column

*n*,*m*Number of rows, columns

- \(Cell_{ij}\)
Cell in row i and column j

- \(I_{cell},V_{cell}\)
Cell current, voltage

- \(I_{ij},V_{ij}\)
Current, voltage in row i and column j

- \(\theta _{ij}\)
\(Cell_{ij}\) temperature

- \(S_{ij}A\ldots X\)
Switch A...X in row i and column j

- \(Cb_{i}\)
Balancing capacitor for row i

- \(S_{ij}+\) and \(S_{ij}-\)
Balancing switches in row i and column j

- \(a\ldots y\)
Element node a...y

- \(I_{applc}\)
Current specification

- \(I_{bat}\)
Current deliver by the battery

- \(I_{n}\)
Cell nominal current

- \(\theta _{a}\)
Ambient temperature

*SoC*Fundamental cell State-of-Charge

*SoH*Fundamental cell State-of-Health

*DoD*Deep of Discharge ( \(DoD=1-SoC\) )

*ESR*Cell Equivalent Series Resistance

*OCV*Cell Open-Current Voltage

- \(R_{w},C_{w}\)
Cell First order Thevenin model parameters

*Q*Cell instantaneous electrical charge

- \(Q_{o}\)
Cell maximum charge, aka operational charge

- \(Q^{*}\)
Cell theoretical charge, aka cell capacity

- \(L_{f}\)
Cell announced lifetime, in number of cycles

*RUL*Cell Remaining Usefull Life, in number of cycles

- \(A_{d}\)
DoD influence parameter for cell aging

- \(A_{t}\)
Temperature influence parameter for cell aging

## 1 Introduction

*n*rows and

*m*columns structure battery, noted (

*n*,

*m*). For a \(Cell_{ij}\), placed in the \(i^{th}\) row and the \(j^{th}\) column, switch \(S_{ij}A\) is used to connect the cell in series on the same column with those in the upstream and downstream rows. The switches \(S_{ij}A\) and \(S_{ij}Z\) when they are respectively OFF and ON, allow to disconnect \(Cell_{ij}\) and replace it by a short circuit. The switches \(S_{ij}B\), \(S_{ij}C\), \(S_{ij}D\),..., \(S_{ij}X\) make possible to distribute the upstream current flow from node

*a*to different cells of the downstream row by nodes

*b*to

*x*[5] or allow a parallel cell association. Finally, the switches \(S_{ij}+\) and \(S_{ij}-\) ensure the balancing function, which will be presented at point 4.1. In normal operation, these two switches are open. Temperature of each cell is noted \(\theta _{ij}\). Usually, there are three major families deployed to built a battery: series-parallel architecture (SP), parallel-series (PS) and reconfigurable architecture. This latest requires the most switches [6].

While many articles speak of continuous modeling or neural systems to represent the behavior of batteries, few resort to discrete systems by a PN. This article presents a discrete model describing the behavior of a single cell as well as a multi-cellular battery. Thereby in this study, the BMS supervision part is described by a Petri Net (PN). A PN is a more general model than an automaton for describing a discrete event system (DES). It is valid whatever the type of DES. In particular, it allows to model and control the behaviors of parallel, distributed, synchronized, communicating systems and sharing the same resources. The battery corresponds to this description since several paths of electric current circulate into the matrix through the cells distributed in rows and columns. Events (opening or closing switches) are synchronized and controlled by the supervisor. The supervisor forbids all configurations that do not meet the specification or that lead to accelerated cell aging. So, since it is a matter of controlling the opening and closing of switches, it is possible to resort to discrete modeling. That’s why in [8], a PN model is used to model a photovoltaic panels association on a micro-grid, wind turbines and batteries. The states being associated with the source and battery states, so as to optimize the electrical flows on the network. Previously, PNs have already been used to model single-source system operation modes, depending on demand and energy stored amount in [9]. Among the different modeling possibilities, a PN associates formal semantics and visual representation with precise syntax and a graphical language. The operation of the system described by a PN is thus visualized by a synthetic structured and compact representation. Formally, a PN has properties such as liveliness, boundary, persistence, and re-initialization [10]. Dedicated mathematical tools ensure to test its properties. By including its BMS and switches, a battery can be considered as a DES. A discrete model to describe its operation can be established. The transitions correspond to switch commutations. Then, an algorithm can be implemented to control the cell matrix so as to modulate the energy stored in the battery.

The objectives of the application part of this paper are limited to compare three different battery architectures, regarding their performances on battery lifespan and to evaluate them. To do that, PN is presented and used as a tool for comparing hardware architectures and cell control logic. Depending on the state of the cells, the battery architecture and the needs of the external load, the BMS can activate or not all or part of the cells as their commandability is allowed by the switches. Whatever the battery architecture and structure, it is relevant to have an unique PN model of the battery that can be used to model its behavior. Likewise, a discrete model of each control law implantable in the BMS permits the study of their effects on the battery behavior. To compare the performances of each control law associated with each architecture, two performance parameters are used: the lifetime before failure and the remaining life potential in non-defective cells. After this analyse, we consider the command by determining whether to use an optimization algorithm for the cell activation improves performances or not. Literature presents many formal models describing a battery [11, 12, 13]. A discrete model by a Colored High Level Petri Network appears relevant to describe the cell functioning and their states since their sequential operation [14]. Furthermore, it can take into account aging phenomena. To date, the PNs have been used to model many sequential or random systems, but not to model the behavior of the cells, both in charge and in discharge. It is therefore in this work to achieve a discrete model that can be reused in the context of a programmable logic controller for running some cells and batteries.

In a first step, part 2 of this article is about presenting the continuous behavioral model of a cell and its equivalence in a discrete representation. Then, the following part shows how, from a cell PN model, a battery model can be built. After, Part 4 is devoted to the definition of different control laws, before presenting the simulations carried out in part 5.

## 2 Examples of equations

A cell has electrical characteristics such as its terminal voltage and the current flowing therethrough. Moreover, to describe the electric charge contained in the cell, two parameters are commonly used: the State of Charge (SoC) and the State of Health (SoH), respectively representing the amount of charge in the cell and the degree of wear. These parameters are related to the electric charge contained at the moment considered in the cell as well as its initial maximum electrical charge and its maximum electric charge at the moment considered. In addition, its temperature is a major parameter, particularly affecting its aging [15]. The behavior of the cell is modeled in the discrete event system in part 2.3, after explain it by a continuous electrical model and an aging model.

### 2.1 Continuous models

*ESR*). The currently market-leading technology for batteries is the lithium-ion because of its good features in energy density and longevity. Indeed, for a cell, the open-circuit voltage (

*OCV*) is different according to the electrode nature. A cell provides a current \(I_{cell}\) with a voltage \(V_{cell}\). However, this model is too simplified to accurately reflect the cell electrical behavior. The second-order model adds to the series resistor and the ideal voltage generator a parallel association of a resistance \(R_{W}\) and a capacitance \(C_{W}\), so as to take the relaxation phenomena into account [23]. Indeed,

*OCV*is not directly measurable on cell terminals before a time corresponding to the return to thermodynamic equilibrium [24]. These components, shown in Fig. 3b, corresponding for \(C_{W}\) to the double-layer capacitance between the electrodes and the electrolyte and for \(R_{W}\) to a resistance related to the charge transfer between them.

*T*with respect to the maximum electric charge that it can contain at this time

*T*, as depict by Eq. (1), when \(Q\left( T\right)\) is the instantaneous charge at time

*T*and \(Q_{0}\left( T\right)\) is the maximum electric charge or the operational charge at same time.

*ESR*in two, according to the \(I_{cell}\) current direction. By convention, in generator mode, the \(I_{cell}\) current is rated positive when the cell operates in discharge mode. It is negative when cell operates in charge mode. However, these models require higher computation times [26] and provide a SoC estimation precision which is often not necessary. Moreover, they present a static nature, not reflecting the aging phenomena. The cell aging is measurable by its SoH which represents the amount of electrical charge that a cell is able to store at a time

*T*with respect to its initial maximum electrical charge \(Q^{*}\) as shown in the Eq. (2). \(Q^{*}\) is the cell theoretical electrical charge, also call capacity. As they are reports of the same quantities, SoC and SoH varies from 0 to 1.

*j*ranging from 1 to 3 can be determined experimentally [28].

### 2.2 Aging continuous models

*ESR*increase. Aging mechanisms are mainly related to the materials used to make the electrodes, the electrolyte composition and the operating conditions. Aging is thus accentuated by the following parameters:

the operating anf the storage temperature of the cell;

the current flowing in the cell;

the charge variation during a cycle (discharge then recharge).

*ESR*(which will grow as they age) and on the capacity (which will decrease as aging). At the end, the aging is measurable by monitoring SoH or \(Q_{0}\) as performed on a lithium-ion cell \((LiFePo_{4}\)) example with a 60 Ah capacity, fully discharged in each cycle, under a \(50\,{^\circ }\hbox {C}\) temperature, corresponding to an operating temperature in a closed box, given in Fig. 4 [32]. This example demonstrate that its lifespan is equal to 400 cycles if it is specified that the cell have to be able to store at least 80% of its

*Q**.

Current influence is mainly felt for high currents, higher than the nominal current [33]. The nominal current is worth in literal (in Ampere) the cell capacity value (expressed in Ampere-hour). That is, for example a nominal current \(I_{n}\) equaling 1.2*A* for a cell having a 1200 mAh capacity. The simulations will be performed considering that the battery is not requested beyond its rated current \(I_{n}\). This means the current is nearly always lower than the nominal current, which means that only two parameters are to be considered for the cell aggravating aging rate: the temperature, which influence will be represented by a parameter \(A_{t}\) and the DoD, modeled by a parameter \(A_{d}\). The respective influence shapes are given in Fig. 5a for DoD influence and Fig. 5b for temperature.

*eV*[35]. The ambient temperature \(\theta _{a}\) is considered here at \(25\,{^\circ }\hbox {C}\). The \(A_{t}\) parameter evolves according to Eq. (6) with

*K*as Boltzmann constant (\(K=8,617.10^{-5}eV/K\)).

### 2.3 Cell PN model

*life*boolean token. Depending on the state of the battery (\(\ll relax \gg\), \(\ll charge \gg\), \(\ll deharge \gg\)) the corresponding transition is fired. The token life goes through the transition to return to the \(\ll cel \gg\) state. Instead of being returned as it is, it can be modified by adding a function on the return arc of each transition. The cell can run in a relax mode, if its current \(I_{cell}\) is equal to zero, in charging mode if its current is negative (energy is supplied to the battery) or in discharge mode (write here as \(\ll decharge \gg\)) if its current is positive.

*QoM*, an initial value. For example,

*QoM*set here at 1200 for a \(Q_{0}=1200\) mAh cell. In this net, only one \(\ll cycling \gg\) transition is sufficient. At each end of cycle, the transition is fired. At first, it is considered a decrease of one unit on the operational charge at each cycle. Also, the Fig. 7 PN models its decay. In the Fig. 6 scheme, it is necessary to add a \(\ll cycling \gg\) transition. The token contained in the \(\ll cell \gg\) state must thus comprise several color fields: a boolean color field for the #

*life*data and an integer for the #

*Qo*data field.

*iDt*value.

*Qstar*is the initial value, equal to

*QoM*. In the relaxation phase, it is decremented to simulate self-discharge.

*Qmax*and

*Qmin*. Temperature evolution is current related [37] and integrate an equilibrium return when cell is in rest. The linear relations between the temperature evolution, related to the current value, diminished of a return to thermodynamic equilibrium at the ambient temperature are then the object of functions allowing to calculate it from the current. Current have to be specified in another field of the token

*i*.

*t*, to lighten the writing.

Thermal simulation is important for determining the storage system performance [38, 39]. There are various models that have been published for the SSEEs [40, 41, 42, 43, 44, 45, 46]. For the present study, the model is limit to a simple growth and decrease of the cell temperature according to the electrical intensity crossing a cell [37, 47]. In the same way, since this study must mainly validate the use of a PN in order to describe a cell functioning, the heat transfer phenomena between cells [48, 49] which take place in the heart of a battery are not taking into account. Then, at each end of cycle, the RUL, represented by a \(\ll Rest \gg\) field, must be reduced. In the same way, two other fields must be created to describe the evolution of the ESR and the OCV, as indicated previously: the fields \(\ll Rs \gg\) and \(\ll V\gg\).

*din*11 equation described in formula (8), relating to a single cell, when

*Qomax*is equal to \(Q^{*}\),

*Qoal*1 is around \(Q^{*}\). \(t_{a}\) is the ambient temperature global variable. In the single cell PN, only one switch is modeled. Its ON/OFF state conditions the color \(\ll actif \gg\) (

*true*or

*false*). Other cells can be defined by this way.

*T*1, a recharge phase (

*T*2) and then a relaxation phase

*T*3, with same duration [50]. An example of \(I_{applic}\) cycling is presented in Fig. 12 for a 1200 mAh cell. The considered cycle allows to discharge two-thirds of the capacity cell under its nominal current [51] for

*T*1. The three states \(\ll Dtdecharge \gg\), \(\ll Dtcharge \gg\) and \(\ll Dtrelax \gg\) represent the three cycle phases, respectively discharge, charge, relaxation. Each state is connected to a corresponding transition: \(\ll cycldec \gg\), \(\ll cyclchg \gg\) and \(\ll cyclrlx \gg\). At any moment, only one transition is firable. When the cycle is complete, the transition \(\ll cycling \gg\) increment a counter recording the sudden cycle number, initialized to 0. The battery output current should be worth \(I_{applic}\) for the mission to be fulfilled. It consists of the sum of internal currents, depending on the architecture.

## 3 Battery model

According to the general electric scheme of an element around a cell (Fig. 1), a battery can respect three ways of manufacturing: a cell parallel-series association (PS), a series-parallel association (SP) or a reconfigurable architecture. One example of a reconfigurable architecture is used in this article: the C3C architecture (a cell to three cells) [6]. After presenting the battery architectures, these are translate into a PN model.

*n*several cells in series by connecting the node

*a*of \(cell_{i+1,j}\) to the node

*y*of the \(cell_{i;j}\) to let the battery provides output voltage equal to \(n.V_{cell}\). Then this stack of cells can be associated

*m*times in parallel with others, to increase the current to \(m.I_{cell}\). This reach the series-parallel (SP) architecture. An example with \(n=2\) rows and \(m=4\) columns is presented in Fig. 13. A single switch in series is sufficient to isolate or disconnect a column. In consequence, only that of the first row \(S_{1j}A\) has been useful.

*a*and all nodes

*y*. Then the stacks of cells can be connected in series to increase the voltage. Figure 14 shows an example of parallel-series (PS) architecture with the same structure as in the previous figure. Each cell \(cell_{ij}\) must have its switch \(S_{ij}A\) to be isolate from the rest of the matrix. Apart from the balancing circuits, these two architectures only need \(S_{ij}A\) switches.

*i*and column

*j*is associated by its node

*a*upstream with the three cells from the row \(i-1\) and columns \(j-1\),

*j*and \(j+1\). Similarly, this cell is associated downstream with the cells of the row \(i+1\) and the same three columns by their nodes

*b*,

*c*and

*d*. An example with the same \(\left( 2,4\right)\) structure as before is shown in Fig. 15. At the \(Cell_{ij}\) output, the flow can either be connected directly to the same column cell, or by the switch \(S_{ij}B\) on the previous column cell, or by the switch \(S_{ij}C\) on the next, respecting the principles described in Fig. 1. The C3C architecture needs \(S_{ij}A\), \(S_{ij}B\) and \(S_{ij}C\) switches but no balancing switches.

*n*: cell location in the matrix. It is unusual to see batteries of more than ten rows and columns. Beyond that, the batteries consist mainly of associated cell packs, connected together. So just two digits are enough to code the battery topology. For example, \(n=23\) for the cell in the third column of the second row. To simulate a single cell, it is sufficient to use one token with \(n=11\);

*V*: *OCV*, with 3900 (mV) as initial value for a fully charged lithium-ion cell;

*i*: current flowing into the cell. Initially, the cell will be considered as entering the discharge phase;

*Rs*: *ESR*, initially having a random value of 20 and \(25 (m\varOmega )\) for a lithium-ion cell;

*Qo*: the operational charge, worth a random value of \(\pm 20\) (mAh) around mean global variable \(Q^{*}=1200\) mAh;

*Qt*: instantaneous charge, worth *Qo* at the beginning;

*Qmax*: variable used to determine the DoD, initialized to zero at the start and at each cycle change. This value corresponds to the electrical charge for \(SoC=SoC_{end\_of\_cycle}\);

*Qmin*: another variable measured on a cycle to determine the DoD during the cycle, initialized at *Qo* and updated with *Qmax*;

*t*: cell temperature, initialized at room temperature, so at 2500 (in hundredth of Celcius degres) with an initial random between 0 to \(+1000 (c\,{^\circ }\hbox {C})\) added;

*Rest*: RUL, equal at the beginning to \(L_{f}\), explains in cycle number;

*Ad*: aging parameter associated with the DoD, equal to 1 at the beginning, calculated at the end of each cycle according to *Qmax* and *Qmin*;

*At*: aging parameter associated with the temperature, initialized to 1;

*actif*: boolean variable that specifies whether the cell is used (*true*) or idle, unconnected (*false*);

*life*: boolean variable, *true* if the cell is usable, or *false* if either *Qt* or *Rest* has dropped to 0.

*actif*(

*true*/

*false*). The other switches implicitly occur in the functions determining the values of the token integer fields (

*V*,

*I*,

*Rs*,

*Qt*,

*Qmax*,

*Qmin*). We now have a battery PN model whose the values of the different field of the token represent all the modeled aspects of the cells. The PN formalization has advantages, such as the liveliness or the blockage of the system in case of short circuit or cycles in the net. In addition, by using a PN with implicit time, rather than continuous, any punctual defects (imperfections in the current \(I_{bat}\) profile) do not disturb the model operation. On the other hand, by using the same PN for the behavior of the cells and the battery, the computation is carried out globally, on all the data of each cell.

## 4 Control laws

The PN with these control laws is intented to replace the BMS to control the cells. To control the switches, control laws can be established. Some, such as balancing, which uses the + and - switches, are today implemented in industrial solutions. Others, conventional, such as inclusion of additional cells used in redundancy, are not. Still others of an innovative nature, such as activation of the cells according to criteria based on their fundamental states, can be envisaged.

### 4.1 Different control laws

the balancing;

the redundancy cell management;

the reconfiguration with an optimization algorithm according to the cell SoH criterion.

*ESR*, even if the cells are from the same batch, some will be recharged faster than others. As the BMS monitors the cell SoCs, when it detects excessive imbalance, if battery includes balancing circuits, it can homogenize the charges contained in the cells. To do this, in a SP architecture, it first connects the capacity \(Cb_{i}\) in parallel to the \(Cell_{aj}\) most charged cell thanks to the \(S_{aj}+\) and \(S_{aj}-\) switches. Then, it connects this capacity to the least charged \(Cell_{bj}\) cell with the \(S_{bj}+\) and \(S_{bj}-\) switches. This cell to cell technique can improve the charges and consequently improve the battery lifespan [56]. The same principle of balancing is represented between a complete row to another in PS architecture. As already indicated, the C3C does not include balancing circuits.

The second studied control law is the redundancy management. A battery may comprise more cells than necessary to provide its nominal power, the cells being solicited under their nominal current. As explain before, this nominal current corresponds to this which completely discharges the cell in one hour (SoC falling from 1 to 0). To improve the battery reliability in the case of cell failure, cells can be added redundantly. These redundant cells are only used when an initially active cell fails. A cell failure corresponds, in this model, to a cell completely empty (SoC = 0) or too old (\(\hbox {SoH} = 0.8\) or \(Rest = 0\)).

Control law comparison

Balancing | Redundancy | SoH-based algorithm | |
---|---|---|---|

SP | Usual | One column | best \(\sum _{i=1}^{n}SoH\) |

PS | Possible | One cell per row | \(min\left\{ SoH_{j=1}^{m}\right\}\) at rest |

C3C | Whatever cell 2 cell | One column | \(min\left\{ SoH_{j=1}^{m}\right\}\) at rest and best Cells combination |

### 4.2 Battery complete model

To simulate options such as using part of the redundant cells and the control laws, it is necessary to add states and transitions to the PN, as well as to define functions.

By principle, when a cell is redundant, it is always initially be the last cell of a row. If redundancy is not actived, the battey operating with the only basic cells, the cell #*life* field is initialized to *false*. Whether redundancy is enabled or not, the end of mission for a battery is reflected by the presence of two cells of the same row with a #*life* data in *false*.

*Batt*, as specified by the formula (11). The whole strength of the colored PN model is summarized in the fact of being able to synthetically create a multi-colored token list. Each of the colors represents a physical variable. All the colors of the tokens are thus treated at the same time.

*Bat*24 representing this structure is given by the formula (12), with

*din*11 to

*din*24 cells have different initial values, in respect with the CPN-Tools spelling.

*archi*data in the #

*archi*field is associated with a SP architecture, 2 for a PS and 3 for a C3C. The applied current in the discharging and in the charging phases is given by the value of the #

*Iapplic*data. It can vary from once the cell nominal current to

*m*times. When the last column is used, the #

*mactif*data is

*true*. The #

*algo*r data allows to determine if the optimization algorithm is used or not. It making \(\ll algo \gg\) state. Thus, a battery operating with a redundant column used in a conventional way have #

*mactif*at

*true*and #

*algo*at

*false*. Finally, if the battery includes balancing circuits (between cells of the same column in SP, between the rows in PS, complementary between non-active cells in C3C), the #

*equil*data is set to

*true*.

When the state \(\ll Valeq \gg\) status token allows balancing (define by #*equil* field with an initial marking at *true*), the *bezeq* function, in which *paseq* is the balancing step, examines the cells’ *Qt* levels and use the subfunction *level*. According to the architecture, if an imbalance appears, the cells are balanced and the *Nbeq* counter is incremented.

Algorithm 1 *Bezeq* function, to define if the battery need balancing

Algorithm 2 *level* function

To allow an arbitration between the control laws and the mission profile, the associated transitions (\(\ll equal \gg\), \(\ll arbitrageBMS \gg\), \(\ll arbitrage2 \gg\), \(\ll algoSoH \gg\)) have a high priority in front of the others.

## 5 Simulations

A simulation example with this model is presented in this part.

### 5.1 Simulation set

*Q*, the current

*I*and the time into account. A \(\varDelta T\) time implied during a transition firing makes possible to translate a current in an electric charge. Also, the proposed model indirectly considers the time in the transitions. For this modeling, only integer variables are used, to avoid rounding errors during the divisions.

*T*1,

*T*2 and

*T*3, resp. charge, discharge and rest times) is fixed at three times 40 minutes if the cell is requested under its nominal current. Solution deployed in the Fig. 16 PN allows to simulate a phase in four steps. The draw of the three associated transitions thus implies a ten minutes \(\varDelta T\). For a \(Q^{*}=1200\) mAh cell, the electric charge variation during \(\varDelta T\) will be 200 mAh. During the first cycle, cell is discharged of 800 mAh. The three successive states \(\ll Dtdecharge \gg\), \(\ll Dtcharge \gg\) and \(\ll Dtrelax \gg\) represent the three cycle phases. During one \(\varDelta T\) time, under the nominal current, the temperature variation under load is set at \(+1\,{^\circ }\hbox {C}\) in charge and \(+6\,{^\circ }\hbox {C}\) in discharge. During relaxation, for simplicity, it is considered a linear decrease in temperature of \(-7\,{^\circ }\hbox {C}\) for this same \(\varDelta T\) instead of an exponential equilibrium thermodynamic equilibrium time constant.

### 5.2 Simulation of a (2, 4) battery under different architectures

The model has been tested under different battery structures. The smaller size of the matrix that compares the architectures requires four columns because in order the C3C to function normally, three path possibilities must leave from any cell while having an inactive cell in each row. It is also necessary to have at least two rows to ensure balancing.

Simulation starts with new cells whose characteristics are different from a few per cents around their initial capacity and their ESR (#Qt and #Rs datas). The \(\ll cell \gg\) state is initialized by the variable *Bat*24, given by the formula (12) to simulate a (2, 4) battery with \(n=2\) and \(m=4\) structure. A dozen simulations is performed on this network by modifying the initial marking data. By operating the battery under its nominal current \(3.I_{cell}\), it is possible to simulate the following variants, for SP and PS: without and with redundancy, with balancing circuits, with redundancy and algorithm optimization; for C3C: with or without active balancing. The parameter chosen to optimize the choice of active cells is SoH. Indeed, in the improving operating dependability perspective, the SoH appears as the most suitable parameter. The simulation stops when the battery can no longer meet the demand in current. The token representing the battery changes from its initial state, of the type shown in Fig. 16, to its final state. In this one, the number of cells with a #life data in true is insufficient. Transition fires simulates the charge and the discharge of the cells.

*th*column cells are used in redundancy. In the Table, the added \(+bal\) says that balancing circuits are activated. The added \(+algo\) means that the optimization algorithm is deployed.

PN simulation results: battery lifespan in cycle numbers

PS | PS+R | PS+bal | PS+algo | SP | SP+R | SP+bal | SP+aglo | C3C | C3C+bal |
---|---|---|---|---|---|---|---|---|---|

153 | 160 | 201 | 301 | 159 | 164 | 201 | 313 | 320 | 332 |

These results show that non-balancing conventional architectures have quite similar performances (153 cycles for PS, 159 for SP architecture). The battery lifespan is improved with the balancing method, both SP and PS (201 cycles both). Even if in this latter case, the solution is deployed little. On the other hand, the solution with conventional redundancy, namely that the spare cell is connected in place of the first cell to fail (often by aging), improves availability just a little: 160 instead of 153 in PS, 164 instead of 159 in SP. This is because this new cell must provide high currents to supplement the aging of other base cells to which it is connected. This confirms why this improvement is not used in batteries. While in theory it can improve reliability, the complexity of the relationships between cell characteristics means that adding redundant cells does not improve lifespan a lot. Finally, an optimization algorithm use to manage dynamically the redundancy significantly improves the performances (301 cycles for PS, 313 for SP and 320 for C3C architecture). Regardless of the architecture, the lifespan is significantly increased, approximately doubled for this structure. For the C3C architecture, balancing between idle cells can also help improve the lifespan, extend to 332 cycles instead of 320. The results obtained with this simplified model, using a PN, are similar to those obtained using a formal model described in Matlab [5], in particular the achievements provided by each control law optimization solution are of the same order.

### 5.3 Contribution of the discrete model

In addition to prove the interest of the balancing method and of the use of an optimization algorithm in the solutions comprising redundant cells, the model allows to classify the different architecture and command combinations according to the quality of the management. At the end of simulation for each architecture, the sum of the lifespan remaining on the critical row is recorded. The critical row has two faulty cells and is responsible for stopping the simulation. With the PS architecture, the sum of the remaining values in the four cell #*Rest* fields reaches 255 cycles. In SP architecture, these same cells only contain a total of 55 cycles. In C3C this total is reduced to 42 remaining life cycles. These values are only valid for the presented simulation and are not generic. With other initial conditions, the cells could have aged differently. This value represent no-used time lost in the battery. It nevertheless appears that the C3C architecture optimizes both the battery aging and the cell agings. Furthermore, to highlighting easily this phenomena by simply read the data contained in the tokens, the discrete model also allows, as expected, to follow in the various fields of tokens the evolution of each cell parameter.

## 6 Conclusions

A battery is a made up of cells and switches. It can thus be modeled by a PN model as any discrete event system to be implanted in the BMS, which integrates the main parameters, such as the temperature, this latter parameter contributing largely to the cell aging. The model proposed here starts from a single cell and is extended to a battery. The PN model is used as a tool to compare different architectures and control laws. The multi-colored token list used allows to treated each color of each token at the same time. It can simulate any type of architecture as well as several control strategies such as redundancy, balancing or resource management according to an optimization algorithm. In the latter case, the model used demonstrates that the cost of adding redundant cells is profitable in terms of lifespan when an optimization algorithm is used to manage the cells to be active. The results obtained on a minimal structure allow to confirm that complexity between cell characteristics means that adding redundant cells does not improve lifespan. It justify the actual choice on balancing. On the other hand, optimizing the activation of the cells in a battery comprising redundant cells, makes it possible to significantly improve the lifespan. The model also highlight the new C3C solution homogenize the cell agings in a battery. In this way, in a second life use perspective [57, 58], all the cells in the same batteries have a homogeneous SoH, which will simplify the reuse.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest. This study was not specifically funded.

## References

- 1.International Electrotechnical Commission (2011) Electrical Energy Storage, white paper. IEC, Geneva, Switzerland. ISBN 978-2-88912-889-1Google Scholar
- 2.Kim H, Shin KG (2012) Dependable, efficient, scalable architecture for management of large-scale batteries. IEEE Trans Ind Inf 8(2):406–417MathSciNetCrossRefGoogle Scholar
- 3.Ota J, Sato T, Akagi H (2016) Enhancement of performance, availability, and flexibility of a battery energy storage system based on a modular multilevel cascaded converter (MMCC-SSBC). IEEE Trans Power Electron 31(4):2791–2799CrossRefGoogle Scholar
- 4.Ci S, Lin N, Wu D (2016) Reconfigurable battery techniques and systems: a survey. IEEE Access 4:1175–1189CrossRefGoogle Scholar
- 5.Savard C, Venet P, Niel E, Pietrac L, Sari A (2018) Comparison of battery architecture dependability. Batteries 4(3):31CrossRefGoogle Scholar
- 6.Savard C, Sari A, Venet P, Niel E, Pietrac L (2016) C3C: a structure for high reliability with minimum redundancy for batteries. In: 17th IEEE international congress of industrial technology, Taipei, Taiwan, pp 281–286Google Scholar
- 7.Hoque M, Hannan M, Mohamed A (2016) Optimal CC-CV charging of lithium-ion battery for charge equalization controller. In: International Conference on Advances in Electrical, Electronic and Systems Engineering (ICAEES)Google Scholar
- 8.Khyzhniak T, Kolesnyk V (2013) Modeling of power-supply subsystems of microgrid using Petri nets Electronics and Nanotechnology (ELNANO). In: IEEE 33th international scientific conferenceGoogle Scholar
- 9.Lu D, Fakham DH, Zhou T, Francois B (2010) Application of Petri Nets for the energy management of a photovoltaic based power station including storage units. Renew Energy 35(6):1117–1124CrossRefGoogle Scholar
- 10.Boufaden A, Pietrac L, Gabouj S (2005) L’usage des reseaux de Petri dans la theorie de controle par supervision. Insa Lyon Notes, FrenchGoogle Scholar
- 11.Rezvanizaniani S, Liu Z, Chen ZY, Lee J (2014) Review and recent advances in battery health monitoring and prognostics technologies for electric vehicle (EV) safety and mobility. Journal of Power Sources 256:110–124CrossRefGoogle Scholar
- 12.Pelletier S, Jabali O, Laporte G, Veneroni M (2017) Battery degradation and behaviour for electric vehicles: review and numerical analyses of several models. Transp Res Part B Methodol 103:158–187CrossRefGoogle Scholar
- 13.Yoo K, Banerjee S, Kim J, Dutta P (2017) A review of lithium-air battery modeling studies. Energies 10(11):1748CrossRefGoogle Scholar
- 14.Savard C, Iakovleva EA (2019) A suggested improvement for small autonomous energy system reliability. Batteries 5:29CrossRefGoogle Scholar
- 15.Smith K, Neubauer J, Wood E, Jun M, Pesaran A (2013) Models for battery reliability and lifetime. Applications in Design and Health ManagementGoogle Scholar
- 16.Smith K, Rahn C, Wang C (2010) Model-based electrochemical estimation and constraint management for pulse operation of lithium ion batteries. IEEE Trans Control Syst Technol 18(3):654663CrossRefGoogle Scholar
- 17.Berrueta A, Urtasun A, Ursúa A, Sanchis P (2018) A comprehensive model for lithium-ion batteries: from the physical principles to an electrical model. Energy 144:286–300CrossRefGoogle Scholar
- 18.Tippmann S, Walper D, Balboa L, Spier B, Bessler WG (2014) Low-temperature charging of lithium-ion cells part I: electrochemical modeling and experimental investigation of degradation behavior. J Power Sources 252:305–316CrossRefGoogle Scholar
- 19.Dong GZ, Zhang X, Zhang CB, Chen ZH (2015) A method for state of energy estimation of lithium-ion batteries based on neural network model. Energy 90:879–888CrossRefGoogle Scholar
- 20.Wang YJ, Yang D, Zhang X, Chen ZH (2016) Probability based remaining capacity estimation using data-520 driven and neural network model. J Power Sources 315:199–208CrossRefGoogle Scholar
- 21.Wang QK, He YJ, Shen JN, Ma ZF, Zhong ZFGB (2017) A unified modeling framework for lithium-ion batteries: an artificial neural network based thermal coupled equivalent circuit model approach. Energy 138:118–132CrossRefGoogle Scholar
- 22.Khumprom P, Yodo NJE (2019) A data-driven predictive prognostic model for lithium–ion batteries based on a deep learning algorithm. Energies 12(4):660CrossRefGoogle Scholar
- 23.Chan HL (2000) A new battery model for use with battery energy storage systems and electric vehicles power systems. Proceedings of the conference on IEEE power engineering society winter meeting, pp 470–475Google Scholar
- 24.Urbain M (2009) Modelisation Electrique et Energetique des Accumulateurs Lithium-ion, Estimation en Ligne du SoH. Ph.D. dissertation. Institut National Polytechnique de Lorrain, FrenchGoogle Scholar
- 25.Chen KH, Ding ZD (2015) Lithium-ion battery lifespan estimation for hybrid electric vehicle. The 27th Chinese Control and Decision Conference (2015 CCDC), Qingdao, China, pp 5602–5605Google Scholar
- 26.Yu J (2015) State-of-health monitoring and prediction of lithium–ion battery using probabilistic indication and state-space mode. IEEE Trans Instrum Meas 64:2937–2949CrossRefGoogle Scholar
- 27.Liu X, Qin SH, He Y, Zheng XX, Cao CR (2016) SOC estimation of the lithium-ion battery with the temperature-based Nernst model. In: IEEE 8th international power electronics and motion control conference (IPEMC-ECCE Asia)Google Scholar
- 28.Lu L, Han X, Li J, Hua J, Ouyang M (2013) A review on the key issues for lithium–ion battery management in electric vehicles. J Power Sources 226:272–288CrossRefGoogle Scholar
- 29.Savoye F (2012) Impact des impulsions periodiques de courant sur la performance et la duree de vie des accumulateurs lithium-ion et consequences de leur mise en oeuvre dans une application transport. Ph.D. dissertation. Universite Claude Bernard—Lyon 1, FrenchGoogle Scholar
- 30.Bourlot S, Blanchard P, Robert S (2011) Investigation of aging mechanisms of high power Li–ion cells used for hybrid electric vehicles. J Power Sources 196:6841–6846CrossRefGoogle Scholar
- 31.Mahalakshmi S, Datchanamoorthy S (2015) Prediction of wireless sensor battery life. IEEE Autotestcon, National Harbor, MD, pp 138–145Google Scholar
- 32.Riviere E, Venet P, Sari A, Meniere F, Bultel Y (2015) LiFePO4 battery state of health online estimation using electric vehicle embedded incremental capacity analysis. IEEE vehicle power and propulsion conference (VPPC). Montreal, QC, p 2015Google Scholar
- 33.Wang J, Liu P, Hicks-Garner J, Sherman E, Soukiazian S, Verbrugge M, Tataria H, Musser J, Finamore P (2010) Cycle-life model for graphite–LiFePO4 cells. J Power Sources 196:3942–3948CrossRefGoogle Scholar
- 34.Soussens M, Laulheret R, Cabardaye A (2014) Modeles de degradation des batteries d’accumulateurs. CNES Notive, FrenchGoogle Scholar
- 35.Okubo M, Tanaka Y, Zhou H, Kudi T, Honma I (2009) Determination of activation energy for Li–Ion diffusion in electrodes. J Phys Chem 113:2840–2847CrossRefGoogle Scholar
- 36.Lievre A, Sari A, Venet P, Hijazi A, Ouattara-Brigaudet M, Pelissier S (2016) Practical online estimation of lithium–ion battery apparent series resistance for mild hybrid vehicles. IEEE Trans Veh Technol 65:4505–4511CrossRefGoogle Scholar
- 37.Damay N, Friedrich G, Forgez C (2016) Modelisation thermique en vue du dimensionnement d’un pack batterie, avec prise en compte du couplage electro-thermique pour la caracterisation des pertes electriques. Symposium de Genie Electrique, French. Grenoble, FranceGoogle Scholar
- 38.Mahbubul IM, Khan MMA, Ibrahim NI, Ali HM, Saidur R (2018) Carbon nanotube nanofluid in enhancing the efficiency of evacuated tube solar collector. Renew Energy 121:36–44CrossRefGoogle Scholar
- 39.Al-Waeli AHA, Sopian K, Chaichan MT, Kazem HA, Hasan HA, Al-Shamani AN (2017) An experimental investigation of SiC nanofluid as a base-fluid for a photovoltaic thermal PV/T system Author links open overlay panel. Energy Convers Manag 142:547–558CrossRefGoogle Scholar
- 40.Ali M, Khan SA, Sheikh NA, Gilani SI, Rashid TU (2017) Performance analysis of a low capacity solar tower water heating system in climate of Pakistan. Energy Build 143:84–99CrossRefGoogle Scholar
- 41.Hijazi A (2010) Modelisation electrothermique, commande et dimensionnement d’un systeme de stockage d’energie par supercondensateurs avec prise en compte de son vieillissement: application la recuperation de l’energie de freinage d’un trolleybus. Ph.D. thesis. Universit Claude Bernard - Lyon 1, FrenchGoogle Scholar
- 42.Damay N (2013) Modelisation thermique d’une batterie Li-ion prismatique de grande capacite et validation experimentale. JCGE 2013 Conference, FranceGoogle Scholar
- 43.Giegerich M, Koffel S, Filimon R, Grosch J, Fhner T, Wenger M, Gepp M, Lorentz V (2013) Electrothermal modeling and characterization of high capacity lithium-ion battery systems for mobile and stationary applications. IECON 2013—39th annual conference of the IEEE industrial electronics society, Vienna, pp 6721-6727Google Scholar
- 44.Shin D, Poncino M, Macii E (2014) Thermal management of batteries using a hybrid supercapacitor architecture. Design, Automation and Test in Europe Conference and Exhibition, DresdenGoogle Scholar
- 45.Jaguemont J, Dube Y, Bouton L (2016) Characterization and modeling of a hybrid-electric-vehicle lithium–ion battery pack at low temperatures. IEEE Trans Veh Technol 65:1–14CrossRefGoogle Scholar
- 46.DeMar P (2012) Thermal runaway—How to halt it and recover from it and How to detect it and prevent it. Intelec 2012, Scottsdale, AZGoogle Scholar
- 47.Faerdella R, Barruet C, Grenouilloux C (2016) Cyclage en temprature des profils complexes FIDES. 20me Congrs de matrise des risques et de sret de fonctionnement, Saint Malo, FrenchGoogle Scholar
- 48.Ali HM, Arshad W (2017) Effect of channel angle of pin-fin heat sink on heat transfer performance using water based graphene nanoplatelets nanofluids. Int J Heat Mass Transf 106:465–472CrossRefGoogle Scholar
- 49.Abdelhedi R, Ammari A, Lahyani A, Venet P (2016) Optimal power sharing between batteries and supercapacitors in electric vehicles. In: 7th International conference on sciences of electronics, technologies of information and telecommunication (SETIT)Google Scholar
- 50.Sato K, Yachi T, Takeuchi A, Babasak T (2011) Battery management for fuel cell/battery hybrid power supply system. In: IEEE 33rd international telecommunications energy conference (INTELEC), Amsterdam, NetyherlandsGoogle Scholar
- 51.Omar N, Daowd M, Hegazy O, Mulder G, Timmermans JM, Coosemans T, Van den Bossche P, Van Mierlo J (2012) Standardization work for BEV and HEV applications: critical appraisal of recent traction battery documents. Energies 5:138–156CrossRefGoogle Scholar
- 52.Savard C, Venet P, Pietrac L, Niel E, Sari A (2018) Increase lifespan with a cell management algorithm in electric energy storage systems. In: 19th International conference on industrial technology. Lyon, France, pp 1748–1753Google Scholar
- 53.Savard C, Niel E, Venet P, Pietrac L, Sari A (2017) Modelisation par un graphe de flots d’une architecture alternative pour les systemes de stockage multi-cellulaire de l’energie electrique.
*SEEDS Days—JCGE 2017*, French. Arras, FranceGoogle Scholar - 54.Savard C (2017) Amelioration de la disponibilite operationnelle des systemes de stockage de l’energie electrique multicellulaires. Ph.D. dissertation. Universite de Lyon, INSA de Lyon. FrenchGoogle Scholar
- 55.Jin F, Shin K (2012) Pack sizing and reconfiguration for management of large-scale batteries cyber-physical systems (ICCPS). IEEE/ACM Third International Conference. Beijing, China, pp 138–147Google Scholar
- 56.Beirao M, do Rosario-Calado M, Pombo J, Mariano S (2016) Balancing management system for improving Li-ion batteries capacity usage and lifespan. IEEE 16th international conference on environment and electrical engineering (EEEIC)Google Scholar
- 57.Martinez-Laserna E, Sarasketa-Zabala E, VillarealI I (2018) Technical viability of battery second life: a study from the ageing perspective. IEEE Trans Ind Appl 54:2703–2713CrossRefGoogle Scholar
- 58.Quinard H, Redondo-Iglesias E, Pelissier S, Venet P (2019) Fast electrical characterizations of high-energy second life lithium-ion batteries for embedded and stationary applications. Batteries 5:33CrossRefGoogle Scholar