Supercapacitor Energy Storage Matlab

Mihitha Maithripalaa, Zongli Lina

Virginia, Charlottesville, 22904, VA, U.S.A.

Abstract

The increasing deployment of distributed Battery Energy Storage Systems (BESSs) in modern power grids necessitates effective coordination strategies to ensure state-of-charge (SoC) balancing and accurate power delivery. While distributed control frameworks offer scalability and resilience, they also raise significant privacy concerns due to the need for inter-agent information ex- change. This paper presents a novel privacy-preserving distributed control algorithm for SoC balancing in a networked BESS. The proposed framework includes a distributed power allocation law that is designed based on two privacy-preserving distributed estimators, one for the average unit state and the other for the average desired power. The average unit state estimator is designed via the state-decomposition method without disclosing sensitive internal states. The proposed power allocation law based on these estima- tors ensures asymptotic SoC balancing and global power delivery while safe- guarding agent privacy from external eavesdroppers. The effectiveness and privacy-preserving properties of the proposed control strategy are demon- strated through simulation results.

Keywords:

Power delivery, battery energy storage systems, state-of-charge, distributed control, privacy preservation, dynamic average consensus,

1. Introduction

Battery Energy Storage Systems (BESSs) are increasingly essential for the operation of modern smart grids due to their ability to buffer the variabil- ity of renewable energy, support grid stability, and improve energy efficiency [1, 2, 3]. In most cases, a BESS consists of several battery units located at

Arxiv:2508.19345V2 [Eess.Sy] 5 Feb 2026

different sites, forming a multi-agent network. The distributed deployment of BESSs on a household, industrial, or community scale allows localized control, resilience to disturbances, and flexible energy management. How- ever, such systems introduce challenges in the coordination of energy storage resources while maintaining operational goals such as state-of-charge (SoC) balancing and total power delivery/tracking [4, 5]. SoC balancing is crucial for extending battery life, ensuring uniform aging across units, and prevent- ing overcharge or deep discharge conditions .

Centralized control approaches rely on global knowledge of all unit states, which limits scalability, increases communication overhead, and is prone to a single point of failure . In contrast, distributed control methods, partic- ularly those based on dynamic consensus or distributed optimization, have been extensively explored for scalable and reliable SoC balancing [6, 8, 9, 10, 11]. These methods enable agents to coordinate through local communica- tion with their neighbors, supporting robust performance even under partial connectivity or information loss.

Numerous decentralized control strategies have emerged in response to the growing complexity of power system applications, each tailored to specific objectives such as voltage regulation, power sharing, or balanced SoC. Many of these approaches rely on consensus convergence techniques and graph- theoretic connectivity to achieve coordination across the network [12, 13, 14]. As the implementation of distributed BESS control gains momentum, addressing diverse objectives under varying communication and operational constraints remains a key area of focus.

Despite their advantages, distributed control frameworks raise serious concerns about privacy. Many algorithms require agents to share internal information, such as SoC levels, power references, voltage values, or other state variables, with neighboring agents. This opens the door to inference attacks that can compromise the privacy of participating units [15, 16].

Two common adversarial models emerge in this setting. The first involves external or internal eavesdroppers who monitor communication channels to capture exchanged data and infer sensitive internal states or usage patterns.

External eavesdroppers are typically malicious third parties outside the sys- tem, while internal eavesdroppers may include compromised nodes within the network that have been taken over by an attacker. The second model involves honest-but-curious agents, who follow the protocol correctly but attempt to infer private information about their neighbors using the data received dur- ing coordination [15, 17].

2

enough communication samples, an attacker could reconstruct the trajec- tory of a neighboring unit’s SoC or estimate its power output. These threats are particularly relevant in collaborative, yet competitive environments, such as peer-to-peer energy trading platforms or energy communities, where par- ticipants aim to maximize their own benefit without disclosing operational strategies [15, 18].

Several approaches have been developed to address these privacy threats. Encryption-based methods, such as homomorphic encryption, enable com- putations on encrypted data without requiring prior decryption. This allows internal agents to process the data without accessing the underlying infor- mation. However, these methods are computationally intensive and often unsuitable for real-time applications [19, 20]. Differential privacy techniques address privacy by injecting calibrated statistical noise into shared data, pro- viding rigorous privacy guarantees. Nevertheless, they introduce trade-offs between privacy levels, accuracy, and convergence speed [21, 22, 23].

As

an alternative, methods such as state-decomposition have gained attention for their ability to preserve privacy while maintaining algorithmic perfor- mance [16, 18, 24]. This approach partitions the state of each agent into an observable (shared) component and a hidden (private) component. Only the observable part participates in distributed computation, whereas the private component remains hidden to safeguard sensitive information, preventing adversaries from inferring private information through shared data.

State-decomposition enables accurate algorithmic convergence while in- herently shielding sensitive information from exposure.

It Has Been Suc-

cessfully applied in dynamic average consensus and static average consen- sus [18, 24], and is now gaining interest in distributed optimization of energy systems [15, 16]. By embedding privacy into the control architecture itself, these methods inherently protect information without sacrificing performance that aligns well with distributed real-time control requirements.

In this work, we propose a novel privacy-preserving distributed control framework for SoC balancing in networked BESSs. We design a power allo- cation algorithm based on two estimators, the average unit state estimator and the average desired power estimator. Each battery unit independently estimates the average unit state and average desired power, and contributes to local power allocation. The proposed method enables accurate SoC bal- ancing and power tracking. The algorithm is lightweight, scalable, and well suited for implementation over an undirected communication graph network.

The contributions of the paper can be summarized as follows,

3

• We develop a privacy-preserving dynamic average consensus-based es- timator that leverages state-decomposition to protect the privacy of both individual battery internal states and their average. The proposed state-decomposed estimator safeguards the internal states of individual batteries against external eavesdroppers. Additionally, the algorithm enhances privacy by ensuring that each battery’s estimated average converges to a predetermined scaled version of the true average.

• We design a privacy-preserving distributed estimator for the average desired power, which achieves consensus over a scaled version of the true average. This approach prevents external observers from inferring global power demand while allowing each battery to recover the correct value locally.

• We propose a distributed power allocation law for BESSs that achieves SoC balancing and desired power tracking using the above privacy- preserving distributed estimators.

• We provide a theoretical analysis establishing convergence, stability, and privacy guarantees of the proposed framework and demonstrate its effectiveness through numerical simulations.

The remainder of the paper is structured as follows. Section 2 provides the preliminaries, including a description of the communication network and an attack model relevant to dynamic average consensus algorithm. Section 3 formulates the baseline control problem for coordinated SoC balancing and power tracking without privacy considerations. Section 4 introduces the pro- posed privacy-preserving distributed control algorithm, which incorporates estimators for both the average unit state and the average desired power. Sec- tion 5 presents simulation results that validate the effectiveness and privacy protection achieved by the proposed strategy. Finally, Section 6 concludes the paper.

2.1. Communication Network

We consider a BESS consisting of N networked battery units. Figure 1 provides an illustrative example to help to understand the BESS setup in a microgrid. The interactions among the battery units are represented by

4

Figure 1: The illustration of a microgrid including a BESSs. a graph G = (N , E), where N = {1, 2, . . , N} represents the set of nodes, each corresponding to a battery unit, and E ⊆N × N denotes the set of edges, each representing a communication link between units. If (i, j) ∈E, it means that communication from unit i to unit j is permitted. In such a case, unit i serves as an in-neighbor of unit j, while unit j acts as an out-neighbor of unit i.

The set of all in-neighbors of unit i is defined as Ni = {j ∈N | (j, i) ∈E}. A graph G is classified as undirected if the presence of an edge (i, j) ∈E automatically implies that (j, i) ∈E as well.

In the graph G, a path from node i1 to node ik is defined by a sequence of directed edges {(i1, i2), (i2, i3), . . , (ik−1, ik)}, where the nodes i1, i2, . , ik are all unique. The graph G is considered connected if every pair of distinct nodes can be linked through such a path.

The adjacency matrix corresponding to the graph G is given by A = [aij] ∈ RN×N, where each entry aij is equal to 1 if there exists a directed edge from unit j to unit i (i.e., (j, i) ∈E) and i̸ = j, and aij = 0 otherwise. If G is undirected, then aij = aji. The Laplacian matrix of the graph G is defined as L = [lij] ∈RN×N, where lij = −aij for i̸ = j, and lii = PN

K=1,K̸=I Aik. Its

eigenvalues are denoted by λ1, λ2, . . , λN. If G is connected and undirected, then L satisfies the following properties : λ1 = 0 < λ2 ≤· · · ≤λN,

Nδ = 0, It Holds That

δTLδ ≥λ2∥δ∥2. (Let 1N denote the column vector with N elements, all equal to 1). We impose the following assumption on the communication graph G.

5

Assumption 1. The communication graph G = (N , E) is undirected and connected. 2.2. Privacy Definition and Attack Model for Dynamic Average Consensus

Algorithm

This work examines a BESS consisting of N battery units that seek to collectively estimate the average of time-varying internal states xi(t) ∈R using the dynamic average consensus (DAC) algorithm:

J=1 Xj(T), Β > 0 Is A Design

parameter, and aij ∈{0, 1} represents the communication connexity between battery unit i and j. While this method ensures consensus, it requires each battery unit to share it’s estimate ˆxa,i(t), which implicitly encodes informa- tion about the battery unit state xi(t) and its derivative ˙xi(t).

In this work, we consider an external eavesdropper who has knowledge of the communication network (A, β) and access to all transmitted information ˆxa,i(t). The concern is that the eavesdropper can exploit these data to infer a battery unit’s private information xi(t) and ˙xi(t).

To demonstrate the vulnerability of the DAC scheme, we present the observer-based attack model from . Let variables vi(t), ξi(t), and φi(t) represent the eavesdropper’s reconstructions of ˆxa,i(t), xi(t), and ˙xi(t), re-

(2E)

where k1, k2, k3, k4 ∈R are positive design parameters, and ˆφ′

I(T) And Zi(T)

are auxiliary variables. Under the assumption that xi(t), ˙xi(t), ¨xi(t) ∈L∞, it has been shown that this observer guarantees uniformly ultimately bounded (UUB) estima- tion errors, and asymptotic recovery is possible if ¨xi(t) ∈L2. This confirms that the conventional DAC algorithm is vulnerable to privacy leakage unless privacy-preserving mechanisms are incorporated.

3. Problem Formulation

This study focuses on a distributed battery network comprising N coor- dinated units, where each unit is indexed by i ∈N = {1, 2, . . , N}. The SoC dynamics of each energy storage device are modeled using the Coulomb counting approach. Accordingly, the SoC of the ith battery unit at time

(3)

where Si(0) represents the initial SoC, Ci denotes the battery capacity, and the output current is represented by ii(t). The sign of ii(t) determines the operational mode of the battery: a positive current (ii(t) > 0) indicates discharging, while a negative current (ii(t) < 0) signifies charging. Differen- tiating equation (3) with respect to time results in:

Ci

ii(t).

(4)

The output power pi(t) of battery unit i is expressed as

(5)

where Vi(t) denotes the voltage output of the battery unit. Similar to the cur- rent, positive power (pi(t) > 0) corresponds to discharging, whereas negative power (pi(t) < 0) corresponds to charging.

Bidirectional DC–DC converters in battery energy storage systems are typically designed to regulate a constant output voltage . Therefore, we

7

simplify the model by assuming that each battery unit maintains a constant output voltage, i.e., Vi(t) = Vi. Under this assumption, substituting equation

Civi

pi(t).

(6)

This equation describes the dynamics of the SoC, taking into account both its output power and capacity. The aim of this work is to develop a privacy-preserving distributed power allocation strategy that ensures SoC balancing across battery units while meeting the required power demand in both charge and discharge operations.

The following are the control objectives for the BESS. Problem 1: Let the BESS comprise N interconnected battery units, where the SoC dynamics are governed by equation (6). The goal is to design distributed control laws for managing charging and discharging power such

That:

1) The SoC balancing among the battery units is achieved in steady state

T→∞|Si(T) −Sj(T)| ≤Ǫs,

i, j ∈N . 2) In steady state, the total charging/discharging power pΣ(t) follows the desired power p∗(t) within a predefined accuracy ǫp ≥0, i.e.,

I=1

pi(t). To solve this problem, we first analyze the power allocation problem in a non-privacy-preserving setting in Section 3.1, establishing the fundamen- tal principles and control laws necessary for effective SoC balancing. Subse- quently, in Section 4 we will introduce privacy-preserving mechanisms, ensur- ing that the distributed power allocation strategy maintains confidentiality while still achieving the desired system objectives.

Before analyzing the non-privacy-preserving case, as explored in Refer- ences , and , we first establish a set of mild assumptions concerning

8

the total power requirement and the extent to which individual battery units have access to this information. Assumption 2. The desired power p∗(t) is bounded such that p ≤|p∗(t)| ≤¯p and its rate of change satisfies | ˙p∗(t)| ≤ψ, with p, ¯p, and ψ being positive constants.

Assumption 3. The desired power p∗(t) is known by at least one battery unit in the BESS. 3.1. Power Allocation Law Based on Average Estimators in a Non-Privacy-

Preserving Setting

This section revisits the distributed power allocation algorithms intro- duced in and introduces a new battery average unit state estimator using the conventional dynamic average consensus algorithm.

The state of battery unit i ∈N is defined to facillitate power allocation

Xc,I(T) = Civi(1 −Si(T)),

(charging mode). Here, xi(t) denotes the amount of electrical energy that battery unit i is capable of storing during charging or delivering during discharging. As a result of physical limitations, positive constants a1 and a2 exist such that

T ≥0,

i ∈N .

(7)

From the SoC dynamics in equation (6), the expression for ˙xi(t) can be

(

˙xd,i(t) = CiVi ˙Si(t) = −pi(t), (discharging mode), ˙xc,i(t) = −CiVi ˙Si(t) = pi(t), (charging mode).

(8)

The power allocation for the ith battery unit is governed by the following control law.

J=1 Xc,J(T)P∗(T),

(charging mode).

9

To simplify the representation, we introduce the average unit state as

Pa(T) = 1

N p∗(t).

(11)

Using these definitions, the power allocation law in (9) can be reformu-

Pi(T) = Xi(T)

xa(t)pa(t).

(12)

Individual battery units may not have access to xa(t) and pa(t), as these are global parameters within the BESS. Therefore, these values must be locally estimated to enable decentralized implementation.

All individual battery units employ a dynamic average consensus scheme, as proposed in , to construct a distributed estimator for the average unit

(13)

where the local estimate of xa(t) is represented by ˆxa,i, and β > 0 is intro- duced as a design parameter. Note that although equation (13) tracks xa(t) with a steady-state error, which can be made arbitrarily small by tuning the value of β.

In a similar manner, the following distributed estimator is constructed to estimate the average desired power at battery unit i.

(14)

where ˆpa,i denotes the local approximation of the average desired power pa(t), and κ > 0 represents a design parameter.

10

Using these estimates, we refine the distributed power allocation strategy given in (9). The updated power allocation law given by

2 > 0 Is Added To Ensure That The

denominator does not become zero, making the power allocation law feasible to implement. Before proceeding to discuss the results related to distributed power al- location algorithm (15), we first review the convergence analysis of the esti- mators defined in (13) and (14).

Lemma 1. There exists a positive constant γs > 0 such that, for ev- ery β > 0, the estimate ˆxa,i(t), generated by the estimator (13), converges exponentially to a neighborhood of xa(t), that is,

= Γs < ∞,

where λ2 represents the smallest positive eigenvalue of the Laplacian matrix

L.

Lemma 2. : There exists a positive constant γp > 0 such that, for every κ > 0, the estimate ˆpa,i(t), generated by the estimator (14), converges exponentially to a neighborhood of pa(t), given by

T→∞Sup |ˆPa,I(T) −Pa(T)| ≤Ψγp

κ . Based on Lemmas 1 and 2, the following results on the distributed power allocation algorithm (15) were established in .

Theorem 1. : Given that the values of Ci and Vi are available, and con- ditions stated in assumptions 1, 2, and 3 are met, the distributed power allo- cation law (15) successfully solves Problem 1. Specifically, for any predefined accuracy levels ǫs, ǫp > 0, there exist sufficiently large parameters β, κ > 0 such that, for all i ∈N , both objectives of Problem 1 are achieved.

11

4. Privacy-Preserving Power Allocation Algorithm Design If eavesdroppers deduce private information via interception, the system’s security could be compromised. Disclosing this information may enable ad- versaries to launch attacks, increasing electricity generation costs, or even causing a power system outage.

In the networked battery system we are considering in this paper, each battery unit power pi(t) is private information that should not be leaked. Considering equation (8), we can conclude that pi(t) can be inferred if the attacker is able to infer ˙xi(t). We know that xi(t) (for discharging xd,i = CiViSi(t), for charging xc,i = CiVi(1 −Si(t))) is also private information for each battery unit that should not be leaked.

It Is Assumed That The

eavesdropper is aware of the communication network and can observe all information transmitted among the battery units. Therefore, information known to the eavesdropper includes ˆxa,i, ˆpa,i, A, β, and κ.

As discussed in Section 2.2, when dynamic average consensus is achieved using (13), the time-varying internal states xi(t) and their derivatives ˙xi(t) can be inferred by the external eavesdropper through the use of an observer model, assuming that xi(t), ˙xi(t) and ¨xi(t) are bounded. Therefore, we can- not guarantee the privacy of the networked battery system.

In the following subsections, we first introduce a privacy-preserving algo- rithm to estimate the scaled average unit state ηxa(t) of the networked bat- tery system. Next, we discuss how the leader-following consensus algorithm in equation (14) is modified to preserve the privacy of the average desired power pa(t). Finally, in the last subsection, these estimators are utilized to develop the power allocation algorithm.

4.1. Privacy-Preserving Distributed Average Unit State Estimator In this section, motivated by the schemes in [18, 24], we decompose the distributed estimator state ˆxa,i, the state of the battery unit xi, and its

I (0) = 2Ηxi(0),

where η > 0 with η̸ = 1 is a predefined scaling constant introduced as part

I (T) + ˙Xβ

i (t) = 2η ˙xi(t).

12

It follows that the sum of the decomposed states evolves consistently with

I=1

xi(t) = ηxa(t). Thus, the consensus value converges to a scaled version of xa(t). This scaling provides an additional layer of privacy, protecting the privacy of not only the individual states xi(t) and their derivatives ˙xi(t), but also the average state xa(t). In Theorem 2, we will prove that sub-states ˆxα

A,I(T)

convergence to a neighborhood of ηxa(t).

A,I(T) Serves As A Substi-

tute for the original state ˆxa,i(t). This includes responsibility for interaction among battery units and is the only state information of a battery unit i that is communicated to its neighbors j.

The Sub-State ˆXβ

a,i(t) is involved in the distributed update process by

A,I(T), While Remaining Hidden From Neighbors

of agent i. Therefore, only available data to the eavesdropper is A, β, and

ˆXα

a,i(t). Figure 2 presents a visual representation of state-decomposition in a net- work. After incorporating the decomposition, we reformulate the average

I (0),

i ∈N .

(16C)

4.2. Privacy-Preserving Distributed Average Desired Power Estimator We use (14) to estimate a scaled version of the average desired power pa(t), thereby enhancing its privacy.

13

shared over the network and thus is accessible to an external eavesdropper. Since ˆpa,i(t) converges to the true average desired power pa(t), this exposes sensitive information.

To mitigate this risk, we modify the estimator such that the consensus is performed with respect to a scaled version of the average desired power. Specially, the average desired power pa(t) is known by at least one designated battery unit, which injects a scaled version σpa(t) into the network.

All

battery units know the scaling factor σ > 0 with σ̸ = 1, but the eavesdropper does not. Therefore all other battery units can recover the true average

!

.

(17)

Through this scaling approach, the shared local estimates ˆpa,i(t) converge to σpa(t) rather than the pa(t). As a result, even though an eavesdropper can access all ˆpa,i(t), it cannot infer the actual average desired power without the knowledge of σ.

T→∞Sup |ˆPa,I(T) −Σpa(T)| ≤Ψγp

κ . This guarantees exponential convergence of the estimates to a neighborhood of the scaled version of the average desired power while preserving the privacy of the average desired power.

Assumption 4. All communication links in the network are encrypted dur- ing the initialization phase of the algorithm, enabling the scaling parameters η and σ to be securely shared among all battery units. These parameters are known to all battery units but are inaccessible to any external eavesdrop- per. Moreover, the scaling parameters η and σ are reinitialized each time the algorithm is executed and can also be refreshed following network reconfigu- ration events (e.g., unit addition or removal), thereby preventing long-term inference and further enhancing privacy protection.

4.3. Power Allocation Algorithm Based on Privacy-Preserving Estimators With the privacy-preserving estimators, it is ensured that ˆxα

A,I(T) Converges

to a neighborhood of ηxa(t), and ˆpa,i(t) converges to a neighborhood of σpa(t).

14

Figure 2: Explanation of state-decomposition process: (a) Original state before decompo- sition. (b) Decomposed state. Since the scaling factors are known to all battery units within the system, the original average state xa(t) and the desired average power pa(t) can be recovered by dividing the respective estimates by η and σ.

Therefore, the power allocation algorithm in (15) for the battery unit i is modified to incorporate privacy-preserving estimators as follows,

,

(charging mode).

(18)

Theorem 2. Consider the proposed decomposition framework in (16a), (16b), and (16c). Let assumption 1 hold. The tracking deviations ˆxα

Pn

j=1 xj(t) remain ultimately bounded, and the bounds on them can be made arbitrarily small by tuning the value of the design parameter β. Proof. The average state estimator given in (13) can be represented in matrix

˙ˆXa(T) = −ΒlˆXa(T) + ˙X(T),

ˆxa,i(0) = xi(0).

15

As shown in , for a given undirected connected graph, there exists a positive constant γs > 0 such that, for every β > 0, the estimate ˆxa,i(t) produced by the estimator in (13) converges exponentially to a bounded

= Γs < ∞,

where λ2 denotes the smallest positive eigenvalue of the Laplacian matrix L, which characterizes the connectivity of the graph. Consequently, (16a) and (16b) can be reformulated in matrix form, anal- ogous to (19), but incorporating a modified Laplacian matrix to account for

T .

The modified decomposition-based Laplacian matrix is defined as

,

where IN ∈RN×N denotes the identity matrix of dimension N. It is noted that the new Laplacian matrix L′ remains symmetric, as L cor- responds to the Laplacian of an undirected connected graph. Consequently, by Lemma 1, dynamic average consensus can be achieved.

′

2 represents the smallest positive eigenvalue of the Laplacian matrix

L′ And Γ′

s > 0 is a scalar constant.

I=1

xi(t) = ηxa(t).

2

.

A,I(T) In (16A) And (16B) Converge To A

neighborhood of a scaled version of the average consensus value corresponding to the original states, and the bounds on the steady-state errors can be made arbitrarily small by tuning the value of β.

Remark 1. The second smallest eigenvalue of the Laplacian matrix of a graph characterizes the convergence rate of consensus algorithms . Since intro- ducing the parameter η in the state-decomposition mechanism does not alter the Laplacian, the convergence rate remains identical to that of the conven- tional decomposition case. Furthermore, by Theorem 2, it is straightforward to see that the tracking errors are ultimately bounded in the same manner as in the conventional state-decomposition case.

17

Theorem 3. Let the BESS be composed of N interconnected battery units with known values of Ci and Vi, and assume that the conditions specified in Assumptions 1, 2, and 3 are satisfied.

The Distributed Power Alloca-

tion law (18), combined with the privacy-preserving estimators (16a), (16b), and (17), solves Problem 1. Specifically, for any predefined accuracy levels ǫs, ǫp > 0, there exist sufficiently large parameters β, κ > 0 such that, for all i ∈N , both objectives of Problem 1 are achieved.

Proof. We analyze the discharge phase and note that the charging phase follows similarly.

Σ

−pa(t).

T→∞Sup |Ex,I(T)| < A1

2η.

A,I(T) ≥A1Η −A1

2 . By Lemma 2, ep,i(t) converges exponentially to zero, leading to

T→∞Sup |Ep,I(T)| ≤Ψγp

σκ .

≤Nψγp

σκp . The steady-state discharging power corresponding to the ith unit is

Xd,I(T)

xa(t) + ex,i(t)(pa(t) + ep,i(t)).

Xa(T)

pa(t).

Xa(T)

−1.

Thus, In Steady State,

ξ−≤ξi(t) ≤ξ+. Choosing sufficiently large β and κ minimizes these bounds.

R(T) = Pa(T)

xa(t). Expressing pi(t) as a function of r(t) and ξi(t), we have pi(t) = r(1 + ξi(t))xd,i(t).

Civisi(T), It Follows

˙si(t) = −r(t)(1 + ξi(t))si(t).

Wij = 1

2(si(t) −sj(t))2.

Its Derivative Is

˙Wij = −r(t)(si(t) −sj(t))((1 + ξi(t))si(t) −(1 + ξj(t))sj(t)).

We Observe That ˙Wij < 0 If

(si(t) −sj(t))((1 + ξi(t))si(t) −(1 + ξj(t))sj(t)) > 0.

Deployment In Out-Of-Position Situations

D. Bendjaballah1, A. Bouchoucha1, M. L. Sahli1,2* and J-C. Gelin2

Abstract

Side-impact collisions represent the second greatest cause of fatality in motor vehicle accidents. Side-impact airbags have been installed in recent model year vehicle due to its effectiveness in reducing passengers’ injuries and fatality rates. In meeting these requirements, simulations of folding and deploying airbags are very useful and are widely used. The paper presents a simulation method for the deploying airbags using three materials in different working conditions. Finite element analysis is primarily used to evaluate this concept. In these simulations, the gas flow is described by the conservation laws of mass, momentum, and energy. The numerical results indicate that the FE method in this paper is capable of capturing airbag deploying process accurately.

Keywords: Airbag simulations, Out-of-position, Crash, Modeling, Out-of-position

Background

The passive safety of cars has become a very high prior- ity issue for the automotive industry. Today, there are not only one or two airbags in a car; certain models have ten times more than that. With the increasing usage of airbags, the number of accidents where the airbag itself can cause an injury to the occupant also increases

(Augenstein Et Al. 2003; Gabauer And Gabler 2010;

Audrey et al. 2011). As is well known, safety belts are also now devices designed to provide protection to the users of vehicles during crash events, minimizing the loads necessary to adapt their movement to the move- ment of the car (Freesmeier and Butler 1999; Schmitt et al. 1997). In general, the seat belt is designed to restrain the occupant in the vehicle and prevent the

Occupant From Having Harsh Contacts With Interior

surfaces of the vehicles. The airbag acts to cushion any impact with vehicle structure and has positive internal pressure, which can exert distributed restraining forces over the head and face. As a safety component of auto- mobile, an airbag decreases occupants’ injury likelihood effectively in case of an accident (Ruff et al. 2007). These safety elements can reduce the death rates on the roads, and its protection effects have been widely approved (Crandall et al. 2001; Teru and Ishikawa 2003). With computational tools such as finite element methods designed for dynamic contact problems, crashworthiness simulations can now be used with reliable accuracy to evaluate occupant protection in various collision condi- tions with safety metric/parameters such as acceleration, head injury criteria, intrusion distance, intrusion vel- ocity, and neck forces (neck injury risk or whiplash).

Thus, new types of airbag products are being developed to handle different collision scenarios.

Become Standard Equipment On Most New Passenger

vehicles (Braver and Kyrychenko 2004; Teng et al. 2007; Yoganandan et al. 2007). The airbag cushion is com- posed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. There- fore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Generally, air bags are designed to deploy in a crash that is equivalent to a vehicle crashing into a solid wall at 8 to 14 mph.

Air bags most often deploy when a vehicle collides with another vehicle or with a solid object like a tree. There are various types of airbags: frontal, side-impact, and curtain airbags. In general, the passenger side airbags are usually larger than the driver airbags (see Fig. 1).

Besançon, France

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Bendjaballah et al. International Journal of Mechanical

Doi 10.1186/S40712-016-0070-2

Extensive studies have shown that the airbag deploy- ment in load cases consists of two occupant loading phases: a punch-out effect where the airbag bursts out of its container with the airbag and airbag module cover accelerating towards the occupant and a second loading phase during which the airbag is taking on its deployed shape and volume (membrane-loading effect). Bankdak et al. (2002) developed an experimental airbag test system to study airbag-occupant interactions during close proximity deployment. The results provided insight for simulating the effect of inflation energy and mass flow on target response. Bedard et al. (2002) found that while left-side (driver-side) impacts accounted for only 13.5% of all crashes, the fatality rate among these

Crashes Was 68.3% In Comparison To Front Impact

(48.3%), right-side impact (31.3%), and rear impact (38.4%). These studies underscore the importance of oc- cupant safety during side-impact collisions. In the last years, the current market requested to reduce the time and cost airbag development. In order to achieve this result, virtual simulations play an important role since they allow to minimize the number of experimental tests (Pei et al. 2013; Cao et al. 2014). Several simulation models of airbag were established (Wang et al. 2007). It is feasible to optimize the parameters of airbag deploy- ment using simulation technology. Experimental and numerical studies have quantified injury risks to close- proximity occupants from deploying side airbags. These studies have focused on the prevention of the most ad- verse effects of airbag deployment (Duma et al. 2003).

Other studies have proposed airbag characteristics to minimize particular biomechanical responses (Haland and Pipkorn 1996). In a more recent study, Marklund and Nilsson (2003) compared deformation patterns with experimental data as well as the computational costs associated with three different airbag deployment simu- lation methods; they concluded that the SPH method is relatively inexpensive and produces incremental deform- ation patterns that compare most closely to the experi- mental results. The process of inflation of an airbag is one of the determining factors in saving lives. The duration from the initial impact of the crash to the full inflation of an airbag is about 40 ms, and during this time, the airbag goes from being in a folded state to a fully inflated state, with a high internal pressure. After achieving this state, the airbag begins to deflate, thus providing a nice cushion for the body impacting it.

Ideally, the person in the crash should come into contact with the airbag at this time. In the present study, a large volume passenger side airbag model is developed to handle different collision scenarios. The main aim is evaluate the performance of deploying of passenger side airbag using finite element methods (FEM).

Materials

The tensile specimens were made in different airbags (P: Peugeot, R: Renault, and VW: Volkswagen) with a length of 200 mm long and a width of 40 mm. Table 1 shows the mechanical properties of the airbag.

Tensile Tests

To determine the mechanical properties of the material of airbag used in the test pieces, tensile tests were performed on Lloyd EZ20 universal testing machine in Constantine. These tests were conducted using rect- angular samples. The axial force and axial displacement acquired during a test are converted into stress and the strain in order to be used for the fabric material model.

The continuous recording of the stress-strain data was performed during both the load and unload phases. A minimum of five samples were made in order to check the repeatability of the measurements. All the data was collected by using a PC-based data acquisition system and analyzed by commercial software. The picture frame test device that is made for this study is shown in Fig. 2.

Fig. 1 a Frontal and side airbags. b Oblique view of facet occupant model in sitting posture following airbag deployment (Lim et al. 2014)

0.150

Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

Page 2 Of 9

Figure 3 shows the stress-strain relationship of the airbag sample under axial tensile loads. The results are showing a linear increase in extension with the increas- ing stresses. This is an expected output and it confirms with the theoretical behavior of a sample subjected to tensile stress. The rupture strain values for different airbags (R/P/VW) were 0.322, 0.441, and 0.472, respect- ively. The measured elastic parameters (i.e., Young’s modulus E and initial yield strength) and Poisson’s ratio are summarized in Table 2. The tensile tests of the woven fabrics can show differences on mechanical prop- erties because woven fabrics can resist in-plane shear loads once the yarn lock-up angle has been reached. The differences of material property on material direction can affect the shape of fully deployed bag (see Fig. 3b).

Theoretical Background

Numerical simulations of airbags use very complex and techniques such as an orthotropic model to identify the mechanical behaviors during the airbag inflation and the fluid mechanics (gas flow) to describe the inflator gas flow (pressure gradient) and improve the representation of the pressures within the airbag. To model the airbag as an orthotropic model, three material constants have to be provided. Assuming a plane stress condition, the

Ð1Þ

where σ is the normal stress and τ is the shear stress, the subscript refers to the principal material directions, i.e., the fill and warp directions. Also, ε and γ are the strain components. The material elastic constants Qij are

Ð2Þ

where E1 and E2 are the Young’s modulus in the fill and wrap directions and G12 is the shear modulus of the fabric material. νij is the Poisson ratio of the material.

The gas exerts a pressure load on the airbag causing it to expand. This expansion puts the airbag under tensile stress lowering the expansion rate. In this study, heat conduction and heat transfer is not taken into account.

Fig. 2 A photograph of Lloyd EZ20 universal testing Fig. 3 Stress versus strain using Lloyd EZ20 machine for a three different airbags at 0° and 90° and b VW airbag test specimens at

Different Angles

Table 2 Physical and mechanical properties of the airbag

Page 3 Of 9

In the deployment of an airbag, an inflator supplies high velocity gas into an airbag causing it to expand rapidly. The gas inside the airbag is assumed to be ideal, to be of constant entropy, and to satisfy the equation of state:

Ð3Þ

Here p, ρ, and e are respectively the pressure, density, and specific internal energy, and γ is the ratio of the heat capacities of the gas. The gas flow is described by the conservation laws for mass, momentum, and energy that

Ð4Þ

here, V is a volume, A is the boundary of this volume,

N Is The Normal Vector Along The Surface A, And U

denotes the velocity vector in the volume. Applying Bernoulli’s equation in the case of an ideal gas with

Ð5Þ

Here, the subscript ex denotes quantities at the throat of the tube. Furthermore u, p, and ρ denote the quan- tities inside that part of the tube that is supplying mass.

Materials And Boundary Conditions

The airbag system mainly consists of three parts: the airbag itself, the inflator unit, and the crash sensor or diagnostic unit. Thus, to study the behavior of the airbag using FE simulations, we need to have an FE model of the airbag in the folded position. A FE model of the airbag was used to simulate the test condition as shown in Fig. 5. LS-DYNA® material model FABRIC (MAT_34) is used to simulate the airbag material. It is a variation of the layered orthotropic material model. Additionally, in the LS-DYNA® material model, fabric leakage can be accounted for. However, for this CAB material, the leak- age is almost negligible and therefore no leakage is specified. The mechanical properties can be determined from the physical test. Typical material properties for airbag fabrics are taken as given in Chawla et al. (2004a) (Table 3). These properties are used to simulate inflation process of airbag (see Table 1). The car dashboard is modeled as the rectangular thin plate using a MAT_RI-

Gid Material, And The Degrees Of Freedom Are Con-

strained in all the directions. The similar properties of thermoplastic polymer are assigned for contact purposes. The porosity of the fabric is assumed zero. The nitro- gen gas is taken for inflating the airbag. Properties of nitrogen gas and initial bag conditions are shown in Table 4. The example on which we perform the study is a typical passenger side airbag. The geometric de- tails have been measured from a commercially avail- able airbag. The initial state of the airbag is a closed rectangular whose sides are to be finished to 482 × 635 mm2 and is shown in Fig. 4.

Table 3 Material properties of airbag and rigid plate used in FE

–

Table 4 Initial values used for FE simulation of the swelling of

3.33 × 10−4

Fig. 4 The initial airbag geometry in the form of a rectangular Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

Related Journal Articles & DOI Links

Selected peer-reviewed publications relevant to 12 Lead ECG Acquisition. Click the DOI to access the full paper (may require institutional access).

Why Choose Us?

Bangalore guidance for robotics, Spectre and autonomous systems projects.