Sssc Static Synchronous Series Compensator Matlab

Economic Valuation And Optimal Deployment Of

Static Synchronous Series Compensators for U.S.

Power System Expansion

Abstract—Flexible AC Transmission Systems (FACTS), par- ticularly Static Synchronous Series Compensators (SSSC), can improve network transfer capability and complement restricted transmission expansion. Evaluations of FACTS within large-scale, real-world power system planning are currently lacking. This paper develops a capacity expansion model for the contiguous U.S. power system toward 2050, incorporating SSSC-modified linear power flow equations and accounting for impedance feed- back in transmission expansion. Cost-optimal system expansion leverages widespread nationwide SSSC deployment on small-to- medium capacity lines and reduces the number of corridors to be reinforced. Overall, SSSCs reduce annualized system costs by $1.9 billion or decrease transmission expansion requirements by 20%. The most advantageous deployments achieving benefit- cost ratios of 59 concentrated in the Midwest, facilitating the delivery of central U.S. wind power to eastern load centers.

The value proposition of SSSCs is robust to cost sensitivities and potential competition from HVDC network expansion, and increases under higher demand growth and more stringent decarbonization policies. These findings provide a blueprint for leveraging SSSC deployment in the U.S. power system.

Index Terms—Capacity expansion, FACTS, grid-enhancing technologies, power system planning, Static Synchronous Series

B

Set of buses.

B

Index of buses.

C

Set of independent cycles in the AC network.

C

Index of independent cycles in the AC network.

G

Set of generators and electrolyzers.

G

Index of generators and electrolyzers.

I

Set of DC lines.

I

Index of DC lines.

I′

Reverse-direction counterpart of DC link i.

K

Set of periods.

K

Index of periods. Wei Ai and Michael T. Craig thank the U.S. National Science Foundation for funding under Grant #2142421.

Michael T. Craig is with the School for Environment and Sustainability,

L

Set of AC transmission lines.

ℓ

Index of AC transmission lines.

M

Set of piecewise-linear AC loss segments.

M

Index of piecewise-linear AC loss segments.

N

Index of fixed-point iterations.

S

Set of storage units.

S

Index of storage units.

T

Set of snapshots.

T

Set of snapshots within each period.

T

Index of snapshots.

Set Of Components Of The Corresponding Type

incident to bus b, e.g., G(b) and S(b).

0

Initial or existing value.

Char

Charge.

Dis

Discharge.

Electro

Electrolysis.

Fix

Fixed cost or fixed AC line capacity.

Idle

Idle state.

Pu

Per-unit.

Sto

Energy storage.

Var

Variable cost.

A

Network incidence matrix entry.

Α

Slope of AC loss-envelope segment.

Β

Intercept of AC loss-envelope segment.

C

Cycle incidence matrix entry.

Annualized Cost Coefficient (Slight Abuse Of No-

tation; c also denotes the cycle index in Sets).

D

Annual electricity demand.

D

Electricity demand.

E

Energy capacity.

E

State of charge.

Ε

Fixed-point iteration convergence tolerance.

Ε

Reserve margin requirement.

Η

Efficiency.

F

Transmission capacity.

F

Transmission capacity vector.

F N,Fix

Fixed AC line capacity to linearize iteration n.

F

Upper bound on transmission capacity.

F

Active power flow.

H

Transmission line length.

I

Maximum operating current.

L

Ohmic loss.

P

Power capacity.

P

Upper bound on generation capacity.

P

Dispatch, charging power, or discharging power.

¯P

Availability factor.

Qsssc

Three-phase SSSC reactive power capacity.

Three-Phase Reactive Power Exchanged Between

the SSSC and AC line.

˜Qsssc

Auxiliary SSSC control variable.

R

Deliverable reserve.

R

Resistance vector of AC line.

R

Resistance of AC line.

U

Scenario-specific transmission expansion limit.

V Sssc

Maximum SSSC injected series voltage.

W

Snapshot weight.

ˆX

Natural reactance vector of AC line.

X

Effective reactance of AC line.

Xsssc

SSSC-induced series reactance on AC line.

ˆX

Natural reactance of AC line.

E

LECTRIFICATION and large loads are pushing electric- ity demand upward, while wind and solar deployment is shifting generation toward resource-rich regions that are often far from load centers. Transmission expansion therefore increasingly serves multiple system functions simultaneously: it improves access to low-cost generation resources to meet growing demand –; smooths variability over wider ge- ographic areas , , ; and enhances resource adequacy during stressed conditions , , . At the global scale, IEA estimates that meeting climate goals requires adding or refurbishing more than 80 million kilometers of grids by 2040—an amount comparable to the entire existing global grid . Recent national-scale U.S. transmission studies find that transmission capacity must at least double by 2050 to achieve cost-optimal deep decarbonization .

Yet new transmission lines face long development timelines because siting, environmental review, cost allocation, inter- jurisdictional coordination, and local opposition frequently delay or derail projects , . In the U.S. and Europe, new overhead lines often take more than a decade to deliver . This challenge makes it increasingly important to use the transmission network more efficiently. Flexible AC Transmis- sion Systems (FACTS) based on power electronic devices can improve network transfer capability by redirecting flows away from congested lines and toward underloaded lines . They have been locally deployed to manage cross-border flows at major interties and enhance the transfer capability of existing corridors –.

However, the value, scale, and geographic pattern of FACTS deployment remain unclear in large-scale power system plan- ning, notwithstanding previous FACTS-transmission integrated planning studies. Related studies fall into two broad cate- gories. The first category develops formulations and algorithms for FACTS modeling and planning but validates them only on stylized systems, including equivalent-phase-shift model-

Ing , Ac Power Flow-Based Modeling , Resilience-

constrained planning , and distributionally robust plan- ning . The second category evaluates FACTS-transmission planning methods on practical systems. Zhang et al.

formulated a security-constrained multi-stage transmission ex- pansion problem with FACTS and proposed a decomposition approach for large systems, achieving a speedup of approx- imately 11× to 18×. Tests on the Polish 2383-bus system demonstrated that FACTS can lower total system cost by 1.1%.

Li et al. developed a two-stage robust model for coor- dinated wind generation, transmission, and FACTS expansion that accounts for ramping uncertainty and construction periods.

Tests on the Gansu provincial system in China showed that the proposed decomposition method reduced solution time by approximately one order of magnitude. Wu et al.

proposed an MILP formulation for transmission expansion planning with FACTS that directly models FACTS-induced flow changes and is tighter than the big-M method. The for- mulation achieved a 4× speedup and 40× fewer branch-and- bound nodes, and FACTS deployment reduced total system cost by 2.5% on the Texas test system.

These studies use large test systems to validate algorithmic scalability rather than to provide actionable planning insights. As such, the planning configurations are often arbitrary and overly simplified. Specifically, two of these studies , consider only transmission expansion, while the third considers only transmission and wind generator expansion.

Furthermore, across all of these studies, the number of mod- eled operational time periods is limited to a few isolated snapshots, rendering the models unable to capture the temporal dependencies of system operation as well as the variability of demand and renewables. In addition, AC transmission capac- ity expansion changes line impedance, creating a nonlinear coupling between transmission expansion and power flow that must be addressed to maintain physical consistency –.

This issue has not been jointly addressed alongside FACTS investment in all aforementioned studies. To yield actionable insights into the value and deployment of FACTS, a compre- hensive assessment must capture the full complexity of power system planning by representing diverse technology portfo- lios, existing brownfield capacities, impedance consistency, renewable resource heterogeneity, chronological variability, and credible demand and policy scenarios.

To address this research gap, we develop a brownfield, single investment period power system capacity expansion model for the contiguous U.S. toward 2050. This model co- optimizes the investment and operation of generation, storage, transmission, and FACTS. We consider Static Synchronous Series Compensators (SSSC) as a representative FACTS, as it features better performance than Continuously Variable Series Reactors (CVSR) and Thyristor Controlled Series Capacitors (TCSC) , can be modularized and easily installed , . Our main contributions are twofold. First, we develop a capacity expansion model that supports the investment and operational optimization of SSSCs while enforcing impedance consistency under transmission expansion. Second, we provide the first comprehensive, national-scale assessment of SSSC value within a power system planning framework under mul- tiple scenarios, allowing us to quantify system cost and trans-

3

mission expansion savings, identify cost-optimal deployment patterns, and delineate high-value deployment regions.

A. System Configuration And Input Data

The capacity expansion model is developed based on PyPSA-USA . Our model simultaneously optimizes the contiguous U.S. power system consisting of three AC sub- networks, the Eastern Interconnection, the Western Intercon- nection, and ERCOT, which are linked exclusively via DC ties.

The integrated network comprises 133 buses, 294 AC lines, and 15 DC lines. The model is initialized using the 2024 fleet of power generation and battery storage assets, curated from the Public Utility Data Liberation project . Transmission topology and transfer capacities are sourced from the ReEDS interface transfer limits dataset , with network impedance parameters derived from the TAMU synthetic grid . AC capacity expansion is restricted to existing corridors. Total demand comprises three components: exogenous non-AI de- mand, exogenous AI data center demand, and endogenously optimized flexible electrolysis demand. State-level load curves are from the NREL Electrification Futures Study and the EPRI Powering Intelligence 2026 report . State-level demand is further disaggregated to ReEDS zones based on population distribution. Meteorological inputs are based on 2012 weather data, with renewable energy capacity factors computed using Atlite . Techno-economic parameters for generators and batteries follow the 2040 projections from

The 2024 Nrel Annual Technology Baseline ; Those

for thermal energy storage are adopted from , , and electrolyzer parameters from . Expansion costs for AC and DC transmission are obtained from ReEDS . The cost of SSSCs is estimated based on project data submitted by Smart Wires, Inc. to CAISO . The total three-phase rated reactive power capacity of SSSC is 76.41 MVAr, and the total installed cost reported ranges from $4.0M to $5.4M, yielding a unit cost of around 60–80 $2022/kVAr. All costs are converted to 2022 US dollars.

To ensure computational tractability, our model approxi- mates full-year operations using four selected periods at three- hour temporal resolution. These periods, identified using the Python package tsam , comprise two typical periods to represent spring and fall (three-week) and two extreme periods to represent summer and winter (six-week). The six-week duration is chosen because indicates that thermal energy storage—the storage technology with the longest duration (under 100 hours) modeled in this study—does not provide flexibility over monthly timescales. Seasonal flexibility is implicitly captured by flexible electrolyzer operation to meet annual load requirement .

B. Mathematical Formulation

The standard capacity expansion model without SSSC in-

(1A)

s.t.

ℓ,T = 0,

∀c ∈C, t ∈T.

(1X)

Problem (1) seeks optimal investment decisions including generation capacity Pg, AC and DC transmission capacity Fℓ and Fi, and storage power and energy capacity P char

S ,

Es, informed by operational decisions including generation and electrolysis dispatch pg,t; storage charging, discharging,

S,T, Es,K,T; Ac And Dc Power Flows

fℓ,t and fi,t with associated ohmic losses lℓ,t; and deliverable reserves Rs,t, Rℓ,t, Ri,t from storage, AC lines, and DC lines. Equation (1a) is the optimization objective, which mini- mizes total annualized system cost from generation and elec- trolysis capacity, generation dispatch, AC and DC transmission capacity, and storage power and energy capacity. Equation (1b)

4

enforces nodal energy balance. The AC incidence matrix

Aac

ℓ,b ∈{−1, 0, +1} equals −1 if line ℓoriginates at bus b and +1 if it terminates there. The DC incidence matrix

Adc

i,b ∈{−1, 0, ηi} equals −1 if link i originates at bus b, and ηi if it terminates there. Equation (1c) imposes the annual national flexible electrolysis load.

Equations (1d)–(1g) impose nonnegativity and upper or lower capacity limits for generation, storage, AC and DC trans- mission. Equation (1h) is the scenario-specific total transmis- sion expansion limit, where DC corridors are counted at half of the sum over directional links because each physical corridor is represented by two unidirectional links. Equation (1i) enforces equal capacity on each DC link and its reverse counterpart.

For transmission lines, F includes both existing and expanded capacity, which does not affect optimality. Equations (1j)–(1n) bound dispatch from generators, elec- trolyzers, storage units, and DC lines. Equations (1o)–(1q) enforce storage energy dynamics, storage energy capacity limits, and cyclic consistency within each period. Equation (1r) approximates quadratic AC ohmic losses over fℓ,t ∈[−Fℓ, Fℓ].

This range is partitioned into segments m ∈M, each defined by the interval [Fℓ,m, Fℓ,m+1], where the boundary breakpoints

= Fℓ. The Segment

coefficients are computed from least squares fitting:

2 Df. (2)

Equation (1s) enforces AC thermal limits. Equation (1t) imposes a computationally efficient cycle- based linear power flow formulation , . For a con- nected network (B, L), graph theory guarantees the existence of a cycle basis C of cardinality |L| −|B| + 1 . The corresponding cycle incidence matrix C ∈{−1, 0, 1}|L|×|C| encodes the orientation of each edge relative to each cycle:

0

otherwise.

(3)

The contiguous U.S. transmission network comprises three AC sub-networks, each forming an independent connected component with its own cycle basis C(n). The overall cycle

= |L| −|B| + 3.

The cycle-based formulation is equivalent to the standard B-

ℓcℓ,C(Θi −Θj) = 0 Around Each Independent

cycle, which—together with the nodal balance (1b)—uniquely determines all branch flows . Its computational advantage is twofold: compared with the B-θ formulation, it eliminates the |B| auxiliary voltage-angle variables and enforces KVL directly on the branch flows; compared with the PTDF for- mulation, it preserves matrix sparsity and avoids the dense |L| × |B| distribution-factor constraints .

Equations (1u)–(1x) impose nodal energy reserve margin requirements and bound deliverable reserves from storage, AC and DC lines, where ϵ > 0 is the reserve margin requirement.

A. Sssc-Modified Linear Power Flow Equation

SSSCs control the active power flow by injecting a control- lable series voltage orthogonal to the line current . Under

,

∀ℓ∈L, t ∈T.

(4)

Because the SSSC is connected in series with the transmis- sion line, its injected-voltage capability is tied to the line’s

,

∀ℓ∈L.

(5)

Under varying line flow conditions, the SSSC reactive power injection is bounded by the product of the maximum injected

,

∀ℓ∈L, t ∈T.

(6)

Substituting (4) into (1t) and introducing the auxiliary variable

Sssc,ℓ,

∀ℓ∈L, t ∈T.

(8)

Equations (7) and (8) provide a linear model of SSSC oper- ation. Another linear model of SSSC based on an effective power injection formulation is also reported in .

B. Nonlinearity And Fixed-Point Iteration

Expanding AC line capacity Fℓintroduces nonlinearity into the capacity expansion problem through two mechanisms. The first is impedance feedback –: line impedance scales

(9)

so that the loss-envelope coefficients (αℓ,m, βℓ,m) in (1r) and

In (7) Become Nonlinear Functions

of Fℓ. The second is the line capacity-SSSC coupling: F pu

), Tolerance Ε = 10−3

Output: Optimal investment and operation decisions

N ←N + 1

7: return optimal investment and operation decisions from

The Last Lp Solve

appears in the denominator of the SSSC term in (7), an additional nonlinearity absent in conventional transmission expansion.

Both nonlinearities are resolved by a fixed-point iteration. We denote the objective expression in (1a) by objbase, and the SSSC-augmented objective adds the annualized SSSC investment cost to it. At iteration n, the capacity vector

L

is treated as a parameter to linearize (1r) and (7); the resulting LP (10) is solved for the optimal capacity Fn,∗

L ;

and the linearization point is updated via Fn+1,fix

L .

The procedure terminates when consecutive iterates satisfy

∥≤Ε ∥F0

L∥and is detailed in Algorithm 1.

(10A)

s.t. Constraints of problem (1) except (1r) and (1t),

!

= 0, ∀c ∈C, t ∈T.

A. Scenario Design

Given various uncertainties, we optimize capacity expansion across scenarios of demand, decarbonization, transmission ex- pansion, and SSSC deployment and cost (Table I). We compare outcomes under each scenario with and without SSSCs.

The model incorporates current decarbonization policies, including the federal Clean Air Act (CAA) and state-level Technology Capacity Targets (TCT), Renewable Portfolio Standards (RPS), and Clean Energy Standards (CES) , along with mandatory retirement of all oil- and coal-fired power plants by 2050. Two additional decarbonization scenar- ios are considered: a “Deep” scenario, which restricts natural gas generation to units equipped with carbon capture and storage (CCS) and caps their output at 10% of total generation

80 $2022/Kvar , 15-Year Cost Recovery Period

† Indicates the baseline scenario. (consistent with the core scenarios in ), and an “Aggressive” scenario mandating 100% zero-carbon generation.

For transmission expansion, our “Mid” scenario limits the annual expansion rate to 1.8 TW-miles—the observed annual maximum since 2014, consistent with the National Transmis- sion Study . The “Low” scenario halves this rate, while the “High” scenario sets a limit of 3.6 TW-miles, representing the annual maximum since 2009 or the expansion rate with reconductoring . DC expansion is permitted on 15 existing corridors in the baseline; the sensitivity scenario additionally allows expansion on 40 potential corridors identified from to examine the interactions between DC expansion and SSSC.

The lower bound of the SSSC cost estimate is adopted as the baseline to account for potential cost reductions of this technology and the upper bound is used for a sensitivity test.

The specific settings for the baseline scenario across the

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

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