Hybrid electric vehicle power management strategy including battery ...

Hybrid electric vehicle power management strategy including battery lifecycle and degradation model François Martel, Yves Dubé, Loïc Boulon, Kodjo Agbossou Université du Québec à Trois-Rivières (UQTR), Institut de Recherche sur l’Hydrogène (IRH) Trois-Rivières (QC), Canada [email protected], [email protected], [email protected], [email protected]

Abstract-The following work is concerned with the inclusion of battery degradation mechanisms in power management strategies applied to current-generation hybrid electric vehicles (HEV). Using a real, physical electric vehicle as a basis for validation, a simulation model was realized to describe the behavior of the HEV and its components, including a lead-acid battery bank, an internal combustion engine (ICE) generator and a polymer electrolyte membrane fuel cell (PEMFC). This model is to be used in conjunction with a newly developed battery degradation model and included as part of a power management strategy aimed at reducing operating costs via an associated function, with the objective of studying the effects of battery lifecycle in such a strategy. As a first step towards the realization of this goal, this paper is aimed at describing the model of the HEV and its validation.

I.

INTRODUCTION

The world of today is on the verge of an energy crisis; a multitude of signs, such as climate change and tense political situations, are indicative that our current energy harvesting and transformation methods are reaching a critical point and must be revised [1]. As such, many alternative energy sources are currently being developed, along with a need to efficiently manage these increasingly diverse and complex energy sources. One particular area of research to benefit from power management is concerned with Hybrid Electric Vehicles, or HEVs. To this end, the Hydrogen Research Institute (HRI) recently acquired a commercial, battery-powered electric vehicle, the Némo (Fig. 1), and had it modified into an HEV. The main problem with the Némo Electric Vehicle (EV), in its original form, is that it relies solely on its lead-acid battery bank as a power source: because of their high sensitivity to deep discharge phenomena, these batteries often underperformed, as their operating range required an almost complete discharge over the course of an average day of use. In order to solve this problem, the Némo was retrofitted with on-board power sources, which could be used to recharge the battery bank on-the-fly between expected recharge intervals on the public power grid, providing an amount of control over their depth of discharge. However, this method raised an additional problem, as the fuel consumed to maintain the battery charge is very expensive. The main objective of the project became the following: how can the operating cost of the HEV be minimised,

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Fig. 1. The modified Némo hybrid electric vehicle, shown here with its onboard PEMFC and hydrogen gas cylinder. Also of note is the electrical outlet on the lower right, indicating plug-in recharge capabilities.

considering fuel expenditure from both auxiliary sources, grid recharge intervals and, more importantly, battery degradationrelated costs? While much work has been done towards power management in HEVs [2] [3], little effort has been deployed to include battery degradation in these strategies: the most common practice is to set fixed boundaries to battery state-ofcharge, based on manufacturer’s recommendations, and operate within that range [4] [5]. The main target of this work is therefore to explore the effects of this parameter when applied to power management strategies. This issue was first addressed via simulation, using MATLAB/Simulink software and the physical Némo HEV as an experimental platform for validation. While simulation models for batteries found in literature are relatively scarce [6] [7], an accurate and easily modifiable method was adapted [8], and a few references aimed at describing battery ageing and degradation [9] [10] were examined and used to build a practical degradation model. The power management strategy explored here is aimed at reducing operating costs of the HEV and is expected to both minimise fuel consumption and extend battery lifetime. This paper begins with a review of relevant background information concerning HEVs, followed by a detailed presentation of the models realised throughout the project. These models will then be put to use in a power management

process, aimed at reducing vehicle operating costs, followed by a discussion on the results observed. II.

BACKGROUND

A. The Némo EV The Némo EV, in its original form, was conceived as a small, low-speed vehicle (see Table I); it is meant to be used in industrial settings, in an indoor/outdoor utility role. Aside from its relative small size, its main defining attribute is its speed, which is limited to 40 km/h [11]. This is important to consider, as driving cycles most commonly used in vehicle simulation models cannot be applied directly [12]. From a research standpoint, the possession of a physical vehicle comes with both advantages and limitations. It provides an essential real-life platform onto which models and experiments can be validated; on the other hand, the vehicle comes already equipped with most components (Fig. 2), each requiring careful characterization in order to provide meaningful test results. In addition, research strongly indicates that lead-acid batteries might have been a poor choice given the task they were meant to accomplish, resulting in accelerated degradation [13] [14].

TABLE I ORIGINAL NÉMO SPECIFICATIONS

Physical Dimensions L 3,48m W 1,52m H 1,90m Wheels 175/70R13 Weight 896 kg Load capacity 453 kg Performance Top speed (limited) 40 km/h Acceleration (0-40 km/h) 6,5s Autonomy 115 km Powertrain Motor ACX-2043, 4,8 kW Transmission ratio 12,44:1 Batteries Battery pack 9 x 8V Battery type Deep cycle lead-acid Battery charger 1,3 kW III.

MODELS

B. The modified Némo HEV As mentioned before, the Némo EV was modified to become the HEV used for this study. Fig 3 shows the architecture resulting from these modifications. Included were an on-board Polymer Electrolyte Membrane Fuel Cell (PEMFC) and an Internal Combustion Engine (ICE) electric generator. This seemingly redundant design was done to provide a wider array of options, as it is explicitly meant to be used as a test bench for future research. Of particular note is the battery charging circuit built in the vehicle, which was only meant to interface with the power grid. It is correspondingly limited in its power transmission capabilities and becomes a bottleneck for the PEMFC. This was done for simplicity’s sake, as the fuel cell acquired outputs 110 VAC, much like the power grid.

The approach proposed by this paper is the modeling of the Némo HEV on a MATLAB/Simulink platform, to be validated using the actual Némo vehicle. As a general rule, models were made to be as accurate as possible, while also maintaining reasonable computing times. The finished model, on average, computes 250 times faster than real time, or about 4 minutes for a full 16h driving and recharge cycle. The model was of aimed at replicating the actual Némo HEV. As such, its many parameters were fitted to duplicate its performance characteristics, found in Table I, as well as measured battery, motor, and drivetrain data; ICE and PEMFC models were built around available manufacturer’s specifications [15] [16]. A general block diagram of the simulation model is found in Fig. 4. It includes a physical model of the HEV to describe its behaviour in real-world conditions. A second module describes the power and electrical elements of the vehicle,

Fig. 2. Original Némo EV architecture

Fig. 3. Modified Némo HEV architecture

such as its battery bank, generator and PEMFC [17], as well as all intermediate components like converters and control elements. Finally, the main power management and necessary data acquisition components were added to the ensemble. A. DC motor The actual Némo uses a 3-phase AC motor for propulsion. However, since it is much simpler to describe and performs similarly, a DC motor model was chosen and implemented in the simulation. This component is well-documented [18]; its basic equations are represented by (1) and (2) below.

di V R Kφ = − i− ω L dt L L

(1)

b dω Kφ i− ω = J J dt

(2)

Where V describes motor tension and corresponding current load i, L and R are motor coil inductance and resistance, respectively, J is rotor inertia, b viscous friction, Kφ is the motor emf constant, and ω is rotor angular speed. This motor model outputs current load and rotor speed in response to voltage input and resistive torque computed by the physical model, as seen in Fig. 4. B. Physical model This model represents the mechanical aspects of the vehicle through (3-7), namely its mass inertia Fi, wheel inertia torque Tw aerodynamic drag FD, axle bearing friction FF and gravitational pull FG, with all linear forces (noted Fx) translated into torques Tx via (8). These elementary equations [19] [20] aim to describe the response of the vehicle to speed and acceleration generated by the traction motor. The model also includes mechanical transmission reduction ratio and efficiency, based on available engineering data [11].

Fi = ma

(3)

1 Tw = α ⋅ mr 2 2

(4)

FD =

1 2 ρv C D A 2

(5)

FF = mgμ

(6)

FG = mg × sin(θ )

(7)

T = Fr

(8)

Where m is vehicle mass, a is linear acceleration, α is wheel angular acceleration, r is wheel radius, ρ is air density, v is linear speed, CD is vehicle drag coefficient, A is vehicle frontal area, g is gravitational acceleration, μ is bearing friction coefficient and θ is vehicle angle relative to a flat surface. C. Lead-acid batteries As part of the main focus of this study, the model for leadacid batteries was the object of much of the effort put in the project, both in modelling and literary research, which is relatively lacking on the subject [21] [22]. The chosen lead-acid battery model is shown in Fig. 5. It describes battery voltage output in relation to current, temperature and state of charge, and is modeled using the equivalent circuit approach [8]. The main hurdle in using this model resides in the fact that it requires 19 parameters, all of which need to be determined experimentally. A recommended protocol suggesting how to do so is found in literature as well [23] and goes a long way Torque resistance

Driver input

Speed demand

Power demand

Motor tension

DC motor

HEV physics

Torque, speed

Power management

Power demand

ICE generator

PEMFC

Power grid

Battery SOC

Fig. 4. Simplified model block diagram

Charge current

Battery tension Batteries

Current load

to simplify the procedure. Other studies were found to have used and applied this procedure, with excellent results [24]. Large modifications were made to the original model, in particular to its battery capacity evaluation component (11), which affects the behaviour of most of the electrical components of the circuit, and, of course, the all-important state-of-charge (SOC) (9) parameter.

SOC = 1 −

Qe C (0, θ )

(9)

dQe = −I m dt

(10) ε

⎛ θ ⎞⎟ K c C 0* ⎜1 + ⎜ −θ ⎟ f ⎠ ⎝ C ( I ,θ ) = 1 + ( K c − 1)( I / I *)δ

(11)

Where Qe is battery charge, C(I,θ) is battery capacity dependent on I, current load, and θ, temperature, θf is electrolyte freezing point, C0* ant I* are chosen nominal capacities and current load, respectively. Kc, ε and δ are experimental constants. Equation (11) is used to compute battery capacity based on two variables, battery current and temperature. High discharge currents and low temperatures adversely affect available battery capacity; its dependence on these two parameters is well-documented [26] and, incidentally, battery manufacturers provide extensive data on these phenomena. Using data extracted from Fig. 6-7 to determine capacity, the model was modified to circumvent (11) entirely. In addition to providing added reliability, it also eliminates the need to identify 3 of the original 19 experimental parameters, as well as the reliance on nominal current I*, which limits the range of application of the model to comparable values. Battery capacity-current curves typically include current measurements to provide useful readings, in most cases up to C100, or the current necessary for a 100-hour discharge time. Such low currents are unlikely to be encountered in realworld conditions; this is further evidenced by the

Fig. 5. Battery equivalent electric network [25]

manufacturer’s choice of the more-realistic C20 for nominal capacity values [27]. Extrapolation of these results draws an exponential curve, as shown in Fig. 6, where capacity tends towards unreasonably large values (up to infinity) when discharge current moves closer to 0, a phenomenon which the authors of the original model took special care to eliminate [8]. As such, the capacity-current curve was limited at values extrapolated linearly above the more reliable C20 values.

D. Battery degradation model Battery degradation is at the heart of the problem originally posed by the Némo, and as such was given a lot of attention. Literature concerned with modelling on the subject is also lacking, as the research behind it is still at an early stage [9]; however, some studies [10] [28] [29] indicate that modelling of this complex phenomenon can be broken down in two major approaches, physico-chemical and weighted Ah. Weighted Ah: battery total energy throughput is computed and compared with energy consumption during use. The rate of energy consumption is modulated, or “weighted”, by a variety of stress factors relating to operating conditions, such

Fig. 6. Battery capacity versus discharge current [27]

Fig. 7. Battery capacity versus temperature [26]

as SOC, discharge current, temperature, etc. The weighted Ah approach was deemed more appropriate to the project, both for its relative simplicity and fast computing times. It was further split into two distinct components: 2a) Lifetime prediction: this part of the model computes the remaining lifetime of the battery, using the method generally described by (12), until battery end-of-life is reached at 80% nominal battery capacity, according to conventional manufacturer’s practices [27]. LT =

Ahactual Ahnom

(12)

Where LT is lifetime spent, ranging from new (0) to endof-life (1), Ahactual is spent battery energy and Ahnom is total battery lifetime available energy. The model first needs a means to calculate the energy “spent” during battery use, Ahactual. This is simply done by integrating battery discharge current, much like the battery model (13).

Ahactual = ∫ I disch arg e

This has the effect of accelerating the available energy consumption proportionally to DOD. FDOD = 1 +

( Ahnom − Ahactual ) Ahnom

(15)

(13)

I weighted = I actual × FDOD

Total available battery energy throughput, Ahnom, is then determined using another piece of widely available manufacturer’s data concerned with expected lifecycle versus depth-of-discharge (DOD), shown in Fig. 8. Lead acid batteries are especially sensitive to deep discharge cycles; their expected lifetime is strongly tied the depth at which they are cycled (discharging, followed by a full charge) during their use. The data presented by Fig. 8 is then put through a simple formula (14) to determine total available battery energy, throughout its lifetime, according to its usage. Throughput = (Cnom × DOD) × LC F , DOD

Fig. 8. Battery lifecycles versus depth of discharge [27]

(14)

Where Cnom is nominal battery capacity, DOD is the considered depth of discharge and LCF,DOD is the lifecycle count to failure at the corresponding depth of discharge. The results of this are presented in Table II. Common practice is to compute an average for all calculated throughput and set this value as available energy, Ahnom. While this is an acceptable simplification for some battery models [9], in our case the energy values vary by as much as 33%, so basing the model on an average value is a strong assumption. What is instead proposed is setting the nominal available energy of the battery, Ahnom to the highest calculated value, and using (15) to determine a variable weighing factor that grows proportionally with deepening battery discharge. The battery discharge current is then multiplied by this factor (16).

(16)

Where FDOD is the weighing factor according to depth of discharge, Iactual is the current load experienced by the battery and Iweighted is the weighted current used by (13) to represent accelerated degradation, which becomes (17).

Ahactual,weighted = ∫ I weighted

(17)

2b) Performance degradation: as battery lifetime diminishes over time and discharge cycles, so does battery capacity; this module integrates the continuous loss of capacity into the main battery model to account for TABLE II CALCULATED BATTERY THROUGHPUT

Depth of discharge (%) 5 10 20 30 40 50 60 70 80 90 100

Number of cycles to failure 15000 7000 3300 2050 1475 1150 950 780 675 590 500

Calculated throughput (Ah) 137250 128100 120780 112545 107970 105225 104310 99918 98820 97173 91500

performance losses associated with ageing and degradation. This model uses the values computed by (12) as a basis. This “spent lifetime” is used in (18-19) and translated into lost battery capacity, which is subtracted from the nominal capacity value used to determine SOC by the main battery model. As such, battery behaviour is continuously updated to reflect its degrading condition; in addition, reduced capacity means depth-of-discharge is reached faster as days go by, accelerating degradation even further.

Cdeg = Cnom − Closs

⎛ I weighted ∫ Closs = (Cnom − (Cnom × 0.8)) × ⎜ ⎜ Ahnom ⎝

(18)

⎞ ⎟ ⎟ ⎠

(19)

Where Cdeg is degraded battery capacity, Closs is lost battery capacity, and 0.8 represents end-of-life point of 80% nominal capacity. It is crucial to note that this approach to battery lifetime modelling has severe limitations. The mechanisms responsible for battery degradation are not limited to DOD; they include corrosion, sulfation, electrolyte stratification, active material degradation, mechanical stress and water loss. These many mechanisms are themselves influenced by a variety of stress factors such as discharge rate, vibration, time and temperature, in complex and non-linear ways [9] [10] [29]. Most research already done using this method is based around the determination of multiple weighting factors meant to represent each of these mechanisms. To reach precise results by this method means performing many time-intensive tests; even then, a slight change in operating conditions can mean that even the best models will over-predict battery life by a factor of 2 or more [9]. Since the only reliable data available was the DOD versus lifecycles curve, it was chosen as the base of the only weighing factor determined by the model. One way to circumvent these shortcomings is to target the lifetime that was observed by the Némo manufacturer during its initial testing phase, which was discovered to be around 3 months under conditions similar to those proposed by the model. This information will be used as experimental lifetime data. Assuming that the vehicle is accurately modeled, a weighing factor resulting in a similar battery lifetime will provide a basis for further study.

E. Driving cycle The basis for the selected cycle was the commonly-used Urban Dynamometer Driving Schedule (UDDS), which aims to simulate abrupt stop-and-go driving conditions encountered in densely populated urban centers [12]. To comply with the 40 km/h speed limitation of the vehicle, the UDDS had to be scaled down, so that the 90 km/h

top speed demanded by the cycle did not exceed the HEV’s limits. The results of this scaling are shown in Fig. 9. In our particular case, zipping around university campus, a reasonable emulation of our targeted operating conditions, confirmed these choices as appropriate, as both driving behaviour and speeds observed were within the range of the modified UDDS. Finally, since this work revolves around battery discharge and their ensuing degradation, the 1370 seconds-long UDDS was found to be too brief to provide an appropriate measure of battery depletion. The solution proposed was simply to loop the cycle end-to-end to represent an average 8 hour day of work. Also included in this modified “work shift” cycle were an hour-long “lunch break” and two 15 minute pauses, corresponding to average industrial working schedules. While the addition of these might seem superfluous, in reality, long pauses do have an impact on battery behaviour. These breaks also bring the possibility of recharging the batteries through the power grid, as the vehicle is presumably parked for the duration. IV.

SIMULATION AND RESULTS

The ultimate goal of this work is to use the models described above and apply them to a power management strategy aimed at reducing operating costs, as well as prolonging battery lifetime to acceptable levels. As a first step towards this goal, the costs associated with battery degradation and fuel consumption were computed according to different loading scenarios. Since the HEV is configured in series, all power comes primarily from the battery pack, with secondary sources being used for recharge. As such, the main control mechanism available to manage power and reduce battery discharge lies in the determination of the best windows of opportunity, both in length and timing, during which recharge should take place. The function to be minimized is simply the total cost of operation of the vehicle (20).

Fig. 9. Scaled-down UDDS

t1

Ctotal (t0 , t1 ) = ∫ ( f cons × f cos t ) + Bcos t + Gcos t t0

(20)

Where Ctotal is total cost, t0 and t1 are the time intervals for recharge beginning and end, fcons is fuel consumption rate, fcost is fuel cost, Bcost is battery degradation cost and Gcost is grid energy cost. It would be an inacceptable oversight, given our present focus on lifecycle costs, to ignore PEMFC degradation, as fuel cells typically last less than 5000 hours depending on operating conditions [30]. However, as their cost is an order of magnitude above lead-acid batteries [15], an economic comparison between the two is a futile exercise. As such, our focus being mainly on the study of battery degradation costs, recharging options were explored using the ICE generator.

A. Recharge scenarios These different scenarios were established to evaluate recharge strategies best suited for the Némo HEV, with minimal costs as an objective. 1) Scenario 1: as a baseline comparison point, a simulation run under conditions comparable to those experienced by the original vehicle, with power provided exclusively by leadacid batteries, followed by a full recharge using the power grid after the driving cycle ends. 2) Scenario 2: adds recharge intervals on the power grid, which is the least expensive power source available to vehicle users. However, since vehicle must be stationary, intervals were set during breaks included in the driving cycle. 3) Scenario 3: similar to Scenario 2, with additional recharge to be provided by the ICE generator during vehicle use, with recharge intervals optimized to minimize costs. The Ctotal score results of these simulation scenarios are shown in Fig. 10. The difference observed between Scenarios

Fig. 10. Comparative results for HEV operating costs

1 and 2 reveals that daily operating costs are reduced by 30% with frequent grid recharge, while simultaneously prolonging their lifetime by 46% (Fig. 11). Also of note is the fact that these recharges are brief enough that additional grid recharge costs are almost negligible. This shows that a simple change in user habits, in this case plugging the vehicle to the power grid during breaks, can significantly reduce operating costs and extend battery lifetime. The difference between Scenarios 2 and 3 is even more telling, as costs are reduced to a minimum equal to 22% less. A breakdown of the total cost indicates that the vast majority of this reduction comes from the lessened battery degradation by an impressive 87% margin, enough to compensate for the added fuel costs. This shows without a doubt that, depending on operating conditions, battery degradation has a profound impact on the operating costs of HEVs and cannot be ignored. Closer examination of the battery state of charge progression during a full driving cycle (Fig. 12) shows that two main recharge intervals take place, the first lasting 39 minutes, and the second 41 minutes, for a total recharge time of 1 hour 20 minutes per 9 hour day cycle. This demonstrates that, while battery degradation is greatly diminished, it still provides the majority of the power required by the vehicle; had the results shown the majority of the vehicle’s driving power coming from the ICE generator, the entire point of using an HEV would have been negated. Finally, Fig. 11 demonstrates a major benefit of the recharging schedule performed in Scenario 3, the greatly extended lifetime of the batteries used in the HEV. While Scenario 2 provided significant improvements over the 3month durability of the original Némo, this new approach all but eliminates the problem, with an expected battery lifetime of over 3 years. This shows just how much depth of discharge has a strong impact on lead-acid battery lifetime.

Fig. 11. Comparative results for battery lifetime prediction

[3] [4]

[5] [6] [7] [8] [9] [10] [11] Fig. 12. Scenario 3 battery SOC profile, 1 day

V.

CONCLUSION

This paper has detailed a model built to accurately represent the behavior of a real, physical HEV, the Némo, with the inclusion of a battery degradation model made to study the effects of such a phenomenon when used in a costreduction focused power management strategy. Models are shown to behave in accordance to measurement and available specifications. The simulation leads to the following conclusions, which may be applied to HEVs in general: 1) Battery degradation has a significant impact on the operating costs of HEVs. Economic studies concerned with battery-equipped vehicles would be impaired if they chose to ignore this phenomenon, which is especially prevalent because of the particular conditions found in a HEV. 2) Effective power management strategies can be used to prevent battery degradation. This is particularly true for leadacid batteries, which are sensitive to depth of discharge, an aspect that is easily manageable with secondary power sources. 3) Battery degradation modeling is a complex and resource-intensive endeavor. While relatively simple in concept, it is greatly limited by the time and expense necessary to quantify its parameters with any measure of precision. ACKNOWLEDGMENT This work was supported by the Natural Sciences and Engineering Research Council of Canada and the Agence de l’efficacité énergétique du Québec.

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

[29]

REFERENCES [1] [2]

Rifkin, J. (2002), “The Hydrogen Economy”, J. Tarcher/Penguin Group (USA) Chau, K.T., Wong, Y.S (2002), “Overview of power management in hybrid electric vehicles”, Energy Conversion and Management 43, 1953–1968

[30]

Chan, C.C., (2007) “The State of the Art of Electric, Hybrid, and Fuel Cell Vehicles”, Proceedings of the IEEE | Vol. 95, No. 4, April 2007 Xiangjun, L, Liangfei, X, Jianfeng, H, Xinfan, L, Jianqiu, L, Minggao O. (2009), “Power management strategy for vehicular-applied hybrid fuel cell/battery power system”, Journal of Power Sources 191, 542– 549 Liangfei, X., Jianqiu, L., Jianfeng, H., Xiangjun, L., Minggao, O. (2009), “Adaptive supervisory control strategy of a fuel cell batterypowered city bus”, Journal of Power Sources 194, 360–368 Medora, N. Kusko, A, (2006), “An enhanced dynamic battery model of lead-acid batteries using manufacturer’s data” Tremblay, O., Dessaint, L., Dekkiche, A., “A Generic Battery Model for the Dynamic Simulation of Hybrid Electric Vehicles” Electrical Engineering Department, Ecole de Technologie Superieure M. Ceraolo (2000), “New dynamical models of lead-acid batteries”, IEEE Transactions on power systems, Vol. 15, No. 4, November 2000 Bindner, H., Cronin, T., Lundsager, P., Manwell, J., Abdulwahid, U., Baring-Gould, I., “Lifetime Modelling of Lead Acid Batteries”, Riso National Laboratory (2005) Uwe Sauer, D., Wenzl, H. (2008), “Comparison of different approaches for lifetime prediction of electrochemical systems—Using lead-acid batteries as example”, Journal of Power Sources 176), 534–546 Némo manufacturer’s website http://www.nev nemo.com/ Nemo_Presentation/ Accueil_Nemo.html US Environmental Protection Agency, (EPA) website http://www.epa.gov/nvfel/testing/dynamometer.htm Ruetschi, P. (2004), “Aging mechanisms and service life of lead-acid batteries”, Journal of Power Sources 127, 33-44 Divya, K.C., Østergaard, J., (2009) “Battery energy storage technology for power systems—An overview”, Electric Power Systems Research 79, 511–520 Mobixane PEMFC manufacturer’s website, online catalogue and specs: http://www.axane.fr/gb/products/mobixane/mobixane.html Honda internal combustion engine generator website: http://powerequipment.honda.ca/generators Barbir, F., (2005), “PEM fuel cells, Theory and practice”, Elsevier Academic Press, p. 1-16 Dorf, R., Bishop, R., (2002), “Modern Control Systems, 8th edition”, Addison-Wesley, p. 52-55 Beer, F., Johnston, R., (1999), “Vector mechanics for engineers: Dynamics”, McGraw-Hill Munson, B., Young, D., Okiishi, T., (2002), “Fundamentals of fluid mechanics”, John Wiley & Sons, p.573-592 Achaibou, N., Haddadi, M., Malek, A. (2008), “Lead acid batteries simulation including experimental validation”, Journal of Power Sources 185, 1484–1491 Linden, D., Reddy, T., (2002), “Handbook Of Batteries, 3rd Edition.” McGraw-Hill, New York, pages 23-44 to 23-53 Ceraolo, M., Barsali, S., (2002), “Dynamical models of lead-acid batteries: implementation issues”, IEEE Transactions on energy conversions, Vol. 17, No. 1, March 2002 El Kadri, K., “Plate-forme véhicule hybride: ECCE et ses différents composants”, Doctoral thesis Jackey, R.A., (2007), “A Simple, Effective Lead-Acid Battery Modeling Process for Electrical System Component Selection”, Copyright © 2007 The MathWorks, Inc Technical manual for Lifeline® batteries manufactured by Concorde Battery Corporation, Document No. 6-0101, Revision B, September 9, 2009 US batteries website http://www.usbattery.com/ Wenzl, H., Baring-Gouldb, I., Kaiserc, R., Yann Liawd, B., Lundsagere, P., Manwellf, J., Ruddellg, A., Svobodah, V., (2005), “Life prediction of batteries for selecting the technically most suitable and cost effective battery”, Journal of Power Sources 144, 373–384 Schiffer, J., Uwe Sauer, D., Bindner, H., Cronin, T., Lundsager, P., Kaiser, R., (2007), “Model prediction for ranking lead-acid batteries according to expected lifetime in renewable energy systems and autonomous power-supply systems”, Journal of Power Sources 168, 66–78 Schmittinger, W., Vahidi, A., “A review of the main parameters influencing long-term performance and durability of PEM fuel cells” Journal of Power Sources 180 (2008) 1–14