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Energy Conversion and Management 76 (2013) 437–452

Contents lists available at ScienceDirect

Energy Conversion and Management

journal homepage: www.elsevier.com/locate/enconman

Optimal energy management of HEVs with hybrid storage system

E. Vinot ?, R. Trigui

French Institute of Sciences and Technology for Transport, Development and Networks (IFSTTAR), 25 Avenue F. Mitterrand, 69675 Bron, France

a r t i c l e

i n f o

a b s t r a c t

Energy storage systems are a key point in the design and development of electric and hybrid vehicles. In order to reduce the battery size and its current stress, a hybrid storage system, where a battery is coupled with an electrical double-layer capacitor (EDLC) is considered in this paper. The energy management of such a con?guration is not obvious and the optimal operation concerning the energy consumption and battery RMS current has to be identi?ed. Most of the past work on the optimal energy management of HEVs only considered one additional power source. In this paper, the control of a hybrid vehicle with a hybrid storage system (HSS), where two additional power sources are used, is presented. Applying the Pontryagin’s minimum principle, an optimal energy management strategy is found and compared to a rule-based parameterized control strategy. Simulation results are shown and discussed. Applied on a small compact car, optimal and ruled-based methods show that gains of fuel consumption and/or a battery RMS current higher than 15% may be obtained. The paper also proves that a well tuned rule-based algorithm presents rather good performances when compared to the optimal strategy and remains relevant for different driving cycles. This rule-based algorithm may easily be implemented in a vehicle prototype or in an HIL test bench. ? 2013 Elsevier Ltd. All rights reserved.

Article history: Received 21 January 2013 Accepted 25 July 2013

Keywords: Hybrid electric vehicle Hybrid storage systems Battery and electrical double-layer capacitor coupling Energy management

1. Introduction Environmental issues are pushing the transportation sector to improve the ef?ciency of road-vehicles. Hybrid Electric Vehicles (HEVs) have a high potential to reduce fuel consumption and emissions. Due to their ability to recover kinetic energy while braking and to operate the engine in a more ef?cient area, CO2 emissions can be reduced [1–3]. The energy management and component sizing are critical factors to achieve a high energetic performance (fuel consumption). The cost and lifetime of batteries is a negative aspect, which prevents HEVs from being competitive in the market. To reduce the battery size and to avoid high battery current stress, a Hybrid Storage System – association of an electrical double-layer capacitors (EDLC [4–6]) and batteries - can be used. In such a con?guration, the battery can be designed to supply the energy while the EDLC is used for high power operations [7–10]. It is well known that EDLC associated with lead-acid batteries increase the regenerative braking capability of the storage system [7,11], thus, the fuel economy could be increased. At the same time it is well documented that associating EDLC with lead-acid battery can increase the battery lifetime. Depending on the application, a gain higher than 30% may be observed [12–15]. Due to an extended lifetime and a better fuel economy, a storage system associating

? Corresponding author. Tel.: +33 472142403.

E-mail address: emmanuel.vinot@ifsttar.fr (E. Vinot). 0196-8904/$ - see front matter ? 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.enconman.2013.07.065

lead-acid battery and EDLC can be a viable economic alternative to Li-ion or NiMh batteries [12–14]. The case of Li-ion associated with EDLC seems less convenient. The regenerative braking capability can be increased and it can provide bene?ts in terms of lifetime [5]. Nevertheless, the economic aspect of such an association seems currently not viable [16,17] even if the reduction of the EDLC costs may change this statement [5]. The method proposed in this paper can be applied to different battery technologies, but the proposed example concerns only lead-acid and EDLC association (Section 4). Off-line energy management optimization for HEVs with one electric energy source (a battery or EDLC) has been a major ?eld of research in the last ten years and two well-known methods are commonly used: Dynamic Programming [18] and Pontryagin’s minimum principle [19]. Although these methods can only be used in off-line simulation (drive cycle known in advance), they have two main advantages: (i) Evaluation of the maximum potential fuel economy of hybrid power-trains. (ii) Enabling studies on optimal component sizing for the considered hybrid architectures [20]. Methods that are implementable in real time have also been developed, and some of them are based on results from these two off-line optimization methods. The results achieved with these methods are always sub-optimal.

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Standard HEVs with two power sources have one degree of freedom to provide the required power output to the wheels. For HEVs with a Hybrid Storage System (HSS), an additional degree of freedom is introduced by adding a third power source. The energy management is therefore more complex and the optimal instantaneous power split between the three sources (fuel tank, battery, EDLC) is more dif?cult to be found. Some rule-based methods suggest the use of the EDLC for high power and/or high frequency output [7–10,21–23], while the battery power is limited and kept at low frequency as long as possible. However, to our knowledge, no previous research has studied the maximum potential of this hybrid architecture under optimal energy management. The ?rst idea to solve the optimization problem is to try an extension of the well-known methods previously developed for the conventional HEV case. For dynamic programming, this means adding a second dimension to the battery State of Charge (SOC) graph that represents the EDLC Open Circuit Voltage (OCV) graph. This is numerically dif?cult because the computational cost grows exponentially with the number of dimensions. The Pontryagin’s minimum principle theory, on the contrary, is more ?exible because of its low computational effort. In addition, the optimization problem can easily be formulated. In the following we will use this last approach that will be explained in detail until its application to the case of an HEV with HSS. As a result, the optimal energy management is found considering two objective functions to be minimized. These are the fuel consumption and the use of battery represented by the Root Mean Square (RMS) of the battery current. The Pareto front depending on these two costs is computed. Users may then choose a point on this front depending on the weight applied to fuel consumption and battery RMS current. In order to compare this optimal approach with possible real time performance, a rule-based energy management strategy was developed. A parametric study was carried out to improve the results of this method. Finally, a comparison between the optimal off-line solution and the best-found implementable rule-based management strategy is shown.

the maximum potential gains with respect to the objective function are determined. On the other hand, if no or only partial knowledge of future driving conditions is available, online energy management laws or instantaneous optimization algorithms can be applied. However these can only lead to suboptimal solutions. In a global optimization approach, the variation of the stored energy between the beginning and the end of the driving cycle has to be taken into account for all additional power sources. In our case, for consumption evaluation, this variation is chosen to be equal to zero as the considered vehicle is a non Plug-in HEV that uses a charge sustaining strategy for the batteries. Meanwhile, the developed algorithm can be extended to vehicles with other kinds of energy sources and could be applied to vehicles with energy depleting operation. 2.2. Optimization problem in backward approach In this section we will de?ne the optimization problem for the HEV case. The aim of this optimization is generally to minimize the global fuel consumption of the HEV for a known driving cycle. Knowing that the fuel cost depends on the torque of the Internal Combustion Engine (ICE) and its rotational speed the cost criterion for a driving cycle can be represented by:

" # n X J ? min C i ?T ICE ; XICE ? ? T s

i? 1

?1?

2. Of?ine energy management optimization of HEVs with HSS 2.1. Global optimization problem de?nition Generally, an HEV architecture (Fig. 1) is composed of a fuel tank, one or more additional power sources and a drive-train, which consists of an ICE, electrical machines (EM), clutches, gears, etc. The goal of the optimization is to ?nd the operation of the different power sources which minimizes a given criterion, often the fuel consumption. If the optimization is performed for an entire driving schedule, assumed to be known in advance, it is called global. Considering a driving cycle, such an algorithm determines the optimal energy management strategy for a given vehicle con?guration. As a result,

where (TICE, XICE) are the ICE torque and speed, Ci(TICE, XICE) is the fuel consumption for the time interval between t = i * Ts and t = (i + 1) * Ts. Ts is the sampling time and n the number of samples in the cycle. Knowing the driving cycle in advance, a backward approach (Fig. 2) can be used. Given the torque and speed of the wheel, calculating upstream through the drive-train, the operation of the engine and the electrical sources are calculated [20,24–25]. In such an approach, for each time step, we know the driving conditions which are de?ned by the wheel torque Tr and the wheel rotation speed Xr. To satisfy these driving requirements, the control unit has to make two decisions: (1) The power split ratio between ICE, electrical sources and mechanical brakes (BR). (2) The transmission ratio between the output shaft and the ICE, (in conventional vehicles this corresponds to the gear ratio). In an optimal strategy, the mechanical brakes come into action only when the EM torque is at its maximum braking capacity, and cannot satisfy the required torque Tr to decelerate the vehicle. In addition, power conservation laws and power-train constraints imply that: – TICE (ICE torque), TBR (Braking torque), TEM (Electrical Machine torque), are linked to Tr (Required torque). – XICE (ICE speed), Xw (wheel speed), XEM (Electrical Machine speed) are linked by the gear ratio and other potential coupling relations [25,26]. – TEM is linked to the electrical power Pelec provided at the output of the electrical source. In the following, a parallel HEV topology is considered and the gear will be imposed at each time step of the driving cycle. As for a given wheel condition (Tr, Xw) the ICE torque TICE depends only on Pelec (cf. Section 4.2 Eqs. (34) and (35)), and the ICE speed xICE is ?xed by the driving cycle and drive-train characteristics, the fuel cost consequently depends only on Pelec.

Fuel tank Electrical Sources 1 Electrical Sources 2

4 0 0

0

Energy/power Sources

4 0 0

Drivetrain : - engine -electrical machine - gear box - clutch - gear …

0

Fig. 1. General architecture.

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439

C (TICE, Ω ICE)

Gear Box

Chassis model

V

Pelec P0elec Pelec

Power train

Eelec Strategy

Fig. 2. HEV backward principle.

k

Therefore, the optimization problem may be expressed as follows:

to the HEV with a hybrid storage system. Speci?cally the utilized battery and EDLC models are described. 2.3.1. General principle Let us consider a system de?ned by the following state equation:

J ? min

" # n X C i ?P elec ?

i?1

? 2?

with a constraint on the amount of electrical energy stored Eelec: Eelec end ? Eelec ini ? DEelec . It should be noted that for series architectures and power split architectures, like the Toyota Hybrid System, the gearbox ratio may be replaced by the speed of the electric generator, which is then treated as a local variable of optimization [20]. This optimization problem can be solved in two ways. The ?rst method is the dynamic programming, which is based on the Hamilton–Bellman–Jacobi’s (HBJ) functional equation, or more generally Bellman’s principle of optimality [18,20]. The second method is based on Pontryagin’s minimum principle [27–30]. In previous works, these methods have already been applied to HEV. For example, dynamic programming has been used to compute optimal energy management strategies for different hybrid power-train architectures with batteries in of?ine simulation [18,31]. Pontryagin’s minimum principle has mainly been implemented for real-time management of parallel hybrid architectures [19,32,33] and fuel cell vehicle with battery or electrical doublelayer capacitor [34]. In these cases, only sub-optimal results can be achieved. In the scope of this paper the application of this principle with two control variables is not investigated as Lagrange multipliers (key variables to be set in the Pontryagin’s minimum method) are dif?cult to predict. This paper focuses on rule-based management easily implementable in real time while estimating their ef?ciency compared to the optimal energy management computed off-line using the Pontryagin’s minimum method. This paper highlights three speci?c points of application of the Pontryagin’s principle: The consideration of a hybrid vehicle with two electrical sources: electrical double-layer capacitor and battery. The resolution of the problem by modeling the components using look-up tables rather than analytical models. The determination of Pareto optimal front between two objectives: fuel consumption and battery RMS current.

_ ?t ? ? x

dx?t ? ? a?x?t ?; u?t ?; t ? dt Z

tf

?3?

and the functional J to be minimized:

J ?u? ?

g ?x?t?; u?t ?; t?dt

?4?

t0

where x(t) is the vector of state variables and u(t) the vector of control variables. Imposing constraints on ?nal time and ?nal state (as in our case) the Pontryagin’s minimum [27] principle can be used in the following way: Given the augmented functional H, also called Hamiltonian:

H?x?t?; u?t ?; p?t ?; t ? ? g ?x?t?; u?t ?; t? ? pT ?t? ? a?x?t?; u?t ?; t?

the necessary conditions for u to be the optimal control are:

?t ?;p?t ?;t ? _ ? t ? ? @ H ?x ?t ?;u ?i? x @p

?

?5?

*

_ ?t ? ? ? @ H?x?t?;u@x?t?;p?t?;t? ?ii? p ?iii? at each step of time and for all admissible control u?t ? H?x?t ?; u? ?t ?; p?t?; t ? 6 H?x?t ?; u?t ?; p?t?; t ?

Here p(t) is usually referred to the Lagrange multiplier and has to be determined in order to respect the constraints on the systems (cf. Section 3). pT(t) is the transpose of p(t). The following points should be noted concerning these conditions: – The ?rst condition (i) represents the system equation (Eq. (3)). – Together with the constraints of the system, the second condition (ii) allows the determination of the Lagrange multipliers (cf. Section 3.2). u?t ?;p?t ?;t ? – The third condition (iii) may be expressed as @H?x?t?;@ ? 0, if u the partial derivative with respect to u exists. Various works, where models with look-up tables were used, have implemented map ?tting to ?nd derivable functions in order to use this criterion [19–32]. However, this is not necessary; the original condition (iii) with its inequality can be used by ?nding the minimum of the functional H with an iterative process for example.

?

2.3. Problem solving using Pontryagin’s minimum principle In this part the Pontryagin’s minimum principle is applied. First the main principles of this method are presented and then applied

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– If the second partial derivative of H with respect to u exists, 2 then the condition @ H?x?t?;@uu?t?;p?t?;t? > 0 is suf?cient to guarantee ? that u causes a local minimum of H. – A key point of this method is if p(t) is known, the global minimization problem is reduced to a minimization of a local functional H. The determination of p(t) will be performed iteratively to respect the constraints (cf. Section 3.2). 2.3.2. Hybrid electric vehicle with hybrid storage system The derived method will now be applied to a hybrid drive-train where an electrical double-layer capacitor is coupled with a battery (Fig. 3). In this con?guration at least one inverter is highly recommended to couple the two electrical sources. It seems that a bidirectional DC/DC converter placed on the EDLC side is a good solution [5–36], in terms of cost and ef?ciency. As the ef?ciency of such a device is relatively high and quite constant the corresponding losses are represented here by a constant ef?ciency. 2.3.2.1. General electrical sources model. Fig. 4 shows the model of the electrical sources used in this paper (in the case of a battery). It is composed of an electrical voltage source and a series impedance (Sections 2.3.3.1 and 2.3.3.2) to take into account ohmic losses, faradic ef?ciency and possibly more complex phenomena (battery relaxation period. . .): U0batt: Open circuit voltage. Ebatt, P0batt: Stored energy and power of the perfect electrical source. Ubatt, Pbatt: Voltage and power at the output of the electrical source. In the case of a battery, the SOC (in%) is commonly de?ned as [37]:

?100 SOC ? 3600 ? C

Ibatt

losses

U0batt,P0batt Ebatt

Fig. 4. Electrical source model.

Ubatt,Pbatt

It should be noted that with Eqs. (7) and (8) a SOC constraint can be de?ned using the energy constraint:

DSOC ? f ?DEbatt ?

?9?

Thus, in the following, the SOC constraint is satis?ed while solving the problem using the amount of stored energy Ebatt. In the same way, a constraint on the EDLC open circuit voltage can easily be de?ned using the EDLC amount of stored energy EEDLC as:

Eedlc ?

1 2 CU 2 0edlc

?10?

2.3.2.2. Pontryagin’s minimum principle application in HEV with HSS. Considering the case of an HEV (Fig. 3), the control variables u(t) are represented by the electric power vector Pelec which components are Pbatt and PEDLC. Ebatt and EEDLC make up the state variable vector x(t), from here on denoted Eelec. All the parameters used in the following equation are de?ned in the nomenclature table in Appendix A. The system’s state equations may be expressed as:

_ elec ?t? ? P0elec ?Eelec ?t?; P elec ?t ?; t? E

with

R

batt

gf Ibatt dt

Eelec ? ?Ebatt P0elec ? ?P0batt Pelec ? ?Pbatt ( )

Eedlc ? P0edlc ? Pedlc ?

with gf ? 1 if Ibat > 0

where Ibatt is the battery current, gf is the faradic ef?ciency, and Cbatt the battery capacity in A h. In this paper, we also use the amount of stored energy Ebatt:

_ batt ? P 0batt ?Ebatt ?t ?; Pbatt ?t ?; t ? ? ?g ?Ebatt ? ? Ibatt ?t ? ? U 0batt ?Eelec ? E f _ edlc ? P 0edlc ?Pedlc ?t?; Eedlc ?t ?; t ? ? ?Iedlc ?t? ? U 0edlc ?t ? E

Ebatt ?

Z

?gf U 0batt Ibatt dt

? 8?

C(TICE,ΩICE)

Gear Box

Chassis model

V

I batt P0batt, U0batt Ubatt

Pbatt Pelec

Power train

Ebatt I edlc Strategy Pedlc

k

P0edlc, U0edlc,

Uedlc

E edlc

Fig. 3. Battery and UC coupling.

E. Vinot, R. Trigui / Energy Conversion and Management 76 (2013) 437–452

441

Eelec ?t ? ?

Z

0

t

P0elec ?t?dt ? Iedlc ?

Z

0

t

?gf ?Eelec ? ? Ielec ?t ? ? U 0 ?Eelec ?dt

Rbatt_discharge Ibatt U0batt,P0batt Ebatt Rbatt_charge Ubatt,Pbatt

with Ielec ? ?Ibatt U 0 ? ?U 0batt

U 0edlc ?

gf ? ?gfbatt 1?

The cost function to be minimized could be simple, such as fuel consumption, or could consist in a complex sum, for example fuel consumption plus emission plus battery RMS current. To minimize the global fuel consumption, taking into account the battery RMS current a weighting method is used. The objective to be minimized is:

Fig. 5. Model of battery.

J?

Z

0

T

C ?Pbatt ; Pedlc ? ? K ibatt ? I2 batt ?Ebatt ; P batt ? dt

?13?

2.3.3. Battery and electrical double-layer capacitor model In this section the partial derivative terms for ideal battery power, battery current squared and the ideal power of the EDLC in Eq. (15) are derived using the following battery and EDLC models. 2.3.3.1. Battery model. The battery model (Fig. 5) consists of an equivalent electric circuit with an open circuit voltage U0batt, an internal serial resistance Rbatt, and a faradic ef?ciency gf.U0batt, gf and Rbatt depend on the SOC (and thus the energy stored in the battery) and on the current using an experimental look up table. The temperature is ?xed for all the driving cycle but is chosen at the beginning of the cycle. In fact different look-up tables depending on the SOC and current can be used for different temperature. Nevertheless, no thermal model is used, and the parameters remain constant vs. temperature during all the driving cycle. This assumption seems acceptable as long as the driving cycle is short enough compared to the temperature rising time. In the same way, ageing can be taken into account if the battery has been characterized at different states of age [40]. It is noted that the dependence of U0batt, gf and Rbatt on SOC, battery current, temperature and life cycle are not included in the equation to avoid too complicated expressions. As an example, a Lead Acid (Orbital from Exide) 40 A h battery is presented as it is the battery used in the case study (Section 4). The OCV is measured after 25 min rest, and the resistance is determined using temporal method of identi?cation, i.e. the voltage drop during current pulse at different SOC. Fig. 6 shows the measurement of the OCV and the charge/discharge resistance at 25 °C. Fig. 7 shows the OCV partial derivative of OCV along the stored energy. The data are presented as a function of the energy stored in the battery, since this is the state variable of our problem. It is clear that other battery types such as Li-ion or Ni-Mh for example can be used. The method remains perfectly valid without any change if the battery model is the same. This method can also be adapted for other more accurate models [5,41–43]. In this case, the derivative of the battery power ideal sources and battery current squared along Ebatt has to be calculated. Using the model presented Fig. 5 and Ohm’s law the battery power can be represented by:

where C(Pbatt, Pedlc) is the fuel consumption which depends only on the electrical power composed of battery and EDLC power. Minimizing the battery RMS current is equivalent to a minimization of the integrated square of the battery current along the cycle (Eq. (13)). Moreover the weighted objective can be proven to be Pareto optimal [38,39]. In the case Kibatt = 0, fuel consumption alone is minimized. On the other hand, if Kibatt is high enough, only the battery current is minimized and the battery is often not used at all. In our case the Hamiltonian is de?ned by:

_ H?Eelec ; Pelec ; k? ? C ?P batt ; Pedlc ? ? K ibatt ? I2 batt ?Ebatt ; P batt ? ? p?t ? ? Eelec H?Ebatt ; Eedlc ; Pbatt ; P edlc ? ? C ?Pbatt ; Pedlc ? ? Kibatt ?Ebatt ; P batt ? ?p1 ?t? ? P0batt ?Ebatt ; P batt ? ?p2 ?t? ? P0edlc ?Eedlc ; Pedlc ? ?14?

with p?t ? ? ?p1 ?t? p2 ?t??. The conditions of the Pontryagin’s minimum principle are represented by:

elec ;P elec ? _ ? t ? ? ? @ H ?E ?ii? p @ Eelec 8 @ I 2 ?E ;P ? Ebatt ;P batt ? <p _ 1 ?t ? ? p1 ?t? @P0batt@?E ? K ibatt batt @ Ebatt batt batt batt ) :p _ ?t ? ? p ?t? @P0edlc ?Eedlc ;Pedlc ?

2

2

2

@ Eedlc

?15?

?iii? at each step of time and for all admissible controls ?Pbatt ?t? Pedlc ?t??

? H?Ebatt ; Eedlc ; P? batt ; P edlc ? 6 H?Ebatt ; Eedlc ; P batt ; P edlc ?

Depending on the battery and EDLC models, the partial derivatives are then constructed and iteratively determined as presented in Section 2.3.3. In discrete time (8.ii) becomes:

_ ?i? p

p?i? ? p?i ? 1? Ts

?16?

Pbatt ? U 0batt ? Ibatt Pbatt ? ?U 0batt ? Rbatt ? Ibatt ?Ibatt Rbatt ? Rbatt ch arg e if Ibatt < 0 with Rbatt ? Rbatt disch arg e if Ibatt > 0

And therefore:

This leads to:

p1 ?i? ?

p1 ?i?1??K ibatt

@ I2 ?Ebatt ;Pbatt ? batt Ts @ Ebatt

1?

@ P 0batt ?Ebatt ;Pbatt ? Ts @ Ebatt

p2 ?i? ?

p2 ?i?1? 1?

@ P0edlc ?Eedlc ;Pedlc ? Ts @ Eedlc

Ibatt ?

q?????????????????????????????????????? 1 U 0batt ? U 2 0batt ? 4P batt Rbatt 2Rbatt

?19?

Then (p1(0) and p2(0)) have to be ?xed in order to respect the constraints on the EDLC open circuit voltage and the battery SOC, (cf. Section 3.2). The expression of partial derivative terms @ I2 batt =@ Ebatt , @ P 0batt =@ Ebatt and @ P 0edlc =@ Eedlc used in Eqs. (15) and (17) depends on the model used for the battery and EDLC and are presented in the following section.

In addition

P0batt ? ?gf Pbatt ? Rbatt I2 batt ? ?gf U 0batt ? Ibatt

?20?

Using Eqs. (19) and (20) the battery power of the ideal sources P0batt can be computed by:

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13.2

12 11

x 10-3

charge discharge

13

open circuit voltage in V

10 12.8 12.6

resistancein ohm

9 8 7 6 5

12.4

12.2 4 12 0 50 100 150 200 250 300 350 400 450 3 0 50 100 150 200 250 300 350 400 450

battery energy in Wh

battery energy in Wh

Fig. 6. Open circuit voltage (left) and resistance (right) of a prismatic NiMh battery element.

6 5.5 5

x 10-3

Iedlc

Redlc Irx Ux Rx Icx Cx

dU0batt/dEbatt

4.5 4 3.5 3 2.5 2 1.5 1 0 50 100 150 200 250 300 350 400 450

Uedlc,

U0edlc,P 0edlc Eedlc

C

Fig. 8. Model of UC.

battery energy in Wh

Fig. 7. OCV partial derivative along stored energy.

P0batt ? ?

gf

2Rbatt

U2 0batt ? U 0batt

q?????????????????????????????????????? U2 0batt ? 4P batt Rbatt

I2 batt

U0batt, gf and Rbatt are functions which vary with the SOC. Once @ gf =@ Ebatt , @ Rbatt =@ Ebatt and @ U 0batt =@ Ebatt are determined this expression is easily implementable with existing software.

?21?

The partial derivative of P0batt and with respect to Ebatt can then be expressed as a function of @ U 0batt =@ Ebatt (Fig. 7), @ gf =@ Ebatt and @ Rbatt =@ Ebatt .

@ P0batt @ Ebatt

? ? 2R f

q?????????????????????????????????????? 0batt 0batt 2U 0batt @@U ? @@U U2 batt0 ? 4P batt Rbatt E E batt batt batt # @ U 0batt U2 @g 0batt @ Ebatt Rbatt ? ? 2R1 Rgf @ ? @E f ? p????????????????????????? @E 2

g

U 0batt ?4P batt Rbatt

batt batt batt batt

q?????????????????????????????????????? ? U ? U2 U2 0batt 0batt 0batt ? 4P batt Rbatt

g @ Ebatt ? ? 2R f U 0batt p????????????????????????? batt 2 U2 ? 4P batt Rbatt 0batt q?????????????????????????????????????? ? 2R1 U 0batt ? U 2 2 0batt ? 4P batt Rbatt batt q?????????????????????????????????????? Rbatt 0batt U 0batt ? U 2 ? @@U ? R1 @ 0batt ? 4P batt Rbatt @ E Ebatt batt batt # @U @R

4Pbatt

@ Rbatt

?22?

@ I2 batt @ Ebatt

2.3.3.2. EDLC model. The EDLC model is presented in Fig. 8. A simple model composed of one main capacity in series with a parallel RxCx and main resistor REDLC is used [44]. No self-discharge is considered as the time constant of this phenomenon is high compare to the time of the driving cycle used in this study. The parameters have been determined using temporal identi?cation with pulse current method. REDLC is determined by the voltage drop. The RxCx represent exponential behavior with time constant around 0.1 s. These parameters are independent of the charge of the EDLC and of the current. A constant DC/DC ef?ciency gDCDC is considered. As for the battery, more accurate models can be used as long as the derivative of the EDLC power ideal sources along Eedlc can be calculated. For example, the DC/DC converter ef?ciency may be function of Eedlc. In order to use Eq. (11) P0EDLC has to be expressed as a function of EEDLC and PEDLC. Using the model Fig. 8 and the Kirchhoff and Ohms laws we calculate:

x Icx ? C x dU dt x IRx ? U Rx

? ? p?????????????????????????? ? p????????????????????????? 2 2

batt

U 0batt

0batt @ Ebatt

2Pbatt @ Ebatt

U 0batt ?4P batt Rbatt

U 0batt ?4P batt Rbatt

?23?

Iedlc ? Icx ? IRx

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443

For a given EDLC current IEDLC, the current ICX in the capacitor Cx is easily derived from Eq. (23):

140

with

sx ? Rx ? C x

?24?

vehicle speed in km/h

t x0 Icx ? ? U ? Iedlc ? e?sx Rx

120 100 80 60 40 20 0 0 200 400 600 800 1000 1200

Then the power in the main capacity C can be expressed as the sum of the power in each component:

P0uedl ? U 0edlc Iedlc ?

Pedlc ? Redlc I2 edlc ?

U2 x ? U x Icx Rx

! ?25?

Thus using Eqs. (24) and (25) the EDLC current is de?ned by:

q????????? Iedlc ? ? s??????????????????????????????????????????????????????????????????????????????????????????????????????????? q????????? 2 2Eedlc 2Eedlc ?t =sx ? U x0 ? e ? U x0 ? e?t=sx ? 4P edlc ?Redlc ? Rx ?1 ? e?t=sx ? ? C C 2?Redlc ? Rx ?1 ? e?t=sx ?? ?26?

time in s

Fig. 9. NEDC cycle.

With the following de?nitions of Re and U0:

Re ? ?Redlc ? Rx ?1 ? e?t=sx ? q?????????? edlc U 0 ? 2E0 C

@ P 0edlc @ Eedlc

can be expressed as:

0 1

The pure electric mode is treated separately. In this case the global electrical power is imposed and the resulting Hamiltonian can be represented by a curve (red1 line Fig. 11). The two minima of the surface (hybrid mode) and the curve (pure electric mode) are identi?ed and compared. The minimum of the two is chosen and with this the operating mode and the share of power between ICE, Battery and EDLC is speci?ed. 3.2. Initial value determination for Lagrange multipliers Using the Hamiltonian previously de?ned (Eq. (14)), two Lagrange multiplier values have to be identi?ed in order to respect the two constraints on battery and EDLC OCV variation. One way to de?ne the initial values p1(0) and p2(0) is to ?x their ratio and then iteratively locate the p1(0) value such that the constraint on the battery SOC variation is ful?lled. Repeating this process for different ratios allows us to choose the ratio which minimizes the EDLC OCV variation as well. It should be noted that usually the battery energy is far greater than the EDLC energy. Therefore it is energetically more important to satisfy the constraint on the battery SOC than the constraint on the EDLC OCV. In Fig. 12 an example of this process is shown. Here p1(0) has been identi?ed so that the SOC deviation on the whole cycle is zero. For varying ratios p2(0)/p1(0) the ?gure shows the EDLC and battery SOC variation SOC (i.e. the soc difference between the initial and ?nal SOC). EDLC SOC is de?ned as 100% corresponds to maximum EDLC OCV. In order to respect a zero SOC deviation of the EDLC, the choice of p2(0)/p1(0) in this case is about 0.97, which leads to p1(0) = 3.180 and p2(0) = 3.326. 4. Results and discussion 4.1. Case study To illustrate the previous method, the case of a parallel mild hybrid architecture (Fig. 13) with two clutches is presented. The vehicle is a small compact car with the power-train characteristics presented Table 1. This architecture was tested in our laboratory on HIL test bench [46] in the scope of a project in collaboration with Valeo [47]. The components are those of a Renault Clio small compact car. This project tends to prove that a conventional architecture with only a boosted starter-alternator motor presents good performances in fuel economy. Only one battery block of 12 V was used to simulate a conventional architecture with one lead-acid starter battery. Even if the components are now no longer available in our test bench we disposes of a validated simulation model of this architecture and its components.

?t =sx 1 2 ?2t =sx C B U2 ? 2Re ?g Pedlc ?t=sx sign?Pelec? ? 2 U x0 e 0edlc ? 2U 0edlc U x0 e C @ P 0edlc 1 B DC =DC ? B1 ? U x0 e ? C ?? ? s??????????????????????????????????????????????????????????????????????????????????????????????? B C @ Eedlc 2U 0edlc Re C @ A 2 U 0edlc ?U x0 e?t=sx ? U 0edlc ? ? 4Re ?g Pedlc ?sign?Pelec?

DC =DC

?28?

where sign(Pelec) is the sign of the electrical power. With the derived equations the optimization problem can be solved using Pontryagin’s minimum principle. 3. Numerical resolution of the problem As presented in the previous section, applying Pontryagin’s minimum principle, two actions to solve the problem have to be performed in combination: At each time step, the minimum of the Hamiltonian with respect to battery current and EDLC current has to be found. The initial values of the Lagrange multipliers have to be identi?ed to take into account the constraints on battery SOC and EDLC open circuit voltage (Section 2.3.1). 3.1. Minimum of Hamiltonian To illustrate the described method, a parallel mild hybrid architecture with two clutches is used (Fig. 13). For the vehicle subsystems modeling, the components of the VEHLIB library [45] were used. Fuel consumption and machine losses are modeled by experimentally identi?ed look-up tables. The NEDC cycle (New European Driving Cycle, Fig. 9) is a common reference in Europe and was chosen here for its simplicity in analyzing results. For a given vehicle con?guration (described in Section 4), Fig. 10 shows the shape of the Hamiltonian at one instant of the NEDC cycle (here t = 940 s). As it can be seen, it is possible that the Hamiltonian, which is a function of battery and EDLC current, contains several local minima. In order to ?nd the global minimum a matrix approach was used; battery and EDLC current are sampled so that a grid is obtained on which the minimum is found (with a precision relative to the grid steps).

1 For interpretation of color in Figs. 11, 17, 18, 20 and 25, the reader is referred to the web version of this article.

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x 10

3.5

3

2.5

Hamiltonian

2

1.5

1

0.5

0 4800

0

-4800

-24000

-12000

0

12000

24000

Battery Power

EDLC Power

Fig. 10. Hamiltonian in hybrid mode at time 940.

x 10 2

4

1.9

1.8

Hamiltonian

1.7

1.6

1.5

1.4 4800

0

-4800

-24000

-12000

0

12000

24000

Battery Power

EDLC Power

Fig. 11. Hamiltonian in electric mode at time 940.

In the scope of this paper, a pack of EDLCs, and a bidirectional buck DC/DC converter [5,35,36] are added to this architecture. In fact the resulting architecture remains a conventional vehicle with slight modi?cations. Thus this system may be easily implemented on the base of an existing vehicle. A pack of 18 EDLCs was used because this is an available size for off-the-shelf EDLCs components from Maxwell. Obviously the proposed method can be applied to other component sizes or vehicle con?gurations.

From the driving cycle (Fig. 9), the wheel speed (xw), is known at each instant. The wheel torque (Tw) is then calculated using the vehicle model:

dxw T w ? J v eh ? Tf dt

?29?

where Tw is the wheel torque, Tf is the load torque calculated from the resistant forces, and xw is the wheel speed. Jveh is the overall inertia of the vehicle brought back to the wheels:

J v eh ? M v eh R2 w ? 4J w

?30?

4.2. Backward model of a parallel HEV architecture This section presents the model of the components used in the simulation and the linked equation to calculate in a backward way the electrical power required on the electrical network.

where Mveh is the global weight of the vehicle, Rw is the wheel radius and Jw the inertia of one wheel. Tf is calculated using chassis model i.e. aerodynamic coef?cient and rolling resistance of the tire:

T f ? Rw ? ? a ? a r ? V 2 ? b r V ? c r

?31?

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40 30 20 10 0 -10 -20 -30 0.85 battery EDLC

Δ SOC

The clutch model assumes that the torque Tcl is completely transmitted when the clutch is locked and that there is no speed sliding between the primary and the secondary speed of the clutch (xpc and xsc). In our case study, there is no gear between ICE and clutch i.e. they are on the same shaft. Thus, if the ICE minimum speed is higher than the speed imposed on the clutch by the wheel speed and gear ratio there is sliding. Otherwise, there is no sliding and no losses in the components. xICE is then de?ned by:

xICE ? xpc ? min?xICE

min ;

xsc ?

?33?

The speed of the electrical motor xmot is calculated using the gear ratio of the coupling device. The torque Tmot is determined using the look up table of the machine losses depending on the control variable Pelec and the speed.

0.9 0.95 1 1.05 1.1

T mot ?

Pelec ? losses?Pelec ; xmot ?

ratio p2/p1

Fig. 12. Battery and EDLC SOC variation in%.

xmot

?34?

The ICE torque is:

?T mot ?xmot ? T ICE ? T cl ? Rcoup gsign T mot coup

?35?

Table 1 Vehicle parameters. Vehicle weigh ICE power EM power Ratio (EM speed/ICE speed) Battery type Battery max/min current EDLC type Ultracapacitor max/min voltage Ultracapacitor max/min current 1073 kg 54 kW@4000 rpm 15 kW @ 4000 rpm 2 Lead acide (orbital from exide) 40 A h 300/?105 A Maxwell 144 F 48/22 V 500/?500 A

where gcoup is the ef?ciency of the coupling device and Rcoup is the ratio. Sign(Tmot?xmot) is the sign of the electrical machine power. The fuel consumption C(Pelec) is then deduced using a look up table [25]. 4.3. Comparison between HSS solution and battery only solution A study has been performed to compare the performance of the hybrid con?guration with battery and EDLC to that of a hybrid vehicle that uses only a battery. This allows us to show the advantages of the HSS system. The comparison is carried out using the models described above and the Pontriagyn’s minimum principle method applied to each con?guration. Fig. 14 shows the Pareto optimal front of the two storage con?gurations (fuel consumption vs. RMS battery current) for the NEDC cycle and a real-life urban driving cycle. The Pareto front is obtained by varying the weighting factor of battery RMS current (Eq. (13)). Globally, better fuel consumption is obtained for similar battery current in the case of the HSS. For a given RMS battery current, a gain in fuel consumption between 15% and 25% can be noted in the case of HSS. Similarly, the RMS current can be reduced with a compromise of higher fuel consumption.

where Rw is the wheel radius, V the vehicle speed, a, the aerodynamic coef?cient of the chassis, and ar, br, cr the rolling resistance coef?cients. The torque relation on the shaft of the clutch (Tcl) is:

T cl ?

Tw

?T w ?x w ? sign?T ?x ? gsign Rtrans ggb w w Rgb trans

?32?

where gtrans is the ef?ciency of the transmission (axle plus gear) and Rtrans is the transmission ratio. Sign(Tw?xw) is the sign of the wheel power. ggb and Rgb are the ef?ciency and ratio of the gear box. One ef?ciency value and one ratio are affected to each gear number.

C(TICE ,ΩICE )

TICE ΩICE

Coupling

Clutch

Gearbox Gear Τ cl , Ωsc k

Chassis model

P0batt

Pbatt

Pbatt

Pelec

0

Τ cl , Ωpc Tmot Ωmot

Vveh

Ebatt

Pedlc P0edlc

Eedlc

Strategy

Fig. 13. Parallel two clutches architecture.

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Pareto front on nedc drive cycle

4 5 BATT + UC BATT+EDLC UC EDLC 3.8 4.8 4.6 4.4 4.2 4 3.8 3.6 3.4 3 0 20 40 60 80 100 120 3.2 0

Pareto front on urban drive cycle

BATT+EDLC BATT + UC UC EDLC

f uel consumption in l/100 km

3.6

3.4

3.2

f uel consumption in l/100 km

20

40

60

80

100

battery rms current in A

battery rms current in A

Fig. 14. Pareto front in NEDC and urban driving cycle with and without EDLC.

It has been shown in previous studies that HSS can increase the lifetime of lead-acid batteries by 30% and more [12–14]. However, this depends strongly on the technology and on the battery usage. It is therefore dif?cult to predict the lifetime gain accurately. Considering the potential gains in fuel economy and lifetime, solutions such as HSS systems that combine lead-acid batteries with EDLC are worth studying as an alternative to Li-ion solutions [12–14]. The rest of the paper deals with a good and simple way to implement an energy management law of the overall system in the vehicle that can give fuel economy results close to the expected optimal one. 4.4. Comparison with parameterized rule-based method The previously presented optimization method can be applied only off-line because the computation effort is too high to be

implemented on-line. To implement a real time method, we propose here a rule-based method. The parameters of this method are tuned to obtain an ef?cient control and the results are compared with the optimal method.

4.4.1. Presentation of the rule-based method Fig. 15 presents the rule-based strategy principle. It consists in two main steps. In the ?rst step the operating mode (electric or hybrid) and the required electrical power Pelec are ?xed. This part is comparable to a load-following charge-sustaining strategy as in the parallel hybrid vehicle case [48]. In the second step the power share between battery and EDLC, i.e. the battery target power Pbatt, is de?ned. Various management strategies of hybrid storage systems have been proposed [7–

Step 1 : setting of required electrical power

Required Gear box torque Required speed Vehicle Speed Vveh

Motor speed

Vehicle model

×

Required Battery power

+

Required System power

Hyb/elec choice

Required Engine power

Required Motor model Electrical Tmax + limitation Power Pelec

Engine model + limitation Tmax Engine Torque Ω TICE

SOC + condition SOC Vehicle speed Hybrid time Engine Speed ΩICE

+ -

Cou pling

Required Motor torque

Ω Available Battery+ EDLC power

SOC

System losses

Step 2 : setting of battery target power

Target batt power Ptarg_batt

First order

1

1+ j.f/ fc

+

Required UC power Preq_EDLC min

+ EDLC power -

Battery power Pbatt

Available EDLC power

Fig. 15. Rule-based management strategy.

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9,21,22]. In this paper, a low-pass ?rst order ?lter approach is used [21,35]. To apply such a strategy in real time, a bidirectional DCDC converter placed between the battery and the EDLCs is used. Then the control management (developed below) provides a EDLC required current (or power) which is performed using a closed loop control on the DCDC converter duty cycle.

4.4.1.1. Step 1. Given the current speed and the desired speed, the torque and power at the gear box is calculated using the developed vehicle model [48]. Depending on the SOC of the battery, a desired battery power (negative or positive) Preq_batt is de?ned and added to the required vehicle power. This vehicle is operated in hybrid (ICE on) or electrical mode (ICE off, where only battery and EDLC provide power). The choice between the two modes is made depending on two different parameters: the required power versus battery SOC curve and the vehicle speed. In general operation, a function of power versus battery SOC is given (ICE switch on power, Fig. 16a). If the total required vehicle power is lower than this curve for the current SOC the vehicle is operated in electrical mode. The hybrid mode is used if the required vehicle power exceeds this curve. However if the vehicle speed is higher than the maximal electrical vehicle speed, the vehicle is forced into hybrid mode. If the mode is changed from hybrid to electric mode, the time passed in hybrid mode ?rst has to be validated. This constraint is imposed to reduce frequent changes between modes. Knowing the required system power and operating mode (electrical/hybrid), the engine torque is calculated (zero in electric mode), and the electrical motor torque can be evaluated. Thus, using motor losses and performances model, the required electrical power Pelec is known. 4.4.1.2. Step 2. Once the required electrical power is ?xed, the second step has to decide the power sharing between battery and EDLC. Here a ?rst order low pass ?lter with a cut-off frequency fc is used on Pelec to compute the battery target power Ptarg_batt. Taking into account the available EDLC power, the EDLC is then used to provide the difference between Ptarg_batt and Pelec. If the desired EDLC power exceeds the available EDLC power (maximum/minimum current and/or voltage) the battery is used to provide the necessary additional power to ful?ll Pelec. Overall, with this strategy the battery satis?es the mean required power while high frequency power peaks are provided by the EDLC. Moreover, if Preq_batt is correctly tuned (Section 4.4.2) the charge sustaining mode is guaranteed. This behavior can be ensured using a speci?c battery power vs. SOC characteristic. As seen in Fig. 16 such a characteristic would

discharge the battery (positive Preq_batt) when the SOC is high, and charge the battery (negative Preq_batt) if the SOC is low. With this approach, at high SOC, the battery tends to discharge in boost mode (battery and engine provide power to the wheels). The boost mode is not necessarily a very ef?cient operation. We can inhibit this mode by setting the high threshold (Preq_max) to zero. At low SOC battery tends to be recharged using the engine. Note, if the required power cannot be provided by the engine alone, the electric motor has to be operated in boost mode (within the scope of the system capability). 4.4.2. Parametric studies of the rule-based method One problem using a rule-based approach is to ?nd good values for the different control parameters in order to achieve an ef?cient control. Here the characteristics of the required battery power vs. SOC (Fig. 16a) and ICE switch on power vs. SOC (Fig. 16b) will be de?ned using seven parameters (Table 2). The cut-off frequency of the low-pass ?lter is used as a parameter as well. A parametric study has been performed on 8 parameters (Table 2). A range and step size were chosen for each parameter. For each combination of these, an iterative method was used to ?nd the initial battery SOC and EDLC open circuit voltage that lead to charge sustaining operation (DSOC < 0.1% and DU0EDLC < 1 V). Figs. 17 and 18 show the fuel consumption vs. RMS battery current for the resulting points, which respect the described constraints. The different colors in Fig. 17 represent groups with the same cut-off frequency. To simplify the ?gure, only selected values of cut-off frequency are presented. In Fig. 18, points with the same color represent results with equivalent required battery power (Preqmin) selected values. Fig. 17 shows that the RMS battery current is highly in?uenced by the cut-off frequency as the clouds of points goes to smaller battery current when the cut-off frequency decreases. Note that under 0.0005 Hz the cut-off frequency seems to have no effect. As seen in Fig. 18 the required battery power Preqmin has a strong in?uence on fuel consumption. Values in the range of 2600 and 3900 W of required battery power seem to be signi?cant for the minimization of the fuel consumption. Higher values are not considered because the maximum battery capability in the regeneration phase is 1300 W and therefore these values would saturate the required power. The other parameters appear to have only minor in?uence (Preqmax, SOCminpreq, SOCmaxpreq SOCminhyb, SOCmaxhyb) or they affect the fuel consumption as well as the RMS current. For example the hybrid mode required power (Phybmax) presents good compromises for all of its values (Fig. 19). It is noted that for RMS current values higher than 60 A the fuel consumption increases. This may be explained by an increase in battery losses considering that in charge sustaining strategy the

Required Battery power Preq max SOC max Preq 0 SOC min Preq

ICE switch on power

Phyb max

SOC 100 %

SOC 0 SOC min Hyb SOC max Hyb 100 %

Preq min

(a)

Fig. 16. Parameterized management curves.

(b)

448 Table 2 Parameters and their values. Parameters Elec/hyb power high threshold Elec/hyb soc low threshold Elec/hyb soc high threshold Battery required power high thresholds Battery required power low thresholds Battery required soc low thresholds Battery required soc high thresholds Low pass ?lter cut off frequency Symbols Phybmax SOCminHyb SOCmaxHyb Preqmax Preqmin SOCmiripreq SOCmaxPreq fc

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fuel consumption vs Ibat rms, color : hybrid start on power

3.7

Min value 0 0 60 0 0 0 60 0.002 Max value 25,000 50 100 5200 2500 50 100 0.02 step 2500 10 10 1300 1250 10 10 0.002 units

3.65

W % % W W % % Hz

15000 10000 5000

fuel consumption in l/100 km

3.6 3.55 3.5 3.45 3.4 3.35 3.3 3.25 3.2 20 30 40 50 60 70 80 90 100

fuel consumption vs Ibat rms, color : cutoff frequency

3.7 3.65 0.0005 0.002 0.006 0.014

battery rms current in A

Fig. 19. Fuel consumption vs. battery RMS current for different hybrid mode required power.

fuel consumption in l/100 km

3.6 3.55 3.5

3.8 clouds of point 3.7 minimum points pareto optimal point 3.6 3.5 3.4 3.3 3.2 3.1

3.4 3.35 3.3 3.25 3.2 20 30 40 50 60 70 80 90 100

battery rms current in A

Fig. 17. Fuel Consumption vs. battery RMS current for different cut-off frequency.

fuel consumption in l/100 km

3.45

3

0

20

40

60

80

100

battery rms current in A

fuel consumption vs Ibat rms, color : required battery power

3.7 3.65 -3900 -2600 -1300

Fig. 20. Fuel consumption vs. battery RMS current.

fuel consumption in l/100 km

3.6 3.55 3.5 3.45 3.4 3.35 3.3 3.25 3.2 20 30 40 50 60 70 80 90 100

energy of the vehicle is ?nally only provided by fuel. Moreover, in the optimal management (Fig. 14), the Pareto front starts at 40 A and points with higher RMS current and small fuel consumption do not exist (become not optimal). 4.4.3. Comparison between optimal control theory and rule-based method 4.4.3.1. Fuel consumption and battery RMS current comparison on the NEDC cycle. As previously stated, the fuel consumption may not be the only objective to minimize. Using a weighted, two criteria objective function (Eq. (13)) allows to determine the Pareto optimal front optimized for fuel consumption and RMS battery current (cf. Section 2.3.2.2). Fig. 20 shows the fuel consumption vs. battery RMS current for the result obtained by the rule-based parametric studies. The points marked in red where highlighted as they represent the most interesting tradeoff between fuel consumption and battery RMS current. The actual Pareto optimal front obtained using optimal control methods is shown here by the black diamond markers.

battery rms current in A

Fig. 18. Fuel consumption vs. battery RMS current for different battery required power.

E. Vinot, R. Trigui / Energy Conversion and Management 76 (2013) 437–452 Table 3 Points of the Pareto front. Optimal control Kibatt 0 0.01 0.05 0.5

449

mance, keeping in mind that a rule-based method is an online method where the cycle is not known in advance.

Fuel consumption in 3.16 3.17 (+0.3%) 3.22 (+1.9%) 3.27 (+3.5%) 1/100 km Battery rms current in A 41.0 33.1 (?19%) 24.0 (?41%) 15.4 (?62%) Kibatt: weighting factor of the objective function (cf. part 2.3.2.2)

It becomes obvious for both cases that the point of minimum consumption is probably not the point of the most interest. A small compromise in fuel consumption allows for a drastic reduction in the battery RMS current. For the optimal control case, an increase in fuel consumption of 0.3% decreases the battery RMS current by 19% (Table 3). This tradeoff is even more important when using the rule-based method, where 0.5% increases in fuel consumption allows to decrease battery RMS current by 40% (Table 4). The rule-based method (even though improvable), shows good performances compared to optimal control. The fuel consumption is only 2–3% higher, which can be considered as a good perfor-

4.4.3.2. Strategy and battery current stress comparison. To show the effect of the strategy on the battery current stress, this section presents comparisons of two rule-based strategies for two set of parameters (Figs. 21 and 22). Then a comparison of two strategies obtained using optimal control theory is presented (Figs. 23 and 24). Figs. 21 and 22 show the differences in terms of strategies of power sharing and battery current stress for two sets of parameters of the rule based method (Table 2): – Strat 1: set corresponding to the point of minimum of fuel consumption (?rst column of Table 4). – Strat 2: set corresponding to the point where a strong reduction of the RMS battery current was found when fuel consumption increased by a small amount (third column of Table 4). Fig. 21 shows the development of the battery and EDLC current and battery SOC for the two strategies over the NEDC cycle.

Table 4 Minimum points (red square) using rule-based method. Rule based control Fuel consumption in 1/100 km Battery rms current in A Cut-off frequency in Hz 3.245 58.4 0.06 3.246 (+0.03%) 47.2 (?19%) 0.04 3.26 (+0.5%) 35.1 (?40%) 0.0005 3.34 (+2.9%) 27.5 (?53%) 0.0001 3.52 (+8.5%) 21.7 (?63%) 0.0005 3.59 (+10.5%) 21.2 (?64%) 0.0005

battery current in A

400 200 0 -200

strat 1 strat 2

0

200

400

600

800

1000

1200

strat 1 strat 2

EDLC current in A

200 100 0 -100 -200 0 200 400 600 800 1000

1200

battery SOC in %

60 55 50 45

strat 1 strat 2

0

200

400

600

800

1000

1200

NEDC cycle speed in km/h

150 100 50 0

0

200

400

600

800

1000

1200

time in s

Fig. 21. Battery and EDLC current and battery SOC on NEDC cycle, rule-based method.

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3

4 3.9

clouds of point minimum points pareto optimal point nedc rule based recall

Amps hours per class of current

2 1 0 -1 strat 1 -2 -3 -4 strat 2

fuel consumption in l/100 km

3.8 3.7 3.6 3.5 3.4

-125 -100 -75 -50 -25

0

25 50

75 100 125 150 175

current in A

Fig. 22. Battery current solicitation on NEDC cycle, rule-based method.

3.3 10

20

30

40

50

60

70

80

battery rms current in A

Fig. 25. Validation of rule-based method.

61 60 59

58 57 56 55 54 53 52 0 200 400 600 800 1000 1200 strat opt 1 strat opt 2 NEDC cycle

These ?gures highlight the effect of the cut-off frequency of the ?rst order ?lter (Fig. 15, Section 4.4.1) on the battery current. The battery current peak (Fig. 21) and the corresponding battery current stress (Fig. 22) are drastically reduced when the cut-off frequency is small (0.0005 vs. 0.06 Hz). In the same time, the EDLC current does not change drastically. This explains a smoother SOC evolution associated to higher fuel consumption (8.5%). However, analyzing the SOC (Fig. 21) the overall strategy does not changed so much. The electric mode is mostly used in the ?rst 800 s of the cycle and for the rest of the cycle the engine is used to charge the battery to achieve charge sustaining operation over the cycle. Figs. 23 and 24 show the SOC and the battery current stress for the NEDC using optimal control with the two different control objectives:

battery SOC in %

time in s

Fig. 23. Battery SOC on NEDC cycle, optimal control method.

– Strat opt 1: objective is to minimize fuel consumption (Kibatt = 0). – Strat opt 2: objective is a weighted function of consumption and battery current (Kibatt = 0.25). These two strategies are meant to correspond to Strat 1 and Strat 2 with the rule-based method while using optimal control theory. Looking at Fig. 23, it can be seen that introducing a weighting factor of battery current in the objective function results in a very different strategy for short time windows as well as long time intervals. With Strat opt 2, high battery currents especially in the negative range, are drastically reduced and absorbed by the EDLC. At the same time, the maximum SOC variation is reduced from 5% with Strat opt 1–2% with Strat opt 2. In fact taking into account the battery RMS current (Strat opt 2) the time passed in pure electrical mode is reduced, from 490 s to 398 s. In both cases the optimal control method uses a smaller SOC amplitude than the rule-based method, which operates with a SOC variation of 11%. 4.4.3.3. Validation of a rule-based method on realistic urban driving cycle. The parametric study (Section 4.4.2) allows choosing sets of parameters which present the best trade-off between fuel consumption and battery current. However this set of parameters is determined on one cycle (NEDC in this case) and does not necessarily remain relevant for another driving cycle. A ?rst validation is thus necessary using these sets of parameters to simulate an ur-

2.5 2 strat opt 1 strat opt 2

Amps hours per class of current

1.5 1 0.5 0 -0.5 -1 -1.5 -2 -125 -100 -75 -50 -25 0 25 50 75 100 125 150 175

current in A

Fig. 24. Battery current solicitation on NEDC cycle, optimal control method.

In Fig. 22 the battery current stress i.e. the distribution of battery Ampere–hours (A h) for the two strategies is shown. The amount of A h given by the battery for different levels of battery current can be seen.

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ban cycle and compare the results to optimal results and the parametric study performed on the NEDC cycle. Fig. 25 shows a comparison on urban driving cycle of the optimal method and the rule-based method: – Black diamond shaped points are the Pareto optimal front obtained by the optimal presented method. – Blue stars represent the points obtained making a parametric study (cf. Section 4.4.2) on urban cycle using the rule-based control. – Green squares are some of the best points obtained in the parametric study. – Red diamonds are the points obtained using the sets of parameters corresponding to some best point of parametric study performed on NEDC cycle (Fig. 20 and Table 4). It clearly appears that the sets of parameters determined on NEDC cycle remain globally relevant on urban driving cycles. An over fuel consumption of 2% is observed for the worst set of parameters. It is also observed that the rule-based method presents good performances for this driving cycle too (fuel consumption 3% higher compared to optimal points). This tends to show that this rule-based method may be relevant to unseen realistic driving cycle.

Appendix A. Nomenclature Battery P0batt Pbatt Ebatt Ibatt U0batt Ubatt Rbatt SOC Power of the ideal battery; without any losses Power of the battery Energy stored in the battery Battery current Battery open circuit voltage Battery voltage Battery resistance State of charge Faradic ef?ciency of the battery

gf

5. Conclusion In this paper energy management strategies of a hybrid vehicle with hybrid storage system were discussed. First the application of the Pontryagin’s Minimum Principle for such an architecture was presented. Then a rule-based method, easily applicable on-line, is proposed and compared to the off-line optimal method. The Pontryagin’s Minimum Principle was applied to the problem considering two state variables. As a result we identi?ed the Pareto optimal front between the two objectives: fuel consumption and battery RMS current. The utilized model implements look-up tables to simulate the electrical machine losses and fuel consumption. Therefore a non-analytical method has to be applied to ?nd the Hamiltonian minimum. A rule-based method, using two control levels, is proposed. A load-following charge-sustaining strategy, usually applied in parallel hybrid vehicles, is coupled with a low-pass ?rst-order ?lter approach. This ?ltering step identi?es the split of power between the battery and EDLC. To achieve an ef?cient control with the rule-based method, an iterative study has been performed to choose the more appropriate values of the parameters. Once tuned, this strategy shows overall good performances compared to the optimal control law (2–3% higher fuel consumption). The results show high potential to reduce the battery RMS current using EDLC. Moreover this study shows the importance of taking into account Pareto optimality (fuel consumption vs. battery current) and not to minimize for fuel consumption only. In fact the Pareto front is relatively ?at in the minimum fuel consumption area. Thus a good compromise may be found when increasing fuel consumption by a small amount while signi?cantly decreasing the battery RMS current and thus probably the battery ageing. Continuing this work, three main areas will be explored in the future: Performing experiment on real vehicle or HIL test bench. Extension of results to other representative driving cycles Implementation online of Pontryagin’s Minimum Principle with estimation in real time of the Lagrange multipliers.

Electrical double-layer capacitor (EDLC) P0edlc Power of the ideal EDLC; without any losses Pedlc Power of the EDLC Eedlc Energy stored in the EDLC I1edlc EDLC current Iedlc EDLC current after DC/DC U0edlc EDLC open circuit voltage Uedlc EDLC voltage Redlc EDLC main resistance C EDLC main capacity Rx EDLC parallel resistance Cx EDLC parallel capacity sx EDLC time constant 1/RxCx Ux EDLC voltage on Rx Ux0 EDLC initial voltage on Rx ICx Current in Cx capacity gDC/DC DC/DC converter ef?ciency Electrical network P0elec Vector [P0batt P0edlc] Pelec Vector [Pbatt Pedlc] Eelec Electrical stored energy; vector [Ebatt Eedlc] Ielec Vector [Ibatt Iedlc] U0 Vector [U0batt U0edlc]

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