- Email: [email protected]
RULE-BASED ENERGY MANAGEMENT STRATEGIES FOR HYBRID VEHICLE DRIVETRAINS: A FUNDAMENTAL APPROACH IN REDUCING COMPUTATION TIME
RULE-BASED ENERGY MANAGEMENT STRATEGIES FOR HYBRID VEHICLE DRIVETRAINS: A FUNDAMENTAL APPROACH IN REDUCING COMPUTATION TIME T. Hofman, M. Steinbuch ∗ R.M. van Druten, A.F.A. Serrarens ∗∗ ∗
Technische Universiteit Eindhoven, Dept. of Mechanical Engineering, Control Systems Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands [email protected] ∗∗ Drivetrain Innovations B.V., Horsten 1, 5612 AX Eindhoven, The Netherlands
Abstract: The hybrid vehicle control problem at the highest level is termed the Energy Management Strategy (EMS). This paper presents a new, and simple RuleBased (RB) EMS, whereby maximum power level of the electric machine during pure electric driving is the control design variable. This maximum power level determines the overall power -, and efficiency specifications of the electric machine. A RB EMS consist of a selection of driving modes. The RB EMS is compared with the strategy based on Dynamic Programming (DP), which is inherently optimal for a given cycle. The RB method proposed in this paper requires 3000 times less computation time with the same accuracy (±1%) as DP. The RB strategy in this paper is a semi-empirical EMS with which the generic component specifications for the secondary power source (battery, power electronics, electric machine), primary power source (engine) and transmission technology can be obtained. Provided these generic specifications, a technology designer can quickly specify the hybrid technologies. In this way, control, optimization and component design are merged in a single framework. Keywords: Automotive control, energy management, optimization, (hybrid) power systems
1. INTRODUCTION Hybridization implies adding a Secondary power source (S) to a Primary power source (P) in order to improve vehicle characteristics. The major desirable improvements lie in the vehicle’s fuel economy, emissions, comfort, safety, and driveability. The fuel consumption of a vehicle can be reduced by down-sizing the engine, which results
in less idle fuel consumption, and internal friction losses. A second, though complementary method is recuperation of the brake energy, and re-using this stored energy when momentary fuel costs are high. In fact, it is instrumental in optimizing the power flows between P, S and the vehicle wheels. Besides the fuel saving itself, the Energy Management Strategy (EMS) plays an important
740
role in an effective usage of the drivetrain components. A commonly used technique for determining the globally optimal EMS is Dynamic Programming (DP) (A. Sciaretta and Onder, 2003), (M. Koot and Steinbuch, 2005). Using DP it is rather straight forward to handle non-linear constraints (Bellman, 1962). However, a disadvantage of this technique is the relative long computation time. Furthermore, it is inherently non-causal, and therefore not real-time implementable.
• The influence of design specifications for the main hybrid drivetrain components, and the drive cycle on the fuel economy can be investigated rather quickly, and with sufficient accuracy as will be explained in the following sections; • Intuitively, fuel saving methods such as Engine on/off modes during stand-still, brake energy recovery, and motor only driving can be explicitly implemented in the EMS.
In order to tackle these drawbacks, which is the aim of this paper, a new, and simple semiempirical solution for the EMS control problem is introduced. The EMS as discussed in this paper is an off-line EMS, but it can easily be converted into on-line, and even implementable EMS in combination with several proven techniques as are discussed in literature (G. Paganelli and Santin, 2002), (S. Delprat and Rimaux, 2002), (A. Sciaretta and Guzzella, 2004), (G. Rizonni and Calo, 2004), (Chen and Salman, 2005), (M. Koot and Steinbuch, 2005).
It is shown, that these fuel saving methods are also resulting from DP, which gives us sufficient confidence in the near optimality of the proposed semi-empirical solution. In this way the RB EMS can also be used as part of a hybrid drivetrain design tool, which is currently under development by the author. Finally, the results, and conclusions are described in Section 4.
1.1 Outline of paper In literature (C. Lin and Peng, 2001), (Sciaretta and Guzella, 2003) more heuristic control strategies can be found, whereby the thresholds for mode switching are optimized, or calibrated by using DP for a certain driving condition. Thereby, the power-split ratio between the electric machine power, or torque, and the vehicle power demand for each driving mode is optimized. It was found, that not all of the available electric motor power during pure electric driving on the typical drive cycles is used, while during braking the power is generated to the maximum available electric generative power. In this paper, a novel RB EMS is proposed. Using the proposed RB EMS a physical background for not using the potentially available electric motoring power is given. The main differences between the EMS proposed in this paper, and the sub-optimal strategies as discussed in literature are: • The EMS consists of a combination of RuleBased (RB) selection of driving modes, in which the system input variable is a function of the vehicle power demand as will be discussed in Section 2; • The propulsion power level of the electric motor during pure electric driving is used as a mode-switch variable; • The numerical threshold values for mode switching do have a physical meaning. The last two differences will be explained in more detail in the Section 3. The method has the following main advantages:
2. HYBRID DRIVING MODES A hybrid drivetrain can be operated in certain distinct driving modes. In Fig. 1, a block diagram is shown for the power distribution between the different energy sources, i.e., fuel tank with stored energy EF , S with stored energy ES , and the vehicle driving over a drive cycle represented by a required energy EV . The efficiencies of the fuel combustion in the engine, the storage and electric motor S, and the Transmission (T) are described by the variables ηP , ηS , and ηT respectively. The energy exchange between the fuel tank, S and the vehicle can be performed by different driving modes (depicted by the thick lines). The engine power at the crank shaft is represented by Pp . The power demand at the wheels (Pv ), and the power flow to and from S (Ps ), which is equal to the power set-point u, determines which driving mode is active. The following operation modes are defined: M: Motor only mode, the vehicle is propelled only by the electric motor and the battery storage supply S up to a certain power level u∗max , which is not necessary equal to the maximum power umax . The engine is off, and has no drag -, and idle losses. BER: Brake Energy Recovery mode, the brake energy is recuperated up to the maximum generative power limitation umin , and stored into the accumulator of S. The engine is off, and has no drag -, and idle losses. CH: Charging mode, the instantaneous engine power is higher than the power needed for driving. The redundant energy is stored into the accumulator of S. MA: Motor-Assisting mode, the engine power is lower than the power needed for driving. The engine power is augmented by power from S.
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E: Engine only mode, only the engine power is used for propulsion of the vehicle. S is off, and generates no losses. During the M, and the BER mode the engine is off, and as a consequence uses no fuel. This is also referred to as the Idle-Stop (IS) mode. Since, the electrical loads in vehicles are expected to increase in the near future it may be important to define more hybrid (charging) modes. However, the auxiliary loads are not considered in this paper, but is referred to (J. Kessels and De Jager, 2005).
Fig. 1. Power flows for the different hybrid driving modes. S connected at the pre-transmission side.
3. THE RULE-BASED EMS The EMS is optimized focussing on the low efficiency zones of the engine. Thereby, operation points can be found in certain distinct driving states, or modes. The modes represented as operation areas in a static-efficiency engine map separated by two iso-power curves are shown in Fig. 2. In the same figure the styled Optimal Operation Line (OOL), and the Wide-Open Throttle (WOT) torque line are depicted. The OOL connects the engine operation points with minimum specific fuel consumption. The engine is assumed to be operated at the OOL whenever it is used, that is in E, CH, and the MA modes. The iso-power curve shown represented by the variable u∗max , separates the M mode from the CH mode, and the E mode. The dotted iso-power curve separates the operation points of the engine during the CH, the E, and the MA mode. Following from the EMS calculated with DP the u∗max appeared to be constant, whereas the dotted iso-power varies with the vehicle power demand given certain driving conditions.
(3)
g3,4 := xmin ≤ x(t) ≤ xmax ,
(4)
with the relative energy change 4x(tf ) = x(tf ) − x(0). A technique providing a global optimal solution to the EMS problem is DP. Thereby, the finite horizon optimization problem is translated into a finite computation problem (Bellman, 1962). Note that in principle the technique results in an optimal solution for the EMS, but that the grid step size also influences the accuracy of the result.
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280 300 260
Engine torque in [Nm]
300
280
M
40
M 280
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0 28
(2)
CH/E
260
h1 := x(tf ) − x(0) = 0, g1,2 := umin ≤ u(t) ≤ umax ,
60
MA/E
0
where m ˙ f is the fuel rate in [g/s]. The state variable x(t) is equal to the stored secondary energy Es , and the control input u(t) is equal to the secondary power flow Ps . The main constraints are energy conservation balance of x over the drive cycle, constraints on the power u, and the energy x.
80
28
The optimization problem is finding the optimal control power flow u(t) given a certain power demand at the wheels Pv , while the cumulative fuel consumption Mf is minimized subjected to several constraints, i.e., Z tf J(x, u) = min m ˙ f (x, u)dt, (1) u 0 s. t. h = 0, g ≤ 0.
100
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2.1 The Energy Management Optimization Problem
WOT 24 0
280
OOL
120
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1.6 l SI Engine, BSFC in [g/kWh] 140
300
40
0
20 00 5600 1000
2000
300
300
u*max
400 400 500 500 600 3000 4000 5000 Engine speed in [rpm]
400 6000
7000
Fig. 2. Contour plot of the Brake-Specific Fuel Consumption (BSFC) in [g/kW h] as a function of the engine output torque and speed. OOL = Optimal Operation Line for minimum BSFC, WOT = Wide-Open Throttle torque. In order to fulfill the integral energy balance constraint over the drive cycle, the energy required for the M and the MA mode needs to be regenerated during the BER mode, or charged during the CH mode. To explain the basic principles of the RB EMS, which is a trade-off between energy balance, and fuel consumption, consider the following two cases, with two different umin whereby the energy recuperated during the BER mode for supplying the energy during the M mode over a complete drive cycle is: (i) not sufficient, and (ii) more than sufficient. i The additional required energy for the M mode has to be charged during the CH mode resulting in additional fuel cost.
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ii The redundant energy of the BER mode can be used for motor-assisting during the MA mode resulting in additional fuel savings. Both cases are schematically shown in the Fig. 3. Referring to case I, if u∗max is lowered, then the additional fuel cost becomes lower due to decrease of the required charging energy. However, the fuel saving due to the M mode is also reduced, and vice-versa, if u∗max is increased. The same holds for case II: The fuel saving during the MA mode is increased if u∗max is lowered, but the fuel saving due to the M mode is reduced.
3.1 The M, and the BER mode The optimal power during the M, and the BER mode becomes, sgn(−Pv )
u∗I = −Pv · ηS
,
(5)
whereby the power set-point is limited between the following constraints, umin ≤ 0 ≤ u∗I ≤ u∗max ≤ umax .
(6)
Braking powers larger than the maximum charging power are assumed to be dissipated by the wheel brake discs. If only the M, and the BER mode are utilized, then the relative energy 4xI (tf ) at the end of the drive cycle becomes, Z tf 4xI (tf ) = u∗I dt. (7) 0
Fig. 3. Energy balance, and fuel consumption. For both cases, additional charging during driving, and using for motor-assisting can be beneficial, if the energy is charged at a lower driving power, and this energy is used for motor-assisting at a higher driving power. However, the additional fuel saving is relatively small, because the drive energy at higher powers is relatively small. This is shown in the Fig. 4 by the histogram of Pv , and the fuel rate at the OOL as a function of the engine power, which is equal to the drive power assumed. The largest relative fuel saving is realized by applying the M, and the BER mode, because the energy for electric driving is obtained without any additional fuel cost, and the engine operation at relative low efficiency is eliminated. In addition, the largest drive energy is at low drive powers. A trade-off between the M, the CH, and the MA mode, can be found by changing the design variable u∗max . This will be explained in the following sections. Histogram for the NEDC
350
Fuel rate at the OOL
300
* umax
150
2
CH
MA 1.5
M
100
1
50 0
0.5
0
5
10
15 20 25 Drive power, Pv [kW]
30
35
3.2 The MA, the CH, and the E mode In this section, the optimal power for charging, or discharging during the MA, and the CH mode is calculated. Accordingly, the optimizing steps for calculating the minimum fuel consumption, and energy balance conservation over the drive cycle are explained. The fuel rate can be written as the sum of the fuel rate change only depending on the drive power demand Pv , and charging, or discharging power uII . E
2.5
Drive energy
200
Fig. 5. Energy balance, and fuel consumption during the BER/M, and the CH/MA modes.
m ˙ f (u, λ) = m ˙ f,0 + 4m ˙ f (uII , λ) . |{z} | {z }
3
Fuel rate [g/s]
Time [s]
250
3.5
This energy has to be counterbalanced with the relative energy 4xII (tf ) at the end of the cycle during the MA, and the CH mode as is shown in Fig. 5.
0 40
Fig. 4. The histogram for the vehicle power demand, and the fuel rate at the OOL as a function the engine power.
(8)
CH/M A
The fuel rate change 4m ˙ f for a certain accumulator power u 6= 0 is, sgn(uII )
4m ˙ f (uII , λ) = λ(uII , Pv ) · uII · ηS
,
(9)
with variable λ representing the equivalent accumulator energy to fuel factor which is a function of uII , and Pv . It can be seen with Eq. (9), that generally the λ is smaller for charging than for discharging if ηS < 1, and is equal for charging, and discharging if ηS = 1, or, λ(uII > 0) ≤ λ(uII < 0) if ηS ≤ 1 ∀ t. A decision variable Γ representing the fuel rate profits is introduced in order to determine at
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∗
∗
Γ(uII , λ ) = (λ − λ) · uII ·
sgn(uII ) ηS
≥ 0 ∀ uII . (10) Thereby, the optimal accumulator power u∗II is the power which maximizes Γ given a certain value for λ∗ , u∗II = arg max (Γ|λ∗ ), (11)
3 ∆ xI (BER/M) ∆ x (CH/MA)
2.95
II
∆x
2.9 2.85 lv
λ ⋅ h [−]
which drive power demand Pv , and at which power set-point uII the energy needs to be recharged during the CH mode, or discharged during the MA mode. The decision variable is a function of the accumulator power uII , and a certain reference value of the energy to fuel ratio λ∗ . The λ∗ determines if charging (λ < λ∗ ), or discharging (λ > λ∗ ) is beneficial, and is used to balance the energy quantities during the MA, and the CH mode over the drive cycle. The decision variable is,
2.8 Optimal value for λ
∆ xI
∆ xII
2.75 2.7 2.65 2.6 −2
−1.5
−1
−0.5
0
0.5
1
1.5
∆ x(tf) [MJ]
Fig. 6. Optimization of λ for 4x(tf ) = 0.
uII
whereby the power set-point is limited between the following constraints, umin ≤ 0 ≤ u∗II ≤ umax
(12)
Then 4xI (tf ) is discharged (charged) at vehicle power demands where the fuel savings (costs), i.e., 4m ˙ f are maximum (minimum). If u∗II = 0, then the vehicle is propelled by the engine only that is the E mode. The total energy flow out of S during the CH, and the MA mode is Z tf 4xII (tf ) = u∗II dt. (13) 0
The total optimal power set-point becomes, u∗ = u∗I + u∗II .
(14)
Obviously, as example if the drive power demand over the drive cycle is known a-priori, it is possible to calculate λ∗ such that the relative energy at the end of the drive cycle becomes exactly zero, i.e., 4x(tf ) = 4xI (tf ) + 4xII (tf ) = 0,
Fig. 7. Block diagram representing the optimization scheme for determining u∗max off-line. solved by iteratively updating λ∗ until 4xI = −4xII , while at each time point the Γ is maximized. This is respectively shown by iteration loop 2, and 1. During this process, the minimum cumulative fuel consumption is determined by changing umax such that, u∗max = arg min (Mf ).
(17)
umax
This is depicted by iteration loop 3 in Fig. 7.
(15)
∗
In Fig. 6, the optimal value for λ is shown where by the energy used during the BER/M mode are in balance with the energy used during the CH/MA mode. The parameter hlv represents the lower heating value for petrol. Concluding, the total fuel consumption calculated with the proposed RB EMS can be stated as, Z tf ∗ ∗ Mf (u , λ ) = m ˙ f (u∗ , λ∗ )dt (16) 0
3.3 Numerical optimization routine for calculating u∗max (off-line) In the Fig. 7, a block diagram is shown of the optimization routine suggested in this study. The problem is, that in order to calculate the optimal power set-point u∗II with Eq. (11), the λ∗ has to be known beforehand. This causality conflict is
4. SIMULATION RESULTS 4.1 Model The vehicle parameters are summarized in the Table 1. Furthermore, the fuel map of the 1.6 [l] Table 1. Relevant vehicle parameters Parameter Curb mass Air drag coefficient Frontal area Roll resistance coefficient Regenerative brake fraction
Value 1390 0.35 2.11 0.9 1
Unit [kg] [−] [m2 ] [%] [−]
SI engine is shown in Fig. 2, and the engine is assumed to be operated at the OOL. The optimal engine angular speed ωp∗ is prescribed for every engine power demand Pp . The transmission is ideally assumed, i.e., with infinitely transmission ratios, and without any losses. Furthermore, the
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fuel rate as a function of engine power at the OOL is approximated by m ˙ f (t) = φ(ωp∗ , Pp ) ≈
i=2 X
14
13 kW 12
ci (ωp∗ , Pp ) · Ppi (t) (18)
Table 2. +: active mode, -: inactive mode BER + + +
MA + +
CH + +
*
7 kW 6
4 kW
4
1 kW
In this section the fuel consumption results are discussed for different constant values for ηS ∈ {30, 100%}, and power constraint values umax = −umin ∈ {1, 13} [kW ] for S. The fuel consumption is calculated with DP, and the RB EMS. In the Table 4.2, the fuel saving results (percentage of the base line fuel consumption) as an average of 35 simulations are summarized. The computation time for 35 simulations (NEDC) with the DP algorithm was 56 hours on a Pentium IV, 2.6 GHz, with 512 M B of RAM. In contrary to the RB EMS the total simulation time was less than 1 minute. The average error between the RB EMS, and DP is 0.6%. The small error between DP and the RB EMS is mainly caused by numerical rounding errors within DP due to grid inaccuracy. Additional charging, and motor-assisting results overall in a small average fuel saving improvement of 0.4%. In the Fig. 8, the results for different M + + +
8
2
4.2 Results, and Conclusions
EMS RB RB DP
umax increase umax [kW]
by neglecting the higher order terms (i > 2) in Pp . The fit coefficients are ci (ωp∗ , Pp ) > 0 for i ∈ {0, 1, 2}. The used drive cycle is the NEDC, and the fuel consumption of the base line vehicle without a S is 375.6 [g].
10 kW
10
i=0
Average fuel saving 26.10% (98.02 [g]) 26.20% (98.43 [g]) 26.05% (97.83 [g])
optimization values of u∗max as a function of ηS for different values of umax calculated with the RB EMS are shown. It can be seen that the sensitivity of u∗max to ηS increases more than linear with increase of umax . Furthermore, the connection of S at the engine-side, or the wheel-side of the T, and ηT , or the parameters describing engine efficiency have influence on u∗max , but can be investigated with this RB EMS very quickly, and with sufficient accuracy. REFERENCES A. Sciaretta, L. Guzzella and C.H. Onder (2003). On the power split control of parallel hybrid vehicles: from global optimization towards real-time control. J. of Automatisierungstechnik 51(5), 195–205. A. Sciaretta, M. Back and L. Guzzella (2004). Energy management strategies for vehicular electric power systems. J. of IEEE-Trans. on Control Systems Technology 12(3), 352–363. Bellman, Richard E. (1962). Dynamic programming. Princeton University Press.
0 30
40
50
60
ηS [%]
70
80
90
100
Fig. 8. Optimal u∗max as a function of ηS for different umax . C. Lin, J. Kang, J.W. Grizzle and H. Peng (2001). Energy management strategy for a parallel hybrid truck. In: Proc. of the American Control Conference. Arlington. pp. 2878–2883. Chen, J. and M. Salman (2005). Learning energy management strategy for hybrid electric vehicles. In: Proc. of the Symposium IEEEVehicle Propulsion & Power. Chicago, USA. pp. 427–432. G. Paganelli, S. Delprat, T.M. Guerra J.M. Rimaux and J.J. Santin (2002). Equivalent consumption minimzation strategy for parallel hybrid powertrains. In: Proc. of the Symposium IEEE-Vehicular Transportation Systems. Atlantic City, USA. pp. 2076–2081. G. Rizonni, P. Pisu and E. Calo (2004). Control strategies for parallel hybrid electric vehicles. In: Proc. of Symposium IFAC-Advances in Automotive Control. Salerno, Italy. pp. 508– 513. J. Kessels, P. van den Bosch, M. Koot and B. De Jager (2005). Energy management for vehicle power net with flexible electric load demand. In: Proc. of 2005 IEEE Conf. on Control Applications. Toronto, Canada. pp. 1504–1509. M. Koot, J.T.B.A. Kessels, B. de Jager W.P.M.H. Heemels P.P.J. van den Bosch and M. Steinbuch (2005). Energy management strategies for vehicular electric power systems. J. of IEEE-Trans. on Vehicular Technology 54(3), 771–782. S. Delprat, T.M. Guerra and J. Rimaux (2002). Optimal control of a parallel powertrain: From global optimal to real time control strategy. In: Proc. of the Symposium IEEEVTC. pp. 2082–2088. Sciaretta, A. and L. Guzella (2003). Rule-based and optimal control strategies for energy management in parallel hybrid vehicles. In: Proc. of the 6th Int. Conf. on Engines for Automobile. Capri.
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Technische Universiteit Eindhoven, Dept. of Mechanical Engineering, Control Systems Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands [email protected] ∗∗ Drivetrain Innovations B.V., Horsten 1, 5612 AX Eindhoven, The Netherlands
Abstract: The hybrid vehicle control problem at the highest level is termed the Energy Management Strategy (EMS). This paper presents a new, and simple RuleBased (RB) EMS, whereby maximum power level of the electric machine during pure electric driving is the control design variable. This maximum power level determines the overall power -, and efficiency specifications of the electric machine. A RB EMS consist of a selection of driving modes. The RB EMS is compared with the strategy based on Dynamic Programming (DP), which is inherently optimal for a given cycle. The RB method proposed in this paper requires 3000 times less computation time with the same accuracy (±1%) as DP. The RB strategy in this paper is a semi-empirical EMS with which the generic component specifications for the secondary power source (battery, power electronics, electric machine), primary power source (engine) and transmission technology can be obtained. Provided these generic specifications, a technology designer can quickly specify the hybrid technologies. In this way, control, optimization and component design are merged in a single framework. Keywords: Automotive control, energy management, optimization, (hybrid) power systems
1. INTRODUCTION Hybridization implies adding a Secondary power source (S) to a Primary power source (P) in order to improve vehicle characteristics. The major desirable improvements lie in the vehicle’s fuel economy, emissions, comfort, safety, and driveability. The fuel consumption of a vehicle can be reduced by down-sizing the engine, which results
in less idle fuel consumption, and internal friction losses. A second, though complementary method is recuperation of the brake energy, and re-using this stored energy when momentary fuel costs are high. In fact, it is instrumental in optimizing the power flows between P, S and the vehicle wheels. Besides the fuel saving itself, the Energy Management Strategy (EMS) plays an important
740
role in an effective usage of the drivetrain components. A commonly used technique for determining the globally optimal EMS is Dynamic Programming (DP) (A. Sciaretta and Onder, 2003), (M. Koot and Steinbuch, 2005). Using DP it is rather straight forward to handle non-linear constraints (Bellman, 1962). However, a disadvantage of this technique is the relative long computation time. Furthermore, it is inherently non-causal, and therefore not real-time implementable.
• The influence of design specifications for the main hybrid drivetrain components, and the drive cycle on the fuel economy can be investigated rather quickly, and with sufficient accuracy as will be explained in the following sections; • Intuitively, fuel saving methods such as Engine on/off modes during stand-still, brake energy recovery, and motor only driving can be explicitly implemented in the EMS.
In order to tackle these drawbacks, which is the aim of this paper, a new, and simple semiempirical solution for the EMS control problem is introduced. The EMS as discussed in this paper is an off-line EMS, but it can easily be converted into on-line, and even implementable EMS in combination with several proven techniques as are discussed in literature (G. Paganelli and Santin, 2002), (S. Delprat and Rimaux, 2002), (A. Sciaretta and Guzzella, 2004), (G. Rizonni and Calo, 2004), (Chen and Salman, 2005), (M. Koot and Steinbuch, 2005).
It is shown, that these fuel saving methods are also resulting from DP, which gives us sufficient confidence in the near optimality of the proposed semi-empirical solution. In this way the RB EMS can also be used as part of a hybrid drivetrain design tool, which is currently under development by the author. Finally, the results, and conclusions are described in Section 4.
1.1 Outline of paper In literature (C. Lin and Peng, 2001), (Sciaretta and Guzella, 2003) more heuristic control strategies can be found, whereby the thresholds for mode switching are optimized, or calibrated by using DP for a certain driving condition. Thereby, the power-split ratio between the electric machine power, or torque, and the vehicle power demand for each driving mode is optimized. It was found, that not all of the available electric motor power during pure electric driving on the typical drive cycles is used, while during braking the power is generated to the maximum available electric generative power. In this paper, a novel RB EMS is proposed. Using the proposed RB EMS a physical background for not using the potentially available electric motoring power is given. The main differences between the EMS proposed in this paper, and the sub-optimal strategies as discussed in literature are: • The EMS consists of a combination of RuleBased (RB) selection of driving modes, in which the system input variable is a function of the vehicle power demand as will be discussed in Section 2; • The propulsion power level of the electric motor during pure electric driving is used as a mode-switch variable; • The numerical threshold values for mode switching do have a physical meaning. The last two differences will be explained in more detail in the Section 3. The method has the following main advantages:
2. HYBRID DRIVING MODES A hybrid drivetrain can be operated in certain distinct driving modes. In Fig. 1, a block diagram is shown for the power distribution between the different energy sources, i.e., fuel tank with stored energy EF , S with stored energy ES , and the vehicle driving over a drive cycle represented by a required energy EV . The efficiencies of the fuel combustion in the engine, the storage and electric motor S, and the Transmission (T) are described by the variables ηP , ηS , and ηT respectively. The energy exchange between the fuel tank, S and the vehicle can be performed by different driving modes (depicted by the thick lines). The engine power at the crank shaft is represented by Pp . The power demand at the wheels (Pv ), and the power flow to and from S (Ps ), which is equal to the power set-point u, determines which driving mode is active. The following operation modes are defined: M: Motor only mode, the vehicle is propelled only by the electric motor and the battery storage supply S up to a certain power level u∗max , which is not necessary equal to the maximum power umax . The engine is off, and has no drag -, and idle losses. BER: Brake Energy Recovery mode, the brake energy is recuperated up to the maximum generative power limitation umin , and stored into the accumulator of S. The engine is off, and has no drag -, and idle losses. CH: Charging mode, the instantaneous engine power is higher than the power needed for driving. The redundant energy is stored into the accumulator of S. MA: Motor-Assisting mode, the engine power is lower than the power needed for driving. The engine power is augmented by power from S.
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E: Engine only mode, only the engine power is used for propulsion of the vehicle. S is off, and generates no losses. During the M, and the BER mode the engine is off, and as a consequence uses no fuel. This is also referred to as the Idle-Stop (IS) mode. Since, the electrical loads in vehicles are expected to increase in the near future it may be important to define more hybrid (charging) modes. However, the auxiliary loads are not considered in this paper, but is referred to (J. Kessels and De Jager, 2005).
Fig. 1. Power flows for the different hybrid driving modes. S connected at the pre-transmission side.
3. THE RULE-BASED EMS The EMS is optimized focussing on the low efficiency zones of the engine. Thereby, operation points can be found in certain distinct driving states, or modes. The modes represented as operation areas in a static-efficiency engine map separated by two iso-power curves are shown in Fig. 2. In the same figure the styled Optimal Operation Line (OOL), and the Wide-Open Throttle (WOT) torque line are depicted. The OOL connects the engine operation points with minimum specific fuel consumption. The engine is assumed to be operated at the OOL whenever it is used, that is in E, CH, and the MA modes. The iso-power curve shown represented by the variable u∗max , separates the M mode from the CH mode, and the E mode. The dotted iso-power curve separates the operation points of the engine during the CH, the E, and the MA mode. Following from the EMS calculated with DP the u∗max appeared to be constant, whereas the dotted iso-power varies with the vehicle power demand given certain driving conditions.
(3)
g3,4 := xmin ≤ x(t) ≤ xmax ,
(4)
with the relative energy change 4x(tf ) = x(tf ) − x(0). A technique providing a global optimal solution to the EMS problem is DP. Thereby, the finite horizon optimization problem is translated into a finite computation problem (Bellman, 1962). Note that in principle the technique results in an optimal solution for the EMS, but that the grid step size also influences the accuracy of the result.
260
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24 0
280 300 260
Engine torque in [Nm]
300
280
M
40
M 280
260
0 28
(2)
CH/E
260
h1 := x(tf ) − x(0) = 0, g1,2 := umin ≤ u(t) ≤ umax ,
60
MA/E
0
where m ˙ f is the fuel rate in [g/s]. The state variable x(t) is equal to the stored secondary energy Es , and the control input u(t) is equal to the secondary power flow Ps . The main constraints are energy conservation balance of x over the drive cycle, constraints on the power u, and the energy x.
80
28
The optimization problem is finding the optimal control power flow u(t) given a certain power demand at the wheels Pv , while the cumulative fuel consumption Mf is minimized subjected to several constraints, i.e., Z tf J(x, u) = min m ˙ f (x, u)dt, (1) u 0 s. t. h = 0, g ≤ 0.
100
260
2.1 The Energy Management Optimization Problem
WOT 24 0
280
OOL
120
260
260
1.6 l SI Engine, BSFC in [g/kWh] 140
300
40
0
20 00 5600 1000
2000
300
300
u*max
400 400 500 500 600 3000 4000 5000 Engine speed in [rpm]
400 6000
7000
Fig. 2. Contour plot of the Brake-Specific Fuel Consumption (BSFC) in [g/kW h] as a function of the engine output torque and speed. OOL = Optimal Operation Line for minimum BSFC, WOT = Wide-Open Throttle torque. In order to fulfill the integral energy balance constraint over the drive cycle, the energy required for the M and the MA mode needs to be regenerated during the BER mode, or charged during the CH mode. To explain the basic principles of the RB EMS, which is a trade-off between energy balance, and fuel consumption, consider the following two cases, with two different umin whereby the energy recuperated during the BER mode for supplying the energy during the M mode over a complete drive cycle is: (i) not sufficient, and (ii) more than sufficient. i The additional required energy for the M mode has to be charged during the CH mode resulting in additional fuel cost.
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ii The redundant energy of the BER mode can be used for motor-assisting during the MA mode resulting in additional fuel savings. Both cases are schematically shown in the Fig. 3. Referring to case I, if u∗max is lowered, then the additional fuel cost becomes lower due to decrease of the required charging energy. However, the fuel saving due to the M mode is also reduced, and vice-versa, if u∗max is increased. The same holds for case II: The fuel saving during the MA mode is increased if u∗max is lowered, but the fuel saving due to the M mode is reduced.
3.1 The M, and the BER mode The optimal power during the M, and the BER mode becomes, sgn(−Pv )
u∗I = −Pv · ηS
,
(5)
whereby the power set-point is limited between the following constraints, umin ≤ 0 ≤ u∗I ≤ u∗max ≤ umax .
(6)
Braking powers larger than the maximum charging power are assumed to be dissipated by the wheel brake discs. If only the M, and the BER mode are utilized, then the relative energy 4xI (tf ) at the end of the drive cycle becomes, Z tf 4xI (tf ) = u∗I dt. (7) 0
Fig. 3. Energy balance, and fuel consumption. For both cases, additional charging during driving, and using for motor-assisting can be beneficial, if the energy is charged at a lower driving power, and this energy is used for motor-assisting at a higher driving power. However, the additional fuel saving is relatively small, because the drive energy at higher powers is relatively small. This is shown in the Fig. 4 by the histogram of Pv , and the fuel rate at the OOL as a function of the engine power, which is equal to the drive power assumed. The largest relative fuel saving is realized by applying the M, and the BER mode, because the energy for electric driving is obtained without any additional fuel cost, and the engine operation at relative low efficiency is eliminated. In addition, the largest drive energy is at low drive powers. A trade-off between the M, the CH, and the MA mode, can be found by changing the design variable u∗max . This will be explained in the following sections. Histogram for the NEDC
350
Fuel rate at the OOL
300
* umax
150
2
CH
MA 1.5
M
100
1
50 0
0.5
0
5
10
15 20 25 Drive power, Pv [kW]
30
35
3.2 The MA, the CH, and the E mode In this section, the optimal power for charging, or discharging during the MA, and the CH mode is calculated. Accordingly, the optimizing steps for calculating the minimum fuel consumption, and energy balance conservation over the drive cycle are explained. The fuel rate can be written as the sum of the fuel rate change only depending on the drive power demand Pv , and charging, or discharging power uII . E
2.5
Drive energy
200
Fig. 5. Energy balance, and fuel consumption during the BER/M, and the CH/MA modes.
m ˙ f (u, λ) = m ˙ f,0 + 4m ˙ f (uII , λ) . |{z} | {z }
3
Fuel rate [g/s]
Time [s]
250
3.5
This energy has to be counterbalanced with the relative energy 4xII (tf ) at the end of the cycle during the MA, and the CH mode as is shown in Fig. 5.
0 40
Fig. 4. The histogram for the vehicle power demand, and the fuel rate at the OOL as a function the engine power.
(8)
CH/M A
The fuel rate change 4m ˙ f for a certain accumulator power u 6= 0 is, sgn(uII )
4m ˙ f (uII , λ) = λ(uII , Pv ) · uII · ηS
,
(9)
with variable λ representing the equivalent accumulator energy to fuel factor which is a function of uII , and Pv . It can be seen with Eq. (9), that generally the λ is smaller for charging than for discharging if ηS < 1, and is equal for charging, and discharging if ηS = 1, or, λ(uII > 0) ≤ λ(uII < 0) if ηS ≤ 1 ∀ t. A decision variable Γ representing the fuel rate profits is introduced in order to determine at
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∗
∗
Γ(uII , λ ) = (λ − λ) · uII ·
sgn(uII ) ηS
≥ 0 ∀ uII . (10) Thereby, the optimal accumulator power u∗II is the power which maximizes Γ given a certain value for λ∗ , u∗II = arg max (Γ|λ∗ ), (11)
3 ∆ xI (BER/M) ∆ x (CH/MA)
2.95
II
∆x
2.9 2.85 lv
λ ⋅ h [−]
which drive power demand Pv , and at which power set-point uII the energy needs to be recharged during the CH mode, or discharged during the MA mode. The decision variable is a function of the accumulator power uII , and a certain reference value of the energy to fuel ratio λ∗ . The λ∗ determines if charging (λ < λ∗ ), or discharging (λ > λ∗ ) is beneficial, and is used to balance the energy quantities during the MA, and the CH mode over the drive cycle. The decision variable is,
2.8 Optimal value for λ
∆ xI
∆ xII
2.75 2.7 2.65 2.6 −2
−1.5
−1
−0.5
0
0.5
1
1.5
∆ x(tf) [MJ]
Fig. 6. Optimization of λ for 4x(tf ) = 0.
uII
whereby the power set-point is limited between the following constraints, umin ≤ 0 ≤ u∗II ≤ umax
(12)
Then 4xI (tf ) is discharged (charged) at vehicle power demands where the fuel savings (costs), i.e., 4m ˙ f are maximum (minimum). If u∗II = 0, then the vehicle is propelled by the engine only that is the E mode. The total energy flow out of S during the CH, and the MA mode is Z tf 4xII (tf ) = u∗II dt. (13) 0
The total optimal power set-point becomes, u∗ = u∗I + u∗II .
(14)
Obviously, as example if the drive power demand over the drive cycle is known a-priori, it is possible to calculate λ∗ such that the relative energy at the end of the drive cycle becomes exactly zero, i.e., 4x(tf ) = 4xI (tf ) + 4xII (tf ) = 0,
Fig. 7. Block diagram representing the optimization scheme for determining u∗max off-line. solved by iteratively updating λ∗ until 4xI = −4xII , while at each time point the Γ is maximized. This is respectively shown by iteration loop 2, and 1. During this process, the minimum cumulative fuel consumption is determined by changing umax such that, u∗max = arg min (Mf ).
(17)
umax
This is depicted by iteration loop 3 in Fig. 7.
(15)
∗
In Fig. 6, the optimal value for λ is shown where by the energy used during the BER/M mode are in balance with the energy used during the CH/MA mode. The parameter hlv represents the lower heating value for petrol. Concluding, the total fuel consumption calculated with the proposed RB EMS can be stated as, Z tf ∗ ∗ Mf (u , λ ) = m ˙ f (u∗ , λ∗ )dt (16) 0
3.3 Numerical optimization routine for calculating u∗max (off-line) In the Fig. 7, a block diagram is shown of the optimization routine suggested in this study. The problem is, that in order to calculate the optimal power set-point u∗II with Eq. (11), the λ∗ has to be known beforehand. This causality conflict is
4. SIMULATION RESULTS 4.1 Model The vehicle parameters are summarized in the Table 1. Furthermore, the fuel map of the 1.6 [l] Table 1. Relevant vehicle parameters Parameter Curb mass Air drag coefficient Frontal area Roll resistance coefficient Regenerative brake fraction
Value 1390 0.35 2.11 0.9 1
Unit [kg] [−] [m2 ] [%] [−]
SI engine is shown in Fig. 2, and the engine is assumed to be operated at the OOL. The optimal engine angular speed ωp∗ is prescribed for every engine power demand Pp . The transmission is ideally assumed, i.e., with infinitely transmission ratios, and without any losses. Furthermore, the
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fuel rate as a function of engine power at the OOL is approximated by m ˙ f (t) = φ(ωp∗ , Pp ) ≈
i=2 X
14
13 kW 12
ci (ωp∗ , Pp ) · Ppi (t) (18)
Table 2. +: active mode, -: inactive mode BER + + +
MA + +
CH + +
*
7 kW 6
4 kW
4
1 kW
In this section the fuel consumption results are discussed for different constant values for ηS ∈ {30, 100%}, and power constraint values umax = −umin ∈ {1, 13} [kW ] for S. The fuel consumption is calculated with DP, and the RB EMS. In the Table 4.2, the fuel saving results (percentage of the base line fuel consumption) as an average of 35 simulations are summarized. The computation time for 35 simulations (NEDC) with the DP algorithm was 56 hours on a Pentium IV, 2.6 GHz, with 512 M B of RAM. In contrary to the RB EMS the total simulation time was less than 1 minute. The average error between the RB EMS, and DP is 0.6%. The small error between DP and the RB EMS is mainly caused by numerical rounding errors within DP due to grid inaccuracy. Additional charging, and motor-assisting results overall in a small average fuel saving improvement of 0.4%. In the Fig. 8, the results for different M + + +
8
2
4.2 Results, and Conclusions
EMS RB RB DP
umax increase umax [kW]
by neglecting the higher order terms (i > 2) in Pp . The fit coefficients are ci (ωp∗ , Pp ) > 0 for i ∈ {0, 1, 2}. The used drive cycle is the NEDC, and the fuel consumption of the base line vehicle without a S is 375.6 [g].
10 kW
10
i=0
Average fuel saving 26.10% (98.02 [g]) 26.20% (98.43 [g]) 26.05% (97.83 [g])
optimization values of u∗max as a function of ηS for different values of umax calculated with the RB EMS are shown. It can be seen that the sensitivity of u∗max to ηS increases more than linear with increase of umax . Furthermore, the connection of S at the engine-side, or the wheel-side of the T, and ηT , or the parameters describing engine efficiency have influence on u∗max , but can be investigated with this RB EMS very quickly, and with sufficient accuracy. REFERENCES A. Sciaretta, L. Guzzella and C.H. Onder (2003). On the power split control of parallel hybrid vehicles: from global optimization towards real-time control. J. of Automatisierungstechnik 51(5), 195–205. A. Sciaretta, M. Back and L. Guzzella (2004). Energy management strategies for vehicular electric power systems. J. of IEEE-Trans. on Control Systems Technology 12(3), 352–363. Bellman, Richard E. (1962). Dynamic programming. Princeton University Press.
0 30
40
50
60
ηS [%]
70
80
90
100
Fig. 8. Optimal u∗max as a function of ηS for different umax . C. Lin, J. Kang, J.W. Grizzle and H. Peng (2001). Energy management strategy for a parallel hybrid truck. In: Proc. of the American Control Conference. Arlington. pp. 2878–2883. Chen, J. and M. Salman (2005). Learning energy management strategy for hybrid electric vehicles. In: Proc. of the Symposium IEEEVehicle Propulsion & Power. Chicago, USA. pp. 427–432. G. Paganelli, S. Delprat, T.M. Guerra J.M. Rimaux and J.J. Santin (2002). Equivalent consumption minimzation strategy for parallel hybrid powertrains. In: Proc. of the Symposium IEEE-Vehicular Transportation Systems. Atlantic City, USA. pp. 2076–2081. G. Rizonni, P. Pisu and E. Calo (2004). Control strategies for parallel hybrid electric vehicles. In: Proc. of Symposium IFAC-Advances in Automotive Control. Salerno, Italy. pp. 508– 513. J. Kessels, P. van den Bosch, M. Koot and B. De Jager (2005). Energy management for vehicle power net with flexible electric load demand. In: Proc. of 2005 IEEE Conf. on Control Applications. Toronto, Canada. pp. 1504–1509. M. Koot, J.T.B.A. Kessels, B. de Jager W.P.M.H. Heemels P.P.J. van den Bosch and M. Steinbuch (2005). Energy management strategies for vehicular electric power systems. J. of IEEE-Trans. on Vehicular Technology 54(3), 771–782. S. Delprat, T.M. Guerra and J. Rimaux (2002). Optimal control of a parallel powertrain: From global optimal to real time control strategy. In: Proc. of the Symposium IEEEVTC. pp. 2082–2088. Sciaretta, A. and L. Guzella (2003). Rule-based and optimal control strategies for energy management in parallel hybrid vehicles. In: Proc. of the 6th Int. Conf. on Engines for Automobile. Capri.
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