Dynamic Programming Technique in Hybrid Electric

Vehicle Optimization

Rui Wang

Department of Electrical and Computer Engineering

North Carolina State University

Raleigh, NC 27606

rwang11@ncsu.edu

Srdjan M. Lukic

Department of Electrical and Computer Engineering

North Carolina State University

Raleigh, NC 27606

smlukic@ncsu.edu

Abstract— Hybrid electric vehicle (HEV) is a type of vehicle

which combines a conventional internal combustion engine (ICE)

propulsion system with an electric propulsion system. HEV is

intended to achieve either better fuel economy than a

conventional vehicle, or better performance. HEVs have been

gaining popularity given that they are an effective solution to

reducing fuel consumption and emissions. However, its potential

in fuel economy is hardly fully explored by existing control

strategies based on engineering intuition. Dynamic programming

(DP) technique is an effective tool to find the globally optimal use

of multiple energy sources over a pre-defined drive cycle. As a

global optimizing algorithm, DP ensures to converge to the global

optimum. Even though DP is an off-line algorithm, the results

can serve as a benchmark to evaluate and improve an existing on-

line algorithm. In this paper, the procedures for implementing

DP to three typical HEV powertrains are explained in detail. Also,

the cost function of DP is discussed. In the case study of Toyota

hybrid system, a simplified vehicle model is given and validated.

Then DP is applied to this model and the effect of cost function on

fuel economy and battery state of health (SOH) is discussed.

Comparing to the simulation results over UDDS cycle obtained

from the Prius model in Advisor, the DP results over the same

drive cycle shows a 30% potential improvement in overall cost,

which converts the electricity cost into fuel cost. In addition,

based on the DP results, a lookup table based real-time control

strategy is developed. This control strategy results in an

improvement of 27% of overall cost, which is very close to the

ideal case.

Keywords-dynamic programming; hybrid electric vehicle;

control strategy; drive cycle prediction.

I. INTRODUCTION

Hybrid electric vehicle (HEV) is a type of vehicle which has more than one propulsion power resources, usually combining a conventional internal combustion engine (ICE) with an electric propulsion system. HEVs have been gaining popularity given that they are an effective solution to reducing fuel consumption and emissions. The fuel-efficiency on 2010 US model Toyota Prius is above 50 miles per gallon, however, it’s interesting to explore if there is still improvement on fuel economy by more advanced control strategy. To answer this question, a global optimization control algorithm is needed. In recent studies, dynamic programming (DP) technique has been demonstrated effective to find the best fuel economy over certain drive cycles [1-5]. As a global optimizing algorithm, DP calculates every possible combination of engine and battery power at each step, ensuring that the algorithm converges to the global optimum. With the results obtained by DP, the current implementable control strategy, such as rule based strategy, can be evaluated and furthermore optimized.

However, the detailed procedures of DP were not introduced in [1-5], and the cost function didn’t consider the cost on battery state of health (SOH).

In this paper, DP was respectively applied to three typical HEV powertrains – series, parallel, and series-parallel – and the detailed procedures were given via three flow charts. Referring to the work in [1] the vehicle model simplification was extended to the three typical powertrains based on the control methods (power control or speed/torque control). The case study selected Toyota Prius as it is currently the most popular commercial HEV model. The simulation results showed a 30% potential improvement in overall cost, which converts the electricity cost into fuel cost. Then a real-time control algorithm was developed based on DP results, and the improvement in overall cost was 27%, very close to the optimal results.

This paper is organized as follow. In section II DP is explained in detail, and how it can be used in HEV control strategy optimization is introduced in Section III. Section IV discusses the procedures of implementing DP to three typical hybrid powertrains– parallel, series, and series-parallel. Case study of Toyota Prius HEV is shown in section V, where DP is implemented and real-time control algorithm is developed.

II. DYNAMIC PROGRAMMING

Dynamic programming deals with the situations in which decisions are made at each stage, with the objective to minimize (maximize) a certain mathematical expression of cost function [6]. Specially, a deterministic discrete-state and finite-state system can be expressed as

( ) ( )

and the cost function is of the form

( ) ∑ ( ) ( )

The dynamic optimization problem is over the controls , to minimize/maximize cost function J subject to constraints. Dynamic programming is an effective tool to solve general dynamic optimizing problems mentioned above. It can handle the constraints and obtain a globally optimal solution for nonlinear systems.

The DP technique is based on Bellman’s Principle of Optimality, stating that the optimal policy can be obtained if

single-stage sub-problem involving only the last stage is solved first, and then sub-problem involving last two stages, last three stages,…, etc. until the entire problem is solved step by step. In this case, the overall dynamic optimization problem, taking minimizing problem for example, can be expressed as

( ) ( ) ( )

( )

[ ( ) ( )]

( )

The preceding expression proceeds backwards in time. However, a deterministic finite-state problem is equivalent to a shortest path problem and can be solved by either backward or forward DP algorithm [6]. The forward DP can be expressed as

( ) ( ) ( )

( )

[ ( ) ( )] ( )

At step k+1, ( ) is the minimum cost at state

among the costs for every admissible route from ,

which can be explained in Figure 1. After ( ) for

every is calculated, the DP continues to the next step

k+2, k+3, … until step N.

. . .. . . k k+1

Jk(xj)

Jk+1(xi, k+1) = min[gk+1(xi, k+1, uk, j+1, i)+Jk(xj+1,k),

gk+1(xi, k+1, uk, j, i)+Jk(xj,k),

gk+1(xi, k+1, uk, j-1, i)+Jk(xj-1,k),

gk+1(xi, k+1, uk, j-4, i)+Jk(xj-4,k)]

Jk(xj-1)

Jk(xj-4)

Jk(xj+1)

step

state

Jk+1(xi-6, k+1) = min[gk+1(xi-6, k+1, uk, j-4,i-6)+Jk(xj-4,k),

gk+1(xi-6, k+1, uk, j-8,i-6)+Jk(xj-8,k),

gk+1(xi-6, k+1, uk, j-10,i-6)+Jk(xj-10,k)]

Jk(xj-8)

Jk(xj-10)

Figure 1. Forward dynamic programming

III. DP PROGRAMMING IN HEV OPTIMIZATION

DP has been widely used in HEV optimization. The model of an HEV is identical to (1) where is the state vector including vehicle speed, SOC, engine speed, motor speed, etc. while is the control vector that covers gear number, generator torque, engine torque, motor torque, etc. The optimization goal is to find the control at each step to minimize a cost function (7). Constraints (8) are necessary during the optimization, and it is explained in Section IV how they are met.

( ) ( )

[ ( ) ( )] ( )

subject to

( )

( ( )) ( ) ( ( ))

( ( ) ( )) ( ) ( ( ) ( ))

( ) (8)

If we concern more than the fuel economy, for example we try to minimize the emissions or reduce the gear changing at the same time, the cost function can be updated by adding more variables multiplied by weights. For a PHEV, it is better to minimize the overall cost while the battery peak power is limited in consideration of battery SOH. The overall cost function can be expressed in (9)

( )

[ ( ) ∑ ( ) ( )

( )]

( )

[ ( ) ∑ ( ) ( )

( )] ( )

where ( ) relates to the mass of emission, such as NOx

and HC, at step k with the weight , ( ) relates to battery

current at step k, equals to absolute value of gear

number at step k-1 minus gear number at step k. At step N,

one additional penalty term is added where

equals initial SOC 0 minus final SOCN. The coefficient is to convert electricity cost to equivalent fuel cost. Note that

this is only feasible for a PHEV, and for a standard HEV, should be substantially large and the penalty term should be

( ) where SOCN* is the desired final SOC.

IV. DETAILED DP PROCEDURES

A. Parallel Model and DP Procedures In this section, a typical parallel powertrain, an integrated

motor assistant (IMA) system, is discussed. For a parallel hybrid electric vehicle, the engine couples to the wheels whenever the car runs. Hence the speed of engine/motor set (rad/s) can be taken as a known parameter by (10) as long as transmission ratio and vehicle speed v (m/s) are given.

( )

The simplified IMA system model can be described as below.

1) Engine model: The engine is modeled as a black box with inputs of engine speed (rad/s) and engine torque (Nm), and output of fuel mass flow rate (kg/s) [7]. Then the engine can be model by (11) and (12).

( )

( )

( )

( )

2) Battery model: The static equivalent circuit battery model in [1] is utilized as shown in (13), with input of motor torque (Nm), and output of SOC.

√

( ) ( )

( ) ( )

start

initialization

stop

No

Find SOCmin<SOCk <SOCmax;Calculate required torque at wheels Treq;

j = 1;

-Tm_max < Tm <Tm_max

JK+1(SOCK+1) = J(SOCK+1);

Yes

Calculate SOCK+1 and J(SOCK+1)

J(SOCK+1) < JK+1(SOCK+1) ?

Yes

k < length(v)?

k = 1

j < length(SOCk)?j = j +1

No

Calculate ωe, ωm, Tm_max

k = k + 1;Jk = J1; J1 = 1e6*ones(n,1);

SOCk = SOCk+1;Yes

gear < gearmax ?

No

Yes gear = gear + 1

Engine_on=1; Te_max = f1(ωe);

Te = Te_min;

No

Te < Te_max ?

No

Te = Te + ΔTe

Yes

Tm = f2(Te, Treq, Gear ratio);

No

ωe_min< ωe <ωe_max?

SOC=SOCk(j);gear = 1;

Te = 0; Engine_ on=0;

Yes

Engine_on==1?

Yes

No

Figure 2. Flow chart of DP procedures for parallel powertrain

3) Vehicle dynamics: The vehicle is modeled as a point-mass and only longitudinal dynamics is considered [8].

( )

( )

If time interval of 1 second is considered, and assume remain constant in this interval, the discrete model for (14) can be expressed as

[ ( ) ( )] ( )

[ ( ) ( )]

( )

where ( ) ( )

The procedures for implementing DP to a parallel powertrain can be summarized in figure 2.

The advantage of the algorithm above is that it calculates only | | times with 1 Nm increment in inner loop for each . But the calculated values of admissible may not fall exactly on grid points, and quantization and interpolation are needed, hence the sacrifice of accuracy is unavoidable. However, if the increment of SOC is substantially small, the error can be negligible. One improvement can be obtained by inversely calculate for each admissible pair of SOCk and SOCk+1 using (13) while and are within the feasible regions. However,

as the density of SOC grid is much larger, at each step the calculation burden is much heavier.

B. Series Model and DP Procedures In a typical series HEV powertrain, the engine is

mechanically decoupled with the powertrain. As a result, both engine speed and torque can to be controlled at each step, and given a power requirement the best operation point can be found out to minimize fuel consumption (figure 3). Thus, engine power Pe can be selected as the control variable, which differs from Te for parallel HEV. The simplified series HEV model can be described as below.

1) Engine model: On the engine map, several power

contour lines are drawn and for each single [ ] the minimum fuel consumption point can be found

(Fig.3). The fuel consumption rate can be constructed as a

function of in form of (16).

( )

Figure 3. Engine fuel consumption map

2) Battery model: As power control is adopted, the battery

model should better use battery power Pb as input. Thus the

battery model in (17) is preferable referring to [1], [3].

√

( ) ( )

( )

3) Vehicle dynamics: As power control instead of torque control is used, the power balance equation is preferable for vehicle dynamics.

[ ( ) ( )] [ ( ) ( )]

( )

where represents the efficiency of generator and power electronics, and represents the efficiency of motor and power electronics. The procedures for implementing DP to a series powertrain can be summarized in Figure 4.

start

initialization

stop

No

Find SOCmin<SOCk <SOCmax;Calculate required power at wheels Preq;

j = 1;

JK+1(SOCK+1) = J(SOCK+1);

Calculate SOCK+1 and J(SOCK+1)

J(SOCK+1) < JK+1(SOCK+1) ?

Yesk < length(v)?

k = 1

j < length(SOCk)?j = j +1

No

k = k + 1;Jk = J1; J1 = 1e6*ones(n,1);

SOCk = SOCk+1;Yes

No

Pe < Pe_max ? Pe= Pe + ΔPeYes

No

SOC=SOCk(j);Pe = Pe_min;

Pbat = Preq – Pe*η1

Figure 4. Flow chart of DP procedures for series powertrains

Engine

Generator Motor

Final Drive

Wheel Wheel

Ring GearCarrier

Sun Gear

Battery

DC Bus

Figure 5. Prius powertrain

The advantages and disadvantages are similar to those discussed in section IV-A. The improvement can also be to inversely calculate for each admissible pair of SOCk and

SOCk+1 using (17) while and ( ) are

within the feasible regions, with a cost on computation burden.

C. Series-parallel Model and DP Procedures Series-parallel powertrain combines the advantages of both

series and parallel powertrains, with compromise on manufacturing cost and control complexity. In this section, a typical series-parallel powertrain, Toyota Hybrid System (THS) for Prius (Fig.5), is discussed.

Sun Gear

Carrier

Tc Tr

Ts Ring Gear

1 a) lever model for torque analysis

Sun Gear

Carrier

Ring Gear

ωs

ωc

ωr

1

b) lever model for speed analysis

Figure 6. lever model for plenary gear set

The simplified series-parallel HEV model can be described as below.

1) Planetary gear set: Planetary gear set can be modeled

as an ideal “lever”[9] as shown in Fig. 6, where

( <1) is the gear ratio.

{

( )( )

( )

( )

( ) ( )

and at each step are known for a given drive

cycle, so there are five unknowns ( ) and

three equations in (19) with degree of freedom equal to 2.

On the other hand, power control is still admissible.

{

( )

at each step is known for a given drive cycle, so there

are five unknowns ( ) and three equations in (20)

with degree of freedom equal to 2.

2) Engine model: Engine model is identical to either the one in section IV-A for speed/torque control or that in section IV-B for power control.

3) Battery model: Battery model for speed/torque control

is identical to (13) except that the term ( ) is

replaced by ( )

( ). For

power control, it is identical to (16) except that the term is

replaced by ( ).

start

initialization

stop

No

Find SOCmin<SOCk <SOCmax;Calculate Tm_max, ωm and Treq;

j = 1;

Te = Te_min;Tg_max = f3(ωg);

Yes

-Tm_max < Tm <Tm_max && -Tg_max < Tg <Tg_max?

JK+1(SOCK+1) = J(SOCK+1);

Calculate SOCK+1 and J(SOCK+1)

J(SOCK+1) < JK+1(SOCK+1) ?

Yes

i < length(v)?

k = 1

j <= length(SOCk)? j = j +1

No

Yes

ωe = ωe_min;

k = k + 1;Jk = J1; J1 = 1e6*ones(n,1)

SOCk = SOCk+1;Yes

ωg_min< ωg <ωg_max?

ωe < ωe_max ?

No

Yes ωe = ωe + Δωe

Te_max = f1(ωe);ωg = f2(ωm, ωe, kp);

No

Te < Te_max ?

No

Te = Te + ΔTe Yes

Tm = f4(Te, kp);Tg = f5(Te, kp);

No

No

Yes

Figure 7. Flow chart of speed/torque control based DP procedures for series-parallel powertrains

4) Vehicle dynamics: For speed/torque control force balance equation (21) is used while for power control power balance equation (22) is used.

[ ( ) ( ) ( )]

[ ( ) ( )]

( )

[ ( ) ( ) ( )] [ ( ) ( )]

( )

The procedures for implementing DP to a series-parallel powertrain can be similar to that in IV-A or IV-B based on the control method. For speed/torque control, the control variables

can be any pair of the five unknowns in (19) with one torque variable and one speed variable. As the cost function relates to engine speed and torque directly, and may be good choices. For power control, the control variables can be any pair of the five unknowns in (20). and may be good choices. The flow chart of speed/torque control based DP procedures for series-parallel powertrain is shown in Figure 7.

V. CASE STUDY

Prius owns the most market share of commercial HEVs, with around 50 MPG (miles per gallon) on the 2009 US model. In this section DP is implemented to Prius model, which is obtained from Advisor 2002 [10], to find the potential of fuel economy improvement.

A. Vehicle Parameters: The vehicle parameters are shown in Table I, where M

is the mass of the vehicle, Rw the wheel radius, io the gear ratio of final drive, ρ the gear ratio of planetary gear set, fr rolling friction coefficient, S the area of frontal surface, Cd the air drag coefficient, ρa the density of air, and Qb the battery maximum capacity. All these parameter are obtained from the Prius model in Advisor2002.

TABLE I

VEHICLE MODEL PARAMETERS

Parameter value unit

M 1365 kg Rw 0.287 m

io 3.93 -

ρ 30/78 - fr 0.015 -

S*Cd 1.746*0.3 m2

ρa 1.2 Kg/ m3 Qb 6 Ah

B. Vehicle Model The engine is modeled as a static map with inputs

engine speed and engine torque and output fuel consumption. The engine map is obtained from Advisor2002.

In the same way, the generator and motor are respectively modeled as static maps with input torque and speed and output efficiency. Note that the efficiency here is the overall efficiency considering the efficiency of power electronics.

The battery model is identical to (13) and the parameters of resistance and capacity are from Advisor2002. The vehicle dynamics is identical to (19) and (21).

C. Model Validation As we take the simulation results in Advisor as

reference, the simplified vehicle model (SVM) is validated in such way: 1) run the Prius model in Advisor over UDDS drive cycle, 2) obtain the operation points at each step, and 3) implement the same operation points in SVM to compare the simulation results.

The fuel consumption of the engine over UDDS cycle is 427.9416g (48 MPG) compared to 436.7761g (47 MPG) given by Advisor, with an error of -8.8345g. The error is due to the cold start correction in Advisor model and not in SVM. However, as the error is small, it can be negligible.

The electric system model, which contains generator, motor and battery, is validated via the input power of generator and motor, which is calculated from the operation points (speed and torque) of the generator and motor in Advisor. Columbic effect is considered when battery is charged and the comparison of battery power is shown in Figure 8. The battery performance in SVM is similar to that in Advisor.

Thus, we are confident that the two models are comparable.

Figure 8. The battery model comparison

D. Simulation Results and DP Algorithm Improvement 1) Improvement in cost function: The UDDS drive cycle is

used to evaluate the algorithms mentioned in Section III. The initial and final SOC are set to 0.5, and at first only the fuel economy is optimized. The results are shown in Figure 9.

a) Battery SOC

b) Engine power

c) Battery power

Figure 9. Simulation results on Prius powertrain

It is observed that the engine turns on and off very frequently, the battery peak power is large, and the brake energy is 100% regenerated. To make the model more feasible, the penalty terms for engine on/off and battery current are added and the engine brake is considered. The cost function is updated as

( )

[ ( )

( )] ( )

where I is the battery current and is the absolute value of current engine on/off status (1 for ‘on’ and 0

for ‘off’) minus that of last step. After many trails, the parameter of is set to 0.0003 and is set to 0.1. Then the simulation results are shown in Figure 10 and Table II.

a) Battery SOC

b) Engine Power

c) Battery power

Figure 10. Simulation results with updated cost function

2) Test with PHEV: It is also interesting to study what is the improvement in driving cost with the technology of PHEV. If the initial SOC is set to 0.9, and the cost of electricity and fuel are transferred into dollars as mentioned in (9), the simulation results are shown in Figure 9. To increase the battery power range, the in (22) is set to 0.0002.

a) Battery SOC

b) Engine power

c) Battery power

Figure 11. Simulation results of charge depleting

The power of battery is in reasonable range in Fig.11, and the overall cost is reduced by 34%. However, as this drive

cycle is only 7.4 miles, the engine doesn’t work much, which means for a longer drive distance, the 119.9 MPG is not possible. On the other hand, however, this test uses the same battery capacity, which means in a PHEV with larger battery capacity the high MPG can last for a longer drive cycle. In sum, PHEV has a lower overall cost than a standard HEV anyway, while the battery SOH is guaranteed.

3) Test with same conditions as in Advisor: The final test uses the same initial and final SOC values as Prius model in Advisor, and the simulation results are shown in Figure 12 and Table II. The battery power is in reasonable range as well as engine power.

a) Battery SOC

b)

Engine power

c) Battery power

Figure 12. Simulation results with same conditions as in Advisor

TABLE II

SIMULATION RESULT COMPARISON

MPG Improvement Cost Decrease

Advisor model (SOC0 = 0.7, SOCN = 0.5)

48.4 0 $0.4884 0

DP fuel only

(SOCN=SOC0) 74.6

9 54.3% $0.3291 32.6%

DP consider engine on/off and battery life

(SOCN=SOC0=0.5) 64.3 32.87 % $0.3822 21.74 %

DP consider engine on/off and battery life (SOC0 = 0.9, SOCN =0.5)

119.9

147.78% $0.3215 34.18 %

DP consider engine on/off and battery life

(SOC0 = 0.7, SOCN = 0.5) 87 79.8% $0.3382 30.75%

E. On-line Control Strategy based on DP Results DP algorithm is based on a given drive cycle, which is

unknown in real cases. In addition, the calculate burden is still too heavy to implement in real time. However, with the optimal operation points obtained via DP, an implementable online control strategy can be developed with quasi-optimal performance.

Figure 13. Relationship between PSR and demanded power in DP results

Figure 14. Relationship between PSR and demanded torque in DP results

In [1] the author uses a simple power-split algorithm. The optimal power split ratio (PSR), which equals engine power over demanded power, is a function of demanded torque at the input shaft of transmission. In this case, for any given demanded torque the optimal PSR can be found and hence the optimal engine power. However, this study is for a parallel powertrain where the engine is mechanically coupled to the wheels and the engine speed is dependent on the vehicle velocity and gear ratio of the transmission. In a series-parallel powertrain the engine speed is independent and there is not a strong relationship between the engine power and demanded power (or torque), as shown in Figure 13 and Figure 14.

If one more variable, the vehicle velocity, is added, the 3D plot is shown in Figure 15. The discrete operation points are not able to build a 2D lookup table, so linear interpolation is used and the lookup table is shown in Fig. 16. This lookup table can be implemented in real-time control. The simulation results are shown in Figure 17 and Table III. The initial and final SOC are identical in all the cases in Table III to make the results comparable.

From Table III, the updated rule-based control algorithm is pretty close to the optimal results obtained by DP. Note that the battery power is not well limited especially for regeneration, so the actual final SOC should be lower and hence the total cost should be higher. However, this algorithm works fine enough for real time control, though some more constraints are needed.

In summary, the rules for finding optimal engine power are more complicated in series-parallel powertrains than in parallel ones. This study shows that 2D lookup table works fine in one particular drive cycle but may not be optimal in some others. Also, a more general algorithm should take SOC into consideration, and the optimal real-time algorithm can be better designed. Anyway, this study has proven that this method (from DP to real-time algorithm) is admissible and effective if the rules are wisely designed.

Figure 15. Relationship among Pe, Preq and v in DP results

Figure 16. 2D lookup table based on DP results

a) Battery SOC

b) Engine power

c) Battery power

Figure 17. Simulation results with new rule-based control algorithm

TABLE III SIMULATION RESULT COMPARISON

MPG increase Total cost

decrease

Advisor model 48.4 0 $0.4884 0

New rule-based 80.66 66.65% $0.3524 27.85%

DP 87 79.8% $0.3382 30.75%

VI. CONCLUSIONS

Dynamic programming is very powerful tool to obtain global optimization results, which can be used to evaluate an existing control strategy for an HEV. The procedures of DP for three typical powertrain topologies are introduced in detail, and based on this study the readers without a background in DP can implement it easily.

The case study on Prius model shows that the optimal fuel consumption results have some unreasonable factors, such as frequent engine on/off switching and large battery discharge current. Hence the cost functions are updated by adding the consideration of SOH, engine on/off, and equivalent cost of electricity with respect to fuel cost. The potential of overall cost improvement is around 30%. The plug-in powertrain can further improve the equivalent cost by around 10% for a short drive cycle.

The new rule-based control algorithm is developed based on the optimal operation points. The simulation results show a pretty close performance to the optimal case with 3% difference. The further work should improve the real-time control strategy with DP results considering SOC and some other factors.

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