A national laboratory of the U.S. Department of Energy

Office of Energy Efficiency & Renewable Energy

National Renewable Energy Laboratory 

Innovation for Our Energy Future 

Dynamic Programming Applied to 

Investigate Energy Management 

Strategies for a Plug-in HEV 

M.P. O’Keefe and T. Markel 

Presented at the 22nd International Battery, Hybrid and Fuel Cell 

Electric Vehicle Symposium and Exhibition (EVS-22) 

Yokohama, Japan 

October 23–28, 2006 

Conference Paper 

NREL/CP-540-40376 

November 2006 

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Dynamic Programming Applied to Investigate Energy 

Management Strategies for a Plug-In HEV1

Michael Patrick O’Keefe 

National Renewable Energy Laboratory 

Tony Markel 

National Renewable Energy Laboratory

Abstract 

Plug-in hybrid electric vehicles (PHEVs) are an advanced dual-fuel powertrain technology that 

combine features of the battery electric vehicle (BEV) and hybrid electric vehicle (HEV). One of the 

fuels of the PHEV is electricity which is supplemented by another fuel (typically gasoline). The 

gasoline consumption for a PHEV is distance dependent based on the vehicle control strategy. In this 

paper, we explore two basic control concepts applied to a PHEV: an “electric vehicle centric” control 

strategy and an “engine-motor” blended control strategy. A near optimal control solution is derived 

using the dynamic programming optimization algorithm. Based on comparison with the dynamic 

programming results, we show that for urban driving, a PHEV should typically operate closer to an 

“electric vehicle centric” control strategy to provide consistently high fuel savings. We also show that 

PHEVs with smaller motors and lower power-to-energy ratio batteries can save nearly the same 

amount of fuel as a full-size PHEV—but perhaps at a reduced cost. 

Keywords: Plug-In Hybrid, Hybrid Strategy, Energy Efficiency, Modeling, Dynamic Programming 

1 Introduction 

1.1 Plug-In Hybrid Electric Vehicles 

Plug-in hybrid electric vehicles (PHEVs) are a dual-fuel technology capable of transforming the 

transportation energy infrastructure away from non-renewable, high-carbon fuels to more 

environmentally responsible options. One of the PHEV fuels is electricity. The other fuel could be one 

of any number of options, although gasoline is considered here. 

PHEVs can deliver performance equivalent with today’s modern vehicles. Furthermore, compared 

with other technology options, the PHEV does not suffer from some of the infrastructure issues (e.g., 

fuel cell vehicles) nor the limited range issues (e.g., battery electric vehicles) exhibited by other 

technologies. These positive benefits are the result of both efficient delivery of fuel-energy from the 

tank to the wheels and, more importantly, a transition from conventional transportation fuels to 

electricity. This is possible because PHEVs exhibit aspects of both battery electric vehicles (BEVs) 

and hybrid-electric vehicles (HEVs): 

 

1

 This work has been authored by an employee or employees of the Midwest Research Institute under Contract 

No. DE-AC36-99GO10337 with the U.S. Department of Energy. The United States Government retains and the 

publisher, by accepting the article for publication, acknowledges that the United States Government retains a 

non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, 

or allow others to do so, for United States Government purposes. 

1
• A large energy storage unit capable of being recharged from the electrical utility grid and 

supplying net motive energy over a significant distance 

• A hybrid powertrain typically using an internal combustion engine with an electrical motor 

By blending aspects of the BEV with conventional HEVs, one can gain many of the advantages of a 

BEV while eliminating several disadvantages. Because fewer batteries are needed than a full BEV, the 

PHEV comes with a reduced cost penalty versus a BEV of equivalent performance. Furthermore, the 

PHEV has no range penalty and charging times are much shorter than an equivalent BEV. In contrast 

to an equivalent HEV, fuel consumption is further reduced since fuel energy is supplied from both 

electricity and liquid fuel as opposed to just liquid fuel as is the case for conventional HEVs. 

PHEVs work well for vehicles that operate where relatively short trips comprise the bulk of distance 

traveled. By recharging between these short trips, a large portion of the motive energy can come from 

the electrical grid as opposed to gasoline or other fossil fuels. The transition from today’s petroleumbased transportation fuels to electricity opens up many opportunities. By recharging the vehicle’s 

batteries overnight, electrical utilities can increase their operating efficiency. Furthermore, due to the 

difference between peak capacity and base-load capacity, power utilities will typically have enough 

excess capacity at night to recharge a large number of vehicles before having to add new capacity. If 

done correctly, this can lead to reductions in carbon dioxide (CO2) and other greenhouse gas emissions. 

Furthermore, if effort is made to transform the power plants within the national electrical grid to use 

more renewable sources of energy, the benefits of renewable energy can be brought to the 

transportation sector through PHEVs. 

However, PHEVs do have some challenges for commercialization. Chief among these are cost which 

is largely connected with batteries and battery life, in addition to the added cost of PHEV power 

electronics and powertrain components. 

In this paper, we will explore two separate PHEV architectures. Both vehicles yield equivalent 

performance and have the same electrical capacity in the vehicle energy storage system. The first, 

referred to as the full-size PHEV, contains a high-power motor and energy storage unit with smaller 

internal combustion engine (ICE). The second vehicle, referred to as the half-size PHEV, uses a lower 

power motor with lower power energy storage and larger ICE. The second vehicle represents a lowercost PHEV solution because it deemphasizes the (expensive) electric powertrain components such as 

high-powered batteries and emphasizes ICE technology. 

In addition to the vehicle architectures, we also discuss two different PHEV energy management 

strategies: an “electric vehicle centric” approach and a blended-control approach. In the “electric 

vehicle centric” approach, the motor and batteries attempt to meet all traction demand electrically, 

with the engine only helping when the motor is not powerful enough. This strategy uses electricity 

whenever possible. The blended control approach attempts to spread the electrical consumption over a 

larger distance by blending engine power with motor usage at times when the system is more efficient. 

In order to compare these control paradigms, the dynamic programming algorithm, which can 

determine the near-optimal solution of any control problem, is used. 

1.2 The Dynamic Programming Method 

Dynamic programming is a numerical technique that can be applied to any problem that requires 

decisions to be made in stages with the objective of finding a minimal penalty decision pathway [1]. 

“Penalty” used in this sense refers to a quantitative measure of the undesirable outcomes of a decision. 

Dynamic programming combines knowledge of the immediate penalty of the decision at hand with 

knowledge of future penalties that arise as a result of the immediate decision. This algorithm has been 

applied with success to HEVs in the past [2], though the authors are unaware of any application to 

PHEVs to date. 

2
Dynamic programming requires the definition of a discrete time dynamic system (DTDS) and a 

penalty function. Because the dynamic programming algorithm is quite computationally intensive, a 

fast computational model is desired for the DTDS. In the context of this paper, the DTDS is a vehicle 

model which calculates the change in state (battery state-of-charge) resulting from a given control 

setting (the engine shaft out power) over a time-step along a duty cycle. The duty cycle, the vehicle’s 

commanded speed versus time, is treated as deterministic for this study. The algorithm proceeds from 

the end of a duty cycle to the beginning, calculating the penalty of possible control settings at each 

time step. Because knowledge of the duty cycle is required beforehand, the dynamic programming 

algorithm cannot be implemented in actual control systems in real life. However, outputs from the 

dynamic programming algorithm can be used to formulate and tune actual controllers. 

The penalty function used in this study attributes a penalty for using fuel, not meeting the specified 

duty cycle speed-time trace, and for not holding end state-of-charge at a reasonable level. 

The dynamic programming algorithm as used in this study only explores a subset of the entire design 

space. Because of this, the control cannot be said to be “optimal,” only “near-optimal”. 

There are two main reasons to employ the dynamic programming method in this study: 

• To compare PHEV architectures under “near-optimal control” 

• To gain insights into what the “optimal” control is for a PHEV under various circumstances. 

By comparing all PHEV architectures under an “optimal control,” control itself is eliminated as a 

design variable from the problem. 

2 Analysis Overview 

This study compares the energy implications for two PHEV architectures over multiple urban duty 

cycles (driving patterns) and multiple distances using a “near-optimal” control strategy derived via the 

dynamic programming algorithm. Both vehicles contain enough electrical energy to drive 

approximately 32 km (5.5 kWh usable capacity). The selection of this capacity and specific sizes is 

based on a cost benefit analysis of PHEVs conducted by Simpson [3]. Both vehicles in this study use a 

parallel hybrid design where the engine and/or motor can contribute to tractive effort at any time. 

Gasoline is assumed for the liquid fuel in this study. However, it is important to note that PHEVs 

could use other fuels such as diesel, ethanol (E85), or even hydrogen if the ICE is properly designed to 

handle the given fuel. 

2.1 Vehicle Platform, Performance, and Assumptions 

The vehicle platform used for this study is a mid-size sedan with performance requirements specified 

so as to be competitive in the North American marketplace. The specific requirements are given below 

in Table 1. 

Table 1: Mid-Size Sedan Performance Requirements and Platform Assumptions 

Attribute Value 

Top Speed

top speed to be maintained 

177 km/hr (110 mph) 

Full Acceleration

time from 0 km/hr to 96.6 km/hr (60 mph) 

8.0 seconds 

Passing Acceleration

time from 64.4 km/hr (40 mph) to 96.6 mph (60 mph) 

5.3 seconds 

Hill Climbing

grade (percent rise over road-surface run) to climb with engine at 66% of 

rated power 

6.5% grade @ 88.5 

km/hr (55 mph) 

3
Range

maximum distance traveled starting fully fueled 

643.7 km (400 miles) 

Glider Mass

the mass of the vehicle minus the powertrain

905 kg 

Cargo Mass

the mass of cargo carried while meeting performance constraints

136 kg 

Accessory Loads

the accessory loads assumed for the PHEV

0.7 kW electric average 

4.0 kW electric peak 

Transmission Efficiency

efficiency of the mechanical gearing between motor/engine and wheels

85% 

Electrical Generation Efficiency for Accessories

efficiency of generating the power to electrical accessories

85% 

These requirements imply minimum component sizes. The peak accessory loads are assumed to be 

engaged for purposes of calculating the hill climbing and top speed power requirements. All other 

calculations (including fuel consumption calculations) assume average accessory loads. The resulting 

component requirements for a conventional vehicle and two PHEVs are given in Table 2. 

Table 2: Component Sizes and Weights Used in Study 

Component Conventional 

Vehicle 

Full-Size 

PHEV 

Half-Size 

PHEV 

Spark Ignited Internal 

Combustion Engine 

121.7 kW peak 

238.6 kg 

80.1 kW peak 

171.3 kg 

99.2 kW peak 

202.2 kg 

Electric Motor and 

Inverter 

NA 44.1 kW peak 

45.1 kg 

21.4 kW peak 

33.0 kg 

Battery Pack NA 5.50 kWh usable 

47.2 kW peak 

11.8 kWh full capacity 

94.4 kg 

5.57 kWh usable 

23.9 kW 

11.9 kWh full capacity 

79.6 kg 

Gasoline and Tank 509 kWh 

49.1 kg 

396 kWh 

38.2 kg 

416.7 kWh 

40.2 kg 

Transmission 167.9 kg 176.0 kg 176.8 kg 

Support Structure 68.3 kg 78.8 kg 79.8 kg 

Glider Mass 905.0 kg 905.0 kg 905.0 kg 

Cargo Mass 136.0 kg 136.0 kg 136.0 kg 

Total Vehicle Mass 1428.9 kg 1508.8 kg 1516.6 kg 

Tested Mass 1564.9 kg 1644.8 kg 1652.6 kg 

Degree of 

Hybridization 

0% 35.51% 17.75% 

There are some subtleties in Table 2 that should be pointed out. First, note that advanced battery 

specifications are assumed. Next, the type of battery used in the full size PHEV is different from that 

used in the half-size PHEV. If one examines both batteries, you will quickly see that the usable 

capacity of both packs is nearly the same (the slight difference is due to the difference in weight and 

requirement for both vehicles to drive the same range). However, the weights of both packs and the 

pack powers are different. This arises from a difference in battery pack power-to-energy ratio. The 

energy density of batteries differs by power to energy ratio. Low power-to-energy ratio batteries also 

tend to be slightly less expensive as a technology. For more detail on how cost and weight of each 

component interact with vehicle requirements, see the paper by Simpson [3]. 

The degree of hybridization of both PHEVs appears in Table 2. This percentage is the ratio of the 

motor power to the engine plus motor power. The degree of hybridization of the “half-size” PHEV is 

half that of the full-size PHEV, hence the name. 

4
2.2 Vehicle Model 

The vehicle model is used as the discrete time dynamic system (DTDS) by the dynamic programming 

algorithm. The model takes in a single control setting for each time step—the desired ICE shaft-out 

power for that time step. Based on the ICE shaft output power, the motor will either accept or transmit 

power so as to satisfy the tractive effort and accessory loads required for the given time step. The 

model also contains state information. The only state variable is the battery state-of-charge at the 

beginning of a time step. Based on the tractive effort required during the time step (defined by the duty 

cycle) and the control setting for ICE power, the battery state-of-charge will change. 

Because the dynamic programming method is computationally expensive (requiring many model 

evaluations), the vehicle model used in this study has been constructed to contain only the minimum 

required detail so as to be quick. For example, components in the powertrain use models of power and 

efficiency as opposed to torque, speed, and efficiency. 

A schematic of the powertrain layout and a listing of component efficiencies by output power are 

given for the ICE and motor/inverter components of the full-size and half-size vehicles in Figure 1. 

Internal Combustion Engine for a Full-Size PHEV Internal Combustion Engine for a Half-Size PHEV 

Motor & Inverter for a Full-Size PHEV Motor & Inverter for a Half Size PHEV 

Internal Combustion Engine for a Conventional Vehicle 

Powertrain Layout 

Figure 1: Component Efficiency Maps by Shaft Output Power and Powertrain Layout

5
The model is written in an open-source object oriented programming language called Python and uses 

the following open source modules: Numeric Python, Scientific Python, and Matplotlib [4, 5, 6]. The 

model does not at this time include details related to engine start-stop (i.e., durability constraints, 

vibration, emissions considerations). 

2.2 Duty Cycles Examined 

The scope of this study is limited to urban driving. Two cycles, the Urban Dynamometer Driving 

Schedule (UDDS) and the Los Angeles 1992 (LA92) cycle are used to represent aggressive and 

passive urban driving. A simplification of the UDDS cycle has been substituted for the real UDDS in 

the interest of simulation time. The simplified UDDS cycle approximates the full UDDS using fewer 

time-steps, which aids in speeding up the dynamic programming algorithm. This is a great time saver 

during dynamic programming runs using higher design space resolution. Graphs of the time-speed 

traces appear below in Figure 2. 

Urban Dynamometer Driving Schedule 

(UDDS) Los Angeles 1992 (LA92) 

Urban Dynamometer Driving Schedule 

Approximation (UDDS-a) 

Close-up of UDDS Cycle Close-up of UDDS-a Cycle 

Figure 2: Drive Cycles Used in this Study 

3 Analysis Results 

The gasoline fuel consumption predicted by the models over the LA92 and UDDS-a cycles is given 

below in Table 3. A lower heating value of 32 MJ/liter of gasoline is assumed. The values given for 

fuel economy for this mid-size sedan are reasonable for the North American market. The charge 

sustaining fuel consumption numbers for the PHEVs represent a best case scenario for hybrid fuel 

savings. Fuel savings from the PHEV relative to the conventional vehicle arise from three main 

sources: regenerative braking, engine-off operation for the PHEVs, and more efficient operation of the 

internal combustion engine by supplementing with the traction motor and battery system. 

Table 3: Gasoline Fuel Consumption for Conventional Vehicle and Charge Sustaining PHEVs 

CYCLE Conventional Vehicle Full-Size PHEV 

charge sustaining 

Half-Size PHEV 

charge sustaining 

LA92 3512 J/m 

10.98 L/100 km 

21.43 mpg 

1789 J/m 

5.59 L/100 km 

42.07 mpg 

1862 J/m 

5.82 L/100 km 

40.42 mpg 

UDDS-a 3508 J/m 

10.96 L/100 km 

21.46 mpg 

1535 J/m 

4.80 L/100 km 

49.03 mpg 

1522 J/m 

4.76 L/100 km 

49.45 mpg 

6
Highway Federal 

Emissions Test (US 

EPA) 

2199 J/m 

6.87 L/100 km 

34.23 mpg 

NA NA 

In Table 3, the values given for the highway federal emissions test (US EPA) are for the conventional 

vehicle only and are used here to aid the reader in gaining a feel for the range of fuel consumption. 

Figure 3 shows two of the output graphs resulting from the dynamic programming algorithm being 

applied to the PHEV models. Here, the (near-) optimal engine command is shown versus the tractive 

effort required between the tires and the wheel. This power measured at the tire-road interface is 

referred to as the “roadload.” Two cycles are run in Figure 3: five UDDS-a cycles to the left (29.4 

miles/47.3 km) and four LA92 aggressive urban driving cycles (39.3 miles/63.2 km) to the right. 

The engine is not producing power (zero load) for a large portion of the duration of both cycles. When 

the engine is transmitting power, it appears to be supplementing the traction motor as evidenced by 

operating commands falling below the line y=x/0.85, where y is the engine command and x is the 

roadload (i.e., relating roadload to power at the engine shaft via the constant transmission efficiency of 

Table 1). This engine operation chosen by dynamic programming is a rather complex blending 

algorithm that minimizes fuel consumption within the constraints given. 

We quickly see that duty cycle does make a difference when we contrast the results of the dynamic 

programming algorithm run over the UDDS-a with those of the LA-92 cycle (Figure 3). Due to the 

repetition and simplification of the UDDS-a, there is not as much point scatter. We do see similar 

trends of increased engine usage with roadload. However, we do not see the same amount of point 

scatter at low engine commands and low roadload as we do for the LA-92. 

0

10

20

30

40

50

60

-60 -30 0 30 60

Roadload (kW)

HALF UDDSax5

FULL UDDSax5

Engine Command (kW)

0

10

20

30

40

50

60

-60 -30 0 30 60

Roadload (kW)

FULL Size

HALF Size

Exact

Requirement

Engine Command (kW)

Figure 3: Engine Command versus Roadload over UDDS-a x5 and LA92 x4 

The engine commands versus time for the full-PHEV over five UDDS-a cycles and five LA92 cycles 

appear in Figure 4. There are some similarities in the approaches taken by the dynamic programming 

algorithm in both cases. First we see a strong emphasis on engine commands in the 20-kW to 40-kW 

region repeating each cycle iteration (i.e., five times over the five cycle sequence). This is a sensible 

approach. If it is known ahead of time that engine power must be used to supplement the electric 

capacity of the ESS, it is best to use the engine when the engine is most efficient. As can be seen from 

Figure 1, the engine is most effective when at higher power loadings (peak efficiency between 30-50 

kW). 

Another way to help understand more in-depth what the dynamic programming algorithm is doing is 

to examine the energy storage system state-of-charge or capacity versus distance. These data inform us 

of how the battery system is being drawn down—aggressively discharged or at a reduced discharge 

rate (due to a blending of battery energy with engine power). 

Figure 5 (left) shows the energy storage system discharge over three dynamic programming runs: a 

run of one UDDS-a cycle, a run of three UDDS-a cycles, and a run of five UDDS-a cycles. The fullsize PHEV is used on all of the runs. Note that before the vehicle reaches the “electric vehicle centric” 

range of about 40 km (for the UDDS), the drawdown is completely along an “electric dominant” 

7
drawdown path. As distance increases over this “electric vehicle centric” drawdown path, the slope of 

discharge with distance decreases (i.e., a longer distance is required to discharge the same amount of 

battery energy). Minimum gasoline consumption is obtained by spreading the battery energy out over 

the entire distance of the cycle (such that the desired end state-of-charge is reached when the desired 

range is reached). 

0

10

20

30

40

50

60

0 2000 4000 6000 8000

Time (s)

UDDSax05

En

gine Command (kW)

0

10

20

30

40

50

60

0 2000 4000 6000 8000

Time (s)

LA92x05

En

gine Command (kW)

Figure 4: Engine Commands for the UDDS-a and LA92 by Time 

The right-hand side of Figure 5 shows the same state-of-charge curve for the LA-92 cycle. This time, 

the PHEV simulations optimized by dynamic programming over 1 through 5 cycles are shown. In 

addition, a charge sustaining (CS) run over one LA-92 distance and a run over five LA-92 cycles 

where the vehicle first has an “electric vehicle centric” discharge followed by charge sustaining 

operation are shown. In the “electric vehicle centric” case (FULLx05 EV followed by CS operation), 

the vehicle discharges its electrical energy as fast as possible and then goes into charge sustaining 

operation. This case provides an interesting contrast to the dynamic programming run over the same 

distance (FULLx05). 

UDDS-a 

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60 70

Distance (km)

FULLx01

FULLx03

FULLx05

UDDSa

State of Charge

LA92 

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80

Distance (km)

State of Charge

FULL x05 EV 

followed by

CS operation

FULL x04

FULL x03

FULL x01cs

FULL x02

FULL x01 (bold)

FULL x05

Figure 5: SOC by Distance over UDDS-a and LA92 cycles 

In the right-hand-side of Figure 5, for the first couple of LA92 cycles, the vehicle has not yet reached 

its “all-electric range”. As such, the PHEV can operate in an “electric vehicle centric” mode relying 

almost exclusively on battery energy. For distances below the “all-electric range”, the dynamic 

programming algorithm operates the PHEV with an “electric vehicle centric” discharge where possible 

because this type of operation minimizes gasoline consumption. When the vehicle is asked to operate 

beyond a range that can be supplied exclusively by battery energy (i.e., beyond the all-electric range), 

the dynamic programming algorithm blends engine operation with battery discharge to minimize 

gasoline usage while still driving the requested distance. 

In contrast to the dynamic programming strategies for distances above the “all-electric range”, the 

“electric vehicle centric” strategy draws down as fast as possible and then goes into a charge 

8
sustaining mode. Note that charge sustaining operation is around 5% of full (usable) SOC. The results 

in Figure 5 were exclusively for the “full-sized” PHEV. Let us examine the effect that component 

sizing and powertrain architecture can have on consumption rates. 

Figure 6 shows the consumption rates of gasoline and electricity for the full-size and half-size PHEVs 

over repeats of the UDDS-a and LA92 cycles. The values for electricity consumption for both full and 

half-size architectures are quite similar after all-electric range is exceeded (around 40 km on the 

UDDS-a). This is because both vehicles discharge all of their capacity (albeit in different ways) over 

the same distance. Electricity consumption also begins to decrease after all-electric range is exceeded. 

This is because a fixed electrical capacity is being spread out over longer and longer distances. In 

contrast, gasoline consumption increases after exceeding all-electric range. Note that the full-size 

PHEV does not require gasoline at any point below its all-electric range for the UDDS-a cycle. On that 

same UDDS-a cycle, full-size gasoline consumption rates exceed half-size PHEV gasoline 

consumption rates at higher distances. This is not the case under more aggressive driving such as what 

is seen on the LA92. The UDDS-a results are due to the smaller motor in the half-size PHEV being 

better utilized and thus more efficient. Note that fuel consumption rates are not very different between 

the full and half-size PHEVs for most distances. Full-size and half-size electricity consumption rates 

are nearly identical after all-electric range is met. The largest disparity is under aggressive urban 

driving prior to all-electric range being met. However, in terms of absolute fuel usage, the fuel usage 

of the half-size PHEV is still only about 1/9th the consumption rate of a conventional vehicle over the 

same cycle. 

UDDSa 

0.0

1.0

2.0

3.0

4.0

0 20 40 60 80

Distance (km)

FULL Electric

FULL Gasoline

HALF Electric

HALF Gasoline

Consum

ption 

(liters gasoline/100 km, 

kWh electricity/10 km)

LA92 

0.0

1.0

2.0

3.0

4.0

0 20 40 60 80

Distance (km)

FULL Gasoline

FULL Electricity

HALF Gasoline

HALF Electricity

Consum

ption 

(liters gasoline/100 km, 

kWh electricity/10 km)

Figure 6: Fuel Consumptions of PHEVs over UDDS-a and LA92 Cycles 

Figure 7 shows us one of the key areas of impact for PHEV technology—gasoline fuel usage by 

distance. The set of graphics in Figure 7 are run for the half and full-size PHEV vehicle over the 

UDDS-a (mild urban driving) cycle. Each curve on the figure is the result of a dynamic programming 

run optimized for the given distance. That is, for example, the curve labeled “HALFx3” is the fuel 

consumption minimized by the dynamic programming algorithm to have the lowest net fuel 

consumption after 3 UDDSa cycles back-to-back. This distance aspect is key to understanding what is 

going on with the dynamic programming control of PHEVs. In this case, the dynamic programming 

algorithm has a priori knowledge of the cycle and distance to be run and shows us the optimum 

control under that circumstance. For distances above the “all-electric range,” the operation is a 

“blending strategy” that blends engine usage with motor usage so as to minimize fuel consumption 

while meeting the drive cycle trace. This blending is in addition to that required due to component size 

limitations such as when an acceleration event requires more power than the traction motor can handle 

in isolation and thus the engine assists. 

9
HALF-SIZE 

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80

Distance (km)

Consumption (liters)

HALFx05 CS

HALFx05

HALFx04

HALFx03

HALFx02

HALFx01

FULL-SIZE 

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80

Distance (km)

Consumption (liters)

FULLx05 CS

FULLx05

FULLx04

FULLx02

FULLx03

FULLx01

Figure 7: Gasoline Consumption of Half and Full-Size PHEVs over UDDS-a Cycles 

What is key to note here is that for both the half and full-size PHEVs, a fuel consumption optimally 

blended to have minimum fuel consumption at a given target distance does not necessarily have the 

minimum consumption at other distances. Figure 8 further expands upon this point by displaying 

gasoline consumption results for the LA92 contrasted with an “electric vehicle centric” control 

strategy. All runs change to charge sustaining operation after reaching their target distances. The 

charge sustaining control is optimized by dynamic programming to minimize fuel. 

HALF-SIZE 

0

1

2

3

4

5

0 25 50 75 10

Distance (km)

0

HALFx03

HALFx04

HALFx05

HALFx05 EV

Cycle End Points

Gasoline Consumption 

(liters)

FULL-SIZE 

0

1

2

3

4

5

0 25 50 75 100

Distance (km)

FULLx03

FULLx04

FULLx05

FULLx05 EV

Cycle End Points

Gasoline Consumption 

(liters)

Figure 8: Gasoline Consumption over LA92 for Half- and Full-Size PHEV 

Figure 8 shows the results of dynamic programming runs applied to multiple distances of the LA92 

driving cycle. The designation “FULLxN” refers to the full-size PHEV optimized by dynamic 

programming to run for a number of LA92 cycles equal to N. Dynamic programming for N cycles 

operates the vehicle to have near minimum fuel consumption for the given distance. 

Figure 8 also contains the results of the PHEV run in an “electric vehicle centric” mode (labeled “EV” 

in the figure) where the engine is commanded to be off unless doing so would cause the vehicle to not 

meet the cycle speed-time trace. Thus, simply put, “use electricity to power the vehicle if at all 

possible and supplement with the engine if the motor does not have enough power to meet the cycle 

speed-time trace.” When the vehicle has run its target distance, the PHEV begins to run in charge 

sustaining mode. That is, the PHEV begins to run similar to a conventional HEV where no significant 

net discharge of the energy storage system occurs over time or distance. This can be seen clearly for 

the “FULLx03” case where the vehicle has one of the lowest energy consumptions of the “blended” 

modes until one LA92 distance and then changes over to charge sustaining mode. 

As can be seen from Figure 8, gasoline fuel consumption for the “electric vehicle centric” control is 

nearly equivalent in all cases to the dynamic programming control set to minimize fuel consumption 

over a specific distance. This is because we are continually running the same cycle back-to-back and 

thus it doesn’t matter if one uses the electrical energy all at once at the beginning or spread out 

10
throughout the cycle. The equivalency between strategies would not necessarily be the case for varied 

driving patterns though the effect of driving pattern variation has not been examined in this study. 

Now, let us consider these dynamic programming results in the context of what can be practically 

implemented in real life. There are two general strategies that have been presented: a “blended” 

engine/motor strategy and an “electric vehicle centric” strategy. The “blended” mode is what the 

dynamic programming algorithm optimized to give minimum fuel usage for a given distance most 

closely resembles. The engine is run at the most opportune times to minimize fuel consumption. In the 

“electric vehicle centric” strategy, focus is given to supplying transportation energy with electricity 

and the engine is only used when necessary to meet vehicle performance constraints. 

Note that for distances below the PHEV’s electric range, the dynamic programming algorithm chooses 

an “electric vehicle centric” strategy as this minimizes fuel usage for any range below the electric 

range. That is, if we know we will be running less than the all-electric range of the vehicle, the 

“electric vehicle centric” strategy is near optimal. For distances longer than the PHEV’s electric range, 

a blended strategy is chosen by dynamic programming. However, let us consider the “cost of being 

wrong” in terms of choosing a control strategy for trip distances above the electric range of the vehicle. 

Figure 8 shows us that a vehicle with a “blended” strategy optimized for an intended 80 km of intense 

urban driving would use more fuel than the “electric vehicle centric” strategy all the way up to above 

70 km of driving distance. It should be noted that the “electric vehicle centric” control strategy 

benefits from a charge sustaining control optimized by dynamic programming. However, so does the 

actual blending mode control. Thus, even if we imagine adjusting the fuel consumption rates upwards 

during both blended charge depleting operation and charge sustaining operation, the point is still clear: 

a vehicle operating in blended mode that deviates from the target distance uses more fuel than a 

vehicle using an “electric vehicle centric” approach. Therefore, the “electric vehicle centric” mode is 

essentially the optimal fuel consumption case over most of the distance. In contrast, the “blended” 

mode is optimal for specific distances but non-optimal for others. 

4 Conclusions 

This study shows that a half-size PHEV using a smaller motor and low power-to-energy ratio batteries 

has nearly the same fuel consumption of a full-size PHEV (and in some cases, can have an even lower 

fuel consumption), but uses components that can be of lower cost. The biggest disparity in gasoline 

fuel consumption rates is at low distances (below the all-electric range of the PHEV) over aggressive 

driving cycles where the engine is often forced to assist when meeting roadloads. Even so, under these 

conditions, the absolute fuel usage is low (~1/9th the fuel consumption of a conventional vehicle for 

the case of the LA92 cycle). 

Furthermore, this study shows that under optimal control, a blended control strategy uses 

approximately the same amount of fuel as an “electric vehicle centric” approach for known target 

distances (and constant driving patterns). However, the penalty for “guessing wrong” on the target 

distance and type of travel can be high for a blended strategy. That is, a control strategy optimally 

blended to have minimum fuel consumption at a given target distance does not necessarily have the 

minimum consumption at other distances. Thus, it is typically better to run with an “electric vehicle 

centric” control strategy that emphasizes using electricity to supply vehicle power demand to the 

extend possible within the limits of the motor size. 

This work was conducted over urban driving cycles for PHEVs with specific energy capacities. 

Further work should cover other energy capacity sizes and types of driving including the effect of 

varied driving patterns along a given trip. Additionally, the details of engine on/off including cranking 

energy and emissions implications should be addressed. 

11
References 

[1] Bertsekas, Dimitri P., Dynamic Programming and Optimal Control, Athens Scientific, 2000 

[2] Lin, Chan-Chiao; Peng, Huei; and Grizzle, Jessy W., Control System Development for an AdvancedTechnology Medium-Duty Hybrid Electric Truck, Society of Automotive Engineers 2003-01-3369, 2003 

[3] Simpson, Andrew, Cost-Benefit Analysis of Plug-in Hybrid Electric Vehicle Technology, EVS-22, 2006 

[4] Python Programming Language, http://www.python.org 

[5] Numeric Python and Scientific Python, http://www.scipy.org 

[6] Matplotlib Python Extension, http://matplotlib.sourceforge.net 

Authors

Michael Patrick O’Keefe, Senior Research Engineer, National Renewable Energy Laboratory, 

1617 Cole Blvd; Golden, CO 80401 USA; Phone: 303.275.4268; Fax: 303.275.4415; e-mail: 

michael_okeefe@nrel.gov. 2000-Present NREL; 1997-2000 MS Mechanical Engineering and 

MS Inter-engineering: Technical Japanese, University of Washington, USA; 1998 Nippon Steel 

Corporation, Japan; 1992-1996 BS Mechanical from Northern Arizona University, USA 

Tony Markel, Senior Research Engineer, National Renewable Energy Laboratory, 1617 Cole 

Blvd; Golden, CO 80401 USA; Phone: 303.275.4478; Fax: 303.275.4415; e-mail: 

tony_markel@nrel.gov. 1996-Present NREL; 2002-2005 MS Mechanical Engineering from 

University of Colorado, USA; 1995-1996 Argonne National Laboratory; 1991-1995 BSE 

Mechanical Engineering from Oakland University, USA. 

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Dynamic Programming Applied to Investigate Energy Management 

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This paper explores two basic plug-in hybrid electric vehicle energy management strategies: an electric vehicle 

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