Optimal Energy Management in Hybrid Electric Trucks Using Route Information

Optimal Energy Management in Hybrid Electric Trucks Using Route Information — To benchmark a hybrid vehicle’s Energy Management Strategy (EMS) usually a given, often certified, velocity trajectory is exploited. In this paper it is reasoned that it is also beneficial to optimize the velocity trajectory. Especially optimizing the vehicle braking trajectories, through maximization of energy recuperation, results in considerable fuel savings on the same traveled distance. Given future route (target velocities as function of traveled distance/location), traffic, and possibly weather information, together with the vehicle’s road load parameters, the future power request trajectory can be estimated. Dynamic Programming (DP) techniques can then be used to predict the optimal power split trajectory for the upcoming route, such that a desired state-of-charge at the end of the route is reached. The DP solution is re-calculated at a certain rate in order to adapt to changing conditions, e.g., traffic conditions, and used in a lower level real-time EMS to guarantee both battery state-of-charge as well as minimal fuel consumption. Advances in Hybrid Powertrains Évolution des motorisations hybrides IFP International Conference Rencontres Scientifiques de l’IFP 104 Oil & Gas Science and Technology – Rev. IFP, Vol. 65 (2010), No. 1


INTRODUCTION
A hybrid vehicle employs several power converters instead of one.The main advantage of hybrid electric vehicles is that the vehicle's kinetic and potential energy can be (partially) recovered and stored during braking or driving down hill.The stored energy can be re-used at a later time to provide propelling power to the vehicle.Second advantage of the hybrid structure is that the engine can temporarily generate energy for storage when this is beneficial, or visa versa.The supervisory control algorithm, dealing with the balanced generation and re-use of the stored energy, such that fuel consumption is minimized, is called Energy Management Strategy (EMS).
During the past years, several useful contributions have been made regarding EMS for hybrid vehicles, see, e.g., [1,2] for an overview.Given prior knowledge of the velocity trajectory, the problem of finding the optimal power split, can be addressed as a problem of minimizing the fuel consumption over a power trajectory as function of time, which turns out to be a nonlinear non convex constrained optimization problem.The objective outlined in this paper is to find a real-time implementable EMS, which has no exact knowledge of the future power trajectory, but, still minimizes fuel consumption.The outcome of this EMS depends upon the requested power trajectory, which is directly related to route and traffic characteristics, vehicle road load parameters, acceleration and deceleration paths.
In several publications it is attempted to adapt the realtime implementable EMS for the current estimate of the future requested power trajectory.The strategies in [3][4][5][6] use drive pattern recognition, and make use of characteristic vehicle operating parameters to chose from a set of representative driving patterns and adjust the EMS accordingly.In [7,8] the special class of problems for vehicles operating in a fixed-route service, where past velocity information of the future route is available, are described.By comparing the current velocity with past information a prediction of the future velocity trajectory can be obtained.With the increasing use of on-board Geographical Information Systems (GIS), [9] proposes to use the future route information to estimate the requested power trajectory.The relation of vehicle mass and road load parameters with the requested power trajectory is not used, however.In [10] it is suggested to estimate the future power trajectory based upon route target velocities, but, fixed values for acceleration and deceleration are used.
All methods discussed above try to optimize the power split for an estimated velocity trajectory, while the authors of this paper concluded in previous work [11] that the decelerations in the route, determining the amount of recoverable energy, have considerable influence on the fuel consumption.This holds especially for commercial vehicles (trucks) because of the large variability of vehicle mass; a truck can be loaded or unloaded changing its mass by a factor 2-3 for distribution trucks.
The main contribution of this paper is to present an EMS, which not only tries to optimize the use of recovered energy, but also to optimize the velocity trajectory on the future route to increase the amount of recoverable energy.This is achieved by modifying the idea of [9,10], to use future route GIS information.The future power trajectory can be estimated using route (velocity-distance), weather conditions and vehicle information (estimated mass and road load).Using this data, the deceleration paths that obtain maximal recoverable energy in the route can be computed and become available for the driver.This approach is particularly relevant for trucks, as their acceleration-deceleration behavior is fairly predictable [11,12], while the mass and road load parameters can vary significantly.
Dynamic Programming (DP) is used to calculate, iteratively with relatively slow update rate, an optimal power split trajectory based upon a power request trajectory which is estimated from an on-board navigation system possibly augmented with actual traffic information (congestion, slow traffic, detour, etc.).Besides, the battery state-of-charge can be constrained to reach a desired level at the end of the route.This could be a relevant feature for plug-in hybrid vehicles, or to initiate an electric (emission free) driving mode at a certain point in the route, for example when entering a city shopping area or an indoor distribution center.Information from this predicted optimal power split trajectory is then used in a lower level real-time EMS.
The remainder of this paper is organized as follows: first, the vehicle dynamics are discussed; Section 2 discusses the vehicle mass and road load parameter variations, in Section 3 the construction of a future power trajectory is outlined; Section 4 describes the proposed EMS; in Section 5 a numerical example is presented; finally, we summarize with conclusions and give an outlook of future work.

VEHICLE MODEL
In this paper the performance of a medium sized heavy-duty parallel HEV is compared with a conventional truck with the same engine size.The vehicle model takes into account the vehicle longitudinal dynamics, and static nonlinear maps describing the efficiency of combustion engine, electric machine, and battery.The engine and electric machine are situated in front of a six speed automated gearbox and run with the same rotational speed.The gear selection strategy is obtained from the gearbox manufacturer and not further discussed in this paper.Finally, the driver is modeled as a velocity controller, controlling the vehicle velocity towards a set point, using a proportional controller; Here P req is the power request of the driver, K d is the proportional feedback gain, v is the vehicle forward velocity, v target is the vehicle target velocity, P max is the maximum available drive power as function of the crankshaft rotational velocity ω, and P min is the maximum available brake power.
The combined maximum power of the electric machine and engine is limited to the maximum power of the engine, such that the hybrid vehicle performance corresponds to the conventional vehicle performance.Note this leads to conservative results, since engine downsizing is possible.The vehicle's road load force is described by; In which m is the vehicle mass, v is the vehicle forward velocity, β is the road angle, g is the gravitational constant, c rol is the rolling resistance coefficient, c 0 is the velocity independent drag force, c 1 is the drag coefficient linear related to vehicle velocity.c 0 and c 1 model the drive-line losses as a drag force at the vehicle wheel.c air is the aerodynamic coefficient, and v wind is the wind force.This model only holds for vehicle forward velocities.The longitudinal vehicle dynamics are described by a force balance at the vehicle wheel; Here m e is the effective vehicle inertia including the rotational inertia of the drive-line (a constant value for m e is used, hereby disregarding inertia fluctuations due to gear setting), F is the resultant drive/brake force of the engine, electric machine, retarder, exhaust brake and/or service brakes.
The prime mover of the truck is a diesel engine, with a maximum power of 136 kW.The engine is modeled as a power converter; see Figure 1, relating the engine output power P ICE to fuel rate ṁ f .The different lines show the (nonlinear, non convex) influence of rotational velocity.Besides, for any rotational velocity, the engine is bounded by a maximum torque, see Figure 3.
The hybrid truck has an electric machine as secondary power converter, with a maximum power of 44 kW.The electric machine is also modeled as a power converter, relating the electric power P b and mechanical power P EM , see Figure 2. The electric machine can work both as a motor and as a generator.At low rotational speeds the electric machine is limited by maximum torque, while at higher rotational speeds the electric machine is limited by maximum power, see Figure 3.
The lithium-ion battery used in the model has a maximum capacity q max of 9 MJ.The state-of-charge S OC(t) is the defined as the electrical charge stored in the battery q(t) divided by the maximum capacity; S OC(t) = q(t) q max (4) from the battery, P b is the electrical power going in/out the electric machine.The losses during charging differ from the losses during discharging.Thermal and transient effects are not considered, nor the influence of state-of-charge.

VEHICLE MASS AND ROAD LOAD PARAMETER VARIATION
Our future power trajectory estimation relies on a physical model of the vehicle's longitudinal dynamics.The vehicle dynamics and road load parameters of a heavy-duty vehicle can vary significantly.Obviously, the vehicle mass can vary, by a factor 2-3, due to varying freight; and the road load varies, due to road conditions, ageing and weather conditions such as wind velocity, wind direction, rain and temperature.Therefore, determination of the future power trajectory requires good online estimation of the vehicle mass and road load parameters.
Vehicle mass estimation has been an important research topic, during the last years, in the automation of different vehicle control systems such as anti-lock brake controllers, and gearshift selection in Automated Manual Transmissions (AMTs).Therefore, several contributions can be found concerning the estimation of vehicle mass, see, e.g., [13,14].Nevertheless, the simulation model used in this study does not contain a mass estimator, it is assumed the vehicle mass is exactly known.The integration of a mass estimator in the simulation model is part of the future work.In contrary to mass estimation, the estimation of road load parameters has been studied to a lesser extend.For that reason, the parameter variations are discussed below.
The parameter c rol is related to tire rolling losses.The rolling resistance of a tire is a complex function influenced by tire material and construction, thread profile, tire pressure, the vertical load, road surface roughness, rotational velocity, wheel alignment, tire wear and temperature.Experiments show that: -the rolling resistance increases approximately proportional to the load perpendicular to the road surface [15]; -the rolling resistance has a small increase with increasing rotational velocity [16]; -the rolling resistance decreases with increasing temperature, the sensitivity for temperature influence (heat generation) is strongly related to the thread compound [15].A higher tire temperature will cause a higher inflation pressure which stiffens the tire; this has a positive effect on the rolling resistance.The rolling resistance of a cold tire compared to a fully warmed up tire can be 15 to 20% higher; -the rolling resistance decreases with increasing tire pressure [15].A tire that is inflated towards a pressure of 1 bar below the nominal pressure has a 3% higher rolling resistance; -the rolling resistance increases with increasing road surface texture.Average rolling resistance differences of 5% can be expected, between steel drum surface and 3M safety-walk paper, and 30% between smooth steel drum and asphalt [17].The rolling resistance increases approximately proportional to the road surface mean profile height; -tire thread profile has considerable influence on the rolling resistance: a truck drive-axle tire has a rolling resistance 5 to 15% higher than a steering-axle tire.
Besides, the thread depth has considerable influence upon the rolling resistance: a worn tire with no thread dept has a rolling resistance up to 30% lower than a brand new tire.
Besides, the tires generate rolling resistance due to cornering.On a city route the rolling resistance due to cornering is expected to be significant.
The parameters c 0 and c 1 are related to the drive-line losses, including the losses in suspension, wheel bearings, differential, and gearbox.The total differential and gearbox torque losses are build-up of: gear losses, bearing losses, plunging losses and sealing losses.Experiments show that: -the best way to model drive-line losses is as a torque loss, rather than an efficiency of power throughput: The losses depend approximately affine on torque and nonlinearly on velocity.Therefore, efficiency as function of power throughput will approach zero for a small absolute value of power; -the drive-line losses can decrease with 50% due to temperature influences; -the gearbox losses depend on the selected gear.
The wind velocity and direction v wind can vary significantly (0-30 m/s) and can have considerable influence on the total road load force of heavy-duty vehicles.In this paper it is assumed that the wind velocity and direction can be obtained from weather information services.The vehicle aerodynamic drag coefficient c air varies for wind velocity and direction, [18] (pp.7-8) suggests a method to implement the coefficient variation.Besides, c air of heavy-duty vehicles can vary heavily due to different cargo and to air density fluctuations.
In the remainder of this paper it is assumed that both the current vehicle mass, and road load force are perfectly estimated.The discussion above shows that the road load parameters can vary widely, due to different operating conditions.The influence of the parameter variations on the EMS performance will be a topic for future research.Construction of the velocity path, based upon the mass and road load estimation, is discussed in the next section.

BUILDING A PREDICTION FOR THE POWER TRAJECTORY USING ROUTE INFORMATION
The real-life driving behavior of trucks during acceleration is to use the full power of the vehicle, even for an unloaded truck [11,12].The acceleration rate depends on the powerto-weight ratio of the vehicle.The power-to-weight ratio of trucks shows a large variation due to varying freight weight.Unlike accelerations, the decelerations happen to be driver dependent.
GPS readings in combination with the on-board GIS system, and possibly other sensors as radar or cameras, can be used to obtain route segments of constant road elevation and maximum velocities as function of traveled distance, vlim (x).As also outlined by [10] (pp.77-79) the maximum velocities in the route are determined by velocity limits, maximum cornering velocity or maximum velocities prescribed by traffic, road and/or weather conditions.
At every point at which vlim (x) changes to a higher value, full throttle can be applied until the velocity reaches vlim .Using the vehicle model of Section 1, a velocity trajectory during the accelerations can be constructed vacc (t).When vlim (x) changes to a lower value, the brake pedal position can be controlled to a required deceleration trajectory.Given the vehicle hybrid system characteristics and road load parameters, a driver independent distance-based deceleration can be constructed that maximizes the recoverable energy on a route, that is the electric machine generates always at its maximum torque/power bound and friction brakes are not used.This is assumed to be optimal because the efficiency of the electric machine is optimal close to its operating bounds.The optimal deceleration rate, at vehicle starting velocity v, is given by; Here, i gb is the current gearbox ratio, i f d is the final drive ratio, R is the wheel radius, me is the estimated effective vehicle mass, Frl is the estimated vehicle road load force described by Equation (2) and T EM_ max is the maximum torque of the electric machine at the prescribed rotational velocity.The electric machine bounds lead to deceleration rates that are not smooth, and change for different gear ratios.Clearly, a larger electric machine leads to larger optimal deceleration rates.By integrating (5) we obtain the optimal velocity trajectory during braking vdec (t).The velocity-distance trajectories vacc (x) and vdec (x) can easily be calculated from the velocity-time trajectories.Given a segment starting velocity, a velocity limitation and a velocity at the end of the segment, a velocity trajectory can be constructed, see Figure 5, by computing; In Figure 6, it can be seen that an empty truck drives a particular route faster than a loaded truck; optimizing the route for fuel economy, comes with the cost of longer traveling time.Velocity-distance trajectory with optimal decelerations for both empty and loaded vehicle.Velocity-time trajectory with optimal decelerations for both empty and loaded vehicle.

GIS, GPS, routeplanner
Trajectory builder (Sect.Energy management topology.The blocks Radar, GIS and Parameter estimator are not explicitly explained in this paper and therefore indicated with dashed lines.The block Trajectory-builder is treated in Section 3, the blocks Driver and Vehicle in Section 1.The remaining blocks Dynamic Programming and Real-time EMS form the EMS and are discussed in Section 4. Legend: vlim are the predicted maximum velocities in the route, m is the estimated vehicle mass, Frl is the estimated/predicted road load force, Preq (t) is the predicted load trajectory, S OC end is the battery state-of-charge at the end of the route, S OC initial is the actual state-of-charge at start of the DP calculation, P req_initial is the actual power request at the beginning of the DP calculation, λ is the predicted Lagrange multiplier, Ŝ OC(x) the predicted state-of-charge trajectory, K is the feedback gain, s(x) the equivalence factor, v target is the actual target velocity, η v is the velocity error, P req is the actual power request, P EM is the electric machine power, P ICE the engine power, P brake is the service brake power, S OC the actual battery state-of-charge and v the current vehicle velocity.
More comments on the velocity trajectory construction can be found in [19].
The estimated velocity-time trajectory can be used to construct a future power request; Here, Frl (v) is the predicted road load force where the future velocities, road grade, and wind expectations are incorporated.Preq (t) can be used as input for a DP algorithm to estimate the optimal power split trajectory.

OUTLINE ENERGY MANAGEMENT STRATEGY
We propose to use a hierarchical control structure, see Figure 7.The control structure consist of a trajectory-builder block which receives future maximum velocities vlim (x) from the GIS, GPS readings, as well as a mass m and road load Frl (v) estimation.Based upon these signals the trajectory-builder predicts a future power trajectory Preq (t) using assumptions on acceleration and deceleration behavior, as was outlined in previous sections.The predicted power trajectory Preq (t) is input to a DP algorithm.Given Preq (t), the current battery state-of-charge S OC initial , and current power request P req_initial , the DP algorithm computes the optimal power split trajectory, between the internal combustion engine power P ICE , and the electric machine power P EM , as function of the upcoming route.Hereby fulfilling the equation; To guarantee a charge sustainable solution we require a battery state-of-charge at the end of the route or at an intermediate point (but end of horizon) equal to S OC end .Static maps of internal combustion engine, electric machine and battery, as shown in Section 1, are included in the algorithm.As stated before, the EMS problem of a hybrid vehicle can be formulated as a nonlinear non convex constrained optimization problem over the route, using P EM as the control variable, subject to a battery end-point constraint and several power constraints on the components; min PEM (t) Equation ( 9) and (10a) be solved with DP techniques, the solution of this problem can be found in literature [2,20,21].
The DP algorithm requires a high computational effort.Therefore, the DP calculation is done iteratively, in order to update the estimated power split trajectory to changing conditions as the vehicle is driving along the route.Equation ( 9) can be rewritten, using the method of Lagrange multipliers.
In literature this is often referred to as Equivalent Consumption Minimization Strategies (ECMS).The optimal power split can be found solving; min PEM (t), λ ṁ f uel (P EM,t ) + λP s (P EM ) , subject to Equation (10b-g).The optimization over a trajectory reduces to an optimization only depending on current data; all future trajectory dependency is lumped into the Lagrange multiplier λ, which is obtained from the DP calculation by; Besides, the DP algorithm generates an electric machine power trajectory prediction as function of time, which can be recalculated to a battery state-of-charge trajectory as function of distance: Ŝ OC(x).However, due to wrongly predicted target velocities or poorly estimated road load parameters, λ could deviate from the optimal value.In order to prevent the battery from over/under charging, λ can be substituted, in real-time, by an estimated equivalence factor s(x). Feedback on the battery state-of-charge is proposed to estimate s(x).This technique was also used in [20,21].
Here, K is the feedback gain.The current operating conditions (traffic, weather, route, etc.) dictate a target velocity v target .The driver (or driver aid as adaptive cruise control) operates as a velocity controller, controlling the vehicle velocity towards the target velocity.Output of the driver is a power request P req , see Equation (1), which in practice deviates from Preq .Given P req and s(x) the power split can be determined in real-time by the minimization; min Given the value of s(x), this is a minimization problem that requires virtually no computational effort.

SIMULATION EXAMPLE
The purpose of this simulation example is to show the benefit of vehicle mass and route information, and the necessity of updating for sustainable battery charging.This approach is implemented on one of the routes proposed in the SAE recommended practice for measuring fuel economy and emissions of hybrid-electric and conventional heavy-duty vehicles [22]: the higher-speed operation heavy-duty Urban Discretized UDDS cycle with optimal velocity trajectories (v target ) for both empty and loaded truck.
Dynamometer Driving Schedule (UDDS), see Figure 8 for the velocity-distance trajectory.Two vehicle masses are simulated; an empty truck and a fully loaded truck.
Figure 9 shows the discretized target velocities in the UDDS cycle and the predicted velocity trajectory, using optimal decelerations.Figure 10 shows the same route, however, now it is assumed that we receive information, at point A in the route (at a distance 2.7 km from the start), of traffic congestion later on in the route.Congestion is modeled, arbitrarily, as a lower target velocity and some stops.
The velocity-distance trajectory, together with the predicted road load trajectory is assumed to be perfectly known in this simulation example, and is used as DP algorithm input.Simulation parameters are depicted in Table 1.
The output of the DP algorithm is the predicted state-ofcharge profile Ŝ OC(x), for the original UDDS cycle, see Figure 11; and the UDDS cycle with optimal decelerations, Congested UDDS cycle with optimal velocity trajectories (v target ) for both empty and loaded truck.see Figure 12.Comparing Figure 11 and 12, it can be seen that during the decelerations, significantly more energy is recuperated and stored in the battery.Moreover, the increased kinetic energy of the loaded truck, compared to the empty truck, is exploited in the optimized route.This can also be seen in the results presented in Table 2, comparing the fuel consumption of the empty conventional and hybrid truck on the original route 1a, and optimized route 1b, with the fuel consumption of the loaded conventional and hybrid truck on the original 2a and optimized route 2b.The loaded truck fuel savings in terms of percentage are significantly larger then of the empty truck.Furthermore, both the hybridization of the drive line and use of the route optimization, leads to a fuel saving of 19.0% for an empty truck, and 33.7% for a loaded truck.
In case of the congested route, a new DP calculation is performed, from point A on, using the battery state-of-   Dynamic programming results of the battery state-of-charge trajectory on the UDDS cycle with optimal deceleration trajectories for both empty and loaded vehicle.charge at that particular moment as the new initial value.See Figure 13 for the empty vehicle results and Figure 14 for the results of the loaded vehicle.The new calculated optimal state-of-charge trajectory, from point A on, is shown by the grey solid line.The black line indicates the global optimal solution, calculated if the congestion was known from the start.
It can be seen that the optimal state-of-charge trajectory on the congested route deviates from the initial optimal state-of-charge trajectory.Clearly, the original solution is a suboptimal solution since it deviates from the black line.The dashed line shows the state-of-charge trajectory obtained by using the initial value of λ; that is the λ calculated over the initial route.
The DP results of the empty vehicle see Figure 13, suggest that it is beneficial to deplete the battery during the first part of the congestion and recharge the battery during the second part of the congestion.We remark that this might   be induced by numerical issues during interpolation, rather than on real physical grounds.In future work we will reconsider the use of an engine map as engine model.Especially the non convex character of the measured map is bothersome.
The state-of-charge trajectory obtained using the initial value of λ, shows a different behavior; it remains constant during the congestion parts.Nevertheless, the fuel consumption obtained deviates only marginally from the DP results, see Table 2. From a battery life-time point of view this result could be considered an improvement.
In Figure 14, the battery state-of-charge trajectory using the initial value of λ shows a similar trajectory as the DP results.However, the state-of-charge end-point deviates.The difference in fuel consumption is again marginal.We conclude that updating the route is useful in retaining the battery state-of-charge within the boundaries, and is less useful to obtain optimal fuel consumption.
Table 2 shows the fuel consumption results for the different simulations.First the empty vehicle results are shown 1), secondly the loaded vehicle results 2).The second column provides the fuel consumption, for the conventional vehicle on the different velocity trajectories.The third column shows the fuel consumption, as well as the relative improvement, of the hybrid vehicle for five different situations: the global optimal savings on the original UDDS route; the global optimum calculated with DP for the initial route with optimal braking trajectories; the global optimum calculated with DP for the congested route; the new calculated optimum onwards from point A in the route, see Figure 10, and the solid line in Figure 13, and finally; the result obtained when no adjustment is made and the initial calculated λ is used.The latter result obviously gives a difference in battery stateof-charge, and therefore, the fuel consumption value is corrected for this difference.The global optimum from point A onwards deviates from the total global optimum, the solution is suboptimal as the optimum battery state-of-charge could not anticipate for the congested part of the route before point A. Because of lower average vehicle speed, the congested route is overall more fuel efficient than the initial route.

CONCLUSIONS AND OUTLOOK
The largest benefit of predictive information is that optimal deceleration paths, that maximize the recoverable energy, can be incorporated in the route.The second benefit is that the battery's state-of-charge can remain within the operating bounds and can achieve a desired state-of-charge at the end of the route.The use of GPS or GIS, providing future route information, together with road load force estimation, in the calculation of the optimal EMS, enables adaption of the EMS both for route changes, as well as for vehicle parameters variations.Distance-based trajectories are preferable, since they allow for optimization of the power requested based upon GIS map data, without changing maximum and minimum route vehicle velocities as well as the exact location of full vehicle stops.
Heavy-duty vehicle operation is widely influenced by vehicle parameter variations and changing environmental making a good prediction of the future power trajectory difficult, on the other hand there are an increasing number of sensors that accommodate estimation of these parameters and conditions.Further research should determine the level of accuracy required from GIS and parameter estimation schemes in order to obtain a power trajectory that is useful for fuel consumption minimization.Topic of research is also the fusion of information from the different sensors and information systems, such as radar, GPS, GIS, vision, CANbus, etc. Besides, we will investigate the possibilities of incorporating elevation, weather and traffic light information, including duration of stops, into the prediction.
The discretization of GIS data, as well as the use of this information, by driver aids as adaptive cruise control is a topic of current research.First results are presented in [19].Moreover, the results presented in this paper indicate that the optimized route enables significant fuel savings, however, with a cost in travel time.Future work will focuss on route optimization subject to a time constraint.Allowing the driver to make a balanced choice between fuel savings and travel time.

Figure 1 Figure 2 Figure 3
Figure 1Diesel engine, fuel to mechanical power conversion for different rotational velocities.

Figure 11 Dynamic
Figure 11Dynamic programming results of the battery state-of-charge trajectory on the original UDDS cycle for both empty and loaded vehicle.
Figure 12 congested UDDS cycle with empty vehicle 1c) Congested route.Initial route until point A. 1d) From point A. 1e) From point A, using initial value of λ.

Figure 13 Dynamic
Figure 13Dynamic programming results of the battery state-of-charge trajectory on the congested UDDS cycle with an empty vehicle.

Figure 14 Dynamic
Figure 14Dynamic programming results of the battery state-of-charge trajectory on the congested UDDS cycle with a loaded vehicle.

TABLE 2
Simulation results on: a) the original UDDS cycle with DP, b) the optimized UDDS cycle with DP, c) b + congestion with DP, d) b + congestion with s(x) using K = 0, e) b + congestion with λ determined on b