Depletion of the petroleum resources and the air pollution are the main reasons behind the advent of hybrid vehicles. Tightened emission standards and rising fuel price compelled manufactures to revise conventional vehicles. On the short term range, hybridization of the conventional vehicles powertrain has proven to be the most efficient way. Main benefits of hybridized vehicles are the shifted operating points of the internal combustion engine (ICE) by the control opportunity and the additional operating modes such as recuperation of brake energy and the purely electric mode in the case of hybrid electric vehicle (HEV).
Integration of an additional energy source (battery) leads to an extra degree of freedom (DOF), since the propulsion force can be provided either by the engine or by the electric machine (EM), thus a suitable energy management strategy (EMS) should be used to control this diversity. Development of the energy management strategy is an important task in the manufacture of the hybrid vehicles and the literature is relatively rich in this topic []-[].
In optimal design study of the hybrid vehicles, energy management strategy is usually treated as a dynamic optimization problem and the causality is not a matter of concern. Furthermore, selection of the suitable topology (including transmission) and choosing the appropriate size of the power source components (sizing) are the other layers of optimization problems, present in most of the hybrid propulsion design studies []. These optimizations problems are mainly concerned with reduction of the fuel consumption (FC) and emission while satisfying drivability constraints. Unfortunately dependency of the different layers on each other and complexity of the problem makes it computationally expensive to simultaneously consider all different layers in design study. On the other hand, it is obvious that a trivial design does not succeed to reveal the ultimate potential of hybridization. Hence a bi-level strategy was employed for the selection of the optimal hybridization ratio of a parallel HEV in [] which the inner level was governed by a boundary dynamic programming (DP) [] and the outer level was addressed by simply changing hybridization ratios. Also the optimal cost and fuel selection of HEV topologies is investigated using particle swarm optimization and dynamic programing []. Pontryagin Minimum Principles is the other method used in [] for the optimal control and thus the design parameters. Furthermore a switching topology with automated transmission is considered in order to evaluate influences made on fuel economy by the components sizes and the transmission technology []. Similar investigations are made for optimal sizing of the fuel cell hybrid vehicle in [].
As concluded from literature, the optimal control method is of prime interest for the design study but not limited to. Also optimization of the parameterized rulebased EMS together with the design variables is another approach made in [] to sizing problem of HEVs. Further, in [] a simplified version of the stochastic dynamic programming is used for design study of fuel cell hybrid vehicle.
It is important to employ an optimal method for the energy management and determination of the fuel consumption for every set of design variables, because, sizing and energy management strategy are inherently coupled, as a consequence different selection of the component sizing need different design of the energy management strategy. This mutual influence is emphasized in [] for a combined plant/controller optimization. The optimal approach to EMS in contrary to the heuristic and rulebased methods is capable of seeking the ultimate potential of the efficiency improvements. Therefore a powerful numerical method, deterministic dynamic programming is used in this paper to serve as the energy management strategy. Furthermore, charge sustaining is available using modified algorithm of this method. Equality of the initial and final state value eliminates the battery electrical energy usage effect on fuel economy and makes it possible to have fair comparisons between different sizes of the energy storage system (battery). Further, without imposing drivability constraints, fuel economy of the HEV will be evaluated for the feasible pairs of the ICE and EM peak power. As a consequence, optimal sizing of the power source components regarding defined acceleration performance will be determined for several driving cycles.
Figure (), Schematic view of the considered hybrid electric vehicle topology.
Powertrain Model
The studied topology is a full parallel hybrid electric vehicle (see Figure )). Combustion engine and an electric machine are connected to transmission via a single shaft thus the engine and electric machine speed profile for a given driving cycle is directly given by the driving cycle speed and transmission ratio. As a sequence the only degree of freedom is choice of the demand torque split decided by EMS. This layout makes possible different operating condition including purely electric, engine alone, hybrid traction, recharge, and combined regeneration and mechanical braking. It is assumed that ICE is turned off when not used and that it does not get cold and loose efficiency during shorter periods of nonuse.
Table ()
Vehicle parameters
Value
Description
Parameter
Air Density
Aerodynamic drag coefficient
Effective frontal Area
Dynamic tire & wheel radius
0.01
Rolling resistance coefficient
Vehicle base mass
Transmission ratio =
This quasi static model has a backward causality and considers steady state condition. Vehicle longitudinal dynamic and the battery state of charge are the only dynamics modeled and all dominating phenomena are considered. All the other components are only algebraic relation or quasi static models. Vehicle dynamic is based on a base mass plus variable mass of internal combustion engine, electric machine, and battery which all depend on specific component size. Driving cycle speed and acceleration are input to the model. Therefore tire traction force will be
()
Applying drive train kinematic and lumped efficiencies, demand torque, and rotational speed, on the right side of the electric machine will be
()
A simple shifting strategy simulating driver behavior is used for the selection of the gear number. Transmission has a constant efficiency of. Different circumstances do not allow high regeneration efficiency andis used for compensation of this condition.
Torque Split Factor
The torque split factor determines the electric machine torque, using a linear interpolation between three distinct values of and according to Table (). This is done in order to guarantee a dimensionless control value independent of component peak torque and to minimize numerical problems when using dynamic programming. Further there will no infeasible control value []. Electric machine speed equals to drivetrain demand speed.
Table ()
Control value definition
Control
Vehicle Operation Mode
Maximum recharge
Conventional braking
Partial recharging
Conventional braking
Provide all torque by engine
Conventional braking
Provide all torque by engine & motor
Partial regenerative braking
Provide all torque by motor
Maximum regenerative braking
The combustion engine torque and speed is then given by
()
Internal Combustion Engine
A quasi static method is used for modeling purpose of the engine. The fuel mass flow rate of the naturally aspirating internal combustion engine is determined via engine maps of the form
()
This map is shown on Figure (). Where engine torque and rotational speed equals to transmission input to the torque coupler. These maps are measured in laboratory under standard condition. Further the engine torque satisfies speed dependent inequality. For the optimization purpose, engine maps (torque axis), engine mass and inequality constraint are scaled linearly with respect to maximum power.
Figure (), ICE brake specific fuel consumption as a function of rotational speed and torque.
Figure (), Electric machine efficiency as a function of rotational speed and torque.
Electric Machine
A permanent magnet dc machine is used as the secondary mover in full parallel hybrid electric vehicle. Electric machine power input or output to the battery is determined via efficiency map of the form
()
This map is shown on Figure ().
()
Further the electric machine torque satisfies speed dependent inequality. Electric machine map (torque axis), mass, and inequality constraint are scaled linearly with respect to maximum power.
Battery
The battery input/output power is the total power supplied to (or by) the electric motor. The battery current is calculated using
()
Where the battery open circuit voltage, and the battery internal resistance, are a nonlinear function of the state-of-charge and dependent on the number of cells in series. The battery power is limited to. The battery's state-of-charge, is calculated using
()
Where is the battery capacity, and the battery columbic efficiency is if charging and otherwise. For minimizing the risk of premature ageing, the state of charge is usually bounded.
Battery Scaling
A battery pack can be simply scaled according to its number of cells and the cell capacity. In order to have constant nominal voltage for the battery pack, it is reasonable to choose cell capacity as the only design variable.
()
Further it is assumed that the battery pack has a constant power-to-capacity ratio [], therefore the battery pack power limits are proportional to the scale factorwhile the pack voltage remain the same. Also the battery internal resistance change as its capacity scale change.
()
()
Further hybridization ratio in parallel HEV is defined as
()
Energy management problem
The optimal control problem under study consists in minimizing the fuel consumption of the vehicle along a prescribed vehicle cycle, taking into account physical constraints from battery, engine, and electric machine. The general optimization problem is the following:
()
Where 0 and are respectively the initial and the final times of the prescribed driving cycle, is the torque split factor, is the battery state of charge,) is the instantaneous fuel consumption, is the function that controls the variations of the battery state of charge dynamic.
It should be noticed that usually a terminal cost is added to the accumulative cost function for penalizing final state deviation which is regarded as a soft constraint. But this method of treating final state deviation is not interesting in optimal sizing of HEVs. Whatever the battery capacity is, a small deterioration of the charge sustaining may result in unfair comparisons of different sizes, therefore a modified version of the basic DP is developed to induce hard constraint on final state. Primary investigation on partially fixed final state is made in [] and may be similar in some of the implementation aspects to this study.
Algorithm
This section gives a brief overview of the DDP algorithm used in this paper for optimal control and thus optimal sizing. Two key feature of the DP is an underlying discrete time dynamic system and a cost function that is additive over time, therefore the continuous time model of the battery dynamic (equation (8)) should be discretized. Let the discrete-time model be given by
()
Where k indexes discrete time, is the state of the system, is the control value to be selected at time k, is the disturbance, and is horizon. The cost function is additive in the sense that the cost incurred at time, denoted by accumulates over time, thus the total cost is
()
The sequence is referred to a control policy, for each, the corresponding cost for a fixed final state is
()
For a given initial state , an optimal policy is the one that minimizes this cost; that is
()
In the above equation, is the set of all admissible policies. Consider the proposed problem and the following steps for determining the optimal policy and thus optimal cost.
Exclusion of the infeasible states
Consideration of a hard constraint on final state produce an infeasible region shown on figure that should be computed priorly and omitted from state-time nodes. Finding reachable space of the state trajectories enforces a fixed final state and further minimizes numerical errors while interpolation near unreachable space of the state (Figure ()). Inaccuracy described, is a common phenomenon in basic implementation of dynamic programming. Limit points of the feasible states can be evaluated by finding extreme achievable value of the state governed by the inverse model starting from final time and sequentially going backward until it crosses predefined state bounds []. In this way, feasible states for every stage (time index), are evaluated.
Further infeasible regions are not limited to the reachability of final state value; either decreasing or increasing nature of the state value induced by disturbance signal may produce more infeasible regions near bounds. Such infeasibilities are estimated and omitted in the next step. After evaluation of infeasible regions, one can proceed to find optimal control policies.
Figure (), Schematic overview of an optimal problem solved by dynamic programing algorithm, this figure shows infeasible areas and optimal trajectory of the state.
Backward in time.
For numerical implementation of the DP, the state and control should be discretized considering limit values of states calculated priorly. The dynamic programming technique rests on a very simple idea, the principle of optimality. It suggests that an optimal policy can be constructed in piecemeal fashion, first constructing an optimal policy for the 'tailsubproblem' involving last stages, then extending the optimal policy to the 'tailsubproblem involving last two stage and continuing this manner until consideration of the last stage which is the entire problem. As a consequence, an optimal cost to go function should be introduced and used for the 'tailsubproblem'. This function saves optimal accumulative cost for the state at time for getting to the final state. The way in which optimal control policy is generated has two periods as follows.
Period 1:
Application of a boundary line method together with a fixed final state cancel searching for optimal control values because there is only one possible control value in this stage which will go to the final state. That is
()
Where refer to the optimal control value at time and for the state. Further, optimal cost-to-function should be initiated.
()
Now, we are assured that that every trajectory of the system at time will get to the
Period 2:
Based on principles of optimality [], the optimal cost to go function at every node in discretized state-space should be evaluated by proceeding backward in time:
()
The optimal control and cost-to-go function is given by the argument that minimizes the right-hand side of the equation. In some circumstances, application of the control value, at in system dynamic will result in which does not belong to the feasible states of the time, therefore the corresponding state node should be omitted from the set of feasible states for that time index.
The cost-to-go function used in equation is evaluated only on discretized points in the state space while the output of the model function is a continuous variable in the state space which may does not coincide with the nodes of the state grid. Therefore an appropriate interpolation scheme should be used.
Forwards dynamics
Given initial state value and using optimal control map, one can generate the optimal trajectory and evaluate the corresponding cost.
DP result
One of the optimization problems, present in design study is energy management strategy optimization. A suitable algorithm based on dynamic programming was developed to acts as the mentioned strategy. In order to investigate operation of the proposed strategy, vehicle operation under NEDC (New European Driving Cycle) driving cycle is evaluated. Vehicle under study includes nominal dimensions of the ICE and EM peak power (see Table ()) in this simulation. Furthermore, three values of the battery capacity are used for the evaluation of the optimal trajectories. Initial value of the state-of-charge is and the same value is expected at final time. Also the soc is bounded between and. In the case of DP setting, control input grid is chosen to be. The number of elements in the state grid is and the time step is chosen to be unit.
Figure (), Optimal trajectory of the state-of-charge for (left axis) and vehicle speed during NEDC (right axis).
Generally, two major tasks were considered for the designed energy management strategy. At first, it is expected that the battery initial and final state of charge be identical for different values of the capacity. It can be seen from Figure () that regardless of the size, charge sustaining is accomplished at the accuracy of (). Thus, the electric storage system (battery) is used only as an energy buffer and the vehicle propelling energy is globally provided by fuel. Secondly, reduction of the fuel consumption occurs as a result of the efficiency improvement. Therefore the proposed energy management strategy should try the best to increase efficiency. Concluded from operating diagram of the ICE, Figure (), most of the operating points have clearly moved towards higher efficiency areas. It should be noticed that dictated engine speed and electric path efficiency prevents from shifting rest of the operating points. This task is fulfilled by optimal selection of the operation mode and torque split (Figure ()). The developed optimal method can accomplish expected tasks, whatever the powertrain size and the driving cycle are. In fact, influences of the control strategy on the fuel economy are eliminated and it is possible to focus on the power source sizing.
Figure (), Time spent at different operating points of the ICE during NEDC.
Figure (), Optimal demand torque distribution during NEDC between ICE (-) and EM (--) and the corresponding operation mode during NEDC.
Sizing Problem
Sizing problem of the HEVs generally include three design variables, Peak power of the ICE and EM and capacity of the battery. This paper considers only power sources (ICE and EM) in design study and a simple scheme is used to set battery capacity for a defined pair of the EM and ICE peak power. According to powertrain structure of the HEV, EM power is always supplied by the battery pack, thus battery pack limit power should corresponds to the EM peak power demand on the driver side (motoric and generator mode). Capacity of the battery is then given by the peak power of the EM on the driver side, together with the maximum power-to-capacity ratio of the battery pack. It is important to note that when performing a detailed sizing, the battery capacity should be considered as a separate design variable and needs to be optimized along with the EM and ICE.
Table ()
HEV base components
Specification
Component
1.5 L SI engine
80-Nm/40-Kw
Peak efficiency: 247 g/Kwh
3.5 Kg/Kw
Internal combustion engine
PM machine
300-Nm/30-Kw
Peak efficiency: 91%
3 Kg/Kw (with accessories)
Electric machine []
12 Ah Saft Lithium Ion battery
250 Wh/kg
2.81 Kw/Ah
Battery pack []
Driving cycles: Unfourtunately, the proceeding methodology for minimization of the fuel consumption is cycle oriented. Therefore a comparison between different driving cycles is made in Figure (), and suitable driving conditions are choosen. Four driving cycle have been considerd to estimate the fuel economy. NEDC (New European Driving Cycle) and ECE -15 are used as benchmark to estimate fuel consumption in Europe. US06 and FTP-75 are the other driving cycles derived from real speed mesurement.
Figure (), Driving cycles acceleration root mean square versus average speed
Component size: The design of experiments varies from to with a step of for the ICE and EM peak power. Different characteristics of the desired components are generated by suitable scaling of baseline data (see Table ()). Of course; the proposed optimal energy management strategy is used to compute the minimum fuel consumption for every pair of design variables.
Result
The fuel consumption maps regarding the proposed method are computed according to driving conditions: NEDC in Figure (), US06 in Figure (), ECE-15 in Figure () and FTP-75 in Figure (). Theses map are constructed by some iso-consumption levels. A pure conventional vehicle is also considered to quantize fuel economy improvements (, vertical axis). In order to track some drivability performance of the vehicle, constant power-to-weight lines are drawn on each map. This quantity is strongly related to the acceleration performance of the vehicle. The 'infeasible area' represent the undersized ICE and EM, which cannot fulfill the driving cycle either torque demand (left border) or energy demand (right border).
Figure (), Iso-consumption levels (-) computed on NEDC driving cycle with constant power-to-weight lines (--).
Figure (), Iso-consumption levels (-) computed on US06 driving cycle with constant power-to-weight lines (--).
Figure (), Iso-consumption levels (-) computed on ECE-15 driving cycle with constant power-to-weight lines (--).
Figure (), Iso-consumption levels (-) computed on FTP-75 driving cycle with constant power-to-weight lines (--).
Discussion
Clearly, the fuel consumption decreases as lower acceleration performance (lower power-to-weight) is of interest. This trend is similar in different driving conditions. Further it is obvious from the result that different hybridization ratios along constant power-to-weight lines yield different fuel economies, as a result, a tradeoff between fuel economy and drivability should be made. To conclude, for optimal sizing of the parallel HEV, it is not necessary to search all the possible pairs of ICE and EM peak power. Considering a fixed drivability performance, the above sizing problem can be reduced to a problem with single design variable. Evaluating the fuel consumption of different hybridization ratios with similar driving performance is an appropriate selection of the design variable. Hence, fuel consumption gains () of the vehicle for the hybridization ratios with equal power-to-weight value (are illustrated in Figure (). Hybridization ratios are discretized by the step of. Depending on the driving cycle, order of the magnitude of the fuel consumption reduction changes because the original distributions of the operating speed are different. Figure () shows the optimal size of the ICE and EM for mentioned driving cycle.
Figure (), Effect of hybridization ratio on fuel consumption gain for mentioned driving cycles.
Figure (), Optimal size of the power sources components in mentioned driving cycles.
ICE efficiency repartition and delivered energy are illustrated in Figure () particularly for the NEDC driving cycle. The results show that optimal hybridization ratio is around. The electric component size impacts the control strategy by giving it more or less freedom in shifting the operating points. As a consequence it seems that the optimal hybridization level corresponds therefore to the necessary size of electric components to move the ICE operating points to optimal areas, but this is not the case because of hybridization is enough to shift all the operating point of the ICE to optimal areas. The reason behind reduced fuel consumption gain after of hybridization ratio is the lower amount of the energy delivered by ICE. In fact, upsizing of the EM enables to capture more regenerative energy. When the hybridization ratio goes behind, the fuel consumption gain increases because downsizing of the ICE to this degree weakens the ability of EMS in shifting the operating points to optimal areas. Of course, decreased efficiency of the electric path and increased vehicle weights are another reasons behind this trend. Concluded from above, there is an optimal hybridization ratio which can explore ultimate fuel saving.
Figure (), left axis shows ICE efficiency repartition, optimal area (gray) and miscellaneous area (white), and right axis shows delivered energy by ICE using NEDC driving cycle.
Conclusion
Design study of a parallel hybrid electric was the prime interest in this paper. To achieve that, an optimal energy management strategy based on dynamic programming was formulated. This strategy could tackle the problem of imbalance state-of-charge. To investigate influences made by the power source sizing, fuel economy of the prescribed vehicle was estimated for different driving conditions. Furthermore the optimal size of the ICE and EM which satisfy drivability constraints while maximizing the fuel economy was derived. Design study results show that: (1) sizing variables of the parallel HEV can be reduced to the combination of the hybridization ratio and drivability constraints; 2) optimality lies in downsizing the internal combustion engine and (3) increasing ratio of hybridization without compromising control strategy operation.
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