Effective optimal control strategy for hybrid electric vehicle with continuously variable transmission

Abstract

Hybrid electric vehicles equipped with continuously variable transmission show dramatic improvements in fuel economy and driving performance because they can continuously adjust the operating points of the power source. This article proposes an optimal control strategy for continuously variable transmission–based hybrid electric vehicles with a pre-transmission parallel configuration. To explore the fuel-saving potential of the given configuration, a ‘control-oriented’ quasi-static vehicle model is built, and dynamic programming is adopted to determine the optimal torque split factor and continuously variable transmission speed ratio. However, a single-criterion cost function will lead to undesirable drivability problems. To tackle this problem, the main factors affecting the driving performance of a continuously variable transmission–based hybrid electric vehicle are studied. On that basis, a multicriterion cost function is proposed by introducing drivability constraints. By varying the weighting factors, the trade-off between fuel economy and drivability can be evaluated under a predetermined driving cycle. To validate the effectiveness of the proposed method, simulation experiments are performed under four different driving cycles, and the results indicate that the proposed method greatly enhanced the drivability without significantly increasing fuel consumption. Compared to a single-criterion cost function, the use of multiple criteria is more representative of real-world driving behaviour and thus provides better reference solutions to evaluate suboptimal online controllers.

Keywords

Continuously variable transmission hybrid electric vehicle dynamic programming drivability penalty function

Introduction

Both academic researchers and car manufacturers are continuously attempting to improve the fuel consumption of hybrid electric vehicles (HEVs). The use of continuously variable transmission (CVT) in HEVs has been proven to be a promising solution to achieve these improvements^1,2 because it avoids driving problems such as power interruption caused by multi-gear automatic transmissions. Moreover, with CVT, the speed ratio can be continuously changed according to the working condition to ensure that the power sources always operate close to their most efficient regions.

The design of a CVT-based HEV is a multidomain task, and energy management strategy (EMS) poses one of the major challenges because it greatly impacts the fuel economy by controlling the CVT speed ratio and power distributions among power sources. To address this challenge, a series of important theories and methods have been developed, which can be divided into two categories: heuristic methods and optimization methods. Heuristic methods, such as rule-based and fuzzy logic,^3,4 are rather intuitive, but optimality is not guaranteed. Instantaneous optimization methods such as the equivalent consumption minimization strategy (ECMS),^5,6 model predictive control^7,8 and the stochastic dynamic programming algorithm⁹ use a high-fidelity vehicle model and seek to find the optimal control to minimize the cost function at each instant. These methods can be implemented in a real-time controller but often yield suboptimal results. On the other hand, global optimization methods such as DP are commonly used for solving optimal energy management problems over a finite horizon.^10,11 The main drawbacks of such methods are their high computation cost and the need to know the driving cycle in advance, which make these methods inapplicable in an actual controller. However, the results of global optimization methods can be used as a reference for component sizing and for evaluating real-time strategies.¹²

Driving performances is another major challenge that affects the performance of the control strategies, which is majorly affected by the engine torsional vibrations and dynamic characteristics of the driveline.¹³ Global optimization methods focusing only on the fuel economy may lead to unexpected driveline behaviours and result in poor drivability. Debert et al.¹⁴ analysed this phenomenon and used a quadratic cost function to constrain the high variation of the gear shifts in the DP to enhance drivability. Opila et al.¹⁵ accounted for the drivability metrics in the cost function and used shortest path stochastic dynamic programming (SP-SDP) algorithm to trade off between the fuel consumption and the drivability. Miro-Padovani et al.¹⁶ and Vidal-Naquet and Zito¹⁷ used ECMS-based strategy to online minimize the fuel consumption as well as drivability metrics. However, few literature have addressed the drivability problem due to the CVT dynamics of an HEV. Practically, strategies focused on fuel optimization for a CVT-based HEV can lead to excessive engine on/off events and violent fluctuations of the CVT speed ratio. Engine on/off events often involve mode switching and undesirable pollutant emissions, and a fast change of the CVT speed ratio will cause a sharp variation in the vehicle velocity, which will significantly impact driving comfort and vehicle safety.¹⁸ Therefore, an effective optimization model with both fuel economy and drivability considerations is of great practical significance for improving the overall performance of CVT-based HEVs.

In this article, DP is adopted to find the overall minimization of fuel consumption based on a quasi-static model of a CVT-based HEV. Furthermore, drivability problems arising from the algorithm for solving the optimal control problem are studied. Two penalty terms were then chosen to constrain excessive engine events and the rate of change of the CVT speed ratio. To validate the effectiveness of the proposed method, simulation experiments are performed under standard driving cycles, and the results are compared between basic DP and the proposed method.

HEV powertrain modelling

Vehicle architecture

A typical CVT-based parallel hybrid configuration is depicted in Figure 1. An electric machine (EM) is connected with the engine through a disc clutch in a pre-transmission configuration. An electric oil pump is employed to provide the hydraulic pressure and flow of the CVT hydraulic system in order to meet the pressure requirement during pure electric driving. The battery provides the necessary energy storage for the EM to function as either a motor or generator. Mode transition is realized through the clutch. Engine torque can be delivered through the rotor of the EM and the CVT to the driving wheel when the clutch is engaged, while the HEV can be driven in a purely electric mode when the clutch is disengaged.

Figure 1.

Vehicle configuration.

Vehicle models

There are two different modelling approaches can be adopted in the research: the forward- and the backward-facing modelling. The former requires a ‘high-fidelity’ vehicle model to reveal the dynamic details of the driveline, while the latter is usually adopted to design and test of the high-level control strategies.

Since numerical method is used in this article, the offline computation is very sensitive to the number of system state variables and the resolution of the motion equations of the vehicle. For this reason, a ‘control-oriented’ backward quasi-static model is adopted for the simulation, as depicted in Figure 2. The driving cycle profile provides the set points of velocity $v (t)$ and acceleration $α (t)$ at each time step. According to the torque demand at the wheels $T_{wh}$ , the controller computes the optimal CVT speed ratio $i_{CVT} (t)$ and the torque split factor $s (t)$ , which will determine the torque demand of the engine $T_{Eng}$ and EM $T_{EM}$ .

Figure 2.

Backward-facing modelling of the HEV.

The torque demanded to propel the vehicle is approximated by the sum of the aerodynamic drag force, rolling resistance and longitudinal acceleration resistance. Here, the influence of the slope resistance is ignored, since the vehicle is assumed to be driven on the flat road surface

T_{R} = (\frac{1}{2} ρ_{a i r} C_{d} A v^{2} + m g f_{r} + m a) \cdot r_{w h}

(1)

where C_D is the drag coefficient of wind, A is the frontal area, m is the total mass of the vehicle, $r_{wh}$ is the wheel radius and $f_{r}$ is the rolling resistance coefficient. Considering the engine drag torque and the moment of inertia of both power sources, the required torque at the CVT input shaft is given as follows

\begin{matrix} T_{Eng} + T_{EM} = \\ \frac{T_{R}}{i_{CVT} \cdot η_{CVT} \cdot i_{FD} \cdot η_{FD}} + T_{Eng_drag} (ω_{Eng}) + J_{FW} \cdot {\overset{\cdot}{ω}}_{Eng} + J_{EM} \cdot {\overset{\cdot}{ω}}_{EM} \end{matrix}

(2)

where $i_{CVT}$ and $i_{FD}$ represent the speed ratio of the CVT and final drive, respectively; $η_{CVT}$ and $η_{FD}$ are the efficiencies of the CVT and final drive, respectively; $J_{FW}$ and $J_{EM}$ are the moment of inertia of the fly wheel and EM, respectively; $T_{Eng_drag}$ and $ω_{Eng}$ are the engine drag torque and speed, respectively; and ${\overset{\cdot}{ω}}_{Eng}$ and ${\overset{\cdot}{ω}}_{EM}$ are the rotational acceleration of the engine and EM, respectively.

Internal combustion engine

The engine is modelled as steady-state look-up tables; it is assumed that the engine torque must exactly match the torque demand, so the dynamic characteristics of the engine are ignored here. The universal characteristics of the engine are obtained through bench tests. The fuel flow rate is a function of the engine speed and torque: ${\overset{\cdot}{m}}_{f} = f (T_{Eng}, ω_{Eng})$ . The fuel consumption can be derived from the following equation

P_{Eng} = LHV \cdot {\overset{\cdot}{m}}_{f} \cdot η_{Eng}

(3)

where LHV is the lower heating value of the fuel and $η_{Eng}$ is the mechanical efficiency of the engine. When the clutch is engaged, the engine speed is assumed directly proportional to the wheel speed based on the speed ratio of the CVT and final drive

ω_{Eng} = i_{CVT} \cdot i_{FD} \cdot ω_{wh}

(4)

EM

The EM is also modelled as static look-up tables based on an efficiency map through bench tests. Since the EM is directly coupled to the engine crankshaft, when the clutch is engaged; it rotates at the engine speed because the connection between the engine and the EM is considered to be rigid

ω_{EM} = ω_{Eng}

(5)

Together with the torque required for the EM, the required electric power can be derived from the following equation

P_{EM} = T_{EM} \cdot ω_{EM} \cdot η_{EM}^{k} (T_{EM}, ω_{EM})

(6)

where $η_{EM}$ is the total efficiency of the EM and the controller. When $k = 1$ , the EM works as a generator; when $k = - 1$ , the EM works as a motor.

CVT

Compared with a multi-gear automatic transmission, a standard hydraulic-actuated push-belt CVT has a lower average efficiency owing to the oil pumping torque loss and mechanical belt friction.¹⁹ The CVT model used in this article employs an electric oil pump instead of a mechanical oil pump, which can decouple the flow of oil pump from the engine speed and reduce the energy consumption of the oil pump. The CVT mechanical efficiency is a function of the speed ratio, input speed and torque

η_{CVT_mech} = f (T_{in}, ω_{in}, i_{CVT})

(7)

The power demand of the drive motor can be derived by

P_{EOP_Motor} = \frac{P_{Pump_req}}{η_{EOP_Motor} \cdot η_{Pump}}

(8)

where $η_{Pump}$ and $η_{EOP_Motor}$ are the efficiencies of the oil pump and the drive motor, respectively. Efficiencies are obtained through bench tests. $P_{Pump_req}$ is the power demand of the oil pump and can be given as follows

P_{pump} = \frac{P_{hyd} \cdot Q_{hyd}}{60}

(9)

where $P_{hyd}$ is the required pressure of the hydraulic system and is a function of the CVT input torque and speed ratio, and $Q_{hyd}$ is the flow of the hydraulic system and is determined by the flow demand for speed ratio control, cooling, lubrication and leakage.

Battery

The battery (a lithium-ion battery is considered) is modelled as a static equivalent circuit based on data from an experiment of charging and discharging of the battery. The battery current can be calculated from the following equation

P_{bat} = U_{OC} \cdot I_{bat} - I_{bat}^{2} \cdot R_{int}

(10)

where the battery power $P_{bat}$ is given by the power demand of the EM and the drive motor of the electric oil pump, $U_{OC}$ and $R_{int}$ represent the open-circuit voltage and the internal resistance of the battery, respectively. The value of $I_{bat}$ is positive during discharging and negative during charging and is confined to the maximum charging and discharging current

I_{bat_min} \leq I_{bat} (t) \leq I_{bat_max}

(11)

The battery state of charge (SOC) can be derived using an ampere-hour integration

SOC (t) = SO C_{init} - \frac{1}{C_{nom}} \int_{0}^{t} I_{bat} (τ) d τ

(12)

where $SO C_{init}$ is the initial SOC of the battery, and $C_{nom}$ is the nominal capacity.

Optimization modelling

Problem formulation

Deterministic DP is a useful approach to solve global optimization problems over a finite horizon. Solving a given DP problem is time-consuming because the computational burden increases exponentially with the number of state and control variables, which is a problem that has been termed ‘The Curse of Dimensionality’. Thus, the DP algorithm can only be applied to a confined number of states and inputs.²⁰ The resolution of the grid of state and control variables will also influence the computational time; however, this factor is not within the scope of this article.

For a CVT-based HEV, the SOC of the battery is chosen as the only state variable, $x_{t} = SOC (t)$ . The first control variable is the CVT speed ratio because it is controlled to adjust the working points of the power sources. In addition, instead of considering the driving torques separately, which will increase the computation cost, the torque split factor s is chosen as the second control variable to regulate the torque distribution between the two power sources. $T_{tot}$ denotes the summation of the torque of the engine and EM. Here, suppose $s = T_{EM} / T_{tot} \in (- 1, 1]$ , the torque demand of the EM is $s \cdot T_{tot}$ and the torque demand of the engine is $(1 - s) \cdot T_{tot}$ . The value $s (t)$ corresponds to the operating modes of the HEV and is summarized in Table 1.

Table 1.

Operating modes corresponding to different s(t) values.

s(t)	Operating mode
1	Pure electric drive
(0, 1)	Hybrid mode
0	Engine drive only
(−1, 0)	Engine recharge

Since DP is a numerical method used here to solve a continuous-time control problem, the continuous-time model must be discretized first. The complete discrete-time HEV model can be given as follows

x_{k + 1} = T_{k} (x_{k}, u_{k}) + x_{k}, k = 0, 1, \dots, N - 1

(13)

where state variable $x_{k} \in X$ and control variable $u_{k} = [u_{1}, u_{2}]^{T} \in U_{k}$ . The state $x_{k + 1}$ is obtained by applying the control $u_{k}$ at state $x_{k}$ . Furthermore, the driving cycles are assumed to be known precisely in advance. Since the major concern is the minimization of the total fuel mass consumed, in general, the cost function can be described as follows

J = G (x_{N}) + ϕ_{N} (x_{N}) + \sum_{k = 0}^{N - 1} Δ m_{f} (x_{k}, u_{k}) \cdot T_{s}

(14)

where $Δ m_{f}$ is the mass of the fuel consumed for the time step, $G (x_{N})$ is the final cost, $ϕ_{N} (x_{N})$ represents a constraint on the final state and $T_{s}$ is the time step. The last term on the right-hand side of the equation corresponds to the cumulated fuel consumption over a given driving cycle based on the principle of optimality.²¹

Drivability problems

As mentioned above, the cost function given by equation (14) only focuses on the minimization of the total fuel consumption over a given driving cycle and ignores the driving problems. Figures 3 and 4 depict the engine on/off events and the rate of change of the CVT speed ratio under the urban dynamometer driving schedule (UDDS). One can observe that intensive engine events occurred over a certain period of time, in addition to a violent fluctuation in the rate of change of the CVT speed ratio.

Figure 3.

Engine on/off events under UDDS.

Figure 4.

Rate of change of CVT speed ratio under UDDS.

Engine on/off often involves mode switching and undesirable pollutant emissions. Thus, excessive events should be avoided. In addition, a high rate of change of the speed ratio will have a negative impact on the dynamic behaviour of the driveline.^22,23 Figure 5 shows the driveline dynamics of a CVT-based HEV. In the figure, $j_{in}$ and $j_{out}$ represent the equivalent moment of inertia of the input and output shaft of CVT, respectively and the system is governed by the following equation

\begin{array}{l} \frac{d ω_{o u t}}{d t} [j_{o u t} + i_{C V T}^{2} \cdot (j_{i n} + j_{C V T})] \\ = - (j_{C V T} + j_{i n}) ω_{i n} \frac{d i_{C V T}}{d t} + T_{i n} \cdot i_{C V T} - T_{l o s s} - T_{l o a d} \end{array}

(15)

Figure 5.

Driveline dynamics of a CVT-based HEV.

As the speed ratio changes, the rotational acceleration of the CVT output shaft changes accordingly and is proportional to the input equivalent moment of inertia and speed. A fast change of the speed ratio will cause a sudden change of the acceleration of the CVT output shaft. Such a sudden change, or jerk, will strongly impact the driving comfort and vehicle safety. According to the works by Wei et al.²⁴ and Chen,²⁵ an acceptable jerk of a vehicle is usually less than $2 m / s^{3}$ .

In Figure 6(a) and (b), typical jerks of a vehicle are presented at a speed of 110 km/h with a CVT input torque of 100 N m. When the speed ratio is increased from 1 to 1.5 within 1 s, a high jerk of more than $- 2.5 m / s^{3}$ occurred, which will cause a sudden deceleration at the driving wheels. When the speed ratio decreased from 1.5 to 1 within 1 s, the situation is slightly better. When the rate of change is within $\pm 0.3 / s$ , the jerk will become acceptable.

Figure 6.

Vehicle jerk for (a) an increase and (b) a decrease in the speed ratio.

In order to enhance drivability, a multi-objective optimization method or the introduction of penalty terms can be adopted to find the best compromise between objectives. Here, we choose to penalize each engine event (on or off) and the rate of change of the CVT speed ratio. Thus, the cost function (equation (13)) can be rewritten as follows

J = G (x_{N}) + \sum_{k = 0}^{N - 1} Δ m_{f} \cdot T_{s} + α \sum_{k = 0}^{N - 1} I_{EE} + β \sum_{k = 0}^{N - 1} {(Δ i_{CVT})}^{2}

(16)

subject to

{\begin{matrix} x_{k + 1} = T_{k} (x_{k}, u_{k}) + x_{k} \\ x_{k} \in X \\ x_{N} = x_{0} \\ u_{k} \in U \end{matrix}

(17)

where

u_{k} = {\begin{matrix} u_{1 k} \\ u_{2 k} \end{matrix}} = {\begin{matrix} i_{CVT} (k) \\ s (k) \end{matrix}}

(18)

In the cost function, $I_{EE}$ is an indicator function that is equal to one when an engine on/off event occurs, and the last term of the penalty function tends to penalize a large rate of change of the speed ratio. The speed ratio difference in the cost function is quadratic because a large variation is supposed to be much more expensive than a small variation. $α, β$ are positive weighting factors for the two penalty terms, respectively, and the tuning of the weighting factors involves some trial and error depending on the desired outcome. Note that the penalty term on the constrained final state has been removed here because the SOC of the battery is supposed to be sustained for the HEV at the end of the driving cycle.

Implementation of DP

According to Bellman’s²⁶ principle of optimality, the DP algorithm needs to backward compute the cost-to-go function $J_{k} (x_{k}^{i})$ at every node in the state grid.

End-step cost calculation

J_{N}^{*} (x^{i}) = G_{N} (x^{i})

(19)

Cost calculation from step k = N − 1 to 0

\begin{array}{l} J_{k}^{*} (x_{k}^{i}) = \\ \min_{u_{k} \in U_{k}} {V_{k} (x_{k}^{i}, u_{k}) + J_{k + 1}^{*} (x_{k + 1}^{i}) + α I_{E E}^{i} + β {(i_{C V T_k}^{i} - i_{C V T_k + 1}^{i})}^{2}} \end{array}

(20)

where $V_{k} (x_{k}^{i}, u_{k})$ is the cost obtained by applying $u_{k}$ at state $x_{k}^{i}$ and $J_{k + 1}^{*} (x_{k + 1}^{i})$ is the optimal cost-to-go function from step k + 1 to the end of the driving cycle. The last two terms on the right-hand side of equation (20) represent the constraints on the engine events and rate of change of the CVT speed ratio, respectively. Figure 7 shows the backward calculation process of the DP algorithm in detail. In the figure, $Ω_{k}^{i}$ is defined as the set of states reachable by applying all the possible controls from state $x_{k}^{i}$ , and $V_{k}^{i} (x_{k}^{i}, u_{k}^{m, n})$ is the cost resulting from applying control $u_{k}^{m, n}$ . Note that the optimal cost-to-go function is only evaluated for discrete nodes in the state grid, and the output of the state transfer function (equation (13)) is continuous. Thus, it may not exactly fall on the grid points, such as the red dots in the graph; usually, interpolation methods can be adopted to find the optimal cost-to-go function. In addition, since the calculation of the penalty terms in equation (20) requires knowledge of the optimal controls of the next step $u_{k + 1}^{*} (x_{k}^{i}, u_{k}^{m, n})$ , the optimal control signals of the red dots also need to be interpolated.

Figure 7.

Interpolation diagram during backward calculation.

The output of recursive functions (19) and (20) is a signal map, which stores the optimal cost and control signals for every node in the state grid. During a forward simulation, interpolation is also needed to find the optimal control sequence when the actual state does not coincide with the points in the state grid. Finally, the optimal state trajectory can be obtained from a given initial state $x_{0}$ .

Simulation results

Since weighting factors are not known a priori, it is necessary to first assess the influence of the weighting factors on the fuel consumption and drivability metrics. Simulations have been carried out under standard driving cycles, namely, UDDS, New European Driving Cycle (NEDC), Highway Fuel Economy Test (HWFET) and Worldwide Light-duty Test Cycle (WLTC). These driving cycles will generally test the controller under normal urban driving, highway and aggressive driving conditions. The velocity profiles of the driving cycles are shown in Figure 8. The time step $T_{s}$ for the simulation is 1 s. Both the initial and final SOCs are set to 0.55, and the range is set to [0.4, 0.7]. The system parameters for the simulation are listed in Table 2. For simplicity, criterion as described in equation (14) is referred to as the ‘basic DP’, while criterion with drivability constraints is referred to as the ‘proposed method’.

Figure 8.

Velocity profile for (a) WLTC, (b) NEDC, (c) UDDS and (d) HWFET.

Table 2.

System parameters.

Parameter	Value
Vehicle mass (kg)	1500
Engine placement (L)	1.5
Engine max. power (kW)	78
EM max. power (kW)	26
Battery energy capacity (Ah)	6
CVT speed ratio range (–)	0.442–2.432
Final drive efficiency (–)	0.95
EOP motor max. power (kW)	2.5

EM: electric machine; CVT: continuously variable transmission; EOP: electric oil pump.

Calculations were carried out for $α = 0$ to $α = 2 \times 10^{- 4}$ when $β$ was set to 0 to test the trade-off between fuel consumption and the restriction on the frequent engine on/off events. As shown in Figure 9(a) and (b), one can see a clear trend that as $α$ increases, the number of engine events reduced with the increase in the fuel consumption. This trend is more obvious for WLTC and UDDS cycles. For WLTC cycle, when $α = 0.8 \times 10^{- 4}$ , engine events reduced by nearly 34% at the expense of a 0.8% overconsumption. For UDDS cycle, when $α = 1.5 \times 10^{- 4}$ , engine events dramatically reduced by up to 60% with a 3.9% overconsumption. For NEDC and HWFET cycles, driving conditions are relatively mild and the numbers of engine events for the basic DP are much smaller, engine events reduced by 50% for both cycles at the expense of 0.6% and 1.2% increase in the fuel consumption, respectively. For the proposed method, when $α$ is within [0.5 × 10⁻⁴, 1.5 × 10⁻⁴], it is considered as a reasonable tuning range, since the number of engine events almost reduced to its lowest level and fuel consumption only increased by less than 4% tops.

Figure 9.

Weighting factor $α$ versus (a) fuel consumption and (b) engine events under different standard cycles.

In order to evaluate the rate of change of CVT speed ratio improvement, a criterion $σ$ is defined as follows

σ = \sqrt{\frac{1}{N} \sum_{k = 1}^{N} {(i_{k} - i_{k + 1})}^{2}}

(21)

This criterion evaluates the variance of the rate of change of speed ratio from a global sense; a high value of $σ$ means a severe fluctuation in the rate of change of speed ratio, therefore more negative dynamics, while a lower $σ$ indicates a more smooth control signal and better drivability.

As demonstrated in Figure 10(a) and (b), when $β = 0$ , which means no constraint was imposed, the value of $σ$ for WLTC and UDDS cycles is much bigger than that for NEDC and HWFET cycles, since driving conditions are more intricate, and leads to more frequent and drastic speed ratio changes. After the rate of change of speed ratio constraint was imposed, the value of $σ$ decreased with only a slight increase in the fuel consumption. For the WLTC and UDDS cycle, as expected, a very small weighting factor will cause the drop in the rate of change variance to a large extent. However, a small value of $σ$ only indicates a small variance in the speed ratio and does not guarantee drivability improvement. In Figure 11, target speed ratio signals under UDDS cycle for different values of $β$ are depicted; when $β = 0.3 \times 10^{- 4}$ , the criterion $σ$ reduced by 50% compared to that for the basic DP, and the speed ratio signal was greatly smoothed. However, when $β = 1.5 \times 10^{- 4}$ , the value of $σ$ reduced by 62%. Practically, the resulting speed ratio signal will lead to a poor overall system performance due to the insufficient speed ratio change.

Figure 10.

Weighting factor $β$ versus (a) fuel consumption and (b) criterion $σ$ under different standard cycles.

Figure 11.

Speed ratio signals for different weighting factors $β$ .

In order to further explore the influence of the weighting factors on the fuel economy, a series of calculations were performed under UDDS cycle for $α = 0$ to $α = 2 \times 10^{- 4}$ and $β = 0$ to $β = 2 \times 10^{- 4}$ . As shown in Figure 12, the optimal fuel consumption for the basic DP is 3.98 L/100 km. With the increase in the weighting factors, the cost increases and finally converges to the minimum of the proposed method, that is, 4.2 L/100 km. It is obvious that there is more extra fuel needed to constrain the frequent engine events than to constrain the rate of change of CVT speed ratio since there are more engine recharge modes used to maintain the battery SOC.

Figure 12.

Fuel consumption versus drivability weighting factors under the UDDS cycle.

In this part, comparisons were made between basic DP and the proposed method with weighting factors $α = 1.5 \times 10^{- 4}$ and $β = 0.3 \times 10^{- 4}$ under UDDS cycle. Figure 13 illustrates a clear improvement in the engine on/off events after the constraint was introduced. Figure 14 shows the result of the rate of change of the speed ratio; compared to Figure 4, the rate of change has been confined to a much lower range. Note that at some points where the vehicle velocity is zero, the rate of change is greater than 0.5/s because it is necessary to control the speed ratio up to its maximum in order to meet the torque needs for vehicle launch.

Figure 13.

Comparison of engine events between basic DP and the proposed method.

Figure 14.

Rate of change of CVT speed ratio for the proposed method under the UDDS cycle.

The comparison of the torque split factor is shown in Figure 15(a) and (b). The torque split factor for the proposed method does not change frequently for basic DP, which implies that less mode switching is involved, resulting in better drivability. It is also clear that the introduction of penalty terms leads to greater usage of the engine recharge mode. This trend is also reflected by the battery SOC, as shown in Figure 16. In the basic DP, charge sustaining is realized by the brake energy regeneration during vehicle deceleration; thus, the SOC trajectory of the basic DP tends to be flat relative to that of the proposed method, and the SOCs for both basic DP and the proposed method are perfectly sustained at the end of the driving cycle owing to the hard constraint on the final state. Figure 17 shows a comparison of the cumulated fuel consumption. The proposed method results in a 4.5% increase in the fuel consumption.

Figure 15.

Torque split factor for (a) basic DP and (b) the proposed method.

Figure 16.

Comparison of the SOC between basic DP and the proposed method.

Figure 17.

Comparison of the cumulated fuel consumption between basic DP and the proposed method.

Conclusion

The design of an EMS is a complex task, and DP is a powerful tool to obtain a global optimal solution. In this article, a ‘control-oriented’ quasi-static vehicle model is built and DP is adopted. To avoid the ‘Curse of Dimensionality’ when implementing the algorithm, the battery SOC is chosen as the only state variable and the global minimum of fuel consumption is solved by optimizing the torque split factor and CVT speed ratio.

A single-criterion cost function will lead to excessive engine on/off events and violent fluctuations in the CVT speed ratio signal, which will cause a sharp change of the vehicle velocity. To tackle this problem, a multicriterion cost function is proposed by introducing drivability constraints. By varying the weighting factors, the trade-off between fuel economy and drivability can be evaluated under a given driving cycle.

Simulation experiments have been performed to validate the effectiveness of the proposed method under four different driving cycles. Although the introduction of drivability constraints inherently results in extra fuel consumption, the results indicate that the proposed method effectively restricted the rate of change of the CVT speed ratio and significantly reduced excessive engine events with very little increase in fuel consumption. Compared to a single-criterion cost function, the results with multiple criteria are more representative of real-world driving behaviour and thus provide a better reference solution when evaluating suboptimal online controllers.

Footnotes

Handling Editor: Yanjun Huang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China PN51475151.

ORCID iD

Hangyang Li

References

Hofman

Steinbuch

Druten

et al . Design of CVT-based hybrid passenger cars. IEEE T Veh Technol 2009; 58: 572–587.

Park

Kwon

et al . An energy management strategy for a CVT based parallel hybrid electric vehicle. In: Proceedings of the vehicle power and propulsion conference, Seoul, South Korea, 9–12 October 2012, pp.380–382. New York: IEEE.

Hofman

Steinbuch

Druten

et al . Rule-based energy management strategies for hybrid vehicles. Int J Electr Hybrid Veh 2007; 1: 71–94.

Zhong

Wang

Yang

et al . An optimal torque distribution strategy for an integrated starter–generator parallel hybrid electric vehicle based on fuzzy logic control. Proc IMechE, Part D: J Automobile Engineering 2008; 222: 79–92.

Paganelli

Ercole

Brahma

et al . General supervisory control policy for the energy optimization of charge-sustaining hybrid electric vehicles. JSAE Rev 2001; 22: 511–518.

Musardo

Rizzoni

Guezennec

et al . A-ECMS: an adaptive algorithm for hybrid electric vehicle energy management. Eur J Control 2005; 11: 509–524.

Tang

Guo

Gao

et al . Energy management of a parallel hybrid electric vehicle with CVT using model predictive control. In: Proceedings of the 35th Chinese control conference, Chengdu, China, 27–29 July 2016, pp.4396–4401. New York: IEEE.

Minh

Rashid

AA.

Modeling and model predictive control for hybrid electric vehicles. Int J Automot Technol 2012; 13: 477–485.

Tate

Jr Grizzle

Peng

Shortest path stochastic control for hybrid electric vehicles. Int J Robust Nonlin Control 2008; 18: 1409–1429.

10.

Sundstrom

Ambühl

Guzzella

On implementation of dynamic programming for optimal control problems with final state constraints. Oil Gas Sci Technol 2009; 65: 91–102.

11.

Perez

Bossio

Moitre

et al . Optimization of power management in an hybrid electric vehicle using dynamic programming. Math Comput Simul 2006; 73: 244–254.

12.

Rousseau

Sinoquet

Rouchon

Constrained optimization of energy management for a mild-hybrid vehicle. Oil Gas Sci Technol 2007; 62: 623–634.

13.

Tang

Yang

et al . Novel torsional vibration modeling and assessment of a power-split hybrid electric vehicle equipped with a dual-mass flywheel. IEEE T Veh Technol 2018; 67: 1990–2000.

14.

Debert

Padovani

Colin

et al . Implementation of comfort constraints in dynamic programming for hybrid vehicle energy management. Int J Vehicle Des 2012; 58: 367–386.

15.

Opila

Wang

McGee

et al . An energy management controller to optimally trade off fuel economy and drivability for hybrid vehicles. IEEE T Control Syst Technol 2012; 20: 1490–1505.

16.

Miro-Padovani

Colin

Ketfi-Cherif

et al . Implementation of an energy management strategy for hybrid electric vehicles including drivability constraints. IEEE T Veh Technol 2016; 65: 5918–5929.

17.

Vidal-Naquet

Zito

. Adapted optimal energy management strategy for drivability. In: Proceedings of the vehicle power and propulsion conference, Seoul, South Korea, 9–12 October 2012. New York: IEEE.

18.

Guo

Peng

et al . Research on regenerative braking control strategy of plug-in hybrid electric vehicle considering CVT ratio rate of change. Proc IMechE, Part D: J Automobile Engineering 2017; 232: 1931–1943.

19.

Hofman

Ebbesen

Guzzella

Topology optimization for hybrid electric vehicles with automated transmissions. IEEE T Veh Technol 2012; 61: 2442–2451.

20.

Sundstrom

Guzzella

A generic dynamic programming Matlab function. In: Proceedings of the control applications, St. Petersburg, 8–10 July 2009, pp.1625–1630. New York: IEEE.

21.

Bertsekas

DP.

Dynamic programming and optimal control, vol. 47, issue 6. Belmont, Massachusetts: Athena Scientific, 2000, pp.833–834.

22.

Wang

Frank

AA.

Plug-in HEV with CVT: configuration, control, and its concurrent multi-objective optimization by evolutionary algorithm. Int J Automot Technol 2014; 15: 103–115.

23.

Yang

et al . Research of the influence law of the CVT speed ratio changing rate on the vehicle acceleration. J Mech Transm 2015; 39: 23–26.

24.

Wei

Pisu

Rizzoni

et al . Dynamic modeling of a hybrid electric drivetrain for fuel economy, performance and driveability evaluations. In: Proceedings of the international mechanical engineering congress and exposition, Washington, DC, 15–21 November 2003, pp.443–450. New York: ASME.

25.

Chen

Effect of communication delays on the performance of vehicle platoons. MS Dissertation, University of California, Berkeley, CA, 1995.

26.

Bellman

RE.

Dynamic programming. Princeton, NJ: Princeton University Press, 1957.