# Driving-Cycle-Aware Energy Management of Hybrid Electric Vehicles Using a Three-Dimensional Markov Chain Model

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## Abstract

This study developed a new online driving cycle prediction method for hybrid electric vehicles based on a three-dimensional stochastic Markov chain model and applied the method to a driving-cycle-aware energy management strategy. The impacts of different prediction time lengths on driving cycle generation were explored. The results indicate that the original driving cycle is compressed by 50%, which significantly reduces the computational burden while having only a slight effect on the prediction performance. The developed driving cycle prediction method was implemented in a real-time energy management algorithm with a hybrid electric vehicle powertrain model, and the model was verified by simulation using two different testing scenarios. The testing results demonstrate that the developed driving cycle prediction method is able to efficiently predict future driving tasks, and it can be successfully used for the energy management of hybrid electric vehicles.

## Keywords

Driving cycle prediction Markov chain model Hybrid electric vehicles Energy management## 1 Introduction

- (1)
Most standard driving cycles were developed specifically to evaluate compliance with emissions or fuel consumption regulations. Thus, they are inappropriate for the design process because fuel economy and environmental impacts are not the only design concerns. Furthermore, current standard driving cycles differ significantly in terms of various properties, such as predictable patterns of speed and acceleration, that place varying strains on the drivetrain. Therefore, they cannot represent real-world traffic conditions. For example, one study demonstrated that fuel consumption results from laboratory tests based on standard driving cycles deviate from those of real-world driving conditions by approximately 30% [1].

- (2)
The predictable patterns employed in standard driving cycles enable engineers to build automobiles that pass regulatory tests but do not produce low emissions under real-world driving conditions. This leads to a process called cycle beating, which creates a distorted linkage between government standards, consumers, and manufacturers [2]. In particular, the energy management strategies of hybrid electric vehicles (HEVs) have been validated using only a limited number of driving cycles and therefore run the risk of cycle beating [3, 4, 5].

- (3)
Current driving cycles do not include roadway slope, road type, and other relevant roadway properties that affect the fuel consumption of commercial vehicles with significant dead weight [6]. The influence is more significant for hybrid powertrains because the regenerative braking function activated during downhill driving plays a major role in fuel savings [7].

- (4)
The driving cycles created by government agencies and manufacturers may be excessively complex for simulation and design purposes, which may lead to unacceptably low calculation efficiency during the simulation and design process.

Addressing the above issues requires the development of more sophisticated driving cycle prediction methods that consider roadway slope and road type information, and the application of these methods in the energy management of HEV powertrain systems [8, 9, 10, 11, 12]. Past studies have focused on the development of driving cycle prediction methods. For example, driving cycle prediction has been conducted according to statistical indicators based on correlations between the phase space and phase trajectory of experimental driving data that include velocity, acceleration, and road grade [13]. In this study, correlation coefficients close to 1 were taken as indicative of strong connections between two variables. Then, the Markov Chain Monte Carlo technique was applied to generate the new driving cycle. A Markov chain concept has also been developed that filters high-frequency changes in acceleration and places velocity–time data into a matrix form that can accommodate a large number of velocity states or modes [14]. In addition, two separate Markov chain processes have been adopted for driving cycle prediction. One was employed for generating the road type using random selections of road types from urban, rural, and expressway profiles. The other generated the acceleration based on the road type using predefined velocity bins [15]. A Markov chain process has also been combined with the concept of equivalent driving cycles [16]. The equivalent driving cycles have the same performance in terms of the mean tractive force (MTF). Meanwhile, researchers have explored other rebuilding methods, including discrete cosine transfer. The impact of driving cycles on the performance of energy management algorithms has also been discussed recently [17, 18, 19, 20, 21]. Commonly applied energy management strategies (e.g., dynamic programming) require prior knowledge regarding driving tasks for implementation. Therefore, engineers focus closely on driving cycle identification to distinguish driving tasks [22]. The iterative learning of driving profiles based on comparisons of the velocity and acceleration at each start and stop with values from a database has improved the performance of an HEV running on a fixed route [23]. In addition, a vehicle velocity predictor has been built using a one-dimensional Markov chain of the velocity [24].

However, these past works suffer from deficiencies. In the studies described in [8, 17, 18], the combined states of velocity and acceleration were obtained from the original driving cycle, and the transitions between different states were extracted as a two-dimensional matrix. However, most studies do not account for roadway slope information in driving cycle prediction. Although it is possible to extract another two-dimensional Markov chain for slope, the harmony between the velocity and slope generation is not reasonable because these are separate stochastic processes. Therefore, the widely used Markov chain model needs to be extended to a higher dimension to ensure good performance when including the slope information.

In addition, optimal control methods are fundamental to HEV technology and have become the subject of intense interest recently. The parallel powertrain of HEVs employing both an internal combustion engine and an electric motor as power sources has only a single degree of freedom in terms of the control strategy, which is denoted as the torque (or power) split ratio. Among the existing optimal control methods for HEVs, the equivalent consumption minimization strategy (ECMS) is a particularly promising instantaneous approach derived from Pontryagin’s minimum principle. It can be implemented in real time because the computation time is reasonable. The ECMS calculates the torque split (or power split) for each sample time and determines the output from each power source. The main objective of the ECMS is to reduce the overall energy consumption. The input to the controller is the demand torque from the driver (the driving cycle), and the outputs are the demand torques from the internal combustion engine and electric motor. The control strategy provides an optimal torque split ratio. However, the ECMS employs a design parameter denoted as the fuel equivalence factor, and it plays a significant role in this control strategy. According to existing studies [3, 4], a linkage exists between the equivalence factor and the terminal state of charge (SoC) of the battery in HEVs. As such, the fuel consumption can be maintained close to a minimum if the equivalence factor is well defined. As such, prior knowledge or accurate prediction of the driving cycle can be employed for adaptively tuning the equivalence factor, thereby achieving better control performance in real-time implementations.

To further advance the existing studies, the present work develops a new method to predict the upcoming driving task (i.e., required longitudinal speed of the vehicle) based on historical driving data for HEVs using a three-dimensional stochastic Markov chain model. The proposed three-dimensional Markov chain model adopts vehicle velocity and acceleration as two of the dimensions and augments the model to include roadway slope information as a third dimension in the driving cycle prediction. Because the design optimization through simulation could last for a few days, the possibility of increasing the dimensions of classes is explored to reduce the simulation time. In addition, the developed online driving cycle prediction method is applied based on an HEV powertrain model to verify a driving-cycle-aware ECMS for HEVs.

The remainder of this paper is organized as follows. Section 2 describes the driving cycle synthesis and prediction method. Next, the HEV powertrain model is established in Sect. 3. In Sect. 4, the designed driving cycle prediction approach is implemented in the energy management strategy of an HEV powertrain. Simulation and results are presented in Sect. 5. Finally, Sect. 6 presents conclusions from this work.

## 2 Driving Cycle Prediction

### 2.1 Three-Dimensional Markov Chain Model for Driving Cycle Prediction

*T*or the transition probability matrix

*F*from the database, generating the new sample from

*T*, and smoothing the result for the MTF. These are discussed in detail in the following subsections.

#### 2.1.1 Extracting Transition Matrix *T* or Transition Probability Matrix *F* from the Database

*K*,

*M*, and

*N*classes for slope (indexed as

*k*), acceleration (indexed as

*m*), and velocity (indexed as

*n*), respectively, which are given as follows:

However, it must be noted that while a large number of classes will improve the accuracy of the predicted driving cycle, higher data resolutions will significantly increase the required computational time [21]. Therefore, considering a reasonable trade-off between accuracy and computational time for real-time implementation, the present work adopted \( v_{\text{res}} \), \( a_{\text{res}} \), and \( \theta_{\text{res}} \) values of 2.22 m/s (8 km/h), 0.2 m/s^{2}, and 0.5°, respectively.

*Sn*

_{(k,m,n)}is defined as the state 1 s after the current state.

*S*

_{(k,m,n)}at time

*t*reside within a particular range. For example, the transition matrix

*T*in Fig. 3 indicates that the slope of

*S*

_{(k,m,n)}is located in the range [0,0.5)°, acceleration is in the range [0, 0.2) m/s

^{2}, and velocity is in the range [40, 48) km/h at time

*t*. Then, all possible transitions from

*S*

_{(k,m,n)}can be found from

*T*in the set of next states

*Sn*

_{(k,m,n)}at time

*t*+ 1.

*S*

_{(k,m,n)}can be calculated after set

*Sn*

_{(k,m,n)}is found. Here,

*Sn*

_{(k,m,n)}will be transformed into a three-dimensional matrix that is similar to the current state. At each block of the matrix, the possible next state will be in the defined range that is divided according to equivalent data resolutions. If one sub-matrix is empty, it means that the current state cannot transition to this particular range. Accordingly, the transition probability matrix

*F*can be calculated. First,

*S*

_{(k,m,n)}is defined as follows:

*S*

_{(k,m,n)}to

*Sn*

_{(k,m,n)}, the three-dimensional index transfers to a one-dimensional vector with length

*K*·

*M*·

*N*as follows.

*h*are \( Sn_{(j,h)} \). In the range

*h*, a set

*B*includes a combination of slope, velocity, and acceleration that belong to the same range. The conditional probability of current state

*j*is rewritten as follows:

After this process, a six-dimensional matrix *F* is obtained for all current states, and all terms in each sub-matrix are scaled such that their sum is reduced to 1.

In most related research, the database is collected under real-world driving conditions to ensure that the sub-matrixes in *F* will be as full as possible. Moreover, the generation can be conducted using the resolution boundaries of the next state if the resolutions are sufficiently high [21]. In this study, the database was collected based on the Worldwide Harmonized Light Vehicles Test Procedure Class 3 (WLTP3) cycle that was revised using random slope information within − 10° to 10°. Because the transition in the database is insufficient, and due to the defined resolutions, *T* was used in this research rather than *F*. As a result, the time-variant driving profile becomes a time-invariant statistical model.

#### 2.1.2 Generating the New Sample from Transition Matrix *T*

The decrease in the number of classes increases the differences between velocities in two states. This leads to obvious fluctuations in the time domain performances of the generated velocities and slopes and also results in high-frequency changes in the acceleration. Therefore, the method for generating velocities and slopes was modified to eliminate the effect of these conditions. Meanwhile, the local regression smoothing method was applied to the velocity and slope after each generation.

*t*belonging to

*S*

_{(k,m,n)}, the next transition is obtained from

*T*, and

*A*is the total number of the transitions from

*S*

_{(k,m,n)}. Then, a discrete uniform distribution is calculated, and the probability

*P*

_{j}of each transition is

*µ*∈ (0, 1] generated at each sample time serves to decide the next state, which defines a precise velocity, acceleration, and slope from the database. The value of

*µ*is also generated from a uniform distribution, where, for each

*µ*, there must be an existing

*i*as follows:

*i*is the index of the next step in all possible transitions in the set

*Sn*

_{(k,m,n)}.

#### 2.1.3 Smoothing the Result for the Mean Tractive Force

*F*(

*t*) required to be applied to the wheels by the powertrain during a driving cycle on average. It has been proposed as a tentative parameter for evaluating fuel consumption [25, 26]. Because the powertrain provides no force to the wheels during coasting or braking operations [i.e., \( F(t) \le 0 \)], \( \bar{F}_{\text{trac}} \) is defined only over intervals of time when the powertrain must provide positive power to the wheels [i.e., \( F(t) > 0 \)]. Therefore, \( \bar{F}_{\text{trac}} \) is defined as a subset \( \tau_{\text{trac}} = t \in \left\{ {\tau :F(t) > 0} \right\} \) and is written as

*N*increases because a greater extent of the short-term dynamic variations in the velocity in the span is lost with increasing

*N*. Thus, the span should be sufficiently small to retain the dynamic behavior of the velocity. Therefore, the present work employs

*N*= 8 s to account for 15% error between the predicted speed and the reference speed in the criteria. After these steps are complete, the time-variant driving cycle is transformed into a time-invariant statistical model. An example of acceptably smoothed generation based on the WLTP3 cycle is presented in Fig. 4.

### 2.2 Impact of Prediction Time Lengths on Driving Cycle Generation

As described in the previous section, the proposed method can successfully predict future driving cycles with the same time length as the original cycle. In this section, the use of different driving cycle time lengths is explored for the short-term prediction of future driving cycles with higher dynamics.

*L*of predicted driving cycle and the defined resolutions \( a_{\text{res}} \), \( v_{\text{res}} \), and \( \theta_{\text{res}} \). The value of

*L*is defined by the user. However, if all historical driving data are used to predict the driving cycle, the volume of data will increase rapidly with increasing time, and the time required for calculating the transition matrix will also increase substantially. Furthermore, the driving tasks of vehicles vary according to the road conditions and the behaviors of the drivers. For daily driving, the road conditions and driver behaviors vary relatively little in the short term, as drivers drive in the city or on the freeway for a certain length of time. In addition, the number of states in the transition probability matrix decrease with decreasing

*L*, and this situation will make it difficult for the Markov chain to generate successful prediction results. As such, while a relatively large value of

*L*will preserve the properties of the original driving cycle, the time required for the generation and simulation processes will increase with increasing

*L*. Therefore, we explore the influence of

*L*on the generation process based on the following 11 values:

*L*is the minimum time length needed to successfully retain the properties of the original driving cycle. Therefore, 200 generations were conducted for each value of

*L*to determine its statistical properties.

*L*on the standard deviations of the generated velocities is presented in Fig. 5, while Fig. 6 presents the differences in the maximum velocities obtained. Comparison of Figs. 5 and 6 shows that the overall performance of the compression is nearly stable when

*L*is greater than 50% of the original length. Furthermore, the median difference values of 50% of the compressed cycles are approximately 10%, which is within an acceptable region. The longest cycle generation in this study, which employed a value of

*L*= 2700 s, demonstrates a much more stable performance because most of the differences are less than 10%.

*L*values of 900 s, 1440 s, and 1800 s are presented in Fig. 7. For velocity generations with

*L*< 900 s, the maximum velocity is generally less than 100 km/h. This is due to a transition that began from the idle state. Thus, according to the results presented here, the maximum acceptable compression ratio for the WLTP3 cycle is 50%, which ensures a rapid generation process, while sufficiently retaining the essential properties of the original driving cycle.

## 3 Hybrid Electric Vehicle Modeling

The overall HEV powertrain model was built according to the backward power flow method [25]. The input consists of the acceleration, velocity, and slope information of the driving cycles. The sampling time during testing is the same as the sampling time of the driving cycle (i.e., 1 s). The output is dependent on the variables of interest, that is, fuel consumption and SoC. The components of the HEV include an internal combustion engine, an electric motor, and a battery, which are modeled individually as described below.

### 3.1 Vehicle Model

*F*

_{acc}), gravity (

*F*

_{grav}), rolling resistance (

*F*

_{roll}), and drag on the vehicle (

*F*

_{drag}), which are given as follows:

*m*

_{all}is the total mass of the vehicle;

*g*is the acceleration due to gravity; and the other vehicle parameters

*ρ*

_{air},

*C*

_{r},

*C*

_{d}, and

*A*

_{f}are constants [27, 28, 29, 30, 31]. The value of

*m*

_{all}depends on the masses of the components, including the equivalent inertia

*m*

_{int}, internal combustion engine

*m*

_{ice}, electric motor

*m*

_{em}, and battery

*m*

_{bat}, which are defined as follows:

### 3.2 Internal Combustion Engine Model

*T*

_{ice}) of the engine with different fitting parameter values \( u_{1} \), \( u_{2} \), and \( u_{3} \) for particular ranges of engine speed (

*ω*

_{ice}), engine type, and

*P*

_{ice}as follows [30, 31]:

### 3.3 Electric Motor Model

*ω*

_{em}and output torque

*T*

_{em}, and it can be defined as follows [30]:

*T*

_{em}:

### 3.4 Battery Model

*k*th time period, which are given as follows:

*V*

_{oc}is the open circuit voltage and

*R*is the resistance. The main parameter of the battery is listed in Table 1. In this study, the value of

*Q*is varied according to the vehicle type.

Main parameters of vehicle model

Parameter | Value |
---|---|

Engine maximal power (kW) | 130 |

Engine maximal torque (N·m) | 183 |

Electric motor (kW) | 35 |

Maximal torque (N·m) | 130 |

Battery (A·h) | 6 |

Total mass (kg) | 1590 |

Road resistance (–) | 0.013 |

Drag coefficient (–) | 0.32 |

Drag area (m | 2.2 |

Radius of tire (m) | 0.3 |

## 4 ECMS Design Based on Driving Cycle Prediction

### 4.1 Conventional ECMS

This section briefly discusses the conventional ECMS and its implementation in a hybrid powertrain system. For offline simulation, the driving cycle is provided in advance; thus, the requested propulsion power and engine speed are known.

*P*

_{b}is the output power of the battery, and \( \eta_{\text{mb}} \) is the efficiency of the power converter between the battery and the electric motor.

*T*is one of the design targets, and it is expressed as follows:

### 4.2 Improved ECMS Based on Driving Cycle Prediction

*β*is initialized using historical driving data or prior knowledge of driving cycles. Then, the driving cycle prediction generates a candidate driving cycle based on historical driving data of fixed length. The length is the prediction horizon. Therefore, the value of

*β*is updated by combining past and predicted data as follows [32]:

*β*

_{k}and \( x_{k} \) are, respectively, the SoC and the equivalence factor at the end of the

*k*th horizon, and \( \gamma \) is a constant. The ECMS is updated for each new value of

*β*at the beginning of an iteration, and the vehicle utilizes the updated ECMS until the end of the iteration. As such, the new value of

*β*becomes a piecewise function of time.

The desired outcome is that the prediction reflects the next driving task in the fixed period and the new estimated value of *β* is able to adjust the power or torque split for the upcoming task. However, the prediction is based on historical data and therefore has no strong connection with the upcoming driving task. This condition is a significant influential factor in the ECMS.

## 5 Simulation and Results

### 5.1 Constraints and Testing Scenarios

The improved ECMS was implemented and tested using a luxury hybrid sedan model with an initial SoC of 70% under two distinct driving scenarios. The key parameters of the model are listed in Table 1. In the first scenario, the overall driving task consisted of the HEV following the WLTP3 cycle two times, for a total duration of 3600 s. The second scenario consisted of the WLTP3, NEDC, and JN1015 cycles, for a total duration of 3600 s. For both scenarios, the improved ECMS was tested using a single optimum value of *β* denoted as *β*_{opt}. Moreover, the driving cycle prediction generates a driving route using a prediction horizon of *H* seconds. Then, an equivalence factor *β*_{1} is determined from the prediction cycle to maintain the charge-sustaining mode at the end of the task. The equivalence factor is updated again at *H* seconds as *β*_{2} from the resulting prediction. The respective horizons *H* are chosen to be 900 s and 450 s, such that *β* is updated four times and eight times, respectively. The time length of the historical driving data is 2*H*, and it is compressed by the designed method using a 50% compression rate. Therefore, the historical time lengths are set as 1800 s and 900 s, respectively, to ensure speed and accuracy for the prediction.

Based on the analysis presented here, we concluded that the lengths of the historical cycle DC_{his} (1800 s and 900 s) are reasonable. The purpose of the testing is to demonstrate that the adaptations in the value of *β* allow the hybrid vehicle to follow a control strategy in real time and maintain the battery SoC. No prior knowledge is employed in the testing because the input is the historical driving data.

### 5.2 Online Driving Cycle Prediction

*H*, the historical data DC

_{his}(

*n*) are updated during the estimation of

*β*. Then, the Markov chain-based prediction described in the previous section is used. The updated transition probability matrix ensures that the predicted cycle is similar to the historical driving scenario. As discussed in Sect. 2, a candidate driving cycle is a valid prediction for the next period if all the conditions are satisfied. For example, the green curve in Fig. 9 is a valid prediction based on the historical driving data given as the blue curve.

### 5.3 Testing Results of the Improved ECMS

*β*obtained from the separate predictions are plotted in Fig. 10 for the entire actual driving cycle. See Sect. 4.2 for an explanation of the method used to adjust the value of

*β*within a historical period DC

_{his}. The SoC results are plotted in Fig. 11 and listed in Table 2.

Results of the different energy management strategies for scenario 1

Length (s) | Terminal SoC (%) | Fuel consumption ( | |
---|---|---|---|

Conventional ECMS | – | 70.01 | 4.41 |

Improved ECMS with horizon of 900 s | 3600 | 70.07 | 4.50 |

Improved ECMS with horizon of 450 s | 3600 | 69.96 | 4.44 |

*H*= 450 s. At the same update time, e.g., 900 s and 2700 s, the SoC is closer to 70% for the shorter horizon. This indicates that the improved ECMS can maintain a charge-sustaining mode with a relatively short prediction horizon. In addition, compared with the conventional ECMS, which is able to achieve a global optimal solution but unable to be implemented in a real-world application, the improved ECMS achieves similar fuel economy and real-time implementation capability. This validates the feasibility of the proposed method and its improvement over the conventional ECMS. The detailed SoC results are listed in Table 3.

SoC at each updated time step

Time (s) | SoC (%) horizon 900 s | SoC (%) horizon 450 s |
---|---|---|

450 | – | 69.95 |

900 | 69.88 | 69.91 |

1350 | – | 69.93 |

1800 | 69.84 | 69.85 |

2250 | – | 69.96 |

2700 | 69.77 | 69.91 |

3150 | – | 69.91 |

3600 | 70.07 | 69.96 |

*β*and SoC results for scenario 2 are given in Figs. 12 and 13, respectively. The terminal SoC obtained for the improved ECMS is 70.09%. It is obvious that the final SoC does not maintain the charge-sustaining requirement, and the difference is greater than that obtained for scenario 1. It is also noted that the SoC values obtained by the improved ECMS at update times of 1800 s and 3000 s are very different from the SoC values obtained when employing only

*β*

_{opt}. This was caused by a prediction error. Here, the prediction depends on past driving data, and if the driving conditions change, the estimation of

*β*will be incorrect in that period. However, the next estimation of

*β*obtained under stable driving conditions will correct this error.

From the presented results, we can conclude that the improved ECMS represents a feasible method for addressing the significant drawbacks that exist in both dynamic programming energy management systems and the conventional ECMS. Unlike these methods, the improved ECMS does not require the entire driving cycle prior to initialization. Furthermore, it can adjust the value of *β* in real time and thereby provide more robust performance for hybrid powertrains. While the results indicate that the improved ECMS cannot maintain the battery SoC, this drawback can be considered acceptable because it is fully amendable to solution.

## 6 Conclusions

This paper introduces a three-dimensional Markov Chain method that obtains a stochastic model from the existing driving cycle. The model effectively works with the Monte Carlo method to predict a driving cycle with the same length while retaining similar properties. It is also possible to accurately and efficiently compress the WLTP3 cycle down to 50% of its original length. Moreover, a real-time energy management strategy for HEVs was designed based on the proposed method. The equivalence factor in the ECMS is updated in real time using the driving cycle prediction. The test results validate the feasibility and effectiveness of the energy management strategy for HEVs using the developed driving cycle prediction method.

## Notes

### Acknowledgements

This research was supported in part by the Young Elite Scientist Sponsorship Program (No. 2017QNRC001) of the China Association for Science and Technology and a Start-Up Grant (No. M4082268.050) from Nanyang Technological University, Singapore.

### Compliance with Ethical Standards

### Conflict of interest

On behalf of all authors, the corresponding authors state that there is no conflict of interest.

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