Receding horizon control strategy for an electric vehicle with dual-motor coupling system in consideration of stochastic vehicle mass

Additional degrees of freedom existed in dual-motor coupling system bring considerable challenge to the optimal control of electric vehicles. Moreover, the stochastic characteristic of vehicle mass can further increase this challenge. A receding horizon control (RHC) strategy in consideration of stochastic vehicle mass is proposed in this study to respond to this challenge. Aiming at an electric vehicle with dual-motor coupling, a Markov chain is firstly deployed to predict future driving conditions by a formulated state transition probability matrix, based on historical driving cycles in real-world. Then, future required power is predicted by the predicted driving conditions, stochastic vehicle mass and road gradient, where the stochastic vehicle mass is formulated as stochastic variables in different bus stops. Finally, dynamic programming is employed to calculate the optimal vector of the vehicle within the defined prediction horizon, and only the first control values extracted from the optimal control vector are used to execute real-time power distribution control. The simulation results show that the proposed strategy is reasonable and can at least reduce electric consumption by 4.64%, compared with rule-based strategy.


Introduction
A sharp increase of vehicles is resulting in not only the depletion of oil energy but also the worsening of global environment [1]. Electric vehicles (EVs) characterized by zero-carbon emission and low energy consumption have attracted significant attention all over the world [2]. However, EVs usually have short driving ranges and high cost, compared with conventional vehicles [3]. Therefore, it is of great importance to investigate economical control strategy for EVs [4].
Different configurations of EVs have been proposed and investigated by researchers and automakers presently. The single-motor drive configuration is the most applicable solution for EVs, due to its simple structure and corresponding control strategy [5]. However, the motor power should be large enough to meet the acceleration and climbing requirements of EVs, which may greatly increase design cost and reduce economic performance [6][7]. To improve a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 2. The factors of stochastic vehicle mass and road gradient are considered in the prediction of the future required power, based on predicted driving conditions. Specially, stochastic vehicle mass is formulated as stochastic variables for bus stops; 3. DP is employed to calculate the optimal control vector (constituted by power distribution and shift instruction) for each receding horizon. Here, the optimal control values are the first control values extracted from the optimal control vector.
The remainder of this paper is organized as follows. Section 2 details the EV, including configuration, motor and battery models. Section 3 describes the RHC-based control strategy including Markov chain, DP and controlled models. The results and discussion of the control strategy are illustrated in Section 4, and conclusions are given in Section 5.

Configuration
The configuration of the EV is shown in Fig 1A, which is characterized by a well-designed dual-motor coupling system. As shown in Fig 1B, the dual-motor coupling system is constituted by two motors and one coupler. The coupler has two input shafts, two countershafts, two output shafts, eight constant mesh gears and two synchronizers. Four driving modes including two single-axle driving modes and two four-wheel driving modes can be realized by controlling the two synchronizer sleeves. That is, if the synchronizers 8 and 19 simultaneously slide to left, joint forward driving mode can be realized; if the synchronizer 8 and 19 simultaneously slide to the right, joint backward driving mode can be realized; if the synchronizer 8 slides to right and the synchronizer 19 sliders to left, four-wheel driving mode with low speed ratio can be realized; if the synchronizer 8 slides to left and the synchronizer 19 sliders to right, fourwheel driving model with high speed ratio can be realized. In addition, it is worth noting that the speed ratios of the joint forward and backward driving modes are the same and are deployed as transition driving modes for the four-wheel driving modes. Besides, regenerative braking mode can be realized during deceleration and downhill.  Fig 1A, the configuration of the vehicle is proposed and the vehicle communication is transmitted by controller area network (CAN) bus (CAN_H link and CAN_L link). In Fig 1B, the dual-motor coupling system is provided, and its parts are named as follows:1-The motor 1; 2, 11, 13, 22-The couplings; 3-The first input shaft; 5, 6, 7, 9, 16, 17, 18, 20- The detailed parameters of the vehicle are shown in Table 1. For the coupler, the speed ratios constituted by the gears of 5 and 6, the gears of 16 and 17 are defined as primary gear ratio, denoted by i g1 ; the speed ratios constituted by the gears of 7 and 20, the gears of 9 and 18 are defined as secondary gear ratio, denoted by i g2 . The output torques of the coupling system in different driving modes can be described as where T out1 is the output torque of the joint forward driving mode; T out2 is the output torque of the joint backward driving mode; T out3 is the output torque of the four-wheel driving mode with low speed ratio; T out4 is the output torque of the four-wheel driving mode with high speed ratio; T 1 and T 2 denote the output torques of the motors, respectively; i g1 and i g2 denote the primary and secondary speed ratios, respectively.

Motor model
Since the motors are the same, the speed of any motor can be described as where u denotes the velocity of the vehicle; R tire denotes the tire radius; i 0 and i g denote the speed ratios of the main retarder and the coupler, respectively. Since the motors can provide both driving torque and regenerative braking torque, the power of any motor can be presented by where T m denotes the torque of the motor; η m denotes the efficiency of the motor, which can be obtained by interpolation method with the efficiency map of the motor in Fig 2; P dem denotes the required power of the vehicle.

Battery model
As shown in Fig 3, a Rint model of the battery is employed to simplify the control problem. Based on the internal resistance and open-circuit voltage of the battery (Fig 4), the power and the current of the battery can be described as ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where P bat denotes the battery power; I b denotes the battery current; U oc denotes the open-circuit voltage; R b denotes the internal resistance.

Framework
An RHC based strategy is designed in Fig 5, which is constituted by Markov chain-based prediction, DP model-based optimization and RHC-based control strategy steps.
Step1 (denoted by Markov chain-based prediction) is designed to predict acceleration and velocity trajectories within a defined receding horizon. A state transition probability matrix of acceleration is firstly formulated using a series of driving cycles in a fixed city bus route in realworld. Then, a Markov chain is built to predict future acceleration and velocity trajectories within the defined prediction horizon at every time step, based on the received information from GPS. Step2 (denoted by DP model-based optimization) is designed to calculate the optimal power distribution and the optimal shift instruction based on DP. It is worth noting that two stochastic factors of vehicle mass and road gradient are considered in the model, due to their strong relationship with the required power of the vehicle.
Step3 (denoted by RHC-based control strategy) is designed to realize the real-time control of the vehicle. The first control values of the optimal control vector from DP are extracted to execute the power distribution of the two motors and the shift instruction of the coupler.  Then, the updated SOC and gear position of the vehicle are taken as the initial state of the DP for the next time-step.

Driving cycles.
One of the most key issues in the RHC strategy is the prediction of future driving conditions. Since historical driving cycles in real-world can be obtained from GPS, a Markov chain for future driving conditions prediction within a given prediction horizon can be constructed [27]. As shown in Fig 6, a series of driving cycles constituted by velocity are collected, and the corresponding acceleration sequence can be deduced by where u(k) and u(k−1) denote the current and previous velocities, respectively; t denotes the sampling time of the velocity.

Markov chain.
To predict future driving conditions, a state transition probability matrix should be firstly formulated. However, if it is formulated by velocity, only the number of states is discrete large enough, can prediction precision be satisfied. Therefore, a state transition probability matrix of acceleration is formulated in this paper.
As shown in Fig 7, to ensure the predictive precision of future driving conditions, the state number of the acceleration is designed as 94 and the values are ranged from -2 to 1.5.  Based on the theory of the Markov chain [28], the state transition probability matrix of the acceleration can be described as  (8), the rolling resistance (F f ), the road gradient resistance (F i ) and the acceleration resistance (F j ) have direct relationship with vehicle mass. This will further influence the optimal power distribution of the two motors and the optimal shift schedule of the coupler. Moreover, city buses usually show more stochastic characteristic than passenger cars with respect to the vehicle mass, due to the strong stochastic distribution characteristic of passengers in different bus stops. Therefore, the factor of the stochastic vehicle  mass should be considered in power distribution control. where m and m a denote the curb mass and the passengers' mass, respectively; g denotes the gravitational acceleration; f denotes the coefficient of rolling resistance; α denotes the angle of the road gradient; δ denotes the correction coefficient of rotating mass; a denotes the acceleration of the vehicle.

Required power. As shown in Eq
As shown in Fig 9A, road gradient can be estimated by trip distance and road altitude. Since their signals can be obtained from the GPS/GIS system, the road gradient can be deduced by where H denotes the road altitude; L denotes the trip distance. Since city bus usually has fixed route, the road gradient can be obtained off-line, based on the historical road information, and can be implemented into control strategy in a prior by taking the trip distance as an independent variable. As shown in Fig 9B, the road gradient of the fixed route changes sharply, where the negative values of the road gradient denote downhill, on the contrary, the meaning is just the opposite. As stated above, the stochastic characteristic of vehicle mass and road gradient will greatly influence the optimal power distribution of the motors and the optimal shift schedule of the coupler. Therefore, the stochastic vehicle mass and road gradient should be considered in the required power prediction.
Based on the vehicle longitudinal dynamic equation, the traction of the vehicle can be described as where F res denotes the traction of the vehicle; C D denotes the coefficient of air resistance; A area denotes the frontal area of the vehicle. Then, the required power of the vehicle can be presented by where P dem denotes the required power of the vehicle. 3.3.2. DP model. Generally, the optimal control of the vehicle can be formulated as a discrete dynamic system, where DP can be deployed to solve it. In this paper, the state vector is designed as battery SOC, and gear position of the coupler. The control vector is designed as power distribution and shift instruction of the coupler. The discrete dynamic system can be described as where SOC(k) denotes the battery SOC; g(k) denotes the gear position; β s (k) denotes the shift instruction; Q denotes the rated battery capacity; I b denotes the battery current. The battery is assumed to be fully charged at the beginning of the trip, which implies that the initial SOC is 0.8. β s (k) is designed as 1, 0 and -1 to denote upshift, hold on and downshift, respectively. The optimization objective (C(k)) is designed as minimizing the whole power consumption of the battery, which can be presented by where P dem (k) denotes the required power of the vehicle. Since the problem of frequent shifting of the coupler can greatly reduce the driving performance of the vehicle, it will be restricted in the cost-to-go function through the absolute of the shift instruction multiplied by a weighting factor (denoted by γ). Then, the cost-to go function of the EV can be described as where J F (x 0 ) denotes the cost-to go function; N denotes the length of the receding horizon (60s). To ensure the reliable control of the vehicle, some physical constraints should also be defined as where P bat,min and P bat,max denote the power boundaries of the battery; P m,min and P m,max denote the power boundaries of the motor; n min and n max denote the speed boundaries of the motor; T m,min and T m,max denote the torque boundaries of the motor. Fig 10, only the first control values extracted from the optimal control vector are applied to the controlled model. The initial states of the battery and coupler will be updated by the current output values.

Controlled model.
Based on the first control variables, the powers of the two motors can be described as P m1 ðkÞ ¼ wðkÞ � P dem ðkÞ P m2 ðkÞ ¼ P dem ðkÞ À P m1 ðkÞ � Z T ðkÞ � Z d ðkÞ ð16Þ ( where P m1 (k) and P m2 (k) denote the powers of motors, respectively; χ (k) denotes the optimal control value; η T (k) denotes the efficiency of the transmission shaft; η m (k) denotes the transmission efficiency of the drive axle.
Considering the efficiency of the motors, the power of the battery can be described as

Results and discussion
As state above, the stochastic vehicle mass in different bus stops has strong relationship with the optimal control of the vehicle. To exhaustively evaluate the robustness, application and optimality of the proposed RHC strategy, 46 stochastic variables with respect to bus stops are formulated to describe the stochastic distribution of the vehicle mass in different bus stops.
Here, the stochastic vehicle mass is denoted by the mass of passengers, and assuming that the passenger's mass is around 70kg. Besides, each stochastic variable is designed to 18 levels based on the maximum number of passengers. In this case, the stochastic variables of the bus stops will constitute a huge design space from the viewpoint of design of experiment. To better verify the proposed power distribution strategy, Opt LHD is employed to exhaustively insight into the design space. As a representative method of design of experiment (DOE), Opt LHD has better performance than others to solve the spatial filling and equalization problems. It can better improve Fig 10. The RHC principle. The optimal control vector at time step of k will be obtained by DP, and the first one of the optimal control vector will be extracted to execute the power distribution control. https://doi.org/10.1371/journal.pone.0205212.g010 Receding horizon control strategy for an electric vehicle the sampling precision and robustness than other methods [29]. In specific, a designed n×m matrix X = [X 1 ,X 2 ,� � �X n ] T will be firstly generated by the stochastic LHD, where the vector of x im � denotes experiment analysis and m denotes the number of bus stops. Then a new matrix will be regenerated by elemental exchanging. Finally, the optimal criteria of spatial filling can be achieved by the maximum distance criterion, which can be defined as min 1�i;j�n;i6 ¼j where X i and X j denote sampling points; d(X i ,X j ) denotes the distance between X i and X j . As shown in Fig 11, a series of stochastic distributions of passengers are presented to verify the robustness and application of the proposed control strategy, based on the sampling theory of the Opt, LHD. Here, the same colored line means the same sampling test, and 0 means that only driver on the bus.
In addition, ten compound driving cycles are constructed by three different driving cycles and the stochastic distributions of passengers (Fig 11), based on stochastic combination method. The trip times of the driving cycles are 7451s, 7811s and 8167s, respectively. The velocities are no more than 50km/h, and the distance of the whole trip is about 35km. The rule-based and DP control strategies are also employed to evaluate the proposed RHC strategy. The rule-based control strategy is defined as evenly power distribution of the two motors with a predefined shift schedule. DP is deployed to benchmark other strategies. Besides, the road gradient of the fixed city bus route has been implemented into the strategies in a prior.
As shown in Figs 12-21B, all of the SOC trajectories of the RHC strategy are higher than the rule-based control strategy and lower than the DP control strategy. This implies that the proposed RHC strategy is better than the rule-based control strategy and worse than the DP control strategy. In addition, all of the SOC trajectories of the strategies decline sharply at the first half of the trip due to the larger required power on the uphill way, and they increase gradually at the second half of the trip, due to the frequent regenerative braking on the downhill way. Receding horizon control strategy for an electric vehicle    Table 2, all the terminal SOC values of the RHC control strategy are higher than the rule-based control strategy, and close to the DP control strategy. Since the control strategies have the same rated battery capacity and initial SOC, higher terminal SOC value implies lower electric consumption. Therefore, the RHC strategy has more energy saving potential than the rule-based control strategy and its economic performance approaches to the DP control strategy. Similarly, Table 3 also demonstrates the above conclusion. That is, the electric consumption of the RHC strategy can be at least improved by 4.64%.

As shown in
Furthermore, the electricity consumption (Q c ) of the vehicle can be calculated by where SOC initial denotes the initial SOC; SOC final denotes the terminal SOC; Q denotes the rated battery capacity. The electricity consumptions of the strategies for different compound driving cycles are shown in Table 3.

Conclusions
This paper proposes an RHC strategy for an EV with dual-motor coupling system in consideration of stochastic vehicle mass. The conclusions are summarized as follows.
1. A Markov chain is constructed for the RHC strategy to predict acceleration and velocity trajectories, based on historical driving cycles in real-world. The stochastic vehicle mass is considered in the RHC strategy to improve its control performance.
2. To better evaluate the control performance of the RHC strategy, rule-based and DP control strategies are deployed. Moreover, Opt, LHD is employed to assist verifying the proposed control strategy, by exhaustively probing the design space (constituted by the designed stochastic variables).
3. The simulation results show that the electricity consumption of the RHC strategy closes to the DP control strategy and can be significantly reduced, compared with the rule-based control strategy.
The future work of this research will focus on the product-level application of the RHC strategy, and the implementation of state of health (SOH).