Research on the Energy Management Strategy of a Hybrid Tractor OS-ECVT Based on a Dynamic Programming Algorithm

Abstract

The multi-degree-of-freedom characteristics of the planetary gear electronic continuously variable transmission (ECVT) configuration in series-parallel hybrid tractors impose more complex requirements for energy management strategies under variable load conditions. For a high-power hybrid tractor, this paper takes the hybrid tractor output-split (OS)-ECVT configuration as the research object and describes the principles of stepless transmission and power-splitting within the configuration. In order to improve the fuel economy of high-power hybrid tractors and the running status of power components, an energy management strategy focused on ploughing conditions based on the Bellman minimum dynamic programming (DP) algorithm is proposed in this paper. Second, equivalent fuel consumption is selected as the performance index for energy-saving control, and the solving principle of the energy management strategy based on the dynamic programming algorithm is established to facilitate the resolution process of the energy management strategy. Finally, the energy-saving control simulation is completed under ploughing conditions. The results show that compared with the energy management strategy based on the optimal operating line (OOL), the energy management strategy based on DP fully utilizes the benefits of low-cost electric energy and enables the hybrid power system to have a wider range of stepless transmission performance. In addition, the hybrid power system has the advantages of enhanced decoupling of speed and torque, higher efficiency, and more economical secondary energy conversion. As a result, the whole machine has enhanced power-split performance, greatly improving the running conditions of the power components. The equivalent fuel consumption values of the energy management strategies based on DP and OOL are about 3.1238 L and 4.2713 L, respectively. The equivalent fuel consumption based on DP is reduced by about 26.87%, which effectively improves the fuel efficiency of hybrid tractors.

1. Introduction

With the call for energy conservation and emission reduction, improving the energy efficiency and environmental performance of agricultural machinery has become an important focus, and the shortcomings of new agricultural machinery energy conservation and emission reduction technologies have become the key issues to be solved [1,2,3]. Tractors are widely used in the field of agricultural machinery due to their advantages of strong power and ability to meet the operation requirements under variable field load conditions [4]. With the proposal of sustainable development requirements in the field of agricultural machinery, new energy tractors have been vigorously developed [5]. As a crucial technology for energy savings and emission reductions in new energy tractors, the energy management strategy directly determines the economic efficiency of the tractors [6]. Li et al. [7] took the pure electric tractor as the research object and proposed an energy tube strategy based on a stochastic dynamic programming algorithm and extremal search algorithm that effectively improved the economic efficiency of the pure electric tractor. Li et al. [8] studied the energy management strategy of an electric tractor power system using a dynamic programming algorithm. Compared with a rule-based energy management strategy, the dynamic programming algorithm can effectively reduce the energy consumption of the electric tractor. Pure electric tractors need to be equipped with large-capacity power batteries for operation. Restricted by the current limitations in battery technology, high-power pure electric tractors face challenges in practical application [9]. Hybrid tractors effectively combine the advantages of the engine and the motor, can compensate for the shortcomings of the pure high-power electric tractors related to battery performance, and can meet the needs of high-power operations [10,11]. At present, there are three types of high-power hybrid tractors: serial type, parallel type, and hybrid type [12,13,14]. In the operation of serial-type hybrid tractors, because the engine does not directly participate in the operation, there is still a large demand for power batteries, which increases the volume of hybrid tractor. The parallel-type hybrid tractor can realize the requirement that both the engine and the motor participate in the balanced load of the tractor, but its power distribution and ECVT performance are still poor. The hybrid tractor effectively integrates the advantages of serial-type and parallel-type tractors, especially the multi-node arrangement of the planetary gear configuration, which is conducive to the decoupling of tractor travel speed and PTO output speed [15]; thus, the hybrid tractor has excellent power shunt and stepless transmission performance [16].

Luo [17] took the series-type hybrid tractor as the research object, proposed a fuzzy reasoning energy management strategy under the ploughing condition of heavy soil, and compared it with SOC power-following and fixed-point strategies. The results showed that the fuel economy of the whole tractor was improved by 20.92% and 23.37%, respectively. Yan et al. [18] took the plug-in hybrid tractor as the research object and proposed an adaptive rule-based energy management strategy. Under the high-frequency ploughing resistance of (30 ± 2) cm simulated by the empirical formula, the energy-saving control simulation was completed, and the equivalent fuel consumption was reduced by 8.78%. Xu et al. [19] proposed a multi-operating point fuzzy PID control strategy and completed energy-saving control simulation under ploughing (simulation of combined random numbers with ploughing depths of 12 cm and 20 cm) and transportation conditions. The results show that the multi-operating point fuzzy PID control strategy can achieve significant energy-saving effects. For tandem hybrid tractors, Li [20] proposed a thermostat-based energy management strategy optimized using a nonlinear programming genetic algorithm, which effectively reduced the cumulative fuel consumption of the engine. Francesco et al. [21] took a hybrid tractor equipped with ECVT as the research object, proposed a rule-based energy management strategy, and conducted simulations in simulated acceleration and slope conditions to verify the advantages of the hybrid tractor. Yan et al. [22] designed an energy management strategy based on a dynamic programming algorithm for serial hybrid tractors to improve the running conditions of the engine. At present, energy management strategies can be divided into two categories: rule-based and optimization-based strategies [23]. The above studies on the energy management strategies for hybrid tractors are all rule-based energy management strategies. Although rule-based energy management strategies can determine the optimal decision of the power source by setting simple switching thresholds or judgment conditions, their energy saving potential is relatively limited compared with optimization-based methods [24]. At present, optimization-based strategies have been widely used in hybrid tractors. Zhao et al. [25] took serial hybrid power as the research object, proposed an energy management strategy based on DP-MPC, and completed energy saving control under ploughing conditions (simulated ploughing depth of 20 cm), rotary ploughing conditions, and transportation conditions, thus improving the economic performance of the tractor. Zhang et al. [26] proposed an energy management strategy based on instantaneous optimization, which improved the working interval of the engine and motor of the parallel hybrid tractor. Dou et al. [27] proposed energy management based on a Markov decision process for a coupled shunt configuration under ploughing conditions (ploughing depth of 22 cm and plough width of 25 cm, using the same ploughing resistance equivalent method as in this paper), achieving high efficiency and energy savings for the whole tractor. The above studies on energy management strategies mainly focus on serial and parallel hybrid tractors, with limited research on serial-parallel hybrid tractors. Given the multi-degree-of-freedom characteristics of the planetary gear ECVT configuration, the output speed and output torque of each power component do not exhibit a single, simple linear relationship with the required rotational speed and required torque of the whole machine. Thus, more complex requirements are put forward for the optimal decision-making of an energy management strategy. Although one study in the literature [25] involves the use of a dynamic programming algorithm, its focuses solely on a series configuration and employs a single optimization method. Although a hybrid tractor equipped with ECVT has been studied in the literature [21], its energy management strategy is conventional, which fails to highlight the complex demands of the multi-freedom characteristics on the energy management strategy, and its energy saving potential is limited. Although the literature [27] has conducted some research on the strategy of a coupled shunt configuration, the Markov decision model is complex and requires a large amount of calculations.

Therefore, this paper addresses the following research objectives:

(1) Taking the OS-ECVT configuration of the power output split series-parallel hybrid tractor as the research object, the working principle and dynamic characteristics of the ECVT configuration were expounded;

(2) Under ploughing conditions, an energy management strategy for a hybrid tractor based on a dynamic programming algorithm was proposed to improve the fuel economy of hybrid tractors;

(3) The energy saving comparison strategy was proposed based on the optimal operating line of the engine. The simulation and comparative analysis of energy saving control under ploughing conditions were completed, and the effectiveness of the energy tube strategy based on the dynamic programming algorithm was verified.

2. Hybrid Tractor OS-ECVT Configuration

This paper takes the OS-ECVT configuration of a hybrid tractor as the research object, and the configuration is shown in Figure 1. This configuration was designed by our research group, and its feasibility has been verified [28].

As shown in Figure 1, the transmission part of this configuration includes planetary row K1, planetary row K2, high and low gears, a main reduction gear, a differential gear, and a hub reduction gear. The power components are mainly composed of a diesel engine, motor MG1, and motor MG2. When the tractor is operating, the engine outputs positive power, and the power output of MG1 is coaxially coupled and input to the sun gear of K2. In addition, the positive power output of MG2 is input to the sun gear of K1 and output through the tooth ring of K1. Then, the power is input to the tooth ring of K2, and the power of the engine and MG1 is input to K2. The power is output by the K2 planet carrier, so the entire system exhibits power output split performance. K2 is the epicyclic gear train. Using the multi-degree-of-freedom characteristics of K2, MG2 can adjust the engine speed in real time, so that the whole gear train has ECVT performance. Thus, when the tractor meets the same required speed, MG2 can output different rotated speeds and adjust the engine rotation speed. Therefore, the output speed of the engine can be located in the higher economic rotated speed range, and the whole hybrid power system has the function of speed decoupling. When the tractor meets the same demand torque, MG1 can consume the power of the engine or compensate the output power for the engine, generate electricity or drive, and then adjust the torque operating point of the engine. Therefore, the whole hybrid power system has the function of torque decoupling.

The parameters of the power components are shown in

Table 1. Power component parameters.

Components	Parameters	Units	Values
Diesel engine	Rated power	kW	90
	Rated rotate speed	rpm	2200
	Maximum torque	N·m	520
MG1	Rated power	kW	70
	Rated rotate speed	rpm	2400
	Rated torque	N·m	280
MG2	Rated power	kW	70
	Rated rotate speed	rpm	2400
	Rated torque	N·m	280

.

The configuration consists of four clutches and two brakes. Specifically, C1 is the engine clutch, which determines whether the engine is involved in driving. C2 is the PTO clutch to ensure the normal opening of the PTO mode. C3 and C4 are the low gear clutch and high gear clutch, respectively, which ensure the normal operation of high and low gears. Brake B1 and brake B2 mainly meet the needs of different working modes.

The configuration has four operating modes, namely pure electric mode 1, pure electric mode 2, engine direct drive mode, and hybrid mode. The relationship between the clutches and brakes under these four working modes and different working states is shown in

Table 2. The relationship between the clutches and brakes for each driving mode and the working state of power.

Driving Mode	C1	C2	B1	B2	Engine	MG1	MG2	PTO Work
1	OFF	OFF	ON	OFF	×	×	√	×
2	OFF	OFF	OFF	OFF	×	√	√	×
3	ON	OFF	OFF	ON	√	√	×	√
4	ON	OFF	OFF	OFF	√	√	√	√

1 is pure electric mode 1; 2 is pure electric mode 2; 3 is the engine direct drive mode; 4 is hybrid mode. OFF indicates disconnect. ON indicates conjugation. Here, √ indicates work; × indicates not working.

.

In particular, in the following table, MG1 is mainly used as a generator. However, when it is difficult for the engine to maintain the power needs of the tractor, MG1 can be used as a driving motor output power. At the same time, it can also be driven by the engine for idling. These three states are regarded as MG1 working, because, in these three states, MG1 is performing the torque decoupling function. It is considered not working only when it is completely powered off.

The schematics of pure electric mode 1, pure electric mode 2, engine direct drive mode, and hybrid mode are shown in Figure 2a–d, respectively. In particular, the ability to open the PTO is available in engine direct drive mode and hybrid mode, as shown in Figure 2e,f, respectively.

3. Dynamics Modeling of a Hybrid Tractor

3.1. OS-ECVT Drive System Model

In order to more easily see the principle of the OS-ECVT stepless transmission, the characteristic parameter k of the planetary gear mechanism is defined as follows:

$$k={\displaystyle \frac{z_r}{z_s}}$$

(1)

where k is the characteristic parameter of the planetary gear mechanism, z_r is the number of teeth on the ring, and z_s is the number of teeth on the sun gear.

The motion equation and torque balance equation of the planetary gear mechanism are as follows:

$$n_s+k{n}_r=\left(1+k\right)n_c$$

(2)

$$T_s:T_r:T_c=1:k:-\left(1+k\right)$$

(3)

where n_s is the rotation speed of the sun gear (rpm), n_r is the rotation speed of the ring gear (rpm), n_c is the rotation speed of the planet carrier (rpm), T_s is the sun gear torque (N·m), T_r is the ring gear torque (N·m), and T_c is the planet carrier torque (N·m).

In this paper, the ploughing condition is taken as the working condition of hybrid tractors. The relationship between the output speed of the OS-ECVT configuration and the required speed of the hybrid tractor can be satisfied as follows:

$$v_t={\displaystyle \frac{2\pi r_t{n}_{out}}{i_g{i}_0{i}_h}}$$

(4)

where v_t is the longitudinal driving speed of the tractor (km/h), r_t is the radius of the tractor driving wheel (m), n_out is the speed of configuration output shaft (rpm), i_g is the transmission ratio of the mechanical transmission, i₀ is the transmission ratio of the main reducing gear, and i_h is the transmission ratio of the hub reduction gear.

Among them, the transmission ratio of mechanical transmission i_g, the transmission ratio of the main reducing gear i₀, and the transmission ratio of the hub reduction gear i_h are defined as follows:

$$\left\{\begin{array}{l}{i}_g={\displaystyle \frac{z_{gout}}{z_{gin}}}\\ i_0={\displaystyle \frac{z_{rout}}{z_{rin}}}\\ i_h={\displaystyle \frac{z_{hout}}{z_{hin}}}\end{array}\right.$$

(5)

where z_gout is the number of driven gear teeth of mechanical transmission, z_gin is the number of driving gear teeth of mechanical transmission, z_rout is the number of driven gear teeth of the main reducing gear, z_rin is the number of driving gear teeth of the main reducing gear, z_hout is the number of driven gear teeth of the hub reduction gear, and z_hin is the number of driving gear teeth of the hub reduction gear.

The output speed of the planetary gear mechanism, that is, the output speed of the hybrid tractor OS-ECVT configuration, meets the following requirement:

$$n_{out}={\displaystyle \frac{(1+k_1)n_e+k_2{n}_{MG2}}{(1+k_1)(1+k_2)}}$$

(6)

where n_e is the engine output speed (rpm), n_MG1 is the output speed of MG1 (rpm), n_MG2 is the output speed of MG2 (rpm), k₁ is the characteristic parameter of planetary row K1, and k₂ is the characteristic parameter of planetary row K2.

According to the balance theory of vehicle traction, the tractor needs to balance the friction resistance F_f, air resistance F_w, slope resistance F_i, acceleration resistance F_j, and operation resistance in real time during field operation. The driving force F_t of the hybrid tractor meets the following requirements:

$$F_t=F_f+F_w+F_i+F_j+F_p$$

(7)

Considering that the drive axle is mainly driven by the shaft and gear, the total power output of each power component through the output end of the planetary gear mechanism will be transmitted to the wheel end due to friction, coaxiality, and other reasons, resulting in power loss, that is, efficiency loss. In this paper, the effective efficiency is equivalent to the intershaft efficiency η_t. Friction resistance F_f, air resistance F_w, slope resistance F_i, and acceleration resistance F_j meet the following requirements, separately:

$$\left\{\begin{array}{l}{F}_t={\displaystyle \frac{\left[T_e+T_{MG1}+(1+k_1)T_{MG2}\right]i_g{i}_0{i}_H{\eta }_t}{r_t}}\hfill \\ F_f=mgfcos\alpha \hfill \\ F_w={\displaystyle \frac{1}{2}}{C}_{D}A\rho {v_t}^2\hfill \\ \begin{array}{l}{F}_i=mgsin\alpha \\ F_j=\delta m{a}_t\end{array}\hfill \end{array}\right.$$

(8)

where m is the mass used by the tractor (kg), g is the acceleration of gravity (m/s²), f is the frictional resistance coefficient, α is the slope angle (°), C_D is the wind resistance coefficient, A is the windward area (m²), ρ is the air density (kg/m³), δ is the conversion coefficient of the tractor’s rotating mass and refers to the conversion coefficient needed to convert the inertia moment of the rotating mass into the inertia force of the translational mass, a_t is tractor acceleration (m/s²), η_t is drive axle intershaft efficiency, and F_p is the ploughing operation resistance (N).

In field operation, the tractor runs at a low speed, so the influence of acceleration resistance is generally ignored. Moreover, it is considered that the field environment is relatively smooth, and the output torque of the engine, MG1, and MG2 meets the following requirements:

$$\left\{\begin{array}{l}{T}_{MG1}={\displaystyle \frac{F_t{r}_t}{i_g{i}_0{i}_h\left(1+k_2\right){\eta }_t}}-T_e\hfill \\ T_{MG2}={\displaystyle \frac{k_2{F}_t{r}_t}{i_g{i}_0{i}_h\left(1+k_1\right)\left(1+k_2\right){\eta }_t}}\hfill \end{array}\right.$$

(9)

3.2. Ploughing Operation Model

In this paper, the initial ploughing data were measured with the ploughing platform for ploughing working conditions. Since the plow trolley cannot bear the operation demand of the high-power tractor, only a single plow is used to collect the reciprocating operation speed of the plow trolley. Then, a cycle speed model satisfying the operating characteristics of the tractor plow operation is established based on the stable running speed of the plow trolley. The sampling frequency of the ploughing platform is 2 times/unit time. The upper computer is designed based on LabVIEW. The upper computer transmits the measured running speed of the plough through the sensor to the upper computer, and then the ploughing speed model in the cycle operation period is obtained. The ploughing operation speed model of the acquisition vehicle and the ploughing cycle speed is shown in Figure 3.

The ploughing cycle speed model is shown in Figure 4.

Field ploughing operation resistance is described as follows:

$$F_{pl}=z·b·h·k_0$$

(10)

where F_pl is the theoretical ploughing operation resistance (N), z is the number of ploughshares (pieces), b indicates the ploughshare width (cm), h is the tilling depth (cm), and k₀ is the proportion resistance of soil (N/cm²).

In order to facilitate the study, a sinusoidal curve was used to approximate the relationship between plowing resistance and time for the high-power tractor [27], which is specifically described as follows:

$$F_p=F_{pl}+\mu F_{pl}\sin (t)$$

(11)

where μ is the soil nonuniformity rate, and F_p is the actual approximate ploughing resistance (N).

The ploughing operation parameters are shown in

Table 3. Ploughing operation parameters.

Parameters	Units	Values
Number of ploughshares	pieces	5
Tilling depth	cm	30
Ploughshare width	cm	25
Proportion resistance of soil	N/cm2	8

, and the ploughing operation resistance model is shown in Figure 5.

3.3. Diesel Engine Model

The output power of the diesel engine is related to the output speed and output torque, which meet the following requirements:

$$P_e={\displaystyle \frac{n_e{T}_e}{9549}}$$

(12)

Due to the complexity of the internal working process of the diesel engine, in order to facilitate the research, the output speed, output torque, and fuel consumption of the diesel engine are equivalent to a functional relationship based on the numerical modeling method. The fuel consumption of the diesel engine is quickly solved using the tabular method to complete the solution of engine fuel consumption [29,30,31] as follows:

$$b_e(k)=f_e(n_e,T_e)$$

(13)

$$B_e(k)={\displaystyle \frac{P_e(k)}{1000}}{\displaystyle \frac{b_e(k)}{3600{\rho }_e}}$$

(14)

where P_e is the engine output power (kW), b_e is engine fuel consumption rate (g/(kW·h)), B_e is engine fuel consumption (L), and ρ_e is diesel oil density (kg/L).

Based on a numerical modeling method, the corresponding relationship among engine output speed, output torque, and engine fuel consumption is established. The MAP of the diesel engine built is shown in Figure 6.

3.4. Motor Model

Similarly, the output speed, output speed, and efficiency of the motor are equivalent to a functional relationship [29] that is expressed as follows:

$${\eta }_m=f_m(n_m,T_m)$$

(15)

The output power of the motor meets the following requirement:

$$P_m={\displaystyle \frac{n_m{T}_m}{9549}}$$

(16)

where η_m is the motor efficiency, n_m is the output speed of the motor (rpm), T_m is the motor output torque (N·m), and P_m is the motor output power (kW).

The motor MAP constructed using a numerical modeling method is shown in Figure 7.

3.5. Power Battery Model

The first-order equivalent internal resistance model consists of only one ideal voltage source and one first-order equivalent internal resistance in series with it [32]. The model is simple and can directly describe the dynamic change process of SOC, meet the needs of a quick solution, and facilitate the quick decision of the energy management strategy. The first-order equivalent internal resistance model is shown in Figure 8.

The power balance equation of the power battery is expressed as follows:

$$P_{batt}={\displaystyle \frac{V_{oc}{I}_{batt}-I_{batt}^2{R}_i}{1000}}$$

(17)

where P_batt is the battery output power (kW), V_oc is the open circuit voltage of the battery (V), I_batt is the battery bus current (A), and R_i indicates the internal resistance of the battery (Ω).

Power battery output current meets the following requirements:

$$I_{batt}={\displaystyle \frac{V_{oc}-\sqrt{V_{oc}^2-4000R_i{P}_{batt}}}{2R_i}}$$

(18)

The SOC value of the power battery based on the Ah integral method is obtained using the following equation:

$$SOC\left(k+1\right)=SOC\left(k\right)-{\displaystyle \frac{{\displaystyle \underset{k}{\overset{k+1}{\int }}{I}_{batt}}dt}{3600Q_{batt}}}$$

(19)

where SOC(k) is SOC at time k, SOC(k + 1) is the SOC at k + 1 time, and Q_batt is the power battery capacity (A·h).

The output power of the power battery is related to the output power and working state of the motors MG1 and MG2 and is specifically expressed as follows:

$$P_{batt}={\displaystyle \frac{P_{m1}}{{\eta }_{m1}^{k_1}}}+{\displaystyle \frac{P_{m2}}{{\eta }_{m2}^{k_2}}}$$

(20)

$$\left\{\begin{array}{c}{k}_1=sign\left(n_{mg1},T_{mg1}\right)\\ k_2=sign\left(n_{mg2},T_{mg2}\right)\end{array}\right.$$

(21)

$$k_1, k_2=\left\{\begin{array}{l}1, \mathrm{discharge}\hfill \\ -1, \mathrm{charge}\hfill \end{array}\right.$$

(22)

where P_m1 is the output power of MG1 (kW), and P_m2 is the output power of MG2 (kW).

4. Energy Management Strategy for Hybrid Tractor OS-ECVT

4.1. Energy Management Strategy of Hybrid Tractor OS-ECVT Based on Dynamic Programming Algorithm

The dynamic programming algorithm, derived from the Bellman minimum principle, can perform reasonable actions on complex programming problems and is suitable for solving multi-power source and multi-stage decision problems of hybrid tractors [33]. Although it is difficult for the DP algorithm to meet the requirements of real vehicle applications, it has good applicability in the preliminary verification of configuration potential [34].

First, the working condition sequence is divided by time step ∆t, and the multi-stage decision problem is transformed into a decision problem with N time steps as follows:

$$u_k=\{u_0,u_1,\cdots ,u_{N-1}\}$$

(23)

Diesel and electric energy are consumed in the operation process of hybrid tractors. For the corresponding decision-making process at each stage, the cost of diesel and electric energy should be comprehensively considered to minimize the equivalent cost of the two energy sources. In this paper, the equivalent fuel consumption is taken as the real-time evaluation index of the energy-saving control model, which is expressed as follows:

$$\underset{\begin{array}{l}k\in [1,N]\\ h\in [1,M\}\end{array}}{J(x_{(k,h)},u_k,k)}=B_e\left(x_{(k,h)},u_k,k\right)+{\displaystyle \frac{P_b\left(x_k,u_k,k\right)C_m}{3600C_e}}$$

(24)

The energy management strategy of hybrid tractor OS-ECVT based on DP is divided into two steps, namely reverse iteration and forward optimization. In the reverse iteration step, the optimal arc control sequence of the end operating point is considered first, and the cost of the end operating point is determined as follows:

$$J_N=J(x_N,u_N)$$

(25)

Under the initial condition x₀, the cumulative cost function of iteration at time k is described as follows:

$$J_k=J_N+{\displaystyle \sum _{k=1}^{N-1}J(x_0,u_{(x_0,i)})}$$

(26)

According to the cumulative cost function, under the initial condition x₀, the optimal control decision vector satisfies the following:

$$\underset{k\in [1,N]}{u{(x_0)}_k^{*}}=\arg \min J(x_0)$$

(27)

The optimal arc cost sequence is formed by the decisions satisfying the above conditions, and then the forward optimization is completed by interpolation as follows:

$$\underset{k\in [1,N]}{u_k^{*}}=f(x_0,u{(x_0)}^{*})$$

(28)

The solution steps of the energy management strategy of hybrid tractor OS-ECVT based on dynamic programming algorithm are described as follows.

(1) Under ploughing conditions, the SOC of the power battery was selected as the state variable, and engine torque, MG2 speed, and high and low gear were selected as control variables. Under rotating ploughing condition, the SOC of the power battery was selected as state variable, and engine torque and high and low gears were selected as control variables. Discrete state variables and control variables with step sizes ∆x, ∆T, and ∆n are defined as follows:

$$\left\{\begin{array}{l}{x}_{1i}=[x_{1\min },x_{1\min }+\Delta x,\cdots x_{1\max }]\\ u_{1j}=[u_{1\min },u_{1\min }+\Delta {\rm T},\cdots u_{1\max }]\\ u_{2n}=[u_{2\min },u_{2\min }+\Delta n,\cdots u_{2\max }]\end{array}\right.\left(\begin{array}{l}(i=0,1,\cdots l)\\ (j=0,1,\cdots m)\\ (n=0,1,\cdots o)\end{array}\right)$$

(29)

(2) According to the required speed of the tractor, the output speed of the ECVT configuration in low and high gears is calculated using Equation (3). Then, the required speed of the engine is solved using Equation (4).

(3) The output torque T_out of the ECVT configuration is calculated according to the tractor demand driving force as follows:

$$T_{out}={\displaystyle \frac{F_t{r}_t}{i_g{i}_0{i}_H{\eta }_t}}$$

(30)

(4) Formula (7) calculates the demand torque of MG1 and MG2 according to the tractor demand driving force.

(5) The engine fuel consumption rate is solved using the lookup table method, and then the engine fuel consumption is calculated using Equation (12). Similarly, the efficiency of MG1 and MG2 is solved using the lookup table method. Then, the output power of the power battery is solved using Equation (17), and the output current of the power battery is solved using Equation (15). Finally, the SOC is solved.

(6) Define control variable u_k and state variable x_k to allow for the setting of Ω to ensure that the power components operate within the permissible range:

$$u_k,x_k\in \mathsf{\Omega }=\left\{\begin{array}{l}{n}_{emin}\le n_e\le n_{emax}\hfill \\ T_{emin}\le T_e\le T_{emax}\hfill \\ n_{mg1min}\le n_{mg1}\le n_{mg1max}\hfill \\ T_{mg1min}\le T_{mg1}\le T_{mg1max}\hfill \\ n_{mg2min}\le n_{mg2}\le n_{mg2max}\hfill \\ T_{mg2min}\le T_{mg2}\le T_{mg2max}\hfill \\ SO{C}_{min}\le SOC\le SO{C}_{max}\hfill \end{array}\right.$$

(31)

(7) A penalty function is introduced to punish all points that fail to meet the requirements of the working condition.

(8) The optimal control variable is obtained using Equation (24). By analogy, the optimal control variable corresponding to all initial conditions x₀ at all operating points is solved through the above solution process. Finally, the forward iteration is completed using Equation (25).

In summary, the energy management strategy of hybrid tractor OS-ECVT based on a dynamic programming algorithm is divided into two solution steps: reverse iteration and forward optimization. In the reverse iteration, ploughing condition data are first introduced and divided into N working condition sequences. Then, control variables and state variables are discretized, separately, and the indexes of control variables and state variables are defined to ensure that each discrete variable is traversed in the decision-making of each working condition point. The required engine speed, the required torque of MG1 and MG2, fuel consumption, and SOC are solved using the dynamic dynamics equation. After solving the problem, the penalty function is used to determine the possible output speed and output torque domains of the engine, MG1, and MG2 and to determine whether all discrete state variables and control variables are traversed. If so, the forward optimization is performed, and the decision of all operating points is completed by interpolation method. If not, proceed to the next discrete variable. The energy management strategy solving process of a hybrid tractor OS-ECVT based on a dynamic programming algorithm is shown in Figure 9.

4.2. Energy-Saving Comparison Strategy Design

The line of the tangent point of all engine isopower lines and isofuel consumption lines is the optimal operating line (OOL) of the engine. The energy management strategy based on the engine optimal operating line takes the engine optimal economy curve as the target working interval of the engine. This ensures that the engine runs within an economically optimal interval and effectively improves the fuel economy of the hybrid tractor [35].

In this paper, an energy management strategy based on OOL is proposed for the OS-ECVT system. By comparing the energy saving potential of DP-based and OOL-based energy management strategies, the advantages of DP-based energy management strategies are verified.

Reference [35] pertains to parallel hybrid tractors, where the energy management strategy based on OOL can achieve optimal energy distribution of the power source only by setting corresponding judgment conditions or thresholds. However, this paper uses an ECVT configuration, which has the characteristics of multiple degrees of freedom. Therefore, when ploughing conditions are applied, it is difficult to determine the corresponding decision of all operating conditions only by setting a certain judgment threshold. Thus, it is still necessary to select appropriate control variables under the premise of designing the corresponding rule judgment. Under rotating farming conditions, since the PTO required speed directly determines the engine output speed, the degrees of freedom of the ECVT configuration can be directly defined using the dynamics equation, and the engine output torque can be determined using the table method. Finally, the decision of all working conditions is obtained, so there is no need to take the power component output as the control variable. The specific solution process is described as follows.

Under ploughing conditions, the hybrid tractor is controlled to run in low gear, and the mode switching threshold is set according to the tractor running speed. Finally, the working mode of the hybrid tractor is determined as follows:

$$\left\{\begin{array}{l}{v}_{mthr1}<v_t<v_{osthr},\hspace{1em}\mathrm{pure} \mathrm{electric} \mathrm{mode} 1\\ v_{osthr}<v_t<v_{mthr2},\hspace{1em}\mathrm{hybrid} \mathrm{mode}\end{array}\right.$$

(32)

When the hybrid tractor is in pure electric mode 1, the speed and torque of MG2 are obtained using the dynamics equation according to the required speed and torque. Then, the output power of the power battery is obtained, and finally the power battery SOC is obtained. When the hybrid tractor is in hybrid mode, the MG2 speed is taken as the control variable, and the output speeds of the engine and MG1 are obtained using the dynamics equation. Then, the OOL curve is taken as the target working area, and the output torque of the engine is obtained using the lookup table method. Finally, the output torque of MG1 is solved.

5. Energy Saving Control Simulation and Result Analysis

Under the two strategies, the running speed and torque curves of the engine, MG1, and MG2 are shown in Figure 10. The engine operating point MAP is shown in Figure 11, and the engine power operating point distribution is shown in Figure 12.

As can be seen from Figure 9, compared with the OOL-based strategy, under DP-based control, MG2 can adjust the engine speed operating point in real time with a wider range of speed, thus achieving a wider range of stepless transmission performance. Thus, MG2 has a better speed decoupling function. As can be seen from Figure 10, under the control of the DP-based energy management strategy, the engine continues to run at an idle speed of ~1400 r/min. However, under the control of the OOL-based energy management strategy, the engine has fewer visible operating points on the MAP, which are basically distributed on the OOL curve. The working points of engine speed and torque based on the OOL energy management strategy are dense, and the output speed and output torque of the engine are larger. Further combined with Figure 11, it can be seen that under the control of the OOL strategy, the engine continues to output more power. On the premise that the fuel consumption rate of the engine exhibits minimal differences, the engine based on the OOL strategy needs to consume more fuel to meet the high power output requirements, which increases the fuel consumption of the engine and reduces the economy of the whole tractor. Under the control of energy management strategy based on DP, the engine works more flexibly and can take into account the economy of the whole tractor under the premise of meeting the requirements of the working conditions.

The working point MAP of the motor is shown in Figure 13. The power operating point distributions of MG1 and MG2 are shown in Figure 14 and Figure 15, respectively, and the SOC change curve is shown in Figure 16.

As shown in Figure 13, compared with the energy management strategy based on OOL, MG2 under the control of the DP-based energy tube strategy continuously outputs a larger speed, and MG2 has a higher working efficiency when performing stepless transmission. This intuitively demonstrates that under the control of the DP-based energy management strategy, the hybrid tractor has a wider range of stepless transmission performance, and the performance is better. Combined with Figure 15, it can be seen that under the control of the energy management strategy based on DP, the hybrid power system can give full play to the characteristics of low-cost electric energy, so that MG2 can output more power in real time for stepless transmission, thus making the engine output lower power to meet the requirements of the working conditions, effectively improving the fuel economy of the engine, and thus ensuring the economic performance of the whole tractor. At the same time, under the control of the energy management strategy based on OOL, due to the decoupling of engine output torque and demand torque, it can be seen from Figure 14 that MG1 has poor torque regulation function, causing it to generate power at a high and uneconomical level, which results in inefficient energy conversion. Under the control of the energy tube strategy based on DP, MG1 has a low torque adjustment function. Due to the decoupling between the engine output torque and demand torque, MG1’s torque regulation function is better, and more economical energy conversion is obtained. Combined with Figure 16, it can be seen that although the SOC based on the energy management strategy of OOL continues to rise, the uneconomical conversion power of MG1 has been utilized for power generation. The SOC changes based on the energy management strategy based on DP effectively take into account the fuel economy of the engine. There is no reliance on the engine’s uneconomical power for generation nor excessive dependence on electric energy to meet the tractor’s heavy load operation. The change in the SOC shows a steady downward trend, leading to improved overall tractor economy, with no occurrence of uneconomical secondary energy conversion. The tractor has excellent power-split performance. That is, under heavy load operation and DP-based control, if the SOC shows an upward trend, it is bound to require the engine to output more power, and this power is also uneconomical power, which will reduce the fuel economy of the whole tractor. The equivalent fuel consumption of the OS-ECVT energy management strategy based on DP is about 3.1238 L, whereas that based on OOL is about 4.2713 L. The equivalent fuel consumption of the OS-ECVT energy management strategy based on DP is approximately 26.87% lower than that of OOL strategy. Thus, the energy management strategy based on DP has a remarkable energy-saving effect.

6. Conclusions

This paper takes the OS-ECVT configuration of a high-power hybrid tractor as the research object, analyzes the working principle of this configuration, completes the dynamic modeling of the hybrid tractor, and proposes an energy management strategy for the hybrid tractor OS-ECVT based on the dynamic programming algorithm rooted in the Bellman minimum principle. The energy saving control comparison strategy was developed based on the optimal economic curve, and the energy saving control simulation was completed using the ploughing condition as the decision condition. The following results were obtained:

(1) Compared with the OOL-based energy management strategy, under the control of the DP-based energy management strategy, the hybrid tractor could fully leverage the low cost of electric energy, allowing the hybrid power system to achieve a wider range of stepless transmission performance. The speed decoupling function is better and more efficient.

(2) Under the control of the energy management strategy based on the dynamic programming algorithm, the torque adjustment function of the hybrid tractor is improved, and the phenomenon of economic secondary energy conversion appears, which greatly improves the running status of the power components. The whole tractor has excellent power-split performance.

(3) The equivalent fuel consumption of hybrid tractor OS-ECVT energy management strategy based on DP is about 3.1238 L, and the equivalent fuel consumption of the OS-ECVT energy management strategy based on OOL is about 4.2713 L. The equivalent fuel consumption of the OS-ECVT energy management strategy based on DP is about 26.87% lower than that of the OOL strategy. The energy management strategy based on DP has a remarkable energy-saving effect.

(4) At present, while the development of hybrid tractors lags behind the development of hybrid vehicles, the growing emphasis on green development needs and ongoing policy support are driving increased research in this area. Although hybrid tractors are still in the initial stages of development, their future prospects appear promising. In the future, as the hybrid technology for tractors and low-speed, high-torque motor technology mature, power distribution techniques will enhance the real-time operational efficiency of tractor engines. In addition, the cooperative work of multiple power sources will also allow hybrid tractors to easily meet the high power and heavy load requirements typically found in agricultural operations.