#### Abstract

Research on the efficiency characteristic of the hydromechanical continuous variable transmission (HMCVT) in tractors is key to obtaining optimal transmission, developing control strategies, and assessing efficiency. To ease and improve the accuracy of obtaining the efficiency completely based on test measurements or theoretical calculation, this study proposes a method for building the HMCVT efficiency model. The method is based on an improved simulated annealing (SA) algorithm according to a small amount of test data. The study uses 8 groups of transmission efficiency values under different operating conditions obtained from bench tests. By theoretical analysis of the HMCVT, this study divides the total transmission efficiency into (i) the transmission efficiency from the output power of the power source to the confluence mechanism, (ii) the transmission efficiency of the confluence mechanism, and (iii) the transmission efficiency of the output part after confluence. The formulas for the three parts of transmission efficiencies are then derived. This study improves the SA algorithm and uses it to identify the three key parameters of hydraulic systems of the transmission efficiency calculation model. Research results indicate that the efficiency model built using the proposed method exhibits high accuracy with an error of about 1.90%. The improved SA algorithm can rapidly complete key parameter identification with an error of about 2.16%; when the displacement ratio is 0, the efficiency values at the same stage are approximately equal under different operating conditions. The HMCVT efficiency model can be built rapidly and effectively with only five groups of efficiency measurement tests.

#### 1. Introduction

Facing complicated working requirements, tractors are necessary to complete heavy-load work, such as tillage and soil preparation and taking delivery of goods, as well as small-medium-load work, such as fertilization and sowing [1–3]. Therefore, tractors require as many operating stages as possible. The hydromechanical continuous variable transmission (HMCVT) can realize free changes in the transmission ratio of tractors while inheriting the high efficiency of the mechanical transmission and high power of the hydraulic transmission [4–6]. As use of tractors increases, research regarding the HMCVT becomes significant [7–10].

Research on the HMCVT efficiency characteristic is key to achieving the optimal transmission ratio of variable speed, developing variable speed control strategies, and assessing the efficiency of the HMCVT design. The HMCVT efficiency characteristic is related to its rotating speed, load, and discharge ratio (or the transmission ratio). To obtain the HMCVT efficiency, numerous tests have to be conducted because of the multiple factors involved [11]. The theoretical calculation of the HMCVT may be a quick process, and the HMCVT efficiency may be easily determined under any operating conditions; however, its accuracy is difficult to ensure.

Studies have rarely been conducted on the total efficiency of the HMCVT. Zhang et al. [12] considered the power cycle and efficiency of the hydraulic unit and developed a formula for determining the HMCVT efficiency; however, the method they proposed was complicated, required experimental research on hydraulic systems, and lacked experimental verification of the total HMCVT efficiency. Guangming Wang et al. [13] established an HMCVT model based on Simulation X, calibrated the model according to experimental data, and then built the efficiency model; however, they calibrated the parameters by repeated manual corrections and required building a model by using Simulation X. Zhang [14, 15] and Xu [16, 17] proposed a method for calculating the HMCVT efficiency by combining the power cycle and efficiency of the hydraulic route; however, the model of the hydraulic route efficiency involved 8 unknown parameters that were difficult to determine, and the calculation process was complex.

To address the aforementioned problems, this study proposes a method for developing the HMCVT efficiency model. The method determines 8 groups of transmission efficiency values under different operating conditions by using bench tests. In accordance with the theoretical analysis of the HMCVT efficiency, the method adopts the improved SA algorithm to identify the key parameters of the calculation model and thereby establish the HMCVT efficiency model. Specifically, the method divides the complex transmission efficiency into three parts: the transmission efficiency from the output power of a power source to the confluence mechanism, the transmission efficiency of the confluence mechanism, and the transmission efficiency of the output part after confluence. The method improves the SA algorithm and uses the algorithm to identify the key parameters of the hydraulic system of the transmission efficiency calculation model. The method proposed in this study for building the transmission efficiency model can determine the HMCVT efficiency under any operating conditions and thus provides a basis for the design and research of the HMCVT as well as the research and development of control strategies.

#### 2. Testing of the HMCVT Efficiency Characteristic

##### 2.1. HMCVT Operating Principles

The HMCVT inputs the power output from the power source into the pump-motor hydraulic transmission system and the gear-engagement mechanical transmission system. After coupling of the hydraulic part and the mechanical part of the transmission power via the confluence mechanism, the HMCVT outputs the power [18, 19]. The hydraulic system can realize the continuous variation of the speed ratio in accordance with the changes in the displacement ratio, prompting the HMCVT to realize the continuous variation of the transmission ratio within a certain range. The HMCVT under study has 1 pure hydraulic stage H_{0}, 4 hydromechanical continuous variable speed stages HM_{1}–HM_{4}, 12 gear pairs, and 5 wet clutches c_{0}–c_{4}. Figure 1 illustrates the transmission principles of the HMCVT.

Table 1 lists the HMCVT parameters.

##### 2.2. Measurement Uncertainty of Transmission Efficiency

To meet the required maximum engine speed of < 2400 r/min (Weichai WP6T180E21), we use a JC3A rotational speed and torque sensor (XiangYi Dynamic Test Instruments Co., Ltd.) as the test bench, with a rotational speed of 0–3000 r/min and a torque range of 0–5000 . In accordance with the operations manual provided by the manufacturer, the sensor errors in torque and rotational speed measurement are not greater than 10 and 1 r/min, respectively.

Our analysis indicates that the measurement error of the HMCVT transmission efficiency is determined using the JC3A rotational speed and torque sensor. The measurement uncertainty is the uncertainty of measurement results caused by random changes in factors, such as the experimental conditions during repeated measurement and the accuracy of the measuring instrument [20]. The current study evaluates the measurement uncertainty of the HMCVT efficiency by using two methods: statistical analysis of the observation columns (uncertainty evaluation caused by random effect) and empirical evaluation based on the assumed probability distribution of other information (the uncertainty evaluation caused by the system).

We measure the torque and the rotational speed with the JC3A rotational speed and torque sensor repeatedly (10 times) under similar operating conditions. On the basis of the measurements, we obtain the following data after calculation: the mean of torque measurements is 100.4720 , and the standard deviation is 0.8889 . The mean of rotational speed measurements is 78.6820 rad/s, and the standard deviation is 0.5309 rad/s. The formula for uncertainty evaluation caused by a random effect is as follows [21]: where is the measurement uncertainly caused by a random effect, is the mean of the measurements, is the standard deviation of measurements, is the number of measurements, and is the value of the measurement.

We determine by calculation that the measurement uncertainties caused by the random effect of torque and rotational speed are 0.88% and 0.67%, respectively.

The following is the calculation formula for the uncertainty evaluation attributed to a system effect:where is the uncertainty caused by the system effect, is the maximum measurement effort of the instrument, and is related to the probability statistical distribution law of error with a value of because of the uniform distribution satisfied by the instrument used in the study.

We know by calculation that the measurement uncertainties caused by the system effect of the torque and rotational speed are 0.12% and 0.019%, respectively.

The four aforementioned uncertainties are independent and uncorrelated; thus, the total uncertainty of the HMCVT efficiency measurement is where is the total uncertainty of the transmission efficiency measurement; , , , and are the measurement uncertainties caused by the random effect of torque and rotational speed and the measurement uncertainties caused by the system effect of torque and rotational speed.

Finally, using formula (3), we determine that the total uncertainty of the HMCVT efficiency is 1.12%. The HMCVT efficiency obtained in this study exhibits high accuracy with minimal errors. The errors resulting from the subsequent parameter identification are also small.

##### 2.3. Transmission Efficiency Measurement Testing

We can obtain the total transmission efficiency of the whole system by measuring the rotational speed and torque of the engine output end as well as the rotational speed and torque of the transmission output end by using the bench test. Figure 2 shows the test bench of the HMCVT.

We choose stage HM_{4} for the test. Limited by the test cost and conditions of the test site and considering that this study proposes the method to build a high-accuracy calculation model for the HMCVT efficiency with a small amount of test data, this study only measures 8 groups of HMCVT efficiencies under different operating conditions and ensures the accuracy of each group of test data. Table 2 lists the results of the transmission efficiency tests.

#### 3. New Calculation Model for HMCVT Efficiency

Considering that the bench test uses the HMCVT stage HM_{4}, we only conduct a theoretical analysis of the stage and build a transmission efficiency calculation model of the stage. The calculation models of other stages or other HMCVTs can be obtained in a similar way.

With regard to the HMCVT used in the study, the power of stage HM_{4} after being output from the engine in one route is transmitted to the variable-pump-constant-displacement-motor system and then output into the confluence mechanism via the sun gear of the planetary row* k*_{1}; in the other route, the power of stage HM_{4} is transmitted completely by the gear to the planetary carrier of the planetary row* k*_{1}. This study omits the transmission loss in the clutch. The following is the calculation formula for solving the rotational speed of the planetary carrier, gear ring, and sun gear of the confluence mechanism [22–24]:where is the output rotational speed of the gear ring, is the input rotational speed of the planetary carrier, and is the input rotational speed of the sun gear.

The transmission ratio of Stage HM_{4} is expressed as

According to [11], the formula for determining , the pressure difference between the inlet and outlet oil circuits of the constant displacement motor, iswhere (Mpa) is the pressure difference between the inlet and outlet oil circuits of the constant displacement motor; () is the torque of the motor output end; () is the viscous resistance torque of the motor output shaft; (cm^{3}/r) is the specified discharge of the motor; is the torque of the HMCVT output end; is the viscous damping coefficient; and (r/min) is the engine speed.

The HMCVT has a complex structure. Accordingly, we divide the HMCVT efficiency into three tandem parts in accordance with the transmission method of the engine output power for convenience in calculating the transmission efficiency.

Part 1 is , the transmission efficiency from the output power of the engine to the confluence mechanism. For convenience in calculation, the transmission loss in the hydraulic route is not considered in calculating Part 1; however, it is taken into consideration in Part 2. Therefore, is determined using the following: where is the percentage of the hydraulic route input power in the total output power of the engine and and are the transmission efficiencies of the corresponding gear pairs.

The HMCVT in the study is of the output split type. Thus, , the split ratio of the hydraulic power of the confluence mechanism, is the ratio of the input power transmitted to the planetary row via the hydraulic route to the output power of the gear ring. The HMCVT is determined using the following:where is the connecting characteristic parameter given by . According to the definition of the hydraulic power split ratio , where (W) is the total output power of the engine; and are the transmission efficiencies of the corresponding gear pairs; is the transmission efficiency of the pump-motor system; and (W) is the output power of the gear ring.

By analysis of the input power of the planetary carrier of the confluence mechanism, where is the transmission efficiency of the confluence mechanism.

In accordance with formulas (8), (9), and (10), is derived using the following:

Part 2 is the transmission efficiency of the confluence mechanism. The loss caused by the circulation power is the major consideration in this part of the efficiency. According to the analysis, a hydraulic power split ratio indicates a hydraulic circulation power, whereas indicates that the system is in a hydraulic–mechanical split-transmission state and thus has no circulation power. is given by where is the tandem transmission efficiency of the pump-motor system, gear pair 4, and gear pair 5, and .

The loss of the hydraulic route in Part 1 mainly consists of the transmission losses of gear pair 4 and gear pair 5. For convenience in calculation, we cascade the pump-motor system, gear pair 4, and gear pair 5 into an integral whole and integrate the transmission loss into the calculation model of Part 2.

As shown in formula (12), an important influencing factor of the confluence mechanism transmission efficiency is the transmission efficiency of the pump-motor system. To build a reliable pump-motor hydraulic system model, we use the following formulas [25]:where and (m^{3}/rad) are the specified discharges of the variable pump and the constant discharge motor, respectively; and (rad/s) are the angular velocities of the variable pump and the constant discharge motor, respectively; and () are the shaft torques of the variable pump and the constant discharge motor; is the total leakage coefficient; (m^{3}) is the dynamic viscosity of the hydraulic oil; is the working volume of the hydraulic oil; (Pa) is the bulk modulus of the elasticity of the hydraulic oil; and () are the shaft rotational inertias of the variable pump and the constant displacement motor; is 0.0049 and is 0.0054; and () are the viscous damping coefficients of the variable pump and the constant displacement motor; and are the mechanical friction loss coefficients of the variable pump and the constant displacement motor, which exert insignificant effects and have values equal to 0.01 in this study.

The HMCVT efficiency in this study is not the transmission efficiency at some instant stages but one of the entire system becoming stable under certain operating conditions. Thus, the quantities in formulas (13), (14), and (15) for time differentiation can be disregarded. In this case, the parameters to be identified in the variable-pump-constant-discharge-motor system are , , and .

Formulas (13), (14), and (15) can be used to derive the rotational speed and rotational torque c of the variable pump, as well as rotational speed and rotational torque of the constant displacement motor. The formula for the transmission efficiency of the pump-motor system is given by

Part 3 is , the transmission efficiency output from the confluence mechanism to the output part of the HMCVT. The transmission loss in this part is completely the transmission loss of the gear engagement. The formula for is as follows: where and are the transmission efficiencies of the corresponding gear pairs.

In accordance with the aforementioned formulas, the calculation model of the HMCVT total transmission efficiency is as follows:

The calculation model includes the three parameters to be identified: , , and . The parameters should be identified according to the specific test data to ensure the accuracy of the transmission efficiency calculation model.

#### 4. Parameter Identification Based on Improved SA Algorithm and Result Analysis

##### 4.1. Enhancements in SA Algorithm

The SA algorithm is a heuristic intelligent optimization algorithm with fast convergence [26–28]. The concept of SA was developed from the principle of solid annealing. The following are the general steps of the SA algorithm. First, the following are optimized: initial temperature , , the temperature reduction coefficient in each iteration, the final temperature and , and the initial value of the parameter to be optimized. The objective function needed for algorithm iteration is then determined, and the initial value of the parameter to be optimized is substituted to obtain the initially optimal objective function value. The initial parameter is assigned a certain disturbance value to obtain the new parameter value , which is then substituted in the calculation to obtain the new objective function value. , the difference between the objective function value obtained from the calculation and the optimal objective function value in the previous generation, is determined. If , is accepted; if , is accepted with a probability of , where is the current temperature. The aforementioned process is repeated continuously until the temperature is reduced to the final temperature . The algorithm ends the iteration and outputs the optimal parameter value and the optimal objective function value.

The system with parameters to be identified in this study is complex. To improve the rate and accuracy of convergence of the SA algorithm and prevent prematurity, this study improves the SA algorithm as follows.

(1) Change the end conditions of the algorithm and set a threshold value . If the objective function value is smaller than the threshold value in a generation, the algorithm ends the iteration.

(2) Improve the disturbing function of the SA algorithm by using the non-uniform mutation of the genetic algorithm for reference, aimed at providing the algorithm iteration a large search space in the initial stages and a strong local search ability in the later stages. The new disturbing function is as follows:where is the value of the parameter of the generation to be optimized; is 0.1 times the definitional domain of the parameter to be optimized; is the theoretical maximum number of iterations at temperatures from to .

(3) To enhance the search ability of the algorithm in the evolutionary process, the temperature reduction coefficient is expected to be the maximum value at the stage where the optimal objective function value changes; the temperature reduction coefficient is the minimum value at the stage where the optimal objective function value remains unchanged. The value of is set as follows: where is the difference between the objective function value of the generation and the optimal objective function value of the previous generation.

(4) To improve the parameter identification efficiency and reduce the complexity of the calculation, a new parameter to be identified was introduced by piecewise parameter identification using Magic Formula for [29]. The transmission efficiency of the pump-motor system was introduced to the three aforementioned parameters, increasing the total number of parameters that have to be identified to four. Parameter identification is then divided into two parts: Part 1 to identify the parameter and Part 2 to the identify parameters , , and .

##### 4.2. Parameter Identification Results and Analysis

This study uses the improved SA algorithm for the two-part parameter identification. In Table 2, the data of tests 1, 2, 4, 5, and 6 are used as substitution data in the objective function of optimization algorithm for parameter identification; the data of tests 3, 7, and 8 are used as the data to verify parameter identification results.

The test measurements of 6, 7, and 8 in Table 2 show that when the displacement ratio , the HMCVT total transmission efficiencies at the same stage are almost equal under different operating conditions. When the displacement ratio , the constant displacement motor does not apply work externally; thus, the theoretical transmission efficiency of the pump-motor system is . However, in the numerical calculation model, 0 is a special value causing the failure of many formulas under normal operating conditions. If the displacement ratio , the HMCVT transmission efficiency is also 0 in accordance with formula (12). This result is not consistent with the actual result because although the hydraulic part does not apply work externally, the mechanical part transmits power all the time. In engineering, it is generally considered as ; thus, this study proposes that when the displacement ratio , then the value of is any nonzero positive in the range of . Using formula (12), we know that the transmission efficiency of the confluence mechanism tallies with the actual test data.

The actual transmission efficiency of the gear is about 0.96–0.99. To evaluate the effect of gear efficiency on the HMCVT transmission efficiency model, this study chooses gear transmission efficiency = 1, 0.99, 0.98, 0.97, and 0.96, by using the enumeration method for the parameter identification test. We adopt the improved SA and use* RSQ* (coefficient of determination) as the objective function for the iteration of the algorithm. The closer the value of* RSQ* to 1, the higher the parameter identification accuracy. The formula for* RSQ* is given bywhere is the predicted value of the HMCVT transmission efficiency; is the measured value of the HMCVT transmission efficiency; represents the group of the experimental measurements (displacement ratio, angular velocity of the engine output shaft, and load torque); and is the total number of measured values.

Table 3 lists the HMCVT transmission efficiency results after parameter identification given different gear transmission efficiencies.

Column 2 of Table 3 shows the average error of the transmission efficiencies, obtained after parameter identification with a certain gear transmission efficiency and all test measurement values (8 groups in total). Column 3 shows the average error for the transmission efficiencies obtained after parameter identification, given a certain gear transmission efficiency and the test measurement values involved in parameter identification (tests 1, 2, 4, 5, and 6). Column 4 shows the average error of the transmission efficiency values obtained after parameter identification and the test measurement values not involved in parameter identification (tests 3, 7, and 8).

Given the relatively small error of gear transmission [30, 31], researchers generally consider the loss of gear transmission negligible and consider the gear transmission efficiency to be approximately equal to 1. Parameter identification results indicate that the transmission efficiency of the gear pair largely influences the accuracy of the transmission efficiency model. Table 3 shows that the gear transmission efficiency in the HMCVT efficiency calculation model should be 0.98 or 0.97.

When the gear transmission efficiency is 0.98, the identification results for , , and are , 0.0333, and 0.5613, respectively; when the gear transmission efficiency is 0.97, the identification results for , , and are , 0.0311, and 0.2934, respectively. Table 4 presents a comparison of the estimated values and actual values of the HMCVT transmission efficiency.

Table 4 shows that when the gear transmission efficiency , the estimated value is closer to the actual value (the average value is about 83.4%); when the displacement ratio , this value is used as the calculation parameter of the HMCVT transmission efficiency model. We choose the load power of 40, 60, and 80 kW and then determine the HMCVT total transmission efficiency. The engine speed changes from the minimum stable speed of 750 r/min to the specified speed of 2200 r/min, and the range of the HMCVT transmission ratio when the displacement ratio changes from -1 to +1 under the three operating conditions, as shown in Figures 3–5.

To verify the effect and computational complexity of the improved SA algorithm for parameter identification of the HMCVT transmission efficiency, this study chooses the standard genetic algorithm (GA) and standard particle swarm optimization (PSO) for comparison testing. Both algorithms are heuristic intelligent optimization algorithms that are commonly used in scientific research [32–34]. The test for parameter identification effect chooses the gear transmission efficiency of 0.98. A Lenovo computer with Intel(R) Core(TM) i5-4590 processor @3.30 GHz and 8 GB memory is used in the current study. Both standard PSO and standard GA adopt 20 particles, and the number of iterations is set as 50. Figure 6 shows the test results.

Figure 6 shows that the standard PSO and the standard GA are within the area of local optimum. The globally optimal value is slowly determined as the number of iteration increases. Meanwhile, the improved SA quickly locates the area of the global optimum and finally converges to the global optimal value. Specifically, the improved SA spends 0.21 s on 12 iterations, with a transmission efficiency error of 0.00039; the standard GA spends 0.47 s on 50 iterations, with a transmission efficiency error of 0.0039; the standard PSO spends 0.36 s on 50 iterations with a transmission efficiency error of 0.0042. Therefore, the improved SA used in the study exhibits high convergence accuracy and uses less time for parameter identification.

#### 5. Conclusions

(1) This study proposes a method for building the HMCVT efficiency model. Research results show that the method only requires 5 groups for measurement testing of the HMCVT transmission efficiency under different operating conditions in order to build a model for calculating the HMCVT transmission efficiency quickly and effectively by parameter identification. The estimation error of the transmission efficiency is about 1.90%.

(2) The SA algorithm enhancements contributed by this study lead to desirable results. The two-part parameter identification reduces the complexity of the calculation. The algorithm exhibits no prematurity during parameter identification and can effectively converge to the global optimal solution of the parameter to be identified. The average error of the parameter identification results is about 2.16%.

(3) At displacement ratio , the transmission efficiency of the pump-motor system should be any nonzero positive <. The calculation results of the proposed HMCVT transmission efficiency model and the measurement results of the bench test indicate that, at displacement ratio , the total transmission efficiencies of the HMCVT at the same stage are approximately equal under different operating conditions.

(4) The gear transmission loss is small, but the gear transmission efficiency significantly affects the accuracy of the HMCVT transmission efficiency model when it ranges from 0.96 to 1. The results show that the optimal gear transmission efficiency is 0.98.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This paper is supported by the National Key Research and Development Plan (2016YFD0701103) and Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX17_0647).