Optimized torque-distribution control method (OTCM) is a critical technology for front/rear axle electric wheel loader (FREWL) to improve the operation performance and energy efficiency. In the paper, a longitudinal dynamics model of FREWL is created. Based on the model, the objective functions are that the weighted sum of variance and mean of tire workload is minimal and the total motor efficiency is maximal. Four nonlinear constraint optimization algorithms, quasi-newton Lagrangian multiplier method, sequential quadratic programming, adaptive genetic algorithms, and particle swarm optimization with random weighting and natural selection, which have fast convergent rate and quick calculating speed, are used as solving solutions for objective function. The simulation results show that compared to no-control FREWL, controlled FREWL utilizes the adhesion ability better and slips less. It is obvious that controlled FREWL gains better operation performance and higher energy efficiency. The energy efficiency of FREWL in equipment transferring condition is increased by 13–29%. In addition, this paper discussed the applicability of OTCM and analyzed the reason for different simulation results of four algorithms.
Hybrid wheel loader has raised much attention due to its green technology. It is considered to be the trend of future in the loader field [
Outline of several prototypes.
Manufacturer | Powertrain configuration | Energy storage devices | Energy saving | Ref |
---|---|---|---|---|
Hitachi | Series | Battery | 25%–30% | [ |
John Deere | Series | Battery | 25% | [ |
Joy Global | — | Flywheel | 45% | [ |
Volvo | — | Battery | — | [ |
XCMG | Parallel | Hydraulic accumulator | 54% | [ |
Liu Gong | Series-parallel | Supercapacitor | — | [ |
However, the energy-saving method of these above loaders is energy management strategy [
There are many technologies demanding prompt solution about OTCM for FREWL. Enlightened by the relative research in on-road vehicle field, tire energy dissipation [
In this paper, the proposed OTCM for FREWL is to gain better operation performance and higher energy efficiency. In the primary stage of FREWL dynamics research, it is more urgent to study longitudinal dynamics than to study the lateral stability because FREWL is often operated in low speed. This paper assumes that FREWL only moves in the longitudinal direction. In Section
The distinctive transmission configuration of FREWL takes diesel generating set as main power source. Rectifier converts the alternating current generated by diesel generating set to a direct current which used to drive front motor, rear motor, and working motor. Supercapacitor is used as auxiliary source to effectively use braking energy and control diesel generating set in its high efficiency operating region. So diesel generating set can always operate in its high efficiency region. The transmission configuration of FREWL is shown in Figure
Transmission configuration of FREWL.
A brief summary of the forces and torques in longitudinal dynamics is shown in Figure
Illustration of FREWL forces and torques.
Because FREWL operates in low speed, the influence of air resistance can be ignored. The wheel vertical load is, respectively, given by
The relationship between the longitudinal force and driving torque on each tire is given by
Suppose that longitudinal force and vertical load of the wheels in the same axle are equal; the relationship between the motor driving torque and tire driving torque is given by
OTCM consists of objective function, constraint conditions, and optimization algorithm.
The nonlinear optimization problem about enhancing operating performance can be formulated in this way. Because of the characteristic that the driving torque of both motors can be controlled online, the objective function is that the weighted sum of variance and mean of tire workload is minimal [
From (
In order to improve the energy efficiency of the FREWL while transferring equipment, the objective function is that the total motor efficiency is maximal [
MAP diagram of motor.
With (
The total driving torque should satisfy expected accelerator position firstly, as is shown in
Defining surface of total driving torque of front/rear motor.
Adhesion force is influenced by wheel vertical load and tire-road friction coefficient. Because the influence of the motor inertia moment and wheel inertia moment is tiny, it is appropriate to ignore them. So in pure longitudinal slip condition, the maximum driving torque is limited by
The motor driving torque and speed should be limited as follows:
The range of driving torque-distribution coefficient
Nonlinear optimization algorithms are widely used to solve nonlinear optimization problems [
Quasi-Newton method (QNM) is a special case of Newton method. The objective function is Taylor expanded in second order at
The derivative of (
An approximate matrix
Quasi-Newton method fails to solve the nonlinear constraints optimization problems. Thus Lagrangian multiplier (LM) is introduced into the problem. Inequality constraints are transformed into equality constraints by an auxiliary variable. Then with the original equality constraints, the Lagrangian function is transformed into
The simulation parameters of the QNLM used in this paper are shown in Table
Simulation parameters of the QNLM.
Parameters | Values |
---|---|
Maximum iteration number of QNM | 200 |
Maximum iteration number of LM | 200 |
Penalty factor of LM | 5 |
Termination error of QNM |
|
Termination error of LM |
|
SQP is an efficient method for nonlinear optimization with advantages of high computational efficiency and fast convergent rate. In SQP, positive definite matrix
If the constraints
To some value function
From
The least square multiplier is calculated by
The approximate matrix
The simulation parameters of the SQP used in this paper are shown in Table
Simulation parameters of SQP.
Parameters | Values |
---|---|
Maximum iteration number | 200 |
Iteration number of subproblem | 200 |
Penalty factor | 0.05 |
Termination error |
|
Termination error of subproblem |
|
AGA is another significant and promising variant of genetic algorithms. AGA adjusts probabilities of crossover and probabilities of mutation in order to maintain the genetic model and to accelerate the convergence speed. In AGA, the evolution usually starts from a population which consisted of randomly generated individuals. By Roulette strategy, individual fitness values are evaluated to judge if it agrees with the optimization criterion. The new individuals are generated by the optimal best mutation probability
The simulation parameters of AGA used in this paper are shown in Table
Simulation parameters of the AGA.
Parameters | Values |
---|---|
Lower bound of independent variable | 1 |
Upper bound of independent variable | 0 |
Scale of population | 50 |
Maximum evolution generations | 200 |
Discrete precision of independent variable |
|
Crossover constant |
0.5 |
Crossover constant |
0.9 |
Mutation constant |
0.03 |
Mutation constant |
0.07 |
Particle swarm optimization (PSO) based on natural selection is one of the improved algorithms, which is characterized by iteratively trying to improve a candidate solution. In each iteration, the worst half of the particles in the population is replaced by the best half of the particles while preserving the original historical optimal value. Therefore, it improves the optimization ability and solving speed and significantly reduces the algorithm premature convergence situation.
The inertia weight
The simulation parameters of the PSO-RN used in this paper are shown in Table
Simulation parameters of the PSO-RN.
Parameters | Values |
---|---|
Particle population | 30 |
Acceleration constant |
2 |
Acceleration constant |
2 |
Upper boundary of inertia weight | 0.9 |
Lower boundary of inertia weight | 0.4 |
Maximum evolution generations | 100 |
Dimension of search space | 1 |
Maximum particle velocity |
0.2 |
Minimum particle velocity |
0 |
Maximum particle position |
1 |
Minimum particle position |
0 |
Conditions of traveling, spading, and stacking on bumpy road are common for wheel loader. The paper establishes these three conditions to verify the improvement of the FREWL operation performance through OTCM based on tire workload. Condition 1 simulates the traveling condition, while condition 2 simulates the spading condition. Condition 3 simulates the stacking condition in bumpy road with
Simulation parameters based on tire workload.
Condition 1 | Condition 2 | Condition 3 | |
---|---|---|---|
|
0.8 | 0.2 | 0.2 |
|
0.3 | 0.7 | 0.2 |
|
300 | 300 | 360 |
|
— | 600 | 600 |
|
— | — | 20 |
From Figures
Longitudinal slip ratio of front wheel.
Longitudinal slip ratio of rear wheel.
Another parameter to evaluate the control effect is the driving distance of the controlled FREWL and no-control FREWL on bumpy road in the same time, which is shown in Figure
Driving distance without spading resistance.
Final distance (m) | Distance increase (%) | |
---|---|---|
No-control | 16.92 | — |
SQP | 19.53 | 15.42 |
AGA | 20.08 | 18.68 |
PSO-RN | 19.54 | 15.48 |
QNLM | 19.54 | 15.48 |
Driving distance in condition 1.
Figures
Slip frequency of the front and rear wheels.
Front wheel slip times | Rear wheel slip times | |
---|---|---|
No-control | 4 | 1 |
SQP | 0 | 1 |
AGA | 2 | 4 |
PSO-RN | 0 | 1 |
QNLM | 0 | 1 |
Longitudinal slip ratio of front wheel.
Longitudinal slip ratio of rear wheel.
Figure
Forward driving time with spading resistance.
Driving time (s) | Time decrease (%) | |
---|---|---|
No-control | 15.42 | — |
SQP | 3.98 | 74.19 |
AGA | 8.42 | 45.39 |
PSO-RN | 3.92 | 74.59 |
QNLM | 3.96 | 74.32 |
Driving distance in condition 2.
Condition 3 simulates stacking condition that the FREWL operates on the bumpy road with 20° slope and encounters a continuous spading resistance after 5 s, and is gradually heavy-loaded. Compared to conditions 1 and 2, condition 3 is more complicated. Figures
Longitudinal slip ratio of front wheel.
Longitudinal slip ratio of rear wheel.
Figure
Driving distance in condition 3.
Two most common equipment transferring conditions are driving straightly on the bituminous road and reciprocating on bumpy road. This paper sets two conditions to simulate the improvement of energy efficiency of FREWL by OTCM while transferring equipment. The simulation parameters are listed in Table
Simulation parameters based on total motor efficiency.
Condition 4 | Condition 5 | |
---|---|---|
Motion state | Straight driving | Reciprocating driving |
|
0.8 | 0.6 |
|
0.8 | 0.6 |
|
0 | 10 |
|
100 | 200 |
Figure
Total motor efficiency in straight driving.
Maximum total motor efficiency (%) | Efficiency increase (%) | |
---|---|---|
No-control | 43.63 | — |
SQP | 49.58 | 13.64 |
AGA | 49.57 | 13.48 |
PSO-RN | 50.12 | 14.86 |
Total motor efficiency in condition 4.
Figures
Total motor efficiency in reciprocating driving.
Forward-total motor efficiency (%) | Forward-efficiency increase (%) | Backward-total motor efficiency (%) | Backward-efficiency increase (%) | |
---|---|---|---|---|
No-control | 46.53 | — | 28.61 | — |
SQP | 53.62 | 15.24 | 36.02 | 25.90 |
AGA | 51.68 | 11.07 | 36.94 | 29.12 |
PSO-RN | 52.25 | 12.29 | 34.52 | 20.66 |
Longitudinal speed in condition 5.
Total motor efficiency in condition 5.
Simulation time is an essential factor to affect the practicability of the method. Table
Simulation time of OTCM on different conditions.
QNLM (s) | SQP (s) | AGA (min) | PSO-RN (min) | |
---|---|---|---|---|
Condition 1 | 32.7 | 55.6 | 69.2 | 27.3 |
Condition 2 | 23.5 | 36.1 | 88.3 | 54.8 |
Condition 3 | 32.1 | 41.6 | 64.9 | 29.5 |
Condition 4 | — | 17.9 | 52.7 | 15.7 |
Condition 5 | — | 13.9 | 54.7 | 18.1 |
In this paper, we study the OTCM of FREWL and prove that OTCM can improve the operation performance and energy efficiency of FREWL through five simulation conditions. In addition to five simulation conditions mentioned in this paper, this method is also adaptable to other operation and equipment transferring conditions of FREWL in longitudinal motion. The changed parameters are shown in Table
Parameters changed in other conditions.
|
|
|
|
| |
---|---|---|---|---|---|
Heavy-haul transportation | ✓ | — | ✓ | — | — |
Loading/unloading material | ✓ | — | ✓ | ✓ | ✓ |
Bulldozing | ✓ | — | ✓ | ✓ | ✓ |
Climbing | — | ✓ | ✓ | — | — |
As can be seen from Table
In the simulation case, compared to no-control FREWL, the FREWL controlled by four optimization algorithms have a great increase in the operation performance and energy efficiency. In the OTCM based on tire workload, the QNLM and SQP optimization solutions are almost identical, because their core algorithm is the BFGS algorithm, as shown in (
However, although the AGA has a better performance, it has the longest computing time because the movement of the whole population is more evenly moving to the optimal region. QNLM and SQP compute faster because they apply the traditional BFGS method. Among them, SQP needs to solve a quadratic programming subproblem at each iteration step, so the calculation time is longer than QNLM.
OTCM is a critical technology to improve the operation performance and energy efficiency of FREWL. The driving torque of front motor and rear motor of FREWL can be controlled independently. The objective function minimizes the weighted sum of variance and mean value of tire workload and maximizes the total motor efficiency. The results show that the operation performance and energy efficiency are obviously improved by OTCM. While the FREWL is operating, the frequency of slip is obviously reduced, and the adhesion ability is improved. While the FREWL is driving straightly in equipment transferring, total motor efficiency is improved by 14.86% at most. While the FREWL is driving reciprocally in equipment transferring, total motor efficiency is improved by 29.12% at most. Considering the simulation results and simulation time comprehensively, SQP is the most suitable one of the four optimization algorithms for field test of FREWL.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors acknowledge the funding support from National Natural Science Foundation of China, no. 51375202, and also acknowledge the support of Programs for Science Research and Development of Jilin Province, no. 20160101285JC.