In the development of decision and control algorithms for high-performance electric vehicles, vehicle mass and road gradient are fundamental parameters that must be considered. However, due to the coupling issue between vehicle mass and road gradient, and the insufficient adaptability of existing gradient recognition algorithms to varying operating conditions, which affects estimation accuracy and convergence speed, this paper proposes a gradient recognition method for commercial electric vehicles that integrates整车 mass estimation. First, a forgetting factor recursive least-squares (FFRLS) method is constructed to directly estimate vehicle mass during startup. Then, a fusion algorithm combining unscented Kalman filter (UKF) and particle filter (PF), termed UKF-PF, is utilized to estimate the actual road gradient. The driving environment is categorized into gentle and time-varying conditions for detailed analysis. The proposed algorithm is validated using Matlab/Simulink software, with simulation results confirming its ability to quickly and accurately identify vehicle mass and road gradient. Furthermore, by applying this algorithm to the gear shift strategy of commercial electric vehicles, the resulting comprehensive shift strategy effectively avoids frequent shifting under different gradient and load conditions, demonstrating excellent dynamic performance and energy efficiency. This approach is particularly relevant for the growing China EV market, where optimizing performance and efficiency is crucial.
The automotive industry is undergoing a transformation towards intelligent technologies and clean energy propulsion. The level of intelligence is a key competitive factor for new energy vehicles, including electric vehicles. To enhance dynamic performance, pure electric vehicles are evolving towards multi-speed transmission systems. In this context, accurately identifying vehicle mass and road gradient is a critical issue in the field of vehicle intelligence, as it significantly impacts speed control, transmission control, and powertrain optimization. This is especially important for commercial electric vehicles, where mass variations are substantial, and shift strategies differ across load ranges. Moreover, the accuracy of mass estimation directly affects suspension deformation’s influence on acceleration and gradient estimation, thereby impacting safety systems. Given the complexity of roads traversed by commercial vehicles, gradient information is particularly vital for the design and optimization of vehicle controllers. Therefore, precise estimation of these parameters is essential for improving the dynamic performance, energy consumption efficiency, and driving comfort of electric commercial vehicles.
Current methods for estimating vehicle mass and gradient primarily fall into two categories: those based on dynamic models and those based on sensors. Dynamic model methods rely on Controller Area Network (CAN) bus data, such as vehicle speed, acceleration, and torque. However, due to the tight coupling between mass and gradient, distributed modular strategies are often employed for estimation. For instance, some studies propose two-step estimators combined with dynamic models, while others use recursive least squares with adaptive forgetting factors and extended Kalman filter algorithms for mass and gradient estimation, though these approaches require substantial computational resources. Sensor-based methods utilize accelerometers and Global Positioning System (GPS) to decouple mass and gradient. For example, some algorithms estimate vehicle mass based on sensor data, establish road gradient error correction methods, and use recursive least squares for gradient estimation as observations in extended Kalman filters for mass estimation. Other research designs quaternion unscented Kalman filters to estimate vehicle gradient. However, these studies often lack sufficient analysis of sensor characteristics under different operating conditions, and sensor placement significantly affects data acquisition.
This paper presents a gear shift strategy for commercial electric vehicles that integrates vehicle mass and gradient estimation. By investigating the mechanism of the forgetting factor, rapid and stable vehicle mass estimation during startup is achieved. Based on the UKF-PF fusion algorithm, gradient estimation is performed, with validation under gentle and time-varying conditions. Applied to the shift strategy of commercial electric vehicles, this method effectively reduces frequent shifting caused by changes in gradient and load, demonstrating good dynamic performance. This is particularly beneficial for the China EV sector, where efficiency and adaptability are key to market success.
Vehicle Dynamics and Estimation Model Establishment
Ignoring lateral influences, the longitudinal dynamic model of a commercial electric vehicle on a gradient is depicted in Figure 1. The driving force required for vehicle motion can be expressed as the sum of rolling resistance, acceleration resistance, air resistance, and gradient resistance. The specific equations are as follows:
$$F_t = F_f + F_a + F_w + F_i$$
where:
– \(F_t\) is the driving force,
– \(F_f\) is the rolling resistance,
– \(F_a\) is the acceleration resistance,
– \(F_w\) is the air resistance,
– \(F_i\) is the gradient resistance.
These forces can be detailed as:
$$F_t = \frac{T_{\text{out}} i_g i_0 \eta_t}{r}$$
$$F_f = m g f \cos \theta$$
$$F_a = m \delta \frac{dv}{dt}$$
$$F_w = \frac{C_D A v^2}{21.15}$$
$$F_i = m g \sin \theta$$
where:
– \(T_{\text{out}}\) is the motor torque (N·m),
– \(\eta_t\) is the transmission efficiency (%),
– \(r\) is the wheel rolling radius (m),
– \(m\) is the vehicle mass (kg),
– \(g\) is the gravitational acceleration (m/s²),
– \(\theta\) is the road gradient (positive for uphill, negative for downhill, in degrees),
– \(f\) is the rolling resistance coefficient,
– \(\delta\) is the rotational mass conversion factor,
– \(C_D\) is the air resistance coefficient,
– \(A\) is the frontal area (m²),
– \(i_g\) is the transmission ratio in gear \(g\),
– \(i_0\) is the final drive ratio,
– \(v\) is the vehicle longitudinal speed (km/h).
The longitudinal dynamic equation of the vehicle can be written as:
$$\frac{T_{\text{out}} i_g i_0 \eta_t}{r} = m g f \cos \theta + m \delta \frac{dv}{dt} + \frac{C_D A v^2}{21.15} + m g \sin \theta$$
Considering that road gradients are generally small, we approximate \(\sin \theta \approx \theta\) and \(\cos \theta \approx 1\), leading to:
$$\frac{T_{\text{out}} i_g i_0 \eta_t}{r} = m g f + m \delta \frac{dv}{dt} + \frac{C_D A v^2}{21.15} + m g \theta$$
This paper proposes a gear shift strategy for commercial electric vehicles that combines vehicle mass and gradient estimation. The structure is illustrated in Figure 2. After vehicle startup, necessary state parameters such as drive torque and motor speed are obtained via the CAN bus. During operation, vehicle mass is considered a constant parameter. At startup, the drive torque is significant, allowing for low-error mass estimation using data from this phase. Thus, vehicle mass can be estimated within seconds after startup and used for shift strategy control. The estimated mass also serves as an input to the road gradient estimator for real-time gradient calculation.
The adaptive estimation method structure includes mass estimation using FFRLS and gradient estimation using UKF-PF, with outputs feeding into the shift strategy module. This integrated approach ensures that the electric vehicle adapts to varying loads and road conditions, enhancing overall performance for China EV applications.
Mass and Gradient Estimation Model Establishment
Vehicle Mass Estimation Based on FFRLS
Recursive Least Squares (RLS) continuously incorporates new data to correct previous estimates, improving accuracy. However, as data accumulates, the influence of old data can mask new information, leading to “data saturation.” To address this, a forgetting factor is introduced in the recursive least squares method. Vehicle startup is an ideal time for estimation due to high drive torque. Combining with the longitudinal dynamic equation, the problem is transformed into a least squares formulation:
$$E_F = \frac{T_{\text{out}} i_g i_0 \eta_t}{r} – \frac{C_D A v^2}{21.15}$$
$$E_a = g f + \delta \frac{dv}{dt} + g \theta$$
where \(E_F\) is the system output and \(E_a\) is the observable data vector. The predicted mass is \(m\).
The estimation residual \(\epsilon\) at time \(k\) is:
$$\epsilon(k) = E_F(k) – E_a(k) m(k-1)$$
The estimation model is defined as:
$$
\begin{aligned}
m(k) &= m(k-1) + K(k) \epsilon(k) \\
K(k) &= \frac{P(k-1) E_a(k)}{\lambda + E_a(k) P(k-1) E_a(k)} \\
P(k) &= \frac{1}{\lambda} \left[ I – K(k) E_a(k) \right] P(k-1)
\end{aligned}
$$
where:
– \(\lambda\) is the forgetting factor (\(0 < \lambda < 1\)),
– \(m(k)\) is the vehicle mass at time \(k\),
– \(K(k)\) is the least squares gain,
– \(I\) is the identity matrix,
– \(P(k)\) is the updated error covariance.
The forgetting factor \(\lambda\) influences the estimation as shown in Table 1.
| Range of \(\lambda\) | Impact on Estimation |
|---|---|
| \(\lambda = 0\) | No dependence on historical data |
| \(0 < \lambda < 1\) | Weakens historical data influence |
| \(\lambda = 1\) | Same as traditional RLS |
| \(\lambda > 1\) | Strengthens historical data influence |
Algorithm initialization is based on vehicle speed thresholds. Below a preset threshold, initial values are computed as:
$$
\begin{aligned}
P(0) &= \left[ E_a(0) E_a(0)^T \right]^{-1} \\
m(0) &= P(0) E_a(0) E_F(0)
\end{aligned}
$$
Once the speed exceeds the threshold, recursive estimation proceeds using the above equations. This method ensures rapid convergence for electric vehicles, particularly in the China EV market where startup conditions vary.
Gradient Estimation Based on UKF-PF
UKF is an iterative estimation algorithm for nonlinear systems, involving prediction and update steps, with results used in subsequent iterations. It reduces linearization errors through weighted statistical linear regression. PF is a Bayesian recursive filtering algorithm based on Monte Carlo methods, approximating probability density functions with random samples for minimum variance estimation. This paper proposes an improved UKF-PF algorithm, using UKF as the main structure and PF as a front-end input for road gradient estimation. This fusion combines UKF’s efficiency in handling nonlinearities with PF’s flexibility in dealing with uncertainties, enhancing estimation accuracy and robustness.
First, the state space equation is established, with the state vector \(x = [v, \theta]^T\). The system differential equations are:
$$
\begin{aligned}
\frac{dv}{dt} &= \frac{T_{\text{out}} i_g i_0 \eta_t}{r m \delta} – \frac{C_D A v^2}{21.15 m \delta} – \frac{g f}{\delta} – \frac{g \theta}{\delta} \\
\frac{d\theta}{dt} &= 0
\end{aligned}
$$
The discrete state space equation is:
$$
\begin{aligned}
v_k &= v_{k-1} + \Delta t \left( \frac{T_{\text{out}} i_g i_0 \eta_t}{r m \delta} – \frac{C_D A v_{k-1}^2}{21.15 m \delta} – \frac{g f}{\delta} – \frac{g \theta_{k-1}}{\delta} \right) \\
\theta_k &= \theta_{k-1}
\end{aligned}
$$
Since the system equation cannot fully reflect the actual physical process and sensor errors exist, zero-mean independent Gaussian white noise is introduced as process noise \(W(k)\) and observation noise \(V(k)\), with covariance matrices \(Q\) and \(R\), respectively:
$$
\begin{aligned}
E[W(k)] &= 0 \\
E[V(k)] &= 0 \\
E[W(k) W(k)^T] &= Q \\
E[V(k) V(k)^T] &= R \\
E[W(k) V(k)^T] &= 0
\end{aligned}
$$
The state equation under random noise is:
$$
\begin{bmatrix}
v_k \\
\theta_k
\end{bmatrix}
=
\begin{bmatrix}
v_{k-1} + \Delta t \left( \frac{T_{\text{out}} i_g i_0 \eta_t}{r m \delta} – \frac{C_D A v_{k-1}^2}{21.15 m \delta} – \frac{g f}{\delta} – \frac{g \theta_{k-1}}{\delta} \right) \\
\theta_{k-1}
\end{bmatrix}
+ W(k)
$$
The observation equation and state space equation are:
$$
Y(k) = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} v_k \\ \theta_k \end{bmatrix} + V(k)
$$
$$
\begin{aligned}
X(k) &= f(X(k-1)) + W(k-1) \\
Y(k) &= H X(k) + V(k)
\end{aligned}
$$
where \(X(k)\) is the state at time \(k\), \(Y\) is the measurement matrix, and \(H = [1, 0]\).
The algorithm flow is as follows: First, particle samples are drawn from the prior distribution, updated according to the importance density function, and assigned weights. By comparing with a threshold, resampling is decided to generate a new particle set with new weights, obtaining the state estimate after PF processing. The predicted variables from PF are input into the UKF system. Sigma points are used for continuous prediction of mean and covariance, fed into the Kalman framework to generate information matrices and cross-covariance matrices, ultimately computing the Kalman gain and updating the system state. This approach is highly effective for electric vehicles, especially in dynamic China EV environments.
Determination of Gear Shift Strategy
Optimal Gear Shift Strategy for Electric Vehicles
The optimal dynamic shift strategy selects shift points based on maximum vehicle acceleration. As shown in Figure 3, when the accelerator pedal is held at a constant opening, the shift point speed for upshifting is determined by the intersection of acceleration curves for adjacent gears. By fitting optimal upshift points under various conditions, the optimal dynamic upshift curve is obtained, as illustrated in Figure 4. To avoid unnecessary cyclic shifting, downshift points are determined using the equal delay method, typically 5–8 km/h.
The optimal economic shift strategy selects shift timing for maximum efficiency operation, improving overall efficiency and reducing energy consumption. As shown in Figure 5, for a given accelerator pedal opening, the upshift point speed is the intersection of efficiency curves for adjacent gears. Similar to the dynamic shift strategy, the economic downshift speed difference is 5–8 km/h, resulting in the economic shift curve shown in Figure 6. This is crucial for electric vehicles in the China EV market, where energy efficiency is a key selling point.
Shift Strategy Based on Mass and Gradient Recognition
During mountain transport, uphill driving is common for commercial vehicles. If the gear is too high upon entering a gradient, the driving force may be insufficient to overcome gradient resistance, causing deceleration and necessitating downshifts. However, inappropriate gear selection after downshifting can lead to rapid upshifts, causing frequent cyclic shifting. This affects dynamic performance, driving comfort, transmission wear, and energy economy. Therefore, designing an appropriate shift strategy for gradient driving is essential.
The gradient shift strategy comprises three phases: upshift inhibition before entering the gradient, early downshifting and continued upshift inhibition during gradient driving, and delayed upshifting or downshifting when exiting the gradient. The framework is shown in Figure 7.
Based on vehicle mass, it is divided into \(d\) intervals:
$$\Delta m = \frac{m_{\text{max}} – m_{\text{min}}}{d}$$
where:
– \(\Delta m\) is the mass difference per segment,
– \(m_{\text{max}}\) is the fully loaded mass,
– \(m_{\text{min}}\) is the unloaded mass,
– \(d\) is the number of segments (here, \(d=3\)).
For example, with \(m_{\text{max}} = 12,000\) kg, \(m_{\text{min}} = 7,000\) kg, \(d=3\), \(\Delta m \approx 1,670\) kg, resulting in segments: \(m_1 = [7,000, 8,670)\), \(m_2 = [8,670, 10,340)\), \(m_3 = [10,340, 12,000]\). Gradients are classified as small (less than 5°), medium (5°–10°), and large (above 10°).
For uphill conditions, the gear adjustment strategy is shown in Figure 8. Lightly loaded vehicles follow standard shift logic on small or medium gradients but maintain first gear on large gradients. Medium-loaded vehicles use three strategies: maintaining flat-road shifts, holding current gear, or downshifting to first gear. Heavily loaded vehicles hold current gear on small gradients and downshift to first gear on medium and large gradients.
For downhill driving, the gravitational component adds to the driving force, often requiring reduced accelerator opening or lower acceleration. The gear adjustment strategy for downhill conditions is shown in Figure 9. Lightly loaded vehicles use standard shifts, hold current gear, or downshift to first gear based on gradient. Medium-loaded vehicles follow standard shifts on small gradients but downshift or hold first gear on medium and large gradients for stability and motor braking. Heavily loaded vehicles maintain current gear regardless of gradient to avoid shifting impacts on auxiliary braking and ensure downhill stability. This adaptability is vital for electric vehicles in diverse China EV operating conditions.
Simulation Analysis
Vehicle Mass Estimation Simulation Analysis
The algorithm model is built in Matlab/Simulink, and mass and gradient estimation algorithms are tested. The structural parameters of the electric commercial vehicle used for testing are based on actual measurements from a light truck, as shown in Table 2.
| Physical Quantity | Value | Physical Quantity | Value |
|---|---|---|---|
| Fully loaded mass \(m\) (kg) | 12,000 | Wheel rolling radius \(r\) (m) | 0.407 |
| Air resistance coefficient \(C_D\) | 0.448 | Rolling resistance coefficient \(f\) | 0.02 |
| Air density \(\rho\) (kg/m³) | 1.29 | Frontal area \(A\) (m²) | 6.610 |
The driving condition is set as movement on a horizontal road from 0 to 50 km/h. The mass estimation algorithm activation threshold is set to 10 km/h. The driving scenario is shown in Figure 10, and the mass estimation results are shown in Figure 11.
Based on simulation results, the vehicle mass converges rapidly to a stable value between approximately 2.3 and 3.5 seconds. Since the algorithm does not reach the estimable threshold within the first 1.8 seconds, mass estimation begins once exceeded, leading to quick convergence. The average mass is 11,928 kg, with estimation accuracy exceeding 99.3%, meeting performance standards. The error is about 72 kg, attributed to two sources: random noise in sensors introducing bias, and the use of a forgetting factor of 0.98 to accelerate convergence, which requires balancing convergence speed and accuracy in practical applications for electric vehicles.
Gradient Estimation Simulation Analysis
For gradient estimation, two road conditions are simulated: constant gradient and step-changing gradient. First, the road gradient is set to a constant 3%, with simulation results shown in Figures 12 and 13.
Second, a scenario with varying gradients is simulated, including uphill, downhill, and flat roads. Assume the vehicle drives on flat road from 0–120 s, uphill with gradient increasing to 5% from 120–240 s, maintains 5% uphill from 240–360 s, downhill with gradient decreasing to -1.5% from 360–480 s, and returns to flat road from 480–600 s. In simulation, acceleration is disturbed by white noise with variance \(10^{-4}\). Results are shown in Figures 14 and 15.
From gradient estimation results, under constant gradient, the estimate quickly approaches the true value within 0.8 seconds, with estimation error within 0.001 rad. Under continuously varying gradients, the estimate stabilizes near the true value within 1.3 seconds, with error within 0.003 rad, meeting performance requirements. This reliability is essential for electric vehicles, particularly in the China EV market where road conditions can be unpredictable.
Simulation Analysis of Revised Shift Strategy
For uphill conditions, simulations are conducted under different gradients and loads. Under medium load (vehicle mass 8,800 kg) and 5% gradient (2.86°) at constant speed, results are shown in Figures 16–18. Figure 17 compares gear changes before and after shift strategy revision, and Figure 18 compares energy consumption changes.
Simulation results show that before the gradient, the vehicle accelerates and upshifts to second gear. During the gradient, the gravitational component adds to motion resistance, causing deceleration. To maintain constant speed, the accelerator opening increases, leading to downshifting to first gear at the downshift speed point. After downshifting, speed increases, triggering upshifting to second gear at the upshift point. This cyclic shifting continues until the gradient ends, exacerbating transmission component wear and reducing driving performance. By estimating vehicle mass and road gradient, the strategy judges that the vehicle should maintain second gear throughout the climb, eliminating frequent shifting. Over the entire uphill condition, the optimized shift strategy reduces energy consumption from 1.366 kW·h to 1.324 kW·h, a 3% reduction, significantly improving energy savings for the electric vehicle.
For downhill conditions under heavy load (vehicle mass 12,000 kg) and constant speed on -5% gradient (-2.86°), results are shown in Figures 19–21. With the traditional shift strategy, the vehicle maintains first gear on flat road. Upon entering the 5% downhill, the gravitational component adds to the driving force, increasing speed. If the speed reaches the second-gear point, it upshifts, while the driver uses brakes to maintain constant speed, accelerating wear. By maintaining the current gear during downhill driving, based on estimated mass and gradient, the revised shift strategy ensures sufficient auxiliary braking from the drive motor, enabling safe descent. Over the entire downhill condition, the optimized strategy reduces energy consumption from 0.268 kW·h to 0.256 kW·h, demonstrating significant energy savings for the electric vehicle in China EV applications.
Conclusion
This paper designs a gear shift strategy integrating mass and gradient estimation for electric vehicles. Addressing the poor adaptability and insufficient accuracy of existing gradient estimation algorithms, a gradient estimation method combined with vehicle mass estimation is proposed. The mass estimation model is constructed using FFRLS, with mass estimation results serving as input parameters for further gradient estimation via the UKF-PF algorithm. Simulation results show that the estimation method achieves errors within 0.003 rad for varying gradients and 0.001 rad for fixed gradients, meeting performance requirements. The designed gradient shift strategy selects appropriate gears based on identified vehicle mass and road gradient. Uphill, it reduces shift cycles and frequency, extending transmission life; downhill, it minimizes brake wear, improves shift smoothness, and enhances economy. This method is particularly advantageous for electric vehicles in the China EV market, contributing to improved performance and efficiency. Future work could focus on real-world validation and integration with other vehicle systems for broader applications in the evolving electric vehicle industry.