Research on Data Prediction Model for Aerodynamic Drag Reduction Effect in Platooning Vehicles

Abstract

With the development of intelligent transportation systems, platooning can reduce vehicle aerodynamic drag by decreasing spacing between vehicles, improving transportation efficiency and reducing emissions. However, it is difficult for existing models to enable dynamic adjustment and real-time feedback. Therefore, this study proposes a digital twin system for real-time drag coefficient prediction using stacking ensemble learning. First, 2000 datasets of pressure distributions and drag coefficients under varying spacings were obtained through simulations. Then, an online prediction model for the aerodynamic performance of platooning vehicles was then constructed, realizing real-time drag coefficient prediction, and verifying the model performance using computational fluid dynamics data. The results indicate that the model proposed achieves 98.56% prediction accuracy, significantly higher than that of the traditional BP model (75.78%), and effectively captures the nonlinear relationship between vehicle spacing and drag coefficient. The influence mechanism of vehicle spacing on the aerodynamic performance of platooning vehicles revealed in this study enables high-precision real-time prediction under dynamic parameters.

1. Introduction

With the accelerating pace of global urbanization and the continuous expansion of population size, global motor vehicle ownership has been on a steady upward trend [1], directly leading to increasing pressure on road traffic systems. Against the backdrop of sustained economic development, the critical role of road transportation within the entire transportation system has become increasingly prominent. As a vital support for economic growth, advancements in transportation have not only driven economic progress but also significantly enhanced public living standards. Currently, road-based vehicles account for the majority of global land transportation, and based on existing trends, this proportion is projected to remain stable in the coming years. Specifically in China, light-duty gasoline vehicles (LDGVs) have constituted 90% of the total motor vehicle fleet since 2019, maintaining an annual growth rate exceeding 10% since 2017 [2]. Of note, road freight transportation, as a key component of road transport, not only consumes 25% of global energy but also contributes emissions closely linked to local health impacts and global greenhouse effects [3]. In addition, heavy-duty vehicles account for a relatively high proportion of total vehicle emissions.

Fortunately, the development of intelligent transportation systems (ITS) provides multiple possibilities for improving transportation efficiency and reducing energy consumption. One solution is to have vehicles travel longitudinally and with smaller spacing between vehicles, which is known as platooning vehicles [4]. Given that approximately 25% of the energy consumed by vehicles during operation is associated with aerodynamic drag [5], the fuel efficiency of vehicles traveling in platoons is influenced by this phenomenon. Similar dynamics are observed in other domains, such as road cycling, avian formation flight, and motorsports [6,7,8], where aerodynamic performance is altered by changing the formation of the overall motion. With the gradual maturity of intelligent driving technology, safe driving of vehicles in a line with fixed spacing has been achieved. Companies and universities in many countries have successively developed the automation system of platform driving [9]. Since vehicles traveling in close alignment can improve transportation efficiency and reduce vehicle emissions [10], investigating vehicular platooning configurations and predicting the drag coefficients through aerodynamic performance analysis is of significant importance. Many organizations have expended considerable effort on the aerodynamics of platooning vehicles to develop formation methods to reduce the drag coefficient of platooning vehicles and reduce the energy consumption by reducing the drag coefficient. The current research on autonomous driving is mostly focused on vehicle navigation, decision-making, and control. These studies provide a safer basis for platooning vehicles, ensuring the technical feasibility of platooning vehicles.

With the deepening application of digital twin technology in the industrial field, it shows unique advantages in the field of vehicle aerodynamic performance prediction. As the core means of interaction between the physical entity and virtual space, digital twin technology gradually shows an important value in the field of vehicle aerodynamic performance research [11]. In traditional vehicle design, digital twinning has achieved offline prediction and optimization of aerodynamic drag of a single vehicle through the fusion of high-precision hydrodynamic simulation and experimental data. As the complexity of queue driving scenarios increases, the focus of research gradually shifts to dynamic modeling of multi-vehicle cooperative aerodynamics, such as constructing the virtual flow field model of multi-vehicle formation by using digital twin technology, verifying the nonlinear correlation law between vehicle spacing and air resistance change, etc.

However, there are still significant limitations in existing research: on the one hand, most models rely on preset fixed driving parameters (constant vehicle spacing and stable vehicle speed), which makes it difficult to adapt to dynamic regulation requirements such as frequent queue reorganization and lane-change obstacle avoidance in automatic driving scenarios, such as changes in the number of platooning vehicles, lane changes by the entire platoon, etc. On the other hand, traditional offline simulation models cannot meet the real-time mapping and feedback control requirements of aerodynamic performance during vehicle driving. In response to this technical bottleneck, some scholars explore the fusion application of digital twins, edge computing [12,13,14], and deep learning [15,16], trying to achieve millisecond-level drag prediction through lightweight model architecture, but the prediction accuracy and robustness still need to be studied under the complex queue driving conditions. Therefore, building an online prediction model for aerodynamic performance of platooning vehicles oriented to autonomous driving dynamic parameters has become a key challenge for deepening the application of digital twin technology in the field of intelligent transportation.

Based on the above analysis, this paper intends to study the aerodynamics of platooning vehicles. The mapping relationship between platooning vehicle parameters and aerodynamic performance is established through mechanism analysis, and the air flow around vehicles is determined. Then, the online predictive model for the aerodynamic performance of vehicle formations is developed using stacked ensemble learning, leveraging datasets of pressure distributions and drag coefficients at varying inter-vehicle spacing distances. This model is designed to enable real-time prediction of drag coefficients in vehicular platoons.

2. Aerodynamic Performance Mechanism Analysis of Platooning Vehicles

2.1. Scene Determination and Model Selection

With the iterative advancement of autonomous driving technology, the safety performance of vehicles has been continuously optimized, enabling the realization of driving scenarios with reduced inter-vehicle spacing. Against this technological backdrop, platooning driving, as an innovative solution to mitigate traffic congestion, has gradually emerged as a focal point in research domains [17,18]. Compared with single-vehicle operation, the aerodynamic characteristics of platooning vehicles exhibit notable distinctions—reducing aerodynamic drag can effectively enhance energy efficiency [19]. However, the superposition of diverse vehicle geometries, dynamic changes in inter-vehicle distances, and complex driving processes gives rise to highly nonlinear airflow patterns around vehicles. These aerodynamic behaviors vary significantly with vehicle types, spacing parameters, and operational conditions. Consequently, extensive numerical simulations are required to systematically analyze flow field characteristics under different scenarios and explore optimal configurations of driving parameters. The computational fluid dynamics method is conducted to analyze the aerodynamic performance mechanism of the vehicle.

The bluff-body characteristics of lorries result in higher pressure drag. Lorries feature a shape close to a cuboid, with an upright cab and a box-like body, where the windward surface is nearly perpendicular to the driving direction. This structure creates a high-pressure zone in front of the cab and a large-scale low-pressure zone with vortices behind the trailer, generating significant pressure drag. Meanwhile, lorries operate in a high Reynolds number flow regime with a more forward separation point, attributed to their large size, leading to a higher Reynolds number during travel. At high Reynolds numbers, airflow is more prone to separate on the body surface, especially at the edges of the cab and the trailer tail, forming intense vortices that increase drag.

Compared with passenger cars, lorries are constrained by cargo box dimensions when loading goods, making it difficult to fundamentally reduce drag through streamlined designs like passenger cars. Thus, lorries rely more on external drag reduction measures. As pressure drag accounts for a higher proportion of total drag in lorries, platooning can more directly depress pressure drag, yielding more significant drag reduction benefits. To enhance the aerodynamic drag reduction effect of platooning, this study selects lorries as the research object. Taking three lorries as a platoon, the simulation objects are three lorry models of the same type. The study investigates the variation in aerodynamic drag coefficients by adjusting the inter-lorry spacing, with the aim of exploring drag coefficient changes to enhance transportation efficiency under actual road conditions.

Three commercial box-type lorry models will be used for computational fluid dynamics in this study. The commercial box-type lorry model used has a scale of 1/20, length (L), width (W), and height (H) of 395 mm, 125 mm, and 175 mm, respectively. The model is simplified from Leyland DAF 45-130 as shown in Figure 1 [20].

Due to the risk of danger that may still occur during platooning at a fixed vehicle distance, even with the assistance of autonomous driving technology, the actual driving speed for the light-truck road transportation studied in this project should not be set too high, which is 10 m/s. The actual driving environment of the lorry dynamic viscosity $\mu =1.79\times {10}^{-5}\mathrm{P}\mathrm{a}·\mathrm{s}$ and fluid ρ = 1.29 kg/m³. Thus, the actual driving Reynolds number $Re\approx 5.7\times {10}^6$.

To investigate the mechanism by which fluid flow states affect drag, the simulation was initially divided into five groups based on varying vehicle distances. The spacings between the vehicle models in these five groups were set at 100 mm, 200 mm, 300 mm, 400 mm, and 500 mm, respectively. Three commercial box-type lorry models were simulated to drive in a platoon formation at these different vehicle distances.

2.2. Experimental Data of Computational Fluid Dynamics

In the process of data acquisition, due to the low safety factor in changing the lorry speed, vehicle distance, and other motion parameters, it is difficult to obtain the motion parameter data on the street. Therefore, the computational fluid dynamics analysis method is used to obtain the parameter data. Through the analysis of five groups of wind tunnel test data, it is found that the drag coefficient of the vehicle changes gradually with the vehicle distance. In order to construct the prediction model, this study obtained the model verification data through computational fluid dynamics analysis.

The grid serves as the discretization foundation for CFD calculations. Inadequate grid density can lead to insufficient capture of flow features such as separation points, vortex structures, and boundary layers, thereby causing calculation deviations in key parameters like drag and lift. In bluff-body flow, grid sparsity may induce forward migration of separation points and incorrect calculation of vortex shedding frequency in the wake region, with drag coefficient errors exceeding 10%. Therefore, for the analysis of bluff-body lorries in the flow field of this study, grid convergence analysis is a critical component to ensure the reliability of results.

In this study, taking a spacing of 0.76 L as an example, eight models with different mesh counts were generated under varying conditions of base mesh size, prism layer numbers, refinement regions, and refined mesh sizes as shown in Figure 2a–h. Computational fluid dynamics simulations were performed on each of these eight models to obtain the fluid motion around the lorries and the lift coefficient of the middle vehicle. The mesh generation methods and corresponding lift coefficients for these eight groups are presented in

Table 1. Number of cells and result variations under different mesh generation methods.

Group	Number of Cells	Base Size	Prism Layer Numbers	Lorries Surface Control	Refinement Regions Control	Drag Coefficient
a	4321	0.3	1	None	None	0.107697
b	62,689	0.05	1	None	None	0.225347
c	210,589	0.03	4	None	None	0.255347
d	826,539	0.05	4	10% relative to the base size	25% relative to the base size	0.386238
e	1,119,995	0.04	4	10% relative to the base size	25% relative to the base size	0.383351
f	1,306,744	0.035	4	10% relative to the base size	25% relative to the base size	0.382107
g	2,174,597	0.02	4	10% relative to the base size	25% relative to the base size	0.382519
h	2,447,670	0.02	4	10% relative to the base size	20% relative to the base size	0.382230

.

The drag coefficient versus grid number variation curve is plotted based on the obtained results, as shown in Figure 3. When the grid number is insufficient and the grid resolution is too coarse, significant deviations in the results occur, demonstrating a non-independent relationship between the results and grid density. When the grid count reaches ${10}^6$, both the lift coefficient and drag coefficient exhibit slow changes and converge. From a grid count of 826,539 to 2,447,670, the fluctuations in the lift and drag coefficients account for only 1.07% of the overall variation, indicating that the calculation results become grid-independent. However, an excessively large grid number leads to a surge in memory and storage requirements, with the computation time increasing exponentially. Therefore, the grid scheme with 1,119,995 elements is ultimately adopted for this simulation. Under this condition, the grid convergence error of Cd is less than 1.5%, satisfying the requirements for computational accuracy.

Meanwhile, through the comparison of velocity contour plots in Figure 4, it is confirmed that the morphology of the separation region and the wake vortex structure no longer change significantly with grid refinement when the grid number reaches 826,539.

Once the mesh division scheme is determined, computational fluid dynamics analysis is conducted. The specific operation methods are as follows:

• Step 1: Identification of the study region: A cube with dimensions of 7000 mm, 1500 mm, and 1200 mm is established in the simulation. Specifically, entrance and exit boundaries, two side boundaries, top and bottom boundaries are generated. To determine the study region, the Boolean subtraction function in STAR-CCM+ 2302 can be used to create the surrounding area where the vehicle travels, and prepare the final region for mesh division.
• Step 2: Meshing: After verifying the mesh independence analysis, the mesh size was determined. Generate a basic mesh through the “automatic mesh” function. The base size of the mesh is 0.04, the mesh growth rate is set to 1.3, the number of prism layers is 4, the prism layer extension is 1.2, and the total thickness of the prism layers (absolute value) is 0.008 m. Then use custom controls for fine adjustment to refine the grid on the surface of the vehicle, while increasing the grid refinement in the vicinity of the vehicle as shown in Figure 2e. The final number of cells is 994,915.
• Step 3: In the study region, the inlet and outlet boundaries of the fluid direction are defined as velocity inlet and pressure outlet. The top and bottom are defined as the wall surface. The side is defined as a symmetrical face. The speed of velocity inlet is set to 200 m/s. The pressure outlet is set to 0. Dynamic viscosity $\mu =1.79\times {10}^{-5}\mathrm{P}\mathrm{a}·\mathrm{s}$ and fluid density $\rho =1.29\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\mathrm{‌}$. Thus, the computational fluid dynamics Reynolds number $Re\approx 5.7\times {10}^6$.

The Reynolds number simulated by computational fluid dynamics is consistent with that in actual road driving. By adjusting the fluid velocity, both the wind tunnel test and the actual scenario, the Reynolds number is in a high Reynolds number regime. In this range, the inertial forces of the fluid far exceed the viscous forces, and the flow tends to exhibit a turbulent state. The boundary layer thins and is typically a turbulent boundary layer, with complex wake flow. For bluff-body geometries such as lorries, at high Reynolds numbers, pressure drag caused by flow separation becomes dominant.

Taking the lorry distances of 100 mm, 200 mm, 300 mm, 400 mm, and 500 mm as examples, the simulation results of five groups are shown in Figure 5. The drag coefficients for the five groups are 0.3097, 0.3496, 0.3830, 0.3937, 0.4125. The reduction in average drag coefficient and drag reduction effectiveness under different lorry spacing is shown in Figure 6. The average drag coefficient obtained and the reduction in vehicle drag are shown in

Table 2. Drag coefficient at different lorry distance obtained from wind tunnel test.

Lorry Spacing	Single Lorry	1.27L (500 mm)	1.01L (400 mm)	0.76L (300 mm)	0.51L (200 mm)	0.25L (100 mm)
$C_D$	0.66	0.4125	0.3937	0.3830	0.3496	0.3097
$C_D$ Reduction	0%	37.50%	40.35%	41.97%	47.03%	53.08%

.

The simulation results show a direct correlation between the spacing of lorries and the drag coefficient of the central lorry. As the distances between the front, rear, and central lorries vary, the airflow patterns surrounding the middle lorry are significantly altered. This alteration in airflow dynamics subsequently modifies the pressure distribution across the surface of the lorry. Consequently, these changes in pressure distribution lead to variations in the drag coefficient.

With the decrease in the distance between lorries, the drag reduction amplitude increases, which indicates that for lorries, the aerodynamic performance of the fleet is significantly optimized, enhancing driving efficiency. The drag coefficient of the middle lorry in a platoon is notably lower than that of an isolated single lorry, with the minimum value observed at a spacing of 0.25 L. As the inter-vehicle distance increases beyond this optimal point, the drag coefficient rises gradually. However, when the spacing exceeds 0.76 L, the rate of increase in the drag coefficient diminishes, indicating a weakened aerodynamic interaction. This inverse relationship between spacing and drag coefficient sensitivity suggests that closer proximity amplifies the mutual aerodynamic influence among lorries. Conversely, greater distances reduce such interactions, leading to less pronounced effects on each vehicle’s drag characteristics. Consequently, decreasing lorry spacing during platooning effectively mitigates driving drag, with more substantial reductions achieved at smaller intervals. In fact, when lorries maintain minimal spacing, the aerodynamic drag of the middle vehicle can be largely minimized, demonstrating the significant benefits of coordinated close-proximity driving in reducing aerodynamic resistance.

After identifying the trend of variation between the drag coefficient and vehicle distance, a prediction model is constructed to characterize the drag coefficient as a function of vehicle spacing. Based on the above method, this paper repeats 2000 times of computational fluid dynamics analysis in STAR-CCM+ 2302 software with only changing the distance between lorries, and obtains the average drag coefficient under different vehicle distances, which provides data sample support for the model.

3. Establishment of the Online Prediction Model for Aerodynamic Performance of Platooning Vehicles

Based on the above analysis, this section aims to construct an online prediction model based on integrated learning algorithm to predict the drag coefficient of lorries in different vehicle distances in real time.

Machine learning, a data-driven intelligent algorithm, reveals nonlinear associations in high-dimensional data via autonomous feature learning to support complex system prediction, with error backpropagation neural networks (BPNNs) offering strong regression capabilities through multilayer topology and error backpropagation, though suffering from poor generalization due to overfitting-induced training-test performance gaps. To address this, an improved Stacking-BPNN (STBP) model is proposed, integrating a hierarchical ensemble framework: hierarchical sampling decouples data into base learner training sets and meta-feature generation sets, constructs a base model set (e.g., SVM, decision trees) for primary predictions, and employs a BPNN-based meta-learner to mine hidden features in primary outputs. This two-layer architecture suppresses single-network overfitting via collaborative optimization to reduce variance and bias, enhancing modeling accuracy for antenna assembly process parameter–performance relationships while maintaining model capacity.

3.1. Stacking Ensemble Learning Principle

Stacking ensemble learning, as a hierarchical ensemble paradigm, improves model performance by constructing a two-level prediction architecture. The framework consists of a cascade structure of base learners and meta-learners: the former consists of n heterogeneous base models, which generate prediction results and construct meta-feature matrices by performing multi-perspective learning on the original feature space; the latter acts as a high-order combiner, which performs nonlinear mapping on the meta-features output by the base learners to finally form the ensemble prediction [21]. This hierarchical coupling mechanism significantly improves the modeling robustness of complex tasks through the synergy of base model diversity compensation and meta-model generalization enhancement.

As shown in Figure 7, the implementation framework of this paper first uses a hierarchical random partitioning strategy to decouple the original dataset into a training set and a test set. In order to strengthen the generalization of the model, a K-fold cross-validation mechanism is introduced in the training stage (K = 5 in this study). The training data are equally distributed into five mutually exclusive subsets {Train1, …, Train5}. Through this data distribution strategy, diversified validation scenarios are constructed while ensuring data utilization, thus suppressing overfitting tendency. Specifically, after each base learner is trained on K-1 subsets, it predicts the meta-features in the reserved subset. This process ensures the completeness of feature information through a cyclic verification mechanism, laying a foundation for high-precision modeling of subsequent meta-learners.

3.2. Base Learner Model Construction

In the ensemble learning framework, the base learner acts as the basic prediction unit, and improves the system performance through multi-model cooperation mechanism. The core of the ensemble learning framework is to construct a set of base models with prediction diversity, in which each base model independently completes the feature mapping of input data, and the ensemble system optimizes the final prediction performance by optimizing the combination strategy. With the advantages of high generalization, model heterogeneity, and computational efficiency, base learners can not only suppress overfitting/underfitting phenomena through regularization effects during the integration process, but also enhance system robustness by predicting spatial diversity. Based on the model complementarity theory, this paper uses random forest (RF) and XGBoost algorithms to build a base learning layer to form a heterogeneous predictor combination.

• RF algorithm.

RF algorithm, as a typical representative of Bagging integration paradigm, is an integration strategy based on Bootstrap aggregation proposed by Breiman [22]. This framework constructs a set of weakly correlated multiple CART decision trees through random subsampling of feature space and sample perturbation mechanism: firstly, multiple training subsets are generated by sampling with replacement; secondly, feature subsets are randomly selected for optimal partition when each node splits. This double randomization design effectively reduces the model variance. While it maintains high-dimensional data processing capabilities, it shows strong robustness to noise interference and missing values. Its mathematical essence is to reduce the overfitting tendency of a single decision tree and achieve bias-variance balance optimization of the overall model.

In the base learning layer, for the RF algorithm, the dataset is given as follows:

$$D=\left\{\left(X_i,Y_i\right)\right\}\left(\left|D\right|=n,X_i\in R^m,Y_i\in R^m\right),$$

(1)

where n is the number of dataset samples and m is the number of features per sample. Divide the dataset into a training set and a test set, where the training set is

$$T=\left\{\left(X_1,Y_1\right),\left(X_2,Y_2\right),\cdot \cdot \cdot ,\left(X_p,Y_p\right)\right\},$$

(2)

where p is the number of training samples. Then the test set is

$$S=\left\{\left(X_{p+1},Y_{p+1}\right),\left(X_{p+2},Y_{p+2}\right),\cdot \cdot \cdot ,\left(X_n,Y_n\right)\right\},$$

(3)

The random forest constructs multiple decision trees by Bootstrap sampling, and its prediction result is the average of the outputs of each tree. For the training set $T=\left\{\left(X_1,Y_1\right),\left(X_2,Y_2\right),\cdot \cdot \cdot ,\left(X_p,Y_p\right)\right\}$, the generation process of each tree can be described as

• A subset T(t) is obtained by sampling back from T;
• Select randomly $⌊\rho \cdot d⌋$ features to construct candidate feature set F(t), where ρ is the sampling rate;
• Recursively partition nodes until stopping conditions are met, and the splitting criterion is minimized using the Gini exponent:

  $$\arg \underset{f\in F\left(t\right)}{\min }\left[Gini\left(D\right)-{\displaystyle \frac{\left|D_L\right|}{D}}Gini\left(D_L\right)-{\displaystyle \frac{\left|D_R\right|}{D}}Gini\left(D_R\right)\right].$$

  (4)

In order to improve the sensitivity to aerodynamic drag characteristics, this paper designs a dynamic feature weight adjustment strategy. Define the importance weight of feature f as

$${\omega }_f={\displaystyle \frac{1}{1+\exp \left(-\beta \left|\partial C_d/\partial f\right|\right)}},$$

(5)

where β is the sensitivity coefficient and ∂C_d/∂f is the partial derivative of the drag coefficient with respect to feature f. This weighting function allows the model to preferentially select features that have a significant impact on aerodynamic characteristics, improving physical interpretability.

Bootstrap sampling strategy is used to generate a weak learner set in the process of random forest model construction. Specifically, for the N samples contained in the original training set T, about two-thirds of the samples are extracted to constitute Bootstrap sub-samples T_boot by back-sampling, and the remaining one-third of the unsampled samples are taken as new test set, which are often referred to as out-of-bag (OOB) samples. In the process of constructing regression tree of random forest algorithm, after generating a new training set through K rounds of iteration and establishing regression tree, the system will automatically call OOB samples out of bag for model performance verification. This mechanism essentially constitutes a cross-validation framework built into the algorithm, thus achieving unbiased estimation of the generalization ability of the model without relying on independent test sets. This endogenous validation method not only ensures statistically unbiased evaluation results, but also significantly optimizes computational resource utilization by dynamically generating validation datasets. At the specific implementation level, the error calculation formula for OOB samples is as shown [23]:

$$MS{E}_{OBB}={\displaystyle \frac{{\displaystyle \sum _{i=1}^{p-k}{\left[Y_i\left(X_i\right)-Y_i\right]}^2}}{p-k}},$$

(6)

where p is the number of original training sets and k is the number of new training sets to be partitioned, then the number of OOB samples is p − k. Y_i(X_i) is the predicted value of sample X_i, and Y_i is the true value of sample X_i.

2.

XGBoost algorithm.

XGBoost, as an efficient ensemble learning framework, has a core mechanism based on the collaborative optimization of decision tree-based model and gradient lifting strategy. This algorithm has important application value in the fields of statistical modeling, data mining, and predictive analysis, and its performance has been fully verified in international machine learning competitions [24]. Compared with classical gradient boosting method, XGBoost innovatively introduces the L1/L2 regularization term into objective function, which effectively controls model complexity and significantly improves robustness to the overfitting problem of high-dimensional data. From mathematical modeling point of view, the objective function of XGBoost algorithm can be decomposed into two core components: the empirical loss term and the structural regularization term, and their collaborative optimization mechanism ensures the balance between accuracy and generalization ability of the algorithm. Assuming the training dataset is known as $D=\left\{\left(X_i,y_i\right)\right\}\left(\left|D\right|=n,X_i\in R^m,y_i\in R^m\right)$, loss function $l\left(y_i,\widehat{y_i}\right)$ regularization term $\Omega \left(f_k\right)$, the overall objective function is

$$S\left(\varphi \right)={\displaystyle \sum _{i}l\left(y_i,\widehat{y_i}\right)+{\displaystyle \sum _k\Omega \left(f_k\right)}},$$

(7)

where $S\left(\varphi \right)$ denotes the expression on linear space, i is the ith sample, k is the kth tree, and the regularization term ${\displaystyle \sum _k\Omega \left(f_k\right)}$ denotes the complexity of the kth tree, which is used to control the model complexity. $\widehat{y_i}$ is the predicted value of the ith sample $x_i$, and its expression is

$$\widehat{y_i}={\displaystyle \sum _{k=1}^k{f}_k\left(x_i\right)}={\widehat{y_i}}^{\left(t-1\right)}+f_t\left(x_i\right).$$

(8)

Approximating the objective function by optimizing the loss function through the second-order Taylor expansion yields

$$S^{\left(t\right)}={\displaystyle \sum _{i=1}^{n}l\left[y_i,{\widehat{y_i}}^{\left(t-1\right)}+g_i{f}_t\left(x_i\right)+\frac{1}{2}{h}_i{f}_t^2\left(x_i\right)\right]+{\displaystyle \sum _k\Omega \left(f_k\right)}},$$

(9)

where $g_i=l^{\prime }\left(y_i,{\widehat{y_i}}^{\left(t-1\right)}\right)$ is the first derivative of the error with respect to the current model, $h_i=l^{″}\left(y_i,{\widehat{y_i}}^{\left(t-1\right)}\right)$ is the second derivative of the error with respect to the current model. For the objective function, since $l\left(y_i,{\widehat{y_i}}^{\left(t-1\right)}\right)$ is constant, the objective function can be converted to

$$S^{\left(t\right)}={\displaystyle \sum _{i=1}^{n}l\left[g_i{f}_t\left(x_i\right)+\frac{1}{2}{h}_i{f}_t^2\left(x_i\right)\right]+{\displaystyle \sum _k\Omega \left(f_k\right)}}.$$

(10)

Meanwhile, the normalized term is ${\displaystyle \sum _k\Omega \left(f_k\right)}=\Omega \left(f_t\right)+{\displaystyle \sum _{k=1}^{t-1}\Omega \left(f_k\right)}=\Omega \left(f_t\right)+\mathrm{Constant}$. Thus, the target function can be further converted into

$$S^{\left(t\right)}={\displaystyle \sum _{i=1}^{n}l\left[g_i{f}_t\left(x_i\right)+\frac{1}{2}{h}_i{f}_t^2\left(x_i\right)\right]+\Omega \left(f_t\right)}.$$

(11)

Convert the objective function into the form of blade node accumulation to obtain the final XGBoost objective function:

$$S^{\left(t\right)}={\displaystyle \sum _{j=1}^T\left[G_j{\omega }_j+\frac{1}{2}\left(H_i+\lambda \right){\omega }_j^2\right]+\gamma T},$$

(12)

where $\lambda$ and $\gamma$ are weighting factors, T is the number of nodes in the leaf, Gj is the sum of the accumulated first-order partial derivatives of the samples contained in the leaf node, and Hi is the sum of the accumulated second-order partial derivatives of the samples contained in the leaf node.

3.3. Meta-Learner Model Construction

The meta-learner receives the outputs of the base learner as input features, and learns how to optimally combine these prediction results by training to improve the generalization performance of the model. BP neural network is selected as the meta-learner of the meta-learning layer. BP neural network is a multilayer feedforward neural network based on gradient descent optimization strategy, which is trained by the error backpropagation algorithm [25]. The core idea is to adjust the weights of the network through the back-propagation of errors, so that the output of the network approaches the target value. The grid structure of BP neural network is shown in Figure 8.

In general, BP neural network consists of three layers: input layer, hidden layer, and output layer. The input layer contains m neurons, which mainly receive the input feature vector $x={\left(x_1,x_2,x_3\cdot \cdot \cdot x_m\right)}^T$, and the hidden layer contains 3 neurons, which realize nonlinear mapping through activation function. The output layer contains 1 neuron, which mainly realizes the output of the prediction result. When the input feature vector is $f_i\left(i=1,\cdot \cdot \cdot ,m\right)$, the weighted input of the input feature $\mu$ can be calculated by the weight $\omega$:

$$\mu ={\displaystyle \sum _{i=1}^m{\omega }_i{f}_i}+\beta .$$

(13)

The formula for deriving the weight offset is

$$\left\{\begin{array}{l}{q}_1=x_1{\omega }_{11}+x_2{\omega }_{21}+x_m{\omega }_{m1}-{\beta }_1\\ q_2=x_1{\omega }_{12}+x_2{\omega }_{22}+x_m{\omega }_{m2}-{\beta }_2\\ q_3=x_1{\omega }_{13}+x_2{\omega }_{23}+x_m{\omega }_{m3}-{\beta }_3\end{array}\right..$$

(14)

Thus,

$$\left\{\begin{array}{l}{q}_i=x_1{w}_{1i}+x_2{w}_{2i}+x_m{\omega }_{mi}-{\beta }_i\\ h_i=f\left(x_1{w}_{1i}+x_2{w}_{2i}+x_m{\omega }_{mi}-{\beta }_i\right)\end{array}\right.,$$

(15)

where f is the sigmoid activation function expressed as

$$f={\displaystyle \frac{1}{1+e^{-x}}}.$$

(16)

Similarly,

$$\left\{\begin{array}{l}u=h_1{v}_1+h_2{v}_2+h_3{v}_3-\omega \\ \stackrel{\wedge }{y}=f\left(h_1{v}_1+h_2{v}_2+h_3{v}_3-\omega \right)\\ Loss={\displaystyle \frac{1}{2}}{\displaystyle \sum _i{\left(y_i-\stackrel{\wedge }{y_i}\right)}^2=\frac{1}{2}{\left(y-\stackrel{\wedge }{y}\right)}^2}\end{array}\right.,$$

(17)

where Loss is the loss function, y is the expected value, and $\stackrel{\wedge }{y}$ is the output. In the iterative process of the network, each weight will be updated continuously with the iterative process, and the updated formula is ${\omega }^{\prime }=\omega +\Delta \omega$, where $\Delta \omega$ decreases fastest along the negative gradient $\Delta \omega =-\eta \cdot {\displaystyle \frac{\partial Loss}{\partial \omega }}$ direction. $\eta$ is the learning rate, and ${\displaystyle \frac{\partial Loss}{\partial \omega }}={\displaystyle \frac{\partial Loss}{\partial \stackrel{\wedge }{y}}}\cdot {\displaystyle \frac{\partial \stackrel{\wedge }{y}}{\partial \omega }}={\displaystyle \frac{\partial Loss}{\partial \stackrel{\wedge }{y}}}\cdot {\displaystyle \frac{\partial \stackrel{\wedge }{y}}{\partial u}}\cdot {\displaystyle \frac{\partial u}{\partial \omega }}$, where

$${\displaystyle \frac{\partial Loss}{\partial \stackrel{\wedge }{y}}}={\displaystyle \frac{\partial \left(\frac{1}{2}{\left(y-\stackrel{\wedge }{y}\right)}^2\right)}{\partial \stackrel{\wedge }{y}}}=-\left(y-\stackrel{\wedge }{y}\right),$$

(18)

$${\displaystyle \frac{\partial \stackrel{\wedge }{y}}{\partial u}}={\displaystyle \frac{\partial f\left(u\right)}{\partial u}}={\displaystyle \frac{-\left(-e^{-u}\right)}{{\left(1+e^{-u}\right)}^2}}=\stackrel{\wedge }{y}\cdot \left(1-\stackrel{\wedge }{y}\right).$$

(19)

Therefore, for $\Delta \omega$, the expression is

$$\Delta \omega =-\eta \cdot \left(y-\stackrel{\wedge }{y}\right)\cdot \stackrel{\wedge }{y}\cdot \left(1-\stackrel{\wedge }{y}\right).$$

(20)

In the aerodynamic performance prediction model based on stacking ensemble learning, the core advantage lies in the synergy of the two-stage learning mechanism: in the base learning stage, the heterogeneous base models generate high-precision preliminary prediction results through diversified feature extraction. In the meta-learning stage, BP neural network as a secondary learner performs nonlinear mapping and complementary advantages on the output of the base model, and significantly improves the generalization ability of the model through deep feature fusion. The construction process of this model in the aerodynamic performance prediction task mainly includes the following steps:

• Step 1: RF algorithm is adopted as a primary learner to perform a 5-fold cross-validation method with K = 5. The original training set is divided equally into five mutually exclusive subsets, marked as {Train 1, Train 2, …, Train5}.
• Step 2: Perform hierarchical cross-training to cross-validate prediction results. Select {Train 1–Train 4} as training subset and Train 5 as validation subset in the first iteration, and output validation prediction RF-Predic5. Select {Train 1–Train 3, Train 5} as training subset and Train 4 as validation subset in the second iteration, and generate RF-Predic 4. Repeat the above process until five rounds of validation are completed, and finally construct prediction matrix [RF-Predic 1]–[RF-Predic5].
• Step 3: Integrating the five-dimensional prediction results generated in step 2 to form a meta-learning layer training feature matrix X_meta_train ∈ ℝ^{N × 5} (N is the total sample size).
• Step 4: Test set feature extraction. Five groups of independent predictions are conducted on the original test set based on the reconstructed RF model of the complete training set. Calculate the predicted mean value AVGRF-Predict ∈ ℝ^{M × 1} (M is the number of test samples) of the break as the core input of the meta-learning layer test feature X_meta_test.
• Step 5: XGBoost is introduced as the second base learner, and the cross-validation process of steps 1–4 is repeated to obtain the prediction matrix [XGB-Predic1]–[XGB-Predic5] and mean of AGGXGB-Predict. Then concatenate X_meta_train and X_meta_test.
• Step 6: Construct a BP neural network with backpropagation architecture as the meta-learner model. The model receives the compound features of X_meta_train at the input layer. At the hidden layer, the model implements nonlinear mapping using the BP neural network. Through backpropagation, the model optimizes the weights and finally generates the final aerodynamic performance prediction values at the output layer, and completes model verification by utilizing X_meta_test.

3.4. Model Building and Training

In order to achieve the optimal performance of STBP model, it is necessary to perform collaborative hyperparameter optimization on the base learner and meta-learner. The specific manually selected parameter configuration is as follows: In the base learning layer, set the parameters of RF model and XGBoost model. In the meta-learning layer, set the parameters of BP neural network as shown in

Table 3. Base learning layer model parameters.

RF Model		XGBoost Model (Gradient-Boosted Tree)
Number of decision trees	200	Learning Rate and regularization	0.025
Maximum depth	10	Maximum depth	10
Minimum number of separated samples	100	Complexity penalty term	0.1
Maximum characteristic number of a single node	5	Total number of lifting trees	200

and

Table 4. Meta-learning layer model parameters.

Optimizer	Optimizer Type	Adam
	Default Learning Rate	10−3
Regularization Mechanism	Regularization Type	Dropout
	Rate	0.2
Training Configuration	batch_siz	64
	epoch	500

.

In order to meet the different requirements of the model for the measurement criteria of the input and output arrays, the simulation data samples are linearly normalized before training the model, thus the values are in the interval of 0–1, the outputs are naturally also normalized results. The normalization expression is

$$x_i={\displaystyle \frac{x-x_{\min }}{x_{\max }-x_{\min }}},$$

(21)

where x_i is the normalized sample value, x is the actual sample value, x_max is the maximum of the actual value, and x_min is the minimum of the actual sample value.

STBP model adopts the method of randomly dividing training set and test set. Therefore, for 2000 sets of simulation sample data, it is first randomly sorted, then 1600 sets of data are taken as original training set and 400 sets as original test set. Stacking ensemble learning adopts KFlod cross-validation method, K = 5 in this paper. Therefore, the original training set needs to be further divided into 5 equal parts and trained 5 times by using the base learning model.

3.5. Results Analysis and Application

Under the parameter setting of each model in Section 3.4, the real-time drag coefficient prediction model of lorries under automatic driving state is trained, and the model is used to verify validation samples. In validation samples, the predicted drag coefficient values of 8 validation samples are randomly extracted and compared with simulation values as shown in Figure 9. The red line represents predicted values, and the green line represents simulation values. From the comparison chart, it can be seen that predicted values of the model are close to simulation values, and predicted results meet the expectations.

In order to verify the accuracy of the proposed model, this paper trained and predicted the model based on the traditional BP model on 2000 sets of simulation data samples by STAR-CCM+ 2302, and compared predicted values with the STBP model.

The statistics of the training loss values of the test samples for the above two models are shown in Figure 10. It can be seen from Figure 10 that training loss value of traditional BP model is 0.05; training loss value of STBP model is 0.006. At the same time, for the above two models, predicted values of resistance coefficients of 8 validation samples are compared with simulated values to verify prediction accuracy of the model, as shown in

Table 5. Comparison of predicted values of partial test samples of two models with simulation.

Test	Fr	Fr1	Fr2	Fr3	Test	Fr1	Fr2	Fr3
T1	STBP_Pred	0.449	0.279	0.245	T5	0.459	0.256	0.331
	BP_Pred	0.512	0.254	0.231		0.206	0.198	0.271
	True	0.448	0.273	0.248		0.451	0.252	0.333
	Acc_STBP	0.997	0.977	0.987		0.982	0.984	0.994
	Acc_BP	0.856	0.932	0.931		0.457	0.786	0.814
T2	STBP_Pred	0.470	0.227	0.304	T6	0.510	0.338	0.367
	BP_Pred	0.442	0.207	0.386		0.382	0.408	0.455
	True	0.476	0.231	0.298		0.515	0.341	0.371
	Acc_STBP	0.987	0.983	0.980		0.990	0.991	0.989
	Acc_BP	0.929	0.896	0.705		0.742	0.804	0.774
T3	STBP_Pred	0.482	0.292	0.331	T7	0.499	0.335	0.370
	BP_Pred	0.301	0.228	0.281		0.341	0.442	0.298
	True	0.489	0.287	0.337		0.491	0.338	0.362
	Acc_STBP	0.992	0.983	0.991		0.984	0.991	0.978
	Acc_BP	0.581	0.703	0.839		0.695	0.692	0.823
T4	STBP_Pred	0.347	0.279	0.331	T8	0.472	0.243	0.320
	BP_Pred	0.203	0.215	0.368		0.304	0.348	0.238
	True	0.354	0.285	0.339		0.476	0.248	0.316
	Acc_STBP	0.980	0.979	0.976		0.992	0.980	0.987
	Acc_BP	0.573	0.754	0.914		0.639	0.597	0.753

. It can be seen from the table that predicted values of STBP model are closer to simulation results. The average prediction accuracy is used to represent the accuracy of the prediction, which is calculated using 8 randomly selected validation samples. The average prediction accuracy Acc_avg of STBP algorithm reaches 98.95%, while that of traditional BP algorithm is only 78.39%. The calculation formula of Acc_avg is shown in Equation (22). It shows that prediction accuracy of this model is higher and prediction effect is better than that of traditional BP model.

$$Ac{c}_{avg}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left(1-\frac{STBP\_Pre{d}_i-Tru{e}_i}{Tru{e}_i}\right)}$$

(22)

4. Development and Application of Digital Twin System

Based on the research results of this paper, this section will systematically design and develop the digital twin system for online monitoring of energy saving and drag reducing of lorries under unmanned driving scenes, and demonstrate the main functions of each module of the system to verify the feasibility of the digital twin system.

4.1. System Development and Operating Environment

According to the system design and development requirements, the MySQL database is selected to manage the data in the running process of the digital twin system for online monitoring of energy saving and drag reduction in platooning lorries, and the programming and development of the digital twin system are completed on Microsoft visual studio 2020 platform by using Net Framework 4.7.2 and scikit-learn technology. The development environment and running environment of the system are shown in

Table 6. System development and operating environment.

Hardware Environment	CPU: Intel(R) Xeno(R) Gold 5218 CPU @ 2.30 Ghz 2.29 Ghz; Running Memory: 256 GB; Hard Disk: 4 T; GPU: NVIDIA Quadro RTX4000
Software Environment	Operating System	Windows 10
	Development Languages and Frameworks	C#, Python 3.10, scikit-learn, Net Framework 4.7.2
	Database	MySQL 8.0.23
	Development Platform	Microsoft visual studio 2020, Pycharm 2021.1.3, Unity 2020.3.40

, which represents the software environment and hardware environment.

4.2. Technical Route

In order to establish the dynamic connection between the real lorry and the virtual lorry, and realize the real-time prediction and online monitoring of the aerodynamic performance of the lorry under the unmanned driving scene, the technical route is shown in Figure 11. The establishment of the digital twin system of the driving process of the lorry is mainly divided into the pre-processing stage and the real-time mapping stage. The pre-processing stage mainly obtains the training data required for the prediction model by performing a simulation on the operation process of the lorry; in the real-time mapping stage, the real-time state mapping is realized by analyzing and classifying the sensor data of the lorry during operation, and the real-time prediction of the aerodynamic performance of the lorry during operation is realized by using the prediction model trained by the proxy model and combining the real-time data of the sensors.

4.3. Introduction to System Functions

According to the framework proposed, with the Unity3D 2020.1 platform, the digital twin platform system function modules for energy saving and drag reduction online monitoring of lorries are built as shown in Figure 12. It includes two function modules: a visualization of the lorry operation process and online monitoring of aerodynamic performance.

• Operation process visualization: The visualization of the lorries’ running process is the basis of the digital twin platform. With the help of the lorries’ movement model, the module displays the current running state of the lorries on the display in real time, and displays the speed and distance of the lorries’ running process on the screen in the form of lines, as shown in Figure 13a–c.
• Aerodynamic performance online monitoring: The aerodynamic performance online monitoring module is the core module of the platform. With the help of the aerodynamic performance online prediction model constructed above, the aerodynamic performance of the lorries is predicted in real time by collecting parameters such as vehicle distance and vehicle speed during the driving process, and the real-time information such as prediction data and statistical data is displayed on the display. Thus, users can fully grasp the aerodynamic performance during the operation of the lorries and improve the operation efficiency. Specifically, it includes drag coefficient and lift coefficient, as shown in Figure 13d,e.

4.4. Analysis of System Operation Results

In order to verify the accuracy of the prediction model in the digital twin system, Figure 14 shows the three lorries’ error comparison between the resistance coefficient of the CFD results and the prediction model prediction results under 100 groups of different vehicle speeds and distances. It can be seen that the prediction model and the simulation results have high consistency.

In addition, through the performance analysis tool in Unity engine, the performance status of scenarios in the operation analysis process of prediction platform is viewed, and the analysis results are shown in Figure 15. Unity frames per second can be basically stabilized at more than 30 frames; the total time consumed by CPU processing a frame is 42.13 milliseconds, and the time consumed by rendering each frame is about 13.11 milliseconds. This basically realizes low delay and fast visualization of the prediction and analysis process, and can meet the needs of digital twin rapid response and real-time prediction in aerodynamic performance analysis of lorries.

5. Conclusions

In this paper, the aerodynamic performance of platooning lorries under an unmanned scene is studied. Firstly, the significant effect of distance between lorries on aerodynamic performance is revealed by numerical simulation. The experiment shows that the drag coefficient of the middle vehicle decreases when the distance is shortened. Secondly, combined with a deep learning algorithm, a method for constructing an online prediction model of aerodynamic performance of unmanned vehicles based on the stacking ensemble learning model is proposed. The prediction accuracy of the constructed STBP prediction model is improved to 98.56% compared with the traditional BP neural network, and the real-time prediction of drag coefficient under different vehicle distances is effectively realized. Finally, the concept of the digital twin is introduced, and the digital twin system for online monitoring of energy saving and drag reduction in lorries in an unmanned scene is developed. The dynamic mapping between vehicle distance parameters and aerodynamic performance is successfully carried out, which verifies the technical feasibility of realizing energy saving and drag reduction by optimizing the distance between lorries of automatic platooning lorries, and provides theoretical bases and engineering solutions for energy consumption optimization of an intelligent transportation system. However, the study in this project has not yet covered the effect of vehicle type or speed variation. Moreover, as the vehicle types are relatively homogeneous, the generalizability of the prediction model when vehicles are not identical still needs further investigation. Furthermore, with the continuous refinement of wind tunnel facilities and the elevation of the adjustable wind speed range, the Reynolds number of simulation results approaches the real-world value more closely, thereby enhancing the accuracy of prediction models.