Virtual Calibration of Steady-State Emissions for Heavy-Duty Diesel Engines Based on Regression Models

Abstract

To promote the green and low-carbon transition and achieve sustainable development in the transportation sector, virtual calibration technology was employed for the efficient and precise control of emissions from heavy-duty diesel engines and aftertreatment systems. A data-driven, semi-empirical and semi-physical simulation modeling method was proposed. By constructing core modules based on physical mechanisms and refining empirical parameters using experimental data, the method improves computational efficiency while maintaining the prediction accuracy of key parameters. Additionally, a collaborative architecture combining physical actuators and virtual sensor signals was introduced, laying the foundation for the validity of virtual calibration. By innovatively introducing a closed-loop system with real actuators and virtual sensors, the dynamic response characteristics of the control system are faithfully reproduced, providing a reliable environment for validating the results of virtual calibration. Under steady-state conditions, the results demonstrated an average relative error of 1.7% for brake-specific fuel consumption (BSFC) and 6.1% for NOx emissions. An open-loop system for the virtual calibration testing platform was constructed for steady-state calibration. Using the main injection timing and common rail pressure as independent variables, a D-optimal design was utilized to generate 43 sets of experimental points, from which a polynomial regression model was established (R² ≥ 98%). Under the constraints of NOx and pre-turbine temperature, fuel consumption in the low-load range is reduced by 0.5–3 g/kW·h, aftertreatment NOx emissions are reduced by 0.5–3 g/kW·h, and exhaust temperature is increased by 10 °C. This study establishes a complete development workflow consisting of “operating condition design-virtual optimization-bench validation,” significantly enhancing calibration efficiency and engineering applicability. This method shortens the calibration cycle and reduces the number of physical bench tests, providing the industry with a comprehensive calibration methodology tailored to engine operating conditions that is both reproducible and scalable.

1. Introduction

As the core power source for key sectors of the national economy—including long-haul freight transport, construction machinery, mining, and port operations—the energy efficiency and emission control of diesel engines are directly linked to the achievement of the “dual carbon” goals [1]. In 2023, China’s commercial vehicle production and sales exceeded 4 million units. This massive installed base and the ongoing production activities not only underscore the pivotal role of diesel vehicles in modern transportation systems but also make them a significant source of petroleum consumption and air pollutant emissions. Diesel vehicles dominate aftertreatment NOx and PM emissions, accounting for 83.6% and 94.3% of total vehicle emissions, respectively [2,3]. Therefore, achieving efficient and precise control of emissions from heavy-duty diesel engines and their aftertreatment systems has become a critical engineering challenge for driving the green and low-carbon transition in the transportation sector and realizing sustainable development.

While modern engine certification procedures, such as the Worldwide Harmonized Light Vehicles Test Procedure (WLTP) and Real Driving Emissions (RDE), have significantly narrowed the gap between laboratory results and road performance, standardized test cycles often face challenges in fully encompassing the stochastic and diverse nature of real-world driving scenarios [4]. This discrepancy can lead to performance offsets, such as worsening fuel economy due to mismatches in dynamic control [5] and the dual pressure of simultaneous emissions and fuel consumption optimization [6]. Consequently, engines are required to maintain compliance across an expanded operational envelope that exhibits stronger dynamic characteristics and a higher proportion of low-efficiency ranges, exponentially increasing calibration complexity. Virtual calibration, utilizing high-precision mathematical models of the engine, aftertreatment system, and vehicle, offers a robust solution to these challenges. By enabling the exploration and validation of control parameters within a simulation environment, it significantly reduces reliance on physical prototypes [7,8]. Its core advantage lies in the ability to efficiently traverse vast parameter spaces, identifying global optima that are subsequently validated through targeted physical testing.

Early calibration relied heavily on engineers’ experience and extensive bench testing. As system complexity increased, a systematic approach combining Design of Experiments (DOE), Response Surface Modeling (RSM), and optimization algorithms became the mainstream method. Sung et al. [9] employed orthogonal design and response surface methods to effectively optimize fuel injection parameters, reducing both the number of tests and NOx emissions. Boccardo et al. [10] began integrating simulation tools with optimization algorithms to conduct multi-objective optimization studies in a virtual environment. Calibration research on aftertreatment systems has also focused on key issues such as Selective Catalytic Reduction (SCR) urea injection [11] and Diesel Particulate Filter (DPF) regeneration control [12], with an emphasis on robust optimization under transient conditions such as China Heavy-duty Commercial Vehicle Test Cycle (CHTC) [13]. These studies mark a shift in calibration work from purely trial-and-error approaches toward systematic experimentation and optimization; however, the optimization cycles of traditional methods rely heavily on physical test benches. Each round of DOE requires actual engine operation, which is costly and time-consuming, severely limiting the scope of parameter space exploration and the speed of optimization iterations. DOE serves as the data foundation for building surrogate models. Classical methods such as Latin Hypercube Design (LHS) and D-optimal design perform global or local optimal sampling in the factor space, but they still rely on prior knowledge and may fail in regions of unknown strong nonlinearity [14]. Gottorf et al. [15] demonstrated the significant advantages of adaptive sampling in virtual calibration. To improve sampling efficiency, methods based on Bayesian optimization utilize active learning through sequential sampling to construct high-precision models with fewer simulation runs. The current challenge lies in designing an adaptive DOE strategy that is both efficient and robust to address the high-dimensional, transient, and multi-objective characteristics of engines. Specifically, when dealing with China’s unique high-dynamic, low-load operating conditions, it is crucial to ensure that the sampling points fully capture the characteristics of these critical yet difficult-to-optimize regions.

Models in virtual calibration workflows require the rapid generation of large amounts of data for optimization. While purely data-driven models are fast, they suffer from insufficient training data and poor predictive performance. Consequently, research has shifted toward developing physics-data hybrid surrogate models. Jia et al. [16] employed a physics-based flow path combined with an artificial neural network (ANN) combustion model, significantly reducing the amount of experimental data required. Proxy model acceleration techniques are also widely applied; for instance, using Kriging models or simplified models to replace high-fidelity models during optimization iterations can significantly shorten virtual calibration time. The common goal of these strategies is to improve the model’s initial prediction accuracy, with virtual calibration aiming for a single optimization cycle and minimal validation. The sophistication of optimization algorithms directly determines optimization efficiency. Evolutionary algorithms, due to their powerful global search capabilities, have become the standard tool for multi-objective engine calibration [17]. Bhattacharjee et al. [18] utilized deep reinforcement learning to optimize speed trajectories for heavy-duty trucks, thereby reducing fuel consumption. However, this approach requires a large amount of interaction data, and the safety and stability of the strategy remain challenges for deployment. With the rapid increase in control parameters and the explosion of the optimization space’s dimensionality, all algorithms face declining efficiency.

Traditional manual calibration methods are characterized by long development cycles and high costs, making it difficult to keep pace with the rapid evolution of vehicle technology. Virtual calibration technology, with its outstanding efficiency and precision, has emerged as a key solution to overcoming this bottleneck. Virtual calibration transforms physical trial-and-error into digital-driven global optimization. It not only overcomes the physical limitations of test benches, significantly shortens the R&D cycle and reduces costs, but also utilizes algorithms to precisely capture the global optimal solutions that are difficult to discover through manual calibration in complex variable matrices. Thus, it ensures that the power system maintains extremely high calibration accuracy and engineering robustness even under extreme or complex operating conditions. This paper addresses the calibration requirements for Chinese engine operating conditions by integrating D-optimal experimental design, polynomial regression modeling, and DOE multivariate collaborative optimization methods. It establishes an integrated virtual calibration technology system for engine operating condition construction, steady-state/transient parameter optimization, and thermal management calibration. By deeply integrating engine operating conditions into the entire energy consumption and emissions calibration process, this approach establishes a complete development chain encompassing operating condition design, virtual optimization, and test bench validation, thereby significantly enhancing calibration efficiency and engineering applicability.

2. System Description

2.1. Engine Model

Engine test bench data serves as the foundation for constructing high-precision engine simulation models. To meet modeling requirements, data accuracy must be strictly controlled during testing, and a systematic data quality assurance framework must be established. This paper is based on a three-dimensional testing approach covering 12 typical operating conditions, and implements a four-level data quality inspection process to ensure data physical plausibility, multi-source consistency, temporal continuity, and smoothness. The engine main technical parameters are shown in

Table 1. The basic technical parameters of the high-speed engine.

Project	Value
Engine Configuration	Inline-4, 4-valve
Displacement	4.2 L
Rated Speed [rpm]	2300
Rated Power [kW]	147
Maximum Torque Speed [rpm]	1200~1600
Idle Speed [rpm]	600
Maximum Torque [N·m]	720

.

This paper employs a high-precision, multi-physics coupled simulation model to replace a physical diesel engine. Deployed and run on a real-time processor, this model can accurately simulate the engine’s actual operating conditions, thereby enabling virtual testing and calibration of its performance and emissions [19]. In the virtual testing environment, the engine model is integrated into the real-time processor of a Hardware-in-the-Loop (HIL) cabinet and possesses bidirectional interaction capabilities: on one hand, it receives control signals output by the Electronic Control Unit (ECU); on the other hand, it feeds back simulated engine operating state signals to the ECU, thereby reproducing the dynamic response characteristics of a real engine. This paper constructs a high-fidelity simulation model for the engine and aftertreatment system that integrates physical principles with empirical methods, covering key submodules such as the air–fuel system, in-cylinder combustion and heat transfer, and exhaust pollutants, as shown in Figure 1.

2.1.1. Intake and Exhaust Model

The virtual simulation model of the intake and exhaust system is shown in Figure 2. Parameters such as the intake conditions, air–fuel ratio, and EGR rate of the intake and exhaust system have a significant impact on engine fuel economy and emissions performance. Given the real-time requirements of virtual testing for engine intake and exhaust systems, this paper employs a one-dimensional fluid dynamics approach to construct the intake and exhaust system model. The core components include models for the air filter, exhaust turbocharger, intercooler, EGR (exhaust gas recirculation), and intake and exhaust valves [20].

The air filter is modeled as a simplified throttling element, focusing exclusively on flow resistance while neglecting heat exchange. By defining a reference diameter and a flow coefficient, the gas mass flow rate shown in Equation (3) is determined via the orifice plate equation, which accounts for key thermodynamic and geometric variables including inlet temperature, inlet pressure, and orifice area:

$${\displaystyle \frac{\mathrm{d}m}{\mathrm{d}t}}=\mu \cdot A_{\mathrm{ref}}\cdot p_{\mathrm{in}}\cdot \beta \cdot \sqrt{{\displaystyle \frac{2}{R_{\mathrm{in}}{T}_{\mathrm{in}}}}}$$

(1)

$\mu$ is the flow coefficient; $A_{\mathrm{ref}}$ is the orifice area of the throttling element; $P_{\mathrm{in}}$ is the inlet pressure; $R_{\mathrm{in}}$ is the gas ratio constant; $T_{\mathrm{in}}$ is the inlet temperature; $\beta$ is the flow function, which depends on the specific heat ratio and pressure ratio of the gas passing through the throttling element:

$$\mathsf{\beta }=\sqrt{{\displaystyle \frac{\mathrm{k}}{\mathrm{k}-1}}\left[{\left({\displaystyle \frac{{\mathrm{p}}_{\mathrm{out}}}{{\mathrm{p}}_{\mathrm{in}}}}\right)}^{{\displaystyle \frac{2}{\mathrm{k}}}}-{\left({\displaystyle \frac{{\mathrm{p}}_{\mathrm{out}}}{{\mathrm{p}}_{\mathrm{in}}}}\right)}^{{\displaystyle \frac{\mathrm{k}+1}{\mathrm{k}}}}\right]}$$

(2)

k represents the specific heat ratio; ${\displaystyle \frac{{\mathrm{p}}_{\mathrm{out}}}{{\mathrm{p}}_{\mathrm{in}}}}$ represents the pressure ratio.

2.1.2. Turbocharger Model

The turbocharger model employs the characteristic curve method, based on the performance diagrams of the compressor and turbine [21]. The model accounts for the relationships between rotational speed, flow rate, pressure ratio, and efficiency, and adapts to different operating conditions through environmental correction factors. The compressor power shown in Equation (3) is determined by the mass flow rate of air and the enthalpy difference:

$${\mathrm{p}}_{\mathrm{c}}=\dot{m}\cdot ({\mathrm{h}}_{\mathrm{c}2}-{\mathrm{h}}_{\mathrm{c}1})$$

(3)

${\mathrm{p}}_{\mathrm{c}}$ represents the power consumed by the compressor; $\dot{m}$ represents the mass flow rate of air within the compressor; ${\mathrm{h}}_{\mathrm{c}2}$ represents the enthalpy at the compressor outlet; and ${\mathrm{h}}_{\mathrm{c}1}$ represents the enthalpy at the compressor inlet. The enthalpy difference $({\mathrm{h}}_{\mathrm{c}2}-{\mathrm{h}}_{\mathrm{c}1})$ shown in Equation (4) can be further derived from the gas states at the compressor inlet and outlet:

$$h_{c2}-h_{c1}={\displaystyle \frac{1}{{\eta }_{s_{2}c}}}\cdot c_p\cdot T_{c1}\left[{\displaystyle \frac{p_{c2}^{\frac{k-1}{k}}}{p_{c1}}}-1\right]$$

(4)

${\eta }_{s_2\mathrm{c}}$ is the isentropic efficiency of the compressor; $T_{c1}$ is the compressor inlet temperature; ${\displaystyle \frac{P_{c2}}{P_{c1}}}$ is the compressor pressure ratio; $c_p$ is the average isobaric specific heat of the compressor; $k$ is the specific heat ratio.

The turbine power is calculated in Equation (5):

$${\mathrm{p}}_{\mathrm{t}}=\dot{m}\cdot ({\mathrm{h}}_{t1}-{\mathrm{h}}_{t2})$$

(5)

${\mathrm{p}}_{\mathrm{t}}$ represents the power generated by the turbine; $\dot{m}$ represents the mass flow rate of the exhaust gas within the turbine; ${\mathrm{h}}_{t1}$ represents the enthalpy at the turbine inlet; and ${\mathrm{h}}_{t2}$ represents the enthalpy at the compressor inlet.

The enthalpy difference is shown in Equation (6):

$${\mathrm{h}}_{t1}-h_{t2}={\eta }_{S,T}\cdot C_p\cdot T_{t1}\left[1-{{\displaystyle \frac{p_{t2}}{p_{t1}}}}^{{\displaystyle \frac{k-1}{k}}}\right]$$

(6)

${\eta }_{S,T}$ represents the isentropic efficiency of the turbine; $T_{t1}$ is the turbine inlet temperature; ${\displaystyle \frac{p_{t2}}{p_{t1}}}$ is the static expansion ratio of the turbine; $c_p$ is the average isobaric specific heat of the turbine; and $k$ is the specific heat ratio.

The overall efficiency of the turbocharger is shown in Equation (7):

$${\eta }_{CT}={\eta }_{\mathrm{m},CT}\cdot {\eta }_{s,C}\cdot {\eta }_{S,T}$$

(7)

${\eta }_{CT}$ represents the overall efficiency of the turbocharger; ${\eta }_{\mathrm{m},CT}$ represents the mechanical efficiency of the turbocharger.

2.1.3. Intercooler Model

An intercooler is used to cool the high-temperature air after turbocharging, thereby increasing the air mass flow rate. The model primarily accounts for cooling and pressure drop effects, assuming that the coolant temperature remains constant and that the pressure drop is caused solely by friction [22]. The outlet temperature shown in Equation (8) is calculated:

$$T_{c2}=T_{cool}+{\eta }_{cool}(T_{c2}-T_{liquid})$$

(8)

$T_{c2}$ represents the temperature of the gas after compression by the compressor; $T_{cool}$ represents the temperature of the gas after cooling by the intercooler; ${\eta }_{cool}$ represents the cooling efficiency of the intercooler; and $T_{liquid}$ represents the temperature of the coolant.

2.1.4. EGR

The EGR system recirculates a portion of the cooled exhaust gas back into the intake manifold to lower combustion temperatures and reduce engine-out NOx emissions. Its model and structure, shown in Figure 3, include the EGR valve, intake and exhaust piping, and the cooler. The model accounts for the control characteristics of the EGR rate, the heat transfer efficiency of the cooler, and the pressure losses and thermal inertia in the piping [23].

Like the intercooler, exhaust gas recirculation primarily serves to reduce temperature and create a pressure drop. When exhaust gases from the engine pass through the EGR valve, a partial pressure drop occurs. The pressure drop is shown in Equation (9):

$$p_{\mathrm{in}}-p_{out}={\displaystyle \frac{m^2}{2\cdot {\rho }_{mix}\cdot A^2\cdot {\mu }^2}}$$

(9)

$p_{\mathrm{in}}$ represents the pressure at the EGR inlet; $p_{out}$ represents the pressure at the EGR outlet; $A$ represents the valve cross-sectional area; and $\mu$ represents the flow coefficient. ${\rho }_{mix}$ represents the density of the mix exhaust gas.

The gas passing through the EGR valve is cooled by the cooler; the relationship between the inlet and outlet temperatures is shown in Equation (10):

$$T_{in}=T_{out}+{\eta }_{EGR}(T_{in}-T_{liquid})$$

(10)

$T_{in}$ is the exhaust inlet temperature; $T_{out}$ is the exhaust outlet temperature; ${\eta }_{EGR}$ is the cooling efficiency; and $T_{liquid}$ is the coolant temperature.

2.1.5. Intake and Exhaust Valve Models

The intake and exhaust manifolds are modeled using the filling-and-emptying method, which accounts for the dynamic flow characteristics and thermal inertia of the gas, enabling accurate simulation of pressure wave propagation and temperature changes. The valve model controls the gas flow into and out of the cylinder, without considering heat transfer or pressure losses [24]. The effective cross-sectional area is calculated as shown in Equation (11):

$$A_{\mathrm{eff}}={\mathrm{f}}_{\mathrm{sc}}\cdot {\alpha }_0\cdot {\displaystyle \frac{{\mathrm{d}}_{\mathrm{pi}}^2\cdot \pi }{4}}$$

(11)

where $A_{\mathrm{eff}}$ is the effective flow area of the valve; ${\alpha }_0$ is the valve angle; ${\mathrm{d}}_{\mathrm{pi}}$ is the outer diameter of the valve seat; ${\mathrm{f}}_{\mathrm{sc}}$ is a parameter coefficient shown in Equation (12) used to account for the effects of different valve structures, the number of valves, and other factors on the effective flow area:

$${\mathrm{f}}_{\mathrm{sc}}={\mathrm{n}}_{\mathrm{v}}\cdot {\left({\displaystyle \frac{{\mathrm{d}}_{\mathrm{vi}}}{{\mathrm{d}}_{\mathrm{pi}}}}\right)}^2$$

(12)

${\mathrm{n}}_{\mathrm{v}}$ represents the number of valves; ${\mathrm{d}}_{\mathrm{vi}}$ represents the inner diameter of the valve seat. Valve lift and the flow coefficient are obtained from characteristic curves, and the valve lift value can be selected within the crankshaft angles of −360° to 360° and 0° to 720°:

$${\mathrm{l}}_{\mathrm{lift}}=\mathrm{f}\left[\alpha \right]$$

(13)

$$C_0=\mathrm{f}\left[{\mathrm{l}}_{\mathrm{lift}}\right]$$

(14)

${\mathrm{l}}_{\mathrm{lift}}$ represents valve lift; $\alpha$ represents crankshaft angle; and $C_0$ represents the flow coefficient.

2.1.6. Pollutant Emission Model

The model employs differentiated prediction methods for different pollutants [25]: NOx emissions are predicted using a semi-physical model based on the Zeldovich mechanism, which correlates the maximum in-cylinder temperature, oxygen concentration, and duration of high temperatures, and is directly driven by pressure and temperature data output from the combustion model. CO, THC, and soot: Due to their complex formation mechanisms (involving local oxygen deficiency, wet fuel walls, etc.), data-driven models are employed—requiring the design of multivariate DOE test conditions (covering parameters such as injection timing, EGR rate, and common rail pressure), the collection of emission data across all operating conditions, and the establishment of “input parameter–emission concentration” mapping relationships via polynomial regression or neural networks. To enhance the model’s extrapolation capability and real-time performance, data-driven models must adhere to two key principles: First, prioritize “physically mechanistically related parameters” as inputs (e.g., using intake air oxygen concentration instead of EGR rate, as oxygen concentration directly influences the combustion process). Second, while ensuring accuracy, input variables must be streamlined (e.g., selecting three core parameters: injection timing, common rail pressure, and cylinder temperature), and a polynomial model should be adopted to balance computational speed and accuracy, thereby meeting the real-time requirements of the mean-based model.

2.2. The Test Benches

The main test equipment used in the experiment and its specifications are shown in

Table 2. The test bench equipment of the high-speed engine.

Name		Parameters
Electric Dynamometer	AVL INDY P44	Torque: ±0.3% F.S. Rotational speed: ±1 r/min
Dynamometer Operating	AVL PUMA 1.5.3
Engine Intake System	AVL ACS2400FH	Pressure: ±1 mbar Temperature: ±0.5 °C Humidity: ±3%
Gas Analyzer	AVL AMA i60	±2%
Particle Counter	AVL 489	±10%
Fuel Consumption Meter	AVL 753C/735S	±0.12%
Particulate Matter Sampling	AVL SPC 472	Response time: ≤0.3 s Flow rate: ≤±5%
Real-Time System	CONNECT™ (RT)	/
Coast-Down Evaluation	AVL Coastdown	/

, and the test bench layout is shown in Figure 4.

2.3. Model Accuracy and Validation

Static Model Calibration

As shown in the Figure 5. The model demonstrated high predictive accuracy across key performance metrics. Specifically, torque and brake specific fuel consumption exhibited relative errors within 3% and 1.7%, respectively, validating the robustness of the powertrain and fuel economy simulations. In-cylinder processes were captured with high fidelity, as evidenced by deviations in peak cylinder pressure (≤5%), air mass flow (≤4%), and EGR rate (≤4%). Furthermore, the turbocharger speed simulation proved exceptionally precise, maintaining a deviation below 2%, which provides a reliable basis for the dynamic optimization of the boosting system.

Exhaust temperature is a key input parameter for modeling aftertreatment systems (such as SCR and DPF), and it is essential to ensure that its prediction accuracy meets the requirements for calculating aftertreatment efficiency. As shown in Figure 6, although there is some dispersion between the simulation results and measured data for exhaust temperature (pre-turbo and post-turbo), they generally meet engineering accuracy requirements: the deviation in exhaust temperature is small under full-load conditions, while it is slightly larger under low-load conditions. This is primarily due to fluctuations in combustion heat release at low loads and can be further optimized through fine-tuning of local parameters in the combustion model.

Intake pressure directly determines the engine’s air-charging efficiency and serves as a key validation metric for assessing the compatibility between the intake manifold flow resistance model and the turbocharging system. As shown in Figure 7, the simulation accuracy of intake pressure (post-supercharger and intake manifold) across all operating modes is high, with deviations from experimental values maintained within ±10 kPa: the deviation for post-supercharger pressure is less than 8 kPa, and the deviation for intake manifold pressure is less than 5 kPa. This accurately reproduces the throttling losses in the intake duct and the pressure control characteristics of the supercharging system, providing a reliable basis for calculating charging efficiency.

Exhaust pressure affects the turbine’s power output and exhaust backpressure loss, and serves as the core validation parameter for matching the exhaust duct flow resistance model with the turbocharger. As shown in Figure 8, the simulated and measured values of exhaust pressure (upstream and downstream of the turbine) show good agreement across different modes, with deviations controlled within ±20 kPa: the upstream pressure deviation is less than 15 kPa, and the downstream pressure deviation is less than 20 kPa, ensuring the accuracy of turbine power calculations and the flow resistance characteristics of the exhaust system.

Validation of pollutant emissions (NOx, CO, and THC) centers on the model’s capability to predict formation patterns during combustion, establishing a basis for subsequent calibration and optimization. As illustrated in Figure 9, simulated engine-out NOx concentrations are marginally lower than experimental measurements, with an average relative deviation ≤ 8%. Despite this offset, the model tracks experimental trends across varying load conditions with high fidelity (R² ≥ 0.92). While CO and THC predictions exhibit greater sensitivity under low-load, incomplete combustion regimes (deviations ≤ 20%), the integration of a closed-loop model—incorporating real-time corrections for intake temperature and air–fuel ratio—refines the prediction accuracy to within 15%, satisfying the criteria for emission calibration.

3. Virtual Calibration Test Platform

The virtual calibration test system serves as the core platform for achieving collaborative optimization between real ECUs and virtual models. The model was developed on MATLAB 2024a. Its development must address two key challenges: ensuring the authenticity of ECU signal interactions and reproducing the dynamic response of actuators. We focused on Open-Loop System Construction, as the foundation of a closed-loop system, the core objective of the open-loop system is to establish a communication link between the ECU, the signal interface, and the physical actuators, thereby laying the signal and hardware groundwork for the subsequent integration of high-precision engine and aftertreatment models to form a complete virtual calibration platform. System development follows the engineering logic of requirements analysis, architectural design, modeling and integration, and test verification. The overall process is divided into two phases: open-loop construction and closed-loop expansion. The system architecture is shown in Figure 10.

3.1. Overall Architecture and Development Logic

Traditional Hardware-in-the-Loop test platforms primarily focus on verifying ECU functional logic, but the pure model simulation of actuators used in these platforms has two major limitations: ① The dynamic responses of complex actuators (such as fuel injectors and EGR valves)—including fuel injector inrush current and EGR valve hysteresis characteristics—are difficult to model accurately. ② Pure models cannot reproduce the actual load interaction between the ECU and actuators (such as voltage drops and current fluctuations in drive signals). To address this, this open-loop system proposes a core strategy for the physical implementation of actuators: integrating physical actuators such as electronic throttle valves, EGR valves, fuel injectors, SCR urea nozzles, and injection pumps (Figure 11). This ensures that the drive signals output by the ECU act on real loads, maintaining the physical authenticity and dynamic response characteristics of the control loop, and providing the ECU with an operating environment consistent with that of the actual vehicle.

3.2. System Testing and Validation

A visual test interface was built using ControlDesk (including screens for master control, actuator monitoring, CAN monitoring, etc.), as shown in Figure 12. In conjunction with tools such as INCA 2024 software and an oscilloscope, tests were conducted on signal consistency, stability, and dynamic response to verify whether the ECU can operate normally in an open-loop system.

Data acquisition involved capturing the crankshaft and camshaft signals from the open-loop system, with the ECU-detected engine speed monitored via INCA. As illustrated in Figure 13, the crankshaft signal (60-2 teeth) exhibited no tooth-count recognition errors, while the camshaft signal (4 + 1 teeth) remained correctly phased. Furthermore, the engine speed detected by the ECU showed high precision with an error margin less than 5 r/min. Fuel injection timing remained consistent throughout the test, with no detectable abnormalities.

4. Steady-State Optimization and Calibration

4.1. Calibration Methods and Procedures

Steady-state calibration uses the MAP obtained from the original engine’s World Harmonized Stationary Cycle (WHSC) calibration as the initial reference. It employs a model-based closed-loop multi-coupled calibration method and is optimized within the range covering typical operating conditions in China (1000–1600 r/min, 15–100% load). The process consists of four stages: experimental design, automated testing on a virtual test bench, data analysis and modeling, and numerical optimization and validation, as shown in Figure 14.

4.2. Experimental Design Methods

The D-optimization was employed to design two core calibration variables—main injection timing and common-rail pressure—with the objective of minimizing fuel consumption and NOx emissions within the constraints, resulting in 43 sets of variable combinations, as shown in Figure 15. Three methods including Latin Square Design, Uniform Design and D-Optimization Design are compared. Under the premise of a known regression model structure (e.g., a third-order polynomial), optimal test points are determined by minimizing the volume of the confidence ellipsoid for regression coefficients. This design offers high experimental efficiency and is particularly well-suited for China VI engine calibration (where most responses follow a third-order polynomial distribution). The parameters including RPM, engine torque, rail pressure, and main injection timing are settings in Equation (15):

$$\left\{\begin{array}{c}\begin{array}{c}{x}_1\in X_1\subseteq \left[800,2000\right]\mathrm{r}\cdot {\min }^{-1}\\ x_2\in X_2\subseteq \left[200,1200\right]\mathrm{N}\cdot \mathrm{m}\\ x_3\in X_3\subseteq \left[50,200\right]\mathrm{bar}\\ x_4\in X_4\subseteq \left[0,20\right]\mathrm{deg}\end{array}\end{array}\right.$$

(15)

4.3. Development of the Regression Model

The rail pressure and main injection timing were used as independent variables, employ stepwise polynomial regression, and perform outlier handling and modeling based on a virtual calibration platform. The optimization objective is to minimize fuel consumption, with constraints including engine-out NOx less than 0.46 g/kW·h and pre-turbine exhaust temperature more than 250 °C. Variable 1 represents rail pressure, and Variable 2 represents injection timing.

After adopting the polynomial regression model, the experimental data set was reduced, and the model quality was improved. Therefore, the polynomial model was used during the steady-state calibration process. The model is shown in Equation (16):

$$Y=C+T\times \left(B\times X\right)+\epsilon$$

(16)

$$B=\left[\begin{array}{cccc}\begin{array}{c}{b}_{11}\\ b_{21}\\ \dots \\ b_{n1}\end{array}& \begin{array}{c}{b}_{12}\\ b_{22}\\ \dots \\ b_{n2}\end{array}& \begin{array}{c}\dots \\ \dots \\ \dots \\ \dots \end{array}& \begin{array}{c}{b}_{1k}\\ b_{2k}\\ \dots \\ b_{nk}\end{array}\end{array}\right],X={\left[\begin{array}{cccc}\begin{array}{c}{X}_{11}^1\\ X_{11}^2\\ \dots \\ X_{11}^m\end{array}& \begin{array}{c}{X}_{12}^1\\ X_{12}^2\\ \dots \\ X_{12}^m\end{array}& \begin{array}{c}\dots \\ \dots \\ \dots \\ \dots \end{array}& \begin{array}{c}{X}_{1k}^1\\ X_{1k}^2\\ \dots \\ X_{1k}^m\end{array}\end{array}\right]}^T,T={\left[\begin{array}{c}1\\ 1\\ \dots \\ 1\end{array}\right]}^T$$

(17)

$$\epsilon ~N\left(0,{\sigma }^2\right)$$

(18)

The intersection method was used to optimize an existing regression model: under the premise of aligning with the actual operating conditions of the engine and aftertreatment system and meeting the calibration optimization objectives, it selects data points for combinations of calibration variables. By recombining the results of this data point selection with the variable combinations obtained from the previous simulation tests and re-modeling, the model quality can be effectively improved. Before selecting optimization points, the need for model optimization can be assessed by verifying the validity of the model curves—specifically, whether the measured test data aligns closely with the model trends.

4.4. Calibration and Optimization Results

Model quality was evaluated using the coefficient of determination (R²), F-statistic, and p-value, while the percentage of repeatability across test conditions was introduced to assess system stability. Both the fuel consumption and engine-out NOx models achieved R² values exceeding 98%, with statistically significant regression (p < 0.05), thereby meeting engineering application requirements. Figure 16 shows the evaluation results of the exhaust temperature model. The engine exhaust temperature exhibits a clear linear relationship, and the regression model demonstrates excellent significance. Except for the full-load conditions at 1000 rpm and 1300 rpm, where R² < 91%, R² exceeds 91% at all other operating points. The poor model quality at the full-load points is attributed to fuel supply constraints in the full-load external characteristic; after adjusting the electronic control parameters, the required torque could not be achieved, resulting in fluctuations in the dynamometer.

Figure 17 shows the effects of the calibration variables (injection pressure, injection timing, and EGR rate) on fuel consumption. Across the entire MAP range, at an EGR rate of 4%, rail pressure has a minor effect on fuel consumption, while injection timing has a significant effect; when the EGR rate increases to 13%, the influence of rail pressure on fuel consumption becomes more pronounced; fuel consumption reaches its minimum value (189.3 g/kW·h) at an EGR rate of approximately 8%. This indicates that the effects of injection pressure, injection timing, and EGR rate on fuel consumption are closely coupled. For each operating condition, there exists an optimal calibration result, demonstrating strong variable coupling and the need for multivariate collaborative calibration.

4.4.1. Steady-State Performance Before CASC Cycle Calibration Optimization

As shown in

Table 3. Emission and fuel consumption rate results before and after optimized calibration.

Case	Fuel	NOx	CO	HC	Net Power
	[g/kW·h]	[g/kW·h]	[g/kW·h]	[g/kW·h]	[kW]
Original engine and WHSC	207.60	5.60	0.27	0.13	30.25
Original engine and CAS	212.45	5.89	0.29	0.17	22.66
Optimization and calibration	209.81	6.76	0.27	0.16	22.63

, when using the reference electronic control data for the original engine and aftertreatment system (in mass-production condition), fuel consumption in the CASC cycle was 1.85% higher than in the WHSC, and aftertreatment NOx emissions increased by 5.08%. The increase in fuel consumption is primarily due to the higher proportion of light-load conditions in the CASC operating conditions. The deterioration of fuel economy under light-load conditions is primarily driven by a sharp decline in mechanical efficiency, as fixed frictional losses consume a disproportionate share of the reduced work output, and a degradation in thermodynamic efficiency caused by lower combustion temperatures and increased relative heat losses. These factors, compounded by heightened pumping losses from intake throttling, force the engine to operate on the steep, inefficient left side of the U-shaped BSFC curve; in this state, a significant portion of fuel energy is diverted to overcoming internal resistance rather than producing effective shaft power, resulting in remarkably high specific fuel consumption despite low absolute hourly consumption; while the absolute increase in NOx is not large, the proportional increase is more noticeable due to the lower baseline value; the rise in NOx is mainly attributable to the higher calibrated emissions of the original vehicle’s aftertreatment system within this operating range; changes in CO and THC emissions are not significant.

4.4.2. Steady-State Performance After CASC Cycle Calibration Optimization

As shown in Figure 18, based on the calibrated MAP distribution data, engine fuel consumption decreased by 0.5–3 g/kW·h, with only a slight increase in fuel consumption (0.5–1 g/kW·h) in the high-speed, high-load range. Since the target vehicle model typically operates below 1500 rpm, the slight deterioration in high-speed fuel consumption has no practical impact on the vehicle’s overall fuel economy. After optimization, aftertreatment NOx emissions increased by 0.5–3 g/kW·h in the high-load range and decreased by 0.5–3 g/kW·h in the low-load range. This adjustment was made because, under typical Chinese driving conditions, the engine operates predominantly in the low-load range; reducing NOx in this range helps control overall vehicle emissions. At the same time, fuel consumption in the low-load range was improved, while a slight increase in NOx in the high-load range significantly optimized fuel economy.

Furthermore, based on the CASC cycle results in Table 3, the approximately 10 °C increase in exhaust temperature in the low-load range helps improve SCR conversion efficiency, making emission control more effective and reducing urea consumption. Consequently, compared to the pre-optimization calibration, fuel consumption in the CASC cycle decreased by 1.2%. Although aftertreatment NOx emissions increased by 14.8%, the post-optimization NOx emissions were 6.8 g/kW·h. As long as the aftertreatment efficiency exceeds 95%, emission regulations can be met, and emission control under low-load conditions has been significantly improved.

In summary, by optimizing the emission zones for Chinese engine operating conditions using a model-based calibration method, reductions in fuel consumption and aftertreatment NOx emissions were achieved within the typical operating range. The optimized data demonstrate superior vehicle adaptability compared to the original engine.

5. Conclusions

A comprehensive virtual calibration model for the engine and aftertreatment system was developed and validated through test bench experiments. An open-loop system for the virtual calibration testing platform was constructed for steady-state calibration.

• A data-driven, semi-empirical and semi-physical simulation modeling method was proposed. By constructing core modules based on physical mechanisms and refining empirical parameters using experimental data, the method enhances computational efficiency while maintaining the prediction accuracy of key parameters. Additionally, a collaborative architecture combining physical actuators and virtual sensor signals was introduced, laying the foundation for the validity of virtual calibration. By innovatively introducing a closed-loop system with real actuators and virtual sensors, the dynamic response characteristics of the control system are faithfully reproduced, providing a reliable environment for validating the results of virtual calibration.
• Under the constraints of engine-out NOx and pre-turbine temperature, fuel consumption in the low-load range is reduced by 0.5–3 g/kW·h, NOx emissions are reduced by 0.5–3 g/kW·h, and exhaust temperature is increased by 10 °C. Compared to the pre-optimization calibration, fuel consumption in the CASC cycle decreased by 1.2%. Although NOx emissions increased by 14.8%, the post-optimization NOx emissions were 6.8 g/kWh. As long as the aftertreatment efficiency exceeds 95%, the engine meets emission regulations, and emission control under light-load conditions has significantly improved.
• By applying a model-based calibration method optimized for the emission zones typical of Chinese engine operating conditions, the fuel consumption and lower NOx emissions achieved reduced within the common operating conditions. The optimized data demonstrated superior vehicle adaptability compared to the original engine. This method shortens the calibration cycle and reduces the number of physical bench tests, providing the industry with a comprehensive calibration methodology tailored to engine operating conditions that is both reproducible and scalable.