Transient NOx emission modeling of a hydrogen-diesel engine using hybrid machine learning methods

Abstract

One promising approach to reduce carbon foot print of internal combustion engines (ICEs) is using alternative fuels like hydrogen, particularly by converting medium and heavy-duty diesel engines to dual-fuel hydrogen-diesel engines. To minimize elevated NOx emissions from hydrogen-fueled engine, fast and accurate emission models are essential for engine model-based control and for engine calibration and optimization using hardware-in-the-loop (HIL) setups. In this study, a fast-response NOx emissions sensor is used to measure the transient NOx emissions from a dual-fuel hydrogen-diesel engine. Subsequently, steady-state models (SSMs), quasi steady-state models (QSSMs), and transient sequential models (TSMs) in the form of black-box (BB) and gray-box (GB) models are developed for transient NOx emissions prediction. GB models utilize both information from a one dimensional (1D) physical engine model and experimental data for training, while BB models only use experimental data. SSMs are optimized artificial neural networks (ANNs) trained using steady-state data, QSSMs are optimized ANNs trained using transient data, and TSMs are time-series networks trained using transient data. Long short-term memory (LSTM) and gated recurrent unit (GRU) networks are used as the time-series deep learning networks. The results showed that the 1D physical model has the poorest performance with successive model performance improvement from SSM to QSSM and from QSSM to TSM. The developed BB TSM model in this study can predict transient NOx emissions with an R² value greater than 0.96 at 89,000 predictions per second which makes this model suitable for real-time engine model-based control where computational efficiency is crucial. The developed GB TSM model can predict transient NOx emissions with an R² value greater than 0.97 but it is computationally more expensive. The extra accuracy of the GB TSM models makes them the best choice for HIL setups where more computational power is available, and accuracy is more crucial.

Introduction

According to national inventory of Canada, carbon dioxide $(\mathrm{C}{\mathrm{O}}_2)$ represents more than 80% of all greenhouse gases (GHG) produced by human activities. ¹ Burning fossil fuels in the transportation sector by the internal combustion engines (ICEs) is the second major source of $\mathrm{CO}2$ production which accounts for over 28% of the total $\mathrm{C}{\mathrm{O}}_2$ emissions production. ¹ Medium and heavy duty vehicles are mostly powered by compression ignition (CI) or diesel engines burning diesel fuel. Diesel engines are responsible for 25% of the fuel consumption and $\mathrm{C}{\mathrm{O}}_2$ emissions production in the transportation sector and this share has been constantly increasing in the recent years. ¹ In addition, electrification which is progressing fast in light duty vehicles is more challenging for medium and heavy duty section. ² Increasing the share of $\mathrm{C}{\mathrm{O}}_2$ production, electrification challenges and more stricter emission regulations for medium and heavy duty vehicles makes searching for alternative solutions necessary. Modifying current diesel engines to burn an alternative zero carbon fuel as the second fuel could be a potential solution that can highly reduce diesel engines carbon tailpipe emissions including $\mathrm{C}{\mathrm{O}}_2$ , soot, CO, and unburned hydrocarbons.^3,4

Hydrogen is considered a promising alternative fuel source due to its unique properties. It is a clean energy source as it results in zero carbon engine-out emissions and can be produced using renewable energies. ³ Additionally, its low ignition energy means that it can ignite quickly and burn cleanly, while its rapid flame propagation ensures efficient combustion. ³ Compression ignition engines, also known as diesel engines, have limitations when it comes to running solely on hydrogen fuel. The compression temperature in these engines is usually not high enough to initiate combustion of hydrogen.^3,5 As a result, diesel engines are unable to directly use hydrogen as the sole fuel source. ⁵ However, these engines can be converted with minimum changes to the engine to dual-fuel hydrogen-diesel where the diesel fuel ignites the premixed mixture of hydrogen-air.^5–7

Adding hydrogen to CI engines increases the combustion temperature, resulting in the production of higher levels of nitrogen oxides $\left(\mathrm{N}{\mathrm{O}}_{\mathrm{x}}\right)$ emissions. ³ $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions are harmful pollutants that have negative impacts on human health and the environment. Therefore, converting diesel engines to operate as dual-fuel diesel-hydrogen engines can effectively reduce their carbon footprint, but it may also result in an increase in $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions. ³ To minimize the $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions production in hydrogen-diesel CI engines, precise control of the engine is crucial. Model-based control is a highly effective approach for minimizing emissions while preserving high efficiency.^8,9 For effective model-based control, accurate transient emission models are required.^8,10 In addition, accurate transient emission models can also play a significant role in hardware-in-the-loop (HIL) setups for engine optimization and calibration. These models can serve as virtual engines, reducing the need for costly experimental tests and providing a cost-effective and efficient solution for engine optimization and calibration. ¹⁰

This study aims to address the following research gaps based on the current literature in $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ transient emission modeling of ICEs:

• The current literature is abundant in transient emission studies for traditional SI and CI internal combustion engines (ICEs) fueled by fossil fuels. However, there is a significant lack of research on transient emission modeling for engines utilizing alternative fuels. This gap in the literature is particularly important for engines fueled by hydrogen, as the addition of hydrogen is known to increase combustion temperatures and $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions. To address this deficiency, this study presents a comprehensive investigation into the development of various transient emission models for a dual-fuel hydrogen-diesel engine. Accurate transient $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emission models are crucial for the effective model-based control of dual-fuel hydrogen-diesel engines which are well-suited for medium and heavy duty tasks that are more challenging to electrify compared to light-duty vehicles. ²

• Data-driven modeling of transient emissions can be accomplished through the use of three main methodologies: SSM, QSSM, and TSM. While some studies have compared the performance of two of these methods, there is a lack of research that has investigated and compared the performance of all three models for the same engine and the same dataset. This study compares the performance of SSM, QSSM, and TSM methodologies on both steady-state and transient $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emission data from hydrogen-diesel CI engine which provides insights into the strengths and limitations of each methodology.

• The use of gray-box (GB) emission modeling has been demonstrated to enhance steady-state emission modeling^11–15; however, the application of GB modeling for transient emission modeling remains unexplored in the literature. This study investigates the use of GB modeling technique in the form of SSM, QSSM, TSM, for modeling transient $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions.

• In the context of engine model-based control, having a computationally efficient accurate emission model is crucial. To address this need, this study investigates a range of different architectures and algorithms for time-series TSM modeling and compares their performance in terms of both accuracy and computational cost. By doing so, this study provides valuable guidance on how to choose the most appropriate TSM model-based on the desired trade-off between accuracy and computational cost, taking into consideration the available computational resources. The results of this study can help researchers and practitioners make informed decisions about which TSM model is best suited for their particular application, ensuring that the model is both accurate and computationally efficient.

The paper is organized into sections, beginning with a presentation of background studies on $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions modeling. Then, the experimental setup, physics-based combustion models, ML and DL methods, and feature selection (FS) are explained in detail in the methodology section. The results and discussion section compares the transient $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions prediction results obtained from various models and analyzes the significance of the results and discuss the performance of each model. In the conclusion section, the key insights gained from the study are highlighted. All abbreviations used throughout this paper are defined in the Nomenclature in the Appendix section.

Background NOx emissions modeling studies

Emissions are often modeled through three main methodologies: physical models or white-box (WB), black-box (BB) models, and GB models. Model-based control requires models to be fast enough for real-time emission prediction. This consideration makes complex and accurate physical models not applicable, and only low dimensional and empirical physical models can be used in this application. ¹⁶ BB models are machine learning (ML) algorithms that are trained using experimental data. Simulation data can also be used for BB modeling, but it will lower the accuracy of the model depending on the accuracy of the simulation data. One advantage of BB models is that they are significantly faster compared to physical models. Finally, GB models are a combination of the WB and BB modeling, where a combination of data from a physical model and experimental data is used for training an ML method. GB models can be as fast as low dimensional physical models and more accurate than BB models because they use extra information provided by the physical model.^11–14

ML algorithms can be utilized for transient emission modeling through GB and BB models in three distinct ways, depending on the algorithm type and the source of the training data. The most straightforward approach in transient emission modeling is to utilize steady-state emission models (SSM). These models consist of a classical ML algorithm that has been trained using steady-state emission data, and are now applied for the prediction of transient emissions. Due to the lack of transient emission data in the training process, these models have limited capacity to accurately predict the emissions. GB version of these models can perform better to a certain extent depending on the accuracy of the physical model. This is because the physical model can provide additional information and incorporate certain elements of the transient nature of transient emissions into the ML method. In Norouzi et al. ¹⁷ and Aliramezani et al. ¹⁸ support vector machine (SVM) algorithm was trained using steady-state data for predicting $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions from a diesel engine. Two models are developed, including a low order model with nine features and a high order model with 29 features. Results showed that the more complex high order model performs better in predicting transient $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions. The developed models then were used for model-based control of the engine.

The second approach for utilizing ML in transient emission modeling involves training classical ML algorithms with transient emission data. In this methodology, the emission value at each time step is treated as an individual steady-state case during the training process. These models are referred to as quasi steady-state models (QSSMs) and have the advantage of using the same type of data for both training and testing. However, the disadvantage of these models is that they cannot account for the sequential nature of transient emissions. This means for predicting each case algorithm only relies on the current state and not the previous states, which does not reflect the reality of transient emissions. Similar to SSMs, the performance of QSSMs can be improved through the implementation of a GB version, as the physical model can provide additional information and capture some aspects of the transient behavior. In Jeyamoorthy et al., ¹⁹ QSSM models for $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions are developed using different classical ML methods including artificial neural network (ANN), extreme gradient boosting, random forest, SVM, and decision tree regression using in-cylinder pressure data as the input. Results showed that all of the ML models are accurate $({\mathrm{R}}_{\mathrm{val}}^2>0.89)$ and can be used as the virtual sensors. However, the study had a drawback in that it lacked an assessment of the models on new test data, and only the validation error terms were reported. This could result in a discrepancy between the performance of the models on validation data and on unseen test data. In Brusa et al., ²⁰ $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions data from a spark ignition (SI) engine for six real driving emission (RDE) cycles were used to develop QSSMs using ANN, SVM, Polynomial Regressor, Random Forest, Light Gradient Boosting Regressor ML models. Both steady-state and transient performance of the developed QSSMs were tested using two other RDE cycles and a steady-state map of $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions. Results showed that QSSMs can predict $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions for both steady-state and transient condition with ${\mathrm{R}}_{\mathrm{test}}^2$ greater than 0.9. It was also found that the QSSMs are more accurate in predicting transient emissions compared to steady-state emission, as they were specifically trained using transient emission data.

The third strategy for modeling transient emissions using ML methods involves the use of transient sequential models (TSMs). In this approach, a deep learning (DL) recurrent neural network (RNN) is trained using transient emission data. RNNs have the capability to capture the sequential nature of transient emissions, meaning that the emission value at each time step depends on both the inputs at the current time step and the previous time steps. Two of the most widely used types of RNNs are long short-term memory (LSTM) and gated recurrent unit (GRU). These models can effectively handle the time-series nature of the data and can provide improved predictions of transient emissions compared to traditional models. ⁸ In Moradi et al., ²¹ both QSSM and TSM are developed for predicting transient $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions of a SI engine. Classical ANN and LSTM algorithms were used for developing QSSM and TSM. Results showed that the TSM model outperform the QSSM which is due to the advantage of LSTM model in TSM that enables it to consider sequential behavior of the data in prediction. In Wang et al., ²² RF and SVM algorithms were used to develop QSSM and LSTM algorithm was used to develop TSM for $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions of a CI engine. Results showed that the mean absolute error for the TSM is over 20% less than the QSSMs. In Li et al., ²³ QSSMs and TSMs are developed for predicting $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions of two CI engines. SVM and RF algorithms were used for QSSM development, while the TSMs were developed using LSTM and GRU algorithms. In addition, noise reduction techniques were used to improve model performance. The results showed that the TSMs achieved around 10% lower mean absolute error compared to the QSSMs. Furthermore, the application of noise reduction techniques resulted in a further improvement in the performance of the models. In our previous work, ²⁴ an LSTM TSM model was trained using transient simulation data obtained from a physical model of a CI engine to predict $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions. The resulting TSM model was then utilized to develop a model-based controller for the engine. The limitation of our previous work was relying solely on simulation data as the training and test data for the TSM model. Incorporating extra information from the physical model with the experimentally measured emission values can provide better training for the ML method in the forms of GB models. Figure 1 provides a comprehensive summary of the studies on NOx emission modeling in ICEs. Given the large number of studies on steady-state NOx emission modeling, only those that specifically pertain to dual-fuel hydrogen-diesel engines which are more related to this study are highlighted in the steady-state section of the figure. As it is seen in Figure 1, most of the studies are still using some form of physical models for engine model-based control and using data-driven models is a new approach in this field.

Figure 1.

Prior $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emission modeling studies on internal combustion engines (ICEs): Steady-state and transient studies classified by modeling method (white-box (WB), black-box (BB) and gray-box (GB)), and methodology (steady-state models (SSM), quasi steady-state models (QSSM), and transient sequential models (TSM)). Novelty of this work is identified in the figure in Transient SSM-GB, QSSM-GB, and TSM-GB.^15,17–47

Methodology

This section explains the experimental setup, physical model that is used for GB modeling, and also ML, DL, and FS algorithms that are used in this study.

Experimental setup

The dual-fuel hydrogen-diesel engine used in this study is a medium-duty CI diesel engine^13,14 that has been modified through the addition of a hydrogen injector in the intake manifold. ¹⁵ To accurately measure the emissions from the dual-fuel operation, sensors have been installed on the separate exhaust of the dual-fuel cylinder, while the other cylinders continue to operate on diesel fuel only. The injection timing of the hydrogen is precisely coordinated with the valve timing of the target cylinder to ensure all of the injected hydrogen goes to the target cylinder. ¹³ The experimental setup for the engine, including the NOx emissions measurement sensor are shown in Figure 2. The properties and specifications of the NOx sensor used for measuring transient NOx emissions are provided in Table 1 with more details on the sensor available.^48–51 The engine experimental setup was utilized to measure NOx emissions over 82,300 engine cycles at a constant engine speed of 1500 rpm and without exhaust gas re-circulation (EGR). This dataset includes duration of over 6500 s. Implementing EGR and other after-treatment systems can reduce output $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions and introduce new predictive parameters that should be considered in emission modeling.

Figure 2.

$\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emission measurement setup for dual-fuel compression ignition hydrogen-diesel Engine. SOI, DOI, and ${\mathrm{P}}_{\mathrm{fuel}}$ stands for start of injection, duration of injection, and fuel pressure, respectively.

Table 1.

ECM $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ sensor specification.

Sensor name	ECM-06-05
Range	$\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ : 0–5000 (ppm), λ (Lambda): 0.40–25, AFR: 6.0–364, ${\mathrm{O}}_2$ : 0–25 (%)
Accuracy	$\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ : ±5 ppm (for 0–200 ppm), ±20 ppm (for 200–1000 ppm), and ±2.0% (elsewhere)
Response time	Less than 100 ms
Fuel type	Programmable H:C, O:C, N:C ratios, and ${\mathrm{H}}_2$
CAN	High speed according to ISO 11898
Environment	−55 to +125°C for the module, 950°C (maximum continuous) for the NOx

To control combustion, this diesel engine employs two diesel injection pulses - a short pilot injection and a longer main diesel injection. Figure 3 shows the injection setting in this study. The control injection parameters in the experimental tests are: (i) the crank angle at the start of injection for pulse 1 $\left(\mathrm{SO}{\mathrm{I}}_{\mathrm{D}1}\right)$ , (ii) the crank angle at the start of injection for pulse 2 $\left(\mathrm{SO}{\mathrm{I}}_{\mathrm{D}2}\right)$ , (iii) the duration of injection for pulse 2 $\left(\mathrm{DO}{\mathrm{I}}_{\mathrm{D}2}\right)$ which affects the diesel injection amount, and (iv) the duration of hydrogen injection $\left(\mathrm{DO}{\mathrm{I}}_{\mathrm{H}2}\right)$ which affects the hydrogen injection amount. While the duration of diesel injection pulse 1 was kept constant $\left(\mathrm{DO}{\mathrm{I}}_{\mathrm{D}1}\right)$ , the input ranges for $\mathrm{SO}{\mathrm{I}}_{\mathrm{D}1}$ and $\mathrm{SO}{\mathrm{I}}_{\mathrm{D}2}$ were adjusted to ensure that the diesel injection pulses did not overlap. The inputs and outputs of these experimental measurements are depicted in Figure 4. As seen in Figure 4 the testing conditions were highly transient, with the hydrogen energy ratio varying from 0% to 80%. This setup was also used for NOx emissions measurement of 79 steady-state cases, which are used for calibrating WB model and developing SSM models. The steady-state data are also used for investigating the performance of QSSMs and TSMs in steady-state operating conditions.

Figure 3.

Hydrogen and diesel injection setting in this study. Injection control parameters are shown in red. $\mathrm{SO}{\mathrm{I}}_{\mathrm{D}1}$ is the crank angle at the start of injection for diesel pulse 1, $\mathrm{SO}{\mathrm{I}}_{\mathrm{D}2}$ is the crank angle at the start of injection for diesel pulse 2, $\mathrm{DO}{\mathrm{I}}_{\mathrm{D}2}$ is the duration of injection for diesel pulse 2, and $\mathrm{DO}{\mathrm{I}}_{\mathrm{H}2}$ is the duration of hydrogen injection.

Figure 4.

Dual-fuel compression ignition hydrogen-diesel engine experimental data over 82,300 engine cycles at constant engine speed of 1500 rpm and without EGR. MPRR stands for maximum pressure raise rate.

Physical model

The physical model or WB model for the dual-fuel hydrogen-diesel was created using the dual-fuel combustion model in the GT-Power software. This model is a combination of direct-injection diesel multi-pulse and SI combustion model that is designed for dual-fuel SI-CI combustion. The cylinder contents in the model are divided into three thermodynamic zones with distinct compositions and temperatures. These include an unburned zone, a spray zone with injected fuel and entrained gas, and a spray burned zone with combustion products. The combustion rate is predicted based on the amount of unburned mixture behind the flame front, which is influenced by the sum of the turbulent flame speed (TFS) and the flame area. ML laminar flame speed (LFS) models ⁵² are used for LFS calculation which is then used to calculate TFS.

The dual-fuel combustion model in the GT-Power software has 7 calibration parameters, which are carefully adjusted to optimize the model’s accuracy and predict the combustion rate of the dual-fuel engine with a pilot injection. The first three parameters are related to the simulation of SI combustion, including the turbulent flame speed multiplier, the Taylor length scale multiplier, and the flame kernel growth multiplier. These parameters affect the turbulent flame speed and, in turn, the burn rate and in-cylinder pressure. The remaining four parameters are related to the ignition process of the dual-fuel combustion and include the entrainment rate multiplier, the ignition delay multiplier, the premixed combustion rate multiplier, and the diffusion combustion rate multiplier. A genetic algorithm was utilized for optimal tuning of the calibration parameters based on the experimental in-cylinder pressure traces for the 79 steady-state measurements. ¹⁵ Table 2 presents the validation of the physical model using experimental steady-state data for various parameters. Figure 5 illustrates a comparison between the experimental and physical model in-cylinder pressure trace for three steady-state cases. Further details about the physical model validation can be found in the previous study. ¹⁵

Table 2.

Experimental validation of the physical model for steady-state data: mean absolute error (MAE) for different parameters. ¹⁵

Parameter	MAE
IMEP	6.5%
CA50	0.49°
Peak pressure	1.1%
Net indicated thermal efficiency	6.6%
$\mathrm{NOx}$ emissions	9.8%

Figure 5.

Experimental validation of the physical model in predicting in-cylinder pressure traces for three steady-state cases. ¹⁵

In this study, the physical model is used for GB modeling. To do this, the physical model produces physical outputs for each engine case. The physical outputs are then selected through a combination of expert knowledge and FS algorithms. The selected outputs form the training data for the ML algorithm, resulting in a gray-box (GB) model that is trained using a wider range of input parameters compared to black-box (BB) models. BB models are only trained using four main inputs ( $\mathrm{SO}{\mathrm{I}}_{\mathrm{D}1}$ , $\mathrm{SO}{\mathrm{I}}_{\mathrm{D}2}$ , $\mathrm{DO}{\mathrm{I}}_{\mathrm{D}2}$ , and $\mathrm{DO}{\mathrm{I}}_{\mathrm{H}2}$ ) because their primary application is for engine model-based control, focusing on parameters that the engine control unit (ECU) can regulate, specifically the injection parameters for combustion control.

In contrast, GB models in this study are built using a robust physical model and trained with a broader range of physically calculated input features. This allows GB models to perform better on data outside the training range, as they incorporate physical understanding in output prediction rather than relying solely on extrapolation like BB models. This is one of the main advantages of GB models over BB models.

Machine learning and deep learning methods

ML models are now employed for predicting NOx emissions through three methodologies: SSM, QSSM, and TSM. A fully connected ANN was selected as the classical ML approach for SSM and QSSM, as it has been proven in previous studies^13,15 that optimized ANNs can outperform other classical ML algorithms such as SVM, RF, GPR, and Ensemble of Regression Trees. The hyperparameters of the ANN model, including the number of hidden layers, the number of hidden neurons per layer, and the regularization parameter $\left(\lambda \right)$ were optimized using Bayesian optimization method. This optimization involves three main components: (i) Gaussian model $f(x)$ , (ii) Bayesian update of $f(x)$ , and (iii) the Acquisition function $a(x)$ . The algorithm updates the hyperparameter values at each iteration and obtains a posterior distribution over the functions $Q(f|x_i,y)$ for $i=1,...,N$ , where N represents the number of hyperparameters. The algorithm then identifies the new point x by maximizing the acquisition function $a(x)$ .^15,52

The SSM models were trained and validated using 79 steady-state cases of dual-fuel hydrogen-diesel engine at constant speed of 1500 rpm. For the QSSMs, over 82,300 transient cycles (Figure 4) were used, with 70% of the data designated for training, 15% for validation, and 15% for testing. The QSSMs were trained using the training data and the best models were selected based on their performance on the validation set. Finally, the models were tested using the test data which was not used for training or validation.

TSM models use time series sequential ML algorithms, such as LSTM and GRU networks, which are RNN algorithms that use gates to overcome the vanishing gradient problem. A GRU network ⁵³ has two gates, a relevance gate, and an update gate, along with one memory cell $c^{<t>}$ . The memory cell retains information about previous time steps, and the relevance gate determines how much influence the memory cell should have on the current prediction. The update gate determines whether the memory cell information needs to be updated using a cell candidate ${\tilde{c}}^{<t>}$ . In this algorithm, the activation at each time step $a^{<t>}$ is equal to $c^{<t>}$ . On the other hand, an LSTM network ⁵⁴ is a more complex RNN algorithm with three gates: the update gate, forget gate, and output gate. In this algorithm, $a^{<t>}$ is different from $c^{<t>}$ , and the output gate’s job is to determine $a^{<t>}$ based on $c^{<t>}$ . In an LSTM network, $c^{<t>}$ first enters the forget gate to drop unnecessary information. Then, the update gate adds new information to $c^{<t>}$ using ${\tilde{c}}^{<t>}$ . The key difference between LSTM and GRU is that LSTM can drop and add information to $c^{<t>}$ separately. This capability gives the LSTM algorithm more predictive power for time series, but it also makes it more complex and time-consuming to train since it has more trainable parameters than GRU. In addition, in an LSTM network, $a^{<t>}$ is not similar to $c^{<t>}$ , and the output gate calculates $a^{<t>}$ from $c^{<t>}$ at each time step. Table 3 shows the summary of equations for LSTM and GRU networks.

Table 3.

Equations of LSTM and GRU Sequential models. In the equations, $G_{u,f,o,r}$ shows gates, $b_{u,f,o,r,c}$ shows biases matrices, $W_{u,f,o,r,c}$ shows weight matrices, σ shows the sigmoid activation function, and * shows element-wise multiplication.

Metric	LSTM	GRU
Number of gates/states	3/2	2/1
Gate 1	(Update gate) $G_u=\sigma (W_u[a^{<t-1>},x^{<t>}]+b_u)$	(Update gate) $G_u=\sigma (W_u[a^{<t-1>},x^{<t>}]+b_u)$
Gate 2	(Forget gate) $G_f=\sigma (W_f[a^{<t-1>},x^{<t>}]+b_f)$	(Relevance gate) $G_r=\sigma (W_r[a^{<t-1>},x^{<t>}]+b_r)$
Gate 3	(Output gate) $G_o=\sigma (W_o[a^{<t-1>},x^{<t>}]+b_o)$	-
Cell candidate	${\tilde{c}}^{<t>}=\tanh (W_c[a^{<t-1>},x^{<t>}]+b_c)$	${\tilde{c}}^{<t>}=\tanh (W_c[G_r*a^{<t-1>},x^{<t>}]+b_c)$
Memory cell	$c^{<t>}=G_u*{\tilde{c}}^{<t>}+G_f*c^{<t-1>}$	$c^{<t>}=G_u*{\tilde{c}}^{<t>}+(1-G_u)*c^{<t-1>}$
Activation	$a^{<t>}=G_o*c^{<t>}$	$a^{<t>}=c^{<t>}$

LSTM and GRU networks are employed to construct TSM models for BB and GB NOx emissions. Figure 6 shows the schematic of the BB LSTM and GB GRU TSMs. Similar architectures are also used for the GB LSTM and BB GRU TSMs. As it is seen in Figure 6, the engine simulation model (ESM) acts as an extra layer in the network for the GB models. The TSM network architecture comprises three fully connected layers at the beginning, followed by a time series layer (either LSTM or GRU), a drop-off layer, and an output layer at the end. The drop-off layer randomly deactivates half of the network during each training iteration, thereby reducing overfitting and training computational expenses. This architecture was determined through trial and error and trade-off between complexity and minimizing error. To train these models, MATLAB Deep Learning Toolbox© with the Adam algorithm with mini batch size of 1024 has been used. The training, validation, and test data were similar to those used in QSSMs, and the training process continued until the validation loss consistently started to increase. At that point, the training was halted and the model with the lowest validation loss was chosen

Figure 6.

Transient sequential models architecture. Left: BB LSTM model, right: GB GRU model. ESM stands for engine simulation model, $\mathrm{SO}{\mathrm{I}}_{\mathrm{D}1}$ is the crank angle at the start of injection for diesel pulse 1, $\mathrm{SO}{\mathrm{I}}_{\mathrm{D}2}$ is the crank angle at the start of injection for diesel pulse 2, $\mathrm{DO}{\mathrm{I}}_{\mathrm{D}2}$ is the duration of injection for diesel pulse 2, and $\mathrm{DO}{\mathrm{I}}_{\mathrm{H}2}$ is the duration of hydrogen injection.

Feature selection

The choice of input feature set is a crucial factor in the performance of the emission modeling GB algorithm. ¹³ Previous research^13,15 has highlighted that relying solely on expert physical knowledge may not be sufficient for selecting the most important physical features for emission modeling. Incorporating ML algorithms for FS in conjunction with expert knowledge has been shown to improve the performance of the GB model. To this end, four FS algorithms (F-test, minimum redundancy maximum relevance (MRMR), RReliefF, and least absolute shrinkage and selection operator (LASSO)) were employed, and their effect on the GB model’s performance was evaluated in comparison to the BB model. Here, each of the FS algorithms that are used in this study are briefly described.

F-test algorithm ⁵⁵ evaluates the significance of each predictor, which involves comparing the means of response values across different predictor variable values. The p-values derived from these F-test statistics can then be used to rank the features, with higher scores corresponding to lower p-values. Essentially, an F-test examines whether the response values for different predictor variable values are drawn from populations with the same mean or not, and the resulting scores reflect the negative logarithm of the p-values.

The MRMR algorithm ⁵⁶ aims to identify the most effective set of features that can accurately represent the response variable, while ensuring that the features are both mutually and maximally dissimilar. To achieve this, the algorithm works toward minimizing the redundancy of the selected feature set, while simultaneously maximizing its relevance to the response variable. This is done by quantifying both the redundancy and relevance of the features using mutual information measures, which consider the pairwise mutual information of features as well as the mutual information of a feature with the response variable. The MRMR algorithm ⁵⁶ is specifically designed for regression problems, and its objective is to identify an optimal feature set S that maximizes the relevance $(V_S)$ of S with respect to the response variable y, while minimizing its redundancy $(W_S)$ , as defined through mutual information I.

RReliefF ⁵⁷ is a weight-based algorithm that determines the importance of predictors when the response variable y is a multi-class categorical variable. The algorithm aims to penalize the predictors that generate different values for observations belonging to the same class while rewarding those that generate different values for observations belonging to different classes. Initially, RReliefF sets all the weights for predictors, denoted as $W_j$ , to zero. The algorithm then proceeds iteratively by selecting a random observation $x_r$ , finding the k-nearest observations for each class, and updating the weights for the predictors $F_j$ based on the values of the nearest neighbors $x_q$ . Specifically, the weights of the predictors are updated by decreasing the value of $W_j$ if the predictor $F_j$ generates different values for observations belonging to the same class, and increasing the value of $W_j$ if $F_j$ generates different values for observations belonging to different classes. This algorithm works for categorical and continuous features.

LASSO^13,14 is a type of regression analysis that combines variable selection and regularization to improve the model’s predictive accuracy. Specifically, the LASSO regression incorporates a penalty term, represented by the variable λ, in the cost function to penalize the $l_1$ norm. This approach tends to drive the weights of certain features down to exactly zero, resulting in a sparse solution that automatically selects the most relevant features for the prediction task. In the present study, the degree of sparsity is controlled by the value of λ, which is tuned using a cross-validation approach. Results from using different FS methods are described in the next section.

Results and discussion

In this study, the input feature set for all BB models was comprised of four features: $\mathrm{SO}{\mathrm{I}}_{\mathrm{D}1}$ , $\mathrm{SO}{\mathrm{I}}_{\mathrm{D}2}$ , $\mathrm{DO}{\mathrm{I}}_{\mathrm{D}2}$ , and $\mathrm{DO}{\mathrm{I}}_{\mathrm{H}2}$ . To construct GB models, expert physical knowledge of NOx emissions formation was used to initially identify the most important physical features, which are outputs of the physical model. Then, four FS algorithms (F-test, MRMR, RReliefF, and LASSO) were employed on the transient training dataset to further narrow down the selected features. Table 4 shows the performance of ANN QSSMs that were trained and optimized using the selected features from each FS algorithm, and Figure 7 provides a comparison of the performance of each FS algorithm with the BB model in terms of maximum error improvement, root-mean-square-error (RMSE) improvement, mean absolute percentage error (MAPE) improvement, and $R^2$ improvement.

Table 4.

$\mathrm{NOx}$ emissions prediction of gray-box quasi steady-state models using different feature sets selected by different feature selection methods.

Metric	BB	GB-ES	GB-F-test	GB-RReliefF	GB-LASSO	GB-MRMR
Number of features	4	42	23	10	7	7
${\mathrm{R}}_{\mathrm{test}}^2(\mathrm{\%})$	68.4	78.9	87	83	81.4	69.4
$\mathrm{RMS}{\mathrm{E}}_{\mathrm{test}}(\mathrm{ppm})$	216.8	180.4	140.5	159.5	166.7	216.6
$|\mathrm{MAP}{\mathrm{E}}_{\mathrm{test}}|(\mathrm{\%})$	60.7	20.4	15.5	15.2	18.1	21.5
$|{\mathrm{E}}_{\mathrm{test},\max }|(\mathrm{ppm})$	1374	1156	1051	1211	1395	1646

Figure 7.

Comparison of black-box quasi steady-state model with gray-box quasi steady-state model with feature selection using different algorithms: (a) maximum error and RMSE improvement compared to black-box model and (b) MAPE and $R^2$ improvement compared to black-box model. “par” in the figure refers to the number of features selected by the related algorithm.

The expert selection (ES) of features, which involves using all features selected based on physical knowledge alone, resulted in a weaker performance compared to most other FS algorithms that combined expert knowledge with ML algorithms. The F-test algorithm outperformed the other FS algorithms, with the highest improvement of RMSE, maximum error, and $R^2$ values compared to the BB model. It also has a very similar MAPE value to RReliefF, which has the lowest MAPE. Furthermore, it can be seen that the use of GB modeling significantly improves emission prediction for QSSMs compared to BB models. The GB QSSM, which uses the F-test FS algorithm, outperformed the BB model in terms of MAPE and RSME by over 30% and 40% respectively. This indicates that the GB model is significantly more accurate than BB model. The improved accuracy can be attributed to the physical model being integrated into the GB model, which provides additional information for the ML method and captures the transient nature of the emission. The physical model’s output is dependent on previous states and is fed into the classical ML method, resulting in improved performance on transient data. In contrast, the BB QSSM relies solely on injection inputs from the current time step for prediction and cannot utilize information from previous steps. Furthermore, the NOx emissions prediction by the physical model is among the selected features by all of the algorithms. This shows that although one dimensional (1D) physics-based models may not be sufficiently accurate for direct transient emission prediction, they can still enhance the performance of GB models when used as inputs for ML models. The used features for BB, GB-ES, and GB-F-test are shown in Table 5. Additionally, Figure 8 illustrates the Pearson correlation coefficients of predictors from the GB-F-test feature-set with $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions. The Pearson correlation coefficient measures the linear relationship between two variables, ranging from −1 to 1, where 1 indicates a perfect positive linear relationship, −1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship. It is calculated as the covariance of the two variables divided by the product of their standard deviations, as below:

$$r_{x,\mathrm{N}{\mathrm{O}}_{\mathrm{x}}}=\frac{\sum (x_i-\overline{x})(\mathrm{N}{\mathrm{O}}_{\mathrm{x},i}-\overline{\mathrm{N}{\mathrm{O}}_{\mathrm{x}}})}{\sqrt{\sum {(x_i-\overline{x})}^2\sum {(\mathrm{N}{\mathrm{O}}_{\mathrm{x},i}-\overline{\mathrm{N}{\mathrm{O}}_{\mathrm{x}}})}^2}}$$

(1)

Here, $x_i$ and $\mathrm{N}{\mathrm{O}}_{\mathrm{x},i}$ are the individual sample points, and $\overline{x}$ and $\overline{\mathrm{N}{\mathrm{O}}_{\mathrm{x}}}$ are the means of the predictor variable x and the $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions, respectively.

Table 5.

List of features for gray-box modeling (selected by expert selection and F-test algorithm) and black-box modeling.

Methodology	Features
GB-ES	Indicated Mean Effective Pressure (IMEP), Indicated Specific Fuel Consumption (ISFC), Indicated Efficiency, Maximum Pressure, Crank Angle at Maximum Pressure, Maximum Temperature Motoring, Maximum Temperature, Maximum Temperature (Unburned Zone), Maximum Temperature (Burned Zone), Combustion Start, Combustion Delay, Burned Fuel Fraction, Combustion Efficiency, Ignition Delay, Crank Angle at 2% Burned (CA2), Crank Angle at 50% Burned (CA50), Burn Duration (0%–10%), Burn Duration (10%–75%), Burn Duration (10%–90%), Burn Duration (0%–50%), Burn Duration (0%–90%), Volumetric Efficiency (Air), Total Mass Trapped at Cylinder at Cycle Start, Air Mass Trapped at Cylinder at Cycle Start, Burned Mass Percent at Combustion Start, Fuel Mass Trapped at Cylinder at Cycle Start, Effective Lambda at Cylinder at Cycle Start, Total Mass Trapped at Cylinder at Cycle End, Maximum Injection Pressure, Local Angle at the Start of Diesel Injection (SOI) Pulse 1, Local Angle at the End of Diesel Injection Pulse 1, Diesel Injection SOI Pulse 2, Local Angle at the End of Diesel Injection Pulse 2, Diesel Injection Pulse 2 Mass, Hydrogen Injection Mass, Nitrogen Oxides $\left(\mathrm{N}{\mathrm{O}}_{\mathrm{x}}\right)$ , Nitric Oxide (NO), Carbon Monoxide (CO), Carbon Dioxide $\left(\mathrm{C}{\mathrm{O}}_2\right)$ , Unburned Hydrocarbons, Carbon Content of Unburned Hydrocarbons (HydrocarbonC), Maximum Pressure Rise Rate (MPRR)
GB-F-test	IMEP, ISFC, Maximum Pressure, Maximum Temperature Motoring, Maximum Temperature, Maximum Temperature (Unburned Zone), Maximum Temperature (Burned Zone), Burned Fuel Fraction, Combustion Efficiency, CA50, Burn Duration (10%–75%), Burn Duration (10%–90%), Burn Duration (0%–90%), Volumetric Efficiency (Air), Total Mass Trapped at Cylinder at Cycle Start, Air Mass Trapped at Cylinder at Cycle Start, Burned Mass Percent at Combustion Start, Fuel Mass Trapped at Cylinder at Cycle Start, Effective Lambda at Cylinder at Cycle Start, Total Mass Trapped at Cylinder at Cycle End, Hydrogen Injection Mass, $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ , MPRR
BB	Hydrogen Duration of Injection (DOI), Diesel Pulse 2 DOI, Diesel Pulse 1 SOI, Diesel Pulse 2 SOI

Figure 8.

Pearson correlation coefficients of predictors from GB-F-test feature-set with the output ( $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions).

The F-test algorithm selected almost all of the features chosen by other FS algorithms, and the 23 features it identified were used for QSSMs and TSMs. For SSM, a similar approach was followed, and seven features were selected using the F-test algorithm, as the training dataset for SSM comprised steady-state data.

The number of hidden units in the time series layer is a critical factor that significantly affects the ability of TSMs to capture the sequential nature of the transient emission data. However, a very high number of hidden units can lead to a more complex model that is unsuitable for engine model-based control and might also result in overfitting. To determine the optimal number of hidden units in the time series layer, the BB LSTM architecture (shown in Figure 6) with different numbers of hidden units within the LSTM layer were developed. The training and validation transient NOx emissions dataset were used for this purpose.

As shown in Table 6, increasing the number of hidden units results in a higher number of trainable parameters in the model, which can increase training and prediction time. An increase in the number of hidden units initially leads to a significant improvement in model performance. However, this trend eventually slows down, and after a certain point, increasing the number of hidden units has little to no effect on model performance and may even degrade it.

Table 6.

Effect of hidden unit number on LSTM model performance in NOx emissions prediction. Modeling was carried out using IntelR XeonR CPU E3-1245 V2 @ 3.40 GHz processor and 32 GB RAM.

Dataset	Metric	LSTM1	LSTM2	LSTM3	LSTM4	LSTM5	LSTM6
	Number of hidden units	4	8	16	32	64	128
	Number of model parameters	1100	1400	2500	6100	19,600	71,100
	Prediction per second	82,000	75,000	69,000	59,000	46,000	20,000
Transient (83,000 cycles)	${\mathrm{R}}_{\mathrm{test}}^2(\mathrm{\%})$	95.7	95.7	96.7	96.7	96.7	96.7
	$\mathrm{RMS}{\mathrm{E}}_{\mathrm{test}}(\mathrm{ppm})$	127	93.7	77.4	76.1	72.3	71.9
	$\mathrm{MAP}{\mathrm{E}}_{\mathrm{test}}(\mathrm{\%})$	16.8	12.1	9.1	8	7.8	7.5
	$|{\mathrm{E}}_{\mathrm{test},\max }|(\mathrm{ppm})$	590	439	408	479	397	585
Steady-state (79 cases)	$\mathrm{RMSE}(\mathrm{ppm})$	200.7	200.7	217	170	168	162
	$|\mathrm{MAPE}|(\mathrm{\%})$	24	24	26	20	20	18

The analysis shows that increasing the number of hidden units up to 16 (as shown in Figure 9) improves model accuracy without significantly affecting computational speed. Increasing the number of hidden units from 16 to 64 slightly improves model accuracy but makes the model slower. Increasing the number of hidden units post 64 only results in a slower model without a corresponding increase in accuracy. Considering the trade-off between accuracy and computation cost, a range of 16–64 hidden units appears to be the optimal choice.

Figure 9.

Effect of hidden unit number on the LSTM model’s (a) prediction rate (per second) and (b) RMSE for the test dataset (ppm). The modeling was performed using an Intel^® Xeon^® CPU E3-1245 V2 @ 3.40 GHz and 32 GB RAM.

Based on this analysis, if accuracy is the most critical factor in the accuracy-computational cost trade-off, the recommended choice would be the LSTM network with 64 hidden units. To investigate the performance of other popular time series network (GRU) and the effects of GB modeling on time series networks, GB LSTM model, BB GRU model and GB GRU model (shown in Figure 6) were modeled with 64 hidden units within the time series layer. Table 7 compares the performance of these four models in terms of model complexity, computational cost, and accuracy.

Table 7.

Comparison of time series networks’ performance for transient and steady-state NOx emission modeling. Modeling was carried out using IntelR XeonR CPU E3-1245 V2 @ 3.40 GHz processor and 32 GB RAM.

Dataset	Metric	BB-LSTM	GB-LSTM	BB-GRU	GB-GRU
	Number of hidden units	64	64	64	64
	Number of ML model parameters	19,600	20,000	14,900	15,200
	Prediction per second	46,000	3	89,000	3
Transient (83,000 cycles)	${\mathrm{R}}_{\mathrm{test}}^2(\mathrm{\%})$	96.7	97.1	96.6	96.7
	$\mathrm{RMS}{\mathrm{E}}_{\mathrm{test}}(\mathrm{ppm})$	72.3	67.8	72.5	71.9
	$\mathrm{MAP}{\mathrm{E}}_{\mathrm{test}}(\mathrm{\%})$	7.9	7.4	8	7.9
	$|{\mathrm{E}}_{\mathrm{test},\max }|(\mathrm{ppm})$	397	370	459	399
Steady-state (79 cases)	$\mathrm{RMSE}(\mathrm{ppm})$	168	155	175	206
	$\mathrm{MAPE}(\mathrm{\%})$	20	14	21	21

The GRU model has fewer trainable parameters for the same number of units than the LSTM model because it is a simpler version of the LSTM network. This makes the GRU model faster than the LSTM model. However, GB models are much slower in prediction than BB models because the physical model inside the GB models needs to simulate each case and feed the network with its output, making the model considerably slower.

In terms of accuracy, GB LSTM and GB GRU are more accurate than the corresponding BB models. This demonstrates that GB modeling can improve TSMs. However, compared to QSSMs, the difference in accuracy between GB and BB models for time series networks is much smaller. This is because time series networks are DL algorithms that effectively capture the sequential nature of the transient emission, leaving little room for improvement by using additional information provided by physical models in the form of GB models.

The GB LSTM model has the best steady-state and transient performance among the four models. It has the highest $R^2$ , and the lowest RMSE, MAPE, and maximum error for the transient tests although all other three models are also very accurate. GB LSTM model has also has the best steady-state performance (lowest RMSE and MAPE). The steady-state performance of the other three models are almost similar. The higher accuracy of GB LSTM network makes it the best option for applications where accuracy is much critical than computational cost such as using the model as the virtual engine in HIL setup. For model-based engine control where computational cost is crucial, BB LSTM or GRU network with 64 units can be used. The BB GRU model is slightly less accurate than the LSTM version, but it is almost twice as fast for the same number of hidden units. This model is well-suited for engine model-based control applications that require real-time execution within the engine control unit (ECU) where the computational efficiency is crucial.

In summary, selecting the appropriate TSM depends on the specific application’s requirements. Here for NOx emissions modeling, the GB LSTM model that can predict transient NOx emissions with $R^2>0.97$ and MAPE = 7.4% at three prediction per second is the best option for high accuracy applications like HIL setup. The BB GRU or LSTM networks are more than 10,000 time faster than the GB LSTM model which makes them more appropriate for the applications where computational cost is crucial such as model based engine control. These models can predict transient NOx emissions with $R^2>0.96$ and MAPE = 8% which is slightly less accurate than the GB LSTM model.

To assess the effectiveness of various modeling methodologies for NOx emissions, Table 8 presents a comparison of ESM, SSMs, OSSMs, and TSMs in terms of their computational cost and performance on both steady-state and transient data. Additionally, Figure 10 illustrates the improvement achieved by BB and GB SSMs, QSSMs, and TSMs compared to the ESM in terms of maximum error, RMSE, $R^2$ , and MAPE. In both Table 8 and Figure 10, the BB and GB TSMs are LSTM networks with 64 units. To provide a visual representation, Figure 11 presents the BB and GB predictions for SSM, QSSM, and TSM, as well as the experimental NOx emissions values on test dataset.

Figure 10.

Comparison of the improvement in performance of steady-state models (SSMs), quasi steady-state models (QSSMs), and transient sequential models (TSMs) compared to engine simulation model (ESM). (a) Reduction in the maximum error and RMSE compared to ESM, (b) Reduction in MAPE and improvement in $R^2$ compared to ESM.

Table 8.

Comparison of gray-box (GB) and black-box (BB) models, including steady-state models (SSMs), quasi steady-state models (QSSMs), and transient sequential models (TSMs) for transient and steady-state emission modeling. Modeling was carried out using IntelR XeonR CPU E3-1245 V2 @ 3.40 GHz processor and 32 GB RAM.

		ESM	SSM		QSM		TSM
Dataset	Metric		BB	GB	BB	GB	BB	GB
	Prediction per second	3	400,000	3	280,000	3	46,000	3
Transient (83,000 cycles)	${\mathrm{R}}_{\mathrm{test}}^2(\mathrm{\%})$	16	52	63	68.4	87	96.7	97.1
	$\mathrm{RMS}{\mathrm{E}}_{\mathrm{test}}(\mathrm{ppm})$	631	279.2	228.9	216.8	140.5	72.3	67.8
	$\mathrm{MAP}{\mathrm{E}}_{\mathrm{test}}(\mathrm{\%})$	83.4	30.6	27.3	28.5	15.5	7.9	7.4
	$|{\mathrm{E}}_{\mathrm{test},\max }|(\mathrm{ppm})$	1630	1495	1468	1388	1051	397	370
Steady-state (79 cases)	$\mathrm{RMSE}(\mathrm{ppm})$	292	10	1	652	548	168	155
	$\mathrm{MAPE}(\mathrm{\%})$	35	2	1	50	35	20	14

Figure 11.

Comparison of experimental NOx emissions and predictions by black-box and gray-box models over the test data: (a) steady-state models, (b) quasi steady-state models, (c) transient sequential model (LSTM with 64 hidden neurons). The figure on the bottom displays the zoomed-in review of the region enclosed within the red box in the figure above.

Comparing the performance of these models gives a better understanding of the improvement level achieved between methodologies. The results show that, overall, the performance of the models improves as we move from left to right in Table 8 and Figure 10, and from top to bottom in Figure 11. The GB models perform better than their BB counterparts on both steady-state and transient data which can visually be seen in Figure 11. In terms of transient performance, all of the ML models outperform the ESM, with BB models being considerably faster.

Overall, the ESM has the poorest transient performance compared to all models, while BB models offer improved transient performance and faster computational speed compared to traditional physical models, making them a better choice for model-based control applications. SSMs demonstrate the best steady-state performance since these models are trained on steady-state data. Nevertheless, it is not recommended to use models trained on steady-state data for predicting transient data since SSMs perform poorly in transient situations compared to other ML methods.

In both Table 8 and Figure 10, a significant difference exists between the performance of TSMs and QSSMs, which is more pronounced than in previous studies that investigated both BB QSSMs and TSMs.^21,22 The significant performance difference observed between TSMs and QSSMs in this study (Figure 10) can be attributed to the highly transient nature of the data used for modeling, as shown in Figure 4. The transient dataset used in this study involves four varying injection parameters, which contributes to the complexity of the data and highlights the need for appropriate TSM approaches to achieve accurate results. Figure 11 shows that QSSMs exhibit poorer performance in highly transient regions, where the experimental NOx emissions demonstrate more fluctuations and variations. This supports the conclusion that using time series models is essential when dealing with highly transient data, which includes high gradients of changes in engine parameters over time such as load, injection parameters, and hydrogen energy ratio. In this situation, TSMs perform much better than QSSMs. In less transient situations where changes in engine parameters over time is more gradual, QSSMs (specifically GB QSSMs) can be as accurate as TSMs.

Summary and conclusion

Predictive models for transient NOx emissions from a dual-fuel hydrogen-diesel engine using various modeling techniques and methodologies were developed. Specifically, steady-state models (SSMs), quasi steady-state models (QSSMs), and transient sequential models (TSMs) using black-box (BB) and gray-box (GB) approaches were developed. For the BB models, the machine learning (ML) model is trained solely on experimental data, with diesel and hydrogen injection timing as the inputs. For the GB models, outputs from a one dimensional (1D) engine physical model were augmented to the experimental data to train the ML model. This helped to capture the complexity of the engine’s behavior. To identify the most informative features for the GB models, two steps were performed. First, expert knowledge on NOx emissions formation in internal combustion engines (ICEs) was used to identify the most important physical model outputs. Then the four feature selection (FS) algorithms of F-test, minimum redundancy maximum relevance (MRMR), RReliefF, and least absolute shrinkage and selection operator (LASSO) were applied to further narrow down the selected features. Results showed that all of the FS algorithms outperform only expert knowledge, with the F-test algorithm yielding the best results. Using the selected features from the F-test algorithm, GB SSMs, QSSMs, and TSMs NOx models were developed. The main findings of this study are as follow:

• The data-driven models, including SSMs, QSSMs, and TSMs, outperform the 1D physical engine simulation model (ESM) in transient NOx emission modeling. Furthermore, the QSSMs outperformed SSMs, and TSMs were the most accurate among the three data-driven models.

• GB models outperformed BB models for all methodologies (SSMs, QSSMs, and TSMs), particularly significant improvements in SSMs and QSSMs were found. This is attributed to classical ANN used in SSMs and QSSMs do not account for the effects of previous states on the current state, while the physical model outputs in the GB models can take these effects into account. The difference in performance between GB and BB TSM models; however, is relatively small, as the time series network used in these models is already effective at capturing the effects of previous steps on the current step. Therefore, the addition of extra information provided by GB modeling may not provide significant improvements for TSMs.

• The optimum number of hidden neurons in the time series layer for NOx emissions modeling using the suggested architecture in this study is between 16 and 64.

• The performance difference between QSSMs and TSMs depends on the level of transience in the dataset. For highly transient datasets (for instance in this study the hydrogen energy ratio varies between 0% and 80% and there are high gradients of changes in engine parameters over time), TSMs significantly outperform QSSMs. However, for datasets with lower levels of transience, the performance gap between the two methodologies is narrower.

• Both GRU and LSTM networks demonstrated strong performance in transient NOx emission modeling. Although LSTM is generally more accurate, it is also more computationally expensive. For model-based control purposes that demand real-time execution within the engine control unit (ECU) and prioritize computational efficiency, the GRU model emerges as the optimal choice. It exhibits nearly double the speed of LSTM while utilizing an equivalent number of hidden units. The BB GRU model with 64 hidden neurons, can predict transient NOx emissions with an $R^2$ value greater than 0.96. For applications such as hardware-in-the-loop, where accuracy is critical and computational cost is less important, GB LSTM models, which were the most accurate models developed in this study, can be used. The GB LSTM model with 64 hidden neurons can predict transient NOx emissions with an $R^2$ value greater than 0.97 and better steady-state performance compared to other models. Using IntelR XeonR CPU E3-1245 V2 @ 3.40 GHz processor and 32 GB RAM, the BB GRU model and GB LSTM model with 64 hidden units can predict NOx emissions for 89,000 and three cases in each second for real time usage, respectively.

Potential avenues for future research include investigating $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ emissions modeling in engines equipped with advanced emission control systems such as exhaust gas recirculation (EGR), exploring other emissions like soot in diesel-hydrogen combustion, and applying the models developed in this study for engine model-based control.