Grey-box modelling and identification using physical knowledge and bayesian techniques

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.

Virtual sensors, or soft sensors, have greatly contributed to the evolution of the sensing systems in industry. The soft sensors are process models having three fundamental categories, namely white-box (WB), black-box (BB) and gray-box (GB) models. WB models are based on process knowledge while the BB models are developed using data collected from the process. The GB models integrate the WB and BB models for addressing the concerns, i.e., accuracy and intuitiveness, of industrial operators. In this work, various design aspects of the GB models are discussed followed by their application in the process industry. In addition, the changes in the data-driven part of the GB models in the context of enormous amount of process data collected in Industry 4.0 are elaborated.

Thermal models of buildings are helpful to forecast their energy use and to enhance the control of their mechanical systems. However, these models are building-specific and require a tedious, error-prone and time-consuming development effort relying on skilled building energy modelers. Compared to white-box and gray-box models, data-driven (black-box) models require less development time and a minimal amount of information about the building characteristics. In this paper, autoregressive neural network models are compared to gray-box and black-box linear models to simulate indoor temperatures. These models are trained, validated and compared to actual experimental data obtained for an existing commercial building in Montreal (QC, Canada) equipped with roof top units for air conditioning. Results show that neural networks mimic more accurately the thermal behavior of the building when limited information is available, compared to gray-box and black-box linear models. The gray-box model does not perform adequately due to its under-parameterized nature, while the linear models cannot capture non-linear phenomena such as radiative heat transfer and occupancy. Therefore, the neural network models outperform the alternative models in the presented application, reaching a coefficient of determination R2 up to 0.824 and a root mean square error down to 1.11 °C, including the error propagation over time for a 1-week period with a 5-minute time-step. When considering a 50-hour time horizon, the best neural networks reach a much lower root mean square error of around 0.6 °C, which is suitable for applications such as model predictive control.

• Enhanced SSV airflow rate estimation of empirical models via data-driven modeling • Achieved high accuracy by customizing models individually for each room • Compared gray-box (physics-informed) with black-box (purely data-driven) models • Assessed data efficiency and model generalization using learning curve analysis • Revealed model robustness and key factors affecting airflow by sensitivity analysis Empirical models are widely used in building design and simulation to estimate airflow rates for single-sided natural ventilation (SSV). However, these models often suffer from limited accuracy. Data-driven modeling offers a viable solution to improve model accuracies by customizing models for various practical scenarios. This study investigates the feasibility and performance of two data-driven modeling approaches for SSV: gray-box modeling (physics-informed using polynomial regressions) and black-box modeling (purely data-driven using machine learning regressions). Taking SSV through a solitary bottom-hung window as a case study, training data were acquired from a full-scale test bench via tracer gas measurements conducted over 11 days. The results indicate that both modeling approaches significantly improve accuracy compared to existing empirical models from the literature. Notably, individual models tailored for each test room generally outperform generalized models developed for all rooms. This finding highlights the importance of customizing models on-site in real-world applications. When comparing the two modeling approaches, gray-box models require lower training ratios to achieve good generalization, demonstrating higher data efficiency than black-box models. Furthermore, sensitivity analysis reveals that gray-box models are more robust and reliable during interpolation and extrapolation, making them particularly suitable for scenarios with limited data availability. Incorporating opening area corrections and selecting root mean square error (RMSE) as the objective function are crucial for achieving optimal gray-box model performance. Overall, this study highlights the potential of physics-informed gray-box data-driven modeling as an advanced strategy for enhancing SSV airflow rate estimation, providing valuable insights to guide future research.

Particulate matter (PM) emissions, including soot, pose a significant environmental concern. Traditional emission control approaches, such as engine calibration, often struggle to maintain consistent performance across varying driving conditions, engine speeds, and loads. This inconsistency leads to variations in pollutant output, making it challenging for internal combustion engines to comply with stringent emission standards under real-world driving conditions. Physics-based soot models, grounded in complex mathematical equations, have been developed to capture the mechanisms of soot formation. However, these models often face accuracy limitations and require substantial computational resources, rendering them impractical for real-time emission control. In contrast, Machine Learning (ML) techniques offer a more versatile and efficient approach to soot prediction. This paper introduces a machine learning methodology for predicting Soot Particle Numbers (PN). The approach integrates both physics-based and black-box strategies to create a grey-box model for predicting soot emissions from a 1600 cc Gasoline Direct Injection (GDI) engine across diverse load and speed conditions. The physical model is based on a validated 1D engine model in GT-SUITE, while the black-box model is formulated using multiple ML approaches exclusively on measured data. Ultimately, the grey-box model is developed through ML techniques based on measured data and combustion-related parameters obtained from the 1D engine physical model in GT-SUITE. The grey-box model’s results are compared with those of the black-box model. The ML techniques employed in this study include Gaussian Process Regression (GPR), Support Vector Machine (SVM), Artificial Neural Network (ANN), and an Ensemble of Regression Trees (ERT). The Minimum Redundancy Maximum Relevance (MRMR) and RreliefF feature selection methods are used to identify the most crucial features in soot PN prediction. The results indicate that the grey-box GPR exhibits superior performance for both single and multi-pulse injection strategies, achieving a coefficient of determination (R2) of 0.99.

Integration of an improved transformer with physical models for the spatiotemporal simulation of urban flooding depths

Groundwater pollution mathematical modelling : Improvement or stagnation ?

Lithium-ion batteries exhibit slow voltage dynamics on the minute time scale that are usually associated with transport processes. We present a novel modelling approach toward these dynamics by combining physical and data-driven models into a Grey-box model. We use neural networks, in particular neural ordinary differential equations. The physical structure of the Grey-box model is borrowed from the Fickian diffusion law, where the transport domain is discretized using finite volumes. Within this physical structure, unknown parameters (diffusion coefficient, diffusion length, discretization) and dependencies (state of charge, lithium concentration) are replaced by neural networks and learnable parameters. We perform model-to-model comparisons, using as training data (a) a Fickian diffusion process, (b) a Warburg element, and (c) a resistor-capacitor circuit. Voltage dynamics during constant-current operation and pulse tests as well as electrochemical impedance spectra are simulated. The slow dynamics of all three physical models in the order of ten to 30 min are well captured by the Grey-box model, demonstrating the flexibility of the present approach.

Hierarchical predictive control of integrated wastewater treatment systems

Hybrid emission and combustion modeling of hydrogen fueled engines

Physics-informed machine learning models for ship speed prediction

Enhancing early-stage energy consumption predictions using dynamic operational voyage data: A grey-box modelling investigation

Summary form only given. Identification of non-linear systems is an important problem in many applications. The topic is substantially richer than linear system identification. One reason for this is of course that the problem is significantly more difficult, but also that it has engaged several different research communities. With origins in statistical non-linear and non-parametric regression theory, areas like neural networks and learning theory can now be seen as research fields in their own right. In addition to the control field, many areas like artificial intelligence, pattern recognition, signal processing, oceanography, geology, etc, have developed their own approaches to the problem. This has lead to a very substantial literature on the topic. This talk will not attempt to give any survey of all approaches. It will focus on some core features of the problem, which represent the basic challenges. The foremost problem is the inherent lack of data support to build complex A black box model with n explaining variables (regressors) can be seen as a surface in R/sup n+1/. Even for moderately large n, this is a huge space to fill with observations. The remedy would be to assume or look for sub-structures in the model/data, linearity in certain directions etc. For control applications it is natural to complement the data support with structures based on physical insights, grey-box models. Interfacing physical modeling tools with identification techniques is thus important. Grey-box models, on other hand, typically lead to minimization problem with many local minima. This is another challenge, which possibly can be dealt with using modern computer algebra and optimization techniques.

System identification based on the black box models was utilized for understanding of the transfer function of the foot and ankle exposed to vertical excitation. Various linear polynomial structures and state-space model were tested by comparing the measured acceleration at the base, the Medial Malleolus, and the Tibial Tuberosity respectively for input and output. Identified models are well matched with the measured data when the high model order was selected. The Output Error model has generally shown the best fit to the data, while in several cases the state-space model presents better fit. These results can be used as a starting point for the nonlinear black box model and the grey box model.

This study proposes a methodology for developing deterioration models to estimate the remaining lifetime of a system using physics-informed learning. The approach consists of combining physical knowledge about the system with a set of data obtained from similar systems that have failed in the past to build a set of constraints on the evolution over time of the state of health of the system. The data consists of partial measurements of the system variables, taken from its commissioning to the present. The physical knowledge used comprises a set of differential equations that approximate the dynamics and aging of the system. In contrast to other studies, the physical model is not assumed to be accurate, but is considered to be an approximation of reality. Constraints on the temporal evolution of the state of health derived from this physical knowledge take into account its imprecision and consist of possibility distributions. In this study, the max-max, max-min and min-max regret principles are applied to extract the time evolution of the deterioration rate that best fits the constraints. The effectiveness of the proposed algorithm is evaluated by means of a comparative empirical analysis in different use cases, and the situations in which informed learning improves on purely data-driven algorithms are analyzed.