Grey-box Model Research Articles

State of temperature detection of Li-ion batteries by intelligent gray box model

Li-Ion batteries are among the key enablers of more sustainable use of energy. However, they need to be supervised and undergo continuous maintenance to assure safety and longevity. This paper focuses on the sensorless detection of the State of Temperature (SOT) of the Li-ion batteries during the operational life cycle of the battery irrespective of its state of charge. The paper presents the new Intelligent Gray Box Model (IGBM) to detect the SOT of Li-ion cells: that combines the three most powerful diagnostic tools Electrochemical Impedance Spectroscopy (EIS), Equivalent Circuit Model (ECM), and Artificial Neural Network Classifier (NNC). The work introduces the experimental test bench capable of emulating real-world and embedded constraints to conduct EIS onboard, its data preprocessing, and useful information extraction for the entire frequency spectrum. Furthermore, this paper presents a new hybrid parameter identification that combines the Whale Optimization Algorithm (WOA) and Levenberg Marquardt algorithm (LM) to identify the fractional order ECM. Finally, a neural network classifier is designed, optimized, and compared with different feature scaling techniques to evaluate its accuracy and robustness to detect and classify exact battery temperatures in real time from experimental data.

Learning and optimization under epistemic uncertainty with Bayesian hybrid models

Hybrid (i.e., grey-box) models are a powerful and flexible paradigm for predictive science and engineering. Grey-box models use data-driven constructs to incorporate unknown or computationally intractable phenomena into glass-box mechanistic models. The pioneering work of statisticians Kennedy and O’Hagan introduced a new paradigm to quantify epistemic (i.e., model-form) uncertainty. While popular in several engineering disciplines, prior work using Kennedy–O’Hagan hybrid models focuses on prediction with accurate uncertainty estimates. This work demonstrates computational strategies to deploy Bayesian hybrid models for optimization under uncertainty. Specifically, the posterior distributions of Bayesian hybrid models provide a principled uncertainty set for stochastic programming, chance-constrained optimization, or robust optimization. Through two illustrative case studies, we demonstrate the efficacy of hybrid models, composed of a structurally inadequate glass-box model and Gaussian process bias correction term, for decision-making using limited training data. From these case studies, we develop recommended best practices and explore the trade-offs between different hybrid model architectures.

Grey-box and ANN-based building models for multistep-ahead prediction of indoor temperature to implement model predictive control

Model-based predictive control (MPC) strategies for heating, ventilation, and air-conditioning (HVAC) systems present an opportunity to lower building energy consumption and operational costs. Such approaches rely on the development of a model to precisely forecast building thermal dynamics, such as room air temperature or heating/cooling rate, and make control-related decisions. The control-oriented modeling of building energy systems should be accurate in predicting indoor conditions and present low computational complexity. These features are the key challenge of implementing advanced control methods such as MPC. Extant studies on building modeling for MPC have focused on step-ahead forecasting techniques to forecast building thermal dynamics, while multistep-ahead forecasting is essential. Moreover, machine learning model suitable in case of the domain-based engineering expertise are also not available. To this aim, we perform a comparative analysis of the grey-box model based on a resistance-capacitance (RC) thermal network and a machine learning model composed of an artificial neural network (ANN) for multistep-ahead prediction of building thermal dynamics using current and historical data. Actual experimental data obtained from the Flexible Research Platform (FRP) in Oak Ridge National Laboratory (US) are used for estimation and validation purposes. The average root mean squared error (RMSE) of the grey-box and ANN models are 0.89 °C and 1.02 °C, respectively. The results indicate that the grey-box model outperforms the ANN model in the considered validation periods in terms of accuracy and prediction stability.

Artificial intelligence based prediction of optimum operating conditions of a plate and fin heat exchanger under uncertainty: A gray-box approach

This study is based on gray-box modeling for the prediction of optimum mass flow rates of inlet streams of a Plate and Fin Heat Exchanger under uncertainty. A first principle model of the Plate and Fin Heat Exchanger was developed in Aspen Exchanger Design and Rating. Genetic algorithm was integrated with the first principle model to achieve the highest possible exit temperature of the cold process stream under uncertainty. A dataset of uncertain process conditions and their corresponding optimum inlet flow rates derived through the first principle-genetic algorithm integration was used to develop an artificial neural networks model. The artificial neural networks model was then integrated with the first principle model by replacing the genetic algorithm to form a novel gray-box framework. The proposed gray box model, i.e., artificial neural networks and first principle integration, achieved a higher effectiveness and higher outlet temperature than those derived through the straight run first principle model. The performance of the gray box framework was also comparable to the first principle integrated genetic algorithm approach and significantly minimized the computation time needed for estimating the optimum process conditions. First principle integrated genetic algorithm approach enhanced the effectiveness of the straight run model by 3.05%. Performance of the proposed gray-box model was comparable to the integrated framework of the genetic algorithm with the first principle model but was significantly faster. The developed artificial neural networks model was employed as a surrogate in Sobol and FAST sensitivity analysis framework to identify the impact of input variables on output variables. The proposed gray box based method enhanced the capability of the plate and fin heat exchanger to recover energy from the process stream and its robustness to cope with uncertainty. The proposed approach is suitable for real-time application and would contribute to laying a foundation for petroleum refinery 4.0.

Performance analysis and comparison of data-driven models for predicting indoor temperature in multi-zone commercial buildings

Building thermal models, which characterize the properties of a building’s envelope and thermal mass, are essential for accurate indoor temperature and cooling/heating demand prediction. Because of their flexibility and ease of use, data-driven models are increasingly used. This study compared and analyzed the performance of gray-box (resistance-capacitance) and black-box (recurrent neural network) models for predicting indoor air temperature in a real multi-zone commercial building. The developed resistance-capacitance model served as a benchmark model for which full sets of temporal data and building information were used as inputs. The recurrent neural network models were trained and tested assuming various available types and amounts of temporal data and known building physical information to investigate the effects of data and information availability. Feature importance analysis was conducted to select the key variables for different prediction targets under different scenarios. This research provides guidance in selecting an appropriate building thermal response modeling method based on the measured data availability, building physical information, and application.

Development, calibration, and validation of a novel gray-box energy model for residential split air conditioners

Energy models for vapor compression refrigeration in residential air conditioners have been developed through white-box, gray-box, and black-box methods in decades. However, existing white-box and gray-box models require complicated equations with detailed geometries while black-box models require substantial experimental data. This paper aims to develop and validate a simple gray-box steady-state energy model without the need for detailed geometries, which can accurately predict the cooling capacity and electrical power input based on outdoor and indoor air conditions, and supply airflow rates. First three state variables, including the evaporation and condensation temperatures, and refrigerant mass flowrate, are applied to develop the energy model and are solved by three physical equations, including the energy conservations at the evaporator and condenser, and the refrigerant volume-mass flow correlation of the compressor. Secondly, seven performance property functions related to three state variables and three physical equations are identified and calibrated by simple temperature and power measurements. Finally, field experiments are conducted on a residential air conditioner to calibrate these performance property functions and validate the developed model. The validated results reveal the model can accurately predict the cooling capacity and electrical power input, with the normalized root mean square errors of 2.3% and 0.87% respectively.

Genetic Algorithm Optimized Grey-Box Modelling and Fuzzy Logic Controller for Tail-Actuated Robotic Fish

Genetic Algorithm Optimized Grey-Box Modelling and Fuzzy Logic Controller for Tail-Actuated Robotic Fish

A Gray-Box Surrogate Vehicle Energy Consumption Model Capable of Real-Time Updating

Vehicle energy consumption model, as a function of its operational environment, plays a significant role in real-time optimization of vehicle route and speed for a given pair of origin and destination with a desired arrival time for minimal energy consumption. In this article, a gray-box vehicle energy consumption model is developed based on a high-fidelity vehicle dynamic model with environmental influence based on the kriging modeling method, which includes rolling resistance, aerodynamics, gravity, and energy consumption of air conditioning and heater (HVAC), along with environmental conditions, such as temperature, wind speed, etc. The data-driven model, trained based on Gaussian process assumption, ensures the accuracy of the resulting model with a modeling error below 2.5%. The real-time model updating is based on recursive least-squares optimization using current driving data so that the model used for route and speed optimization represents the current vehicle status. The proposed gray-box model is validated in computer-in-the-loop simulations using SUMO and MATLAB with less than 2% error of energy consumption, which is a significant improvement over the vehicle dynamic model with up to 35% error in certain cases. A case study also indicates energy consumption reduction for vehicle route-speed optimization.

A step towards digital operations—A novel grey-box approach for modelling the heat dynamics of ultra-low temperature freezing chambers

Ultra-low temperature (ULT) freezers store perishable bio-contents and have high energy consumption, which highlight a demand for reliable methods for intelligent surveillance and smart energy management. This study introduces a novel grey-box modelling approach based on stochastic differential equations to describe the heat dynamics of the ULT freezing chambers. The proposed modelling approach only requires temperature data measured by the embedded sensors and uses data from the regular operation periods for model identification. The model encompasses three states: chamber temperature, envelope temperature, and local evaporator temperature. Special attention is given to the local evaporator temperature state, which is modelled as a time-variant system, to characterise the time delay and dynamic variations in cooling intensity.The model has three states, of which a time-variant model with nonlinear input for the local evaporator temperature state is specifically established to adapt to the variation of the cooling intensity at the position of the embedded chamber control probe. Two ULT freezers with different operational patterns are modelled. The unknown model parameters are estimated using the maximum likelihood method. The results demonstrate that the models can accurately predict the chamber temperature measured by the control probe (RMSE <0.19°C) and are promising to be applied for forecasting future states. In addition, the model for local evaporator temperature can effectively adapt to different operational patterns and provide insight into the local evaporation cooling supply status. The proposed approach greatly promotes the practical feasibility of grey-box modelling of the heat dynamics for ULT freezers and is a step forward in future digital operationscan serve several potential digital applications. A major limitation of the modelling approach is the low identifiability, which can potentially be addressed by inferring model parameters based on relative parameter changes.

In situ model fusion for building digital twinning

Building digital twins are crucial in holistic, synthetic, and supervisory digital environments for intelligent and energy-efficient building operations. As buildings are designed, constructed, and operated for long-term periods with uncertainties, it is essential to model, verify, extend, and calibrate the digital twin models in situ over the building life cycle, unlike digital twin modeling in the manufacturing industries. Therefore, this study proposes in situ model fusion techniques for building digital twinning, which include (1) model coupling and (2) model assembly. The first model coupling technique enables nonintrusive in situ verification and calibration for prediction models without the model observations (Y). The second model assembly enables in situ modeling or a more accurate model construction by connecting the verified models through the model coupling technique to the input layer of the target model. In the case studies conducted in a central heating system, the prediction model for the secondary-side return water temperature was modeled and calibrated without relying on the model observation data (Y), thanks to the model coupling technique. The nonintrusive model showed an accuracy with a root mean square error (RMSE) of 0.53 °C, which was comparable to the intrusive gray-box models (around 0.5 °C RMSE) calibrated using intrusive datasets (Y). Additionally, the prediction model could improve the benchmark model's performance from 0.44 °C to 0.26 °C (RMSE) using the model assembly technique. The in situ model fusion would enable and enhance nonintrusive modeling approaches, providing a reliable and extensive model environment for digital twins in the building sector.

Process modelling to facilitate model-based decision-making for resource recovery from urban wastewater - A grey-box approach applied to nanofiltration

Currently, decision support tools (DSTs) for wastewater treatment and resource recovery from wastewater use oversimplified databases evaluate and design of treatment trains. The databases consist of only the average, minimum, and maximum process performances, whereas most processes perform differently depending on the process characteristics and operating conditions. To address this issue of oversimplification, this study demonstrates how a grey-box modelling approach for nanofiltration (NF) can serve as an alternative to extensive databases. The membrane model used in this study is a modified version of the solution-diffusion imperfection model proposed by Niewersch et al. (2020). This model was used to estimate water and solute permeabilities (chemical oxygen demand (COD), total nitrogen (TN), and total phosphorous (TP)) based on flux and solute removal literature data at various transmembrane pressures (TMPs; between 4 and 24 bar) for the two membranes, Dow NF90 and NF270. The estimated parameters were cross-validated to predict flux and solute removal. The validation mean absolute percentage errors were below 20 % in most cases, except for the TN rejection, which was 51 %. The applicability and relevance of the NF model were then evaluated using an optimisation model aimed at meeting recovery targets and simultaneously minimising costs (operational and capital expenditure defined by the membrane area and the pumping power, respectively). The optimisation results showed that the selection of an NF membrane (NF90 or NF270) and the operating condition (TMP) were sensitive to the resource recovery targets. In conclusion, a grey-box model can potentially improve the performance of DSTs for resource recovery from wastewater.

Machine learning modeling of the atomic structure and physical properties of alkali and alkaline-earth aluminosilicate glasses and melts

The first version of the machine learning greybox model i-Melt was trained to predict latent and observed properties of K2O-Na2O-Al2O3-SiO2 melts and glasses. Here, we extend the model compositional range, which now allows accurate predictions of properties for glass-forming melts in the CaO-MgO-K2O-Na2O-Al2O3-SiO2 system, including melt viscosity (accuracy equal or better than 0.4 log10 Pa⋅s in the 10−1-1015 log10 Pa⋅s range), configurational entropy at glass transition (≤ 1 J mol−1 K−1), liquidus (≤ 60 K) and glass transition (≤ 12 K) temperatures, heat capacity (≤ 3 %) as well as glass density (≤ 0.02 g cm−3), optical refractive index (≤ 0.006), Abbe number (≤ 4), elastic modulus (≤ 6 GPa), coefficient of thermal expansion (≤ 1.1 10−6 K−1) and Raman spectra (≤ 25 %). Uncertainties on predictions also are now provided. The model offers new possibilities to explore how melt/glass properties change with composition and atomic structure.

Prediction of ship maneuvering motion with grey-box modelling incorporating mechanism and data

ABSTRACT A grey-box modelling incorporating mechanism and data is presented in this paper to predict the ship manoeuvering motion. The traditional mathematical model in three degrees of freedom is adopted to reveal the known movement mechanism, in which an additional hydrodynamic correction term is developed by Neural Network (NN) based on the data to adaptively estimate the errors induced by the idealised approximation. The KRISO Container Ship (KCS) is taken as the study object and the free-running model tests are carried out to obtain the data for establishing the NN model of hydrodynamic correction terms. The typical manoeuvers (tactical circles, zigzags, Williamson turns) are simulated by the grey-box model and the mathematical model, respectively. The results indicate that the grey-box modelling method proposed in this paper can not only improve the prediction accuracy but also demonstrate a good generalisation performance.

Physics-informed online learning of gray-box models by moving horizon estimation

A simple yet expressive prediction model is an essential ingredient in model-based control and estimation. Models derived from fundamental physical principles may fail to capture the complexity of the actual system dynamics. A potential solution is the use of a physics-informed, or gray-box model that extends a physics-based model with a data-driven part. Learning the latter might be challenging, due to noisy measurements and lack of full state information. This work presents a method based on Moving Horizon Estimation (MHE) for simultaneous state estimation and training of a black-box submodel, such as a neural network. The method can be used in offline training or applied online for adaptation without any prior knowledge than the white-box submodel. We analyze the capabilities of the method in a two degree of freedom robotic manipulator case study, also showing how it can be used for online adaptation to cope with a time-varying model mismatch.

A Modular Approach for Diesel Engine Air Path Control Based on Nonlinear MPC

This article presents a systematic approach to realize highly dynamic control strategies for the air path of diesel engines. It is based on grey-box models of individual air path components, designed to be applied in nonlinear model predictive control (NMPC). Specifically, they are suited for algorithmic differentiation and gradient-based optimization. This modular approach allows to derive models for a variety of complex air path systems and can be identified with a low amount of measurement data. An NMPC structure, which is based on these models, enables the tracking of arbitrary air path reference signals and allows the introduction of further control objectives for overactuated systems. We demonstrate our approach’s general applicability and effectiveness on two different laboratory engines using rapid control prototyping hardware. For a turbocharged light-duty diesel engine with dual-loop exhaust gas recirculation (EGR), we apply the proposed control structure to track intake manifold gas conditions while simultaneously minimizing engine pumping losses. Experimental results show an excellent tracking performance and reduced pumping losses compared to other control strategies. For a turbocharged heavy-duty engine with high-pressure EGR, we experimentally demonstrate a superior tracking performance to that obtained with a reference controller. Due to the modular and systematic approach, the algorithm design is straightforward, and the experimental calibration effort is low.

A Gradient-Descent Optimization Assisted Gray-Box Impedance Modeling of EV Chargers

Extracting an electric vehicle (EV) charger's input impedance with the analytical model (white-box approach) or the frequency sweep (black-box approach) is limited by the parameter confidentiality or the measurement noise, respectively. To overcome these challenges, a Gradient-Descent (GD) optimization based gray-box modeling approach is proposed. To start with, a sensitivity study on the analytical impedance model of an EV charger with a typical controller is carried out to identify the influential frequency range per controller and circuit parameter. On top of that, given an EV charger with unknown control and circuit information, a GD optimization based algorithm for multiple parameter estimation is designed to identify the unknown controller and circuit parameters based on the measured impedance, by assuming the EV charger is using the typical controller. Then, an analytical input impedance of the black-box EV charger can be obtained. Moreover, the low accuracy issue commonly encountered when estimating multiple parameters with GD optimization is mitigated with the proposed algorithm. Compared to pure frequency sweep, the proposed approach achieves a higher accuracy for the coupling impedance and a comparable accuracy for the diagonal impedance. The effectiveness of the proposed approach is validated by experimental results.

Grey-box Model Development and Simulation Study for Model-based Predictive Control of Office Zone in Heating Season

Grey-box Model Development and Simulation Study for Model-based Predictive Control of Office Zone in Heating Season

A Grey-box model approach using noon report data for trim optimization

Trim optimization improves the energy efficiency of ships, thus reducing operational costs and emissions; however, trim tables are only available for a limited number of ships. There is thus a desire to develop additional, more accurate trim tables without the need for expensive model testing. The objective of this research was to develop a method to decrease fuel consumption by trim optimization, by a dynamic shaft power estimation model based on available operational data. A method that uses noon report data and a grey-box modelling approach is proposed. The grey box model consists of a multi-layer feedforward neural network to estimate the required shaft power, using operational parameters and an initial estimate of the required shaft power. A case study is presented for a modern chemical tanker and sea trials have been conducted to validate the results. The method provides correct trim advice for full load conditions; however, the magnitude of the effect is smaller compared to sea trial results. The model is able to estimate the required power with an average accuracy of over 6% for a random subset of the noon report data. Due to challenges inherent to noon reports as a data source, the actual effect of trim and speed have a bigger magnitude than the extracted trend.

Hybrid gray and black-box nonlinear system identification of an elastomer joint flexible robotic manipulator

Series elastic actuators possess several properties that make them widely used in collaborative robots, which play a major role in the current paradigm of industry 4.0. However, the compliant element responsible for those desired properties can also be responsible for the addition of unmodeled nonlinearities in the system. Therefore, the aim of this paper is to propose a novel hybrid model approach and apply it in the modeling of an elastomer-based series elastic joint. The proposed hybrid model combines a phenomenological gray-box model with a black-box Nonlinear Auto-regressive model with Exogenous inputs, which is able to provide the desired physical insight while enhancing accuracy by addressing the unknown nonlinearities. The results showed that the proposed hybrid model is more than 60% more accurate than the phenomenological model, considering the mean square error, and obtained a multiple correlation coefficient up to 0.97, indicating its capacity to be used in the construction of a digital twin of the system.

Analysis and comparison of innovative PEMFC systems coupled with an ASHP for space heating in cold climates

In this study, two integrated systems are proposed and analysed: an air source heat pump (ASHP) coupled with either high-temperature proton exchange membrane fuel cells (HT-PEMFCs) or low-temperature proton exchange membrane fuel cells (LT-PEMFCs), where the exhaust gases from the PEMFCs are used to increase the external air temperature to the evaporator of the ASHP. The objective is to understand how the proposed systems behave in a cold climate region and determine which of the systems is better to couple with an ASHP for domestic space heating purposes in a cold climate. The novelty of the work lies in the idea of recovering heat from the exhaust gases from the fuel reformer and PEMFCs and mixing them with external air, before sending it to the ASHP as the inlet air. Grey-box models are employed to model the PEMFCs, whereas the hypothesis of the chemical equilibrium of reacting species is considered in the reformer model. Steady state analyses are considered to understand the performances of the proposed systems compared to two different types of furnaces: high-efficiency and mid-efficiency ones. The results indicate that the HT-PEMFCs are the best for the case studied here, with performances up to 100% greater than the mid-efficiency furnace and up to 63% more than the high-efficiency one. A scenario analysis on a simple pay back is also carried out, and the return on the investment costs in the 2024 scenario is about 24 years if compared to the high efficiency furnace and 14 if compared to the mid-efficiency one, whereas the corresponding numbers are 19 and 11 years, respectively, for the 2030 scenario.