Grey-box Model Research Articles

Comparison of time-invariant and adaptive linear grey-box models for model predictive control of residential buildings

Model predictive control (MPC) is a promising optimal control technique for activating building energy flexibility using its thermal mass. The performance of the MPC controller is directly related to the accuracy of the model prediction. Grey-box models, based on physical laws and calibrated on measurement data, are commonly used to represent the building thermal dynamics in MPC. Most research works use Linear Time-Invariant (LTI) grey-box models even though weather conditions vary significantly throughout the heating season. This is critical as inaccurate model prediction can lead to a lower performance of the MPC controller. This study introduces two adaptive MPC schemes to overcome this limitation of LTI models. The first one, called the Partially Adaptive MPC, only updates the effective window area of the prediction model. The second one, called the Fully Adaptive MPC, updates all the parameters of the grey-box model. The adaptive MPC performance is compared with MPC using LTI models in two different tests. The simulation-based results show that MPC based on LTI performs well if the control model is trained during a period with similar weather conditions as the period when the MPC will be applied. The Partially Adaptive MPC is unable to deliver satisfactory prediction performance due to the limited number of parameters that are updated. The Fully Adaptive MPC has the best performance compared to the other MPCs, especially as it avoids thermal comfort violations.

Comparative Study-Based Data-Driven Models for Lithium-Ion Battery State-of-Charge Estimation

Batteries have been considered a key element in several applications, ranging from grid-scale storage systems through electric vehicles to daily-use small-scale electronic devices. However, excessive charging and discharging will impair their capabilities and could cause their applications to fail catastrophically. Among several diagnostic indices, state-of-charge estimation is essential for evaluating a battery’s capabilities. Various approaches have been introduced to reach this target, including white, gray, and black box or data-driven battery models. The main objective of this work is to provide an extensive comparison of currently highly utilized machine learning-based estimation techniques. The paper thoroughly investigates these models’ architectures, computational burdens, advantages, drawbacks, and robustness validation. The evaluation’s main criteria were based on measurements recorded under various operating conditions at the Energy Systems Research Laboratory (ESRL) at FIU for the eFlex 52.8 V/5.4 kWh lithium iron phosphate battery pack. The primary outcome of this research is that, while the random forest regression (RFR) model emerges as the most effective tool for SoC estimation in lithium-ion batteries, there is potential to enhance the performance of simpler models through strategic adjustments and optimizations. Additionally, the choice of model ultimately depends on the specific requirements of the task at hand, balancing the need for accuracy with the complexity and computational resources available and how it can be merged with other SoC estimation approaches to achieve high precision.

Low-order gray-box modeling of heating buildings and the progressive dimension reduction identification of uncertain model parameters

Heating buildings have a huge heat capacity and have abilities to accept fluctuating heat supply from renewables. Creating a low-order scalable model that describes the thermal dynamics of coupled heat transfer, charge, and discharge processes among heating buildings is essential to unlocking the thermal storage potential of building mass and achieving space heating decarbonization. White-box models tend to be too complex and black-box models lack scalability. Thus, a low-order gray-box model with uncertain lumped parameters is proposed based on comprehensive heat transfer analyses and necessary assumptions. Thermal mass in heating buildings is divided into four independent heat storage units. The building envelope that can be heated directly or not is regarded as a different heat storage unit. To reduce the search space dimension of model identification, the decision variables are downsized to essential model parameters based on the sensitivity analysis. Moreover, a progressive dimension reduction method is introduced to further improve the probability that the optimization algorithm converges to a global minimum. Case studies show that the calibrated model with global optimal solutions of uncertain model parameters has a good performance on the prediction of the average temperature of indoor air, the average absolute error is only 0.1 °C.

Grey-Box Modeling and Decoupling Control of a Lab Setup of the Quadruple-Tank System

The quadruple-tank system (QTS) is a popular educational resource in universities for studying multivariable control systems. It enables the analysis of the interaction between variables and the limitations imposed by multivariable non-minimum phase zeros, as well as the evaluation of new multivariable control methodologies. The works utilizing this system present a theoretical model that may be too idealistic and based on erroneous assumptions in real-world implementations, such as the linear behavior of the actuators. In other cases, an identified linear model is directly provided. This study outlines the practical grey-box modeling procedure conducted for the QTS at the University of Cordoba and provides guidance for its implementation. A configurable nonlinear model was developed and controlled in a closed loop using different controllers. Specifically, decentralized control, static decoupling control, and simplified decoupling control were compared. The simulation designs were experimentally validated with high accuracy, demonstrating that the conclusions reached with the developed model can be extrapolated to the real system. The comparison of these three control designs illustrates the advantages and disadvantages of decoupling in certain situations, especially in the presence of non-minimum phase zeros.

Indoor Temperature Forecasting in Livestock Buildings: A Data-Driven Approach

The escalating global population and climate change necessitate sustainable livestock production methods to meet rising food demand. Precision Livestock Farming (PLF) integrates information and communication technologies (ICT) to improve farming efficiency and animal health. Unlike traditional methods, PLF uses machine learning (ML) algorithms to analyze data in real time, providing valuable insights to decision makers. Dairy farming in diverse climates is challenging and requires well-designed structures to regulate internal environmental parameters. This study explores the application of the Facebook-developed Prophet algorithm to predict indoor temperatures in a dairy farm over a 72 h horizon. Exogenous variables sourced from the Open-Meteo platform improve the accuracy of the model. The paper details case study construction, data acquisition, preprocessing, and model training, highlighting the importance of seasonality in environmental variables. Model validation using key metrics shows consistent accuracy across different dates, as the mean absolute percentage error on daily base ranges from 1.71% to 2.62%. The results indicate excellent model performance, especially considering the operational context. The study concludes that black box models, such as the Prophet algorithm, are effective for predicting indoor temperatures in livestock buildings and provide valuable insights for environmental control and optimization in livestock production. Future research should explore gray box models that integrate physical building characteristics to improve predictive performance and HVAC system control.

Adaptive control for shape memory alloy actuated systems with applications to human-robot interaction.

Shape memory alloy (SMA) actuators are attractive options for robotic applications due to their salient features. So far, achieving precise control of SMA actuators and applying them to human-robot interaction scenarios remains a challenge. This paper proposes a novel approach to deal with the control problem of a SMA actuator. Departing from conventional mechanism models, we attempt to describe this nonlinear plant using a gray-box model, in which only the input current and the output displacement are measured. The control scheme consists of the model parameters updating and the control law calculation. The adaptation algorithm is founded on the multi-innovation concept and incorporates a dead-zone weighted factor, aiming to concurrently reduce computational complexities and enhance robustness properties. The control law is based on a PI controller, the gains of which are designed by the pole assignment technique. Theoretical analysis proves that the closed-loop performance can be ensured under mild conditions. The experiments are first conducted through the Beckhoff controller. The comparative results suggest that the proposed adaptive PI control strategy exhibits broad applicability, particularly under load variations. Subsequently, the SMA actuator is designed and incorporated into the hand rehabilitation robot. System position tracking experiments and passive rehabilitation training experiments for various gestures are then conducted. The experimental outcomes demonstrate that the hand rehabilitation robot, utilizing the SMA actuator, achieves higher position tracking accuracy and a more stable system under the adaptive control strategy proposed in this paper. Simultaneously, it successfully accommodates hand rehabilitation movements for multiple gestures. The adaptive controller proposed in this paper takes into account both the computational complexity of the model and the accuracy of the control results, Experimental results not only demonstrate the practicality and reliability of the controller but also attest to its potential application in human-machine interaction within the field of neural rehabilitation.

From the Modeling of an Electrochemical YSZ-Based Gas Sensor Used in Electrolysis Mode.

Electrochemical sensors have been used for many decades. However, the modeling of such sensors used in electrolysis mode is poorly documented, especially in the case of multiple gases' parallel actions. These are of great interest since they constitute the first brick to bring information on the natures and concentrations of gaseous mixture compositions, thanks to gray box modeling of sensor arrays, for example. Based on Butler-Volmer's equations, a model assuming parallel reactions at gold cathode has been introduced in this article and confronted with experimental results. The establishment of the model is based on the extraction of three variables: the charge transfer coefficient "α", the reaction order γ, and the reaction constant rate k0. Tests performed without pollutants and with different concentrations of oxygen could be nicely fitted using the model. The influence of the polarization current on the three variables of the model has been evaluated, showing a clear influence on the constant rate and the reaction order. Moreover, increasing the polarization current enabled us to obtain selectivity for oxidant gases. Similarly, the effect of the oxygen concentration was evaluated. Results showed that, in this case, the charge transfer coefficients "α" obtained for oxidant gases are quite different from the ones obtained in the polarization current varying conditions. Therefore, the model will be interesting in situations where polarization current and oxygen content are not varied together. Variation of polarization current can be quite interesting to obtain increased information for multivariate analysis purposes in constant oxygen content situations. Additionally, other parameters have to be considered for applications in which the oxygen content is bound to change, such as exhaust gases or combustion.

A comparison between grey-box models and neural networks for indoor air temperature prediction in buildings

Model Predictive Control has gained much attention due to its potential to improve building operations by reducing costs, integrating renewable energy sources, and increasing thermal comfort. This paper aims to compare the accuracy of grey-box models based on resistance–capacitance (RC) networks and Long-Short-Term Memory (LSTM) neural networks in the prediction of the buildings’ thermal response, which is a key feature for the successful implementation of predictive controllers. Indoor air temperature prediction tests have been performed on simulated and measured data from buildings with different thermal insulation and thermal mass during both heating and cooling seasons. Results show that neural networks have, on average, a better prediction performance than grey-box models. Both modelling approaches are affected by the building characteristics and by the season considered. The grey-box models require less training data, although the latter seems to play a role only in the worse-performing tests. When user setpoint changes in the testing phase, the LSTM neural network shows a significant drop in the root mean square error. In conclusion, although LSTM outperforms grey-box models on average, the reduced training data and higher reliability under normal operating conditions, as well as their linearity, make RC models a strong alternative for predictive controllers.

An integrated grey-box model for accurate ship engine performance prediction under varying speed and environmental conditions

The development of ship engine fault diagnosis has led to an increased interest in predicting marine engine performance under various environmental and operating conditions. However, predicting engine performance using a single model is limited due to the various characteristics of ship engines depending on speed and environmental conditions. To address this issue, we propose a grey box model (GBM)-based ship engine performance prediction framework that combines the white box model and black box model (BBM) appropriately for both low- and high-speed operating conditions. The application of data preprocessing techniques, such as clustering and cleaning, under specific engine and environmental conditions, combined with dimensionality reduction techniques (partial least square and principal component) and the use of WBM/BBM models under classified speed conditions, makes the proposed framework accessible to real-world operators through data-driven approaches and domain knowledge. We demonstrate that the proposed framework improves the robustness, accuracy, and efficiency of engine performance predictions by considering the characteristics of each speed in real-world navigation data, compared to using single models.

Real-time Learning-based Nonlinear Model Predictive Control of a virtual motorcycle employing grey-box modeling through Gaussian processes

Virtual prototyping tools simulations are nowadays largely adopted methods in the automotive industry. However, to achieve effective simulative tests for two-wheeled vehicles, a virtual rider, i.e. a controller for virtual motorcycles, is typically needed due to the inherent instability of the system. Different control strategies have been employed to deal with this control task, and promising results have been obtained using Nonlinear Model Predictive Control (NMPC). Yet, performance of NMPC highly relies on the plant characterization, that can be highly complex to obtain analytically for 2-wheel vehicles. To improve it, learning dynamics approaches can be used within a Learning-based Nonlinear Model Predictive Control (LbNMPC) framework, exploiting the combination of data-driven techniques and NMPC strategy. In this manuscript, we present a tailored real-time capable LbNMPC for a virtual motorcycle that relies on a continuous grey-box model based on Gaussian Processes (GPs), method that has proven to be effective for 4-wheel vehicle applications. To cope with the real-time requirement, a feature selection procedure and sparse GP approximations have been adopted. A comparison with a physics-based model NMPC implementation taken from the literature has been carried out, analyzing the impact of different sparse GP representations. Remarkable improvements in both model accuracy and tracking performance have been obtained, and the generalization properties of the grey-box model have been assessed.

Grey-box modelling and identification of the industrial oven of a shrink tunnel

This paper presents a lumped-parameter grey-box sampled-data state-space model for the industrial oven of a shrink tunnel. The model is derived following the thermal-electrical analogy. A novel discretization strategy is developed to take into account that the sampling time of the system is equal to the lowest period of the pulse-width-modulated voltage signals which drive the heat resistors of the industrial oven. The model parameters are estimated by means of an extensive experimental campaign. Experimental results show that the derived model outperforms state-of-the-art transfer-function models while depending on fewer parameters.

Physics-Informed Online Learning by Moving Horizon Estimation: Learning Recurrent Neural Networks in Gray-box Models*

In Model Predictive Control (MPC) closed-loop performance heavily depends on the quality of the underlying prediction model, where such a model must be accurate and yet simple. A key feature in modern MPC applications is the potential for online model adaptation to cope with time-varying changes, part-to-part variations, and complex features of the system dynamics not caught by models derived from first principles. In this paper, we propose to use a physics-informed, or gray-box, model that extends the physics-based model with a data-driven component, namely a Recurrent Neural Network (RNN). Relying on physics-informed models allows for a rather limited size of the RNN, thereby enhancing online applicability compared to pure black-box models. This work presents a method based on Moving Horizon Estimation (MHE) for simultaneous state estimation and learning of the RNN sub-model, a potentially challenging issue due to limited information available in noisy input-output data and lack of knowledge of the internal state of the RNN. We provide a case study on a quadruple tank benchmark showing how the method can cope with part-to-part variations.

Online Lithium-ion Battery Modeling and State of Charge Estimation via Concurrent State and Parameter Estimation

This paper develops a novel approach for online Lithium-ion (Li-ion) battery model identification and state of charge (SOC) estimation. To account for the SOC-dependent battery dynamics and the static nonlinearity between the open-circuit voltage (OCV) and SOC, we formulate a grey box nonlinear state-space model, in which elements depend on SOC in a polynomial way. For model identification, we propose an online concurrent state and parameter estimation by alternating the recursive least squares algorithm and particle filter; the SOC is computed via Coulomb counting during the modeling. The identified grey box model is then applied for SOC estimation using the particle filter. Simulation with real-world battery measurements demonstrates the effectiveness of the model structure and the estimation approach, which is reflected in accurate terminal voltage estimation and nonlinear OCV-SOC relation, and superior performance regarding SOC estimation compared to state-of-the-art approaches.

Grey-box modeling of Air Handling Units for Analysis and Virtual Sensing

The optimization of control sequences in air handling units (AHUs) presents a significant opportunity for energy savings within HVAC systems. However, many building owners and operators require quantifiable estimates of potential energy savings before committing to retrofitting control systems. Valid estimates of energy savings require system models that consider capacity and limitations of the AHU, but in existing systems, scarce information hinders such modeling efforts. This lack of information complicates AHU modeling and the assessment of alternative control strategies. This paper demonstrates an approach that leverages time-series data from a newly constructed Danish AHU, equipped with multiple sensors for temperature, flow, and pressure, to construct a grey-box model of the unit, including component properties. The estimated parameters for the components are validated against data sheet information, and shows that the estimation procedure is accurate for parameter estimation. To analyze the energy-efficiency of the cooling coil, the model is used to estimate the latent cooling in the cooling coil, as Danish conditions rarely require dehumidification of supply air.

Physics-informed and black-box Identification of robotic actuator with a flexible joint

In robotics, precise models are critical for ensuring safety and functionality. However, acquiring a precise model characterizing a system’s dynamics can be challenging. One of the alternatives to address this issue is system Identification, which aims to obtain models through physical and experimental observations. In this manner, the developments in machine learning algorithms, such as neural networks, have significantly improved the modeling of complex and nonlinear phenomena. In this work, a mass-spring-damper (MSD) system and a low-cost original elastomer-based Series Elastic Actuators (eSEA) assembly are used to evaluate the performance of system Identification models. The black-box models selected are variations of the AutoRegressive Moving Average with eXogenous input (ARMAX) algorithm. The gray-box model aims to estimate the parameters of 4 friction models; the optimization is done utilizing physics-informed neural networks (PINNs). For both case studies, the PINNs outperformed the black-box models. In the didactic example, the parameters obtained are close to the ground truth, and the highest determinant coefficient obtained is 0.99. The friction model that best represents the robotic actuator is the LuGre model, with the parameters obtained using the PINNs, outperforming the best black-box model by lowering the mean absolute error (MAE) by 30.83%. With a determinant coefficient of 0.94, the model shows a high capacity for describing the multiple nonlinearities present in the system.

Gray-Box Modeling for Distribution Systems with Inverter-Based Resources: Integrating Physics-Based and Data-Driven Approaches

Gray-Box Modeling for Distribution Systems with Inverter-Based Resources: Integrating Physics-Based and Data-Driven Approaches

Power Increase due to Marine Biofouling: a Grey-box Model Approach

This paper proposes a grey-box modelling approach to predict marine biofouling growth and its effects on ship performance. The approach combines empirical or experimental-based white-box models with data-driven black-box models. First, a white-box model is built to predict ship resistance considering a bare hull. This prediction is based on calm water resistance, wind, waves, and temperature differences. Subsequently, marine biofouling growth is predicted using an experimental model that estimates the level of roughness on the ship hull. Finally, a deep extreme learning machine is used as a black-box model, employing a feedforward neural network technique. To test the approach, a superyacht case study was selected as a category of vessel heavily exposed to fouling. The study used a 2-year dataset obtained through a collaboration with Feadship. Results showed that the black-box approach outperforms the white-box approach in predictive capabilities. However, when the knowledge encapsulated in the white-box model is included in the grey-box approach, the model shows the highest prediction accuracy achieved by leveraging less historical data. This study demonstrates the potential of the proposed grey-box approach to accurately predict marine biofouling growth and its effects on ship performance, which can benefit ship operators and designers in improving operational efficiency and reducing maintenance costs.

Digital twin of patient in clinical workflow

The concept of a digital twin being made use of in healthcare is an emerging research area. Digital twins in healthcare have the potential to enable more precise and personalised care, especially for patients experiencing chronic conditions. Previous work has identified digital twins as mathematical models. Digital twins have been classified as grey box, surrogate and black box models. Based on this classification, the black box models can handle data intensive and sophisticated problems. This makes black box models a candidate for constructing digital twins of patients. Such digital twins can then assist clinicians in clinical decision making. This has the potential to reduce cognitive burden on clinicians, to increase precision and personalisation of care through enhanced use of data, and to improve patient outcomes and cost implications. However, introducing such digital twins to healthcare would be a significant intervention that would alter traditional clinical workflows. As such, we present in this paper one of the first attempts of conceptual mapping of altered next generation primary care clinical workflows that would allow the incorporation of digital twins of patients in managing chronic conditions.

Diversity, quality, and quantity of real ship data on the black-box and gray-box prediction models of ship fuel consumption

Establishing an efficient and reliable fuel consumption prediction model by using big data from ships facilitates optimized decision-making and ensures the green and intelligent development of ships. As data are crucial in model construction, this study presents a general multi-source data processing method for obtaining high-quality training data. A long short-term memory (LSTM) neural network, suitable for time-series data, was used to develop the fuel consumption black-box model. This was combined with the ship theory to establish a fuel consumption theoretical model, thereby generating the LSTM based gray-box model. We explored the impact of data diversity, quality, and quantity on black-box and gray-box models. Analysis of the navigation data of a passenger ship and meteorological data collected from the European Centre for Medium-Range Weather Forecasts (ECMWF) and MeteoBlue indicated that the combination of variables obtained via the feature selection of the least absolute shrinkage and selection operator (LASSO) statistical method yielded the best overall prediction performance. Moreover, the gray-box model was relatively stable in terms of the changes in effective variables. An analysis of data quality revealed that the systematic processing of outliers, which improves the accuracy of both models by 6.19% compared with direct deletion. Furthermore, the gray-box models use less amounts of data than the black-box models to achieve higher accuracy.

Higher heating value prediction of high ash gasification-residues: Comparison of white, grey, and black box models

The measurement of the higher heating value (HHV) in high-ash solid waste poses persistent challenges due to the inherent limitations of using an oxygen bomb calorimeter. HHV predictive models based on the components analysis were proved to be useful methods. To improve the accuracy and applicability of HHV models, this work selected 100 actual measured data of high ash gasification residues, and compared three types of predictive models from white, grey, and black box by the following procedures: Modeling, External-validation, and Extending study. In the Modeling and External-validation processes, eight models from linear regression, grey model, and machine learning models, were proposed with R2 > 0.95 and MAPE<8.42 %. Compared to existing research, the models proposed in this study provide appealing accuracies. In the Extending study for applicability evaluation, the UA-based models UOGM (1, 5) (R2 = 0.86 and MAPE = 12.96 %) and RL-Ult (R2 = 0.89 and MAPE = 11.52 %) outperformed the other six models by using the eight-expanding data of different solid wastes that collected from the literature. Based on these results, the above two models were applied to predict the HHV of remaining 53 gasifier residues, showing reliable predicting results. This work shows that the developed models have high accuracy for the HHV prediction of high-ash solid wastes.