Grey-box Model Research Articles

Vibration based pavement roughness monitoring system using vehicle dynamics and smartphone with estimated vehicle parameters

Due to the high cost associated with pavement roughness monitoring, researches are going on to develop new techniques of pavement monitoring which are cost effective, frequent and easily implementable. Conventionally, pavement monitoring is performed using road profilers consisting of laser and inertial sensors. Though these profilers provide measurements of high accuracy, the high operating cost of these devices makes it difficult to ensure regular pavement monitoring. With the development of technology, new researches are being performed for the development of low-cost pavement monitoring systems using ordinary vehicle and smartphones. However, in most of the studies, the effect of vehicle suspension system on pavement monitoring was not considered and also direct reconstruction of pavement profile was not performed to compute the International Roughness Index (IRI), widely used for pavement roughness monitoring system. Instead, some correlation based procedures have been followed for measurement of pavement roughness. In this paper, a vibration based method for monitoring of pavement condition is introduced using ordinary vehicle and smartphone as vibration sensors which includes calibration for vehicle suspension system. The whole process can be divided into two stages. The unknown dynamic parameters of the ordinary vehicle are identified first. The vehicle is driven over a speed bump of known dimension and vehicle vertical acceleration is recorded. State-space representation of the vehicle model with unknown parameters is performed and the parameters are estimated using Grey-Box model system identification technique. Secondly, a new pavement profile reconstruction technique is proposed based on inverse formulation of vehicle dynamics. The vertical acceleration of the vehicle with known parameters are collected by driving it over the pavement, the condition of which needs to be monitored. A new mathematical formulation of vehicle dynamics is developed and using the collected acceleration data, pavement profile is reconstructed. Further, International Roughness Index of the reconstructed pavement is calculated. Numerical simulations and field testing of the proposed method are performed for various pavement roughness conditions and vehicle speeds performed for determining its viability for pavement condition monitoring.

Artificial neural network grey-box model for design and optimization of 50 MWe-scale combined supercritical CO2 Brayton cycle-ORC coal-fired power plant

Supercritical CO2 (sCO2) Brayton cycle is a promising technology for coal-fired power generation with high efficiency and compact equipment size. However, its sophisticated construct and high-temperature waste heat rejection require a systematic design to maximize its performance. Herein, we develop a combined sCO2 Brayton cycle-organic Rankine cycle (ORC) design for coal-fired power plant. A novel glass-box model that considers the specific designs of sCO2 boiler, recuperators, coolers, and turbomachinery is formulated to optimize the power plant. A high-accuracy artificial neural network model is also developed to estimate the system’s pressure drop to reduce model complexity. As a result, the glass-box model is reformulated into a grey-box model. The model is applied to three different combined cycles’ design problem to evaluate their performance. Result shows that the grey-box model saves more than 50% of CPU time. With the turbine inlet at 620 °C/25 MPa and the main compressor inlet at 35 °C/7.38 MPa, the proposed combined cycle reaches a thermal efficiency of 45.73%, thereby achieving a 2.75 percentage point improvement compared with the standalone design. Sensitivity analysis is also carried out to evaluate the effects of ORC working fluid, flue gas temperature at the cooling wall outlet, main compressor inlet pressure, and evaporating temperature of ORC, on the system’s performance.

Grey-box modeling and hypothesis testing of functional near-infrared spectroscopy-based cerebrovascular reactivity to anodal high-definition tDCS in healthy humans

Transcranial direct current stimulation (tDCS) has been shown to evoke hemodynamics response; however, the mechanisms have not been investigated systematically using systems biology approaches. Our study presents a grey-box linear model that was developed from a physiologically detailed multi-compartmental neurovascular unit model consisting of the vascular smooth muscle, perivascular space, synaptic space, and astrocyte glial cell. Then, model linearization was performed on the physiologically detailed nonlinear model to find appropriate complexity (Akaike information criterion) to fit functional near-infrared spectroscopy (fNIRS) based measure of blood volume changes, called cerebrovascular reactivity (CVR), to high-definition (HD) tDCS. The grey-box linear model was applied on the fNIRS-based CVR during the first 150 seconds of anodal HD-tDCS in eleven healthy humans. The grey-box linear models for each of the four nested pathways starting from tDCS scalp current density that perturbed synaptic potassium released from active neurons for Pathway 1, astrocytic transmembrane current for Pathway 2, perivascular potassium concentration for Pathway 3, and voltage-gated ion channel current on the smooth muscle cell for Pathway 4 were fitted to the total hemoglobin concentration (tHb) changes from optodes in the vicinity of 4x1 HD-tDCS electrodes as well as on the contralateral sensorimotor cortex. We found that the tDCS perturbation Pathway 3 presented the least mean square error (MSE, median <2.5%) and the lowest Akaike information criterion (AIC, median -1.726) from the individual grey-box linear model fitting at the targeted-region. Then, minimal realization transfer function with reduced-order approximations of the grey-box model pathways was fitted to the ensemble average tHb time series. Again, Pathway 3 with nine poles and two zeros (all free parameters), provided the best Goodness of Fit of 0.0078 for Chi-Square difference test of nested pathways. Therefore, our study provided a systems biology approach to investigate the initial transient hemodynamic response to tDCS based on fNIRS tHb data. Future studies need to investigate the steady-state responses, including steady-state oscillations found to be driven by calcium dynamics, where transcranial alternating current stimulation may provide frequency-dependent physiological entrainment for system identification. We postulate that such a mechanistic understanding from system identification of the hemodynamics response to transcranial electrical stimulation can facilitate adequate delivery of the current density to the neurovascular tissue under simultaneous portable imaging in various cerebrovascular diseases.

Machine learning based digital twin for stochastic nonlinear multi-degree of freedom dynamical system

The potential of digital twin technology is immense, specifically in the infrastructure, aerospace, and automotive sector. However, practical implementation of this technology is not at an expected speed, specifically because of lack of application-specific details. In this paper, we propose a novel digital twin framework for stochastic nonlinear multi-degree of freedom (MDOF) dynamical systems. The proposed digital twin has four modules — (a) a physics-based nominal model, (b) a data collection module, (c) algorithm for real-time update of the digital twin and (d) module for predicting future state. The modules for real-time update and prediction are based on the so-called gray-box modeling approach, and utilizes both physics based and data driven frameworks; this enables the proposed digital twin to generalize and predict future responses. The gray box modeling framework used within the digital twin is developed by coupling Bayesian filtering and machine learning algorithm. Although, the proposed digital twin can be used with any machine learning regression algorithm, we have used Gaussian process in this study. Performance of the proposed approach is illustrated using two examples. Results obtained indicate the applicability and excellent performance of the proposed digital twin framework.

System dynamics kinetic model for predicting biogas production in anaerobic condition: Preliminary assessment.

This preliminary assessment of a grey-box model, was predicated on system dynamics principles and developed using Vensim® DSS software. The purpose is to predict biogas production under anaerobic conditions for energy utilization at the design stage. To describe the process of a developed system dynamics model to predict biogas production under anaerobic conditions. This method involves two-stage kinetics of the biogas production process in anaerobic conditions using the first-order and Gompertz functions. The model is depicted in two parts: causal loop diagram and stock-flow diagram. The causal loop diagram describes the anaerobic digestion process a substrate undergoes for the production of biogas, while stock-flow diagram depicts basic building blocks of the dynamic behavior of an anaerobic digestion process. Primary data is from a laboratory-scale experiment of biogas production using vegetal wastes, while the secondary one is from the literature on studies using similar substrates. Primary and secondary data are used to validate and stimulate the developed model. The kinetic model shows the substrate being reduced exponentially with increasing time; consumption of substrate and production of methane and carbon dioxide follows exponential growth and decay pattern, with carbon dioxide production starting early compared to methane, and was produced at a rate faster due to the strong and resilient characteristics of fermentative microorganisms. Comparing data from empirical and model simulation shows some close relationship, though not too perfectly. Both results reflect signs of inhibitions occurring within the substrates in the digester under anaerobic conditions explaining the low methane yield or instability.

Battery Model Identification Approach for Electric Forklift Application

Electric forklifts are extremely important for the world’s logistics and industry. Lead acid batteries are the most common energy storage system for electric forklifts; however, to ensure more energy efficiency and less environmental pollution, they are starting to use lithium batteries. All lithium batteries need a battery management system (BMS) for safety, long life cycle and better efficiency. This system is capable to estimate the battery state of charge, state of health and state of function, but those cannot be measured directly and must be estimated indirectly using battery models. Consequently, accurate battery models are essential for implementation of advance BMS and enhance its accuracy. This work presents a comparison between four different models, four different types of optimizers algorithms and seven different experiment designs. The purpose is defining the best model, with the best optimizer, and the best experiment design for battery parameter estimation. This best model is intended for a state of charge estimation on a battery applied on an electric forklift. The nonlinear grey box model with the nonlinear least square method presented a better result for this purpose. This model was estimated with the best experiment design which was defined considering the fit to validation data, the parameter standard deviation and the output variance. With this approach, it was possible to reach more than 80% of fit in different validation data, a non-biased and little prediction error and a good one-step ahead result.

Dynamic mode decomposition for nonintrusive and robust model predictive control of residential heating systems

Obtaining accurate models for heating and building systems is crucial for prediction and control in the context of energy efficiency and demand response. Models should be both computationally and data-efficient, as well as easy to implement. This paper therefore introduces a methodology for data-driven modeling and control of residential heating systems and buildings. Dynamic mode decomposition is used to fit a linear state-space model of the building and the heating system. It is shown that this procedure results in prediction accuracy that is akin to the literature on greybox models. In order to cope with the uncertainty around weather predictions, the state-space model is integrated in a robust linear model predictive control framework. The controller exhibits the required energy shifting behavior while only requiring a dataset size on the order of days.

Use of Machine Learning Methods for Indoor Temperature Forecasting

Improving the energy efficiency of the building sector has become an increasing concern in the world, given the alarming reports of greenhouse gas emissions. The management of building energy systems is considered an essential means for achieving this goal. Predicting indoor temperature constitutes a critical task for the management strategies of these systems. Several approaches have been developed for predicting indoor temperature. Determining the most effective has thus become a necessity. This paper contributes to this objective by comparing the ability of seven machine learning algorithms (ML) and the thermal gray box model to predict the indoor temperature of a closed room. The comparison was conducted on a set of data recorded in a room of the Laboratory of Civil Engineering and geo-Environment (LGCgE) at Lille University. The results showed that the best prediction was obtained with the artificial neural network (ANN) and extra trees regressor (ET) methods, which outperformed the thermal gray box model.

An adaptive real-time grey-box model for advanced control and operations in WRRFs.

Grey-box models, which combine the explanatory power of first-principle models with the ability to detect subtle patterns from data, are gaining increasing attention in wastewater sectors. Intuitive, simple structured but fit-for-purpose grey-box models that capture time-varying dynamics by adaptively estimating parameters are desired for process optimization and control. As an example, this study presents the identification of such a grey-box model structure and its further use by an extended Kalman filter (EKF), for the estimation of the nitrification capacity and ammonia concentrations of a typical Modified Ludzack-Ettinger (MLE) process. The EKF was implemented and evaluated in real time by interfacing Python with SUMO (Dynamita™), a widely used commercial process simulator. The EKF was able to accurately estimate the ammonia concentrations in multiple tanks when given only the concentration in one of them. In addition, the nitrification capacity of the system could be tracked in real time by the EKF, which provides intuitive information for facility managers and operators to monitor and operate the system. Finally, the realization of EKF is critical to the development of future advance control, for instance, model predictive control.

Non-linear grey-box modelling for heat dynamics of buildings

This paper introduces a non-linear grey-box (GB) model based on stochastic differential equations that describes the heat dynamics of a school building in Denmark, equipped with a water-based heating system. The building is connected to a local district heating network through a heat exchanger. The heat is delivered to the rooms mainly through radiators and partially through a ventilation system. A monitoring system based on IoT sensors provides data on indoor climate in the rooms and on the heat load of the building. Using this data, we estimate unknown states and parameters of a model of the building’s heating system using the maximum likelihood method. Important novelties of this paper include models of the water flow in the circuit and the state of the valves in the radiator thermostats. The non-linear model accurately predicts the indoor air temperature, return water temperature and heat load. The ideas behind the model lay a foundation for GB models of buildings that use different kinds of water-based heating systems such as air-to-water/water-to-water heat pumps. Such GB models enable model predictive control to control e.g. the indoor air climate or provide flexibility services.

Gray-box model-based predictive control of Czochralski process

The present study proposes a gray-box (GB) model-based predictive control method to produce high-quality 300 mm silicon ingots in the commercial Czochralski (CZ) process. The GB model consists of an energy transfer, hydrodynamic, and geometrical model and a statistical model, predicts three controlled variables, i.e., crystal radius, growth rate, and melt position, and represents the time-varying and nonlinear characteristics of the CZ process. Solving an optimization problem with the GB model requires heavy computational load; therefore, the proposed method derives the prediction model by successive linearization of the GB model to compute optimal manipulated variables in several seconds. The proposed method was compared with the conventional method using PID controllers in disturbance rejection performance through control simulations. The results have demonstrated that the integral absolute error (IAE) of the proposed method was reduced by 60% on average and 89% at maximum even when a plant-model mismatch exists.

Validity and Reliability of Hydraulic-Analogy Bioenergetic Models in Sprint Roller Skiing.

Purpose: To develop a method for individual parameter estimation of four hydraulic-analogy bioenergetic models and to assess the validity and reliability of these models’ prediction of aerobic and anaerobic metabolic utilization during sprint roller-skiing.Methods: Eleven elite cross-country skiers performed two treadmill roller-skiing time trials on a course consisting of three flat sections interspersed by two uphill sections. Aerobic and anaerobic metabolic rate contributions, external power output, and gross efficiency were determined. Two versions each (fixed or free maximal aerobic metabolic rate) of a two-tank hydraulic-analogy bioenergetic model (2TM-fixed and 2TM-free) and a more complex three-tank model (3TM-fixed and 3TM-free) were programmed into MATLAB. The aerobic metabolic rate (MRae) and the accumulated anaerobic energy expenditure (Ean,acc) from the first time trial (STT1) together with a gray-box model in MATLAB, were used to estimate the bioenergetic model parameters. Validity was assessed by simulation of each bioenergetic model using the estimated parameters from STT1 and the total metabolic rate (MRtot) in the second time trial (STT2).Results: The validity and reliability of the parameter estimation method based on STT1 revealed valid and reliable overall results for all the four models vs. measurement data with the 2TM-free model being the most valid. Mean differences in model-vs.-measured MRae ranged between -0.005 and 0.016 kW with typical errors between 0.002 and 0.009 kW. Mean differences in Ean,acc at STT termination ranged between −4.3 and 0.5 kJ and typical errors were between 0.6 and 2.1 kJ. The root mean square error (RMSE) for 2TM-free on the instantaneous STT1 data was 0.05 kW for MRae and 0.61 kJ for Ean,acc, which was lower than the other three models (all P < 0.05). Compared to the results in STT1, the validity and reliability of each individually adapted bioenergetic model was worse during STT2 with models underpredicting MRae and overpredicting Ean,acc vs. measurement data (all P < 0.05). Moreover, the 2TM-free had the lowest RMSEs during STT2.Conclusion: The 2TM-free provided the highest validity and reliability in MRae and Ean,acc for both the parameter estimation in STT1 and the model validity and reliability evaluation in the succeeding STT2.

Comparison of Gray-Box and Black-Box Two-Level Three-Phase Inverter Models for Electrical Drives

The accuracy of the set voltage compared to its reference value has a high impact on the control performance of three-phase electrical drives, especially if the voltage is an input variable of integrated observers (e.g., flux observer). Voltage deviations are generally a disturbance and, therefore, they lead to a less-than-ideal control performance of the drive. In the literature, most approaches only model the basic inverter effects but do not sufficiently address its strongly nonlinear behavior at very high or low duty cycles. To enable accurate voltage estimation in the entire operation range of a given drive, a black-box inverter model utilizing machine learning is presented in this article. By means of artificial neural networks, a direct mapping of available signals in the control framework and the actual inverter output voltage is realized. For comparison, a gray-box inverter model whose parameters are identified by using half-automatized recorded datasets and a particle swarm optimization is derived. Comprehensive experimental investigations prove the effectiveness of both approaches. The gray-box model can estimate the phase voltages per switching period precisely with a root-mean-square error of less than 1.1 V at a 560 V DC-link voltage level, whereas the machine-learning-based black-box approach even achieves an error of less than 0.65 V.

A robust gray-box modeling methodology for variable-speed direct-expansion systems with limited training data

• A robust training method against limited training data is proposed for DX units. • The methodology identifies the most influential parameters to estimate. • More estimation parameters result in degraded parameter accuracy. • Model prediction accuracy is highly dependent on training data size. Compared to physics-based models, the gray-box modeling approach has clear advantages including reduced engineering efforts in the model development phase, higher computational efficiency, improved accuracy and reduced uncertainties. In this paper, we present a gray-box steady-state modeling methodology for variable-speed direct-expansion systems that is robust against limited training data. The methodology incorporates a dual-stage training scheme, where component models are identified separately, and then integrated through continuity constraints to establish a system model trained on a higher hierarch. To further improve model reliability and robustness, the training methodology incorporates two commissioning techniques, i.e., significance ranking and de-correlation, to down-select the estimation parameters according to the quality of the training data. To demonstrate the efficacy, the proposed modeling methodology was applied to a 3-ton variable-speed heat pump. Qualities of the identified models were evaluated in terms of parameter accuracy and model prediction accuracy against variable training data sizes. Robustness of the methodology was demonstrated in achieving reasonable model accuracies with limited training data.

Grey-box modelling of lithium-ion batteries using neural ordinary differential equations

Grey-box modelling combines physical and data-driven models to benefit from their respective advantages. Neural ordinary differential equations (NODEs) offer new possibilities for grey-box modelling, as differential equations given by physical laws and neural networks can be combined in a single modelling framework. This simplifies the simulation and optimization and allows to consider irregularly-sampled data during training and evaluation of the model. We demonstrate this approach using two levels of model complexity; first, a simple parallel resistor-capacitor circuit; and second, an equivalent circuit model of a lithium-ion battery cell, where the change of the voltage drop over the resistor-capacitor circuit including its dependence on current and State-of-Charge is implemented as NODE. After training, both models show good agreement with analytical solutions respectively with experimental data.

Integration of Classical Mathematical Modeling with an Artificial Neural Network for the Problems with Limited Dataset

One of the most common problems in science is to investigate a function describing a system. When the estimate is made based on a classical mathematical model (white-box), the function is obtained throughout solving a differential equation. Alternatively, the prediction can be made by an artificial neural network (black-box) based on trends found in past data. Both approaches have their advantages and disadvantages. Mathematical models were seen as more trustworthy as their prediction is based on the laws of physics expressed in the form of mathematical equations. However, the majority of existing mathematical models include different empirical parameters, and both approaches inherit inevitable experimental errors. Simultaneously, the approximation of neural networks can reproduce the solution exceptionally well if fed sufficient data. The difference is that an artificial neural network requires big data to build its accurate approximation, whereas a typical mathematical model needs several data points to estimate an empirical constant. Therefore, the common problem that developers meet is the inaccuracy of mathematical models and artificial neural networks. Another common challenge is the mathematical models’ computational complexity or lack of data for a sufficient precision of the artificial neural networks. Here we analyze a grey-box solution in which an artificial neural network predicts just a part of the mathematical model, and its weights are adjusted based on the mathematical model’s output using the evolutionary approach to avoid overfitting. The performance of the grey-box model is statistically compared to a Dense Neural Network on benchmarking functions. With the use of Shaffer procedure, it was shown that the grey-box approach performs exceptionally well when the overall complexity of a problem is properly distributed with the mathematical model and the Artificial Neural Network. The obtained calculation results indicate that such an approach could increase precision and limit the dataset required for learning. To show the applicability of the presented approach, it was employed in modeling of the electrochemical reaction in the Solid Oxide Fuel Cell’s anode. Implementation of a grey-box model improved the prediction in comparison to the typically used methodology.

Development of a Framework for Activation of Aggregator Led Flexibility

This paper presents a novel framework architecture for an online, real-time flexibility assessment and activation platform targeted at unlocking the flexibility potential of commercial buildings and smaller industrial sites, thereby enabling greater levels of renewable grid integration. Renewable integration targets in Europe of up to 40% of power generation from renewable sources by 2030 and over 90% by 2050 aim to decarbonize the electrical grid and increase electrification of transport, industry, and buildings. As renewable integration targets increase, participation in flexibility programs will be required from a much greater range of buildings and sites to balance grids hosting high levels of renewable generation. In this paper, an online implementation of a standardized flexibility assessment methodology, previously developed for offline contract negotiations between stakeholders, is modified to automate the assessment. The automated assessment is then linked to an aggregator-based multi-building or site optimization stage, enabling increased participation of multiple buildings and sites. To implement the assessment, models for individual flexible systems were reviewed, selected, and adapted, including physics-based, data-driven, and grey-box models. A review of optimization for flexibility found mixed-integer linear programming to be the optimal approach for the selection of flexible systems for demand response events.

Grey-box model calibration and validation for a continuous horizontal fluidized bed dryer

Fully continuous drying can improve the production flexibility and throughput of tablets manufactured by wet granulation lines. A recently developed prototype incorporates a screw conveyor to a horizontal fluidized bed dryer for pharmaceutical particles. Research on advanced process control strategies led to the publication of a two-phase tanks-in-series framework to describe its dynamics. This paper calibrates the model to replicate the prototype behavior. A grey-box identification methodology estimates the empirical and uncertain parameters using experimental input-output data. The underlying assumptions are cross-validated with time-domain comparisons, output root mean square errors and confidence intervals. Results show that the model reproduces the observed trends in general and the discrepancies result from mechanical issues on the equipment.

How does the control logic influence the establishment of a data-driven chiller model?

The modelling of chiller performance is critical for chiller optimal control, and fault detection and diagnosis (FDD). Different kinds of chiller models including sophisticated mechanistic models (white-box models), purely data-driven models (black-box models like artificial neural networks, ANN), and semi-physical models (grey-box models like empirical equations) have been proposed and tested. Due to the development of machine learning techniques, data-driven models have become more popular recently. The performance of the established data-driven model (accuracy, robustness, generalization) could significantly affect the model application. To enhance the model performance, a lot of studies have been carried out on investigating and modifying the model structure. However, the influence of the data quality on model training has not been sufficiently studied. When adopting historical data to train models, the data distribution is highly correlated to the control logic. So how does the control logic influence the establishment of data-driven chiller models by affecting the data distribution? In this study, experiments are conducted on an air-cooled chiller under model-free stochastic control to acquire rich and variable operational dataset; then the dataset is grouped into three corresponding to different chiller control logics. Finally, three models trained by three training datasets are evaluated, and the results suggest that when establishing data-driven chiller models, preliminary stochastic operation is cost-effective to acquire rich data for robust chiller modelling.

Physical-stochastic continuous-time identification of a forced Duffing oscillator

Despite the simplicity of the Duffing oscillator, its dynamical behaviour is extremely rich. Hence, the Duffing equations are used to describe the dynamic behaviour of many real-world nonlinear systems for a wide range of frequency bands and amplitude of the excitation signal in basic sciences and engineering. For example, the Duffing oscillator has been successfully used to model a variety of physical processes such as stiffening springs, beam buckling, nonlinear electronic circuits, superconducting Josephson parametric amplifiers, and ionisation waves in plasmas etc. Therefore, the identification of the Duffing oscillator model/parameters directly from the measured input–output data is a topic of active research in many scientific fields In this paper, we use the concept of stochastic differential equations (SDEs) to identify a model of the Duffing oscillator. SDE-based grey-box models allow us to capture the underlying mathematical structure describing the physics of the system (e.g. the original Duffing equations) using the drift term and explicitly handling of model uncertainty (or the process noise) using the diffusion term whereas the measurement uncertainty is modelled using the measurement noise term respectively. In this paper, we propose a slight variation of the maximum likelihood estimation framework used for the identification of SDEs based grey-box models yielding improved performance for long-term predictions. The proposed framework is combined with an iterative residual analysis to develop a grey-box model of the forced Duffing oscillator. The benchmark data from the so-called Brussels “Silverbox system”, which is an electrical circuit mimicking the forced Duffing oscillator dynamics is used for the identification purpose. Finally, the identified model performance (the simulation errors) is compared with the existing results available in the literature.