Grey-box Model Research Articles

Physics-integrated hybrid framework for model form error identification in nonlinear dynamical systems

For real-life nonlinear systems, the exact form of nonlinearity is often not known and the known governing equations are often based on certain assumptions and approximations. Such representation introduce model-form error into the system. In this paper, we propose a novel gray-box modeling approach that not only identifies the model-form error but also utilizes it to improve the predictive capability of the known but approximate governing equation. The primary idea is to treat the unknown model-form error as a residual force and estimate it using dual Bayesian filter based joint input-state estimation algorithms. For improving the predictive capability of the underlying physics, we first use machine learning algorithm to learn a mapping between the estimated state and the input (model-form error) and then introduce it into the governing equation as an additional term. This helps in improving the predictive capability of the governing physics and allows the model to generalize to unseen environment. Although in theory, any machine learning algorithm can be used within the proposed framework, we use Gaussian process in this work. To test the performance of proposed framework, case studies discussing four different dynamical systems are discussed; results for which indicate that the framework is applicable to a wide variety of systems and can produce reliable estimates of original system’s states. Apart from this, the algorithm has also been tested for a case where the data has been taken from an experimental setup (Silver box dataset). The results produced further showcase the efficacy of the proposed framework.

Time-dependent solar aperture estimation of a building: Comparing grey-box and white-box approaches

This paper proposes a B-splines integrated method combining in-situ data with grey-box modeling to estimate buildings' dynamic solar gain more efficiently than the conventional white-box model and much more precisely than the classic grey-box model. Solar gain, referring to the overall indoor energy gain supplied by solar radiation, plays a vital role in the indoor energy balance. Estimating dynamic solar gain precisely is essential to building energy optimization, e.g., in model predictive control. However, in almost all existing grey-box modeling works, a constant solar gain factor (solar aperture; gA) is assumed to estimate dynamic solar gain, which almost certainly will result in solar gain prediction errors, especially in buildings with unevenly distributed windows. To fill this gap, this study presents an advanced B-splines integrated grey-box model, using customized B-splines to advance the constant gA assumption toward its nature of time-dependence and precisely characterize the dynamic solar gain conclusively. On-site measured datasets of a portable site office (PSO) representing a ‘simplified’ building, under two scenarios with windows fully or partially uncovered, serve as test cases. To verify the physical interpretation of outcomes estimated by the proposed method, based on the said test cases, the proposed B-splines integrated grey-box model is compared with a classic white-box simulation. It is concluded that the proposed method can reveal the main trends and key dynamic features of solar gain very well, but still has some limitations of quantifying ‘local’ details with acceptable variations. Nevertheless, given that the proposed method merely asks for a very limited amount of low-frequency data, the proposed method is considered as a much more effective alternative to the classic white-box simulation approach, which requires massive and often hard-to-collect input data.

Longitudinal Mode System Identification of an Insect-like Tailless Flapping-Wing Micro Air Vehicle Using Onboard Sensors

In this paper, model parameter identification results are presented for a longitudinal mode dynamic model of an insect-like tailless flapping-wing micro air vehicle (FWMAV) using angle and angular rate data from onboard sensors only. A gray box model approach with indirect method was utilized with adaptive Gauss–Newton, Levenberg–Marquardt, and gradient search identification methods. Regular and low-frequency reference commands were mainly used for identification since they gave higher fit percentages than irregular and high-frequency reference commands. Dynamic parameters obtained using three identification methods with two different datasets were similar to each other, indicating that the obtained dynamic model was sufficiently reliable. Most of the identified dynamic model parameters had similar values to the computationally obtained ones, except stability derivatives for pitching moment with forward velocity and pitching rate variations. Differences were mainly due to certain neglected body, nonlinear dynamics, and the shift of the center of gravity. Fit percentage of the identified dynamic model (~49%) was more than two-fold higher than that of the computationally obtained one (~22%). Frequency domain analysis showed that the identified model was much different from that of the computationally obtained one in the frequency range of 0.3 rad/s to 5 rad/s, which affected transient responses. Both dynamic models showed that the phase margin was very low, and that it should be increased by a feedback controller to have a robustly stable system. The stable dominant pole of the identified model had a higher magnitude which resulted in faster responses. The identified dynamic model exhibited much closer responses to experimental flight data in pitching motion than the computationally obtained dynamic model, demonstrating that the identified dynamic model could be used for the design of more effective pitch angle-stabilizing controllers.

Grey-box modelling and fuzzy logic control of a Leader–Follower robot manipulator system: A hybrid Grey Wolf–Whale Optimisation approach

This study presents the development of a grey-box modelling approach and fuzzy logic control for real time trajectory control of an experimental four degree-of-freedom Leader–Follower​ Robot (LFR) manipulator system using a hybrid optimisation algorithm, known as Grey Wolf Optimiser (GWO) - Whale Optimisation Algorithm (WOA). The approach has advantages in achieving an accurate model of the LFR manipulator system, and together with a better trajectory tracking performance. In the first instance, the white box model is formed by modelling the dynamics of the follower manipulator using the Euler–Lagrange formulation. This white-box model is then improved upon by re-tuning the model’s parameters using GWO-WOA and experimental data from the real LFR manipulator system, thus forming the grey-box model. A minimum improvement of 73.9% is achieved by the grey-box model in comparison to the white-box model. In the latter part of this investigation, the developed grey-box model is used for the design, tuning and real-time implementation of a fuzzy PD＋I controller on the experimental LFR manipulator system. A 78% improvement in the total mean squared error is realised after tuning the membership functions of the fuzzy logic controller using GWO-WOA. Experimental results show that the approach significantly improves the trajectory tracking performance of the LFR manipulator system in terms of mean squared error, steady state error and time delay.

Building Thermal-Network Models: A Comparative Analysis, Recommendations, and Perspectives

The development of smart buildings, as well as the great need for energy demand reduction, has renewed interest in building energy demand prediction. Intelligent controllers are a solution for optimizing building energy consumption while maintaining indoor comfort. The controller efficiency on the other hand, is mainly determined by the prediction of thermal behavior from building models. Due to the development complexity of the models, these intelligent controllers are not yet implemented on an industrial scale. There are primarily three types of building models studied in the literature: white-box, black-box, and gray-box. The gray-box models are found to be robust, efficient, of low cost computationally, and of moderate modeling complexity. Furthermore, there is no standard model configuration, development method, or operation conditions. These parameters have a significant influence on the model performance accuracy. This motivates the need for this review paper, in which we examined various gray-box models, their configurations, parametric identification techniques, and influential parameters.

COBALT: COnstrained Bayesian optimizAtion of computationaLly expensive grey-box models exploiting derivaTive information

Many engineering problems involve the optimization of computationally expensive models for which derivative information is not readily available. The Bayesian optimization (BO) framework is a particularly promising approach for solving these problems, which uses Gaussian process (GP) models and an expected utility function to systematically tradeoff between exploitation and exploration of the design space. BO, however, is fundamentally limited by the black-box model assumption that does not take into account any underlying problem structure. In this paper, we propose a new algorithm, COBALT, for constrained grey-box optimization problems that combines multivariate GP models with a novel constrained expected utility function whose structure can be exploited by state-of-the-art nonlinear programming solvers. COBALT is compared to traditional BO on seven test problems including the calibration of a genome-scale bioreactor model to experimental data. Overall, COBALT shows very promising performance on both unconstrained and constrained test problems.

Influence of data pre-processing and sensor dynamics on grey-box models for space-heating: Analysis using field measurements

A grey-box model is a combination of data-driven and physics-based approaches to modeling. For applications in buildings, grey-box models can be used as the control model in model predictive control (MPC) or to characterize the thermal properties of buildings. In a previous study using data generated from virtual experiments, the influence of data pre-treatment on the performance of grey-box models has been demonstrated. However, field measurement differs from data generated using building performance simulation (BPS). This is because the precision and accuracy, the location, and the dynamics of the sensors could be different. Consequently, this paper extends previous results and conclusions using a real test case of a highly-insulated residential building. The results confirm that data pre-processing has a minimal influence on the identified results (parameter values and simulation performance) for deterministic models. On the contrary, data pre-treatment influences the performance of stochastic models as follows. Firstly, large sampling time (Ts) can cause the parameters to become non-physical and can sometimes reduce the one-day ahead prediction performance. With large Ts, the anti-causal shift (ACS) proves to be beneficial to keep the parameters physically plausible while low-pass filtering can also contribute but to a lesser extent. With large Ts, ACS does not guarantee a higher one-day ahead prediction performance for stochastic models, whereas pre-filtering generally has a positive impact. Secondly, for the stochastic model, the sensor dynamics should be modeled if the sensor has a noticeable time constant to guarantee the physical plausibility of the parameters. Thirdly, the dynamics of the hydronic radiator do not need to be modeled if the time constant in the temperature sensors is larger than the radiator. These findings provide practical guidelines for grey-box modeling of buildings with field measurement data.

An Adaptive Building Skin Concept Resulting from a New Bioinspiration Process: Design, Prototyping, and Characterization

Building envelopes can manage light, heat gains or losses, and ventilation and, as such, play a key role in the overall building performance. Research has been focusing on increasing their efficiency by proposing dynamic and adaptive systems, meaning that they evolve to best meet the internal and external varying conditions. Living organisms are relevant examples of adaptability as they have evolved, facing extreme conditions while maintaining stable internal conditions for survival. From a framework based on the inspiration of living envelopes such as animal constructions or biological skins, the concept of an adaptive envelope inspired by the Morpho butterfly was proposed. The system can manage heat, air, and light transfers going through the building and includes adaptive elements with absorption coefficients varying with temperature. This paper presents the developed framework that led to the final concept as well as the concept implementation and assessment. A prototype for heat and light management was built and integrated into a test bench. Measurements were performed to provide a first assessment of the system. In parallel, geometrical parametric models were created to compare multiple configurations in regards to indicators such as air, light, or heat transfers. One of the models provided light projections on the system that were compared with measurements and validated as suitable inputs in grey-box models for the system characterization.

An Incremental Grey-Box Current Regression Model for Anomaly Detection of Resistance Mash Seam Welding in Steel Mills

Annealing and galvanization production lines in steel mills run continuously to maximize production throughput. As a part of this process, individual steel coils are joined end-to-end using mash seam welding. Weld breaks result in a production loss of multiple days, so non-destructive, data-driven techniques are used to detect and replace poor quality welds in real-time. Statistical models are commonly used to address this problem as they use data readily available from the welding machine and require no specialized equipment. While successful in finding anomalies, these statistical models do not provide insight into the underlying process and are slow to adapt to changes in the machine’s or material’s behavior. We combine knowledge-based and data-driven techniques to create an incremental grey-box welding current prediction model for detecting anomalous welds, resulting in a powerful and interpretable model. In this work, we detail our approach and show evaluation results on industrial welding data collected over a period of 15 months containing behavioral shifts attributed to machine maintenance. Due to its incremental nature, our model resulted in two-thirds fewer rejected welds compared to statistical models, thus greatly reducing production overhead. Grey-box modeling can be applied to other welding features or domains and results in models that are more desirable for the industry.

Performance Comparison of Control Strategies for Plant-Wide Produced Water Treatment

Offshore produced water treatment (PWT) accounts for cleaning the largest waste stream in the offshore oil and gas industry. If this separation process is not properly executed, large amounts of oil are often directly discharged into the ocean. This work extends two grey-box models of a three-phase gravity separator and a deoiling hydrocyclone, and combines them into a single plant-wide model for testing PWT control solutions in a typical process configuration. In simulations, three known control solutions—proportional-integral-derivative (PID) control, H∞ control, and model predictive control (MPC)—are compared on the combined model to evaluate the separation performance. The results of the simulations clearly show what performance metrics each controller excels at, such as valve wear, oil discharge, oil-in-water (OiW) concentration variance, and constraint violations. The work incentivizes future control to be based on operational policy, such as defining boundary constraints and weights on oil discharge, rather than maintaining conventional intermediate performance metrics, such as water level in the separation and pressure drop ratio (PDR) over the hydrocyclone.

Gray-box Soft Sensor for Water Content Monitoring in Fluidized Bed Granulation.

Soft sensors are powerful tools for the implementation of process analytical technology (PAT). They are categorized into white-box (first-principle), black-box (statistical), and gray-box models. Gray-box models integrate white-box and black-box models to address each drawback, i.e., prediction accuracy and intuitiveness. Although they have been applied to various industrial processes, their applicability to water content monitoring in fluidized bed granulation has not been reported. In this study, we evaluated three types of gray-box models, i.e., parallel, serial, and combined gray-box models, in terms of prediction accuracy using real operating data on a commercial scale with two formulations. The gray-box models were constructed by integrating the heat and mass balance model (white-box model) and locally weighted partial least squares regression (LW-PLSR) model (black-box model). LW-PLSR was utilized to cope with collinearity and nonlinearity. In the serial gray-box models, LW-PLSR models adjusted the fitting parameters of the white-box model depending on the process parameters for each query. In the parallel gray-box or combined gray-box models, LW-PLSR models compensated for the output error of the white-box or serial gray-box models, respectively. The results demonstrated that all three types of gray-box models improved the prediction accuracy of the white-box models regardless of the formulation. Besides, we proposed the assessment method based on Hotelling's T2 and Q residual for gray-box models using LW-PLSR, which contributes decision support to select gray-box or white-box model. The accurate and descriptive gray-box models are expected to enhance process understanding and precise quality control in fluidized bed granulation.

When is gray-box modeling advantageous for virtual flow metering?

Integration of physics and machine learning in virtual flow metering applications is known as gray-box modeling. The combination is believed to enhance multiphase flow rate predictions. However, the superiority of gray-box models is yet to be demonstrated in the literature. This article examines scenarios where a gray-box model is expected to outperform physics-based and data-driven models. The experiments are conducted with synthetic data where properties of the underlying data generating process are controlled. The results show that a gray-box model yields increased prediction accuracy over a physics-based model in the presence of process-model mismatch, and improvements over a data-driven model when the amount of available data is small. On the other hand, gray-box and data-driven models are similarly influenced by noisy measurements. Lastly, the results indicate that a gray-box approach may be advantageous in nonstationary process conditions. Unfortunately, model selection prior to training is challenging, and overhead on gray-box model development and testing is unavoidable.

From white-box to grey-box modelling of the heat dynamics of buildings

Identifying the parameters of grey-box models requires enough data collected from sensors installed inside and outside of the building for long enough period of time. Consequently, this process is time consuming, costly especially in large buildings that require more sensors, and can only be conducted after the building is constructed. This paper introduces a procedure for identifying greybox models from white-box models. Following this procedure, grey-box models can be identified using data generated by a white-box model, without any requirement for mounting sensors in a building. This reduces the cost and time of modelling, control design and prediction. The introduced procedure is utilized to find a grey-box model of the heat dynamics of a four-floor building in Spain. Simulation results demonstrate the effectiveness of this procedure.

Adaptive Linear Grey-Box Models for Model Predictive Controller of Residential Buildings

Model predictive control (MPC) is an advanced optimal control technique to minimize a control objective while satisfying a set of constraints and is well suited to activate the building energy flexibility. The MPC controller performance depends on the accuracy of the model prediction. Inaccurate predictions can directly lead to low control performance. Linear time-invariant (LTI) models are often used in MPC in buildings. However, LTI models do not adapt to the weather conditions varying throughout the whole space-heating season, which makes the MPC based on LTI models not perform well over a long period of time. Therefore, this study introduces an adaptive MPC where the parameters of a linear grey-box model are continuously updated in real-time. Two alternative versions of this adaptive control are investigated. The first one, called partially adaptive MPC, only updates the effective window area of the grey-box model, while the second one, called fully adaptive MPC, updates all the parameters of the grey-box model. Results show that the partially adaptive MPC is not able to deliver satisfactory prediction performance. The fully adaptive MPC shows better performance compared to the other models when implemented in a MPC, especially in avoiding thermal comfort violation.

Concise Parameter Identification Method for 1C1R Gray-Box Model Using Thermal Balance Points Associated with Active Cooling/Heating

Gray-box modeling of the thermal environment of buildings is widely used. The simplest gray-box model, called the 1C1R, is particularly important because it provides the first approximation of the thermal characteristics of building fabrics. We propose a fast and robust estimation method in which we introduce an integrated form of the 1C1R model as well as active cooling/heating at night to generate temperature change. The artificial temperature change is utilized to sequentially estimate the model parameters. Numerical experiments showed that our method is not only accurate but also avoids the multi-collinearity problem that conventional approaches have been suffering.

Transfer Function-Based Characterization of the Honey Bee Olfactory System: From Biology to Electronic Circuits

We exemplify an interdisciplinary approach wherein a mesoscopic-scale functional model of a biological system is derived from time-series recordings, yielding transfer functions that can be used to design analog electronic circuits. Namely, sensory processing in the honey bee, a universal model for studying olfaction, is considered. Existing studies have focused on its antennal lobe, wherein only the responses of its functional units, known as glomeruli, have been accessible. Here, high temporal resolution calcium imaging is deployed to track the dynamics of odor-evoked activity beyond this processing stage. The responses in the somata outside of the antennal lobe are recorded, showing for the first time how the glomerular signals are transformed before entering the higher brain centers. A transfer function approach is applied to capture as a “gray box model” the remarkably heterogeneous signal transformations between odor input and glomerular response, and between glomerular signals and somata activity. The somata are tentatively mapped to the glomeruli via Granger causality, while machine learning classification and clustering allow grouping common properties regarding response amplitudes and temporal profiles. The obtained low-order transfer functions display time- and frequency-domain input-output properties closely similar to the biological system. Because transfer functions have universal applicability, once they have been determined, it is readily possible to design corresponding analog electronic circuits, with possible future applications in sensor signal conditioning. To exemplify this, examples based on resistor-capacitor (RC) networks and operational amplifiers are physically built and confirmed to generate responses highly correlated to the initial biological recordings.

Fractional modeling of wind speed turbulence

This paper proposes a method to design a wind turbulence model based on real wind spectral characteristics. It uses models based on Cole-Cole fractional functions to approximate the wind turbulence power spectral density. Von Kármán model is the most commonly used model but originally designed for aircraft in high altitude. Therefore, it is not suitable for systems operating at low altitude, hence the need for more accurate models in these specific conditions. Shaping filters issued from the fractional models are employed to generate realistic random wind speed turbulence from a random white noise input. Then, a grey box model is proposed from physical parameters such as the classic von Kármán ones (mean speed, turbulence intensity and length scale).

Grey-Box Modelling and Virtual Sensors Enabling Continuous Commissioning of Hydronic Floor Heating

Grey-Box Modelling and Virtual Sensors Enabling Continuous Commissioning of Hydronic Floor Heating

Reduction and Observer Design for a Grey-Box Model in Continuous Pharmaceutical Manufacturing

In this contribution, a novel Reduced Order Model (ROM) formulation of the grey-box model proposed in Elkhashap et al. (2020a) for the pharmaceutical continuous vibrated fluid bed dryer (VFBD) is presented. The ROM exploits the H2-norm projection-based model order reduction method after a special solution formulation of the model's infinite-dimensional part. This is mainly by introducing a vector field mapping between the model parts casting the semi- discretized PDE into a bilinear form. The ROM produced is then integrated into an nonlinear Kalman Filtering-based observer design also handling the estimation of the model's algebraic variables. Evaluations of the FOM, ROM, ROM-based observer variants, and the FOM-based observer are performed using Monte-Carlo simulations as well as simulations based on experimental data of the real system. It is shown that the ROM could reproduce the FOM states accurately with a relative mean square error below 0.3 % for the experimental data simulation. This is while reaching a computational-time reduction up to a factor of 40. The ROM-based observer with algebraic states correction is shown (using Monte-Carlo simulations) to be able to converge to the true values for all cases regardless of initialization. Moreover, it is also shown that the performance degradation of the observer due to reduction is practically insignificant. This is while the computational speedup of the observer due to reduction reached a factor of more than third order of magnitude.

Integrating Thermal-Electric Flexibility in Smart Buildings using Grey-Box modelling in a MILP tool

In a smart grid setting, building managers are encouraged to adapt their energy operations to real-time market and weather conditions. However, most literature assume stationary temperature set points for heating and cooling. In this work, we propose a grey-box model to investigate how the energy flexibility of the thermal mass of the building may impact its energy flexibility potential as well as the investment decisions of the energy system within a building, by using an already developed investment decision tool, BUILDing’s OPTimal operation and energy design model (BUILDopt) (Lindberg et al. (2016)). As BUILDopt is a Mixed Integer Programming (MIP/MILP) tool, the flexibility models must be linear as well. We evaluate the energy flexibility potential, here called comfort flexibility, for use cases reflecting different heating systems (electric panel ovens vs. ground source heat pump) and operation (flexible vs. non-flexible). The case study of an Office building is performed, which considers electric specific demand, domestic hot water demand and space heating demand. Real historical data for weather and energy prices from Oslo are used, including grid tariffs related energy and monthly peak power. Most of the savings are obtained through peak load reduction, which can reach up to 13-16%. These and the savings from shifting demand away from peak prices lead to total savings of around 2%. Yet, these actions do not require additional investment in heat supply or storage components, nor in building renovations: only system measurement and control components are needed.