Grey-box Model Research Articles

A Generalized Load Model Considering the Fault Ride-Through Capability of Distributed PV Generation System

Considering the voltage stability problem brought by large-scale distributed PV access to the distribution network, this paper proposes a generalized load model that considers the fault ride-through capability of distributed PV. Firstly, the detailed model of the distribution network is established, and the detailed model is calibrated based on the measured data, the simulation errors are below 1%. And then establish a generalized load model considering distributed PV high and low voltage traversal ability. The sensitivity analysis results are used to rank the parameters to be identified, and the parameters with higher sensitivity are identified. The parameters are obtained from the detailed model and measured data, and four sets of parameters are identified and simulated under different PV penetration rates and fault conditions. The calculated fitting errors are less than 1%. The results show that the generalized load gray box model of the distribution network with distributed PV high and low voltage ride-through capability can reflect the dynamic characteristics of the distribution network well.

Modeling and Aggregation of Electric Water Heaters for the Development of Demand Response Using Grey Box Models

Modeling and Aggregation of Electric Water Heaters for the Development of Demand Response Using Grey Box Models

Experimental parameter identification of scalable white and grey box jounce bumper models for ride comfort simulation

The design of a passenger car’s ride and handling goals requires an enormous amount of hardware and prototypes. To avoid time-consuming testing activities, one crucial trend is the virtualisation of suspension development. Another goal of the virtualisation is the determination of vehicle properties at an early stage of the development process. Highly accurate suspension models are required to transfer the full vehicle’s characteristics into simulation tools. Since the jounce bumper is an important component of the suspension system, modelling its functional behaviour is essential for virtualised chassis development. Therefore, two dynamic modelling approaches are derived and compared in this contribution. One approach is based on local linear model networks and thus can be seen as a grey box model. The other approach consists of white box rheological elements, which allow the physical interpretability of the model parameters. The comparison of the approaches is based on experimental parameter identification, which includes synthetic and real excitations recorded on representative roads. The advantages of the proposed approaches will be discussed in terms of their suitability for the virtual design of the driving behaviour.

A hierarchical framework for minimising emissions in hybrid gas-renewable energy systems under forecast uncertainty

Developing and deploying renewables in existing energy systems are pivotal in Europe’s transition to net-zero emissions. In this transition, gas turbines (GTs) will be central for balancing purposes. However, a significant hurdle in minimising emissions of GTs operating in combination with intermittent renewables arises from the reliance on unreliable meteorological forecasts. Here, we propose a hierarchical framework for decoupling this operational problem into a balancing and emissions minimisation problem. Balancing is ensured with a high-level stochastic balancing filter (SBF) based on data-driven stochastic grey-box models for the uncertain intermittent renewable. The filter utilises probabilistic forecasting and less conservative chance constraints to compute safe bounds, within which a proposed low-level economic predictive controller further minimises emissions of the GTs during operations. As GTs exhibit semi-continuous operating regions, complementarity constraints are utilised to fully exploit each GT’s allowed operational range. The proposed method is validated in simulation for a gas-balanced hybrid renewable system with batteries, three GTs with varying capacities, and a wind farm. Using real historical operational wind data, our simulation shows that the proposed framework balances the energy demand and minimises emissions with up to 4.35% compared with other conventional control strategies in simulation by minimising the GT emissions directly with complementarity constraints in the low-level controller and indirectly with less conservative chance constraints in the high-level filter. The simulations show that the computational cost of the proposed framework is well within requirements for real-time applications. Thus, the proposed operational framework enables increased renewable share in hybrid energy systems with GTs and renewable energy and subsequently contributes to de-carbonising these types of isolated or grid-connected systems onshore and offshore.

Comparison of different control methods on the thermally activated building system (TABS) with large energy flexibility

Thermally Activated Building Systems (TABS) has great energy flexibility potential for the effective demand response, but the inherent large thermal inertia creates challenges for the control. Model Predictive Control (MPC) can improve the control efficiency while achieving certain objectives. However, the modeling mechanism of the TABS thermal behavior can lead to the difference in the controller performances. In this study, the white-box model, grey-box model, and black-box model based MPCs (referred to as W-MPC, G-MPC, and B-MPC) are developed and analyzed in depth for a case TABS with high heat flexibility potential. The performances of MPC strategies are investigated together with the baseline Rule Based Control (RBC) strategy under normal conditions as well as a variety of uncertainty events. In general, MPC strategies show better performance compared to the RBC. The prediction-based control can effectively maintain the indoor temperature within the constraints considering the uncertain disturbances and the gradual outdoor climate change. Moreover, MPC strategies help to increase the cost savings due to the effective load shifting by 30%–50%, and enhance the flexible energy utilization by 14%–29%. In terms of the comparison among the different MPC strategies, W-MPC shows better performance in the room temperature control, which reduces the violation of the indoor temperature constraint by 30% and 15% compared to G-MPC and B-MPC, respectively. G-MPC shows slight superiority concerning the economy and flexible energy usage. The normalized cost saving of G-MPC is approximately 3% and 8% lower compared to the W-MPC and B-MPC, respectively, while the utilization efficiencies of the flexible energy are approximately 6% and 12% higher.

Advancing Structural Health Monitoring in Civil Engineering with Greybox Modelling: A Review

This review explores the application of Greybox Modelling, specifically Physics-Informed Machine Learning (PIML), in the context of Structural Health Monitoring (SHM) within civil engineering. SHM is vital for ensuring the safety and durability of infrastructure namely bridges. Greybox Modelling, merging physics-based principles with data-driven techniques, offers a functional approach for understanding complex structural behaviors. By leveraging existing knowledge of physical laws, this method extracts valuable insights from sensor data. The integration of Greybox Modelling takes SHM a step further. By applying machine learning algorithms with the physics that govern civil engineering structures, Greybox Modelling bridges the gap between data-driven models and physics-based models. This leads more accurate predictions and interpretable outcomes. This review discusses findings from literature, recent studies, that illustrate the effectiveness of Greybox Modelling, specifically PIML in various civil engineering applications. These case studies highlight how this approach can adapt to different structural designs and environmental conditions typically encountered in civil engineering. Moreover, they indicate its potential for damage detection and early warning systems, which enhance the safety and efficiency of infrastructure. The paper also addresses current challenges and outlining future directions in the field.

A grey-box model with neural ordinary differential equations for the slow voltage dynamics of lithium-ion batteries: Application to single-cell experiments

Lithium-ion batteries exhibit a complex, nonlinear and dynamic voltage behaviour. Modelling their slow dynamics is a challenge due to the multiple potential causes involved. We present here a neural equivalent circuit model for lithium-ion batteries including slow voltage dynamics. The model uses an equivalent circuit with voltage source, series resistor, and diffusion element. The series resistance is parameterized using neural networks. The diffusion element is based on a discretized form of Fickian diffusion, parameterized using a neural network and learnable parameters. It is flexible to represent not only Warburg behaviour, but also resistor-capacitor-type dynamics. Mathematically, the resulting model is given by a differential–algebraic equation system combining ordinary and neural differential equations. Therefore, the model combines concepts of both physical theory (white-box model) and artificial intelligence (black-box model) to a combined framework (grey-box model). We apply this approach to a lithium iron phosphate based lithium-ion battery cell. The experimental voltage behaviour during constant-current cycles as well as the dynamics during pulse tests are well reproduced by the model. Only at very high and very low states of charge the simulation significantly deviates from experiments, which might result from insufficient training data in these regions.

Benchmarking Instance-Centric Counterfactual Algorithms for XAI: From White Box to Black Box

This study investigates the impact of machine learning models on the generation of counterfactual explanations by conducting a benchmark evaluation over three different types of models: a decision tree (fully transparent, interpretable, white-box model), a random forest (semi-interpretable, grey-box model), and a neural network (fully opaque, black-box model). We tested the counterfactual generation process using four algorithms (DiCE, WatcherCF, prototype, and GrowingSpheresCF) in the literature in 25 different datasets. Our findings indicate that: (1) Different machine learning models have little impact on the generation of counterfactual explanations; (2) Counterfactual algorithms based uniquely on proximity loss functions are not actionable and will not provide meaningful explanations; (3) One cannot have meaningful evaluation results without guaranteeing plausibility in the counterfactual generation. Algorithms that do not consider plausibility in their internal mechanisms will lead to biased and unreliable conclusions if evaluated with the current state-of-the-art metrics; (4) A counterfactual inspection analysis is strongly recommended to ensure a robust examination of counterfactual explanations and the potential identification of biases.

Enhancing decanter centrifuge process design with data‐driven material parameters in multi‐compartment modeling

AbstractPredicting the separation performance of decanter centrifuges is challenging due to dynamic events within the apparatus. Current methods for designing decanter centrifuges rely on simplified models, often leading to inaccuracies. Consequently, manufacturers must perform time‐intensive pilot scale experiments to derive their own correction factors. Growing computing power sparks interest in alternative modeling strategies. Grey box models (GBM) combine mechanistic white box models (WBM) and data‐driven black box models (BBM), with the optimal structure (parallel or serial) varying by application. For modeling decanter centrifuges, we propose a serial GBM that comprises an artificial neural network that outputs unknown material parameters into a first‐principle multi‐compartment model. Comparing this approach to alternative data‐driven modeling strategies (pure BBM, parallel GBM), we conclude that the serial GBM excels in terms of extrapolation, prediction ability, and transparency while also enabling a better comprehension of the separation process.

Using active cooling/heating for 1C1R grey-box model parameter identification in an actual environment: A proof-of-concept study

A model predictive control (MPC) has been important method for managing both comfort and energy saving in heating, ventilation, and air conditioning (HVAC) problems. However, recent data-driven models for MPC, relying on extensive training data, pose challenges for practical implementation in real-world thermal environments due to their time-consuming nature. The grey-box approach, utilizing a simple RC model as a thermal environment proxy, is well known for its data efficiency. Nevertheless, numerical instability arising from the multi-collinearity of observed data can degrade the model’s time efficiency. To deal with this issue, we propose and implement a rapid and stable parameter identification method that integrates active cooling-heating within the actual thermal environment. Our method requires only a single day of data to estimate basic RC model parameters, allowing for immediate use of the RC model in MPC as soon as an air conditioner is installed. Practical Application Using a grey-box model for MPC in HVAC systems is efficient, but the multicollinearity problem of observed data undermines its time efficiency. Our method deals with this challenge by enabling immediate model parameter identification with just 1 day’s data, incorporating active cooling-heating. We demonstrate it in an actual thermal environment. The high level of time and data efficiency afforded by our method is particularly beneficial for IoT applications like malfunction detection and adaptive room reparameterization associated with room rearrangement.

Air-conditioning load characteristics and grey box predicting model in subway stations

With the continuous expansion of metro, energy conservation of air-conditioning system in subway stations draws the attention of the whole world. Most of the researches are focused on improving the energy efficiency of the system, while the load characteristics of subway stations are paid little attention. The inaccurate load prediction in the designing of air-conditioning system results in redundancy and low operating efficiency. Therefore, accurately predicting the air-conditioning load is crucial for energy saving in subway stations. In this paper, an innovative three-stage subway air-conditioning load test method was proposed and implemented in nine subway stations in Qingdao. Based on the test data, the special air-conditioning load characteristics of subway stations as underground buildings were analyzed. The results showed that electrical equipment load accounted for the largest part, reaching 53.7 %, while only 42 % of the equipment power was converted into air-conditioning load. The positive role of the envelope should not be ignored, especially in the case of entrance and exit walls, which bore almost all of the fresh air load. Based on the three-stage test, a grey box model was established and the error was validated to be within 6 %. With this model, the load of air-conditioning system can be calculated by inputting passenger flow, power of electrical equipment and public area. This paper provides a reference for the design and operation of air-conditioning system in subway stations.

Experimental and simulated evaluation of inverse model for shallow underground thermal storage

This study employs an inverse grey-box (IGB) modeling approach, which combines measured data and the physics of systems to predict the performance of shallow underground thermal energy storage (UTES) with top and side insulation. A simplified IGB model of the shallow UTES was developed using thermal network analysis. Experimental studies were conducted for shallow vertical and horizontal UTES configurations. Detailed models representing the field experiments were developed using the TRNSYS simulation tool and calibrated with the measured data from the experimental setup. The 4 Resistance 2 Capacitance (4R2C) IGB model was trained and tested in MATLAB using field experimental data and the data from the calibrated TRNSYS model. In comparing experimental data with the TRNSYS model for UTES, the prediction of outlet water temperature showed good agreement, with root mean square error (RMSE) and coefficient of variation of RMSE (CVRMSE) values of 0.94 °C and 3.16 % for vertical, and 0.99 °C and 2.97 % for horizontal configurations. The IGB model also aligned well with the experimental data, showing CVRMSE of 7.91 % and 3.17 % for vertical and horizontal systems, respectively. Sensitivity analysis revealed that model performance improves with longer training durations and closer testing times to training periods. However, a convergence point of 20 weeks of training data was achieved for making long-term performance predictions.

Machine-Learning-Based Kinetic-Free Hybrid Gray-Box Model to Predict Sulfuric-Acid-Catalyzed Esterification Reaction.

The sulfuric-acid-catalyzed esterification reaction of 2-butanol and propionic anhydride is a vital industrial process. In this paper, several experiments are conducted via reaction calorimetry to validate that both sulfuric and propionic acids have discernible catalytic effects on the reaction. This finding complicates the accurate description of the reaction kinetics through traditional methods. So this paper turns to a kinetic-free black-box model, Gaussian process regression (GPR) model via 24 experiments, as a more adaptable approach. Besides, the best GPR model is combined with traditional heat balance model to generate a hybrid gray-box model, which can give complete knowledge of reaction process. The hybrid gray-box model finally achieves maximum root-mean-square error (RMSE) of 0.0069 for conversion, 0.6535 for temperature, and 10.6087 for heat flow rate, underscoring its pretty good predictive ability.

A two-step identification framework for building thermal dynamic based on high fluctuation performance data: A case of office with variable refrigerant flow system

Thermal dynamics model plays a crucial role of advanced control in the intelligent building. Grey box method is widely used in the modeling of building thermal dynamics based on the operation data of air-conditioner. The performance of direct-expansion (DX) system fluctuated severely during its operation, which leads to the difficulties in identification and evaluation of the building thermal model. In this paper, a two-step identification framework for second-order thermal resistance-capacity network of building based on DX system data is proposed. A case study is conducted on an office building equipped with variable refrigerant flow system in Foshan, China. The results show that the proposed framework can identify the parameters rationally. The model achieves high accuracy for both indoor temperature (Mean absolute errors of 0.74°C) and cooling load prediction (Mean absolute errors of 4453.86W) in a short time interval of 10 minutes. Additionally, the best complexity of the grey-box model for case building and the data acquisition interval is determined. Finally, an improved set of evaluation metrics for thermal load evaluation is proposed and discussed.

Grey-box modeling for thermal dynamics of buildings under the presence of unmeasured internal heat gains

An accurate grey-box building thermal model is an essential component for prediction-based control of split air conditioners. As the thermal performance of building envelope improves, the internal heat gains play an increasingly noteworthy influence on the building’s thermal dynamics. However, the internal heat gains are usually not measured, which makes the conventional data-dependent identification methods of building thermal models fail. To improve the accuracy of building thermal models under unmeasured internal heat gains, a two-step parameter identification framework for resistance–capacitance (RC) models is proposed in this paper. First, the time-invariant building model parameters are estimated by designing a novel optimization objective function. Next, the time-varying internal heat gains are identified based on the mismatch in predictions of the estimated building model from the first step and the measured temperature. The effectiveness of the method is evaluated using data from a simulation case and an experiment case. The results show that the proposed method outperforms the conventional identification method as well as two other identification methods that handle the unmeasured internal heat gains. Besides, the internal heat gains estimated by the proposed method can capture the trend of the actual internal loads well.

Modeling and nonlinear predictive control of solar thermal systems in district heating

This study presents data-driven modeling and nonlinear model predictive control of solar thermal plants in district heating, for the purpose of operation optimization. The study considers the efficient operation of a solar thermal plant in Hillerød, Denmark. A dynamic model is estimated as a system of stochastic differential equations using grey-box modeling and real-world data. The presented nonlinear model predictive controller design is based on repeated trajectory linearization and the dynamic model. Several objective designs are considered, e.g., maximizing energy or temperature. The study provides simulations for analyzing the model fitness and controller performances. The model is shown to fit the daytime operation of the plant. The designed controllers are shown to improve efficiency, increasing the transported energy up to 28%.

Farm electricity system simulator (FESS): A platform for simulating electricity utilisation on dairy farms

The objective of this paper was to define, validate and demonstrate a model capable of accurately simulating dairy farm electricity consumption across varying herd and parlour sizes, to facilitate research investigating renewable energy systems (RES) and demand side management (DSM). The Farm Electricity System Simulator (FESS) was developed using grey-box modelling techniques utilizing empirical data for parameter tuning. Empirical data were gathered from nine spring calving, pasture based dairy farms located in the Republic of Ireland. A k-means clustering analysis was conducted, separating the farms into three, near homogenous groups, from which representative farms were selected. FESS was trained using 12 months of data from three representative farms using the repeat hold out method for data partitioning with 75 % of data used for training and 25 % used for validation. An optimisation algorithm was used to minimize the error during model training. Through cross-validation, FESS achieved a root mean squared error (RMSE) of 7.65 kWh, mean absolute percentage error (MAPE) of 7.10 %, mean percentage error (MPE) of −0.86 % and a relative prediction error (RPE) of 7.56 % for total daily electricity consumption. Across the three farms, the simulated outputs of FESS achieved an average R2 value of 0.72, demonstrating good agreement with observed data. FESS’s utility was demonstrated by analysing the effects of different electricity pricing structures and on-site solar photovoltaic electricity generation on total farm energy costs. We concluded that FESS simulated on-farm electricity consumption with sufficient accuracy for the intended application. FESS accurately simulated dairy farm electricity consumption across three dairy farms of different herd and parlour sizes while evaluating the effects of demand side management and renewable generation on farm electricity consumption and costs.

Estimation of children's CO2 generation rates in naturally ventilated educational buildings

Indoor CO2 concentration serves as an indicator of indoor air quality, which is mainly influenced by occupants' CO2 generation rate, outdoor ventilation rate and outdoor CO2 levels. Determining the CO2 generation rate is fundamental to design effective ventilation systems that ensure good indoor air quality. The ASTM D6245 standard offers a method to estimate human CO2 generation based on sex, age, body mass, and activity level. However, due to the limited number of studies on CO2 generation rate for children and adolescents in daily scholarly activities, accurated values are difficult to estimate. This research uses a stochastic grey box modelling approach to estimate CO2 generation rates in naturally ventilated schools and compares it with the standard approach. This method can address the complexities and variations in real educational environments. The proposed model was verified through an experimental campaign including 32 classrooms with students aged 4, 10, 13 and 16 years. The model provided a detailed description of the indoor CO2 dynamics in 72% of the studied cases. Estimated CO2 generation rates were found to be in line with the expected standard value of between 1.0 MET to 2.0 MET activity levels (7.07–12.65 L/h for a 4-year-old, 9.63–13.29 L/h for a 10-year-old, 14.54–19.32 L/h for a 13-year-old, and 12.11–20.53 L/h for a 16-year-old). These findings not only bridge the gap in children's CO2 generation rates but also introduce a novel approach to assess human CO2 generation rates under uncontrolled conditions.

Load shifting versus manual frequency reserve: Which one is more appealing to thermostatically controlled loads in Denmark?

This paper investigates demand-side flexibility provision in two distinct forms of manual Frequency Restoration Reserve (mFRR) services and load shifting, and explores which one is financially more appealing to Thermostatically Controlled Loads (TCLs) in Denmark. While mFRR is an ancillary service required for the system and being bought by the system operator, load shifting is an individual act of the TCL in response to the variation of hourly electricity prices. Without loss of generalization, we consider a supermarket freezer as a representative TCL, and develop a grey-box model describing its temperature dynamics using real data from a supermarket in Denmark. Taking into account price and activation uncertainties, a stochastic mixed-integer linear program is formulated to maximize the flexibility value from the freezer. Examined on an ex-post simulation based on Danish spot and balancing market prices in 2022, load shifting shows to be almost as profitable as mFRR provision, although it could be more consequential for temperature deviations in the freezer. This indicates the need for regulatory measures by the Danish system operator to make the attraction of ancillary service provision more obvious for TCLs in comparison to the load shifting alternative.

Process control and energy saving in the ladle stage of a metal casting process through physics-based and ANN-based modelling approaches

The process and temperature control of metal casting applications is of utmost importance both to guarantee the good quality of the final product and also to pursue an energy saving policy. For this purpose, in this paper two different modelling approaches have been proposed to predict the liquid steel temperature inside a ladle for metal casting, shortly before the casting process. The first is a physics-based grey-box model relying on equations for the characterization of the heat transfer mechanisms inside the ladle structure, whereas the second approach relies on artificial neural networks (ANNs). Both methods have been calibrated with experimental data of a case study plant, and subsequently assessed and compared in terms of prediction accuracy. Results show that the physics-based approach is able to predict the casting temperature with a higher mean absolute error (MAE) of 14 °C, whereas the ANNs predictions result to be better, with MAEs around 6 °C. On the other hand, it has been demonstrated that the ANNs approach may lack of reliability, especially if input data strongly differ from the calibration dataset, whereas the physics-based approach results to be more consistent and trustworthy. Finally, an energy analysis is conducted to demonstrate the feasibility of the model in evaluating the potential energy saving compared with situations in which decisions are taken by operators without the aid of a model predictive control.