1. Introduction
Internal air temperature is a critical parameter for the simulation of the indoor environment in buildings. In order to simulate important metrics such as thermal comfort, occupant productivity and energy consumption, accurate internal air temperature models are required.
White box (WB), black box (BB) and grey box (GB) models have been employed in previous studies for internal air temperature prediction. WB models are physics-based and mechanistic in operation. WB models typically utilise a large number of building descriptive parameters such as material properties, spatial dimensions, fenestration orientations and mechanical/thermal system specifications. BB models are data driven and use regression or machine learning algorithms to map the relationship between system inputs and outputs using large amounts of empirical data without the requirement for static building descriptive parameters [
1,
2]. GB models, like WB models, are mechanistic in nature, however, the iterative physics-based operations are simplified and aggregated. Therefore, they are less computationally expensive and require fewer building descriptive parameters [
3,
4]. The most commonly utilised GB models in thermal engineering applications are resistive-capacitive (RC) models [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17] (
Table 1). An RC model uses an electrical analogy to describe model structure and function, where resistors (R) and capacitors (C) are used to simulate the thermal energy flow and storage in the building. RC models typically simplify WB model parameters by lumping parameters together into single resistors or capacitors. The simplification of RC models is achieved through order-reduction [
7,
18] and the lumping of parameters [
4,
14,
16].
Typically, WB models which utilise precise building parameters were found to be reasonably accurate when validated under experimental conditions in building test-cells [
19,
20]. However, WB models were found to be inaccurate when compared to measured data from real buildings [
21]. Applying manual or automated calibration techniques to WB models has been shown to greatly improve accuracy for internal air temperature prediction in real buildings [
22,
23,
24].
Model calibration often requires parameter tuning or selection. Parameter tuning can be performed using manual [
25], automated [
26] and semi-automated [
27] approaches. The manual calibration approach uses human intuition combined with standardised calibration metrics [
28] to minimize prediction error. This approach begins with an initial un-calibrated model populated with design stage building parameters. These parameters are then tuned via a series of revisions to the initial model based on the relative error reduction after each revision. [
29,
30]. The automated calibration approach uses mathematical or statistical methods for parameter tuning [
26]. These methods typically utilise optimisation algorithms to reduce the error between model outputs and measured data [
16]. Semi-automated or mixed approaches to calibration combine intuitive and mathematical methods [
31]. Typically, an initial model is created manually, parameters are selected for tuning and boundary limits are manually selected for each parameter. The algorithm then selects the final values of these parameters through an iterative error minimization process [
8,
32]. The difference in time taken to manually or automatically calibrate models has been shown in some instances to be very substantial [
26,
32]. As manual calibration relies on user intuition, this process can be time intensive (e.g., 40 to 50 h), whereas automated calibration can take over 80% percent less time (e.g., two to seven hours).
In building simulation, WB models are the most widely used models in practice [
33]. Typically, WB models are manually calibrated [
22,
25,
34]. However, previous studies have also used semi-automated calibration procedures with WB models [
32,
35]. BB models are developed solely from empirical data where the internal architecture and coefficients (i.e., number for neurons in the hidden layer of a neural network and the synaptic weightings between neurons) are selected through an automatic procedural method based on globalised error reduction [
36,
37,
38,
39,
40,
41]. For GB models, several studies use manual [
5,
11,
14] and semi-automated calibration approaches [
12]. However, the most common approach is automated calibration [
3,
4,
13,
15,
16,
17]. The majority of studies using RC models for internal air temperature prediction have either relied on synthetic data generated by WB models or a higher order GB model for calibration and validation [
3,
6,
7,
8,
11,
15,
16,
17]. When empirical data are employed for such purposes, it is most commonly recorded data from unoccupied test-cells or rooms [
5,
12,
14]. The number of RC models that have been calibrated and validated using recorded data from real occupied buildings is limited [
10,
13].
Nearly zero energy buildings (nZEBs) are now a legislative requirement for new-builds in many parts of the world. A large amount of literature has identified nZEBs and zero energy or net zero energy buildings as the target for buildings in the future [
42,
43,
44,
45]. Reaching nZEB standards requires improvements in fabric and energy performance when compared to existing buildings. Although energy performance certificates suggest that many nZEBs exist [
46], the sample of well documented case study examples in published literature is limited [
23,
47,
48,
49,
50]. Over 80% of the simulation studies that focused on modelling and performance of actual nZEBs, identified occupant behaviour (OB) as the primary inhibiting factor for accurate simulation and predictions of both energy and internal air temperatures [
23,
49,
51,
52,
53]. This is due to the near stochastic nature of OB in certain conditions. OB has a strong influence on the internal air temperatures in nZEBs [
23], as these buildings are highly insulated and can be thermally decoupled from external climatic conditions. As a result, the internal thermal gains from OB have a more dominant effect on internal air temperature. The observed efficacy of parameter tuning for RC models may be lower if measured empirical data from nZEBs are used (as opposed to synthetic data) due to the included noise from OB. The majority of published research has focused on RC model calibration of air-conditioned buildings [
3,
6,
7,
8,
13,
16,
17]. As these buildings incorporated controlled air-conditioning systems for regulating the internal environment, the influence of OB on air temperature was attenuated, therefore, the model calibration and validation processes are less susceptible to the negative effects of OB noise. Pavlak et al., (2014) trained an RC model on three weeks of synthetic data. Varying levels of noise were added to the calibration dataset to simulate the uncertainty of recorded empirical data. The results of this study showed that least-squares based automatic model calibration error increased relative to noise level and noise type, with high levels of brown noise resulting in a substantial increase in model error. RC models calibrated and validated using automatic calibration algorithms have produced low levels of error with root mean squared error (RMSE), with average of RMSE values of 0.4 °C and a maximum RMSE value of 1.2 °C [
6,
7,
8,
13,
16,
17]. RC model predictions in naturally ventilated (NV) buildings are reported in the literature as being less accurate, with mean absolute errors of 1.0 °C to 1.1 °C and daily maximum errors of 1.8 °C [
14]. Air temperature in NV buildings is much more sensitive to external climatic conditions (external air temperature and wind velocity) than mechanically air-conditioned buildings [
54]. OB also has a strong influence on the operation of NV buildings as many NV systems either fully or partially rely on manual occupant-controlled openings.
Table 1 presents findings from a systematic mapping (from peer-reviewed literature) of published RC models that have been calibrated and/or validated for predicting internal air temperatures in different buildings. This table includes the types of RC models that have been used, the building type (i.e., residential, non-residential or test-cell), the types of calibration data used (i.e., synthetic from other software package or empirical from a real building), the calibration method (i.e., manual, automated or mixed methods), the duration of calibration or validation in days, and the season(s) that were considered during calibration or validation. 40% of the studies identified in
Table 1 used a combined calibration and validation approach. Of the studies that calibrate or validate GB models using measured empirical data, the data requirements for calibration purposes were typically between 6 days to 26 days and depended on the application [
5,
10,
12,
13] with some studies that used larger datasets broken into smaller periods [
7]. Following a review of the RC model studies presented in
Table 1, a number of gaps have been identified. Existing literature on GB model calibration has a limited number of validated examples of thermally decoupled environments such as nZEBs, and these examples are in test-cell environments [
5]. GB models of nZEBs have yet to be calibrated in occupied conditions. The majority of GB models have used synthetic data for parameter tuning. There are few studies of GB model calibration in NV buildings [
5,
14]. While some examples have added noise to synthetic data [
6,
7], calibration or validation with measured empirical data from real buildings is very limited [
10,
13] and non-existent for NV nZEBs.
This paper presents the first example of GB model calibration for internal air temperature prediction in an occupied NV nZEB. The five objectives of this study are, (1) investigate GB model performance for predicting internal air temperatures in a thermally decoupled NV building with different sources of empirical calibration and validation datasets, (2) develop a GB model of a naturally ventilated nZEB, (3) apply an automatic calibration algorithm to select optimal GB model parameters using measured internal air temperature data, (4) compare the validated GB model to a calibrated WB model and, finally, (5) investigate the potential for practical implementation of GB models. In
Section 3 we present the model theory and the calibration and validation approaches. In
Section 4 we analyse the accuracy of the GB model when compared to the WB model and measured empirical data along with an analysis of the parameter selection when using different calibration and validation periods. In
Section 5 we discuss the efficacy of different calibration and validation configurations as well as a practical comparison of WB and GB models.
Section 6 presents the conclusions of this study.
5. Discussion
From the results in
Section 4.1.1 we can see the simple GB model employed in this study was capable of accurately capturing the dynamic characteristics of the internal air temperature profile when calibrated over one week using an automatic algorithm (
Figure 5). However, depending on the week selected as the calibration period (varying season and occupancy levels), the parameters selected by the algorithm varied substantially (
Figure 6), which resulted in varied model performance (generally poor) when validated (
Table 5). Many previous studies utilised only one week or less of empirical data (single season) for model calibration purposes. Using a similar sized single-season calibration period (C1V5) for the RC model and calibration algorithm used in this study did not produce consistently accurate results. When additional weeks (across varying seasons) where added to the calibration period, the consistency of the model’s performance substantially improved. However, the generalisation abilities of the C3V3 GB model where still below that of the WB model. The C3V3 GB model did however produce accuracy levels comparable to that of the WB model (mean RMSE of the GB model was 1.5% higher than mean RMSE of the WB model).
From a practitioner’s perspective, the GB modelling and automatic model calibration techniques were found to be far more straightforward and time efficient methods of simulating internal air temperature in an nZEB, in comparison to a WB model manually calibrated using a piecemeal evidence-based approach. The WB model had a complex structure and accounted for many other factors which interact with internal air temperature (such as relative humidity), while the GB model had a simple RC structure, less interactions and fewer parameters. The manual evidence-based WB calibration method was entirely depended on lengthy human interaction, while the automatic calibration method required minimal human interaction. The mean time for the calibration algorithm to converge on a final set of parameters for the full C3V3 configuration (six individual calibration periods made up of three weeks each) was 148 s using a six core Intel i7 3930 3.2 GHz processor with parallel computing enabled. The authors of this study estimated that by employing the GB modelling method with an automatic model calibration technique, the human labour input to simulating internal air temperature was reduced by approximately 90% relative to WB modelling using a manually calibrated evidence-based approach. The labour reduction applies only to the model development, calibration and validation time and does not include the time required to record and process the empirical data, which would be the same for both methods. This results in a significant decrease in human labour input, albeit with a slight decrease in accuracy and drop in generalisation abilities for this application. In this study, both WB and GB modelling techniques and their corresponding calibration methods were found to possess independent attributes and both styles had unique merits.