Grey-Box Method for Urban Building Energy Modelling: Advancements and Potentials

Abstract

Urban building energy modelling (UBEM) has consistently been a pivotal tool to evaluate and control a building stock’s energy consumption. There are two main approaches to build up UBEM: top-down and bottom-up. The latter is the most commonly used in engineering. The bottom-up approach includes three methods: the physical-based method, the data-driven method, and the grey-box method. The first two methods have previously received ample attention and research. The grey-box method is a modelling method that has emerged in recent years that combines the traditional physical method with the data-driven method while it aims to avoid their problems and merge their advantages. Nowadays, there are several approaches for modelling the grey-box model. However, the majority of existing reviews on grey-box methods concentrate on a specific technical approach and thus lack a comprehensive overview of modelling method perspectives. Accordingly, by conducting a comprehensive review of the literature on grey-box research in recent years, this paper classifies grey-box models into three categories from the perspective of modelling methods and provides a detailed summary of each, concluding with a synthesis of potential research opportunities in this area. The aim of this paper is to provide a foundational understanding of grey-box modelling methods for similar research, thereby removing potential barriers in the field of research methods.

1. Introduction

Building energy modelling (BEM) and urban building energy modelling (UBEM) represent crucial instruments for regulating energy consumption and carbon emissions in buildings. An evaluation of a building’s energy performance can be conducted through the utilisation of energy modelling. In the context of existing buildings, building energy modelling can facilitate an understanding of their current energy performance, thereby informing the necessary renovation and repair work. In the context of new construction, building energy modelling can be employed to optimise energy performance from the design phase onwards. Compared with building energy modelling, urban building energy modelling is more complex because it involves interrelationships and microclimates among buildings [1]. The challenge of balancing computational overhead with accuracy is also a key issue in this field of research [2]. However, due to the scale flexibility of urban building energy models, they can reflect the energy consumption characteristics of several to thousands of buildings at different scales, which helps to reflect the synergistic effect of the complete solution and has high application value. Therefore, it is a hot research issue in the field of building energy nowadays.

As Figure 1 shows, there are two approaches for UBEM: the top-down approach and the bottom-up approach [3]. The former refers to the prediction of energy characteristics of buildings by analysing a large set of statistics reflecting the energy performance characteristics of a certain type of building. This is a statistical method and normally based on socio-econometric, technological, and physical factors [4,5]. The top-down approach usually does not require detailed building information; however, the resulting predictions are comparatively less precise and are unable to account for the impact of sudden shifts brought about by technological advancements [6,7]. A further limitation of this approach is the accessibility of large and accurate data [8].

The bottom-up methodology is more prevalent in the field of engineering. The methods for modelling include the physics-based dynamic simulation method (or white-box model), the data-driven method (or black-box model), and the reduced-order method (or grey-box model, hybrid model).

At the present time, the physics-based dynamic simulation method is the most developed. This method needs climate data, building geometric information, and non-geometric information (schedule, thermo-physical properties of the envelope, etc.) as input, and it calculates the energy consumption with energy equations. The most common computational tools are EnergyPlus and a large number of derivatives using EnergyPlus as the underlying computational engine, including CityBES [11], COFFEE [12], and UMI (1.0) [13], in addition to IDA-ICE and DOE2. This method is highly comprehensible, but the model’s level of development has a significant impact on the results of the calculations, which are greatly reduced when complete modelling information is difficult to obtain [14]. In order to ensure the most accurate simulation results, it is essential to conduct a thorough validation process [15], as the uncertainty introduced by the parameters involved may lead to discrepancies in the results [16]. For a case of a simplified physics model, studies have reported errors of ±10% for cooling and ±20% for heating [17]. Otherwise, the physics-based model computes the building energy consumption by abstracting building geometry as nodes in a thermodynamic network, which is cumbersome and time-consuming for modelling a large number of buildings due to the large number of nodes and equations that need to be solved and the equally high requirement for computing power [3,18,19].

The data-driven method is particularly well suited to the needs of rapid assessment or real-time evaluation. This approach does not require the input of detailed geometric and non-geometric information about the building, and it can significantly increase the speed of model computation. The approach to this method can be classified into two categories of related research: statistical and artificial intelligence approaches [20]. The statistical approach is common in early research [4,19]. Historical statistics can reflect the impact of economic and social factors and people’s behaviour on building energy consumption, which are difficult to characterise in a white-box approach [21,22]. The algorithms used for the statistical approach include regression analysis, conditional demand analysis, and artificial neural networks [19]. The most prevalent of these is the multiple regression analysis, particularly linear regression methods. One example of typical research is a multivariate linear regression (MLR) model that was developed for the New York City LL84 building dataset to predict the relationship between building end-use energy intensity and a range of building characteristics, including age and energy type [23]. The artificial intelligence approach is frequently combined with machine learning technology in order to mine the fundamental relationships and patterns within a dataset, which can then be used to develop a model of urban building energy consumption. This, in turn, reflects the mathematical relationship between energy use and building characteristics that are related to building energy consumption [20]. In addition to regression, the artificial intelligence approach is capable of performing a range of tasks, including classification and clustering [24,25]. The performance of the UBEM developed with the data-driven method is contingent upon the performance of the algorithms employed and the quality of the dataset. Currently, these approaches are widely utilised and have demonstrated accuracy and validity. A variety of algorithms have been used for UBEM, apart from several artificial neural networks [26,27], and comprise an extreme learning machine (ELM) [28], a supported vector machine (SVM) [29], multiple gradients boosting algorithms based on decision trees [30], and hierarchical clustering methods in unsupervised learning [31]. The accessibility and quality of datasets represent significant constraints for the development of black-box methods. At present, open-source urban data is an important basis for relevant research [32,33]. Chen et al. clarified that the level of detail of the information presented could meet the demands of research, but there is still a necessity for further enhancements to be made with regard to the standardisation of data [34]. Data share-ability and data platform development also need to be further improved. Furthermore, the scope of open data that can be employed for data-driven applications is also subject to certain limitations. Currently, the locations mentioned in the study where open urban building energy consumption data are available were mainly in developed countries or regions, including New York [23,35,36], San Francisco [37], Chicago [33], and Hong Kong (China) [30]. Considering that the large amount of non-geometric information, such as user behaviour, implicit in the black-box model can be affected by the development of regional economic levels and climate, the generalisability of the model is limited. In addition, while data-driven models are also capable of predicting the energy consumption characteristics of buildings based on historical data, they can suffer from problems similar to those that exist in top-down approaches that cannot reflect the role of technological breakthroughs. Moreover, complex data-driven models may require significant computational resources in the training phase.

The grey-box model combines the advantages of the black-box and white-box models, which can improve the overall accuracy of the model and produce more reliable results with less input information [38]: fewer model details are required than in white-box models, which reduces the modelling time and accelerates the computation speed; and the ability and generalisability of the model can be effectively improved compared to black-box models. Related studies have also demonstrated the advantages of grey-box models in the field of building energy systems and HVAC control strategies [39,40]. According to statistics in three SCI journals focusing on building energy, Energy and Buildings, Building Simulation, and Energies, in the last five years, more than 30% of the UBEM-related research papers have developed a grey-box method or adopted grey-box tools to assess the energy performance of urban buildings, and in 2023 this proportion even exceeded 40% (shown in Figure 2), which confirms the application value and research potential of the grey-box method. However, compared to black-box and white-box methods, the grey-box method is more ambiguous in definition, and the modelling approach is relatively diverse, and thus may cause confusion to researchers at the method selection stage of the study. Most of the previous reviews of UBEM grey-box approaches have reviewed and summarised one specific modelling approach [41,42,43], lacking an explanation of the different modelling paths as well as a summary of their strengths and limitations. For an important current research area [41], this hardly solves the difficulties for researchers to select a grey-box model at the beginning of the study. Therefore, the aim of this paper is to introduce the different modelling paths and their technical details for the grey-box model of urban building energy consumption, and to summarise its advantages and limitations.

Section 2 of this paper will classify and briefly explain several concepts and modelling approaches present in grey-box modelling, Section 3 will explain the technical details of different grey-box modelling approach types and summarise their advantages and limitations, respectively, and Section 4 and Section 5 will compare, summarise, and discuss the limitations and shortcomings of this paper’s work by illustrating the grey-box modelling approaches of different technical paths.

2. Methodology

2.1. Literature Search Strategy

The field of grey-box methods is currently undergoing rapid evolution, yet it is also characterised by a substantial corpus of original and classic studies that have established the foundations upon which the field is built. Accordingly, the objective of this study is to conduct a comprehensive literature search, initially focusing on literature published since 2018, and subsequently examining classical literature for review based on the citation networks within these texts. During the initial search, the following search query was used:

(TS = (urban building energy modelling) OR TI = (urban building energy modelling)) AND (AK = (hybrid model) OR AK = (data-driven) OR AK = (grey-box)) AND PY = (2018–2024)

(TS: Topic; TI: Title; AK: Author Keywords; PY: Published Year)

Subsequently, the retrieved literature was initially classified according to the modelling approach.

2.2. Preliminary Analysis for the Classification of the Modeling Approach

According to the literature materials, some basic concepts will be clarified. On this basis, a preliminary categorisation based on technical pathways in the modelling process has been made.

The early researchers in this field first proposed the concept of simplified building models [43]. At this stage, there are three main technical paths to building simplified models: executing order reduction techniques on detailed building models [44], building simplified models from known building information [45], and building simplified models directly and adopting inverse methods to determine model parameter values [46]. The objective of all three technical routes is to construct a lumped capacitance model (also referred to as a thermal resistance heat capacity model, RC model), thereby simplifying the calculation process.

In 2013, the concept termed “hybrid model” was proposed by Foucquier et al. [47], which entails integrating the physics-based method with the data-driven method, so that the input of the model is simplified while maintaining its physical interpretation. Three strategies have been put forth for the construction of hybrid models. The first strategy involves the utilisation of machine learning techniques to estimate the parameters of a physics-based model. The second strategy entails the creation of a dataset through the physics-based method, which is then employed to construct a black-box model. The third strategy involves the application of both physical and statistical methods to discrete segments of the model. For example, detailed physical modelling may be employed in specific thermal zones, while statistical data may be utilised in the remaining thermal zones. The term “reduced order model” (ROM) has been employed in the majority of recent reviews [1,8], as the objective of grey-box models is to reduce the complexity of the model to the greatest extent possible while maintaining accuracy.

In this paper, the grey-box modelling method is classified into three categories based on the technical approach of modelling. The portion of the modelling process that is the focus of these three approaches is shown in Figure 3. Approach 1 aims to speed up the modelling process by using statistical knowledge to reduce the need for refinement of the input parameters. Approach 2 improves the calculation process in white-box models with the help of statistical knowledge to speed up the calculation speed or increase the accuracy of the calculation. Approach 3 is more inclined to black-box models, but compared to the traditional black-box models, the datasets built through the white-box method may contain more technical parameters to support the subsequent technical analysis. However, compared to the traditional black-box model, the dataset created by the white-box approach may contain more technical parameters, thus providing more support for subsequent technical analyses. Depending on the mixing process, the three approaches can be referred to as the Simplified Parameter Approach, the Computational Optimisation Approach, and the Data Expansion Approach, respectively.

3. Results

3.1. Review of the Simplified Parameter Approach

The physics-based method requires input parameters such as meteorological data, building geometry data, building thermal performance data, and occupant behaviour and usage data. The difficulty in obtaining complete and accurate input parameters is a recurring issue highlighted in existing research, with the potential to significantly influence the precision of the final outcomes. A statistical approach to defining normative criteria for the calculation process and the parameters of the building and technical equipment represents a viable strategy. Consequently, the overall process of the method may be regarded as a white-box model, but statistical knowledge is included in the parameter input process in order to overcome the difficulties of data accessibility and to increase the speed of modelling. The general approach is shown in Figure 4. This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, and the experimental conclusions that can be drawn.

The simplified parameter approach, in particular, usually involves the creation of a database by collecting the input parameters of the white-box model required for a particular region, thus allowing the user to reduce the difficulty of the data collection phase by calling the parameters in the database. The database will typically contain meteorological data, information regarding the performance of equipment, regional simplified calculation method constants, and other pertinent information. This approach is typically accompanied by the development of software tools. The objective is therefore to reduce the size of the database or to enhance the user-friendliness of the operation. A further solution is the classification or clustering of buildings based on their energy characteristics. This involves analysing the characteristics of the buildings in question and treating those with similar characteristics as a single type. Selected prototype buildings are then extracted from this group, and the modelling parameters of these prototype buildings are extended to include other buildings with similar characteristics. This is also an example of grey-box modelling with simplified parameters.

The research on the establishment of simplified parameter grey-box models through the development of databases is mainly concentrated in the European region, which is inextricably linked to the large amount of data collected in the stock building data surveys led by the governments of various countries after the release of the European Union’s Energy Performance of Buildings Directive (EPBDII: Energy Performance of Buildings Directive). A large number of related integrated calculation tools have been developed, such as the EPA series of tools, SimStadt (version 0.2), City Energy Analyst, and TEASER.

Bart Poel et al. developed software called EPA-ED (version 1.4.10.30) (for residential buildings) and EPA-NR (for non-residential buildings), for the analysis of the energy performance of existing buildings in the EU region. The software is equipped with country-specific databases containing local weather files, building libraries (encompassing thermal performance of the envelope and performance of the systems and equipment, among other factors), and constants of simplified calculation methods for specific regions. Additionally, case studies were conducted in Austria, Denmark, the Netherlands, and Greece [48]. Kaden et al., in the production of Energy Atlas Berlin, combined with geographic information data in the open CityGML format, specified different values of envelope heat transfer coefficients and window-to-wall ratios based on the age of the buildings, thus realising fast energy consumption calculations on a city-wide scale. However, this study is relatively rudimentary and only considers the effect of building age on the thermal performance of the envelope [49]. SimStadt also employs the open CityGML format to realise the geometric modelling from the city-region-buildings-buildings-region-buildings construction level and the modelling parameters required at different levels The database of Simstadt comprises three main libraries: the building type library, which includes physical parameters based on building type and age; the building use library, which primarily encompasses user occupancy and operational parameters; and the energy system and fuel type libraries. Additionally, probabilistic extrapolation of default parameters is available in cases where insufficient parameters are available [50,51]. City Energy Analyst (CEA) is a calculation software developed by ETH Zurich. Compared to EPA and SimStadt, its built-in library is more complete and detailed, initially requiring 26 parameters stored in five databases. The databases include meteorological databases, a city geographic information database (containing information on building characteristics and the surrounding topography), a building prototype database (thermal characteristics of the building envelope structure and specific annual consumption values of different buildings of different ages), a distribution database (mainly occupancy characteristics and HVAC system equipment characteristics), and a building type database (thermal characteristics of the building envelope structure and specific annual consumption values of different buildings of different ages). It also includes the parameters of HVAC system equipment and specific annual energy consumption values of different buildings with different building ages), a distribution database (mainly occupancy characteristics and HVAC system setpoints) and a measurement database (energy consumption values of non-standardised building types), which was further expanded in the subsequent development process by the addition of a metrics database (metrics database, mainly occupancy characteristics and HVAC system setpoints) [52]. The database (metrics database, mainly containing economic and technical indicators) and Decision Database (target database, mainly containing key indicators and weights in the decision-making process) can better guide urban design and regeneration [53]. Compared to EPA and SimStadt, CEA has subsequently been adopted by researchers in different parts of the world, further improving its generalisation performance. Another open-source tool is TEASER, based on the Modelica computational engine, which provides parametric inputs for energy modelling by integrating statistical information on existing buildings and relevant standard codes, with a database covering building form, interior zoning, use functions, and envelope material properties, in addition to a simplified model for speeding up computation using the RC model (the simplified model will be discussed and reviewed in Section 4) [54].

Table 1. Projects using simplified parameter approach.

Name	Library Contains Parameters	Study Case
EPA-ED, EPA-NR	Local meteorological documentation, Enclosure thermal performance, System equipment performance, Local simplified calculation method factors.	EU countries (Austria, Denmark, Netherlands, Greece) [48]
Energy Atlas Berlin	Envelope thermal performance, Window-to-wall ratio.	Berlin [49]
SimStadt	Building Physical Parameters, Occupancy and operation, Energy system and fuel type.	Germany (Ludwigburg [50]), Netherlands (Rotterdam [55])
City Energy Analyst (CEA)	Meteorological data, Building characteristics and surrounding terrain information, Thermal characteristics of the envelope, Characteristic parameters of HVAC system equipment specific annual energy consumption values, Energy consumption values for non-standardised building types, Economic and technical indicators of equipment and systems, Decision-making key indicators and weights.	Switzerland, (Zug [53], Zurich [56]), Singapore [57], Netherlands [56,58], Norway [56]
TEASER	Architectural form, Interior zoning, Functions, Enclosure material properties.	Germany (Bonn [54], Hamburg [59]), Austria (Graz [60])

below lists the library parameters collected and included in the relevant regimentation tools or projects and the areas where they have gained application.

The utilisation of stock databases undoubtedly reduces the necessity for input data refinement while maintaining considerable modelling accuracy. However, the current modelling process of simplifying input parameters with databases relies on a substantial quantity of open-source data; therefore, this modelling approach is not applicable to research in areas where open data is not accessible. Furthermore, the time and effort required to collect and maintain a large-scale database is also considerable.

In regions where is a lack of widely available and detailed data, a potential solution is through the prototype to reduce the work for survey. The prototype method is also applied to reduce the size of the database. In the process of the modelling of CEA and Energy Atlas Berlin, the prototype method has been used to assign the same computational parameters to a class of similar buildings. This method usually requires classification or clustering work on a certain number of building energy features during the extraction process. The primary objective of the prototype extraction process is to establish a set of criteria for classification. These criteria serve to evaluate the extent to which the prototype accurately represents the prevailing characteristics of the class of buildings in question. A normal set of criteria includes date of construction, building function, and climate zone [61,62]. On this basis, some research projects have sought to enhance the criteria for delineation, with the aim of improving the precision of prototype extraction and classification. A typical example of this is the TABULA project [63], which provides a classification of residential buildings in 21 European countries (Austria, Bosnia and Herzegovina, Belgium, Bulgaria, Cyprus, Czech Republic, Germany, Denmark, Spain, France, United Kingdom, Greece, Hungary, Ireland, Italy, Netherlands, Norway, Poland, Serbia, Sweden, Slovenia). based on the region, residential buildings are categorised according to region, age, building structure, building use, and form of energy system. The study of Davila et al. was developed with Boston’s open-source data, the buildings were classified by year of construction and function of use, given building non-geometric information based on relevant standards and construction references, and further consolidated and classified by energy characteristics (peak energy use and sub end-use energy intensities for lighting, appliances, water heating, heating, and cooling) using statistics from the United States Energy Information Administration [64]. In contrast, the study by Pasichnyi et al. employed the type of heating source as the primary criterion for categorising the building prototypes. This is due to the fact that the study area is a cold region, such as Stockholm, where heating energy consumption is a dominant factor [65].

In addition to classification based on pre-set criteria, some studies have employed unsupervised learning techniques to cluster building samples. In comparison with the traditional method, this method can facilitate the discovery of objective data patterns and, to a certain extent, mitigate the potential influence of human subjectivity in classification. The most commonly used unsupervised learning algorithms can be divided into two categories: data transformation and clustering. Principal Component Analysis (PCA) is one of the most common data transformation algorithms, and clustering algorithms can be classified into distance-based clustering and density-based clustering. The former includes K-means, K-Medoide, GMM, PAM, Mean-Shift, Hierarchical clustering algorithms, etc. Current research on prototype extraction using machine learning methods usually starts from building morphological features or building energy features. In studies that used building form as a basis for classification, such as Li et al.‘s study of residential building prototypes in Yuzhong District, Chongqing, China [66], building height, aspect ratio, and compactness were used as morphological indicators, and K-means and K-Medoide algorithms were adopted to execute clustering, respectively, to compare the two parameter combination methods of building height, aspect ratio, and aspect ratio and compactness, and to compare the energy consumption of 321 buildings calculated by prototype extraction and building-by-building simulation methods, respectively. The study concluded that taking the K-Medoide algorithm and using building height and aspect ratio as prototype extraction metrics is a better extraction method, at which point three prototypes were extracted with an error of only 0.03% compared to the building-by-building simulation method. In studies using energy characteristics as clustering attributes, such as Borges et al.’s study of Andorran building prototypes [67], the prototype extraction took a two-stage approach, firstly classifying building types based on building function and heating form, based on which a K-means clustering algorithm was executed based on the intensity of electricity usage collected, and finally 18 prototypes were extracted to represent 1172 residential, commercial buildings. Some studies were conducted with the indicators combining morphology and energy features, such as Ghiassi et al., who raised an indicator system based on building form (building volume, envelope area, building height, compactness), solar gain (weighted window-to-wall ratio), and envelope performance (effective envelope heat transfer coefficient, effective wall heat transfer coefficient, effective roof heat transfer coefficient, effective floor heat transfer coefficient), and operational parameters (year-round occupancy, daytime occupancy ratio, year-round daytime occupancy ratio, year-round nighttime occupancy ratio, weighted internal heat gain intensity, heating season internal heat gain intensity, weighted hourly air change ratio, daily air change ratio). Under these criteria, seven building types were obtained in a study of a neighbourhood in Vienna [68,69]. Tardioli et al. conducted a representative identification of buildings in the city of Geneva [70]; this study considered both morphological and energy characteristics of buildings in order to avoid the influence of building age and function on morphology and energy. Firstly, buildings in the city of Geneva were classified according to their year of construction and their function, and then the number of floors of the building, the building floor area, the building perimeter, the building heating area, the building height, the number of floors, annual energy consumption, energy index, emissions and the number of meters were then used as input parameters for clustering, resulting in the extraction of 67 building prototypes.

The majority of extant studies adopt a two-step approach to the application of machine learning methods for prototype extraction, comprising firstly the artificial classification of building types, typically based on functional classification, energy form, and year of completion, and secondly the implementation of unsupervised clustering. The classification process can effectively reduce the size of the dataset. Furthermore, since the majority of current research employs the same distance-based clustering method (similarity measure) as the clustering algorithm, reducing the dataset size can enhance the speed and accuracy of the clustering process.

UBEM through the prototype method usually involves running simulations on a single prototype building and then performing summation calculations based on the proportions of each type of building within a city or region. Undoubtedly, the amount of work required for modelling is greatly reduced. For example, Shen et al. used the classifications in the US Residential Energy Consumption Survey (RECS) and based them on related studies to build different building form prototypes for five different types of residential buildings (mobile, single detached, single sttached, 2–4 unit flat, and 5 or more unit flat), and analysed the energy consumption and energy saving potential of residential buildings within New York State [71]. Deng et al. used Changsha as an example, extracting building prototypes based on building use and number of floors, and, given building envelope performance parameters based on the age of construction, the energy performance simulation was run on individual buildings using EnergyPlus, and the final summation was performed based on the percentage of different types of buildings in the city [72]. A similar approach was extended to Shanghai and visualisation of electricity and natural gas usage was carried out [32]. In addition to the above prototyping methods, the simulation process is easy and fast. However, the quality of prototype extraction can have a significant impact on the final results, and the key lies in selecting appropriate indicators and making suitable fuzzy abstractions to build the prototype, which usually requires rich relevant experience. In addition, since simulations are usually performed on individual buildings only, the interactions between the building and the environment, e.g., microclimate and daylight shading, are not taken into account, which may have a negative effect on the accuracy of the results.

3.2. Review of the Computational Optimisation Approach

In contrast to the simplified parameter approach, the computational optimisation model is typically a streamlined representation of the energy calculation process, thereby accelerating the calculation. Furthermore, the process of simplifying the model may also result in a reduction of the number of input parameters required. The most prevalent simplified model is the RC model, which employs the lumped parameter approach. This model removes the building from electrical circuits, replacing components with heat transfer capability with resistors, components with heat storage capability with capacitors, the temperature difference between indoor and outdoor with a voltage source, and the solar radiation and other indoor heat with a current source. The RC model is typically named after the thermal resistance and heat capacity in the simplified model. The quantity is designated as the xRyC model (where x represents the number of thermal resistances and y represents the number of thermal capacitances). Additionally, the nomenclature of the z-order model is employed, where z represents the number of thermal capacitances [42]. For the sake of clarity, the nomenclature of the xRyC model is employed uniformly in this paper. In

Table 2. Some common RC models.

Model Name	Simplifying Assumptions and Descriptions	Study Case
1R1C	In the simplest RC model, the following four assumptions were made to build the simplified model [73]: Same initial temperature inside and outside the building; Good mixing and uniform distribution of temperatures in the room; The room thermal resistance and heat capacity are global parameters and are not calculated from the values of each material; No additional heat fluxes from solar radiation, ventilation, infiltration, thermal bridges, etc.	Indoor air temperature in single buildings [74,75,76], Solar heat gain in single buildings [75]
2R1C	Heat transfer between the internal and external surfaces of the wall is considered, and convective heat transfer is included in the thermal resistance term [42], which can also reflect the uneven heating of the external surfaces of the building (e.g., roofs with solar collectors laid [74], following the following two basic assumptions [77]: Heat transfer in building components is a one-dimensional heat transfer process; The thermal resistance and heat capacity in the model can be calculated as global parameters.	Indoor air temperature in a single building [74,78], Concrete embedded tube radiant floor [79], Heat load calculations for single buildings [80]
3R1C	The heat transfer between internal and external surfaces is explicitly considered, the 3 thermal resistances represent the external surface heat transfer, the wall heat transfer and the internal surface heat transfer respectively, which is also the model adopted by ISO [81] and VDI	Usually for engineering applications
3R2C	Based on the 3R1C model, the wall heat storage is split into external and internal surface heat storage. It is one of the most widely used models due to its moderate degree of simplification.	Indoor air temperature in single building [74,82,83,84,85,86], Calculation of annual cooling and heating loads and peak loads in buildings with multiple heat zones [87], Energy performance of urban buildings [88]
5R1C	Ventilation heat transfer, door and window heat transfer are considered to have no heat storage capacity and are abstracted as two thermal resistances. The building envelope is abstracted as an external surface thermal resistance, an internal surface thermal resistance, and a heat storage heat capacity. The indoor heat transfer case is also abstracted as one thermal resistance [89]. This model is also the recommended model for the EN ISO 13790 standard [81].	Heating and cooling loads for individual buildings [89,90,91,92], Peak heating and cooling loads for individual buildings [92], Internal air temperature in a single building [90]
5R4C	It can be disassembled into a 3R2C model and a 2R2C model, with the 3R2C model being used for thermodynamic modelling of the building envelope and the 2R2C model being used for thermodynamic modelling of the building’s interior components and furnishings [93].	Calculation of real-time heat loads in individual building thermal zones [94,95,96,97]

, some common RC models and related research are shown.

In addition to some of the more common RC models mentioned above, a large number of other RC models have been proposed for specific studies, such as the 4R1C model [87], the 4R4C model [98], the 6R2C model [99], the 6R4C model [74], the 7R2C model [92], and the 8R3C model [76]. For an overview and summary of the different types of RC models, Li et al. conducted a detailed systematic review [42], which will not be repeated here.

The modelling process of the RC model shown in Figure 5 usually consists of two steps: firstly, the building is simplified and assumed reasonably according to the research objectives, and the equivalent analogous circuit model of heat transfer process is established; then, based on the established analogous circuit model, the values of thermal resistance and heat capacity parameters in the RC model are calculated according to the test or simulation data; the values of the parameters in the RC model determine the accuracy of the model. In order to ascertain the optimal parameter values, optimisation algorithms or machine learning methods are frequently employed. Optimisation algorithms are applied by global search methods. For example, Harb et al. used global search algorithms to identify the optimal parameter set with the highest accuracy based on the actual indoor air temperature data. This was achieved by defining the upper and lower limits of the values of each parameter and the accuracy of different types of RC. Models for different types of buildings were computed and compared [76], and Hossain et al. employed Bayesian neural networks (BNNs) for training purposes in order to determine the parameter values of RC models, utilising a dataset derived from real thermostat recording data [100].

The RC model is one of the most widely used grey-box models, and many packaged tools have been developed, such as TEASER [49], CitySim [101,102], OpenIDEAS [103], and the Modelica library [104], which have been demonstrated to exhibit exemplary performance in the prediction of dynamic temperatures, the calculation of cooling and heating loads for individual buildings, and other related tasks. Some of the current studies adopted TEASER or Citysim report results at the neighbourhood or city scale [54,105]. But there are still some challenges to extend the RC model from BEM to UBEM. One of the challenges is that different types of buildings may be suitable for different simplified levels [100]; a potential solution is to combine with the prototype method, extending the modelling calculations from a small number of prototype buildings. Another challenge is the problem of solar radiation calculation mentioned in almost all relevant studies. A further notable drawback of the RC model is that when alterations are made to a building or even a component of a building within the model’s scope, the values of resistance and capacitance in the original model should also be recalibrated. This limits the model’s adaptability.

Another approach for the grey-box model focuses on computational process combining the simulation and data-driven approach, as Figure 6 and Figure 7 show. The simulation and data-driven processes may be either serial or parallel. Serial processes are typically employed to bridge the gap between physical and data-driven approaches. In contrast, parallel processes are based on the availability of existing data. In such cases, the white-box approach is used to perform calculations in the parts where the physical approach can be implemented, while the black-box approach is used in the other parts. The two parts are then aggregated to obtain the final result. Li et al. refer to the model created by this method as a hybrid model [106], which does not aim to create a black-box model as an alternative to thermodynamic calculations, but rather to use the black-box model as an adjunct or complement to the white-box model.

A representative study that has adopted a serial structure is the DUE-S series of studies [18,38,107], in which a baseline model of building energy was established by EnergyPlus simulation. The building information required for simulation was simplified, with the degree of simplification based on the trade-off between the level of details (LOD) and the data availability (DA). The objective was to establish a baseline model for the analysis of general patterns of building energy use, rather than precise energy use. The aforementioned trade-offs between LOD and DA are also relevant in this context. The baseline model was established for the purpose of analysing the general pattern of building energy use, rather than the precise energy use. Relevant studies have reported that the direct transfer of the baseline model may result in significant errors [108]. Subsequently, a deep learning approach was employed to train a calibrated model utilising building energy consumption data obtained from simulations and test or reference data as a training dataset. The objective was not to optimise the computational process, but rather to enhance the accuracy of the building energy consumption model with less precise input data. The fundamental premise of this approach is to adopt an artificial intelligence methodology for the calibration of simulation outcomes, thereby reducing the necessity for modelling input parameters through the utilisation of open data on building energy. Similarly, Chen et al. developed a meta-modelling approach for calibrating the simulation data of a white-box model. This approach effectively improves the calibration efficiency and accuracy of the simulation results by building a black-box model as a calibrator [109].

Studies that have adopted a parallel structure include those conducted by Li and Dong et al., in which a thermodynamic model was employed to calculate heating and cooling energy consumption, and a machine learning approach was used to calculate non-air-conditioning energy consumption [110,111]. This approach mitigates the impact of high uncertainty factors inherent to traditional physical models, including room rates, ventilation, and indoor occupant activity. Furthermore, it reduces the number of input parameters required for physical model modelling through machine learning as a statistical method, with the study reporting an improvement in accuracy of approximately 15%. The objective of the parallel structure is analogous to that of the serial structure study, which similarly seeks to enhance the precision of the model with a reduction in the quantity of input data.

The primary challenge associated with this approach persists in the initial stage of acquiring authentic building energy data. However, Nutkiewicz et al. have proposed a solution in the absence of genuine test data, namely the utilisation of officially published guideline reference building energy levels as calibration data [38]. Nevertheless, the precision of this method in comparison to authentic data remains a subject of debate. Furthermore, the optimised calculation method still necessitates the partial or complete calculation of the building energy consumption by physical methods, which presents a significant computational overhead challenge when applied to large-scale studies.

3.3. Review of the Data Expansion Approach

Unlike the grey-box model for optimisation calculations, which combines simulation and data-driven calculations as mentioned above, the data augmentation approach to modelling aims to develop an agent model [106] that can be used as an alternative to physical simulation methods. As shown in Figure 8 below, this modelling approach is also usually a two-step process, where the building energy performance dataset is firstly created by a white-box approach, and then a black-box approach is applied to the dataset created by the white-box approach to create a statistical model. But in the creation of the black-box model, the input parameters are usually the ones that are required in the modelling of the physical approach (meteorological data, geometric and non-geometric information of the building, etc.). However, with the introduction of model interpretation algorithms such as SHAP, the influencing factors of black-box models of building energy consumption can be summarised in a meaningful way [35,112,113,114,115].

The traditional black-box approach relies on open datasets that usually lack detailed technical characteristics, and building age or use function is adopted in related studies to roughly reflect the thermal attributes of buildings. This is shown in

Table 3. Open datasets that used for black-box modelling [116,117].

Country/City	Name of the Dataset	Time Refinement	Data Features
USA	Building Performance Dataset (BPD) [118]	yearly	Building location and climate zone, Building envelope and system type, Building envelope thermal resistance, Building energy intensity, Energy intensity by type of itemised statistics.
New York, USA	LL 84 [119]	monthly	Building codes, Monthly building energy consumption. Often used in conjunction with the PLUTO dataset [120], which contains address information, geometric information, building function, and year of construction/renovation information for New York City buildings
Chicago, USA	Chicago Energy Benchmarking [121]	yearly	Year of data, Building function, Building energy rating, Floor area, Construction year.
Seattle, USA	Seattle’s Building Energy Benchmarking Program [122]	yearly	Building code, Building location, Building type, Year of completion, Number of storeys, Energy intensity, Breakdown of energy consumption by type of building.
Singapore	Listing of Building Energy Performance Data [123]	yearly	Name of the building, Location of the building, Function of the building, Year of completion, Form and technical details of the building air-conditioning system, Energy intensity of the building.
England and Wales, UK	Energy Performance of Buildings Register [124]	monthly	Building location, Building function, Floor area, Building envelope level (level descriptions, not specific values) and technical details of equipment, Breakdown of building energy consumption (including renewable energy) and potential for retrofitting
Ireland	Building Energy Rating [125]	yearly	Building location, Building function, Building area, Building envelope level (heat transfer coefficient U for different parts), Envelope area and technical details of equipment, Building energy intensity and carbon emission intensity.
Hong Kong, China	Energy Audit Form [126]	yearly	Building address, Building name, Energy use index.

below, which demonstrates the basic situation of some current statistical datasets, and three limitations of current open datasets can be seen: spatial limitation: these datasets are usually concentrated in developed countries or regions; temporal limitation: the time granularity of the datasets is coarse, usually month by month or year by year; technical limitations: the datasets usually do not contain or contain only a small number of technically detailed parameters.

Therefore, compared to the traditional black-box model, this way of building a dataset through a white-box approach has the following three advantages:

It overcomes the difficulty of data accessibility, which is important for conducting relevant research in domains where open-source data is scarce.

The input parameters of the model will be more focused on technical features, which will facilitate a more comprehensive understanding of the impact of technical parameters in the design process on the energy performance of buildings among architects and engineers.

The range of input parameters for technical characteristics can cover values that may be reached in future developments, which can reflect the impact of technical developments on energy performance.

The first advantage is obvious, for a considerable number of countries and regions, particularly in developing countries, where there is an acute necessity to enhance the energy performance of buildings. The lack of open statistical data can be a hindrance to relevant research, and the use of modelled data is an alternative. The latter two points are more reflective of the differences and advantages of hybrid modelling over black-box modelling. Firstly, current open datasets are typically not based on a technical perspective. Consequently, input parameters such as building function and year of construction are not as important as technical parameters for the design of refurbished or new buildings. Furthermore, when datasets are constructed using a white-box approach, it is possible to anticipate the development of technical parameters, such as the performance of the envelope, which can reflect the role of technological development in the improvement of energy performance. Furthermore, the temporal resolution of existing datasets is typically annual or monthly, which is conducive to modelling energy consumption at the urban scale. However, when the scale is refined to a regional or neighbourhood level, data with higher temporal resolution, such as daily or hourly, are essential for the design of energy systems, including energy storage facilities and microgrids.

This type of studies aims to reduce the computation cost and improve the efficiency, such as how Ding et al. established a database and analysed with linear regression with EnergyPlus simulation data by extracting building prototypes in the absence of a Chinese open building energy consumption dataset, and analysed the effects of year of completion and land price on building energy consumption [127]. Liu et al. developed a machine learning model based on a MARS model with energy consumption data generated from EnergyPlus simulations. They then analysed the importance of different simulation input parameters in order to propose potential building energy efficiency strategies [128]. Miu et al. determined the value ranges of building envelope performance, operating parameters and indoor temperature ranges based on a synthesis of relevant studies and standards. They also collected meteorological data of Hong Kong since 1989 and adopted a parametric approach to obtain 620,000 parameters. Based on these combinations, they established a dataset and trained a hybrid EP-ANN model for building energy consumption calculations. They then analysed the effect of envelope performance, indoor temperature set point, and outdoor temperature on building energy consumption in Hong Kong based on the results of the calculations [129]. In a recent study, Zhang et al. developed an artificial neural network agent model for replacing the CFD computational process during coupled energy simulations. The study reported a 2/3 reduction in simulation time, which can significantly speed up microclimate simulation calculations and thus improve the computational efficiency of coupled simulations [130]. Vazquez-Canteli et al. constructed two deep neural networks for the prediction of solar heat gain and building heat loss, respectively, on a larger scale. These were then aggregated to calculate building energy consumption. The training data for the deep neural networks were obtained from CitySim, in which simulations were performed on 2620 buildings of varying heights in Austin, USA. The simulation results were formed into a dataset comprising 25 input variables for the building heat loss model and 15 input variables for the solar heat gain model. The most notable contribution of the method is its improvement in computational speed, which enables the return of computational results within 12 s, which greatly improves the computational efficiency of large-scale building energy models [131]. A study by José et al. mixed real and simulated data during the construction of the dataset and found that the addition of the simulated data and variables can improve the prediction accuracy of NARX and RNN neural networks [132]. Westermann et al. trained a residual neural network incorporating a feature learning process to calculate building energy consumption using EnergyPlus simulations and verified the generalisability of the model by applying it to meteorological data from several different climatic zones in Canada, concluding that feature learning can effectively improve the generalisation performance of the agent model [133].

The data expansion approach for grey-box modelling, although it also combines thermodynamic and statistical knowledge, is more oriented towards black-box modelling. This improves the computational efficiency in comparison to white-box methods and adds technical features to the inputs that can be used to explain the role of building attributes in relation to the building’s energy consumption. Nevertheless, the primary issue is the absence of a calibration process. Additionally, it is essential to deliberate on the manner in which the requisite data for AI modelling in the development phase can be balanced with the computational overhead associated with the white-box approach for data acquisition.

4. Discussion

In the above sections, we have reviewed several modelling approaches for establishing grey-box models through relevant researches. As proposed in the methodology, the grey-box model is divided into three paths. But the boundaries between these approaches are in fact quite blurred, for example, the establishment of TESEAR combines the parameter simplification method and the optimisation calculation method.

The basic logistic of the simplified parameter approach is still physics-based, and it is assisted by statistical methods to reduce the parameters needed in the modelling process so that the efficiency of the energy consumption calculation of urban buildings can be improved. With this goal in mind, there are two methods to realise this approach. One method is to collect as much information as possible to build a database, which is usually done with government support, while another method is to carry out a typology study with a small amount of open data and build a prototype of each type, which can be replaced by a small amount of research to obtain the detailed parameters of the prototype. The two approaches can also be combined to reduce the size of the database and the cost of development and maintenance. Nevertheless, since this approach remains contingent upon simulation, the process of reducing the parameters introduces an element of uncertainty into the model. Consequently, it would be prudent to incorporate a validation into this type of research process, with a view to reducing errors and ensuring an accurate reflection of the energy performance of the building.

The focus of the computational optimisation approach to modelling is on the development of new computational methods to replace the original computational engine, and the process of building new computational methods may reduce the need for input parameters, thus further simplifying the parameter input stage at the same time. The most common simplification is the lumped-parameter approach, which is based on a thermodynamic heat transfer network with a clear physical interpretation, and is classified as a white-box model in some studies. However, a large number of studies have been conducted on the optimal values of the parameters in the network through the training of machine-learning or optimisation algorithms for a more accurate parameter value in the RC model, which is a statistical process, and is therefore also a grey-box modelling category. Another approach is to perform simulation and data-driven steps sequentially in the computational process to achieve hybridisation, which is suitable for modelling situations where public open-source energy data is available but lacks the technical details of the parameters. This hybridisation process reduces the need for input parameters and provides model calibration, which is a modelling approach that improves the accuracy of the model.

The fundamental premise of the data expansion approach is the data-driven approach. The main difference is the source of the dataset for modelling. The data expansion approach enables researchers to construct datasets that are more aligned with the specific requirements of their study. Advanced AI technologies are integrated in this approach. For instance, they can analyse technological pathways or crucial influencing factors that enhance the energy performance of a building. Furthermore, this approach is not constrained by data openness and exhibits enhanced generalisability, which are two additional factors that contribute to its popularity among a significant number of researchers.

Of the above three approaches, the modelling approaches’ focus of the first two is to enhance the performance of the UBEM and to improve the efficiency or accuracy, whereas the focus of the latter is mainly on performing technology-oriented analyses. Thus, the first two have a natural advantage in the area of software tools development, while the latter is more suitable for individualised research.

5. Conclusions

Grey-box models have shown greater potential for application than single white- or black-box models due to their combination of thermodynamic and statistical knowledge, and there has been a considerable number of different types of research work in this area in recent years. As proposed in the Methodology Section, based on the modelling approach, the related research classified grey-box models into three categories, which provides a useful framework for related research in this field. However, there are still certain gaps that need to be filled by subsequent research, and these potential research opportunities lie in the following four main points:

The development of separate databases for application in different regions in the parameter simplification methodology, which usually focuses on operational parameters, especially personnel behaviour, is not negligibly affected by the level of economic and social development, different cultural customs and local climatic conditions, e.g., in developing countries and regions, where the energy intensity is significantly lower than in developed countries and regions, and people living in hot areas use air conditioners at a higher operating temperature than those in cold areas. This limits the application of the tools, as most of the existing parameter simplification tools have been developed based on data from developed countries; however, developing countries or regions have more pressing sustainable building development needs and the development of complementary databases or plug-ins for application in these countries or regions is of significant relevance. The development of the databases is also helpful for the validation of the simplified parameter approach and future research on the modelling of the data-driven approach.

Parametric simplification methods with higher generalisability attributes, existing prototype extraction studies have used crude parameters and to some extent lacked consideration of design parameters, which may result in cognitive biases between the building design and building energy domains. In addition, prototype extraction often targets individual buildings, which ignores the influence of the urban fabric on building energy consumption in urban environments, an influence which may need to be first classified and defined at the neighbourhood scale, and subsequently extracted on the basis of building morphology and energy characteristics.

Combining the selection of input parameters for the building in the white-box and black-box optimisation calculation methods, the most important input parameters can be selected in the machine learning step in conjunction with data processing methods, such as feature selection. This can further reduce the workload of data collection in the input phase and thus also provide objective selection criteria for the modelling choices, i.e., at what level of detail a more accurate white-box model can be built, and under what circumstances further data collection or alternative modelling approaches are required.

Extending the calibration process in dataset approaches, existing studies usually lack a model calibration step during the dataset building process, which may compromise the quality of the dataset and thus the accuracy of the agent model. One possible approach is to analyse and correct the model’s own errors during the simulation process by using some samples with open test data (not the research object itself), and apply the results of this calibration to the dataset building process to improve the quality of the dataset.