Alessio Mastrucci
email: mastrucc@iiasa.ac.at

Paula Perez-Lopez

Enrico Benetto

Ulrich Leopold

Isabelle Blanc

Global sensitivity analysis as a support for the generation of simplified building stock energy models

Keywords: Building stocks, Energy modelling, Urban housing, Uncertainty Analysis, Global Sensitivity Analysis, Parameter screening, Sobol' indexes, Surrogate models, Decision support, Sustainable urban planning Geographical Information Systems, Global Sensitivity Analysis

Buildings are responsible for 40% of total final energy consumptions in Europe. Numerous bottom-up models were recently developed to support local authorities in assessing the energy consumption of large building stocks and reduction potentials.

However, current models rarely consider uncertainty associated to building usage and characteristics within the stock, resulting in potentially biased results.

This study presents a generic model simplification approach using uncertainty propagation and stochastic sensitivity analysis to derive fast simplified (surrogate) models to estimate the current building stock energy use for improved urban planning. The methodology includes an engineering-based energy model as input to global sensitivity analysis (GSA) using the elementary effects (EE) screening and Sobol' method for key parameter identification and regression analysis to derive simplified models for entire building stocks.

The application to the housing stock of Esch-sur-Alzette (Luxembourg) showed that the parameters explaining most of the variability in final energy use for heating and domestic hot water are floor area, set-point temperature, external walls U-values, windows and heating system type. Results of the simplified models were validated against measured data and confirmed the validity of the approach for a simple yet robust assessment of the building stock energy use considering uncertainty and variability.

Introduction

The building sector is responsible for 40% of the total final energy consumption in Europe [1] and has been identified as one of the most promising sectors in terms of carbon reduction potential [START_REF] Ma | Existing building retrofits: Methodology and state-of-the-art[END_REF]. In Europe, a wide legislative framework has been developed to promote the reduction of building energy consumption and carbon emissions, including the building energy performance directive [1] and the energy efficiency directive [3]. The latter entails a reduction of building consumption in a stock perspective. Public authorities are therefore in the need to assess the energy demand of buildings at a large scale to support decision in energy planning strategies.

A number of bottom-up building stock energy models have been recently developed to cover this need (see e.g. [START_REF] Swan | Modeling of end-use energy consumption in the residential sector: A review of modeling techniques[END_REF][START_REF] Kavgic | A review of bottomup building stock models for energy consumption in the residential sector[END_REF][START_REF] Mata | Building-stock aggregation through archetype buildings: France, Germany, Spain and the UK[END_REF][START_REF] Österbring | A differentiated description of building-stocks for a georeferenced urban bottom-up building-stock model[END_REF][START_REF] Nouvel | The influence of data quality on urban heating demand modeling using 3D city models[END_REF][START_REF] Dall'o' | A methodology for the energy performance classification of residential building stock on an urban scale[END_REF][START_REF] Theodoridou | Assessment of retrofitting measures and solar systems' potential in urban areas using Geographical Information Systems: Application to a Mediterranean city[END_REF][START_REF] Mastrucci | Estimating energy savings for the residential building stock of an entire city: A GIS-based statistical downscaling approach applied to Rotterdam[END_REF][START_REF] Howard | Spatial distribution of urban building energy consumption by end use[END_REF][START_REF] Reinhart | Urban building energy modeling -A review of a nascent field[END_REF]). The archetypes technique [START_REF] Moffatt | Stock Aggregation. Methods for Evaluating the Environmental Performance of Building Stocks[END_REF] is one of the most common approaches. In this approach, a series of typical buildings are identified to represent specific building cohorts, e.g. depending on the type of building, period of construction, etc. Then, the average energy performance of the individual typical buildings is calculated using an energy model and finally extrapolated to the regional or national level based on the representative weight of the modelled sample.

Bottom-up models represent a valuable tool to assess the energy performance of large building stocks. However, they are still affected by a number of shortcomings. Current models rarely account for the variability of building characteristics and usage within the same building cohort. This might limit the accuracy of the results due to an oversimplification of the building stock complexity. In addition, building stock models rely on a large number of input parameters that are commonly highly uncertain at a large scale due to limited data availability and accuracy. While uncertainty and sensitivity of input parameters have been widely investigated regarding the energy performance of individual buildings [START_REF] Tian | A review of sensitivity analysis methods in building energy analysis[END_REF], only few studies focused on building stocks [START_REF] Booth | Handling uncertainty in housing stock models[END_REF][START_REF] Booth | Decision making under uncertainty in the retrofit analysis of the UK housing stock: Implications for the Green Deal[END_REF][START_REF] Kavgic | Uncertainty and modeling energy consumption: Sensitivity analysis for a city-scale domestic energy model[END_REF][START_REF] Mauro | A new methodology for investigating the costoptimality of energy retrofitting a building category[END_REF][START_REF] Branger | Global sensitivity analysis of an energy-economy model of the residential building sector[END_REF]. Uncertainty propagation and stochastic sensitivity analysis are therefore needed to assess the energy performance prediction and the identification of key parameters at the building stock level [START_REF] Kavgic | Uncertainty and modeling energy consumption: Sensitivity analysis for a city-scale domestic energy model[END_REF][START_REF] Keirstead | A review of urban energy system models: Approaches, challenges and opportunities[END_REF]. Simplified models, also known as surrogate models [START_REF] Aijazi | Comparison of regression techniques for surrogate models of building energy performance[END_REF], based on uncertainty propagation and sensitivity analysis can provide policy makers with fast and reliable estimations to evaluate scenarios and energy efficiency strategies [START_REF] Kavgic | Uncertainty and modeling energy consumption: Sensitivity analysis for a city-scale domestic energy model[END_REF]. Simplified building stock models have therefore the potential of addressing uncertainty propagation by simplifying more complex engineering-based models.

This study aims at developing a methodology for the analysis of variability and uncertainty related to the building stock energy use and the subsequent generation of building cohort-specific simplified energy models, based on key parameters, for decision support in sustainable planning and policies. A generic model simplification approach using a complex engineeringbased energy model in global sensitivity analysis (GSA) is developed to derive the simplified surrogate models.

The methodology consists of the following consecutive steps: 1) building stock characterization, 2) design of the engineering-based reference model, 3) GSA -including a) model input and parameter screening using the elementary effects (EE) method, b) characterization of input parameter probability distributions and Monte-Carlo (MC) sampling, c) calculation of the final energy use by propagation of uncertainties through the reference energy model and validation against measured or statistical energy consumption data, d) computation of Sobol' sensitivity indices at building cohort level for key parameter identification, 4) model simplification for each building cohort using multiple linear regression, 5) final aggregation at the stock level. Simplified models and building stock aggregation are finally presented as an easy-to-use tool to assess the overall building stock energy performance for decision support. This generic simplification modelling approach contributes to improving the archetype technique as it accounts for the full variability and uncertainty linked to input parameters instead of considering one single representative building for every cohort. The simplified models are capable of providing a simple, yet robust, estimation of the energy performance distribution for the different building cohorts and for the entire stock. The approach was tested and validated for the housing stock of Eschsur-Alzette (Luxembourg).

State of the art

Building stock energy models

Building energy models are commonly classified into engineering, statistical and hybrid models [START_REF] Fumo | A review on the basics of building energy estimation[END_REF][START_REF] Foucquier | State of the art in building modelling and energy performances prediction: A review[END_REF]. Engineering (also called physical) models assess the energy use of buildings based on the solving of the equations that describe their physical behaviour and heat transfer. A variety of models exist, ranging from simplified steady-state models to detailed dynamic models [START_REF] Wang | Advances in building simulation and computational techniques: A review between 1987 and 2014[END_REF]. Steady-state models use long time steps (monthly to seasonal) in the calculation of the energy balance and account for dynamic effects of the building mass in storing and releasing heat by empirical gain and loss utilization factors (quasi-steadystate methods). Dynamic models use short time steps to calculate the energy balance (e.g. hourly) and directly account for dynamic effects.

Statistical models estimate the energy behaviour of buildings by using a function deduced from samples of training data without using any explicit heat transfer modelling [START_REF] Foucquier | State of the art in building modelling and energy performances prediction: A review[END_REF]. Regression analysis, conditional demand analysis and neural networks are among the techniques at the basis of statistical models. Hybrid (or "grey box") models aim at coupling engineering and statistical models by using several strategies. They are suitable to analyse the energy behaviour of buildings in case of incomplete or uncertain data [START_REF] Zhao | A review on the prediction of building energy consumption[END_REF]. Surrogate models are a particular case of hybrid models which allow a coherent data-driven model simplification assuring a replication of the more complex, often engineering-based, model accounting for uncertainties [START_REF] Aijazi | Comparison of regression techniques for surrogate models of building energy performance[END_REF].

At the building stock level, bottom-up statistical and engineering methods have been widely applied [START_REF] Kavgic | A review of bottomup building stock models for energy consumption in the residential sector[END_REF][START_REF] Swan | Modeling of end-use energy consumption in the residential sector: A review of modeling techniques[END_REF]. They imply the estimation of the energy behaviour of individual (or groups of) buildings and end-uses and successive extrapolations at the building stock level. Conversely, hybrid building stock models have been rarely developed and have the potential to overcome some of the limits of current modelling approaches in properly taking into account uncertainty and variability related to large building stocks.

Uncertainty analysis

Uncertainty analysis focuses on quantifying uncertainty in model output [START_REF] Saltelli | Global Sensitivity Analysis, The primer[END_REF] and is a major milestone to analyse complex models [START_REF] Keirstead | A review of urban energy system models: Approaches, challenges and opportunities[END_REF]. Uncertainty analysis has been widely applied to study the energy performances of individual buildings [START_REF] Lomas | Sensitivity analysis techniques for building thermal simulation programs[END_REF][START_REF] Wit | Analysis of uncertainty in building design evaluations and its implications[END_REF][START_REF] Macdonald | Applying uncertainty considerations to energy conservation equations[END_REF],

highlighting the importance of including both physical and occupants' behaviour related parameters [START_REF] Silva | Uncertainty analysis of user behaviour and physical parameters in residential building performance simulation[END_REF]. Whilst additional sources of uncertainty arise when the scale of the analysis expands from individual buildings to entire building stocks [START_REF] Booth | Handling uncertainty in housing stock models[END_REF], only a few studies addressed uncertainty at this level [START_REF] Booth | Handling uncertainty in housing stock models[END_REF][START_REF] Booth | Decision making under uncertainty in the retrofit analysis of the UK housing stock: Implications for the Green Deal[END_REF][START_REF] Kavgic | Uncertainty and modeling energy consumption: Sensitivity analysis for a city-scale domestic energy model[END_REF][START_REF] Mauro | A new methodology for investigating the costoptimality of energy retrofitting a building category[END_REF][START_REF] Branger | Global sensitivity analysis of an energy-economy model of the residential building sector[END_REF]. The techniques used for uncertainty propagation at the building stock level include MC sampling [START_REF] Mauro | A new methodology for investigating the costoptimality of energy retrofitting a building category[END_REF][START_REF] Pettersen | Variation of energy consumption in dwellings due to climate, building and inhabitants[END_REF] and factorial sampling analysis [START_REF] Yamaguchi | District level energy management using a bottom-up modeling approach[END_REF]. According to Booth et al. [START_REF] Booth | Handling uncertainty in housing stock models[END_REF], four sources of uncertainty can be discerned in building stock energy models: chance variability (also known as "first order" uncertainty), due to the random variability within a group of homogeneous buildings; heterogeneity due to the variation in characteristics of buildings belonging to different cohorts; parameter uncertainty (similar to "epistemic" uncertainty) due to parameters that could theoretically be measured if sufficient evidence was available ("second order" uncertainty) and assumptions; ignorance due to lack of knowledge. Our study addresses a method to handle parameter uncertainty and heterogeneity in the building stock analysis by applying the concept of building cohorts to isolate homogeneous groups of buildings and subsequently extrapolating results at the building stock level.

Sensitivity analysis

Sensitivity analysis studies how uncertainty associated to model outputs can be apportioned to different sources of uncertainty in the model input, parameters and structure [START_REF] Saltelli | Sensitivity analysis practice: A guide to scientific models[END_REF]. Two main types of sensitivity analysis can be distinguished in building performance analysis [START_REF] Tian | A review of sensitivity analysis methods in building energy analysis[END_REF]: local sensitivity methods and global sensitivity methods (GSA). In local sensitivity methods each input parameter is varied while holding the other constant to study the output variability. Global sensitivity methods assess the significance of input parameters by varying all other model inputs at the same time. Screening methods, such as the EE method [START_REF] Morris | Factorial Sampling Plans for Preliminary Computational Experiments[END_REF], are qualified as global methods and can be used to reduce computational cost in high-dimensional or computationally expensive models [START_REF] Zhao | Reconstructing building stock to replicate energy consumption data[END_REF]. EE typically does not characterise the uncertainty according to their complete probability distribution but rather randomly varies inputs by adding proportions and analyse the effects. This nevertheless, is useful when identifying the most sensitive model inputs for which a successive computationally expensive uncertainty propagation method will have to be performed.

At large scale (urban level or larger), most studies apply local sensitivity analysis because of its ease of use and in results interpretation whereas GSA techniques are rarely implemented due to heavier computation burdens. However, local sensitivity analysis is limited as it does not allow for the exploration of the whole input space and further inclusion of interaction between parameters [START_REF] Keirstead | A review of urban energy system models: Approaches, challenges and opportunities[END_REF].

We propose to use GSA taking advantage of previous studies which explored GSA for the Life Cycle Assessment (LCA) of energy pathways [START_REF] Padey | From LCAs to Simplified Models: A Generic Methodology Applied to Wind Power Electricity[END_REF][START_REF] Marini | Towards prospective life cycle assessment: How to identify key parameters inducing most uncertainties in the future? Application to photovoltaic systems installed in Spain[END_REF][START_REF] Marini | Assessing the prospective environmental impacts of photovoltaic systems based on a simplified LCA model[END_REF][START_REF] Lacirignola | A simplified model for the estimation of life-cycle greenhouse gas emissions of enhanced geothermal systems[END_REF] and other fields [START_REF] Aijazi | Comparison of regression techniques for surrogate models of building energy performance[END_REF][START_REF] Cucurachi | Characterisation factors for life cycle impact assessment of sound emissions[END_REF][START_REF] Azadi | Simulation and life cycle assessment of algae gasification process in dual fluidized bed gasifiers[END_REF]. Advantages of this method lie in its suitability for complex nonlinear and non-additive models and in its higher reliability of results even at the cost of increased computational time [START_REF] Tian | A review of sensitivity analysis methods in building energy analysis[END_REF]. For these reasons, GSA techniques such as EE combined with Sobol' method were demonstrated to be appropriate and more robust than other techniques for building energy performance applications [START_REF] Nguyen | A performance comparison of sensitivity analysis methods for building energy models[END_REF].

Methodology

This section describes the generic model simplification approach covering the following main steps (Fig. 1): 1) building stock characterization based on building cohorts; 2) design of the reference energy model; 3) GSA for the identified building cohorts, including input parameter screening (EE), uncertainty characterization and propagation (MC), identification of key parameters through Sobol' method; 4) simplification of the energy models; 5) aggregation of results at the stock level. This methodology is applicable to the analysis of existing building stocks in the current state. Possible adaptations to consider the future evolution of the building stock are addressed in the discussion (section 5).

Figure 1 Steps of the generic model simplification approach.

Building stock characterization

The first step of the methodology consists in the identification of the targeted building stock and its characterization according to a series of building cohorts. The methodology can be directly applied for existing building stocks and buildings of different functions. Some adaptations of the methodology might be required for the application to new building stocks, e.g. to consider the uncertainty linked to the future evolution of the stock. The division of the building stock into cohorts is advised due to the great variability of characteristics among buildings and the resulting difficulty in having a unique model representing the entire building stock [START_REF] Moffatt | Stock Aggregation. Methods for Evaluating the Environmental Performance of Building Stocks[END_REF]. Grouping buildings by cohorts allows for a more accurate representation of the building stock thus maintaining the differences specific to each category.

Each cohort should be composed by buildings having relatively similar characteristics, for instant sharing the same building type and period of construction. A number of building libraries exist at the country or regional level, e.g. the project TABULA [START_REF]Last consulted[END_REF] provides a standard for a common definition of building typologies in European countries.

Design of the reference energy model

A reference engineering-based model for the calculation of energy performance of buildings should be selected. Steady-state models are easier to implement and computationally lighter, however they are limited in the calculation of peak loads, due to the simplification in accounting for dynamic effects of the building [START_REF] Fumo | A review on the basics of building energy estimation[END_REF]. In contrast, dynamic models provide more accurate estimation of heating and cooling load, at the cost of higher inputs requirement and computation load [START_REF] Wang | Advances in building simulation and computational techniques: A review between 1987 and 2014[END_REF][START_REF] Peuportier | Simulation tool with its expert interface for the thermal design of multizone buildings[END_REF]. Therefore, the choice of the modelling approach highly depends on the goal and scope of the study (e.g. estimation of heating/cooling load, building types, etc.), the possibility of automating simulation runs and the computational capacity.

In order to apply GSA using Sobol' method, a set of independent input parameters for the reference model is required. In the case of building energy models, dependent input parameters are not uncommon. If two parameters are not independent, a possible solution is to keep the first one and introduce the second one by means of its residual variability [START_REF] Padey | From LCAs to Simplified Models: A Generic Methodology Applied to Wind Power Electricity[END_REF][START_REF] Kioutsioukis | Uncertainty and global sensitivity analysis of road transport emission estimates[END_REF].

Alternatively, other sensitivity methods may be applied instead of Sobol' method, for instance the Fourier Amplitude Sensitivity Testing -FAST [START_REF] Saltelli | Global Sensitivity Analysis, The primer[END_REF]. However, the FAST method is not suitable for discrete distributions and it is not able to consider the interaction effects [START_REF] Tian | A review of sensitivity analysis methods in building energy analysis[END_REF].

Global sensitivity analysis

The GSA step can be organized in four sub-steps. First, a screening of the input parameters is performed to select the most significant ones. Second, suitable probability distributions are identified for the selected parameters and a set of n scenarios are stochastically generated by a sampling method, for instance the Monte Carlo technique. Third, the model output (in this case, the energy use of buildings) is computed by running the reference energy model for the n scenarios. Finally, sensitivity indices are estimated for each parameters applying the Sobol' method, leading to the identification of key parameters.

Screening of input parameters

A screening step is suggested to identify the most important input parameters to be included in the GSA. This screening step is necessary as the number of input parameters is commonly high for energy models, regardless of the model type, and might result in unreasonable computation burdens for running the GSA. Another motivation to undertake this screening step is the fact that the boundary values must only be provided for each input parameter instead of the full values distribution required by the GSA which are potentially difficult and costly to determine.

The Elementary Effect (EE) method by Morris [START_REF] Morris | Factorial Sampling Plans for Preliminary Computational Experiments[END_REF] is proposed as the screening method as it is simple but effective in identifying the important parameters among the many that can be contained in a model [START_REF] Saltelli | Global Sensitivity Analysis, The primer[END_REF][START_REF] Mery | An integrated "process modelling-life cycle assessment" tool for the assessment and design of water treatment processes[END_REF][START_REF] Mery | Formalization of a technical procedure for process ecodesign dedicated to drinking water treatment plants[END_REF][START_REF] Andrianandraina | Sensitivity Analysis of Environmental Process Modeling in a Life Cycle Context: A Case Study of Hemp Crop Production[END_REF]. The computation cost for the method is r • (p+1) simulations, where p is the number of factors and r is the number of repetitions for the design. Two sensitivity measures are proposed by Morris: the mean μ estimating the overall influence of the factor on the output and the standard deviation σ assessing the effects due to interactions with other factors and nonlinear effects. The sensitivity analysis measures provided by this method are qualitative as they make it possible to rank input factors but not to quantify their effects on the output [START_REF] Tian | A review of sensitivity analysis methods in building energy analysis[END_REF].

For every building cohort, the EE method is applied to isolate the most influencing parameters to be further characterized by identifying their probability distribution and subsequently introduced into the uncertainty analysis and GSA.

Characterization of the probability distribution of input parameters

This step consists in characterizing the probability distribution of the selected input parameters. Probability distribution can be obtained from different sources depending on the parameter, for instance from a statistical survey [START_REF] Mauro | A new methodology for investigating the costoptimality of energy retrofitting a building category[END_REF] or from previous studies and literature values [START_REF] Heeren | Environmental Impact of Buildings -What Matters?[END_REF]. Once the probability distribution of input parameters has been identified, the Monte-Carlo (MC)

technique can be used to generate a set of random values following the given distributions.

Uncertainty propagation and validation of the energy use of buildings

In this step, the energy model is run for the sample of values stochastically generated in the previous step and a vector of energy use values is generated for each of the building cohorts. Results should be validated by comparison with a sample of measured values, statistical data for the region or country of interest, data from literature, etc. Validation can be performed at different levels, namely at the building level or cohort (aggregated) level, based on the energy consumption data availability.

Identification of key parameters through Sobol' method

The methodology used for the identification of key parameters is based on the Sobol' method. This technique is a variancebased method of GSA that consists in determining a set of indices called "Sobol' indices" or "variance-based sensitivity

indices" [START_REF] Sobol | Sensitivity analysis for nonlinear mathematical models[END_REF][START_REF] Iooss | A review on Global Sensitivity Analysis methods[END_REF]. Given the output Y=f(x) of a deterministic model f (in this case the reference building stock energy model) under the conditions indicated by Sobol' [START_REF] Sobol | Sensitivity analysis for nonlinear mathematical models[END_REF], both the function itself and the corresponding variance can be decomposed in individual summands associated with each of the model inputs (or parameters) and their interactions. Hence, Sobol' indices express the share of the total variance of the output Y that is caused by a given parameter (first-order Sobol' indices) or a combination of parameters (n-order Sobol' indices) [START_REF] Iooss | A review on Global Sensitivity Analysis methods[END_REF]. First-order Sobol' indices are calculated according to Eq. ( 1):

( ) ( ) (1)
Where S i represents the first-order Sobol' index for parameter i, Var i (Y) is the variance due to the individual effect of parameter i, and Var(Y) is the total variance of the output.

Higher-order indices can be calculated in an analogous way, as shown in Eq. ( 2) for second-order interactions:

( ) ( ) (2)
Where S ij represents the Sobol' index for the effect of the interaction between parameter i and parameter j, Var ij (Y) is the variance due to the combined effect of parameters i and j and Var(Y) is the total variance of the output. First-order and totalorder indices were computed in this study.

Model simplification

According to the results of GSA, a set of key variables (generally the parameters with the highest Sobol' indices) are selected to be introduced into a simplified model. The number of selected parameters may vary depending on the objectives of the study. A possible criterion for their selection consists in defining a minimum threshold for the cumulative sum of Sobol' indices and picking up the parameters having the highest indices and contributing together to achieve the previously defined threshold.

A simplified model is generated for every building cohort based on the selected key parameters through multiple linear regression analysis and respective regression coefficients. A regression equation allows the assessment of the energy use for each building cohort based on key parameters only. In case of non-linear relationships or interactions among input parameters, higher-order or interaction terms may be added in the regression equation without increasing complexity of the model.

Nevertheless, we also have to warn for overfitting a regression model if interactions and higher orders do not explain substantially more of the overall variability. The fit provided by the simplified models should be checked against the original results by using suitable measures, e.g. root mean square error (RMSE), mean absolute percentage error (MAPE).

Two possible applications are envisaged for the simplified models (Fig. 2) detailed and archetype-based. In the detailed application, the model of each building cohort is fed with suitable probability distributions of the key input parameters and a distribution of the energy use is obtained as an output.

Figure 2 Overview of the original and simplified model applications for individual building cohorts.

In the case of the archetype-based application, a single set of average or typical values for the identified key parameters is assigned to each cohort in place of their probability distributions. The set of values describes an "archetype" building representative of the cohort. One single calculation is then run for each cohort, i.e. archetype building, by applying the regression equations. In contrast to the detailed application, only an average energy use value is obtained for each cohort instead of a full distribution.

Aggregation at the building stock level

Results are finally aggregated at the level of the building stock and compared with measured or statistical values for validation. The following equation can be applied to obtain the total energy use of the building stock Q E,V,T :

∑ ̅̅̅̅̅̅̅ ( 3
)
where n is the number of building cohorts, N i is the number of buildings belonging to the i-th cohort and ̅̅̅̅̅̅̅ the average energy use per building for the i-th cohort. For the original model and the detailed application of the simplified model, ̅̅̅̅̅̅̅ can be calculated as the median value of the output distribution. The median may be introduced instead of the mean value due to the recurrence of skewed distributions for energy use within the cohorts. Similarly, the first and third quartiles may be calculated as a measure of the deviation from the median energy use for every cohort and for the entire stock using proper error propagation formulas. For the archetype-based application of the simplified model, the values ̅̅̅̅̅̅̅ are directly obtained in output from the regression equations for each of the building cohorts.

SIMPLIFIED MODEL ORIGINAL MODEL

Engineering-based model Regression equation

Regression equation

x 1

Archetype-based application

Detailed application

x 2 x p x 1 x 2 x n Y Y Input Output
All parameters

Key parameters only

Key parameters only

Input Output

The described methodology leads to the assessment of building stock energy use for the current state. For prospective studies focusing on the evolution of the building stock over time, adaptations might be needed to upscale results at the stock level accounting for the uncertainty on future size, composition of the stock and implementation of retrofitting measures.

Application to the case study and results

The generic methodology described above was applied and tested to the housing stock of Esch-sur-Alzette (Luxembourg), composed by more than 6000 residential buildings. The city was selected for its rich housing stock encompassing many building typologies and periods of construction and for the ready availability of quality geo-spatial data for buildings.

Building stock characterization

The classification of the housing stock was performed according to the national classification in the Second Energy Efficiency Action Plan [START_REF]Le Gouvernement du Grand-Duché de Luxembourg, Second National Energy Efficiency Action Plan[END_REF]. Residential buildings are grouped in three types: single-family detached houses (DH), singlefamily row-houses (RH) and multi-family houses (MFH). Four periods of construction were considered instead of the three periods proposed by the National Plan. This choice is motivated by significant differences in the building geometry and distribution of heating systems highlighted by the analysis of the available dataset. Hence, a total of 12 building cohorts are obtained by combining the various housing types and periods of construction.

The geospatial dataset provided by the municipality of Esch-sur-Alzette was used to further characterise the housing stock.

The dataset includes a vector polygon file for the year 2014 containing the building footprints and related attributes, for instance building functions and year of construction. In a previous study [START_REF] Mastrucci | Geospatial characterization of building material stocks for the life cycle assessment of end-of-life scenarios at the urban scale[END_REF], we processed the building data and obtained information on the geometry and housing type of every building by means of Geographical Information Systems (GIS). Table 1 summarises the main characteristics of the housing stock in Esch-sur-Alzette obtained by GIS analysis. With the term floor surface area we refer here to the heated net floor area of the building (named energy reference floor area in [START_REF]Performance énergétique des bâtiments d'habitation et fonctionnels[END_REF]).

Design of the reference energy model

The reference energy model was designed based on the national regulation for the energy performance of buildings in Luxembourg [START_REF]Performance énergétique des bâtiments d'habitation et fonctionnels[END_REF]. This semi-steady model calculates the final energy use intensity per floor surface area for space heating and domestic hot water (DHW) using a monthly time step. Space cooling was not considered as it is significant only to a certain extend for residential buildings in Luxembourg and the National regulation recommends to avoid the installation of air conditioning systems in residential buildings.

The choice of a semi-steady state at a monthly resolution is motivated by the fact that it provides reasonable accuracy and computation load for urban energy scenarios for stocks containing thousands of buildings [START_REF] Nouvel | Combining GIS-based statistical and engineering urban heat consumption models: Towards a new framework for multi-scale policy support[END_REF]. In addition, the exclusive focus on space heating and DHW further justify this choice as dynamic aspects are particularly important for the calculation of cooling loads.

Model description

The model follows a procedure similar to EN ISO 13790 [START_REF]EN ISO 13790:2008 Energy performance of buildings -Calculation of energy use for space heating and cooling[END_REF] for the calculation of energy use intensity for space heating with a monthly time step. The monthly energy demand for space heating is computed by Eq. ( 4):

( ) (4)
where Q h,M (kWh) represents the heat demand for space heating, Q tl,M (kWh) the heat losses for ventilation and transmission in kWh, η M (-) the heat gain utilisation factor depending on the thermal inertia on the building, Q s,M and Q i,M (kWh) the solar and internal heat gains. The final energy use intensity q E,V (kWh/m 2 a) is subsequently calculated by applying proper factors accounting for the efficiency of the heating and DHW systems and provided as a yearly value normalized on the reference floor surface area. Simplifications of the calculation for existing building were adopted in accordance with the regulation and regarded thermal bridges, shading coefficients, DHW demand, efficiency of the heating system and DHW systems. Describing the entire model is out of the scope of this paper and we refer to the cited regulation [START_REF]Performance énergétique des bâtiments d'habitation et fonctionnels[END_REF] for further information.

Model implementation

In order to proceed to the screening of input parameters and to the GSA, the reference model has to be designed with independent input parameters only. A scaling approach was chosen similarly to other studies [START_REF] Padey | From LCAs to Simplified Models: A Generic Methodology Applied to Wind Power Electricity[END_REF], by expressing dependent parameters as a function of input parameters. Dependency was observed for several parameters linked to building geometry (floor surface area, volume, external walls area, etc.), building envelope (windows U-value and g-value) and technical systems (heating system type and efficiency).

Fig. 3 shows the relationship between all parameters considered in this study. Several authors have already investigated the relationship between building geometry-related parameters. Bon [START_REF] Bon | Allometry in the Topologic St ructure of Architectural Spatial Systems[END_REF] and Steadman et al. [START_REF] Steadman | Wall area, volume and plan depth in the building stock[END_REF] examined the relationship between building volume (V) and external wall area (W) and found that it follows the allometric rule W ~ V 0.77 . This relationship can be explained as the building shape distorts to capture as much surface area as possible to get natural light as it increases in size. Analogous relationships were found by GIS analysis for the housing stock of Esch-sur-Alzette (Table 2), with very similar results for RH and MFH building types. A slightly different relationship was found for DH, justified by the lower compactness of buildings belonging to this type. Relationships were also found between volume and footprint area, similarly to Steadman et al. [START_REF] Steadman | Wall area, volume and plan depth in the building stock[END_REF], and between volume and reference floor area [START_REF] Mastrucci | Geospatial characterization of building material stocks for the life cycle assessment of end-of-life scenarios at the urban scale[END_REF]. Windows and heating systems were implemented as discrete variables (Table 3-Table 4). A set of 7 window types were identified according to the national reference [START_REF]Ministère de l'Economie et du commerce extérieur, Leitfaden zur Verwendung von U-Wert-Typologien im Rahmen der Energie-passerstellung für bestehende Gebäude[END_REF] and linked to specific U-values and g-values. Only natural gas and fuel oil heating systems were included in the analysis as other energy carriers represent a minor share in the housing stock. Heating system global efficiency values were assumed based on the national regulation [START_REF]Performance énergétique des bâtiments d'habitation et fonctionnels[END_REF] and depending on the boiler type (conventional, condensing), housing type and heating demand level. We finally implemented the reference energy model in a script written in R [START_REF] Core | R: A language and environment for statistical computing[END_REF] and automated the calculation of the final energy use intensity of buildings.

Calculated result

Global sensitivity analysis

Screening of input parameters

A set of 25 parameters were selected for the initial screening of model inputs (Table 5). Parameters were further classified according to generally accepted groups: building positioning, geometry, envelope, occupants' behaviour and technological systems. Special care was dedicated in considering parameters where information is typically lacking at the building stock level, such as window-to-wall ratio (WWR) and exposition of envelope elements (e.g. presence of attic for the upper closure).

Other studies have highlighted the importance of occupants' behaviour in addition to building configuration and physical parameters in regards to energy use [START_REF] Silva | Uncertainty analysis of user behaviour and physical parameters in residential building performance simulation[END_REF][START_REF] Guerra Santin | The effect of occupancy and building characteristics on energy use for space and water heating in Dutch residential stock[END_REF]. We introduced three parameters related to occupants' behaviour being identified among the most influential for heating energy by other studies [START_REF] Heeren | Environmental Impact of Buildings -What Matters?[END_REF][START_REF] Peuportier | Eco-design of buildings using thermal simulation and life cycle assessment[END_REF]: heating set point, ventilation rate and internal gains.

Boundary values were identified according to the results of the GIS analysis (floor surface area), normative limits and reference calculation values [START_REF]Performance énergétique des bâtiments d'habitation et fonctionnels[END_REF][START_REF]Ministère de l'Economie et du commerce extérieur, Leitfaden zur Verwendung von U-Wert-Typologien im Rahmen der Energie-passerstellung für bestehende Gebäude[END_REF], other studies [START_REF] Merzkirch | Wie genau sind unsere Energiepässe?: Vergleich zwischen berechneter und gemessener Endenergie in 230 Wohngebäuden in Luxemburg Wie genau sind unsere Energiepässe?[END_REF][START_REF] Nouvel | Citygml-Based 3D City Model for Energy Diagnostics and Urban Energy Policy Support[END_REF] and qualified assumptions. A screening test (EE method) was run for each of the 12 building cohorts described in Section 4.1 using the function "morris" in the R package "sensitivity" [START_REF] Core | R: A language and environment for statistical computing[END_REF] and setting the number of repetitions r to 10. For each of the 12 cohorts, parameters were ranked based on the mean μ estimates. In order to select a set of sensitive input parameters for the GSA, we considered the number of times each parameter appeared in the first ten ranking out of the 12 runs and we finally selected the parameters scoring within the first ten ranking for at least 50% of the runs (Fig. 4). While many of the selected parameters are related to the building geometry (A n , WWR) and envelope characteristics (U r , F t,r , U w , U f , n50), the heating set point temperature T i and the type of heating system HS id were also retained. Other parameters related to the occupants' behaviour resulted less influential and were excluded from the subsequent step. The complete results of the screening test are reported in the supplementary material.

Characterization of the probability distribution of input parameters

Probability distributions of the 10 selected input parameters for GSA (Table 6) and were determined in several ways. Floor surface area (A n ) distributions were generated based on the building GIS data: lognormal distributions were fitted for each of the building cohorts and then tested against the original sample by means of the Kolmogorov-Smirnoff test [START_REF] Conover | Practical Nonparametric Statistics[END_REF]. The probability distributions for window-to-façade ratios, indoor set-point temperature and envelope tightness were identified according to relevant literature [START_REF] Mauro | A new methodology for investigating the costoptimality of energy retrofitting a building category[END_REF][START_REF] Heeren | Environmental Impact of Buildings -What Matters?[END_REF][START_REF] Merzkirch | Wie genau sind unsere Energiepässe?: Vergleich zwischen berechneter und gemessener Endenergie in 230 Wohngebäuden in Luxemburg Wie genau sind unsere Energiepässe?[END_REF][START_REF] Nouvel | Citygml-Based 3D City Model for Energy Diagnostics and Urban Energy Policy Support[END_REF]. U-values were assumed to be normally distributed, similarly to Mauro et al. [START_REF] Mauro | A new methodology for investigating the costoptimality of energy retrofitting a building category[END_REF], and centred on typical values for Luxembourg depending on the period of construction [START_REF]Ministère de l'Economie et du commerce extérieur, Leitfaden zur Verwendung von U-Wert-Typologien im Rahmen der Energie-passerstellung für bestehende Gebäude[END_REF]. Cut-offs were applied to U-value distributions in the proximity of normative limits according to the current construction practice. The discrete distributions for the type of heating system and windows were obtained based on statistical data [START_REF] Statec | Statistic Portal Grand Duchy of Luxembourg[END_REF], data on past renovations provided by the municipality and qualified assumptions. A series of 10 000 sample values for every input parameter and for each building cohort was finally generated by MC based sampling.

Uncertainty propagation and validation of the energy use of buildings

Uncertainties were propagated throughout the engineering-based reference energy model using the generated sample of input values. The distributions of the final energy use intensity for space heating and DHW for the several building cohorts were then compared against reference and statistical values to check their reliability (Fig. 5). Results demonstrate a generally good agreement with reference values [START_REF]Le Gouvernement du Grand-Duché de Luxembourg, Second National Energy Efficiency Action Plan[END_REF], being their difference between the calculated median lower than 10% in most cases.

Additional comparison was performed against the distribution of measured final energy use intensity provided for a sample of buildings in Luxembourg by Merzkirch et al [START_REF] Merzkirch | Wie genau sind unsere Energiepässe?: Vergleich zwischen berechneter und gemessener Endenergie in 230 Wohngebäuden in Luxemburg Wie genau sind unsere Energiepässe?[END_REF] showing a good agreement. [START_REF]Le Gouvernement du Grand-Duché de Luxembourg, Second National Energy Efficiency Action Plan[END_REF].

Figure 5 Estimated distribution of final annual energy use intensity for heating and DHW in different housing types and periods of construction. Marked with *, reference values for Luxembourg

The results highlight that the cohorts composed of older buildings have higher energy use intensities and larger distributions due the higher degree of uncertainty and variability (e.g. uncertainty on past renovations, building characteristics and usage, wider range of construction years compared to the other periods of construction, etc.). DH have the higher energy use intensities, followed by RH and MFH, mainly because of the lower degree of compactness of their geometry (Table 2) resulting in higher heat losses per unit of volume.

Identification of key parameters through Sobol' method

After the validation of final energy use results, a GSA was performed for every building cohort and Sobol' sensitivity indices were determined out of 10000 simulation runs for every building cohort. Figs. 67show the results of GSA for the housing stock of Esch-sur-Alzette depending on the housing type and period of construction.

Results demonstrate a significant difference in the first-order sensitivity indexes of input parameters among the different building categories. Overall, results are highly sensitive to geometric characteristics (An), user behaviour (Ti) and heating systems type (HS id ) for most of the building categories. The floor surface A n is significant in particular for MFHs due to the high range of building sizes in the sample. The heating set-point temperature T i has higher sensitivity indices for newer buildings because of the higher set-point values resulting in a larger gradient between indoor and outdoor temperature.

Nevertheless, the high sensitivity indices obtained for the (Ti) confirm the importance of the occupants' behaviour even for older buildings, where the building characteristics can commonly explain a large part of the variability [START_REF] Guerra Santin | The effect of occupancy and building characteristics on energy use for space and water heating in Dutch residential stock[END_REF]. The heating system type has a higher influence on the results for SFHs as the different energy carriers (gas, oil) and boiler types (traditional, conventional) are more evenly distributed whereas conventional gas boilers represent the majority of heating systems installed in MFHs.

Parameters related to the thermal characteristics of the building envelope, such as U-value of external walls and roof and the type of windows, demonstrated a higher influence for older buildings. This can be explained by both higher U-values and larger variability due to different building characteristics and renovation states of buildings. In contrast, more recent buildings are characterized by lower and more uniform values due to the thermal regulation requirements. Other parameters related to the building envelope characteristics have lower sensitivity indices in all simulations, namely F t,r , WWR, U f and n50. Results were similar for total-order indices and are reported in the supplementary material.

Model simplification

A set of key parameters was selected for the generation of the simplified models based on the GSA results. For every building cohort, we set the threshold for the cumulative sum of Sobol' indices to 0.7 and selected the parameters having the highest indices and assuring together the threshold level. A multiple linear regression model was developed for every building cohort to get the final energy use intensity of buildings Y (kWh/m 2 a) expressed as a function of the key parameters x i specific of the cohort:

∑ (5)
where β 0 is the regression intercept and ε the random error term.

For every building cohort, the previously selected independent variables were introduced in the model. A logarithmic transformation was applied to A n being the distribution of this variable lognormal and therefore manifestly skewed. This kind of transformation improves significantly the accuracy of the prediction in the case of variables with skewed distributions, as demonstrated by other authors [START_REF] Kolter | A Large-scale Study on Predicting and Contextualizing Building Energy Usage[END_REF][START_REF] Schüler | Heat demand estimation for different building types at regional scale considering building parameters and urban topography[END_REF]. The variables Gl id and HS id were treated as factorial. Higher order or interaction terms were not introduced in the equations, being the first-order Sobol' indices explicative of the largest part of the variability when compared with the total-order indices. The Ordinary Least Squares technique was used to fit the models and the model assumptions and accuracy of the results was carefully checked by means of proper diagnostics, including tests for heteroscedasticity, independence of errors, multi-collinearity and outliers.

The variables selected for every building cohort and the respective regression coefficients obtained by regression analysis are reported in Table 7. The results proved that an accurate prediction of the building final energy use intensity can be made by selecting only 3 up to 5 variables. Among the predictive variables, A n and T i are important for every building cohort as a consequence of their high sensitivity indices (Section 4.3.4). The variable U w was selected only for buildings constructed until 1968 while Gl id is significant mainly for buildings between 1949 and 1994. Finally, HS id is used as a predictor for DH and RH, regardless of the period of construction, and for MFH built after 1994.

A simplified model was subsequently generated for every building cohort and expressed by the following generic equation:

where β 0… β 5 represent the regression coefficients specific to the considered building cohort (Table 7). Eq. ( 6) can be used to predict the final energy use intensity of buildings for each cohort accounting for the set of key parameters using the detailed or archetype-based applications of the simplified models. showing a good agreement (Fig. 8). RMSE and MAPE were additionally calculated and are reported in the supplementary material. Input parameters and results of the simplified models (archetype-based application) are reported in Table 8.

Comparison of the final energy use intensities with the median values of the distributions obtained using the original model showed a low deviations, being the error lower than 5% for all building cohorts.

Aggregation at the building stock level

Results of the original and simplified models were aggregated at the stock level for the city of Esch-sur-Alzette (Fig. 9) using Eq. ( 3) and data on the number of buildings (Table 1). DH cohort has the lowest total energy use due to its limited number of buildings in the stock. The high energy use values for RH and MFH built before 1949 can be explained by the combination of the large number of buildings and a high energy use intensity. Uncertainty intervals were derived from the computation of first and third quartiles for the original and simplified models (detailed application). They are also larger for cohorts with older buildings as a result of the greater uncertainty and variability depending, for instance, on past renovations and less homogeneous building envelope and heating systems characteristics. Uncertainty intervals become smaller by moving to cohorts including more recent buildings due to lower uncertainty and variability, e.g. due to the alignment with new thermal regulations. The results demonstrated a small difference between the original, detailed and archetype-based applications of the simplified models. Results were finally aggregated at the entire stock level as reported in Table 9. A total final energy use of 232.92 GWh/a with an uncertainty interval ranging from -9.6% to +12.5% was obtained using the original model. Results of the detailed application of the simplified model are very similar, being the difference with the original prediction lower than 1% and the uncertainty intervals comparable with the original model. The prediction given by the archetype-based application of the simplified model is slightly higher than the original prediction being the total energy use 240. [START_REF] Yamaguchi | District level energy management using a bottom-up modeling approach[END_REF] GWh/a and the error around 3%.

The calculated total final energy use values were finally compared with measured values provided by the municipality for validation purpose. The measured consumption falls within the uncertainty range provided by the original model and resulted 5% lower than the predicted value. The same conclusions apply for the detailed application of the simplified model, while the archetype-based application introduces a slightly higher overestimation of the total energy use. Overall, results are considered as satisfactory due to the high level of uncertainty and variability occurring at the level of the entire housing stock.

Discussion

This study introduced a generic approach for the simplification of energy models for building stocks in the current state combining the archetype aggregation approach, GSA to identify the key parameters and regression analysis to get surrogate models per building cohort. The identification of key parameters is important in building stock energy modelling due to the high level of uncertainty and variability. Splitting energy modelling per building cohort was found to be necessary to handle the quite heterogeneous behaviour of each building cohort through tailored cohort model.

The generated simplified models provide a simple but robust tool to stakeholders and decision-makers for assessing both the energy use of each building cohort and the entire building stock energy use without running a full energy model. Compared to current engineering-based models, e.g. archetypes, this methodology makes it possible to fully account for the uncertainty and variability in the stock in contrast with deterministic approaches. Another advantage of this methodology lies in its generality and applicability to several contexts and scales, e.g. ranging from urban to national scale. The methodology can be adapted depending on the goal of the study and used regardless of the type of energy model.

The application to the test case city of Esch-sur-Alzette proved the validity of the approach in assessing the current final energy use of residential buildings for space heating and DHW under uncertainty and variability. Results of the GSA demonstrated that it is possible to cover most of the variability in energy use by considering a limited number of key parameters. The simplified models generated out of the key parameters are denoted by small deviation of the results from the original complete model (less than 1% for the detailed application of the simplified model and around 3% for the archetypebased application for total energy use) and from measured values. The developed simplified models and equations can be further used for similar study in other cities in Luxembourg and neighbouring countries. Nevertheless, the distribution of input parameters, in particular floor surface area, building characteristics and technical systems, might be different from Esch-sur-Alzette and should be carefully verified prior to the application. In all other cases, the same generic methodology can be applied to develop simplified models from the urban to the national scale.

Limitations of this methodology lie in the need for detailed input data on the building stock necessary to generate probability distributions of input parameters. Lacking sufficiently detailed information and making imprecise assumptions might result in inaccurate results [START_REF] Lacirignola | LCA of emerging technologies: addressing high uncertainty on inputs' variability when performing global sensitivity analysis[END_REF]. However, the increasing availability of GIS data on buildings will enhance and facilitate the generation of probability distributions, especially for parameters related to geometry and construction characteristics.

Occupants' behaviour is an important source of uncertainty for the energy use of buildings [START_REF] Silva | Uncertainty analysis of user behaviour and physical parameters in residential building performance simulation[END_REF]. In this study we considered a limited number of parameters, such as heating set point temperature, ventilation rates and internal gains, based on the indications of previous studies [START_REF] Heeren | Environmental Impact of Buildings -What Matters?[END_REF][START_REF] Peuportier | Eco-design of buildings using thermal simulation and life cycle assessment[END_REF] and we confirmed the importance of heating set point temperature as one of the most influential parameters on the energy use for existing housing. Additional parameters, e.g. occupation and operation schedules, should be included in future studies to address occupants' behaviour in a more comprehensive way, especially for new or retrofitted buildings where the occupants' behaviour is expected to have a higher influence on the energy use.

This assessment was limited to the present state of the building stock and serves as a basis for diagnosis of current energy use under uncertainty and for the identification of hotspots. However, uncertainty in the future evolution of the stock should be included to properly address prospective mitigation scenarios. Future developments are envisaged to improve the upscaling methodology so to include the modelling of stock dynamics and are expected in two directions: accounting of construction and demolition activities leading to a different size and composition of the stock; retrofitting of existing buildings to improve the energy performance of existing buildings.

Regarding the Sobol' method, independent input parameters are required. Input parameters in energy models are frequently interrelated, demanding for further pre-processing and adaptation to make the Sobol' method applicable. Nevertheless, the identified relationships of dependence between input parameters can be generalised and further support the development of analogous models for other contexts and building types. Another limitation is represented by the possible high computation burden associated with running multiple energy simulations and GSA, especially if a dynamic energy model is used. This can, nevertheless, be improved through stratification in the sampling to minimise the number of Monte Carlo runs, e.g. Latin

Hypercube Sampling [START_REF] Stein | Large Sample Properties of Simulations Using Latin Hypercube Sampling[END_REF]. Also the parallelisation and usage of distributed computing approaches, such as Apache Spark can be a solution to solve computational burden by using more efficiently cloud computing infrastructures [START_REF] Zaharia | Apache Spark: a unified engine for big data processing[END_REF].

In spite of these limitations, the methodology has proven to be effective in identifying key parameters and generating simplified models for the assessment of building stock energy use for energy planning and policy support.

Conclusion

This paper presented an innovative approach to develop fast simplified models while accounting variability and uncertainty related to the energy modelling of a building stock in the current state. The model simplification is based on the identification of key input parameters using Sobol' indices in a GSA, representative of the stock. This simplified modelling approach helps to support decision making for sustainable energy planning.

The application of the methodology to the existing housing stock of Esch-sur-Alzette (Luxembourg) showed that the most significant input parameters explaining the variability of final energy use are the floor surface area, indoor set-point temperature, U-value of external walls, window type and heating system type. Results of the simplified models at the building stock scale were validated against aggregated measured data and confirmed its validity for a simple yet robust assessment of the final energy use of large building stocks.

The proposed approach and the developed simplified models represent an effective tool to support local authorities in assessing the current energy performance of large building stocks which represent an essential step for the identification of hotspots and consequent mitigation strategies. Its generic formulation allows further development and application to different context and scales, from the urban to the national scale. Future developments include the extension of the approach to the environmental impact assessment of large building stocks from a LCA perspective, the application to other contexts, the refinement of the upscaling methodology and the inclusion of building stock dynamics to assess uncertainty in the development of future mitigation scenarios.

Figure 3

 3 Figure 3 Overview of the reference engineering-based energy model.

  Gr floor area A f Gr floor U-value U f Gr floor exposit. F t,f Window-to-wall WWR Window type

Figure 4

 4 Figure 4 Results of the screening steps: times within the first ten ranking of input parameters out of the 12 runs. The parameters selected for the further GSA step are the ones above the dashed vertical line.

Figure 6 Figure 7

 67 Figure 6 Results of the GSA for DH (left) and RH (right) and different construction periods.

  t,r WWR Gl id U w U r U f n50 T i HS id

Figure 8

 8 Figure 8 Final energy use intensity for heating and DHW of different building cohorts as a result of the original and detailed simplified models. The bars indicate the median values and the whiskers the first and third quartiles.

Figure 9

 9 Figure 9 Total final energy use for heating and DHW and uncertainty intervals for different building cohorts using different models at the stock level.

  model: Detailed app. Simplified model: Archetype-based app.

4. Model simplification 3. Global sensitivity analysis (GSA)

	3.1. Screening of input parameters	3.4. Identification of key parameters through Sobol' method
	3.2. Definition of probability distributions for input	3.3. Calculation of energy use through uncertainty	5
	parameters -MC sampling	propagation -Validation

. Aggregation at the building stock level 1. Building stock characterization 2. Design of the reference energy model

Table 1

 1 Characteristics of the housing stock in Esch-sur-Alzette (Luxembourg) extracted from the database of the city.

	Housing	Period of	Number of	Median floor
	Type	construction	buildings	surface area (m 2 )
	DH	<1949	30	234
		1949-68	38	200
		1969-94	37	226
		>1994	11	270
	RH	<1949	2418	150
		1949-68	1195	141
		1969-94	382	167
		>1994	232	151
	MFH	<1949	910	332
		1949-68	333	413
		1969-94	303	575
		>1994	189	581

Table 2 Relationship

 2

between geometrical parameters estimated for different types of residential buildings in Luxembourg.

Table 3

 3 Input data for windows. Adapted from[START_REF]Ministère de l'Economie et du commerce extérieur, Leitfaden zur Verwendung von U-Wert-Typologien im Rahmen der Energie-passerstellung für bestehende Gebäude[END_REF].

	Window type	Glazing type	Frame type	Construction period	U-value	g-value
	1	Single	Wood	Until 1968	5.00	0.87
	2	Double	Aluminium (no thermal break)	1969-83	4.30	0.75
	3	Double	Aluminium (with thermal break) 1984-94	3.20	0.75
	4	Double	PVC	1969-94	3.00	0.75
	5	Double	Wood	Until 1994	2.70	0.75
	6	Double	PVC -Aluminium	From 1995	1.90	0.60
	7	Double	Wood	From 1995	1.60	0.60

Table 4

 4 Input data for heating systems depending on the heating system type, housing type and building heat demand.

	Adapted from [56].
	Heating system type	Housing type		Building heat demand (kWh/m 2 a)
			≤ 50	100	150	200	≥ 250
				Global heating system efficiency (-)
	1 -Conventional boiler	DH, RH	0.66	0.71	0.74	0.75	0.76

Table 5

 5 Input parameters selected for the screening step.

	Parameter	Cohort	Min	Max	Data source
	Building positioning
	Orient	-	North	South	QA
	e (-)	-	0.04	0.07	Values corresponding to low and
					medium wind exposition [56]
	Building
	geometry
	An (m 2 )	DH <1949	82.0	657.9	GIS
		DH 1949-68	96.9	393.4
		DH 1969-94	119.8	411.8
		DH >1994	148.6	511.5
		RH <1949	38.3	277.6
		RH 1949-68	44.8	267.6
		RH 1969-94	81.4	363.8
		RH >1994	66.9	267.8
		MFH <1949	75.3	753.1
		MFH 1949-68	102.3	949.0
		MFH 1969-94	145.2	1702.2
		MFH >1994	126.0	2042.6
	WWR (-)	<1949; 1949-68	0.15	0.30	[66], QA
		1969-94; >1994	0.20	0.40
	Building envelope

Table 7

 7 Results of the multiple linear regression for the prediction of the final energy use intensity for heating and DHW in different building cohorts.Results of the simplified model (detailed application) were compared with the results of the original model for each cohort

	Housing	Period of		Regression coefficients and respective input parameters	R 2	Sum	of
	Type	construction								Sobol'
			β 0	β 1	β 2	β 3	β 4	β 5		1 st ord.indices
				log(A n )	Gl id	U w	T i	HS id
	DH	<1949	122.65	-38.38	-	64.87	17.71	-37.19	0.775		0.714
		1949-68	173.53	-40.18	-11.38	62.14	18.46	-39.38	0.840		0.864
		1969-94	213.68	-29.81	-14.03	-	14.29	-33.86	0.802		0.978
		>1994	92.81	-16.47	-	-	8.46	-21.91	0.801		0.729
	RH	<1949	133.27	-38.06	-	44.67	14.15	-30.46	0.708		0.765
		1949-68	168.38	-39.17	-7.75	42.82	14.54	-31.17	0.798		0.811
		1969-94	185.28	-28.87	-9.64	-	11.08	-26.32	0.813		0.979
		>1994	94.39	-16.39	-	-	6.70	-18.62	0.834		0.756
	MFH	<1949	116.16	-28.04	-5.37	38.72	11.34	-	0.752		0.754
		1949-68	124.23	-27.05	-6.46	35.10	11.19	-	0.779		0.766
		1969-94	139.16	-20.09	-7.36	-	8.72	-	0.739		0.853
		>1994	78.84	-12.94	-	-	6.16	-10.84	0.888		0.700

Table 8

 8 Results of the archetype-based simplified models for each building cohort and comparison with the original model.

		2 a)	300
		intensity (kWh/m	150 200 250
		use	100
		energy	50
		Final	0	< 1949 1949-68 1969-94 >1994 < 1949 1949-68 1969-94 >1994 < 1949 1949-68 1969-94 >1994
				DH	RH	MFH
				Original model	Simplified model
	Housing	Period of	Input parameters -simplified model	Final energy for heating and DHW
	type	construction		(kWh/m 2 a)

Table 9

 9 Total final energy use for heating and DHW and uncertainty intervals predicted by different models at the housing stock level.

	Model	Total final energy use (GWh/a)	Error with the original
		Inferior limit	Prediction	Superior limit	model prediction (%)
	Original	210.64	232.92	262.14	0.00
	Simplified: Detailed application	211.31	232.85	261.41	0.03
	Simplified: Archetype-based application	-	240.34	-	3.18

Acknowledgements

This study is part of the postdoctoral project "DAEDALUS" supported by the National Research Fund, Luxembourg (FNR), Grant agreements AFR -7579115. MINES ParisTech, PSL Research University is kindly acknowledged for having supported the scientific visit of Alessio Mastrucci which made this work possible at MINES ParisTech, Centre O.I.E. (Observation Impacts Energy), in particular with Prof. Philippe Blanc. We would like to thank the City Council of Esch-sur-Alzette for contributing with relevant discussion and data provision.

Abbreviations

Gl id (-) Discrete nonuniform


Xiaodong Liu

Dongzhou Ning

Hubin Deng
email: hubindeng@yahoo.com.cn

Jinhua Wang

The The The The Application Application Application Application Research Research Research Research of of of of Neural Neural Neural Neural Network Network Network Network in in in in Embedded Embedded Embedded Embedded Intelligent Intelligent Intelligent Intelligent Detection Detection Detection Detection

Keywords: words:Neural Network, Embedded System, Intelligent Detection

Neural network which can adapt the sample data by training has good fault-tolerance and can be used in the field of intelligence widely. In the embedded system, restricted to the resources and the capacity of processor, the neural network application has a series of problems, such as losing timelines and the system could be collapsed easily. This article discusses how to use limited memory, processor and external equipment resources to achieve the neural network algorithm for improving the universality of detection system and adaptive ability in the embedded intelligent measuring system.

Introduction Introduction Introduction Introduction

Most embedded detective devices are developed by using the fixed parameter method, that is, the mapping relationship between test data and test conclusion is fixed. However, in practical application, the input conditions of detection and the testing standards may need to be changed, which may make the existing detection system inadaptable. The neural network owns the ability of self-adaptation and self-learning, therefore, by studying on the sample data, it can determine the mapping relationship between input and output. And if the neural network is applied to the detection system, it may adjust the mapping relationship automatically by training samples, which can make the detection system more flexible and adaptable. Application of the algorithm is shown in the figure 1 [START_REF] Bernard | Adaptive Signal Processing( 自 适 应 信 号 处 理 )[END_REF].

Because of the constraints of the resources of embedded portable device and the capacity of processor, computational complexity should be minimized in the application of neural network algorithm, thus to improve the efficiency by centralizing resources to deal with necessary tasks. embedded system, we should use as simple and effective methods as possible. The following is the description of issues needed to be considered in the design:

The The The The choice choice choice choice of of of of network network network network model model model model

2-layer linear perceptron model and 3-layer BP network model are provided and the user can choose one of them in the network parameter settings in accordance with the complexity of the detection system. According to the Almighty Approaching Theorem, a 3-layer BP neural network with a hidden layer can approximate to any continuous function of bounded domain with arbitrary precision as long as there are enough hidden layer nodes [START_REF] Yubo | Hybrid Neural Network Technology(混合神经网络技术)[END_REF]. Therefore in terms of the function, 3-layer BP neural network can meet most sophisticated detection systems' requirements, while for some detection systems which are simple mapping, linear perceptron model with a small amount of calculation and fast speed can be chosen so as to maximize the efficiency of the systems.

The The The The choice choice choice choice of of of of the the the the number number number number of of of of input input input input and and and and output output output output nodes nodes nodes nodes

The number of the input nodes is determined by testing item. The number of the output nodes is generally determined by the number of the testing conclusions. But if there are a lot of testing conclusions, the number of the output nodes becomes great, then, the amount of calculation also increases. So when there are a lot of testing conclusions, the testing conclusions can be encoded to binary code, then output nodes take the number greater than or equal to log2N, where N is the number of conclusions.

The The The The choice choice choice choice of of of of the the the the number number number number of of of of hidden hidden hidden hidden layer layer layer layer nodes nodes nodes nodes

If the network model is BP network, the number of the hidden layer nodes needs to be set. Generally speaking, if there are too few hidden layer nodes, the network can not study well, and much more training will be needed and, meantime the training will not be highly precise; while too many nodes will bring issues such as a large amount of calculation, long training time and reduced network fault-tolerance capacity. Because there is no corresponding theoretical guidance, most designs are determined by combining the experience and trial calculation. System uses the default settings as the empirical formula (1).

1 1 1 1 + + + + + + + + = = = = m m m m n n n n n n n n

.

(1 1 1 1)

In the formula (1), "n1" is the number of the hidden layer nodes; "n" is the number of the input nodes; "m" is the number of the output nodes.

If the training effect of experience value is not good, you can re-set the number of the hidden layer nodes in the parameter settings and do many experiments in order to achieve faster convergence rate. In order to know the training efficiency, you should output the training number in training process, changes of the network errors, changes of network weights and the training time and so on, so as to a make a report for the network analyses.

The The The The choice choice choice choice of of of of t t t transfer ransfer ransfer ransfer function function function function

In the application of detection, most of the outputs are the field data and test conclusions. The field data are collected by the acquisition system; detection conclusions are obtained from the neural network algorithm. The detection conclusions are indicated with two-value, so the transfer function of output layer generally uses the sigmoid function or the hard limit function. The sigmoid function formula is shown as following .

x x x x e e e e x x x x f f f f

---- + + + + = = = = 1 1 1 1 1 1 1 1 ) ) ) ) ( ( ( ( . ( 2
2 2 2)

The The The The choice choice choice choice of of of of initial initial initial initial weight weight weight weight

Initial weight is generated randomly by the random function, and its value range is (-0.5, 0.5).

The The The The choice choice choice choice of of of of the the the the learning learning learning learning rules rules rules rules

Because of the particularity of embedded systems, we should try to select the learning algorithm with simple calculation and low memory consumption. The perceptron network model structure is simple, and the overall amount of calculation is not large, so the weight correction algorithm of BP model is mainly considered. Conventional BP learning algorithm has large amount of calculation and long training time, and the slope of sigmoid function is close to 0 when the input is large, therefore the increase amplitude of gradient of learning algorithm is very small ,which may make the modification process of network weights almost at a standstill and the training very inefficient.

To reduce the amount of calculation and to improve the efficiency, we may adopt a flexible BP learning algorithm. We only need to consider the symbol of gradient to the error function, rather than the increase amplitude of gradient. The symbol of gradient determines the direction of weight updating, and the weight size change is determined by an independent "update value" (based on the former "update value") [START_REF] Yubo | Hybrid Neural Network Technology(混合神经网络技术)[END_REF].The iterative process of weight correction example is shown below:

) ) ) ) ) ) ) ) ( ( ( ( sgn( sgn( sgn( sgn( ) ) ) ) ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) ) 1 1 1 1 ( ( ( ( t t t t E E E E t t t t t t t t t t t t t t t t ω ω ω ω ω ω ω ω α α α α ω ω ω ω ω ω ω ω ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ × × × × ∆ ∆ ∆ ∆ + + + + = = = = + + + + . (3 3 3 3)

In the formula (3) [START_REF] Yubo | Hybrid Neural Network Technology(混合神经网络技术)[END_REF], Δω (t) is the previous update value, and the initial value Δω (0) is set by the actual application. α is the learning parameter, using variable step-size learning, that is, if it is in two successive iterations, the symbol of the partial derivative of error function to a weight remains unchangeable, then α is 1.2; otherwise α is 0.8 [START_REF] Yubo | Hybrid Neural Network Technology(混合神经网络技术)[END_REF]. It is faster to converge by using this algorithm, which can also be achieved effectively in embedded system.

The The The The preprocessing preprocessing preprocessing preprocessing of of of of i i i input nput nput nput and and and and output output output output data data data data in in in in t t t training raining raining raining sample sample sample sample

In the network learning process, because the transfer function of neuron is a bounded function, while in the detection, as the dimension and unit of input test item data may be different, some values are very large, while some values are very small, which has a great influence on the network.. To prevent some neurons from reaching saturation state, and meantime make the larger input fall in the region where gradient of neuron activation function is large, input and output vectors need to be normalized before the training. Namely:

min min min min max max max max min min min min xi xi xi xi ----xi xi xi xi xi xi xi xi ----xi xi xi xi = = = = i i i i x' x' x' x'

.

(4 4 4 4)

In the formula (3): xi max , xi min are the maximum value and minimum value of each input component of number i neuron; xi, x'i are respectively the former and later input component after pretreatment of number i neuron. The input data values after normalization processing range between 0 and 1. Normalization processing requires a large number of mathematical operations. But if there is no huge difference among the input data values, the normalization is not required to reduce the computational complexity. Therefore, we may use whether to normalize the input data or not as the network training option parameter and make choice before the training.

If the neural network training is unsuccessful, or if the training time is too long, we need to analyze the process of training and to train after readjusting the network parameters. If the training results are satisfactory, we may use the testing sample data to test the function of the network.. Successful testing demonstrates that the construction of the network is completed, and then the network parameters and the weights of each Layer of the network are saved as files, ready for testing procedures.

Input Input Input Input and and and and O O O Output utput utput utput P P P Platform latform latform latform D D D Design esign esign esign

In the application of neural network, we need to input a large number of training data and output the training process and the training results, which requires a higher demand of input and output. To reduce the cost of portable detect device and the burden of volume, we may use two kinds of solution methods. One method is training on a host computer independently, and we download the network weights and network parameters of success training to the target computer in form of files for the detection program to recover the network;The other method is sharing the rich resources of input and output device of remote computer by using the remote training mode. In the above two methods, the training process of the former method is conducted on PC machine, which doesn't take up resources of embedded device, and its advantage is that the high-speed PC-CPU and the large capacity memory can greatly enhance the training speed, while the disadvantage is that the neural network parameter settings and the training are independent of detection system, which is inconvenient for users to modify and update the system; the second method integrates network training and testing, completed by the target machine, with remote computer only serving as sharing input and output devices. Because of limitations of embedded CPU and memory, the training becomes slow, and even collapses when the network structure is complex, or the amount of calculation is huge. But the system is flexible, and more convenient to be maintained and updated.

Because in neural network design, requirements of application have already been considered, and we have adopted more effective and simpler methods in the design of network structure and training algorithm; besides, the network training is the preliminary work of establishing detection system, and it will not be run after the network is built successfully, so its speed does not affect the work efficiency of the detection system. For these reasons, we use the second method to design.

To reduce the development, maintenance and use costs, we use B / S structure in the system , embedding a small WEB server in device, all of the user interface using static or dynamic web pages and being viewed through remote browser, and the data interaction and generation of dynamic web pages being realized by using high efficient CGI programming. The overall structure of the system is shown in figure 2:

Conclusion Conclusion Conclusion Conclusion

In the detection system, using neural network algorithms can increase the system flexibility and self-adaptability. When testing standards or project data change, we can use new training data to train the network again and get the new network parameters for updating system rapidly. Besides, the system is based on remote detection technology, making the system updating more convenient. And that also reduces the requirements of input and output of portable detect device and the cost of hardware and software development; if the test site environment is poor, the remote detection could show more on its advantages.

But the small embedded WEB server in the system may accordingly occupy more resources .Moreover, due to the inherent delay of network communication, the display of testing data and testing results have a certain delay relative to field data, so it is only suitable to the situation in which on-site environmental data are stable. If the on-site environment changes fast, you can add display components in the embedded device, which can have the devices test offline, and neural networks building and the system maintaining and updating could be conducted remotely.

Fig 2

 2 Fig. Fig. Fig. Fig. 1 1 1 1. . . . application of neural network

Fig

  Fig Fig Fig Fig. . . . 2 2 2 2. . . . Overall system structure diagrams