# Numerical model assessment methodology for physical interpretability

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## Existing methods for the estimation of the thermal performance of a building envelope

To estimate the thermal performance of a building from measured data has raised multiple data-driven methods. One may divide them into two main categories: steady-state or dynamic methods. At the same time, the methods may rely on destructive/intrusive or non-intrusive experiments and may be performed in controlled or uncontrolled conditions.
Although the thermal properties of a building envelope may be studied at wall scale with heat flux methods or thermography, and literature is abundant on this matter (Biddulph et al. (2014); Bienvenido-Huertas et al. (2019); Chaffar (2012); Gori et al. (2017); ISO 9869-1 (2014); Rasooli and Itard (2018); Rodler et al. (2019); Yang et al. (2019) just to name a few recent), this section will only detail methods estimating an overall thermal performance of the building envelope.
With the term methods is actually meant the combination of:
• a specific design of experiment (acquisition of Data in Figure 1.1);
• and its numerical methodology to infer the estimation from the collected data:
– the type of model used for the energy balance or for the heat dynamics in the building (Model and parametrization in Figure 1.1);
– as well as the numerical tools to solve the inverse problem (solver and inference in Figure 1.1).
The usual distinction between steady-state and dynamic methods refers to the choice of model and relies on whether the hypothesis of thermal equilibrium of the energy balance equation
is accepted or not. In other words, upon considering the energy balance of the building, one may consider the boundary conditions sufficiently invariant over a certain chosen time span and consider the heat storage in the building elements negligible. At equilibrium, the latter is indeed very low. In steady-state conditions, the energy balance then constitutes a linear regression model, solved through least square methods. On the contrary, dynamic conditions will consider heat storage and variations of boundary conditions non negligible and incorporate ad hoc dynamic terms in the formulation of the model. Let us finally underline that the methods described hereafter are based on or inspired by the physical framework described in the PSTAR procedure Subbarao (1988), which is itself more a proper framework to a thorough analysis of the energy performance than a method.
Very basically, the steady state methods used for parameter interpretation rely on a simplified static energy balance of a building in Equation 1.1 where the heat charging in and out of the building elements Qstorage(t) are considered null.
HTC(Ti(t) To(t)) + Qstoragein(t) + Qstorageout (t) Qsun(t) (1.1)
+Qventilation(t) Qheating(t) + Qground (t) Qinternal (t) = 0
Depending on the measured variables and simplifications, this equation may result in a simple static equation as in Hammarsten (1987), with Equation 1.2. It merely becomes the relationship between the energy consumption Q, the temperature difference between indoors and outdoors Ti To, the solar heat gains through S the solar aperture and I the solar irradiation and Q0 constant indoor heat gains:  Q= Q0 + HTC(Ti To) + SIsol (1.2)
Ferlay (2012) proposes a variation of Equation 1.2 by replacing the temperature difference by degree-days. Degree-days then allows to use monthly averaged energy consumptions, which are easier to collect.
Applications of this steady-state building energy model with among others Day et al. (2004); Fels (1986); Ferlay (2012); Hammarsten (1984, 1987); Lee et al. (2014); Sonderegger (1978) use data of buildings under occupancy, which constitutes a major advantage as it may be easily used by any facility manager and with basic knowledge of the physics nor accurate measurement of the building.
Solving the equation is done through ordinary least square methods, to which Subbarao (1988) suggests using generalized least squares to improve the auto-correlated residuals.
If some of the applications are meant for energy prediction, parameter interpretation should be done cautiously (Hammarsten, 1987). The author underlines that with a static equation such as 1.2, the time step of the data should not be shorter than 24h. Additionally, parameters estimates need to be unbiased, by which the author means parameters to have insignificant covariances. Sources of error for physical interpretation are, according to the author, due to a rough representation of the physics and/or multi-collinearity of the input variables To and Isol and the author proposes to preferably use more advanced models such as dynamic energy-balance models as in Sonderegger (1978).
While the above mentioned applications use data from occupied buildings, i.e. without particular conditions for the measurements, co-heating test method rely on a particular design of experiment (Johnston et al., 2013). As reported in Bauwens and Roels (2014), the usual setting is to uniformly heat a building at a constant temperature, reaching at least a 10K temperature difference between indoor and outdoor, for example 25 C. The building is to be left unoccupied for several days to several weeks as to reduce uncertainties and perturbations. Weather variables are measured, if available: temperatures, wind speed and direction, solar irradiation, relative humidity. The collected data is aggregated with at least 24 h time steps, but measurements are done preferably with lower time steps as to catch high frequency dynamics that would create uncertainties in the data. Suitable starting time for averaging is suggested in Everett (1985) with a start around 6 am, thus asserting the steady state hypothesis. The co-heating is coupled to an estimation of the heat losses by infiltration through a tracer gas test or a blower-door test. Knowing the infiltration losses, it becomes possible to decompose the overall heat transfer coefficient obtained by the direct exploitation of the data into transmission heat transfers by deducing the infiltration heat transfers.
The data from a co-heating test is exploited by a linear regression analysis of Equation 1.2. As suggested by Bauwens and Roels (2014), performing a linear regression analysis instead of simply averaging over the total dataset allows to deal with outliers more easily. Indeed, according to the adaptation of the energy balance equation to the building configuration, it is possible to take more than one confounding factors, like solar gains or wind effect (Jack et al., 2018). Various techniques to plot the results of the regression analysis directly on a graph are available (Bauwens and Roels, 2014; Jack et al., 2018) and make the interpretation of the results and their uncertainty easier.
Let us conclude on the co-heating method by underlining that controlling the indoor temperature holds the advantage to increase the accuracy of the estimation result, see Jack et al. (2018). The same authors established a 8 to 10 % error, pinpointing the residual variability to the difficulty to correctly estimate the solar gains.
As made clear in Bauwens and Roels (2014), the previously mentioned methods are steady-state assumptions, considered true in the conditions in which the data is collected and exploited, of a fundamentally dynamic system. Other methods take advantage of the dynamic nature of the heat transfers in buildings. These methods rely then on a dynamic formulation of the energy balance equation as in Sonderegger (1978), where the state of the dependence of the system at t = tn to t = tn 1 is literally accounted for. The overall advantage of using a dynamical energy balance model is to achieve satisfactory accuracy in fewer days. In overall, methods based on dynamic models use state space models in a statistical form like auto-regressive models or state space models inspired by the heat transfer dynamics like RC models.
Auto-regressive with exogenous inputs (ARX) models are based on an input-output relation: a chosen output is considered function of some influential inputs. For example, as is described in Madsen et al. (2015), the heating power can be considered as a time dependant function of indoor temperature, outdoor temperature and solar radiation, which results in the functional relationship of Equation 1.3 where B is the backshift operator and f (B) a polynomial of order p such as f (B) = 1 + f1 • B + f2 • B2 + ::: + fp • Bp. Put very simply, the backshift operator links, at order p, the variable F at state t to its p previous states, i.e. function of the states t p to t 1. The model order is inferred iteratively aiming at e a Gaussian white noise, which means that the ARX prediction would fit properly the data.
f (B)Fh = w (B)T i + w (B)T e + w sol (B)Isol + e (1.3)
The indoor temperature may also be considered as the output variable. Its dynamic behaviour is then described by Equation 1.4.
f (B)T i = w (B)Fh + w (B)T e + w sol (B)Isol + e (1.4)
Although ARX models have been used for prediction purposes, an overall Heat Transfer Coefficient (HTC) can be estimated from its parameters (Madsen et al., 2015). For example, with the heating power as output, the HTC is estimated by Htot (l ) = l Hi + (1 l )He such that the variance of Htot (l )) is minimal, with Hi = wi(1) and He = we(1) (again see Madsen et al. (2015) f (1) f (1) for the detailed procedure).
In Senave et al. (2019) and Senave et al. (2020a), the authors establish promising performance of ARX models in various occupant-friendly conditions, depending on the availability of measurements of some boundary conditions such as solar irradiation or heat fluxes through poorly insulated ground floor slabs. Error goes from a few percents up to almost 25% in the case of solar irradiation approximations (Senave et al., 2020a). In overall, within a few weeks ARX parameters yield robust estimates of heat loss coefficients.
Next to ARX based methods are methods based on linear state space models, also called RC models (Madsen et al., 2015). Unlike ARX models which is a simple input-output relation, RC models naively approximate the heat transfer physics in the building. Under some assumptions, they can be described by linear terms using time invariant parameters. As such, RC models can be written in a state space representation: a set of first order differential equation that presents convenient properties.
Existing methods using RC models are the Quick U-Building (QUB) method (Alzetto et al., 2018a,b; Mangematin et al., 2012; Meulemans, 2018; Meulemans et al., 2017) or the ISABELE method (Boisson and Bouchié, 2014; Schetelat and Bouchié, 2014; Thébault, 2017; Thébault and Bouchié, 2018). There are also other punctual applications to heat transfer estimation (Rouchier et al., 2019, 2018) or building thermal behaviour prediction (Bacher and Madsen, 2011). These methods can be distinguished by the design of experiment used to control measurement conditions and by the RC models used.
The QUB method relies on a rather short data acquisition duration, compared to the other methods. Performed during one night as to avoid solar irradiation, the heating power is programmed to follow a two phase scenario with first large constant heating power followed by a close to free floating phase, during which the indoor temperature slowly decays as the heating power is set to minimal. The heating power step of the first phase is designed as to result in an exponential curve of temperature. With sufficient duration of each phase, a few hours, it is expected that the largest time constant, the one of interest, is visible in the response of the building.
The data is exploited by the simplified dynamic model of Equation 1.5: P = HLC (Tin Tout ) +C dTin (1.5) dt
This equation can be considered accurate for each phase, and the overall heat loss coefficient is then defined by Equation 1.6, where Pi, DTi and ai are respectively the total power, the inside-outside temperature at the end and the derivative of the indoor temperature at the end of phase i.
Strictly speaking, the QUB method does then not actually make use directly of the RC model but rather uses the derivatives at
HLC = P1a2 P2a1 (1.6)
DT1a2 DT2a1
Results of HLC estimations in various buildings show deviation from target reference value up to 15%. Variable weather conditions such as strong wind and large outdoor temperature variations may lead to larger errors in HLC estimations, as suggested in Alzetto et al. (2018a).
The ISABELE method was developed to characterize the thermal properties of the building envelope when commissioning a newly built building. The goal is to determine heat transfers through the envelope, excluding the air infiltrations. The latter are measured prior to the HTC measurement and is included in the data exploitation.
For the HTC estimation, the ISABELE method also relies on a specific controlled experiment: after a short free floating phase, the indoor temperature is set to follow a long step signal. During the experiment, the window blinds or shutters are closed and the hypothetical ventilation shut. The experiment lasts longer than for the QUB method, as it has been found that more than 2 days led to more robust results (Thébault and Bouchié, 2018).
The collected data is used to fit an RC model, which takes into account the infiltration priorly measured. Although uncertainties of the results are calculated by the calibration method (Nelder-Mead (Boisson and Bouchié, 2014) or MCMC (Schetelat and Bouchié, 2014)), Thébault and Bouchié (2018) propose to correct their widths by propagating through a Monte-Carlo the most influential systematic measurement errors, such as temperature or heat power errors. Systematic errors are indeed not taken into account in the calibration process by any means other than uncertainty propagation, whereas they have a significant influence on accuracy.
Finally, Bacher and Madsen (2011); Rouchier et al. (2019, 2018) also suggest dynamic models to exploit in situ controlled data. Bacher and Madsen (2011) proposes a model selection process to determine what stochastic RC model is best fit for a given set of data. The authors have however heat dynamics prediction in view rather than thermal performance inference. The collected data is that of an unoccupied office building which heat power is controlled to follow a so-called pseudo-random signal.
Rouchier et al. (2018), on basis of a heating power step signal of two days, compare stochastic RC models with deterministic RC models and conclude that parameter estimation with stochastic models is not closer to the target values, but that at least the associated uncertainties are more realistic. In Rouchier et al. (2019), a pseudo-random heating power signal in a 1m3 box serves as training for an on-line algorithm called sequential Monte-Carlo. It provides RC model training with an update of the parameter estimation at each new observation. The authors show how a sequential Monte-Carlo yields similar parameter estimation than a Marginal Monte-Carlo algorithm and that the results are in agreement with the target values for both the HTC and the solar aperture coefficient (i.e. the coefficient bringing correction of the solar irradiation heat gains).
Interestingly, the in-line algorithm revealed the most influential events that enhanced parameter estimation. In particular, it uncovered that the model learns a lot when heating power is finally turned on. This outcome brings into light how useful information in the collected data can be to model training, in particular for dynamic models.
Table 1.1 lists as a conclusion the aforementioned methods. The most accurate estimations of the thermal performance unsurprisingly result from controlled measurements conditions. When the building cannot be left vacant, some existing methods are compatible with uncontrolled conditions. Methods relying on steady-state models require almost no data acquisition material, but need data over several months, seemingly without guarantee of physical interpretability. Methods relying on dynamic models require shorter duration datasets, and seem to reach more satisfactory results.
This section is not viewed as an exhaustive literature review of all methods for characterising the thermal performance of a building. To dig further into that subject, the curious will find reader-friendly sources in Zayane (2011), Bauwens and Roels (2014), Bauwens (2015), Janssens (2016), Thébault (2017) and Raillon and Ghiaus (2018).

### From controlled to uncontrolled conditions: an ill-posed inverse problem

Inverse problems: some definitions

For clarity, let us first define some concepts that will thoroughly be used in this work. Definitions are freely inspired by those in Walter and Pronzato (1997) or Muñoz-Tamayo et al. (2018).
A system S is a part of the universe, chosen more or less arbitrarily, considered as the entity to study and that has interaction as a whole with some external variables. The system is delimited by spatial and temporal boundaries.
The external variables that act on the system and internal state variables are most probably not all observable, i.e. measurable. The observable quantities that influence the system’s behaviour are usually called inputs of the system and noted u. The remaining influential variables can then be considered as perturbations or simply noises. They are not controllable and sometimes difficult to apprehend. The internal and observable quantities may then be called output of the system y. Some other quantities might be of interest, but are not directly observable.
The system and its boundaries as well as the observables and the input variables then drive the construction of a model of the system. A model, as defined by Walter and Pronzato (1997), is « a rule to compute from quantities observed from the system other quantities which are hoped to be close to the actual values as observed in the system ». A model can be understood as a set of equations that describe the physics of interest in the system. The model links the observable inputs with the observable outputs of the system. The model may be linear or non-linear (in its inputs): it is linear if for all l and m, y(l • u1 + m • u2) = l • y(u1) + m • y(u2).
Some models, such as the RC models mentioned earlier in section 1.2, describe quantities of the system by their derivatives with respect to time as a function of their previous state. Such models can be called state space models describing state variables x. RC models are incidently linear in their inputs.
The set of equations describing the system uses parameters usually named p or q . They are scalars (not vectors) and usually time-independent. When all parameters are known, one may speak of model M(q ). Before actual values are attributed to the parameters, one talks about a model structure M(:). A structural property is then a property that is valid for almost all values of parameters (Bellman and Åström, 1970). Linearity is an example of structural property of a model (Muñoz-Tamayo et al., 2018).
Model calibration is the « action of using a numerical routine, algorithm, for finding the value of unknown parameters of a model that best fit an experimental data set » of observable quantities (Muñoz-Tamayo et al., 2018). Model calibration is also called parameter identification (or estimation) and model fitting (Muñoz-Tamayo et al., 2018).
Walter and Pronzato (1997) expressly underline the importance of estimating parameter uncertainty along with parameter identification. Indeed, it would be foolish to consider that given the measurement uncertainties, the result of model calibration would yield a unique set of parameters. On the contrary, there is a set of acceptable models and that translates into parameter estimates and their uncertainty.
Upon developing an appropriate model to fit some collected data, the model can possibly contain more parameters than needed to describe the system output. This is called over-parametrization (Muñoz-Tamayo et al., 2018) and results in the model over-fitting the data (Rouchier, 2017).
When applied to the thermal performance estimation of a building, the system may be the building in its whole, or simply the building envelope. Observable inputs and outputs are for example weather conditions, heating or cooling power, indoor temperatures, wall surface temperatures, etc…
When estimating the overall thermal performance, one is then interested in a quantity that is not directly observable. This whole procedure is called solving an inverse problem. Upon considering any system S, here the building envelope, Tarantola (2005) considers that inverse problems are part of three arbitrary scientific procedures to study such a system and underlines that they obviously are strongly linked:
• parametrization of the system: determining a minimal set of parameters that completely characterize the system from a given point of view. Characterization here means almost perfectly describing the physical phenomena at hand in the system.
• forward modelling: discover the physical laws that allow predictions of the quantity of interest.
• inverse modelling: use actual measurements to infer the actual values of parameters (or variables) of interest.
Let us underline that a distinction is made between forward and inverse modelling, which implies that as the goals are different, the appropriate models to the one or the other might be different as well. In other words, a valid model for forward modelling might be misleading in inverse modelling. The choice of an appropriate model for solving this particular inverse problem is therefore further discussed in section 1.4.

General introduction
1 Estimation of thermal performance from measurement data: an inverse problem
1.1 Introduction
1.2 Existing methods for the estimation of the thermal performance of a building envelope
1.3 From controlled to uncontrolled conditions: an ill-posed inverse problem
1.3.1 Inverse problems: some definitions
1.3.2 Loss of information in uncontrolled conditions
1.4 Appropriate models for thermal performance estimation
1.4.1 From data-driven to physics-driven modelling
1.4.2 RC models: physics-driven simplified models
1.5 Algorithmic and numerical tools for solving the inverse problem
1.5.1 Frequentist approach
1.5.2 Bayesian approach
1.6 Conclusions and orientations of the following work
2 Identifiability and interpretability
2.1 Introduction
2.2 Structural identifiability : a necessary condition
2.2.1 What is structural identifiability?
2.2.2 Principle of a unique input-output expression: the exhaustive summary
2.2.3 Some existing methods for verifying structural identifiability
2.2.4 Why some models for forward problems are unfit for inverse modelling
2.2.5 Application: a set of structurally identifiable state space models
2.3 Practical identifiability
2.3.1 Grasping the necessity of practical identifiability
2.3.2 Assessing practical identifiability
2.4 Threats to physical interpretation and calibration good practice
2.4.1 Threats to physical interpretation from poorly informative data
2.4.2 Enhancing information in data from uncontrolled measurements
2.4.3 And yet not enough for interpretation: workflow for meaningful calibration
2.5 Conclusion and work prospects
3 Numerical model assessment methodology for physical interpretability
3.1 Introduction
3.2 A numerical assessment framework for physical interpretability
3.2.1 Proposition for a numerical assessment framework
3.2.2 A comprehensive building energy model as reference
3.2.3 Case study
3.3 Model assessment and comparison : a quantitative indicator
3.4 Global sensitivity analysis
3.4.1 How to perform global sensitivity analysis
3.4.2 Assessing the influence of weather variables
3.4.3 Influence of variable thermal properties of the envelope
3.4.4 Conclusion on global sensitivity analysis
3.5 Conclusion
4 Repeatability of parameter estimation under variable weather conditions
4.1 Introduction
4.2 Weather conditions influence: state of the art
4.3 The reference model undergoes variable weather conditions
4.3.1 Adaptations of the reference model methodology
4.3.2 Calibration and model validation
4.3.3 Weather variability in a numerical methodology
4.4 Decrease in variability of Req estimation with experiment duration
4.4.1 Variability with a 2-days model training
4.4.2 Minimal measurement duration for model training
4.5 Influential weather variables on an Req estimation
4.6 Discussion
4.7 Conclusion
5 Decomposition of heat losses in a building
5.1 Introduction
5.2 Model assessment framework for heat transfer decomposition
5.2.1 What decomposition can be reasonably expected?
5.2.2 Application of the model assessment framework
5.2.3 State space model selection and validation
5.2.4 Convergence of the sensitivity analysis
5.3 Estimation of the heat losses through ventilation
5.3.1 Variability of parameters Cw, Ci, Ro, Ri, Aw and cv of model TwTi RoRi cv
5.3.2 Sensitivity analysis of parameters Cw, Ci, Ro, Ri and cv
5.3.3 Estimation and physical interpretability of ventilation and infiltration
5.3.4 Conclusions on decomposing ventilative heat losses
5.4 Estimation of heat losses towards unheated neighbouring space
5.4.1 Variability of the estimated parameters of model TwTi RoRiRb
5.4.2 Sensitivity analysis of the thermal resistance estimation
5.4.3 Identifiability and interpretability of the proposed model
5.4.4 Conclusions on identification of heat losses to neighbouring spaces
5.5 Conclusion and Bayesian prospects
General conclusion
Bibliography

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