Assessment of data analysis methods to identify the heat loss coefficient from on-board monitoring data

Energy & Buildings 209 (2020) 109706

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Assessment of data analysis methods to identify the heat loss coefficient from on-board monitoring data Marieline Senave a,b,c,∗, Staf Roels a, Glenn Reynders b,c, Stijn Verbeke b,c,d, Dirk Saelens a,c a

Department of Civil Engineering, Building Physics Section, KU Leuven, Belgium VITO, Unit Smart Energy and Built Environment, Belgium c EnergyVille, Cities in Transition, Belgium d University of Antwerp, Applied Engineering, EMIB, Belgium b

a r t i c l e

i n f o

Article history: Received 24 June 2019 Revised 12 November 2019 Accepted 14 December 2019 Available online 16 December 2019 Keywords: Characterization Physical parameter identification Heat loss coefficient Synthetic Monitoring Data Data Analysis Methods

a b s t r a c t The past decade has seen the rapid development of sensor technologies. Monitoring data of the interior climate and energy consumption of in-use buildings, so-called on-board monitoring (OBM) data, offers the opportunity to identify as-built energy performance indicators, such as the heat loss coefficient (HLC) of the building envelope. To this end, it is important to advance the understanding of the impact of the OBM set-up and the applied data analysis method. This paper uses synthetic OBM data sets, generated from building energy simulations. The level of accuracy achieved with four data analysis methods for characterizing the HLC is investigated. The considered methods are the Average Method, the Energy Signature Method, Linear Regression and ARX modeling. Different cases, representing different building types, are considered in order to gain thorough insight into the physical interpretation of the results. By taking subsets of the original data sets, the sensitivity of the data analysis methods to the availability of specific data is assessed. This theoretical exercise illustrates how, under idealized monitoring circumstances, both linear regression and ARX models can accurately determine the HLC. The latter is able to assess the performance indicator within 5%. However, when subjected to practical limitations regarding the measurement of system inputs, such as unavailable solar or internal heat gains, the characterization results show large variations in accuracy and uncertainty. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The Heat Loss Coefficient or HLC [W/K] quantifies the joint insulation and air tightness quality of a building envelope. It is a crucial parameter of interest for applications such as quality assurance of building construction practice, selection of renovation actions and design of heating systems, including district heating. Several researchers have developed methods to assess the HLC based on a combination of (1) in-situ measurement tests and (2) data-driven modeling [1–5]. By using as-built data, such empirical characterization approaches can for example provide a better understanding on the frequently observed difference between the actual building energy performance and the one that is theoretically calculated according to governing standards [6–10]. Given the relatively high cost and intrusiveness associated with data collection through dedicated heating experiments such as the co-heating test [11] or QUB ∗ Corresponding author at: Department of Civil Engineering, Building Physics Section, KU Leuven, Belgium. E-mail address: [email protected] (M. Senave).

https://doi.org/10.1016/j.enbuild.2019.109706 0378-7788/© 2019 Elsevier B.V. All rights reserved.

test [12], current research [13–16] tries to shift the data collection more towards so-called on-board monitoring (OBM), which denotes the monitoring of the interior climate and energy consumption of in-use buildings via non-intrusive sensors. Gathering qualitative data through OBM is a first step. It is, however, equally important to correctly apply data-driven modeling techniques to obtain accurate HLC estimates. Throughout the years, a wide variety of data analysis techniques, ranging from straightforward steadystate methods to more advanced dynamic methods, have been proposed to analyze the data of controlled heating experiments [3,17–19]. However, it remains to be established whether and how these methods can be extended to OBM-data, which are (for reasons of comfort and convenience) characterized by lesspronounced heat input profiles and are furthermore affected by occupant-induced disturbances. Moreover, to guide the selection of an optimal OBM set-up and data analysis method for each application, insights should be obtained into (1) the data required by the data analysis methods, both with regard to the type and number of variables to be monitored and the frequency and duration of the OBM, and (2) the level of accuracy the data analysis methods

2

M. Senave, S. Roels and G. Reynders et al. / Energy & Buildings 209 (2020) 109706

can provide. The trade-off between these two aspects may be influenced by for example the building type, occupant behavior or climate. This paper will use synthetic data sets, obtained from dynamic building performance simulations, to systematically evaluate the ability of four types of data analysis models (Average method, Energy Signature method, Linear Regression and Auto-Regressive models with eXogenous inputs (ARX models)) to identify the Heat Loss Coefficient from on-board monitoring data. Additionally, the influence of the concerned building type and the available input data (number of variables and length of data set) will be examined. To support a clear comparison, this paper first presents the performance indicator ‘HLC’ in a broader physical context (Section 1.1) and gives a brief overview of the evaluated steadystate and dynamic data analysis methods (Section 1.2). Next, the methodology section focuses on the creation of the synthetic data sets (Section 2.1) and the systematic application of multiple data analysis techniques (Section 2.2). Section 3 then presents and discusses the obtained results. Finally, Section 4 summarizes the main findings and identifies areas for further research.

HLCt = Hin f ;t + Htr;t =

Htr,e;t =



n 



q 

=

Applying the law of conservation of energy, the simplified dynamic heat balance of a single-zone dwelling can be written as Eq. (1.1), with Ci the zone effective heat capacity [J/K], dθ i /dt the change of the reference interior temperature θ i [ °C] over time t, h [W] the net heating power supplied by the heating system, int [W] the internal gains and sol [W] the solar gains through transparent fabric parts. The intended ventilation and infiltration heat exchange are respectively represented by v and inf [W], and tr [W] is the heat transfer by transmission, taking account of longwave radiation exchange and absorbed incident solar radiation at the exterior and interior surface. Since both inf and tr express properties of the building envelope which are proportional to the temperature difference between the exterior and interior environment (θ e -θ i ), several authors [1,20] choose to combine them1 and express them as this temperature difference term times a so-called heat loss coefficient or HLC [W/K] (Eq. (1.2)).

(1.1)

= h;t + int;t + sol;t + v;t + HLCt · (θe;t − θi;t ) (1.2) The heat loss coefficient thus yields the heat transfer coefficients by infiltration Hinf [W/K] and transmission Htr [W/K], both thermal performance indicators of the building fabric, as shown in Eq. (2.1). The first coefficient, Hinf , equals the product of the heat capacity of air per volume ρ a ca [J/(m³K)] and the infiltration air flow rate Qinf [m³/s] (Eq. (2.2)). Htr , on the other hand, embeds the transmission heat loss per degree temperature difference to the exterior environment (Htr,e ), the ground (Htr,g ), unconditioned spaces (Htr,u ) and adjacent buildings (Htr,a ). Htr,e takes the sum over the heat transfer through the n building elements, o linear thermal bridges and p point thermal bridges in contact with the ambient, as shown in Eq. (3) with Ai and Ui respectively the surface area [m²] and thermal transmittance [W/(m²K)] of building element i, and Lj ,  j and Xk respectively the length [m], linear thermal transmittance [W/(mK)] and point thermal transmittance [W/K] of the thermal bridges. Likewise, Htr,g expresses the transmission heat loss from the interior environment to the ground (Eq. (4.1)). However, since the 1 By expressing both heat flow rates in function of the same temperature difference term, the distinction between the equivalent temperature (on which the transmission losses are based) and air temperature (on which the infiltration losses are based) is ignored [44].

(Ai · Ui;t ) +

i=1

Htr,g;t =

(2.1)

 ρa · ca · Qin f ;t + (Htr,e;t + Htr,g;t + Htr,u;t + Htr,a;t )



1.1. The heat loss coefficient

Ci · dθi /dt = h;t + int;t + sol;t + v;t + in f ;t + tr;t

q elements and r and s thermal bridges considered in this term are in thermal contact with the ground instead of the ambient, an additional temperature factor bT,g is included which accounts for the difference between the ground (θ g ) and exterior (θ e ) temperature. Finally, Htr,u and Htr,a follow similar equations as Htr,g (Eqs. 5 and 6), with bT,u and bT,a respectively factoring in the reference temperature of the unconditioned zones (θ u ) and adjacent buildings (θ a ).

o  



L j ·  j;t +

r  



L j ·  j;t +

i=1

j=1

q 

r  

(Ai · Ui;t ) +

i=1

Xk;t

k=1

j=1

(Ai · Ui;t ) +

p 

s 

(3)

 · bT,g;t

Xk;t

Htr,u;t =

 Htr,a;t =

t 



L j ·  j;t +

s 

 Xk;t

k=1

j=1

(Ai · Ui;t ) +

u  

i=1

j=1

w 

x  

i=1

(Ai · Ui;t ) +

(4.1)

k=1

·( (θg;t − θi;t )/(θe;t − θi;t ) )



(2.2)

(4.2)



L j ·  j;t +

v 

 Xk;t

k=1



L j ·  j;t +

j=1

y 

· bT,u;t

(5)

· bT,a;t

(6)

 Xk;t

k=1

1.2. Statistical data analysis methods This section discusses four data analysis techniques that will be applied to characterize the HLC from on-board monitoring data. The so-called average method (Section 1.2.1), linear regression analysis (Section 1.2.2) and the energy signature method (Section 1.2.3) assume that by collecting the temperature and energy consumption measurements over longer periods, the building’s dynamic behavior can safely be ignored. Hence, the heat balance Eq. (1) simplifies to:

Ci · dθi /dt = 0

(7.0)

⇓ h;t + int;t + sol;t + v;t = −in f ;t − tr;t

(7.1)

= HLCt · (θi;t − θe;t ) = HLCt · θie;t (7.2) By contrast, Section 1.2.4 outlines ARX modeling, a method that incorporates dynamic effects. 1.2.1. Average method (Avg.Meth.) This first method stems from a technique proposed by ISO 9869-1 [21] to determine the U-value of a homogenous building element from in-situ measurements. As described in Eq. (8), the thermal transmittance can be obtained by dividing the mean density of heat flow rate through the building element, q [W/m²], by the mean temperature difference between the two environments separated by the building element (here the interior and exterior environment). The suffix tj in Eq. (8) denotes the observation at

M. Senave, S. Roels and G. Reynders et al. / Energy & Buildings 209 (2020) 109706

time tj . By steadily increasing the number of observations n, the estimate is assumed to asymptotically converge to the soughtafter parameter. For building elements, the standard mentions a minimal test duration of 72 h.

U=

n 

qt j /

j=1

n  

θi;t j − θe;t j



(8)

j=1

The heat loss coefficient can be considered as a U-value on building scale that additionally incorporates heat losses by thermal bridging and infiltration. Hence, by analogy with Eq. (8) and considering the steady-state heat balance (Eq. (7)), Eq. (9) presents a method to estimate the HLC. It states that the average HLC for the considered period, which will from now on be referred to as ‘HLC’ without subscript ‘t’, equals the division of the mean sum of the net heat input, internal and solar gains and ventilation heat loss over that period by the mean interior-exterior temperature difference.

HLC =

n  



h;t j + int;t j + sol;t j + v;t j /

j=1

n  

θi;t j − θe;t j



(9)

j=1

1.2.2. Linear regression (LR) As shown in Eq. (10), linear regression analysis seeks to determine a linear relationship between an output or dependent variable2 y and one (n = 1, ‘simple linear regression’) or more (n>1, ‘multiple linear regression’) input or independent variables x.

y=

n 

( ai · xi ) + b

(10)

i=1

Bauwens and Roels [22] explain how this technique can be used to identify the HLC from measurement data of co-heating tests. They distinguish three different model parametrizations (Eqs. (11), (13) and (14)), which will be discussed below. Eq. (11) presents a simple linear regression model which uses the interior-exterior temperature difference θ ie as independent variable explaining the variability of the sum of the heat flow rates. The term ε represents the prediction error or model residual. Since the co-heating test is a controlled experiment, performed in an unoccupied building and with the intended ventilation turned off, Bauwens and Roels only consider the heat flow rates h and sol . However, in order to analyze OBM data, int and v should additionally be incorporated in the regression equation. When fitting model Eq. (11) on the monitoring data, the HLC is expected to be estimated as the coefficient of θ ie , which can be visually interpreted as the slope of the regression line fitted through the [ (), θ ] point cloud.

h;t + int ;t + sol ;t + v;t = HLC · θie;t + εt

(11)

In a practical OBM set-up, data on the total solar gain of a zone (sol ) is not evidently gathered. Instead, it can be derived from summing the solar gains through each of the n transparent parts of its building envelope (Eq. (12)).

sol;t =

n  i=1

sol;i;t =

n  

gi;t · Ai;t · Isol;i;t



(12)

i=1

These element-specific solar gains can in turn be written as the product of (1) the g-value [-] of the glass panes; (2) the element’s effective collecting area A [m²], which equals the element’s total surface area reduced by a frame area fraction and shading reduction factor; and (3) the combined direct and diffuse solar irradiance Isol [W/m²]. A subscript t is used to emphasize that both g and A vary in time as they are function of the angle of incidence of 2

Notation: vectors are written in bold.

3

the radiation. Based on Eq. (12), Bauwens and Roels [22] propose an alternative linear regression model, which uses the incident solar radiation under a certain projection k, Isol;k , as a second independent variable: Eq. (13). The authors state that, due to the strong mutual correlation between the aggregated measurement data for the respective solar irradiation projections, it might not be possible to estimate their gA-coefficients separately. Hence, they simplified Eq. (12) by only adding a single projection k to the parametrization. If such model yields satisfactory results, it even reduces the costs of the monitoring set-up, since the incident solar radiation only has to be monitored at one location instead of every window. The extended linear regression set-up of Eq. (13) allows to assess two performance indicators of the building envelope at once: the heat loss coefficient and an apparent or effective solar aperture coefficient. The latter is denoted as gAl , since it represents a lumped gA-value for all transparent building fabric parts.

h;t + int ;t + v;t = HLC · θie;t − gAl · Isol ;k;t + εt

(13)

By dividing all terms in Eq. (13) by θ ie , a third linear regression approach is deduced (Eq. (14)). Here, the model’s constant intercept term is ought to yield an indication of the HLC, while the slope of the fitted regression line is assumed to represent a lumped gA-value.



 h;t + int ;t + v;t /θie;t = HLC − gAl · Isol ;k;t /θie;t + εt (14)

1.2.3. Energy signature method (ES) The Energy Signature method, which is discussed in more detail in [23,24], can be considered as a special case of the abovepresented linear regression analysis. This method, too, stems from the steady-state heat balance (Eq. (7)) and aims to fit a linear relationship between the measurement data. However, instead of merely relating the interior-exterior temperature difference to the total heat flow rate as Eq. (11) does, the ES method additionally seeks to assess a baseline exterior temperature θ b , for which the building at interior temperature θ i is in thermal equilibrium with its environment, without heating (Eq. (15)). Combining Eqs. (7) and (15), Eq. (16) is obtained, which implies that, if the heating input is larger than zero, the HLC can be estimated by the slope of the regression line fitted on the [h , θ e ] point cloud.

  int;t + sol;t + v;t = HLC · θi;t − θb + εt  h;t =

HLC ·

  θb − θe;t + εt

0 + εt

θe;t < θb i f θe;t ≥ θb if

(15)

(16.1 ) (16.2 )

1.2.4. ARX modeling Auto-Regressive models with eXogenous inputs (in short ‘ARX models’) are discrete time transfer function models that describe the dynamic behavior of systems, such as buildings, using linear difference equations [19,25]. This type of equations incorporates dynamic effects in the model by factoring in past observations of the variables. For example, the basic ARX model in Eq. (17) employs time series data of input variables X and Z to explain the variability of output variable Y. The terms ϕ (B), ωx (B) and ωz (B) are respectively one output and two input polynomials in the backward shift operator B (Eqs. (18)–(21)). Their order, respectively nϕ , nωx , and nωz , determines the number of past observations (socalled ‘lags’) that is taken into account through B, as exemplified by Eq. (21) for the observations of variable Z.

ϕ (B ) · Yt = ωx (B ) · Xt + ωz (B ) · Zt + εt

(17)

ϕ ( B ) = 1 · B0 + ϕ1 · B1 + ϕ2 · B2 + . . . + ϕnϕ · Bnϕ

(18)

ωx (B ) = ωx,0 · B0 + ωx,1 · B1 + . . . + ωx,nωx · Bnωx

(19)

4

M. Senave, S. Roels and G. Reynders et al. / Energy & Buildings 209 (2020) 109706

Fig. 1. The four explored building types (BTs): BT1 and BT2 are both detached dwellings, the floor slab of which is respectively in contact with an unconditioned space and the ground. BT3 and BT4 represent the semi-detached variants of respectively BT1 and BT2. The building fabric of interest is colored dark gray. The dwelling’s interior temperature (further defined) is denoted as θ i , θ e is the exterior temperature, θ g the ground temperature, θ u the temperature of the unconditioned crawl space and θ a is the interior temperature of the adjacent dwelling.

ωz (B ) = ωz,0 · B0 + ωz,1 · B1 + . . . + ωz,nωz · Bnωz

(20)

Bk · Zt = Zt−k

(21)

In the case of a building, the general model structure shown in Eq. (17) can be translated into Eq. (22), with the sum of the heat flow rates included in the heat balance as output, and the interior and exterior temperature as input variables. Similar to the linear regression parametrization in Eq. (13), it is also possible to model the solar irradiation Isol explicitly (Eq. (23)).

  ϕ (B ) · h;t + int ;t + sol ;t + v;t = ωi (B ) · θi;t +ωe (B ) · θe;t + εt 

(22)



ϕ (B ) · h;t + int ;t + v;t = ωi (B ) · θi;t + ωe (B ) · θe;t +ωsol (B ) · Isol ;x;t + εt

(23)

Although the coefficients of ARX models are essentially not directly related to physics, an estimate of the steady-state, averaged HLC can be derived from the steady-state gains of the inputs and output as demonstrated by Bauwens [1] and Senave et al. [26]. The steady-state gains from the variables yield the model’s steady-state behavior and can be obtained by setting B equal to one in Eqs. (18)–(20). 2. Methodology The comparison of the four data analysis techniques will be based on synthetic data derived from building energy simulations. This not only ensures that the theoretical heat loss coefficient is precisely known and that measurement errors can be discarded,

but also gives the opportunity to readily investigate diverse scenarios. Here, the sensitivity of the characterization procedure and outcome to the considered building type will be analyzed. The first part of the methodology description (Section 2.1) presents the four different data sets, and for each data set discusses (1) the performed building energy simulations and (2) the associated reference HLC values. Next, Section 2.2 delineates the different analysis models that are identified on the synthetic OBM data, the performed model validation, and the determination of the HLC. Finally, Section 2.3 describes the criteria used to evaluate the quality of the estimates and the robustness of the models. 2.1. Generation of synthetic OBM data 2.1.1. Building simulations All simulations are executed in TRNSYS 17 [27], a building energy simulation tool which has scored well in analytical, comparative and empirical tests [28–30]. Following Bauwens [1], four commonly observed building types are considered: a detached dwelling with a suspended floor structure and naturally ventilated crawl space (‘BT1’); a detached dwelling with a solid slab-onground floor (‘BT2’); and two semi-detached variants (’BT3’ and ’BT4’), again respectively with suspended floor and slab-on-ground floor. The four BTs are schematically represented in Fig. 1. In each of the four scenarios, the main dwelling (cross-section depicted in solid gray in Fig. 1) is modeled as a simple single-zone box of 7m∗ 5m∗ 3 m (length∗ width∗ height) using TRNSYS Type 56 [31]. It has fully insulated cavity walls (see Table 1) with windows of 4 m² and 1 m² in respectively the south and west façade.3 While both the roof and floor assembly include a concrete slab of 20 cm, 3

Total surface area: glass + frame

M. Senave, S. Roels and G. Reynders et al. / Energy & Buildings 209 (2020) 109706

5

Fig. 2. Setpoint temperature profile for the zone air temperature (top) and internal gain profile (bottom) for the main and adjacent dwelling.

Table 1 Overview of the composition of the building components, with d the thickness and λ the thermal conductivity of the material layers. Building Component

Material layers

d [m]

λ [W/(mK)]

Roof

reinforced concrete slab screed to falls thermal insulation weatherproof membrane inner leaf, brick masonry thermal insulation outer leaf, brick masonry screed reinforced concrete slab brick masonry

0.20 0.12 0.08 0.003 0.14 0.08 0.09 0.08 0.20 0.20

1.70 1.30 0.03 0.23 0.79 0.03 1.49 1.30 1.70 0.79

Exterior Walls Floor Common Wall

only the roof is thermally insulated. The resulting transmission heat transfer rates through the building envelope are discussed in Section 2.1.2. The dwellings are simulated for the moderate, heatingdominated climate of Uccle in Belgium, using a TMY2-weather file. No intended ventilation is foreseen, but a fairly common infiltration rate of 0.35 ACH is modeled. By default, TRNSYS Type 56 takes account of neither wind nor buoyancy pressure, hence this infiltration rate is assumed to be constant throughout the simulation. Space heating is generated using a 100% convective heating system, which is designed to cover the heat load on the coldest day. No active cooling system is foreseen. The interior air is assumed to be perfectly mixed. Its temperature is controlled by the setpoint temperature profile displayed in Fig. 2: on weekdays the main dwelling is only heated during the morning and evening, whereas on Saturday and Sunday the dwelling is heated the entire day. At night, a set-back temperature of 13 °C is imposed. To simulate varying occupant presence, the 100% convective internal gain profile shown in Fig. 2 is applied. To take account of the thermal mass of furniture, the thermal capacity of the zone air was multiplied by a factor 5 [32,33]. The crawl space considered in BT1 and BT3 has the same floor area as the main dwelling (35 m²), but its height is limited to 1.2 m. The above-discussed wall and floor assembly (Table 1) apply

for this construction as well. However, no windows are foreseen and no internal gains or infiltration rate were specified. The space is naturally ventilated at a rate of 1.5 h, meeting the ventilation requirement for cellars in Belgium [34]. The free floating interior temperature of this unconditioned space will further be denoted as θ u . The soil mass used in BT2 and BT4 was modeled with TRNSYS Type 49 [35], considering a soil thermal conductivity of 2 W/(mK) and density of 20 0 0 kg/m³. The average surface soil temperature is set equal to 10 °C, the amplitude of soil surface temperature to 7 °C and the day of minimum surface temperature to day 33 [36,37]. For cases BT3 and BT4, the main building’s northern wall is a common wall consisting of 20 cm brick masonry. The adjacent dwelling is identical to the main one, except that the window of 4 m² is now north-oriented and a slightly different setpoint temperature and internal gain profile is imposed (see Fig. 2). It should be noted that the defined models only represent one specific, highly simplified situation and the results of this characterization exercise should be looked at in this context. The simulation output is generated with a sampling time of 3 min. 2.1.2. Calculation of reference HLC values In order to assess the accuracy of the obtained HLC estimates later-on, a reference value is required. This section explains how this value is calculated for the four simulation models. The building envelope part of which we want to assess the HLC is marked in dark gray in Fig. 1. In BT1 it separates the interior environment at temperature θ i from the exterior environment at temperature θ e and an unconditioned space at temperature θ u . In BT2 its boundary conditions are at temperatures θ i , θ e and θ g . In cases BT3 and BT4, the building fabric of interest is additionally in thermal contact with an adjacent dwelling at temperature θ a . Hence, in each of the four cases the HLC includes a different combination of the five possible components listed in Eq. 2: Htr,e , Htr,g , Htr,u , Htr,a and Hinf . This is visualized in Table 4 in Section 3.1, which gives an overview of the reference HLC and its constituents for each BT: the non-zero heat transfer coefficients are shaded in gray.

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M. Senave, S. Roels and G. Reynders et al. / Energy & Buildings 209 (2020) 109706

The heat transfer coefficient by infiltration is equal to 12.5 W/K for each building type (Eq. (2.2)). The heat transfer coefficients by transmission, on the other hand, are determined in accordance with Eqs. (3)–(6). Since neither the U-values nor the temperature factors bT are direct simulation outputs of TRNSYS, they are calculated manually as described below. As demonstrated in Eq. (24), the U-value of a building element x is the reciprocal of the element’s total thermal resistance Rtot . This term can in turn be split up in (1) the thermal boundary resistance at the internal surface Rsi ; (2) the thermal resistances of the element’s n material layers Ri ; and (3) the thermal boundary resistance at the external surface Rse . Rather than using the conventional average values provided by ISO 6946 [38] for the surface resistances, the actual 3-min values were manually calculated based on the simulation output. Hereto, Eqs. (25)–(27) were used, with θ si;x the interior surface temperature of building element x [°C], θ STAR;i the star node temperature4 of the main dwelling’s interior zone [°C] and comi the heat flow rate from the inside surface, including convection to air and longwave radiation exchange with the other surfaces [W] [31]. hce and hre represent the convective and radiative heat transfer coefficient of the external surface [W/(m²K)] respectively, σ is the Stefan-Boltzmann constant [W/(m²K4 )] and eL the longwave emissivity of the surface, which was set equal to 0.9. The variables θ se;x , θ sky and θ air,e represent the exterior surface temperature of building element x, the sky temperature and the exterior air temperature, respectively (all in [°C]). Finally, vssky,x is the view factor from the external surface of building element x to the sky (set at 0 for floor slabs, 0.5 for walls and 1 for flat roofs). It should be noted that TRNSYS assumes the building elements, and thus their characteristics (e.g. surface temperatures, heat fluxes) to be uniform.



Ux;t = 1 / Rtot;x;t = 1 /

Rsi;x;t +

n 



Ri;x;t + Rse;x;t

(24)

i=1

Rsi;x;t = Ax · (θsi;x;t − θSTAR;i;t )/comi;x;t

Similarly, the exterior temperature is determined as the surface area weighted average of the exterior equivalent temperature (θ equiv,e;x ) of the building elements x exchanging heat with the ambient. As Eq. (29) shows, the exterior equivalent temperature [ °C] takes account of convection to the exterior air and longwave radiation to the exterior environment and sky (combined in como [W]) and solar radiation absorbed at the exterior surface (abso [W]).5

  θequiv,i;x;t = absi;x;t − comi;x;t · Rsi;x;t /Ax + θsi;x;t = θSTAR;i;t + absi;x;t · Rsi;x;t /Ax

(28)

  θequiv,e;x;t = abso;x;t + como;x;t · Rse;x;t /Ax + θse;x;t

(29)

The temperature of the ground θ g is set equal to the ground equivalent temperature θ equiv,g,floor , which is described in Eq. (30). The floor slab’s surface resistance to the ground Rsg is very small, since the exterior surface temperature of the floor θ sg nearly equals the boundary temperature of the soil mass θ boundary;g (Eq. (31)). The temperatures θ u and θ a in bT,u and bT,a (Eqs. (5) and (6)) are respectively calculated as θ equiv,u,floor, the equivalent temperature of the floor slab at the side of the unconditioned space (Eq. (32)) and θ equiv,a,partywall , the equivalent temperature of the party wall at the side of the adjacent dwelling (Eq. (34)). The temperatures θ STAR;u and θ STAR;a in Eqs. (32) and (34) represent the star node temperature of the unconditioned space and the adjacent dwelling [°C], respectively, and the surface thermal resistances Rsu;floor and Rsa;partywall are explained in Eqs. (33) and (35).

θg;t = θequiv,g; f loor;t = como; f loor;t · Rsg; f loor;t /A f loor + θsg; f loor;t Rsg; f loor;t = A f loor ·

θu;t = θequiv,u; f loor;t = θSTAR;u;t + abso; f loor;t · Rsu; f loor;t /A f loor

(25) Rsu; f loor;t = A f loor ·

Rse;x;t = 1/(hce;x;t + hre;x;t )

  θsu; f loor;t − θSTAR;u;t /como; f loor;t

(31)

(32)

(33)

(26)



hre;x;t

  θsg; f loor;t − θboundary;g;t /como; f loor;t

(30)

   4 θse;x;t 4 − vssky,x · θsky;t + 1 − vssky,x · θair,e;t     = σ · eL ·  θse;x;t − vssky,x · θsky;t + 1 − vssky,x · θair,e;t (27)

The heat transfer coefficients to the ground, unconditioned spaces and adjacent dwellings include a temperature factor bT correcting the driving temperature difference (Eqs. (4)–(6)). To determine these factors, θ i , θ e, θ u and θ a need to be assessed first. The interior temperature of the main dwelling, θ i , is taken equal to the surface area weighted average of the interior equivalent temperature of the walls, floor and roof. These latter are obtained from Eq. (28), with absi;x the radiation that is absorbed (or transmitted) at the interior surface of building element x (including solar gains, radiative heat, internal radiative gains and wall gains, except longwave radiation exchange with other walls [31]). In the TRNSYS model used, both the direct and diffuse radiation are distributed by absorptance weighted surface area ratios. 4 In the TRNSYS model used, the interior zone is approximated by a star network. Central to this network is an artificial temperature node denoted by ‘θ STAR ’. This node considers (1) the energy flow by convection between the interior surfaces within the zone and the zone air node and (2) the longwave radiation exchange between the interior surfaces [26,31].

θa;t = θequiv,a;partywal l ;t = θSTAR;a;t +abso;partywal l ;t · Rsa;partywal l ;t /A partywall

Rsa;partywal l ;t = A partywall ·

(34)

  θsa;partywal l ;t − θSTAR;a;t /como;partywal l ;t (35)

The above equations clearly emphasize the fact that the temperature factors, and to a lesser extent also the U-values, are time-dependent. Hence, the following three steps are taken to obtain a steady-state, calculated reference HLC: 1 Using Eqs. (24) to (35), 3-min values of θ i , θ e , θ u and θ a and the U-values of the building elements are derived from the simulation output 2 The 3-min data are resampled to the frequency at which the OBM-data will be fed into the data analysis methods (see further, Section 2.2.2), for example 72-hour averaged values. One exception is made for the ARX models: they are identified on hourly data, but their reference HLC value is derived from 96hdata, to temper the high variance of the temperature factors.

5

como and abso are the variable names used in TRNSYS.

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7

Table 2 Overview of the analysis models that are identified on the OBM data under 3 different set-ups. The first column specifies the name with which the model will be referred to, the second column indicates the type of analysis method used and the third column gives the exact model parametrization. Set-up A: Precise input for the internal gains (int ) and solar gains (sol ) ‘AvgMeth’ ‘LR_θ ie ’ ‘LR_θ i θ e ’ ‘ARX_θ i θ e ’

HLC =

Average Method

n

j=1

Simple Linear Regression Multiple Linear Regression ARX

(h;t j + int;t j + sol;t j ) /

n

j=1

(θi;t j − θe;t j )

h;t + int ;t + sol ;t = HLC · θie;t + εt ϕ (B ) · (h;t + int ;t + sol ;t ) = ωi (B ) · θi;t + ωe (B ) · θe;t + εt with nϕ =nωi =nωe =0 ϕ (B ) · (h;t + int ;t + sol ;t ) = ωi (B ) · θi;t + ωe (B ) · θe;t + εt with (nϕ +nωi +nωe )>0

Set-up B: Monitoring data of Isol available to approximate sol ‘LR_θ ie Isol ’ ‘LR_θ i θ e Isol ’ ‘LR_ratio’ ‘ARX_θ i θ e Isol ’

Multiple Linear Regression Multiple Linear Regression Simple Linear Regression ARX

h;t + int ;t = HLC · θie;t + (−gAl ) · Isol ;south;t + εt ϕ (B ) · (h;t + int ;t ) = ωi (B ) · θi;t + ωe (B ) · θe;t + ωsol (B ) · Isol ;south;t + εt with nϕ =nωi =nωe =nωsol =0 (h;t + int ;t )/θie;t = HLC + (−gAl ) · Isol ;south;t /θie;t + εt ϕ (B ) · (h;t + int ;t ) = ωi (B ) · θi;t + ωe (B ) · θe;t + ωsol (B ) · Isol ;south;t + εt with (nϕ +nωi +nωe +nωsol )>0

Set-up C: No data on internal or solar gains available ‘AvgMeth’ ‘LR_θ ie ’ ‘LR_θ i θ e ’ ‘ES’ ‘ARX_θ i θ e ’

Average Method

HLC =

n

j=1

Simple Linear Regression Multiple Linear Regression Energy Signature ARX

 e

+



(Ai · mean(Ui;t ) )

  (Ai · mean(Ui;t ) ) · mean bT,g;t

g

+



(Ai · mean(Ui;t ) ) · mean (bT,u;t )

u

+



(Ai · mean(Ui;t ) ) · mean (bT,a;t )

n

j=1

(θi;t j − θe;t j )

h;t = HLC · θie;t + εt ϕ (B ) · h;t = ωi (B ) · θi;t + ωe (B ) · θe;t + εt with nϕ =nωi =nωe =0 h;t = HLC · (θb − θe;t ) + εt ϕ (B ) · h;t = ωi (B ) · θi;t + ωe (B ) · θe;t + εt with (nϕ +nωi +nωe )>0

3 By taking the arithmetic mean of the U-values and bT -factors over the considered model training period, a steady-state, averaged HLC is deduced. This is shown in Eq. (36), with  e ,  g ,  u and  a respectively the sum over the elements of the building envelope in contact with the ambient, ground, crawl space and adjacent dwelling.

HLCre f = Hin f +

h;t j /

(36)

a

Eq. (36) implies that the calculated reference HLC varies with the considered data period and the resampling time used during the data analysis. This should be considered when evaluating the characterization results. 2.2. Physical parameter identification 2.2.1. Applied data analysis methods To evaluate the data analysis methods presented in Section 1.2 with regard to the monitoring data they need as input and the accuracy of their results, they are categorized in three set-ups (see Table 2): • Set-up A: Set-up A represents the situation in which all data are readily available: not only are temperature sensors and a heat meter installed to respectively monitor the exterior and interior climate and the dwelling’s net heat input, but also the hard-to-trace internal and solar gains are supposed to be registered. This way, all input data required by the average method (‘AvgMeth’), simple linear regression (‘LR_θ ie ’) and basic ARX model (‘ARX_θ i θ e ’) can be accurately provided.6 Consequently, these models are expected to achieve their best possible results. 6 In this simulation exercise, no intended ventilation heat losses (v ) were modeled.

In addition to the above three data analysis models, a ‘LR_θ i θ e ’ model is formulated. This model will assess the impact of splitting the variable θ ie of the ‘LR_θ ie ’ model into two separate variables θ i and θ e . This split is generally also done in ARX modeling. Since ‘LR_θ i θ e ’ forms the steady-state, ‘zero order’ equivalent of ‘ARX_θ i θ e ’ the same notation with polynomials will be used for the coefficients of the input and output variables. It must however be stressed that for ‘LR_ θ i θ e ’ the order of all polynomials (nϕ , nωi and nωe ) equals zero and hence, ϕ (B)=1, ωi (B)=ωi,0 and ωe (B)=ωe,0 . The Energy Signature model will not be applied for this set-up. As explained in Section 1.2.3, this method assesses a base temperature θ b instead of using the available interior temperature signal like the other data analysis models do. Moreover, it only relates the net heat input to the exterior temperature, without considering the internal and solar heat gains. Therefore, the ES method would not fully exploit the wide range of data available in this set-up as the other models do, and the comparison would not be meaningful. • Set-up B: In reality, monitoring data of the solar irradiation under a certain projection (Isol ) is more likely to be available than observations of the actual solar gains (sol ). Set-up B examines the consequence of this practical limitation on the estimation accuracy. Hereto, three different linear regression models (‘LR_θ ie Isol ’, ‘LR_θ i θ e Isol ’ and ‘LR_ratio’7 ) and one ARX model (‘ARX_θ i θ e Isol ’), all with Isol as input variable, are identified on the monitoring data, and the HLC estimates are compared with those obtained under set-up A. The incident solar radiation on the south façade, Isol;south [W/m²], is considered to be monitored since south is the dominant solar orientation in Belgium and the simulation model’s largest window faces south. • Set-up C: This final set-up analyzes the case in which detailed monitoring data of the internal and solar gains are missing. Instead of, for example, pro-actively selecting data points acquired at night or during cloudy periods, or using simplified information on solar and internal heat gains, as proposed by some authors in the literature 7 Name refers to the fact that in this LR parametrization both the heat flow rates and incident solar radiation are divided by θ ie .

8

M. Senave, S. Roels and G. Reynders et al. / Energy & Buildings 209 (2020) 109706 Table 3 Overview of the resampling time selected for the different set-ups, analysis models, and building types.

[15,16,21,39], set-up C investigates the extreme scenario in which both gains are merely neglected. Hence, via the two extreme set-ups A (accurate knowledge on sol or int ) and C (sol or int assumed equal to zero), it is aimed to gain deeper insight into the impact of the assumptions on sol and int on the HLC estimate. 2.2.2. Model identification and validation Data analysis is performed in R [40]. All analysis models listed in Table 2 are identified on the training data sets of BT1, 2, 3 and 4, which all range from October 1 to April 1 (26 weeks), covering the entire heating season. The data sets consist of six variables, which are selectively drawn upon depending on the considered set-up and method: h , sol , int , Isol;south , θ i and θ e. The heat flow rates and solar irradiation are direct outputs of TRNSYS, the interior and exterior temperatures are deduced from other output data as described in Section 2.1.2. The original data are calculated using 3-min sample periods. For the model identification, the resampling time applied to the original data is both case and method-specific as it is chosen in function of two model validation criteria: • The first criterion is that the model residuals ε t resemble ‘white noise’; a sequence of uncorrelated, zero mean random variables [18]. This is verified through analyzing the autocorrelation function (‘ACF’) and cumulated periodogram (‘CP’) [19]. The autocorrelation of the residuals should be insignificantly (95% confidence interval) different from zero, a condition that can be assessed visually in a plot of the ACF. The cumulated periodogram, on the other hand, is a test in the frequency domain. It verifies whether the variation of the residuals is uniformly distributed among all frequencies, a property that manifests itself as a quasi-straight line that does not exceed the 95% confidence interval (CI) in the CP plot. • The second criterion states that the model coefficients must be statistically significant. In the case of the steady-state models, this rule is imposed on all coefficients, for the ARX models it only concerns the highest lag of the output polynomial and at least one lag of each input polynomial, an approach that is recommended in the statistical guidelines of IEA Annex 58 [18]. This criterion is examined using a marginal t-test with a threshold of 0.1. However, in practice, a threshold of 0.05 seems to suffice in the majority of cases. For the steady-state models, a resampling time (‘RT’) of 12 h is taken as a starting point. It is assumed that this is a minimum resampling time needed to average out the most extreme dynamic effects in the data and to make Eq. (7) applicable. If either of the above criteria is not fulfilled at RT=12 h, the resampling time is increased to the next multiple of 12 h. This process is iterated until the model is deemed valid. In the case of the ARX models, which are able to capture a building’s dynamic behavior, a lower resampling time of 1 h is applied. Here, the RT is kept constant, whereas the model order is increased until both validation criteria are met. As prescribed in the Annex 58 guidelines [18] the order of the output polynomial (nϕ ) is set one higher than that of the input polynomials (nωi , nωe and nωsol ). The ARX models identified on the full 26-week data sets require orders of nϕ ranging between 30 and 46, depending on the considered set-up and building type. The resulting resampling times are displayed in Table 3, per model and data set. The RTs range between 72 and 108 h (respectively 3 and 4.5 days) for the average and linear regression models, with two outliers of 192 and 204 h. The ES method requires a sampling time of 108 h. No clear distinction could be found between the different building types. The fitted linear regression models are furthermore evaluated using the adjusted coefficient of determination R2adj , a variant of

analysis model ↓

data set →

Set-AvgMeth up LR_θ ie A LR_θ i θ e ARX_θ i θ e Set-LR_θ ie Isol up LR_θ i θ e Isol B LR_ratio ARX_θ i θ e Isol Set-AvgMeth up LR_θ ie C LR_θ i θ e ES ARX_θ i θ e

Resampling Time [h] BT 1 BT 2 BT 3

BT 4

72 72 108 1 96 108 96 1 96 96 96 108 1

72 72 192 1 96 108 72 1 108 108 96 108 1

72 72 204 1 / 108 96 1 108 108 108 108 1

72 72 108 1 96 108 72 1 96 96 96 108 1

the standard coefficient of determination R². Similar to R², this metric yields an indication of the proportion of the variance in the dependent variable that can be explained by the fitted model. However, it additionally adjusts for the considered sample size n and the number of independent variables in the regression equation k (Eq. (37)). Both R² and R2adj normally take on a value between zero and one, the better fits being indicated by a value closer to one.





R2adj = 1 − 1 − R2 · (n − 1 )/(n − (k + 1 ) )

(37)

2.2.3. Determination of HLC This section will briefly discuss the approach taken to deduce the HLC from the fitted models coefficients. The HLC estimate of the Average method follows directly from Eq. (9). No uncertainty band is associated with the estimate. It is nevertheless possible to judge the robustness of the model (see next section). The models ‘LR_θ ie ’, ‘LR_θ ie Isol ’, ‘LR_ratio’ and ‘ES’ are fitted by applying the function ‘lm’ (‘linear model’), which uses ordinary least squares, in R [41]. The mean HLC estimate and the associated standard deviation σ are then obtained as the model coefficient of θ ie (‘LR_θ ie ’ and ‘LR_θ ie Isol ’), the model coefficient of θ e (‘ES’), or the model intercept (‘LR_ratio’). The models ‘LR_θ i θ e ’, ‘LR_θ i θ e Isol ’, ‘ARX_θ i θ e ’, ‘ARX_θ i θ e Isol ’ are also fitted using ‘lm’, but here none of the model coefficients directly corresponds to the HLC. Instead, the HLC is inferred as a minimum variance weighted average of the parameters ‘Hi ’ and ‘He ’, which represent the ratio of the steady-state gains of respectively θ i and the sum of the heat flow rates, and θ e and the sum of the heat flow rates (Eq. (38), (39)). For models ‘LR_θ i θ e ’ and ‘LR_θ i θ e Isol ’ Hi simply equals ωi,0 and He equals ωe,0 , since these steady-state models do not include lags (nϕ =nωi =nωe =nωsol =0). The applied weighting is a Lagrange weighting that has previously been used by Jiménez, Madsen and Andersen [42] and Madsen et al. [18] in the same context. The Lagrange multiplier λ considers the variance (‘Var’) and covariance (‘Cov’) of Hi and He (Eq. (40)) and ensures that the parameter of the two with the lowest variance receives the highest weight when they are combined into a single value (Eq. (39)). The 2.5th and 97.5th percentile defining the HLC estimate’s 95% confidence interval are determined using a bootstrap simulation approach: 10,0 0 0 random samples are taken of the fitted polynomials, from which subsequently a range of the HLC is inferred [43].



ωi ( 1 ) = Hi ϕ (1 )





&

ωe ( 1 ) = He ϕ (1 )

HLC = λ · Hi + (1 − λ ) · He



(38) (39)

M. Senave, S. Roels and G. Reynders et al. / Energy & Buildings 209 (2020) 109706

λ=

9

significance levels (5% and 10%) are set.

V ar (He ) − Cov(Hi , He ) V ar (Hi ) + V ar (He ) − 2 · Cov(Hi , He )

(40)







2.3. Criteria for evaluation of the results In the results section, both (1) the quality of the HLC estimates and (2) the robustness of the models are numerically quantified and discussed. The former aspect is judged based on a comparison between the HLC estimate on the one side and the reference value that was implemented in the TRNSYS simulations (HLCref ) on the other side. Two criteria are considered [26]: C1. It is checked if HLCref is included in the estimated 95% confidence interval. It should be noted that the reference value as well as the limits of the 95% CI are rounded to two decimal places before the judgment is passed. C2. It is checked if the maximal deviation of the HLC estimate from HLCref amounts to more than 5% or 15%. These limits were set based on expert judgement. The max deviation is defined as shown in Eq. (41), with HLC2.5; w = 26 and HLC97.5; w = 26 respectively the 2.5th and 97.5th percentile of the HLC estimate for the data set of 26 weeks.



H LC(w+1) − H LCre f ;(w+1) − H LCw − H LCre f ;w H LCw − H LCre f ;w





/ (43)

Fig. 3 illustrates that the four criteria should be considered together. It depicts the HLC estimates obtained by 2 different models (upper and lower half) for the data sets of 1 week up to 26 weeks. The dashed line segments represent the reference value, which is time- and condition-dependent because of the embedded temperature factor bT (see Section 2.1.2). Fig. 3a shows an example of an HLC estimate that does not pass criterion C1. The maximal deviation of the 26-week estimate (criterion C2) is furthermore determined by the 2.5th percentile and only meets the 15% limit (11.9%). Although its estimates are not so accurate and precise, the model is rather robust. From 15 weeks onward, the range of the 95% CI does not change with more than 10%, and from 20 weeks onward, this relative deviation is even lower than 5% (criterion C3). The HLC estimate based on 24 weeks of data still approximates the reference value significantly better (11.7%) than the estimate based on 23 weeks of data (criterion C4). But, from a length of the data set of 24 weeks on, the relative deviation between the number of W/K with which the models underestimate HLCref is limited to 3.7%.

max deviation = max

     

H LC2.5;w=26 − H LCre f ;w=26 , H LC97.5;w=26 − H LCre f ;w=26 / HLCre f ;w=26

Two terms will be used in this context: ‘accurate’ and ‘precise’. An estimate is considered ‘accurate’ when the reference value is included in the estimated 95% CI. The term ‘precise’ is used to indicate that the estimated 95% CI, and hence the uncertainty on the result, is rather small. The quality assessment is performed on the estimates obtained for the full 26-week data set. It is, however, interesting to know whether the same characterization results could have been obtained with a shorter data set. Therefore, a ‘robustness test’ is conducted investigating the extent to which the model improves when more input data is provided. Twenty-five subsets were taken of each of the original 26-week data sets of BT 1, 2, 3 and 4. The length of the subsets ranges from one week (only the first week of the original data set, the first week of October) to 25 consecutive weeks (a one week shorter data set than the original data set). The same resampling time is applied to these data sets as to their original 26-week data set. Hence, in the case of a 108-hour resampling time, the 1-week data set for example only includes one averaged observation per variable. The above-described characterization exercise is repeated on these additional subsets, resulting in 26 estimates – each based on one extra week of data – per method and building type. These values are compared to assess: C3. The length of the data set (expressed in weeks) from which point on adding another week of data does not result in a relative change of the range of the 95% CI of more than 5% or 10%. The change of the range of the 95% CI is determined in accordance with Eq. (42), with HLC2.5;w the 2.5th percentile of the HLC estimate identified on the data set of w weeks.







H LC97.5;(w+1) − H LC2.5;(w+1) − (H LC97.5;w − H LC2.5;w ) /

(H LC97.5;w − H LC2.5;w )

(42)

C4. The length of the data set (expressed in weeks) from which point on the mean estimates appear to converge. Since the reference HLC value itself is subject to change, this criterion is evaluated based on the absolute difference between the estimate and HLCref , as demonstrated in Eq. (43). Again, two

(41)

By contrast, the second model (Fig. 3b) scores well on criteria C1 and C2. The reference value is included in the 95% CI of the 26-week estimate (C1) and the estimate’s maximal deviation is only 4.5% (C2). But, although the estimates based on the shorter data sets also exhibit small maximal deviations, their relative deviations (criterion C4) are rather large. The same holds for the relative changes between the ranges of the estimated 95% CIs (criterion C3). Hence, the second model exceeds the 10% limit for criterion C3 and only meets the 10% limit for criterion C4 at 25 weeks. It will therefore not be considered robust. It is important to note that of the 4 criteria used, criterion C3 is the only criterion which does not require knowledge of the reference HLC value. In this study, the reference HLC value of the building simulation models is known, but this is in general not the case for real-life dwellings. 3. Analysis and discussion The analysis and discussion has been organized in four parts. The first part reports on the reference HLC values calculated according to the procedure outlined in Section 2.1.2. The subsequent three parts present and interpret the HLC estimates obtained under set-up A, B and C. 3.1. Calculated reference values Table 4 gives an overview of the reference values obtained for the heat transfer coefficients, temperature factors and HLC (HLCref ) for the full heating season. The parameters are listed per simulated building type, and are presented as the mean value and standard deviation of the reference values for the different resampling times (Table 3) applied on the data sets. As can be seen, the effect of the different resampling times is only reflected in the HLC itself; for its constituents (Htr , Hinf , bT ) the standard deviation is almost zero (smaller than 0.01). The temperature factors show that both θ g and θ u are on average higher than θ e , whereas θ a is on average higher than θ i .

10

M. Senave, S. Roels and G. Reynders et al. / Energy & Buildings 209 (2020) 109706

Fig. 3. Example of the HLC estimates identified by (a) model LR_θ ie and (b) model ARX_θ i θ e on a data set with a length of 1 week up to 26 weeks for case BT2. The dots indicate the mean estimates, the whiskers the 95% confidence intervals. The dashed line indicates the reference value. The above-discussed evaluation criteria (C2, C3 and C4) are illustrated. Table 4 Reference values for Htr , Hinf , bT and HLC, as determined for the full 26-week period. Listed below are the mean (± standard deviation) of the reference values obtained for the different resampling times applied on the data sets (see Table 3, with the remark that for the ARX models the reference value is derived from 96h-data, see 2.1.2). Building Types BT1: Crawl space

BT2: Ground

BT3: Crawl space + Neighbor

Htr,e [W/K] Htr,g / bT,g [W/K] Htr,u / bT,u [W/K] Htr,a / bT,a [W/K] Hinf [W/K]

41.58 (± <0.01) – 84.43 (± <0.01) – 12.50 (± <0.01)

41.58 (± <0.01) 106.58 (± <0.01) – – 12.50 (± <0.01)

33.91 – 84.90 47.28 12.50

bT,g [-] bT,u [-] bT,a [-]

/ 0.36 (± <0.01) /

0.23 (± <0.01) / /

/ 0.31 (± <0.01) −0.17 (± <0.01)

0.18 (± <0.01) / −0.16 (± <0.01)

HLC [W/K]

84.06 (±0.17)

78.53 (±0.47)

64.50 (±0.08)

57.86 (±0.28)

3.2. Set-up A: Precise input for int and sol Fig. 4 gives an overview of the characterization results obtained by the models of Set-up A for the 26-week period. The results are presented as a mean estimate (dot) with a 95% confidence interval (whiskers), and are compared to the reference values, which in

(± <0.01) (± <0.01) (± <0.01) (± <0.01)

BT4: Ground + Neighbor 33.92 (± <0.01) 106.58 (± <0.01) – 47.32 (± <0.01) 12.50 (± <0.01)

turn depend on the resampling time. Numerical details on the quality of the estimates and the robustness of the models are summarized in Table 5. For each of the building types, Fig. 4 (and Table 5) show similar trends in terms of accuracy and precision of the estimated heat loss coefficients compared to the theoretical values.

M. Senave, S. Roels and G. Reynders et al. / Energy & Buildings 209 (2020) 109706

11

Fig. 4. HLC estimates obtained using the 4 different data analysis methods from set-up A on the 26-week data sets of the 4 building types. The dots indicate the mean estimates, the whiskers the 95% confidence intervals. The dashed line segments represent the calculated reference values. Table 5 Qualitative evaluation of the estimates and models of set-up A using the criteria explained in Section 2.3. The ‘-‘ or ‘/’ signs respectively indicate that the test could not be conducted (AvgMeth) or that the criterion was not fulfilled before the end of the 26-week training period. In the case of the robustness tests, the number indicates the length of the data set (in weeks) from which point on the limit is met. Quality final estimate in CI?

Robustness model

max deviation

Change range CI / added week

Change estimate / added week

<5%

<15%

<10%

<5%

<10%

<5%

BT 1

AvgMeth LR_θ ie LR_θ i θ e ARX_θ i θ e

– Yes Yes Yes

Yes No No Yes

Yes Yes Yes Yes

– 15 11 22

– 20 20 /

24 24 24 /

/ 24 24 /

BT 2

AvgMeth LR_θ ie LR_θ i θ e ARX_θ i θ e

– No Yes Yes

Yes No No Yes

Yes Yes Yes Yes

– 15 22 /

– 20 25 /

24 24 22 25

24 24 / /

BT 3

AvgMeth LR_θ ie LR_θ i θ e ARX_θ i θ e

– Yes Yes Yes

Yes No No Yes

Yes Yes Yes Yes

– 15 11 /

– 20 20 /

24 24 24 /

/ 24 24 /

BT 4

AvgMeth LR_θ ie LR_θ i θ e ARX_θ i θ e

– Yes Yes Yes

Yes No No Yes

Yes Yes Yes Yes

– 15 15 /

– 20 24 /

24 24 24 25

24 24 / /

The results of the two linear regression models and the ARX model are accurate, in the sense that the reference value is included in the estimated 95% CI for each building type (except for LR_θ ie in BT2). The ARX model is the most precise of the three, with a maximal deviation from HLCref of up to 4.5% (comparison made between the 4 BTs). LR_θ i θ e follows next, with a max deviation of up to 8.5%. The maximal deviation of the LR_θ ie model is a little higher, and varies around 10.4%.

The results in Fig. 4 were obtained for a data set of 26 weeks. However, Table 5 shows that a training period of 15 weeks usually suffices to obtain a LR model that is rather robust with regard to the uncertainty of the estimate: from this number of weeks onward, the range of the 95% CI does not change with more than 10% when an additional week of data is added. This limit decreases to 5% from a period of 20 weeks onward. The mean estimates, on the other hand, do not seem to converge up until

12

M. Senave, S. Roels and G. Reynders et al. / Energy & Buildings 209 (2020) 109706

Fig. 5. HLC estimates (top) and linear regression fit (bottom) identified by model LR_θ ie on the data set of BT1 resampled to respectively 72-hourly, 144-hourly and 204-hourly values. The dashed line indicates the reference value of the HLC.

Fig. 6. HLC estimates obtained using the 4 different data analysis methods from set-up B on the 26-week data sets of the 4 building types. The dots indicate the mean estimates, the whiskers the 95% confidence intervals. The dashed line segments represent the calculated reference values. Note that no valid model could be obtained for LR_θ ie Isol on data of BT2.

a period of 24 weeks. The ARX model does not score well on the robustness tests. The high precision of its estimates (e.g. range of 95% CI is 3.4 W/K for 26-week estimate BT1) and the small deviation between its mean estimates and the reference value (e.g. 0.4 W/K for 26-week estimate BT1) make it highly unstable, compared to the other models. The Avg.Meth. does not assess the uncertainty of its estimate. The deviation of the mean estimates from HLCref only ranges between 1.0% and 2.7%. Linear regression methods are commonly evaluated based on their score for R2 or R2adj . A sufficiently high score for this goodness-of-fit measure could be assumed a necessary condition for a proper model structure and HLC estimate. Fig. 5 challenges this. It displays the HLC estimates and regression fits obtained by

identifying model LR_θ ie on the data set of BT1 resampled to 3 different frequencies. The model fitted at the 72h-values satisfies both validation criteria (residuals resemble white noise and model coefficients are significant, see Section 2.2.2), and its HLC estimate was hence accepted (see also Fig. 4). However, the fit only receives an R2adj score of 0.2, signaling a very weak predictive power. The resampling time needs to be increased to 8.5 days before a value of 0.7 is obtained. Nevertheless, the HLC result based on 72h-values already embedded the reference value and the mean estimate only improves with 2.5% towards the 204h-result. In summary, the results indicate that, given the same 3-min data set, the ARX models are able to infer the most accurate and precise HLC estimate for this reference set-up. Nevertheless,

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13

Table 6 Qualitative evaluation of the estimates and models of set-up B using the criteria explained in Section 2.3. Note that no valid model could be obtained for LR_θ ie Isol on data of BT2 (‘-‘). The ‘/’ sign indicates that the criterion was not fulfilled before the end of the 26-week training period. In the case of the robustness tests, the number indicates the length of the data set (in weeks) from which point on the limit is met. Quality final estimate in CI?

max deviation

Robustness model Change range CI / added week

Change estimate / added week

<5%

<15%

<10%

<5%

<10%

<5%

LR_θ ie Isol BT LR_θ i θ e Isol 1 LR_ratio ARX_θ i θ e Isol

No No Yes Yes

No No No No

No No Yes Yes

15 11 10 25

15 / 20 25

/ / / /

/ / / /

LR_θ ie Isol BT LR_θ i θ e Isol 2 LR_ratio ARX_θ i θ e Isol

– No Yes Yes

– No No No

– No No Yes

– 18 10 22

– 18 20 /

– / / /

– / / /

LR_θ ie Isol BT LR_θ i θ e Isol 3 LR_ratio ARX_θ i θ e Isol

No No Yes Yes

No No No No

No No No Yes

15 11 10 /

15 18 15 /

/ / / /

/ / / /

BT LR_θ ie Isol 4 LR_θ i θ e Isol LR_ratio ARX_θ i θ e Isol

No No No No

No No No No

No No No Yes

15 18 10 /

15 18 15 /

/ / / 25

/ / / /

these models score relatively bad on the model validation criteria regarding robustness (C3 and C4). In addition, criterion C3 is the only criterion which can be tested in real life cases where the reference HLC value is unknown. Among the steady-state models, which require a higher resampling time, preference should be given to model LR_θ i θ e , which for this case guarantees that the reference value is included in the estimated 95% CI and has a reasonable maximal deviation. 3.3. Set-up B: Monitoring data of Isol available to approximate sol As can be expected, forcing the models to approximate the solar gains based on a constant, lumped gA-value and the monitored irradiation on the south façade results in a loss of accuracy. Fig. 6 shows how the mean estimated HLCs of LR_θ i θ e Isol , LR_θ ie Isol , LR_ratio and ARX_θ i θ e Isol differ with 13.3%, 10.7%, 7.4% and 3.7%, respectively, from the reference HLC (average taken over the four building types). Comparing the estimates of models LR_θ ie Isol , LR_θ i θ e Isol and ARX_θ i θ e Isol and their equivalents under set-up A (LR_θ ie , LR_θ i θ e and ARX_θ i θ e ), it can be seen that the deviation from HLCref on average increased with 6.1%, 10.5% and 3.1%, respectively. The characterization moreover yields a higher uncertainty, manifested by an on average 1.8 times larger range of the 95% CI (taking all building types and the three equivalent methods into account). Table 6 shows how the max deviation of the results is never within the 5% limit, and even the 15% limit is exceeded in 10 out of the 15 cases. LR_ratio is the only linear regression model that contains the reference value in its 95% CI. However, the estimates of this model consistently have the highest uncertainty (up to 15.4 W/K, which equals 19.1% of the mean estimate). In general, the results for the detached dwellings (BT1 and 2) and those with the suspended floor (BT1 and 3) are more uncertain. It is apparent from Table 6 that a longer monitoring period results in a more precise result for the linear steady-state models: the range of the 95% CI does not alter with more than 10% from 10 to 18 weeks of data on, and the relative deviation is limited to 5% for 15 to 20 weeks of data. Deviations of the mean estimate of more than 10% occur up until the addition of the 26th week. Again, the ARX model does not pass the robustness tests.

3.4. Set-up C: No data on internal or solar gains available This final set-up demonstrates the impact of either forgetting to take account of the internal and solar gains during data analysis, or assuming it to be safe to neglect them when no monitoring data is available. Fig. 7 illustrates how this approach results in a significant underestimation of the heat loss coefficients for all modelling approaches. The maximal deviation of the estimates from the reference value is for all cases greater than 15%, hence the quality assessment part is omitted in Table 7. Obtained deviations amount up to 28.4% and 28.9% for the Average and ARX model respectively, 32.7% and 32.4% for LR_θ ie and LR_θ i θ e and 53.7% for the AvgMeth. Regardless of their lack of estimation accuracy, the models can be considered robust, both with respect to the mean estimate and the confidence interval. Table 7 illustrates that a training period of less than 20 weeks would have resulted in nearly the same results. Comparing the modeling approaches, it can be seen that the mean estimates of the AvgMeth, LR models and ARX models are fairly consistent. Although the ES models equally meet the validation criteria set in Section 2.2.2, the results of these models are in general more inaccurate and unprecise: the 97.5th percentiles (upper whiskers) underestimate HLCref with minimally 16.1% and the uncertainty on the estimates is moreover quite large with ranges of the 95% CI amounting up to 21.4 W/K. The robustness tests (Table 7) do not indicate that the models can be significantly improved by using a larger training period. Hence, the cause of the deviations must rather be sought in the model structure itself. Since the ES model does not utilize data on sol or int , even when it is available (see model parametrization Table 2), it was not included in set-up A. As its estimates align more closely with the estimates of the AvgMeth, LR and ARX for set-up C than for set-up A, it indeed appears that the model, which focuses on determining a base temperature, is not capable of properly accounting for the solar and internal gains. The obtained mean estimates for the base temperature range between 18.1 °C and 20.3 °C, depending on the building type considered. Compared to the other data analysis models in set-up C, the ES method moreover does not use the available interior temperature signal, which may explain the higher uncertainty of the estimates. It should furthermore be

14

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Fig. 7. HLC estimates obtained using the 5 different data analysis methods from set-up C on the 26-week data sets of the 4 building types. The dots indicate the mean estimates, the whiskers the 95% confidence intervals. The dashed line segments represent the calculated reference values. Table 7 Evaluation of the robustness of the models of set-up C using the criteria explained in Section 2.3. The ‘-‘ or ‘/’ signs respectively indicate that the test could not be conducted (AvgMeth) or that the criterion was not fulfilled before the end of the 26-week training period. In the case of the robustness tests, the number indicates the length of the data set (in weeks) from which point on the limit is met. Robustness model

Robustness model

Change range CI / week

Change estimate / week

Change range CI / week

Change estimate / week

<10%

<5%

<10%

<5%

<10%

<5%

<10%

<5%

AvgMeth LR_θ ie BT LR_θ i θ e 1 ES ARX_θ i θ e

– 15 15 16 12

– 23 23 20 /

6 11 11 11 6

16 16 16 25 22

AvgMeth LR_θ ie BT LR_θ i θ e 3 ES ARX_θ i θ e

– 15 6 16 12

– 23 20 20 20

6 10 10 19 10

16 16 16 / 13

AvgMeth LR_θ ie BT LR_θ i θ e 2 ES ARX_θ i θ e

– 9 9 18 10

– 20 24 20 21

6 11 11 11 10

11 11 11 / 22

AvgMeth LR_θ ie BT LR_θ i θ e 4 ES ARX_θ i θ e

– 8 15 18 15

– 20 23 20 21

6 6 10 19 10

11 11 16 / 13

noted that the estimates of the ES method for the semi-detached dwellings (BT3 and BT4) are in better agreement with those of the other models. For the ES method, the absolute difference between its estimates for the detached and semi-detached dwellings (BT1 and 3, and BT2 and 4) is also smaller than for the other methods. These observations seem to imply that there is an issue related to the constant intercept term in the ES model parametrization that does not occur for semi-detached dwellings, where the transmission heat transfer to adjacent dwellings is non-zero. Further research should be undertaken to gain a deeper understanding. Looking at the results from the point of view of the building types, it can be seen that the AvgMeth, LR and ARX models on average underestimate the HLCref of BT1, BT2, BT3 and BT4 with 24.0%, 26.4%, 27.4% and 30.6%, respectively (max deviation). These figures correspond more or less with the ratio

( (int;t + sol;t )/(int;t + sol;t + h;t )) which on average equals 21.8%, 23.1%, 26.8% and 29.1% for building types 1 to 4, respectively.8 3.5. Summary In conclusion, we observe that when 26 weeks of accurate data on the net heating input, interior and exterior temperature and the internal and solar gains is available, both the linear regression models and ARX model are capable of accurately identifying the heat loss coefficient. For applications demanding a higher accuracy, 8 Considering the 26-week data sets, resampled to 96h-values for the cases with the suspended floor (BT1 and 3) and 108h-values for the cases with the slab-onground floor (BT2 and 4), as determined in Table 3.

M. Senave, S. Roels and G. Reynders et al. / Energy & Buildings 209 (2020) 109706

the ARX method is recommended as its maximal deviation from the reference value does not exceed 5%. The simplest data analysis technique, the Average method, scores surprisingly well (max error of 2.7%), yet it does not provide a confidence interval. The error and uncertainty induced in the model by approximating the solar gains by a constant multiplied with the solar irradiation under one projection inevitably propagates to the HLC estimate. The max deviation of the estimates from the reference value (considering all 4 models and BTs) amounts up to 24.5% (compared to 11.9% for set-up A), the linear regression models LR_θ ie Isol and LR_θ i θ e Isol cannot accurately assess the HLC and the range of the 95% CI on average nearly doubles compared to set-up A. Again, the ARX model seems the best option to exploit the available data to the fullest. Robustness tests indicate that from about 20 weeks on the uncertainty of the LR estimates does not further reduce with more than 5%. Neglecting solar and internal gains, even the ARX model is proven unable to infer a reasonable HLC estimate. Its minimal deviation ranges from 15.8% to 22.4% for the 4 building types. The Energy Signature method is not preferred in case monitoring data of the interior temperature is available. No clear dependency was found between the considered building typology and the applicability of the analysis methods. 4. Conclusion This paper presents a theoretical exercise exploring the suitability of different data analysis methods to identify the heat loss coefficient (HLC) of the building envelope from on-board monitoring (OBM) data. Using building energy simulations, synthetic OBM data sets are generated for four different building types. By taking subsets of these original data sets and applying the Average method, linear regression analysis, the Energy Signature method and ARX modelling to them, the sensitivity of the HLC estimate to (1) the availability of specific input data and (2) the applied data analysis method is assessed. First, the paper establishes that the parameter of interest, the HLC, includes a time-varying temperature factor, making it dependent on the monitoring period and the applied (re)sampling time. Next, both the steady-state linear regression and the dynamic ARX method are shown capable of accurately assessing the HLC from data sets including 26 weeks of monitoring data of all variables present in the heat balance equation. Whereas the first analysis method requires a resampling time of 72 to 204 h in order to satisfy the statistical validation criteria, the ARX model can be fitted on hourly values and moreover obtains the most precise estimate (maximal deviation of only 4.5%). Robustness tests indicate that from a data set of 15 weeks onward, the uncertainty on the linear regression estimates usually does not decrease with more than 10% when another week of data is added. The mean estimates on the other hand, tend to show more fluctuations. Using the Average method, the HLC is inferred within 2.7%. Since monitoring the actual solar gains for all windows is often not feasible in practice, four alternative model parametrizations are put forward. The approximation error associated with solely using the incident solar radiation on the south façade however, has a negative impact on the HLC estimates, with underestimations up to 24.5%, and an on average 1.8 times larger 95% confidence interval. For this set-up, applying an ARX model appears to be the only guarantee that the HLC estimate is within 15% of the actual reference value. As could be expected, all investigated models significantly underestimate the reference HLC, when the characterization is solely based on the net heating input and interior and exterior temperature, thus ignoring the harder-to-trace internal and solar gains.

15

By performing the exercise in parallel on four building typologies and comparing the results, no clear dependency could be determined between the characterization accuracy and the considered building type. Based on these results, it seems essential to explore how to monitor and model the solar and internal heat gains in a practical and reliable way. Future research should furthermore broaden the scope of this preliminary exercise, by examining a larger set of buildings and analyzing the sensitivity of the HLC estimate to other input variables. For example, an in-depth study is required on the influence of the interior climate in the examined building and its adjacent buildings on the characterization process. The range of investigated data analysis methods could also be extended, for instance with stochastic state space models. Declaration of Competing Interest We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). He/she is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author. Acknowledgement The authors gratefully acknowledge the Research Foundation Flanders (FWO) and the Flemish Institute for Technology (VITO) for funding this research (application number 1131918N). References [1] G. Bauwens, In situ testing of a building’s overall heat loss coefficient PhD thesis, KU Leuven, Belgium, 2015. [2] R. Jack, D. Loveday, D. Allinson, K. Lomas, First evidence for the reliability of building co-heating tests, Build. Res. Inf. (2017), doi:10.1080/09613218.2017. 1299523. [3] S. Roels, P. Bacher, G. Bauwens, S. Castaño, M.J. Jiménez, H. Madsen, On site characterisation of the overall heat loss coefficient: comparison of different assessment methods by a blind validation exercise on a round robin test box, Energy Build. 153 (2017) 179–189, doi:10.1016/J.ENBUILD.2017.08.006. [4] F. Alzetto, D. Farmer, R. Fitton, T. Hughes, W. Swan, Comparison of whole house heat loss test methods under controlled conditions in six distinct retrofit scenarios, Energy Build. 168 (2018) 35–41, doi:10.1016/J.ENBUILD.2018.03.024. [5] S. Thébault, R. Bouchié, Refinement of the isabele method regarding uncertainty quantification and thermal dynamics modelling, Energy Build. 178 (2018) 182–205, doi:10.1016/J.ENBUILD.2018.08.047. [6] P. de Wilde, The gap between predicted and measured energy performance of buildings: a framework for investigation, Autom. Constr. 41 (2014) 40–49, doi:10.1016/J.AUTCON.2014.02.009. [7] J. Wingfield, M. Bell, D. Miles-Shenton, T. South, B. Lowe, Evaluating the impact of an enhanced energy performance standard on load-bearing masonry domestic construction - Understanding the gap between designed and real performance: lessons from stamford brook, London, 2011.

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