Towards the characterization of the heat loss coefficient via on-board monitoring: Physical interpretation of ARX model coefficients

Abstract

This paper explores the concept of characterizing the as-built Heat Loss Coefficient (HLC) of buildings based on-board monitoring (OBM), via energy consumption and temperature sensors, and time series analysis. It is examined (1) how the coefficients of different Auto-Regressive with eXogenous inputs (ARX) models can be interpreted and (2) whether these conclusions are sensitive to the building envelope assembly or the applied indoor temperature profile. The paper presents a theoretical case study whereby detailed building energy simulations are used to accurately map the impact of physical phenomena on the characterization process. The simulation models and boundary conditions are composed to focus on the link between the estimated ARX-coefficients and the physical driving forces for transmission heat loss to the ground and the exterior environment. The results show how the various ARX model coefficients are linked to specific components of the HLC (e.g. heat transfer through the walls and roof or through the slab-on-ground floor) and to what extent they are affected by the selection of input variables. By monitoring the ground temperature, the transmission heat losses can rather accurately be assigned to either the slab-on-ground or the walls and roof. Without this measurement data, the uncertainty on the estimates increases (ranges of the 95% confidence interval of up to 35% of the mean estimate). Modeling the ground heat losses by a constant intercept term leads to underestimations of the reference HLC of up to 59%, whereas adding heat flux sensors to monitor the transmission heat losses to the ground to the measurement setup allows to assess the transmission heat transfer coefficient to the exterior environment HLC e within 2%.

Sensitivity of Characterizing the Heat Loss Coefficient through On-Board Monitoring: A Case Study Analysis

“…In the case of the latter two the temperature ratios b T are also taken into account, as shown in Equations (25) and (26) with j the rooms where the temperature was monitored (see Figure 2) and k the building components separating these room interiors from the ground, cellar, attic, or exterior environment. In reality these ratios are time dependent (see further, Section 4.2), but for the calculations in Equations (25) and (26) the constant values listed in Table 1 are used.…”

“…By 2020 the European Commission expects 72% of all European consumers to have a smart meter for electricity, while about 40% will have a smart meter for gas [21]. Hence, current research [22][23][24][25][26][27][28] investigates whether these smart meters, optionally combined with sensors from building automation systems, could open the way for a more practical and cost-effective approach for as-built HLC characterization. However, an assessment based on monitoring data of the energy consumption and indoor climate of in-use buildings, which will further be referred to as 'on-board monitoring' (OBM) data, faces major challenges.…”

“…However, what is particularly striking is the estimate's large 95% CI (range of 214.6 W/K). Senave et al [26] analyzed the capability of ARX models for HLC characterization, using synthetic OBM data generated via simplified simulation models. They showed that, when transmission heat losses to the ground are not explicitly modeled, as is the case here, setting the intercept term in the ARX model structure equal to zero gives more precise and accurate HLC estimates.…”

“…To test the ARX models for overfitting, the models are challenged to predict the interior temperature (or Φ h in case of package 1) for 2 weeks in both the training and validation period, using one-step ahead prediction [26]. The normalized RMSE (nRMSE (%)) [26] between the predicted and measured interior temperature is then compared for both periods to verify if they are of the same order of magnitude. In this case, the maximum difference between the nRMSEs proves to be 32.7%, which was considered to be acceptable.…”

“…In the case of the LR models, the coefficient of the temperature difference term represents the HLC, for the ES model, this is the coefficient of θ e . Finally, for the ARX models, the HLC estimate needs to be derived from the steady-state gains using Lagrange weighting, as shown in Equation (27) for model Equation (19) and Equation (28) for model Equations (20) and (21) [26,39,44,51]. The steady-state gains of the variables are their polynomials with the backshift operator B set equal to 1 (e.g., ω e (1)).…”