(1) The document describes using ARX models to identify physical parameters for building energy performance assessment based on in-situ measurements. (2) It finds that the best model uses nighttime indoor temperature, outdoor temperature, and heating power data to estimate the building's heat loss coefficient. (3) Applying the model to different occupancy periods, it estimates the heat loss coefficient as 63.2 W/K with a standard deviation of 5.1 W/K.

1. DTU Compute, Technical University of Denmark 07 August 2015
ARX MODELS FOR BUILDING ENERGY
PERFORMANCE ASSESSMENT BASED ON
IN-SITU MEASUREMENTS
IEA EBC Annex 71
Common Exercise 1, Subtask 3
Physical Parameter Identification
Christoffer Rasmussen (chrras@dtu.dk), Peder Bacher, Henrik Madsen

2. DTU Compute, Technical University of Denmark 07 August 2015
MODEL
ARX model as used in Annex 58, Subtask 3, Part 2
• Output: Indoor temperature
• Input: Heating power
Outdoor temperature
Solar radiation
Annex 71, CE1 ST3 – October 23-25, 20172
𝜙 𝐵 𝑇$
%
= 𝜔( 𝐵 Φ$
(
+ 𝜔+ 𝐵 𝑇$
+
+ 𝜔,-. 𝐵 𝑄$
,-.
+ 𝜖$

3. DTU Compute, Technical University of Denmark 07 August 2015
DATA REQUIREMENTS FOR THE GOOD FIT
High signal-to-noise ratio
– Daily mean outdoor temperature below 5 ºC
Minimal disturbances in data from occupants
– Out-of-the-house vacation:
• No disturbances
• Limited periods throughout the year.
– Out-of-the-house weekdays:
• No disturbances
• Only applicable if occupants actually leave the house regularly for several hours
– Night-time:
• Few disturbances
• New data accessible every night (normally)
Annex 71, CE1 ST3 – October 23-25, 20173

4. DTU Compute, Technical University of Denmark 07 August 2015
MODEL
ARX model as used in Annex 58, Subtask 3, Part 2
ARX model as used in Annex 71, Common Exercise 1, Subtask 3
• Output: Indoor temperature
• Input: Heating power
Outdoor temperature
• Data: Only night-time
• Time resolution: 10 minutes
Annex 71, CE1 ST3 – October 23-25, 20174
𝜙 𝐵 𝑇$
%
= 𝜔( 𝐵 Φ$
(
+ 𝜔+ 𝐵 𝑇$
+
+ 𝜔,-. 𝐵 𝑄$
,-.
+ 𝜖$
𝜙 𝐵 𝑇$
%
= 𝜔( 𝐵 Φ$
(
+ 𝜔+ 𝐵 𝑇$
+
+ 𝜖$

5. DTU Compute, Technical University of Denmark 07 August 2015
DOWN-WEIGHTING DAY-TIME DATA
Annex 71, CE1 ST3 – October 23-25, 20175
051015
Heat consumption, day and night
Energy[kW]
18:00 00:00 06:00 12:00 18:00 00:00 06:00
Day (weight = 0)
Night (weight = 1)

6. DTU Compute, Technical University of Denmark 07 August 2015
ACF AND CCF ISSUES
N Time Residual
⋮ ⋮ ⋮
35 2014/01/01 06:50 +0.05
36 2014/01/01 07:00 –0.02
37 2014/01/01 19:30 –0.01
38 2014/01/01 19:40 +0.04
⋮ ⋮ ⋮
Annex 71, CE1 ST3 – October 23-25, 20176
𝛾33 𝜏 = Cov 𝑋 𝑡 , 𝑋 𝑡 + 𝜏
𝜌33 𝜏 =
𝛾33 𝜏
𝜎3
=

7. DTU Compute, Technical University of Denmark 07 August 2015
SEPARATING SPACE HEATING & DHW
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Summer data
Water consumption [m3
]
Heat[kW]
Hot water and space heating
Hot water
Space heating
0510152025
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
All data
Water consumption [m3
]
0510152025 Annex 71, CE1 ST3 – October 23-25, 20177

8. DTU Compute, Technical University of Denmark 07 August 2015
SEPARATING SPACE HEATING & DHW
Annex 71, CE1 ST3 – October 23-25, 20178
0515
Energy[kW]
00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 24:00
Hot water production
Space heating
0.000.020.04
Waterconsump.[m3
]
00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 24:00
20.021.5
Indoortemp.[°C]
00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 24:00

9. DTU Compute, Technical University of Denmark 07 August 2015
CASE
DIFFERENT OCCUPATION PERIODS
Annex 71, CE1 ST3 – October 23-25, 201712
Different occupation periods
Sample period [nights]
HLC[W/K]
335075100125150
1 3 5 7 9 11 13 15 17
Estimate (occupation period 2)
95 % CI
Assumed 83.4 (±16.3) W/K

10. DTU Compute, Technical University of Denmark 07 August 2015
CASE
DIFFERENT OCCUPATION PERIODS
Annex 71, CE1 ST3 – October 23-25, 201713
Different occupation periods
Sample period [nights]
HLC[W/K]
335075100125150
1 3 5 7 9 11 13 15 17
Estimate (occupation period 1)
Estimate (occupation period 2)
95 % CI
Assumed 83.4 (±16.3) W/K
63.2 (±5.1) W/K

11. DTU Compute, Technical University of Denmark 07 August 2015
RESIDUAL ANALYSIS
Annex 71, CE1 ST3 – October 23-25, 201714
0 10 20 30 40
−0.050.05
ACF (Residuals)
ACF
0 10 20 30 40
−0.060.04
CCF (Residuals & Outdoor temperature)
CCF
0 10 20 30 40
−0.060.04
CCF (Residuals & Heating power)
Lags
CCF

12. DTU Compute, Technical University of Denmark 07 August 2015
RESIDUAL ANALYSIS
Annex 71, CE1 ST3 – October 23-25, 201715
20.521.522.523.5
One−step predictions
Indoortemp.(°C)
Measured
Predicted
−0.050.000.05
Residuals
ε
Night 1 Night 2 Night 3 Night 4 Night 5 Night 6 Night 7 Night 8 Night 9 Night 10 Night 11 Night 12 Night 13 Night 14 Night 15 Night 16

13. DTU Compute, Technical University of Denmark 07 August 2015
RESIDUAL ANALYSIS
Annex 71, CE1 ST3 – October 23-25, 201716
0.0 0.1 0.2 0.3 0.4 0.5
0.00.20.40.60.81.0
Frequency
Cumulative periodogram
−3 −2 −1 0 1 2 3
−0.050.000.05
Q−Q plot
Theoretical quantiles
Samplequantiles
Histogram
Sample quantiles
Density
−0.10 −0.05 0.00 0.05 0.10
051015

14. DTU Compute, Technical University of Denmark 07 August 2015
CONCLUSIONS
Data
– Obtain good time series where input and output is well-exercised.
– Prober filtering of domestic hot water consumption.
– Get representative indoor temperatures.
– Model internal heat gains from occupants. E.g. through CO₂ measurements.
Assumed and estimated HLC
– Air leakages.
– Thermal bridges.
– All heat is assumed to be transferred to the air.
– Heat capacity of building not taken into account
Annex 71, CE1 ST3 – October 23-25, 201717

15. DTU Compute, Technical University of Denmark 07 August 2015
PERSPECTIVE
Evaluation of building performance and occupants effect on energy consumption
1. Apply occupant models to estimate if apartment is empty or not.
2. Use periods where apartment is un-occupied and of occupants are sleeping, instead of
night-time only, to include solar radiation in model.
3. Use fitted model on all data and estimate the effect of the occupants on the energy
consumption.
Annex 71, CE1 ST3 – October 23-25, 201718

16. DTU Compute, Technical University of Denmark 07 August 2015
MODEL MATRIX
Annex 71, CE1 ST3 – October 23-25, 201719
HLC estimate and
standard deviation
obtained from best
model:
63.2 (±5.1) W/K
10 min data of:
Indoor temperature
Outdoor temperature
Heating power

17. DTU Compute, Technical University of Denmark 07 August 2015
MODEL MATRIX
Annex 71, CE1 ST3 – October 23-25, 201720
HLC estimate and
standard deviation
obtained from best
model:
63.2 (±5.1) W/K
10 min data of:
Indoor temperature
Outdoor temperature
Heating power