1. Lumped parameter models for energy auditing of existing buildings
Mariangela BENEDETTELLI, Berardo NATICCHIA, Luigi RIDOLFI, Massimo VACCARINI
DICEA Department, Univerisità Politecnica delle Marche, Ancona, Italy,

2. SYNOPTIC OUTLINE OF THE CONTENTS
Context
Energy Audit
Less
demanding
process
Reduced-order modelling
To derive
reduce order
models
Grey-box modelling
Experimental application
Case study
Testing
Results
Conclusions
Future developments

3. defined as DETAILED AUDIT
ENERGY AUDIT
An Energy Audit is a basic and systematic tool to support the decision-making in the area of energy management of an existing building in order to promote energy saving opportunities, considered as the most appropriate ones through a cost/benefits analysis, but also related to comfort and living conditions.
a substantial time to set the data and provide the detailed dynamic model of the audited object;
a considerable computational effort is necessary to run the large number of simulations;
The «investment-grade» Level-3 ASHRAE audit,
defined as DETAILED AUDIT
it does not guarantee an accurate prediction of the energy performance, due to the uncertainty in model inputs, especially concerning existing buildings.

4. REDUCED-ORDER MODELLING
The usefulness of the Reduced-order Modelling to do assessments in the first stage of the energy audit process.
DEVELOPMENT OF A PROCEDURE FROM SURVEYS,
IN CASE THERE IS NO ENERGY MODEL AVAILABLE FOR THE AUDITED OBJECT
Fields of construction
Reliability and usefulness of reduced-order models in:
simulations, advanced controls, real-time predictions, and energy management strategies.
Structural damages
Retrofit at city scale
Safety
Occupancy dynamics
Facility
management
Application of Reduced-order Models

5. Approaches to derive Reduced Models
SUGGESTED BY LITERATURE…
As defined in literature, Grey-box Models are the result of a combination of:
a prior knowledge of the system dynamics
and
statistical methods
For the structure
For the estimation of the unknown parameters
Effective to achieve a suitable description of thermal response of buildings in a short time

6. GREY-BOX MODELLING
A dynamical system is characterized by its state variables. The state-space representation provides a mathematical description to analyse dynamical systems with multiple inputs and outputs.
Regarding grey-box models, the state-space form derives from the physical laws, which are formulated by these equations:
𝒅𝑿 𝒕 = 𝑨 𝜽 𝑿(𝒕)+𝑩 𝜽 𝑼(𝒕 )𝒅𝒕+𝝈 𝜽 𝒅⍵(𝒕)
𝒀(𝒕)=𝑪 𝜽 𝑿(𝒕)+𝑫 𝜽 𝑼(𝒕)+𝜺(𝒕)
The states of the system correspond to the temperatures of the different building components. 𝑈(𝑡) is a vector containing the measured inputs of the system (heating and ventilation system, solar gains, outdoor temperature, etc.). The measured output 𝑌(𝑡) is function of the states 𝑋 𝑡 and the inputs 𝑈(𝑡). ⍵ is an n-dimensional standard Wiener process and 𝜺 is an l-dimensional white noise process representing the measurement error.
The unknown parameters 𝜃 are estimated using the maximum likelihood estimation (MLE).

7. THE STRUCTURE OF THE GREY-BOX MODELS
The structures for modelling the heat dynamics are derived from the analogy with electric circuits.
The energy-balance equations associated to each node of the RC-network are connected to the principles of heat transfer and they are obviously related to the state space form:
1°-order model
2°-order model
Increasing
number of parameters and variables
3°-order model
For every system of differential equations, the output equation is represented by the discrete time measurement equation:

8. Case study: Experimental set up
In order to develop an estimation procedure of the models, we realized a first controlled situation in our laboratory.
The entire laboratory simulates the outdoor environment temperatures. The changing room was the test- room, in which an electric heater of 400 W generates the heat flux. This heat input was provided at regular intervals, programmed with a sequence of PRBS signal, by means of a timer. The floor of the changing room has been covered with polystyrene sheets and bricks to insulate it from the ground.
A Hobo Zw Series Wireless Network has been installed in the testing area to collect the temperatures of the various environments.

9. Case study: Data generation
Extract of the horizontal section
Envelope Data
Example of the second-order modelling:
The applied data covered a period of about 18 days, from 2nd to 20th July The number of total observations was equal to 27131, corresponding to an acquisition per minute. Prior to estimating the model parameters, the data were resampled at a sampling time of 15 minutes, based on the smallest period of the electric heater that is 30 minutes.

10. Case study: Results THE BEST DATA PERIOD THE BEST MODEL
Model used: 3R3C;
Data set: four period implemented for the ML estimation: 3-10 July, 8-15 July, 3-15 July, 3-20 July;
Tool: CTSM toolbox, implemented in the statistical software R (by Madsen, Ctsm-r User Guide, ).
By the comparison of the root of the mean squared errors for the 1-step prediction and simulation between the three the identified models, resulted that:
the best data set corresponded to the period 3-15 July.
THE BEST MODEL
Iterative model estimation, beginning from the simplest model;
Tool: CTSM in the statistical software R;
ML identification method;
Validation of the estimation by means of three values: Pr, dF/dPar, dPen/dPar.
The estimated results of the 2nd-order model better fitted with the measured plots, than the other models.
Estimate
Std. error
t value
Pr(>|t|)
dF/dPar
dPen/dPar
Ti0
2.9692e+01
4.9569e-02
5.9901e+02
0.0000e+00
e-05
Te0
2.9224e+01
5.7916e-02
5.0458e+02
e-06 Ce
7.4006e+00
9.8259e-01
7.5317e+00
9.7255e-14
e-05 Ci
1.7640e-01
1.6073e-03
1.0974e+02
5.2610e-05
e11
e+00
4.8150e-02
e+02
e-06
p11
e+01
1.6143e-01
e+01
8.7844e-05
p22
e+00
8.6550e-02
e+01
2.1533e-05
Rea
1.4677e+00
1.8143e-01
8.0896e+00
1.3323e-15
e-05
Rie
3.4447e+00
4.8751e-02
7.0658e+01
e-05
DATA ACQUISITION
Start: 03/07/2015
End: 15/07/2015
RMSE 1-step prediction
0.06 °C
RMSE in simulation
0.16 °C
- Pr(>t) = The fraction of probability of the corresponding t-distribution outside the limits set by the t value.
- dF/dPar = Derivative of the objective function with respect to the particular initial state or parameter.
- dPen/dPar = Derivative of the penalty function with respect to the particular initial state or parameter.

11. Case study: Results VALIDATION OF THE REDUCED MODEL
Simulation of the test-room indoor temperature, performing on the tested reduced-order models with their estimated values in CTSM-R;
Environment of simulation: Dymola 2015 with Modelica Standard library (version 3.2.1)
Dataset: July, corresponded to the not used 1/3 of the acquisition.
The 2nd-order model
RMSE °C
The 1st-order model
RMSE °C
The 3rd-order model
RMSE °C

12. CONCLUSIONS FUTURE DEVELOPMENTS
The first order model resulted as non-representative of the heat dynamics of the test room;
The third order model, which fitted better, resulted over-parameterized for the parameters estimation;
The second order model was considered the best trade off between the complexity and the number of parameters, with the lowest RMSE value in simulation;
The best data set corresponded to the period 3-15 July, suggesting that data with adequate dynamics contribute to the strength of the identified models;
The minimum length of the data set (about 2 weeks) resulted reasonable to be collected in the preliminary stage of an energy process.
FUTURE DEVELOPMENTS
Future works will be developed to test other more complex cases, comparing with real conditions;
Further research will be to generalize as much as possible the model identification procedure in order to provide an efficient support tool for the energy audits of buildings.

13. THANKS FOR YOUR ATTENTION
Mariangela BENEDETTELLI, DICEA Department, Univerisità Politecnica delle Marche, Ancona, Italy,