Data-Driven Virtual Replication of Thermostatically Controlled Domestic Heating Systems
Abstract
:1. Introduction
2. Methodology
2.1. Demand-Side Model
- Time-lagged (n) indoor temperatures (${T}_{t-n}^{i}$) to characterize the inertia of the building.
- Low-pass filtered outdoor temperature (${T}^{e,lp}$) to characterize the heat loses through the envelope of the building due to changes in the outdoor temperature.
- Raw outdoor temperature (${T}^{e}$) to consider fast changes in indoor temperatures due to changes in the daily minimum and maximum temperatures.
- Heat consumption of the boiler (${\Phi}^{h}$) to characterize the increase in the indoor temperature due to the operation of the heating system.
- Solar direct normal irradiance (${I}^{sol,lp}$), interacting with the Fourier series of the solar azimuth (${S}^{az,fs}$) and of the solar elevation (${S}^{el,fs}$) to characterize the solar gains of the building.
- Wind speed (${W}^{s,lp}$), interacting with Fourier series of the wind direction (${W}^{d,fs}$) and the temperature difference between indoors and outdoors ($\Psi ={T}^{i}-{T}^{e}$) to characterize the heat losses due to air leakage and convection effects through the envelope.
2.2. Supply-Side Model
- Time-lagged (n) heat consumption (${\Phi}_{t-n}^{h}$) to consider how the boiler was performing in the last time steps.
- Raw data of the outdoor temperature (${T}^{e}$) to consider the variation of the coefficient of performance of the boiler due to changes in the outdoor temperature.
- ${T}^{e,lp}$ is the low-pass filtered version of the outdoor temperature. It represents the temperature of the building envelope.
- As in the demand-side model, the solar direct normal irradiance (${I}^{sol,lp}$) interacts with the Fourier series of the solar azimuth (${S}^{az,fs}$) and of the solar elevation (${S}^{el,fs}$).
- $\Psi $ as in the case of the demand side model, it is the temperature difference between indoors and outdoors.
- Wind speed (${W}^{s,lp}$) interacts with Fourier series of the wind direction (${W}^{d,fs}$) and the temperature difference between indoors and outdoors ($\Psi ={T}^{i}-{T}^{e}$).
2.3. Transformation of Input Variables
2.3.1. Low-Pass Filter
2.3.2. Fourier Series
2.4. Models Coupling
2.5. Model Training and Parameter Optimization
2.6. Evaluation of Potential Energy Savings
3. Case Study
Case Study Datasets
Climate Data
4. Results
4.1. Detailed Model Validation in One Household
4.2. Model Validation in a Larger Population of Households
4.3. Assessment of Potential Energy Savings
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
Nomenclature
$AMI$ | Advanced Metering Infrastructure |
$ANN$ | Artificial Neural Network |
$ASHRAE$ | American Society of Heating, Refrigerating and Air-Conditioning Engineers |
$ARMAX$ | AutoRegressive Moving Average with eXogenous |
$ARX$ | AutoRegressive with eXogenous |
$BaU$ | Business as Usual |
$BES$ | Building Energy Simulation |
$CVRMSE$ | Coefficient of Variation of the Root Mean Squared Error |
$ECM$ | Energy Conservation Measures |
$EU$ | European Union |
$HVAC$ | Heating, ventilation and air conditioning |
$IoT$ | Internet of Things |
$IT$ | Information Technology |
$LPF$ | Low-Pass Filter function |
$MAPE$ | Mean Absolute Percentage Error |
$NLME$ | Non Linear Mixed Effect |
$RC$ | Resistor and Capacitor |
$RMSE$ | Root Mean Squared Error |
$SH$ | Space Heating |
$SVM$ | Support Vector Machines |
$TS$ | Time Series |
Appendix A
Algorithm A1. Forecasting algorithm and coupling of supply-side and demandside models | |
Input: | Trained supply-side model; trained demand-side model; autoregressive orders ${n}_{\gamma \left(B\right)}$, ${n}_{{\beta}_{t}\left(B\right)}$, ${n}_{{\beta}_{e}\left(B\right)}$, ${n}_{{\beta}_{p}\left(B\right)}$, ${n}_{\varphi \left(B\right)}$, ${n}_{{\omega}_{h}\left(B\right)}$, ${n}_{{\omega}_{e}\left(B\right)}$, ${n}_{{\omega}_{p}\left(B\right)}$; number of harmonics ${n}_{har,az}$, ${n}_{har,el}$ and ${n}_{har,wd}$; the smoothing parameters of the low-pass filter ${\alpha}_{e}$, ${\alpha}_{s}$ and ${\alpha}_{w}$; initial indoor conditions (${T}^{i}$), weather conditions during the whole evaluation period (outdoor temperature ${T}^{e}$, wind speed ${W}^{s}$, wind direction ${W}^{d}$, solar irradiance ${I}^{sol}$, and solar position ${S}^{az}$, ${S}^{el}$), the space heating consumption few timesteps before the period to be evaluated the hysteresis of the thermostat h and, finally, the setpoint temperature (${T}_{t}^{s,sim}$) to apply during the evaluation period |
Output: | The predicted heat consumption ($\widehat{{\Phi}^{h}}$) and the predicted indoor temperature ($\widehat{{T}^{i}}$) considering a setpoint temperature schedule (${T}^{s,sim}$) during a period $ts\in [0,j]$. |
Column | Conditions | Column | Conditions |
---|---|---|---|
${T}_{t-k}^{e}$ | $k\in \mathbb{N}\wedge k\le max({n}_{{\beta}_{e}\left(B\right)},{n}_{{\omega}_{e}\left(B\right)})$ | ${T}_{t-k}^{e,lp}$ | $k\in \mathbb{N}\wedge k\le max({n}_{{\omega}_{p}\left(B\right)},{n}_{{\beta}_{p}\left(B\right)})$ |
$\widehat{{T}_{t-k}^{i}}$ | $k\in \mathbb{N}\wedge k\le max\left({n}_{\varphi \left(B\right)}{n}_{{\omega}_{e}\left(B\right)}\right)$ | ${\Psi}_{t-k}$ | $k\in \mathbb{N}\wedge k\le {n}_{{\beta}_{p}\left(B\right)}$ |
$\widehat{{\Phi}_{t-k}^{h}}$ | $k\in \mathbb{N}\wedge k\le max({n}_{{\omega}_{h}\left(B\right)},{n}_{\gamma \left(B\right)})$ | ${S}_{t}^{az,fs,{h}_{sin}}$ | ${h}_{sin}\in \mathbb{N}\wedge 1\le {h}_{sin}\le {n}_{har,az}$ |
${S}_{t}^{az,fs,{h}_{cos}}$ | ${h}_{cos}\in \mathbb{N}\wedge 1\le {h}_{cos}\le {n}_{har,az}$ | ${S}_{t}^{el,fs,{h}_{sin}}$ | ${h}_{sin}\in \mathbb{N}\wedge 1\le {h}_{sin}\le {n}_{har,el}$ |
${S}_{t}^{el,fs,{h}_{cos}}$ | ${h}_{cos}\in \mathbb{N}\wedge 1\le {h}_{cos}\le {n}_{har,el}$ | ${W}_{t}^{d,fs,{h}_{sin}}$ | ${h}_{sin}\in \mathbb{N}\wedge 1\le {h}_{sin}\le {n}_{har,wd}$ |
${W}_{t}^{d,fs,{h}_{cos}}$ | ${h}_{cos}\in \mathbb{N}\wedge 1\le {h}_{cos}\le {n}_{har,wd}$ | ${I}_{t}^{sol,lp}$ | - |
${W}_{t}^{s,lp}$ | - | ${T}_{t}^{s,sim}$ | - |
Algorithm A2: Genetic Algorithm for the optimization of the auto regressive orders (${n}_{*\left(B\right)}$), the low-pass filter (${\alpha}_{*}$), and the number of harmonics (${n}_{har,*}$) to be considered in the transformation of the input variables | |
Input: Hourly space heating consumption, indoor and set point temperature of the thermostat and historical weather of the location of the household during a period where the boiler is operating. At least 3 months of data are required. | |
Output: Find the optimal auto regressive orders ${n}_{\gamma \left(B\right)}$, ${n}_{{\beta}_{t}\left(B\right)}$, ${n}_{{\beta}_{e}\left(B\right)}$, ${n}_{{\beta}_{p}\left(B\right)}$, ${n}_{\varphi \left(B\right)}$, ${n}_{{\omega}_{h}\left(B\right)}$, ${n}_{{\omega}_{e}\left(B\right)}$, ${n}_{{\omega}_{p}\left(B\right)}$; optimal number of harmonics ${n}_{har,az}$, ${n}_{har,el}$ and ${n}_{har,wd}$; and optimal smoothing parameters of the low-pass filter ${\alpha}_{e}$, ${\alpha}_{s}$ and ${\alpha}_{w}$ | |
DEFINE a test set and a training set (15% and 85%, respectively); DEFINE a cross-validation with 8 folds from the training set. Randomly select, for each of the folds, a set of 80% of the days for training and 20% for validation; SET the value ranges, levels and type of variables of the parameters to optimize; DEFINE an encode–decode technique to convert each single combination of parameters to a Reflected Binary Code (RBC) representation, taking into account the allowed ranges or levels assigned to each parameter; INITIALIZE population with random candidate RBC representations, also called chromosomes; EVALUATE the related cost of each chromosome using Algorithm A3. In this step, $\omega $, $\beta $, $\varphi $ and $\gamma $ ARX-models coefficients are estimated using the least squares method; | |
Algorithm A3: Cost evaluation of each chromosome |
Input: A chromosome which contains an RBC representation; training set; test set; and the description of the cross-validation folds. |
Output: The cost related to the input chromosome |
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Parameter | Type | Values Range | Number of Levels | Weights Distribution * | Optimal Value for Household in Study |
---|---|---|---|---|---|
${n}_{\gamma \left(B\right)}$ | integer | $\mathbb{N}\in [1,4]$ | 4 | uniform | 1 |
${n}_{{\beta}_{t}\left(B\right)}$ | integer | $\mathbb{N}\in [0,7]$ | 8 | uniform | 5 |
${n}_{{\beta}_{e}\left(B\right)}$ | integer | $\mathbb{N}\in [0,3]$ | 4 | uniform | 3 |
${n}_{{\beta}_{p}\left(B\right)}$ | integer | $\mathbb{N}\in [0,3]$ | 4 | uniform | 0 |
${n}_{\varphi \left(B\right)}$ | integer | $\mathbb{N}\in [1,16]$ | 16 | uniform | 13 |
${n}_{{\omega}_{h}\left(B\right)}$ | integer | $\mathbb{N}\in [0,3]$ | 4 | uniform | 1 |
${n}_{{\omega}_{e}\left(B\right)}$ | integer | $\mathbb{N}\in [0,3]$ | 4 | uniform | 1 |
${n}_{{\omega}_{p}\left(B\right)}$ | integer | $\mathbb{N}\in [0,3]$ | 4 | uniform | 0 |
${n}_{har,az}$ | integer | $\mathbb{N}\in [1,3]$ | 3 | uniform | 2 |
${n}_{har,el}$ | integer | $\mathbb{N}\in [1,3]$ | 3 | uniform | 1 |
${n}_{har,wd}$ | integer | $\mathbb{N}\in [1,3]$ | 3 | uniform | 1 |
${\alpha}_{e}$ | float | $\mathbb{R}\in [0.00,0.99]$ | 20 | exponential | 0.891 |
${\alpha}_{s}$ | float | $\mathbb{R}\in [0.00,0.70]$ | 14 | exponential | 0.252 |
${\alpha}_{w}$ | float | $\mathbb{R}\in [0.00,0.90]$ | 18 | exponential | 0.824 |
$mod{e}_{{I}_{sol}}$ | discrete | ** | 3 | uniform | linear depending solar position |
$mod{e}_{{W}_{s}\times \Psi}$ | discrete | *** | 3 | uniform | linear depending wind direction |
h | float | $\mathbb{R}\in [0.25,1]$ | 4 | uniform | 0.5 |
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Mor, G.; Cipriano, J.; Gabaldon, E.; Grillone, B.; Tur, M.; Chemisana, D. Data-Driven Virtual Replication of Thermostatically Controlled Domestic Heating Systems. Energies 2021, 14, 5430. https://doi.org/10.3390/en14175430
Mor G, Cipriano J, Gabaldon E, Grillone B, Tur M, Chemisana D. Data-Driven Virtual Replication of Thermostatically Controlled Domestic Heating Systems. Energies. 2021; 14(17):5430. https://doi.org/10.3390/en14175430
Chicago/Turabian StyleMor, Gerard, Jordi Cipriano, Eloi Gabaldon, Benedetto Grillone, Mariano Tur, and Daniel Chemisana. 2021. "Data-Driven Virtual Replication of Thermostatically Controlled Domestic Heating Systems" Energies 14, no. 17: 5430. https://doi.org/10.3390/en14175430