# Baseline Energy Use Modeling and Characterization in Tertiary Buildings Using an Interpretable Bayesian Linear Regression Methodology

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## Abstract

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## 1. Introduction

#### 1.1. M&V Background

#### 1.2. Bayesian Paradigm in M&V

#### 1.3. Multilevel Models

#### 1.4. Present Study

- The model is explainable and its coefficients have a clearly interpretable meaning, a feature which is highly valued by investors in building energy renovation and other stakeholders of the industry.
- As the model is probabilistic, uncertainty is estimated in an accurate, automatic, elegant and intuitive way.
- The model coefficients and target variables are expressed in terms of probability distributions instead of point estimates, providing a range of additional actionable information to stakeholders.
- The provided uncertainty ranges are characterized by dynamic locally adaptive intervals, reflecting how the uncertainty is not symmetrically distributed around the mean of the predictions, and that the values can vary depending on the distribution of the explanatory variables.
- The methodology provides extra useful information for stakeholders, such as the typical consumption profile patterns of the building, as well as the probability distributions of the heating and cooling change-point temperatures and of the heating and cooling linear coefficients.
- Since Bayesian models are fit to provide reasonable results even when the training data available are scarce, the methodology is well-fit to solve M&V problems, where it is not always feasible to obtain long training time-series with high granularity. Furthermore, the model also has high applicability since the data required is limited to hourly electricity consumption and outdoor temperature values, which are fairly easy to obtain.

## 2. Methodology

#### 2.1. Bayesian Methodology

#### 2.1.1. Data Pre-Processing

- The original frequency of the consumption data is resampled (aggregated) to have one value per 3 h (8 values per day);
- For each day in the time-series, the consumption values ${Q}^{abs}$ are transformed into daily relative values ${Q}^{rel}$. ${Q}_{t}^{rel}=\frac{{Q}_{t}^{abs}}{{\sum}_{t\in day}{Q}_{t}^{abs}}$;
- A matrix of relative consumption values is generated, having as rows the days of the time-series and as columns the 8 parts of the day defined in point 1;
- The values in the matrix are transformed with a normalization between 0 and 1. This enables more accurate predictions with the clustering algorithm. ${Q}_{day,dh}^{norm,rel}=\frac{{Q}_{day,dh}^{rel}-min\left({Q}_{dh}^{rel}\right)}{max\left({Q}_{dh}^{rel}\right)-min\left({Q}_{dh}^{rel}\right).}$

#### 2.1.2. Regression Model

- ${\alpha}_{k\left[i\right]}$ is the intercept of the model, one for each profile cluster k detected,
- ${f}_{dh,i}={\sum}_{p=1}^{n}{\delta}_{k,p\left[i\right]}sin\left(2\pi p\frac{{h}_{d,i}}{24}\right)+{\gamma}_{k,p\left[i\right]}cos\left(2\pi p\frac{{h}_{d,i}}{24}\right)$ represents the effect of the hour of the day ${h}_{d}$, following a Fourier decomposition with n harmonics. ${\delta}_{k,p\left[i\right]}$ and ${\gamma}_{k,p\left[i\right]}$ are the linear coefficients that mark the weight of each hour on the final electricity consumption; one for each profile cluster k detected and for each harmonic p.
- ${\beta}_{c,j\left[i\right]}$ and ${\beta}_{h,j\left[i\right]}$ are the coefficients that represent the piece-wise linear temperature dependence of the model, one for each day-part j previously defined.
- $({T}_{c{p}_{h}}-{T}_{o,i})$ and $({T}_{o,i}-{T}_{c{p}_{c}})$ are the difference between the outdoor temperature and the change-point temperatures detected by the model for heating and cooling.
- ${d}_{h,k}$ and ${d}_{c,k}$ are logical variables making sure that the temperature-dependent term is only evaluated when the outdoor temperature is above/below the cooling/heating change-point temperature:$\begin{array}{cc}{d}_{h}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}({T}_{c{p}_{h}}-{T}_{o,i})>0\hfill \\ 0\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}({T}_{c{p}_{h}}-{T}_{o,i})\le 0\hfill \end{array}\right.\hfill & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{d}_{c}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}({T}_{o,i}-{T}_{c{p}_{c}})>0\hfill \\ 0\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}({T}_{o,i}-{T}_{c{p}_{c}})\le 0.\hfill \end{array}\right.\hfill \end{array}$
- $de{p}_{h,k}$ and $de{p}_{c,k}$ are two logical coefficients that are automatically optimized by the model and that mark whether or not a certain profile cluster k has heating or cooling dependence.

#### 2.1.3. Prior Probability Distributions

#### 2.1.4. Pooling Techniques

#### 2.1.5. Posterior Estimation Methods

#### 2.1.6. Uncertainty Intervals

#### 2.2. Model Comparison

#### 2.2.1. Model Comparison Phases

#### Phase 1

#### Phase 2

#### Phase 3

#### Phase 4

#### 2.2.2. Comparison Metrics

#### CV(RMSE)

#### Coverage

## 3. Case Study

- the buildings belong to 19 different sites across North America and Europe, with energy meter readings spanning two full years (2016 and 2017);
- there are five main primary use categories: education, office, entertainment/public assembly, lodging/residential, and public services;
- the weather data provided includes information about cloud coverage, outdoor air temperature, dew temperature, precipitation depth in 1 and 6 hours, pressure, wind speed and direction;
- for most of the buildings, additional metadata such as total floor area and year of construction are available.

## 4. Results

#### 4.1. Model Comparison

#### 4.2. Individual Buildings

## 5. Discussion

- The use of regularizing priors based on building physics knowledge improves the model both in terms of CV(RMSE) and coverage;
- The addition of a regression term to take into account the wind speed did not improve the model predictive capabilities;
- The use of MCMC sampling techniques to estimate the posterior distribution yields comparable results to the variational inference method, despite being characterized by a more than 30-fold increase in computational time;
- When looking at the ADVI case, partial pooling has an almost negligible effect on the prediction accuracy, providing only a very modest median CV(RMSE) and coverage improvement, and a slightly worse adjusted coverage;
- In the NUTS case, the partial pooling regression seems to have a stronger impact, improving the CV(RMSE), but at the same time reducing the coverage of the model;
- The computational requirements of the partial pooling regression are comparable to the ones of the no pooling case.

## 6. Conclusions and Future Work

- The explainability of the model and the interpretability of its coefficients even for non-technical audiences;
- An elegant, efficient, dynamic and coherent estimation of uncertainty, that makes it apt to be used in financial risk assessments of retrofit strategies;
- The ability to provide a detailed building energy use characterization that can employed by energy managers to improve the performance of their facilities;
- High scalability to big data problems, because of the low computational complexity and the limited data requirements.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ADVI | Automatic Differentiation Variational Inference |

CV(RMSE) | Coefficient of Variation of the Root Mean Square Error |

HDI | Highest Density Interval |

HMC | Hamiltonian Monte Carlo |

IQR | Interquartile Range |

M&V | Measurement and Verification |

MCMC | Markov Chain Monte Carlo |

NUTS | No-U-Turn Sampler |

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**Figure 4.**Coverage and adjusted coverage for the 95% HDIs obtained in Phases 1, 2 and 3 of model comparison.

**Figure 11.**Posterior distributions estimated for the heating and cooling change-point temperatures for the building Rat_health_Gaye.

**Figure 12.**Electricity time-series vs. outdoor temperature values for the building Rat_health_Gaye. The heating and cooling change-point temperatures estimated by the model are shown in red and blue with the corresponding 94% HDIs.

**Figure 13.**Test year metered electricity time-series (in black) and model predictions (in red) for Rat_health_Gaye.

**Figure 14.**July 2017 metered electricity time-series (in black) and model predictions (in red) for Rat_health_Gaye.

**Figure 16.**Posterior distributions estimated for the heating and cooling change-point temperatures (upper panels) and for the heating and cooling linear dependence coefficients (lower panels) in Rat_education_Royal.

**Figure 17.**Electricity time-series vs. outdoor temperature values for the building Rat_education_Royal. The heating and cooling change-point temperatures estimated by the model are shown in red and blue with the corresponding 94% HDIs.

**Figure 18.**Test year metered electricity time-series (in black) and model predictions (in red) for Rat_education_Royal.

Phase | Prior | Wind Speed Feature | Posterior Estimation | Pooling |
---|---|---|---|---|

1 | Uninformative | No | ADVI | No pooling |

2 | Regularizing | No | ADVI | No pooling |

3 | Best according to previous results | Yes | ADVI | No pooling |

4 | Best according to previous results | Best according to previous results | ADVI, NUTS | Partial pooling, No pooling, Complete pooling |

**Table 2.**Overview of the sites from which the meter data were collected and number of buildings having electricity meter readings for each site.

Site | Actual Site Name | Location | Buildings |
---|---|---|---|

Panther | Univ. of Central Florida (UCF) | Orlando, FL | 105 |

Robin | Univ. College London (UCL) | London, UK | 52 |

Fox | Arizona State University (ASU) | Tempe, AZ | 137 |

Rat | Washington DC—City Buildings | Washington DC | 305 |

Bear | Univ. of California Berkeley | Berkeley, CA | 92 |

Lamb | Cardiff—City Buildings | Cardiff, UK | 146 |

Eagle | Anonymous | N/A | 106 |

Moose | Ottawa—City Buildings | Ottawa, Ontario | 13 |

Gator | Anonymous | N/A | 74 |

Bull | Univ. of Texas—Austin | Austin, TX | 123 |

Bobcat | Anonymous | N/A | 35 |

Crow | Carleton Univ. | Ottawa, Ontario | 5 |

Wolf | Univ. College Dublin (UCD) | Dublin, Ireland | 36 |

Hog | Anonymous | N/A | 152 |

Peacock | Princeton University | Princeton, NJ | 45 |

Cockatoo | Cornell University | Cornell, NY | 117 |

Shrew | UK Parliament | London, UK | 9 |

Swan | Anonymous | N/A | 19 |

Mouse | Ormand Street Hospital | London, UK | 7 |

**Table 3.**Results obtained in Phase 4 of model comparison. Median CV(RMSE), coverage and adjusted coverage are shown, as well as the average computational time required to estimate the model for a single building of the dataset.

Model | CV(RMSE) (%) | Coverage (%) | Adjusted Coverage | Computational Time |
---|---|---|---|---|

NP ADVI | 18.95 | 92.54 | 2.03 | 15 s |

PP ADVI | 18.93 | 92.61 | 2.04 | 16 s |

CP ADVI | 28.62 | 93.25 | 3 | 9 s |

NP NUTS | 20.34 | 90.82 | 2.06 | 8 min |

PP NUTS | 18.95 | 89.94 | 1.87 | 9 min |

CP NUTS | 27.56 | 91.18 | 2.57 | 1.5 min |

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**MDPI and ACS Style**

Grillone, B.; Mor, G.; Danov, S.; Cipriano, J.; Lazzari, F.; Sumper, A. Baseline Energy Use Modeling and Characterization in Tertiary Buildings Using an Interpretable Bayesian Linear Regression Methodology. *Energies* **2021**, *14*, 5556.
https://doi.org/10.3390/en14175556

**AMA Style**

Grillone B, Mor G, Danov S, Cipriano J, Lazzari F, Sumper A. Baseline Energy Use Modeling and Characterization in Tertiary Buildings Using an Interpretable Bayesian Linear Regression Methodology. *Energies*. 2021; 14(17):5556.
https://doi.org/10.3390/en14175556

**Chicago/Turabian Style**

Grillone, Benedetto, Gerard Mor, Stoyan Danov, Jordi Cipriano, Florencia Lazzari, and Andreas Sumper. 2021. "Baseline Energy Use Modeling and Characterization in Tertiary Buildings Using an Interpretable Bayesian Linear Regression Methodology" *Energies* 14, no. 17: 5556.
https://doi.org/10.3390/en14175556