# Robust Building Energy Load Forecasting Using Physically-Based Kernel Models

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Building Energy Load Forecasting Algorithm Using Gaussian Process

#### 3.1. Gaussian Process Regression

#### 3.2. Covariance Function Modeling

#### 3.2.1. Kernel Types

- 1.
- Periodic Function:$$\begin{array}{cc}\hfill K(x,{x}^{\prime})& ={\theta}_{1}^{2}\times exp\left(-\frac{2{sin}^{2}\left(\frac{\pi |x-{x}^{\prime}|}{{\theta}_{2}}\right)}{{\theta}_{3}^{2}}\right)\hfill \end{array}$$

- 2.
- Squared Exponential:$$\begin{array}{cc}\hfill K(x,{x}^{\prime})& ={\theta}_{1}^{2}\times exp\left(-\frac{{(x-{x}^{\prime})}^{2}}{2{\theta}_{2}^{2}}\right)\hfill \end{array}$$

- 3.
- Matern Kernel:$$\begin{array}{cc}\hfill K(x,{x}^{\prime})& =\frac{{2}^{1-\nu}}{\Gamma (\nu )}{\left(\frac{\sqrt{2\nu}(x-{x}^{\prime})}{l}\right)}^{\nu}{K}_{\nu}\left(\frac{\sqrt{2\nu}(x-{x}^{\prime})}{l}\right)\hfill \end{array}$$

- 4.
- Linear Kernel:$$\begin{array}{cc}\hfill K(x,{x}^{\prime})& ={\theta}_{1}^{2}\times (x-c)\times ({x}^{\prime}-c)\hfill \end{array}$$

- 5.
- Random Noise Kernel$$\begin{array}{cc}\hfill K(x,{x}^{\prime})& ={\theta}_{1}^{2}{\delta}_{x,{x}^{\prime}}\hfill \end{array}$$

#### 3.2.2. Long-Term Forecasting

#### 3.2.3. Short-Term Forecasting

## 4. Evaluation

#### 4.1. Experimental Setup

#### 4.1.1. Electricity Consumption Data of Carnegie Mellon University

#### 4.1.2. Cooling and Lighting Load Data of Y2E2 Building in Stanford University

#### 4.1.3. Benchmark Methods

#### 4.2. Results and Discussion

#### 4.2.1. Long-Term Forecasting under Varying Duration of Training Data

#### 4.2.2. Long-Term Forecasting under Varying Prediction Horizon

#### 4.2.3. Short-Term Forecasting

#### 4.2.4. Long-Term Forecasting with Predicted Inputs

#### 4.2.5. Short-Term Forecasting with Predicted Inputs

#### 4.2.6. Impacts of Different Kernels

- 1.
- Matern kernel for Y2E2 Building’s Cooling Load Forecasting

- 2.
- A combination of Matern and Linear Kernel for Y2E2 Building’s lighting load forecasting

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Different 5-day loads and temperatures. (

**a**) Aggregate Electricity Load and Temperature of Carnegie Mellon University; (

**b**) Cooling load and Temperature for Y2E2 building in Stanford Campus; (

**c**) Lighting load and Temperature for Y2E2 building in Stanford Campus.

**Figure 2.**Different loads for a short duration of time. (

**a**) Aggregate Electricity Load of Carnegie Mellon University; (

**b**) Cooling load for Y2E2 building in Stanford Campus; (

**c**) Lighting load for Y2E2 building in Stanford Campus.

**Figure 3.**A Daily Load Profile for each of the dataset. (

**a**) CMU electricity load; (

**b**) Y2E2 cooling load; (

**c**) Y2E2 lighting load.

**Figure 4.**Distribution of the prediction MAPE over different weeks for varying duration of training data using GPR. (

**a**) CMU electricity load; (

**b**) Y2E2 cooling load; (

**c**) Y2E2 lighting load.

**Figure 5.**Mean of the MAPE of prediction over different weeks for varying duration of training data using GPR, SVR, RF and ARIMA. (

**a**) CMU electric load; (

**b**) Y2E2 cooling load; (

**c**) Y2E2 lighting load.

**Figure 6.**Distribution of the prediction MAPE over different duration of training data for varying duration of prediction horizons using GPR. (

**a**) CMU electricity load; (

**b**) Y2E2 cooling load; (

**c**) Y2E2 lighting load.

**Figure 7.**Mean of the MAPE of prediction over duration of training data for varying duration of prediction horizon using GPR, SVR, RF and ARIMA. (

**a**) CMU electricity load; (

**b**) Y2E2 cooling load; (

**c**) Y2E2 lighting load.

**Figure 8.**Short Term Forecasting of the Y2E2 lighting load every 30 min period using a linear kernel in GPR. (

**a**) Actual v/s Predicted Load for a 20 h period; (

**b**) Predicted (blue dashed) and Actual (orange solid) load for a 20 h period.

**Figure 9.**A smooth predicted (blue dashed curve) and a jagged actual (orange solid curve) Temperature curve for two different days in Stanford, CA. (

**a**) Day 1; (

**b**) Day 2.

**Figure 10.**Mean of the MAPE of prediction when using Actual Temperature Input v/s Predicted Temperature Input in the GPR model for all the three loads. (

**a**) CMU electricity load; (

**b**) Y2E2 cooling load; (

**c**) Y2E2 lighting load.

**Figure 11.**Actual Y2E2 lighting load (orange solid line), Short Term Forecast of the Y2E2 lighting load using Actual Temperature Input (blue dashed line) and Short Term Forecast of the same load using predicted Temperature Input (red dashed-dotted line) for a 20 h period.

**Figure 12.**MAPE for forecasting a Y2E2 cooling load chilled water consumption using Squared Exponential Kernel and Matern Kernel. (

**a**) Squared Exponential Kernel; (

**b**) Matern Kernel.

**Figure 13.**MAPE for forecasting the Y2E2 lighting load using our long-term load forecasting kernel (Section 3.2.2) and a combination of a linear and matern kernel. (

**a**) Our long Term Load Forecasting Kernel in Section 3.2.2; (

**b**) Combination of Matern and linear kernels for Long Term Forecasting.

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## Share and Cite

**MDPI and ACS Style**

Prakash, A.K.; Xu, S.; Rajagopal, R.; Noh, H.Y.
Robust Building Energy Load Forecasting Using Physically-Based Kernel Models. *Energies* **2018**, *11*, 862.
https://doi.org/10.3390/en11040862

**AMA Style**

Prakash AK, Xu S, Rajagopal R, Noh HY.
Robust Building Energy Load Forecasting Using Physically-Based Kernel Models. *Energies*. 2018; 11(4):862.
https://doi.org/10.3390/en11040862

**Chicago/Turabian Style**

Prakash, Anand Krishnan, Susu Xu, Ram Rajagopal, and Hae Young Noh.
2018. "Robust Building Energy Load Forecasting Using Physically-Based Kernel Models" *Energies* 11, no. 4: 862.
https://doi.org/10.3390/en11040862