# Heat Load Profiles in Industry and the Tertiary Sector: Correlation with Electricity Consumption and Ex Post Modeling

^{*}

## Abstract

**:**

^{2}is 0.94). The ex post model architecture makes the model suitable for anomaly detection in energy monitoring systems.

## 1. Introduction

#### 1.1. Operation of Energy Networks

#### 1.2. Potential and Feasibility Studies on Renewable Heating Systems

#### 1.3. Design of Renewable Heating Systems

#### 1.4. Model Predictive Control

#### 1.5. Anomaly Detection

#### 1.6. Load Profile Prediction for Individual Consumers in Recent Literature

^{2}of 0.64. Heidari and Khovalyg [20] use a feed forward ANN as a baseline model and compare it to a Long Short-Term Memory (LSTM) model, an attention-based LSTM model, and an attention-based LSTM model using decomposed data for domestic hot water demand prediction. Compared to the feed forward ANN, the three LSTM-based models yield a 25%, 28%, or 41% reduced Mean Absolute Error (MAE).

#### 1.7. Implications from Literature and the Objective

- Analysis of the correlation between heat and electricity consumption for consumers from industry and the tertiary sector:No studies on the correlation between heat or natural gas and electricity consumption on the level of individual consumers from industry and the tertiary sector could be identified in the literature. However, Lauterbach et al. [11] provided a comprehensive overview of the heat-consuming processes commonly used in industry in the temperature range of up to 200 °C, which was later supplemented by Wolf et al. [4] and Arpagaus et al. [31]. For many of these processes, it can be assumed that there is a correlation between electricity and heat demand, e.g., for processes such as drying or washing, where heating and the operation of electric motors are required simultaneously. In contrast, some processes, such as cooking, are expected to require only heat. Finally, there are also processes that only require electricity. Therefore, the present study is the first to investigate the relationship between measured heat and electricity load profiles systematically for a broad range of different consumers from the industrial and tertiary sectors. As an important precondition to the model development, this study examines whether there is a universal correlation pattern between natural gas and electricity consumption or whether the observed correlations are specific to individual consumers. A universal correlation pattern would suggest that a universal heating load profile model could be developed for all consumers or a group of consumers. In contrast to that, consumer-specific correlations without a universally observable pattern, would point to the need for individually trained heat load profile models for each consumer.
- The development of a generally applicable heat load profile model for consumers from industry and the tertiary sector:No study of a heat load profile models valid for consumers from different industries could be found, other than the authors’ own previous work. In their previous publications, the authors presented a method to predict normalized heat load profiles with a resolution of one day for individual consumers from industry and the tertiary sector. The accuracy of this method is sufficient for applications, such as preliminary design or potential studies for renewable heating systems. The present study aims to further increase the accuracy of this method to be sufficiently accurate for more demanding applications, such as anomaly detection. For this purpose, data-driven black-box models will be developed that evaluate commonly available and previously ignored information, such as electricity load profiles.Electricity load profiles are selected as an input to the heat load profile model for the following reasons: The minimum threshold for online metering of energy consumption load is significantly lower for electricity consumption compared to natural gas consumption in Germany. Additionally, all consumers connected to a public grid in Germany will be equipped with a digital measurement of electricity consumption by 2032. Therefore, it can be assumed that electricity load profiles are available much more frequently than gas load profiles. At the same time, it is reasonable to assume that machine learning methods can be used to automatically extract important information on the user behavior of a particular consumer from the electricity load profiles.

## 2. Database

## 3. Methods

#### 3.1. Pre-Processing

#### 3.2. Model Development

#### 3.2.1. Ex Ante 1

_{hl}), a linear function represents the baseline heat demand in summer, e.g., due to water heating or other processes that are independent of the ambient temperature. For ambient temperatures below T

_{hl}, a second regression line represents the heat demand, which increases linearly with decreasing ambient temperatures, e.g., due to space heating or heating processes that are dependent on the ambient temperature (drying, ventilation systems). Since wd and wknd-clusters are different, the heat demand must be calculated for wd and wknd separately [28].

${\mathrm{b}}_{\mathrm{h}}$ | y-axis intercept of space heating line (-) |

${\mathrm{b}}_{\mathrm{w}}$ | y-axis intercept of domestic hot water (process heat) line (-) |

${\mathrm{m}}_{\mathrm{h}}$ | slope of space heating line (-) |

${\mathrm{m}}_{\mathrm{w}}$ | slope of domestic hot water (process heat) line (-) |

${\mathrm{Q}}_{\mathrm{d}}/{\mathrm{Q}}_{\mathrm{d}}\left(8\xb0\mathrm{C}\right)$ | normalized daily heat consumption (-) |

${\mathrm{T}}_{\mathrm{amb}}$ | daily mean ambient temperature (°C) (insert unitless) |

${\mathrm{T}}_{\mathrm{hl}}$ | heating limit temperature (°C) (insert unitless) |

#### 3.2.2. Ex Ante 2

#### 3.2.3. Ex Ante 3

#### 3.3. Correlation of Heat and Electricity

#### 3.3.1. Ex Post 1

#### 3.3.2. Ex Post 2

#### 3.4. Evaluation of Correlations and Models

- Reduced computation time:The overall sum of trained models in this study is high. For ex ante 3 to ex post 2, various regression algorithms are trained and compared with each other. Since this is done for each consumer individually, this results in a five-digit number of training runs. A more complex validation method, for example, a blocked cross-validation, would result in a higher number of training runs corresponding to the number of blocks.
- Stationarity:Bergmeier and Benitéz [36] emphasized the importance of an adequate control for load profile stationarity before cross-validation. According to Cryer and Chan [37], a load profile is stationary if the joint distribution ${\mathrm{Y}}_{\mathrm{t}},{\mathrm{Y}}_{{\mathrm{t}}_{2}},\dots ,{\mathrm{Y}}_{{\mathrm{t}}_{\mathrm{n}}}$ is the same as the joint distribution of ${\mathrm{Y}}_{\mathrm{t}-\mathrm{k}},{\mathrm{Y}}_{{\mathrm{t}}_{2}-\mathrm{k}},\dots ,{\mathrm{Y}}_{{\mathrm{t}}_{\mathrm{n}}-\mathrm{k}}$ for all choices of time points ${\mathrm{t}}_{1},{\mathrm{t}}_{2},\dots ,{\mathrm{t}}_{\mathrm{n}}$ and all choices of time lag k. For annual natural gas load profiles with high seasonality, as examined in this study, the same distribution of two blocks is only given if both blocks cover at least a whole year. Only blocks of at least one year ensure that all occurring operation modes (e.g., normal operation, maintenance, holidays) are covered by each block.
- Robust sample size:For small sample sizes, accuracy metrics, such as R, can significantly deviate from the true value. Schoenbrodt and Perugini [38] showed that a sample size of 362 is required to ensure an accuracy of R in the range of ±0.10 with a 90% confidence interval. This minimum sample size corresponds to the sample size used for the last-block evaluation (365 days). Cross-validation or blocked cross-validation with more than two blocks would result in smaller sample sizes and therefore less robust sample sizes.

^{2}) is used. R

^{2}is usually defined to be between 0 and 1. Statistic parameters, such as the median or the quartiles of R

^{2}, therefore evaluate the overall strength of a correlation for a group of consumers without considering their sign. The function to calculate R

^{2}implemented in Scikit-Learn [34] and used in this study employs an alternative definition of R

^{2}(2) that can also lead to negative values if the sum of squared residuals is larger than the total sum of squares. However, negative values of R

^{2}are equally to be interpreted as zero values.

^{2}can still take values close to zero, even if the prediction is sufficiently accurate for the intended applications. Therefore, the standard deviation of the residuals (σ) is used as another metric for model accuracy evaluation.

${\mathrm{R}}^{2}$ | coefficient of determination (-) |

SSR | sum of squared residuals (-) |

SST | total sum of squares (-) |

$\mathrm{y}$ | value (-) |

$\overline{\mathrm{y}}$ | mean of values (-) |

$\hat{\mathrm{y}}$ | prediction of values (-) |

## 4. Results

#### 4.1. Ex Ante Models

^{2}for the ex ante 3 model separately for the four wd-clusters. The results of the other two ex ante models are similar, except that the overall accuracy of ex ante 1 is significantly poorer. For those consumers with a strong dependency on ambient temperature (wd-cluster 2 and 3), the ex ante model achieves a median of R

^{2}greater than 0.8. For wd-cluster 1, the cluster with only a small dependence on ambient temperature, the distribution of R

^{2}is much broader. For wd-cluster 0, the median of R

^{2}is at least 0.2 lower than the other clusters. The interquartile distance is two to three times higher than the other clusters.

#### 4.2. Correlation of Heat and Electricity

^{2}is 0.33. Three examples of high, medium, and low correlation are presented below.

_{wknd}= 0.49). The consumption of a few wknd days is in the range of wd. When only wd are considered, a high negative correlation can be detected (R = −0.91).

^{2}only for wd is even slightly poorer than in general for all types of days (R

^{2}= 0.33; R

^{2}

_{wd}= 0.28). In contrast, the correlation between electricity and heat consumption is significantly higher, if only non-seasonal consumers (wd-cluster 0) are considered (R

^{2}

_{wd,cl 0}= 0.42). More than two thirds (69%) of the consumers in wd-cluster 0 show a strong correlation. For all other clusters, most consumers show a weak to medium correlation.

^{2}only for wd for all clusters is 0.26. If only consumers from wd-cluster 0 are considered, R

^{2}is 0.45.

#### 4.3. Ex Post 1

^{2}for the residual prediction for 10 algorithms from Scikit-Learn with standard hyperparameters, which yield the highest median of R

^{2}for the 82 consumers examined in this study. For the algorithm with the highest median of R

^{2}(NuSVR), the plot also shows the distribution of R

^{2}after randomized hyperparameter tuning (NuSVR-tuned). Overall, differences between the algorithms are small. However, the variance of R

^{2}for each algorithm between the 82 consumers is large. For each of the algorithms, R

^{2}ranges roughly from −1.0 to 0.8. Negative R

^{2}values occur when the sum of squared model residuals is larger than the total sum of squares (see Section 3.4). With standard hyperparameters, NuSVR yields the highest median of R

^{2}. Therefore, the NuSVR algorithm is selected to be optimized but the randomized hyperparameter tuning of the NuSVR algorithm (NuSVR-tuned) does not lead to a significant improvement. While the maximum R

^{2}and the lower quartile are slightly increased, the upper quartile and the median of the tuned NuSVR algorithm are poorer than of the standard NuSVR algorithm.

^{2}of the NuSVR residual prediction separated by the different clusters. Only for the wd-clusters 0 and 1, R

^{2}values of more than 50% are achieved. This corresponds to the results of the correlation analysis (Figure 14). Consumers from these clusters, especially from wd-cluster 0, show by far the strongest correlations between heat consumption and ex ante 3 residuals.

^{2}is narrower and shifted towards higher values. For all wd-clusters, the median of R

^{2}is greater than 0.8.

^{2}of the ex post 1 sub-model’s residual prediction is below 0.5 for more than 75% of the consumers, but the complete ex post 1 model only leads to a poorer heat prediction for 13% of the consumers, compared to the ex ante 3 model (Figure 18). However, for most consumers, only small improvements of less than 5 percentage points of σ and R

^{2}are observed. On average, σ is improved by 4 percentage points and R

^{2}by 12 percentage points. The maximum decrease in σ and R

^{2}is 3 percentage points and 5 percentage points, respectively.

#### 4.4. Ex Post 2

^{2}, but to a slightly poorer minimum prediction accuracy compared to a window length of 2 days or 14 days.

^{2}of the ex post 2 prediction is larger than 0.8 (Figure 20). For most of these consumers, R

^{2}is even larger than 0.9. The distribution of R

^{2}is broader for wd-cluster 1. With one exception, however, for all consumers from wd-cluster 0, the R

^{2}is greater than 0.6.

^{2}of about 5 percentage points. For most consumers, the improvement is in the range of up to 20 percentage points of σ or 40 percentage points of R

^{2}.

#### 4.5. Model Comparison

^{2}(b) values of each consumer. The distribution of both metrics is successively improved from ex ante 1 to ex post 2. While the improvement from ex ante 1 to ex post 1 is smooth, the ex post 2 model stands out clearly, especially in reducing poor predictions. R

^{2}of all models except ex post 2 ranges from negative values (not shown in Figure 24) to about 0.9. In contrast, R

^{2}is below 0.6 only for one consumer for the ex post 2 model.

^{2}due to the overall relatively small fluctuations in heat consumption.

## 5. Discussion

#### 5.1. Ex Ante Models

^{2}is still higher than 0.7 for the majority of the consumers. Especially for those consumers strongly dependent on the ambient temperature (wd-clusters 2 and 3), the overall accuracy is sufficient for less demanding applications, such as preliminary design studies. In contrast, strong unexplained variances in heat load often occur for consumers from wd-clusters 0 and 1, making the ex ante models less applicable for these consumers.

#### 5.2. Correlation of Heat and Electricity

#### 5.3. Ex Post 1

#### 5.4. Ex Post 2

#### 5.5. Model Comparison

## 6. Conclusions

^{2}of 0.84, the NuSVR algorithm showed slight advantages in overall accuracy compared to 51 other shallow learning algorithms from the Python-based Scikit-Learn library [34]. Based on the literature review, the LSTM algorithm is selected for the deep learning model. In addition to electricity consumption, the LSTM algorithm also evaluates the heat consumption of the last 7 days prior to the predicted day and achieves by far the highest accuracy with a median of R

^{2}of 0.94.

## 7. Directions of Future Work

- If the difference of actual and predicted load profile exceeds a defined threshold, an anomaly is indicated. This threshold must be determined.
- Only one classification algorithm uses the same inputs as the ex post 2 model and additionally evaluates the actual heat consumption.
- A two-step algorithm first uses the ex post 2 model to predict heat consumption. In the next step, another algorithm could detect an anomaly based on actual and predicted heat consumption.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

ANN | artificial neural network |

b | y-axis intercept (-) |

CHP | combined heat and power |

HLNUG | Hessian State Agency for Nature Conservation, Environment, and Geology |

HVAC | heating, ventilation, and air conditioning |

k | time lag (d) |

lin | linear |

m | slope (-) |

MAE | mean absolute error |

MPC | model predictive control |

n | number (-) |

Q | natural gas consumption, heat demand (kWh) |

R | Pearson correlation coefficient (-) |

R^{2} | coefficient of determination (-) |

RSME | root mean square error (-) |

SLP | standard load profile |

SSR | sum of squared residuals (-) |

SST | total sum of squares (-) |

SVM | support vector machines |

T | temperature (°C) |

TRY | test reference year |

wd | working day |

wknd | weekends and holidays (idle days) |

XGBoost | extreme gradient boosting |

y | value |

$\overline{\mathrm{y}}$ | mean of values |

ŷ | prediction of value |

Greek symbols | |

σ | standard deviation (-) |

Subscripts | |

amb | ambient |

d | day, daily |

h | (space) heating |

hl | space heating limit |

w | hot water or process heat |

wd | working day |

wknd | weekends and holidays (idle days) |

## Appendix A

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**Figure 1.**Boxplot of annual natural gas and electricity consumption sorted by industry in logarithmic scale. The dashed lines represent the official thresholds for online load measurement. Industries are assigned according to Eurostat’s classification method [32].

**Figure 3.**Linear cluster regression functions for working days (

**a**) and weekend days (

**b**) [28]. Q/Q

_{wd}(8 °C) is the daily heat demand normalized to the mean heat demand on working days with a mean ambient temperature of 8 °C.

**Figure 4.**Frequencies of working day clusters in the secondary industry sector (manufacture and assembly of goods) [28]. Industries are assigned according to Eurostat’s classification method [32]. Consumers operating a CHP plant are excluded from the cluster analysis due to a nonlinear correlation between natural gas consumption and heat demand.

**Figure 5.**Scheme of training of the ex post 1 model (rectangles: input feature; hexagon: algorithm; oval: output feature).

**Figure 6.**Scheme of the ex post 1 model architecture (rectangle: input feature; hexagon: algorithm; oval: output feature).

**Figure 7.**Scheme of the ex post 2 model architecture (rectangle: input feature; hexagon: algorithm; oval: output feature).

**Figure 8.**Boxplot of the distribution of R

^{2}for ex ante 3 prediction, separate for each wd-cluster.

**Figure 9.**Histogram of the distribution of R for the correlation between heat and electricity consumption (all types of days, 82 consumers).

**Figure 10.**Example consumer 1: (

**a**) normalized heat load profile, (

**b**) normalized electricity load profile, and (

**c**) normalized heat consumption versus normalized electricity consumption (the same graphics of all consumers can be found in the supplementary materials [40]).

**Figure 11.**Example consumer 2: (

**a**) normalized heat load profile (

**b**), normalized electricity load profile, and (

**c**) normalized heat consumption versus normalized electricity consumption (the same graphics of all consumers can be found in the supplementary materials [40]).

**Figure 12.**Example consumer 2: (a) normalized heat load profile (

**b**), normalized electricity load profile, and (

**c**) normalized heat consumption versus normalized electricity consumption (the same graphics of all consumers can be found in the supplementary materials [40]).

**Figure 13.**Histogram of the distribution of R between heat consumption and electricity consumption for the wd-clusters 0 (

**a**), 1 (

**b**), 2 (

**c**), and 3 (

**d**) (only wd; 82 examined consumers).

**Figure 14.**Histogram of the distribution of R between heat consumption residuals and electricity consumption for the wd-clusters 0 (

**a**), 1 (

**b**), 2 (

**c**), and 3 (

**d**) (only wd; 82 examined consumers; the x-axis is reversed to ease comparison with Figure 13).

**Figure 15.**Boxplot of the distribution of R

^{2}for the residual prediction sub-model using different algorithms.

**Figure 16.**Boxplot of the distribution of R

^{2}for NuSVR residual prediction separately for each wd-cluster.

**Figure 17.**Boxplot of the distribution of R

^{2}for the ex post 1 prediction, separate for each wd-cluster.

**Figure 18.**Histogram showing the distribution of the improvement of σ (

**a**) and R

^{2}(

**b**) by the ex post 1 model compared to ex ante 3 model.

**Figure 19.**Boxplot of the distribution of R

^{2}for the ex post 2 prediction for different time window lengths. To save computation time, the optimal window length was determined only for a random sample of 20 consumers.

**Figure 20.**Boxplot of the distribution of R

^{2}for the ex post 2 prediction, separate for each wd-cluster.

**Figure 21.**Histogram showing the distribution of the improvement of σ (

**a**) and R

^{2}(

**b**) by the ex post 2 model compared to the ex ante 3 model.

**Figure 22.**Overview on the architectures of the LSTM algorithm (ex post 2). The bars visualize the shares of the different LSTM types. The crosses visualize the number of trained parameters for each of the different architectures.

**Figure 23.**Histogram of the distribution of the residuals of all developed prediction models for all of the 82 consumers (bin size: 0.01).

**Figure 24.**Boxplot of the distribution of σ (

**a**) and R

^{2}(

**b**) of load prediction for all developed models.

**Figure 25.**Comparison of real and predicted load profiles for example consumer 1. (

**a**) Load profiles as time series for ex ante 3. (

**b**) Predicted versus real load for ex ante 3. (

**c**) Load profiles as time series for ex post 1. (

**d**) Predicted versus real load for ex post 1. (

**e**) Load profiles as time series for ex post 2. (

**f**) Predicted versus real load for ex post 2 (the same graphics of all consumers can be found in the supplementary materials [40]).

**Figure 26.**Comparison of real and predicted load profiles for example consumer 2. (

**a**) Load profiles as time series for ex ante 3. (

**b**) Predicted versus real load for ex ante 3. (

**c**) Load profiles as time series for ex post 1. (

**d**) Predicted versus real load for ex post 1. (

**e**) Load profiles as time series for ex post 2. (

**f**) Predicted versus real load for ex post 2 (the same graphics of all consumers can be found in the supplementary materials [40]).

**Figure 27.**Comparison of real and predicted load profiles for example consumer 3. (

**a**) Load profiles as time series for ex ante 3. (

**b**) Predicted versus real load for ex ante 3. (

**c**) Load profiles as time series for ex post 1. (

**d**) Predicted versus real load for ex post 1. (

**e**) Load profiles as time series for ex post 2. (

**f**) Predicted versus real load for ex post 2 (the same graphics of all consumers can be found in the supplementary materials [40]).

**Table 1.**Statistics of the load profile database of 82 industrial consumers located in Hesse, Germany, for 2018 and 2019.

Natural Gas | Electricity | |
---|---|---|

Resolution | 1 mean value per hour | 1 mean value per 15 min |

Median consumption | 4.0 GWh/a | 8.4 GWh/a |

Mean consumption | 11.8 GWh/a | 39.6 GWh/a |

Max. consumption | 152.8 GWh/a | 1279.2 GWh/a |

Min. consumption | 0.55 GWh/a | 0.18 GWh/a |

Lower threshold for online measurement | 1.5 GWh/a [7] | 0.1 GWh/a [6] |

Consumers below measuring threshold | 12 | 0 |

**Table 2.**R

^{2}and σ of the ex ante models for consumers from databases of a previous study (797 heat load profiles) [28] and of this study (82 heat and electricity load profiles).

Ex Ante 1 | Ex Ante 2 | Ex Ante 3 | Ex ante 2 (Previous Study) | Ex ante 3 (Previous Study) | |
---|---|---|---|---|---|

Mean of R^{2} | 0.28 | 0.63 | 0.64 | 0.71 | 0.79 |

Median of R^{2} | 0.53 | 0.72 | 0.75 | 0.83 | 0.88 |

σ | 0.43 | 0.32 | 0.32 | 0.24 | 0.21 |

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## Share and Cite

**MDPI and ACS Style**

Jesper, M.; Pag, F.; Vajen, K.; Jordan, U. Heat Load Profiles in Industry and the Tertiary Sector: Correlation with Electricity Consumption and Ex Post Modeling. *Sustainability* **2022**, *14*, 4033.
https://doi.org/10.3390/su14074033

**AMA Style**

Jesper M, Pag F, Vajen K, Jordan U. Heat Load Profiles in Industry and the Tertiary Sector: Correlation with Electricity Consumption and Ex Post Modeling. *Sustainability*. 2022; 14(7):4033.
https://doi.org/10.3390/su14074033

**Chicago/Turabian Style**

Jesper, Mateo, Felix Pag, Klaus Vajen, and Ulrike Jordan. 2022. "Heat Load Profiles in Industry and the Tertiary Sector: Correlation with Electricity Consumption and Ex Post Modeling" *Sustainability* 14, no. 7: 4033.
https://doi.org/10.3390/su14074033