# Clustering and Support Vector Regression for Water Demand Forecasting and Anomaly Detection

## Abstract

**:**

## 1. Introduction

- completely data-driven; it considers as input only historical water demand data;
- completely independent of the data source and therefore directly applicable to urban water demand (SCADA data) as well as individual customer consumption (AMR data);
- based on two-stage learning: (i) identifying and characterizing typical daily consumption patterns and (ii) dynamically generating a set of forecasting models for each typical pattern identified in the previous stage. This approach deals with the nonlinear variability of water demand at different levels, automatically characterizing periodicity (e.g., seasonality) and behaviour-related differences of different types of days and hours of the day;
- able to provide reliable forecasts of the urban water demand (SCADA data) in the short term in order to support optimization of operations, in particular pump schedule optimization;
- able to detect possible anomalies in typical water consumption behaviour at the individual customer level (AMR data) associated with metering faults, possible frauds and cyber-physical threats.

#### State of the Art of Water Demand Forecasting

## 2. Materials and Methods

#### 2.1. SCADA Data

- 149,639 junctions
- 118,950 pipes
- 26 pumping stations
- 501 wells and well pumps
- 33 storage tanks
- 95 booster pumps
- 36,295 valves
- 602 check valves
- total base demand 7.5 ± 4.2 m
^{3}/s.

_{1}, x

_{2}, …, x

_{n}} consisting of n vectors, one for each day in the observation period, where each vector x

_{i}is a set of 24 ordered values that are the hourly volume of water delivered in the i-th day. As a first step, a preliminary pre-processing of the retrieved data was performed to evaluate data quality with respect to outliers and missing values.

#### 2.2. AMR Data

^{j}= {x

^{j}

_{1}, x

^{j}

_{2}, …, x

^{j}

_{n}}, where j is the index identifying the j-th AMR and n is the number of days available for that AMR. Even for AMR data, each x

^{j}

_{i}, i = 1, …, 24 in the dataset is a 24-dimensional vector where every component is the hourly water consumption value. The available set on AMR data is small in that it was a piloting activity performed in the EU project ICeWater, and it consisted of 26 AMRs with 110 vectors of 24 hourly water consumption values in each one. Each AMR is devoted to collecting the consumption data of a customer, which in the case of Milan is a building, so different types of patterns can be observed according to residential, non-residential and mixed types of buildings. In particular, 19, 5, and 2 out of the 26 AMRs are associated with residential, non-residential and mixed water usage patterns, respectively.

#### 2.3. Time Series Clustering

- Type 1: similarity in time. The goal is to cluster together series that vary in a similar way at each time step. In this case, time series can be clustered by capturing repetitive behaviours occurring always at the same time step or in the same time window (e.g., peak/burst hours).
- Type 2: similarity in shape. The goal is to cluster together time series having common shape features e.g., common trends occurring at different times or similar sub-patterns.
- Type 3: similarity in change. The goal is to cluster together time series that vary similarly from time step to time step. In this case, the data are clustered with respect to the variations between two successive time stamps.

- ${\overline{C}}_{k}$ is the centroid of the k-th cluster.
- $\overline{X}$ is the mean vector of the whole dataset.

_{m}is the number of days in that month. Thus, the new dataset consists of M time-series data where M is the number of months. A first clustering is performed on this new dataset in order to identify k

_{1}clusters corresponding to seasonality (i.e., months characterized by similar average daily patterns). Cluster assignment at this first level is used to label the original time-series dataset; then, according to the attached labels, k

_{1}sub-datasets are selected from the original dataset in order to perform clustering on each of them. The best k

_{2}

^{q}is selected for each sub-dataset, with q = 1, …, k

_{1}. A schematic representation of the process in provided in Figure 2.

#### 2.4. Support Vector Regression Based Demand Forecasting

^{i}for all the data in D and, at the same time, is as “flat” as possible. The easiest solution is a linear function in the form:

^{i}, making f(x) completely independent of the dimensionality p of the input; it depends only on the number of Support Vectors (x

^{i}such that the associated Lagrangian multiplier is not zero). As f(x) is described in terms of dot products between data, it is not necessary to compute w explicitly, an important consideration when formulating the extension to the nonlinear case.

- Y—time-series of observed water demand (at any forecast periodicity),
- Y
_{t}—water demand observed at the time t, - Ŷ—time-series of forecasted water demand (at any forecast periodicity),
- Ŷ
_{t}—water demand forecasted at the time t, and - N—time-series length;

## 3. Results and Discussion

#### 3.1. SCADA Data and Urban Demand Forecasting

_{1}= 3) and two different types of day for each season (k

_{2}

^{i}= 2 with i = 1, …, k

_{1}), when k

_{1}= 1, …, 6 and k

_{2}

^{i}= 1, …, 4 were used.

_{1}and k

_{2}are summarized in the following Table 1 and Table 2.

_{1}= 3 clusters at the first level can be identified as

- “Fall-Winter”: from November to March
- “Spring-Summer”: from April to June and from September to October
- “Summer break”: July and August

#### 3.2. AMR Data and Anomaly Detection

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The urban water distribution network in Milan. Highlighted, in the South, the Pressure Management Zone (PMZ) named “Abbiategrasso”, where a pilot zone was identified for the installation of Automatic Meter Reading (AMRs).

**Figure 2.**A schematic representation on the two-level clustering procedure that is the first stage of the proposed approach.

**Figure 3.**A schematic representation of the second stage of the approach: learning a pool of Support Vector Machine (SVM)-based regression models for each cluster to predict hourly water demand.

**Figure 4.**A schematic representation of how forecasting is generated online according to the results obtained from the two stages of the approach: clustering and SVM learning.

**Figure 5.**A graph showing the cluster assignment over the observation period. Although the final cluster label is used, it is easy to also identify the three clusters obtained at the first level; cluster1 & cluster2, cluster3 & cluster4, and cluster5 & cluster6.

**Figure 6.**The k = 6 typical water demand patterns identified through the two-level clustering procedure. The cluster assignment is the same as the previous Figure 1.

**Figure 7.**Case 1—a residential customer: results from clustering (dotted line is the pattern associated with weekends and holidays).

**Figure 8.**Case 2—a non-residential customer: results from clustering (dotted line is the pattern associated with Sunday, Monday, and holidays).

**Figure 9.**Case 3—a non-residential “mixed” customer: results from clustering (dotted line is the pattern associated with weekends and holidays).

**Figure 10.**Mean Absolute Percentage Error (MAPE) over the observation period for the three cases (red, blue and green are residential, non-residential and non-residential mixed customers, respectively).

**Figure 14.**Case 2—actual consumption versus forecast for the day with the lowest (

**a**) and highest (

**b**) MAPE.

**Figure 16.**Case 3—actual consumption versus forecast for the day with the lowest (

**a**) andhighest (

**b**) MAPE.

**Table 1.**Silhouette and Calinski-Harabasz measures for different k

_{1}(i.e., the number of clusters at the first step of the proposed approach).

k_{1} | Silhouette | Calinski-Harabasz |
---|---|---|

2 | 0.51 | 2.96 |

3 | 0.74 | 3.15 |

4 | 0.49 | 2.12 |

**Table 2.**Silhouette and Calinski-Harabasz measures for different k

_{2}(i.e., the number of clusters at the second step of the proposed approach), where k

_{1}= 3 has been selected.

k_{2} | Silhouette | Calinski-Harabasz |
---|---|---|

2 | 0.70 | 192.78 |

3 | 0.62 | 54.35 |

4 | 0.50 | 36.68 |

**Table 3.**Mean Absolute Percentage Error (MAPE) for the best and the worst forecasts in each cluster with standard deviation.

Size | Best | Worst | |
---|---|---|---|

Cluster 1 | 67 | 0.79% ± 0.59% | 6.11% ± 2.95% |

Cluster 2 | 46 | 1.57% ± 1.18% | 14.33% ± 11.68% |

Cluster 3 | 30 | 0.84% ± 0.66% | 8.48% ± 3.53% |

Cluster 4 | 31 | 1.71% ± 2.56% | 12.84% ± 7.53% |

Cluster 5 | 54 | 1.31% ± 0.93% | 7.85% ± 13.26% |

Cluster 6 | 127 | 1.10% ± 0.85% | 6.54% ± 3.46% |

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Candelieri, A. Clustering and Support Vector Regression for Water Demand Forecasting and Anomaly Detection. *Water* **2017**, *9*, 224.
https://doi.org/10.3390/w9030224

**AMA Style**

Candelieri A. Clustering and Support Vector Regression for Water Demand Forecasting and Anomaly Detection. *Water*. 2017; 9(3):224.
https://doi.org/10.3390/w9030224

**Chicago/Turabian Style**

Candelieri, Antonio. 2017. "Clustering and Support Vector Regression for Water Demand Forecasting and Anomaly Detection" *Water* 9, no. 3: 224.
https://doi.org/10.3390/w9030224