# Analysis and Prediction of Dammed Water Level in a Hydropower Reservoir Using Machine Learning and Persistence-Based Techniques

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## Abstract

**:**

## 1. Introduction

- We show that the long-term persistence of the total amount of dammed water at Belesar reservoir has a clear yearly pattern, which supports the application of persistence-based approaches such as typical year or AR models.
- We show how ML techniques are extremely accurate in short-term prediction, involving exogenous hydro-meteorological variables.
- We evaluated the performance of a number of the ML regressors such as multi-layer perceptrons, support vector regression, extreme learning machines, or Gaussian processes in this problem of short-term prediction.
- We finally show how the seasonal pattern of dammed water is very accused in this prediction problem; thus, it can be exploited to obtain better prediction mechanisms, mainly in short-term prediction.

## 2. Data and Methods

#### 2.1. Data Description and Methodology

#### 2.2. Detrended Fluctuation Analysis for Long-Term Persistence Evaluation

- (1)
- First, the periodic annual cycle of the time series is removed, by the procedure explained in detail in [27]. The process consists in standardizing the input time series ${x}_{i}$ of length N as follows:$${x}_{i}^{\prime}=\frac{{x}_{i}-{\overline{x}}_{i}}{{\sigma}_{i}},$$
- (2)
- Then, the time series profile ${Y}_{j}$ (integrated time series) is computed as follows:$${Y}_{j}=\sum _{i=1}^{j}{x}_{i}^{\prime}.$$The profile ${Y}_{j}$ is divided into ${N}_{s}=\left(\right)open="\lfloor "\; close="\rfloor ">\frac{N}{s}$ non-overlapping segments ${Y}_{j}^{s}=\left\{{Y}_{j}^{k}\phantom{\rule{0.277778em}{0ex}}\right|\phantom{\rule{0.277778em}{0ex}}1\le k\le {N}_{s}\}$ of equal length s.For each segment ${Y}_{j}^{k}$, the local least squares straight-line ${Z}_{j}^{k}$ is calculated, which measures its local trend. As a result, a linear peace-wise function ${\tilde{Z}}_{j}^{s}$ compounding each linear fitting is obtained:$${\tilde{Z}}_{j}^{s}=\left[\begin{array}{ccccc}{Z}_{j}^{1}& \cdots & {Z}_{j}^{k}& \cdots & {Z}_{j}^{{N}_{s}}\end{array}\right],$$
- (3)
- Then, the so-called fluctuation as the root-mean-square error from this linear piece-wise function ${\tilde{Z}}_{j}^{s}$ and the profile ${Y}_{j}$ is obtained, varying the time window length s:$$F\left(s\right)=\sqrt{\frac{1}{N}\sum _{k=1}^{N}{({\tilde{Z}}_{j}^{s}-{Y}_{j}^{s})}^{2}}.$$At the time scale range where the scaling holds, $F\left(s\right)$ increases with time window s as power law $F\left(s\right)\propto {s}^{\alpha}$. Thus, the fluctuation $F\left(s\right)$ versus the time scale s would be depicted as a straight line in a log-log plot. The slope of the fitted linear regression line is the scaling exponent $\alpha $, also called correlation exponent. If this coefficient $\alpha =0.5$, the time series is uncorrelated, which means there is not long-term persistence in the time series. For larger values of $\alpha $, the time series are positively long-term correlated, which also means the long-term persistence exist across the corresponding scale range.

#### 2.3. Auto-Regressive Moving Average Models

#### 2.4. Typical Year Prediction Approach

#### 2.5. Machine Learning Regression Techniques

`R`language in the cases of ELM and SVR, where the libraries used are:

`elmNNRcpp`[45] for ELM;

`e1071`[46] for the implementation of the SVR algorithm; and the

`Python`library

`scikit-learn`MLP and GPR [47,48].

#### 2.5.1. Support Vector Regression

#### 2.5.2. Multi-Layer Perceptrons

`scikit-learn`implementation of the multi-layer perceptron with the SGD training algorithm.

#### 2.5.3. Extreme-Learning Machines

- 1.
- Randomly assign input weights ${\mathbf{w}}_{i}$ and the bias ${b}_{i}$, where $i=1,\dots ,\tilde{N}$, using a uniform probability distribution in $[-1,1]$.
- 2.
- Calculate the hidden-layer output matrix $\mathbf{H}$, defined as follows:$$\mathbf{H}={\left(\right)}_{\begin{array}{ccc}g({\mathbf{w}}_{1}{\mathbf{x}}_{1}+{b}_{1})& \cdots & g({\mathbf{w}}_{\tilde{N}}{\mathbf{x}}_{1}+{b}_{\tilde{N}})\\ \vdots & \cdots & \vdots \\ g({\mathbf{w}}_{1}{\mathbf{x}}_{N}+{b}_{1})& \cdots & g({\mathbf{w}}_{\tilde{N}}{\mathbf{x}}_{N}+{b}_{\tilde{N}})\end{array}}N\times \tilde{N}$$
- 3.
- Calculate the output weight vector $\beta $ as follows:$$\beta ={\mathbf{H}}^{\u2020}\mathbf{T},$$

#### 2.5.4. Gaussian Processes for Regression

## 3. Experiments and Results

#### 3.1. Experimental Design

#### 3.2. Long-Term Reservoir Level Analysis

#### 3.3. Short-Term Reservoir Level Prediction

#### 3.3.1. Standard Data, Random Partitioning

#### 3.3.2. Standard Data, Temporal Partitioning

#### 3.3.3. Seasonal Data, Temporal Partitioning

#### 3.3.4. Discussion on Short-Term Prediction Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Geographical location of the meteorological stations (blue dots) and the Belesar reservoir (red dot). (1) Belesar reservoir; (2) Sarria river (Pobra de S. Xulian); (3) Neira river (O Paramo); (4) Miño River (Lugo); (5) Narla river (Gondai); (6) Ladra river (Begonte); (7) Miño River (Cela); (8) Miño River (Pontevilar); and (9) Azúmara river (Reguntille).

**Figure 2.**Time series of weekly water level at Belesar reservoir: (

**a**) complete time series (since 1964); and (

**b**) time series since 2009, when data from upstream measuring stations are available.

**Figure 3.**Example of a support vector regression process for a two-dimensional-regression problem, with an $\u03f5$-insensitive loss function.

**Figure 6.**AR(9) and typical year persistence-based long-term predictions for Belesar reservoir water level.

**Figure 7.**Results of the different ML regression techniques for different variables considered (datasets) in the case of standard data (all seasons) and random partitioning: (

**a**) the RMSE results; and (

**b**) the MAE results.

**Figure 9.**Results of the different ML regression techniques for different variables considered (datasets) in the case of standard data (all seasons) and temporal partitioning: (

**a**) the RMSE results; and (

**b**) the MAE results.

**Table 1.**Descriptive statistics for the dependent variable (water level at Belesar reservoir) for both problems presented in this paper: long-term prediction and analysis (time series since 1964) and short-term prediction (time series since 2009).

Time Series Descriptive Statistics | Water Level Data for Short-Term Prediction Since 2009 (hm${}^{3}$) | Water Level Data for Long-Term Analysis Since 1964 (hm${}^{3}$) |
---|---|---|

Minimum | 77.36 | 33.21 |

1st Quartile | 225.92 | 244.61 |

Median | 374.92 | 391.64 |

Mean | 354.42 | 371.80 |

3rd Quartile | 491.16 | 511.54 |

Maximum | 560.68 | 653.62 |

Standard Deviation | 147.19 | 161.00 |

**Table 2.**Input variables included in each of the datasets used in the experimental evaluation of the ML regression techniques for short-term analysis and prediction of dammed water level at Belesar.

Dataset | Variables Included | Number of Variables |
---|---|---|

1 | Upstream and tributaries’ flow | 19 |

2 | Upstream and tributaries’ flow & Precipitations | 28 |

3 | Upstream and tributaries’ flow, Precipitations & Reservoir output | 29 |

4 | Upstream and tributaries’ flow, Precipitations, Reservoir output & Snow data | 31 |

MLP |
---|

Transfer function: {Logistic, Hyperbolic Tangent, ReLU} |

Architecture: {25, 50, 75, 100, 125, 25:25, 50:50, 75:75, |

50:50:50, 75:50:25} |

${\mathit{L}}_{\mathbf{2}}$: $0.0035$ |

Learning rate: {$0.0001$, $0.00085$, $0.0016$, |

$0.00235$, $0.0031$, $0.00385$, $0.0046$} |

Training algorithms: |

{Stochastic Gradient Descent: $\mu $ = {$0.9$, $0.85$, $0.7$}, |

Adam: ${\beta}_{1}$ = $0.9$, ${\beta}_{2}$ = $0.999$ } |

SVR Linear |

C: $\{0.25,0.5,1,2,4,...,512\}$ |

$\u03f5:$$\{0,0.25,...,0.2\}$ |

SVR rbf |

C: $\{0.25,0.5,1,2,4,...,512\}$ |

$\u03f5:$$\{0,0.25,...,0.2\}$ |

$\gamma :$$\{\frac{1}{N},\frac{1}{N}+0.1,...,0.4\}$, |

(where N is the number of input variables) |

GPR |

Kernel: {Dot Product, Radial, |

Dot Product + White Noise} |

$\alpha :$$1\times {10}^{-6}$ |

ELM |

Transfer function: {ReLU, Sigmoid, Radial} |

Number of layers: 1 |

Number of neurons: {50, 75} |

**Table 4.**Results of the different ML regression techniques for different variables considered (datasets) in the case of standard data (all seasons) and random partitioning. Boldface stands for the best value found in a dataset, and italic for the second best.

Dataset | Model | RMSE (hm${}^{3}$) | MAE (hm${}^{3}$) |
---|---|---|---|

1 | ELM | 33.68 | 19.61 |

1 | SVM (lin) | 35.85 | 22.76 |

1 | SVM (rbf) | 24.63 | 16.37 |

1 | MLP | 24.86 | 16.45 |

1 | GPR | 32.51 | 21.10 |

2 | ELM | 33.17 | 20.22 |

2 | SVM (lin) | 30.52 | 18.49 |

2 | SVM (rbf) | 30.63 | 22.42 |

2 | MLP | 22.94 | 16.28 |

2 | GPR | 36.91 | 21.57 |

3 | ELM | 32.50 | 17.93 |

3 | SVM (lin) | 28.72 | 15.58 |

3 | SVM (rbf) | 30.35 | 20.80 |

3 | MLP | 20.20 | 14.26 |

3 | GPR | 34.27 | 18.12 |

4 | ELM | 31.13 | 17.99 |

4 | SVM (lin) | 27.65 | 15.04 |

4 | SVM (rbf) | 29.79 | 20.24 |

4 | MLP | 20.44 | 14.04 |

4 | GPR | 36.22 | 18.29 |

**Table 5.**Results of the different ML regression techniques for different variables considered (datasets) in the case of standard data (all seasons) and temporal partitioning. Boldface stands for the best value found in a dataset, and italic for the second best.

Dataset | Model | RMSE (hm${}^{3}$) | MAE (hm${}^{3}$) |
---|---|---|---|

1 | ELM | 24.27 | 17.48 |

1 | SVM (lin) | 19.96 | 16.18 |

1 | SVM (rbf) | 19.34 | 14.55 |

1 | MLP | 21.66 | 17.24 |

1 | GPR | 21.74 | 18.28 |

2 | ELM | 24.86 | 18.31 |

2 | SVM (lin) | 25.35 | 20.79 |

2 | SVM (rbf) | 22.56 | 16.46 |

2 | MLP | 23.42 | 17.38 |

2 | GPR | 24.43 | 19.28 |

3 | ELM | 21.59 | 14.56 |

3 | SVM (lin) | 16.44 | 11.28 |

3 | SVM (rbf) | 19.28 | 12.09 |

3 | MLP | 21.12 | 15.17 |

3 | GPR | 17.00 | 11.99 |

4 | ELM | 22.87 | 15.10 |

4 | SVM (lin) | 18.60 | 12.74 |

4 | SVM (rbf) | 21.88 | 14.25 |

4 | MLP | 20.37 | 15.19 |

4 | GPR | 18.65 | 13.40 |

**Table 6.**Best results obtained for the case of temporal partitioning. Only the best algorithm in each season and dataset are detailed in this table.

Season | Model | Dataset | RMSE (hm${}^{3}$) | MAE (hm${}^{3}$) |
---|---|---|---|---|

Spring | SVR (Linear) | 3 | 17.41 | 13.94 |

Summer | ELM | 1 | 7.83 | 5.73 |

Autumn | Gaussian Process | 3 | 14.40 | 11.01 |

Winter | SVR (RBF) | 1 | 22.14 | 15.39 |

Average Metrics | 15.45 | 11.52 |

**Table 7.**General comparison of the experimental results in the short-term prediction of water level at Belesar reservoir. Boldface stands for the best value found in a dataset, and italic for the second best.

Experiment | RMSE (hm${}^{3}$) | MAE (hm${}^{3}$) |
---|---|---|

Standard data, random partitioning | 20.20 | 14.26 |

Standard data, temporal partitioning | 16.44 | 11.28 |

Seasonal data, temporal partitioning | 15.45 | 11.52 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Castillo-Botón, C.; Casillas-Pérez, D.; Casanova-Mateo, C.; Moreno-Saavedra, L.M.; Morales-Díaz, B.; Sanz-Justo, J.; Gutiérrez, P.A.; Salcedo-Sanz, S.
Analysis and Prediction of Dammed Water Level in a Hydropower Reservoir Using Machine Learning and Persistence-Based Techniques. *Water* **2020**, *12*, 1528.
https://doi.org/10.3390/w12061528

**AMA Style**

Castillo-Botón C, Casillas-Pérez D, Casanova-Mateo C, Moreno-Saavedra LM, Morales-Díaz B, Sanz-Justo J, Gutiérrez PA, Salcedo-Sanz S.
Analysis and Prediction of Dammed Water Level in a Hydropower Reservoir Using Machine Learning and Persistence-Based Techniques. *Water*. 2020; 12(6):1528.
https://doi.org/10.3390/w12061528

**Chicago/Turabian Style**

Castillo-Botón, C., D. Casillas-Pérez, C. Casanova-Mateo, L. M. Moreno-Saavedra, B. Morales-Díaz, J. Sanz-Justo, P. A. Gutiérrez, and S. Salcedo-Sanz.
2020. "Analysis and Prediction of Dammed Water Level in a Hydropower Reservoir Using Machine Learning and Persistence-Based Techniques" *Water* 12, no. 6: 1528.
https://doi.org/10.3390/w12061528