# Vibration Prediction and Evaluation System of the Pumping Station Based on ARIMA–ANFIS–WOA Hybrid Model and D-S Evidence Theory

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## Abstract

**:**

## 1. Introduction

## 2. Calculation Method and Procedure

#### 2.1. Autoregressive Integrated Moving Average Algorithm

_{t}is the white noise sequence, θ

_{i}is the moving average coefficient, p is the order of autoregression and q is the order of moving average.

- (1)
- The time-series data of vibration response are obtained and preprocessed. This may involve removing any outliers or missing data points, as well as normalizing the data.
- (2)
- The stationarity test is conducted after the data preprocessing step. If the test fails, differential processing shall be carried out until it passes the stationarity test.

_{t}follows an AR(p) process, the model used for the unit root test can be represented by Equations (2)–(4).

_{t}} is stationary, and the alternative hypothesis is that the sequence {y

_{t}} is non-stationary. The principle is to remove the intercept term and trend term from the residual estimate sequence {${\widehat{e}}_{r}$} and construct the LM statistic. The basis for the existence of a unit root in the original sequence is whether there is a unit root in the test {${\widehat{e}}_{r}$}.

_{0}is the residual spectral density when f = 0, and S(t)

^{2}is a consistent estimate of the residual variance. The stationarity of the sequence can be determined by comparing with the critical value.

- (3)
- The autocorrelation and partial correlation coefficients of the series are calculated to determine the order of the ARIMA model. Information criteria, including Akaike’s Information Criterion (AIC) and Bayesian Information Criterion (BIC), are adopted to select the optimal ARIMA model.$$AIC=-2\mathrm{log}\left(L\right)+2K$$$$BIC=-2\mathrm{log}\left(L\right)+K\mathrm{log}\left(T\right)$$
- (4)
- A residual sequence independence test is conducted. The Durin–Watson test, also known as DW test, is used to test for first-order autocorrelation of residuals in regression analysis, especially in the case of time series. Assuming the residual is e, the equation for the autocorrelation of each residual is e
_{t}= ρe_{t−}_{1}+ V_{t}. The null hypothesis for the test is ρ = 0, and the alternative hypothesis is ρ ≠ 0. The test statistic d is shown in Equation (9).$$d=\frac{{\displaystyle \sum _{t=2}^{T}{\left({e}_{t}-{e}_{t-1}\right)}^{2}}}{{\displaystyle \sum _{t=1}^{T}{e}_{t}^{2}}}$$

#### 2.2. Adaptive Network-Based Fuzzy Inference System

- (1)
- If x
_{1}is A_{1}and x_{2}is B_{1}, then y = p_{1}x_{1}+ q_{1}x_{2}+ r_{1}; - (2)
- If x
_{1}is A_{2}and x_{2}is B_{2}, then y = p_{2}x_{1}+ q_{2}x_{2}+ r_{2}.

_{i}(i = 1, 2) is the precise input of node i. A

_{i}(or B

_{i}) is the fuzzy subset corresponding to x

_{i}. µ

_{Ai}and µ

_{Bi}are the membership functions of A

_{i}and B

_{i}, respectively. {δ

_{i}, b

_{i}, c

_{i}} are antecedent parameters, whose values are related to the shape of the membership function.

_{i}, q

_{i}, r

_{i}} are the subsequent parameters, which will be constantly adjusted during the training process.

- (1)
- Grid partition algorithm

- (2)
- Subtractive clustering algorithm

_{i}of the data point x

_{i}is defined as [38]:

_{a}represents the influence radius of the data point density range. Obviously, the more data points within the influence radius, the greater the density D

_{i}, and the greater the probability that the data point will become the cluster center.

_{i}, and its density is defined as D

_{Xi}. The density of the remaining data points is then adjusted based on D

_{Xi}as follows.

- (3)
- Fuzzy C-means algorithm

_{1}, x

_{2}, …, x

_{n}} into k fuzzy groups and determines the cluster center {c

_{1}, c

_{2}, …, c

_{k}} for each fuzzy group based on the minimum cost function. The membership value of the jth data point x

_{j}to the ith cluster center c

_{i}is denoted as u

_{ij}, and it ranges from 0 to 1. The sum of the entire membership matrix is 1 after data normalization, namely:

**U**is the membership matrix, c

_{i}is the ith cluster center point, d

_{ij}is the Euclidean distance from the jth data point x

_{j}to the ith cluster center c

_{i}and m is the weighted index and ranges in [1, +∞).

_{j}(j = 1, 2, …, n) is brought into Equation (18) to solve the necessary conditions for J to reach the minimum value.

_{i}(Equation (20)) and membership degree u

_{ij}(Equation (21)) is obtained by taking the derivative of cost function J with respect to c

_{i}and u

_{ij}, respectively. The FCM clustering algorithm iteratively solves the problem. The cost function J stops when it becomes less than the threshold or reaches the maximum number of iterations, resulting in the determination of the final clustering center c and membership matrix

**U**.

#### 2.3. Optimization Algorithms

**X**and

**X*** are the current position and optimal position vector of the whale, respectively. t is the number of iterations.

**A**and

**C**are vector factors and are expressed as follows.

**r**

_{a}and

**r**

_{c}are random vectors between 0 and 1.

**X**

_{rand}(t) is a randomly selected search agent position vector.

_{1}and c

_{2}are non-negative constant learning factors and r

_{1}and r

_{2}are random numbers between 0 and 1.

#### 2.4. Calculation Procedure of ARIMA–ANFIS–WOA Hybrid Model and Evaluation Criterion

_{t}= {y

_{t}} is assumed as the actual time-series data. ${\widehat{Y}}_{t}$ is the final prediction result of the hybrid prediction model. ${\widehat{Y}}_{t}$ is expressed as follows.

_{1t}and f

_{2t}represent the prediction results of the first and second prediction model at time t, respectively. m is the maximum prediction time. w

_{1}and w

_{2}are the weight coefficients of the first and second prediction results, respectively.

**A**and

**C**are updated. Then, the probability p value discrimination is conducted. If p ≥ 0.5, spiral motion will be performed according to Equation (26). Otherwise, if |

**A**| > 1, random search will be performed according to Equations (28) and (29). If |

**A**| ≤ 1, prey surrounding will be conducted according to Equations (22) and (23). Next, the fitness f is calculated and compared with the optimal fitness f

_{best}. If f < f

_{best}, the position is updated, and the next iteration is proceeded until the optimal solution is achieved. Otherwise, the next iteration proceeds without updating the position.

_{k}is the predicted value of the model, ŷ

_{k}is the mean value of the predicted value and μ is the mean value of the sample data.

#### 2.5. D-S Evidence Theory

_{i}(A

_{j}) of the evidence m

_{i}for the proposition A

_{j}could be referred to Equation (38).

_{i}represents the monitoring value of the evidence m

_{i}. a

_{ij}

_{1}and a

_{ij}

_{2}denote the lower and upper limits of the range for proposition A

_{j}, respectively. These limits are determined based on threshold parameter results. The average value of the range of proposition A

_{j}is represented by u

_{ij}, as shown in Equation (39).

_{i}(A

_{j}) obtained from Equation (38).

_{i}′(A

_{j}) is the normalized probability distribution value.

## 3. Prediction Results of the Vibration Responses and Evaluation

#### 3.1. Data Source and Collection

_{2}= 3.25m, single unit design flow Q = 30m

^{3}/s, lift H = 0.96m, rated speed n = 105rpm and total installed capacity 6250 kW.

#### 3.2. Prediction Results of ARIMA Model

#### 3.3. Prediction Results of ANFIS Model

#### 3.4. Prediction Results of ARIMA–FCM–ANFIS–WOA Hybrid Model

#### 3.5. Prediction Evaluation of Different Models

#### 3.6. Vibration Prediction Results Based on ARIMA-FCM-ANFIS-WOA Hybrid Model

## 4. The Safety Early warning Study of the Pumping Station Based on the D-S Evidence Theory

#### 4.1. Evaluation Criteria for Vibration Response of the Pumping Station

^{2}. The stress control value for the concrete structure is 17.5 MPa, and for the metal structure, it is 175 MPa. Based on relevant literature [49], the criteria for evaluating the vibration response of pumping stations using D-S evidence theory are defined and presented in Table 4. The limit values for extremely unsafe level IV, unsafe level III, relatively safe level II and safe level I are 90%, 80~90%, 70~80% and 70% of the allowable value, respectively.

#### 4.2. Identification Framework for Vibration Safety Warning of Pumping Station

_{1}, A

_{2}, A

_{3}, A

_{4}, A

_{s}and A

_{u}within the set Θ. The evidence set Θ consists of L1, L2, L3, L4 and L5, which correspond to extreme response point data for displacement, velocity, acceleration, first principal stress and effective stress, respectively. The basic probability distribution function, m

_{1}, m

_{2}, m

_{3}, m

_{4}and m

_{5,}represents the supporting probability set for each level within Θ. Figure 14 depicts the D-S evidence theory matrix, where each line represents the support probability of the corresponding evidence for different operation levels of the pumping station, with a sum value of 1. Column j indicates the support probability for a specific level of pumping station operation. A high value indicates a high probability for this level.

^{2}and 0.13 MPa, respectively. The effective stress of the blade is 24.80 MPa. The basic probability distribution value of each piece of evidence is calculated and normalized according to Equations (38) and (40), as shown in Table 5.

#### 4.3. Multi-Source Information Fusion Results

_{41}and m

_{42}is m

_{4}{A

_{1}, A

_{2}, A

_{3}, A

_{4}, A

_{s}, A

_{u}} = {0.6919, 0.0600, 0.0107, 0.0107, 0.2230, 0.0036}. The vibration data of measurement points L1, L2, and L3, as well as the stress data after the first-level fusion, were then subjected to second-level fusion. The final result of the information fusion is M{A

_{1}, A

_{2}, A

_{3}, A

_{4}, A

_{s}, A

_{u}} = {0.9462, 0.0240, 0.0000, 0.0000, 0.0297, 0.0000}, the belief measure Bel{A

_{1}, A

_{2}, A

_{3}, A

_{4}, A

_{s}, A

_{u}} = {0.9462, 0.0240, 0.0000, 0.0000, 1.0000, 0.0000} and the plausibility measure Pl{A

_{1}, A

_{2}, A

_{3}, A

_{4}, A

_{s}, A

_{u}} = {0.9759, 0.0537, 0.0000, 0.0000, 1.0000, 0.0000}. They are listed in Table 6 and Figure 15.

_{1}) = 0.9462, indicating that the trust interval for the safe operation status of the pumping station is [0.9462, 1.0000]. It demonstrates that the pumping station’s safety status under design operating conditions is very good. Based on the upper and lower limits of the trust interval, the uncertainty interval is only 0.0538, indirectly proving the high reliability of the D-S evidence theory for the evaluation of the pumping station’s safety status. Bel(A

_{s}) = Pl(A

_{s}) = 1, meaning that the supporting evidence interval for the pumping station’s safety warning model reaches 1, indicating that the pumping station’s operating status is safe. Considering the complexity and fuzziness of the safety influencing factors of the pumping station, the D-S evidence method integrates different types of evidence information, including the displacement, velocity, acceleration and stress indicators, of the pumping station’s vibration response. It quantitatively displays the probabilities and degrees of trust of each proposition, providing reliable references for the evaluation and decision making of the pumping station’s operation status.

## 5. Conclusions

- (1)
- Vibration prediction research was performed using the ARIMA model. The model order was determined based on the minimum sum of AIC and BIC information. The prediction results for effective stress indicate a good fit for most time points, but there are cases where the curve abruptly changes, leading to relatively large prediction errors.
- (2)
- The prediction study involved using various fuzzy structures for the combined ANFIS prediction model. GA was employed to optimize the important model parameters, including the time lag, the number and type of the membership function in the GP–ANFIS model, the influence radius in the SC–ANFIS model and the weighted exponent in the FCM–ANFIS model. The prediction results demonstrate that the FCM–ANFIS model has better accuracy and efficiency compared to the GP–ANFIS model and SC–ANFIS model.
- (3)
- In the research on vibration prediction using the hybrid model, the weight coefficients were derived to integrate intermediate results using the WOA algorithm. The research findings indicate that the ARIMA–FCM–ANFIS–WOA model has higher prediction accuracy compared to the hybrid model using GA and PSO algorithms. The hybrid model exhibits higher accuracy than the single model ARIMA and the combined model ANFIS. The vibration responses of the concrete structure and metal unit comply with the vibration standard and do not exceed the specified amplitude in the vibration specifications.
- (4)
- A safety early warning model was developed using the D-S evidence theory to assess the safety of the pumping station. Displacement, velocity, acceleration and stress indicators were selected as the fusion indicators for analyzing the vibration data. The results indicate a confidence interval of [0.9462, 1.0000], suggesting excellent pump operation status in the design condition. The D-S evidence method provides a quantitative display of the probability and confidence of each proposition, making it highly credible for evaluating and making decisions regarding the operating state of the pumping station.
- (5)
- The proposed framework for predicting and providing warning for the vibration responses can be applied to the real-time operation and management of pumping stations. The prediction model exhibits strong generalizability and computational efficiency. However, there is room for improvement in predictive performance, particularly in areas with significant data fluctuations. Comparing the proposed models with existing benchmark datasets or alternative methods will also be considered in our future research to support the validity and significance of the predictions. In addition, the safe operation of pumping stations is influenced by numerous factors that interact in a complex manner. The models developed for pumping station safety warnings in this research do not account for all these factors and are relatively simplistic. Further research should be conducted to incorporate multiple factors and develop pumping station safety warning models that rely on more accurate predictions.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Flow chart of pumping station vibration response safety early warning based on D-S evidence theory.

**Figure 10.**Fitting results of the correlation coefficient R based on (

**a**) GP–ANFIS model; (

**b**) SC–ANFIS model; (

**c**) FCM–ANFIS model.

**Figure 11.**Prediction results of the effective stress based on different fuzzy structures of ANFIS model.

**Figure 13.**Prediction results of the concrete structure of pumping station (ARIMA–FCM–ANFIS–WOA model). (

**a**) Displacement; (

**b**) velocity; (

**c**) acceleration; (

**d**) the first principal stress.

**Figure 15.**Support probability distribution of data fusion of monitoring point of the pumping station at (

**a**) the original status; (

**b**) the first-level fusion; (

**c**) the second-level fusion.

Prediction Model | Parameter Optimization Interval | Optimal Parameter Setting | |
---|---|---|---|

GP–ANFIS model | Time lag τ | {1Δt, 2Δt, 3Δt, 4Δt, 5Δt, 6Δt } | 4Δt |

Membership function number | {2, 3, 4, 5, 6} | 2 | |

Membership function type | {Triangular, Bell, Trapezoid, Gaussian} | Gaussian | |

SC–ANFIS model | Influence radius IR | [0.20, 0.90] | 0.2272 |

FCM–ANFIS model | Weighted exponent m | [1, 9] | 3.8268 |

Prediction Model | Parameter | Value |
---|---|---|

FCM–ANFIS | Input membership function type | Gaussian |

Output membership function type | Linear | |

Fuzzy structure | Takagi-Sugeno | |

Fuzzy rule number | 10 | |

Maximum number of epochs | 1000 | |

Initial time step | 0.01 | |

Time step reduction rate/growth rate | 0.9/1.1 | |

ARIMA–FCM–ANFIS–WOA | Number of iterations | 100 |

Number of whales | 100 |

**Table 3.**Prediction results of the effective stress of the blades using different prediction models.

Model | Training Set | Test Set | ||||||
---|---|---|---|---|---|---|---|---|

RMSE (MPa) | MAE (MPa) | SD (MPa) | R | RMSE (MPa) | MAE (MPa) | SD (MPa) | R | |

ARIMA | 0.1465 | 0.0215 | 0.1465 | 0.9263 | 0.0850 | 0.0072 | 0.0852 | 0.9831 |

GP–ANFIS | 0.0900 | 0.0081 | 0.0901 | 0.9816 | 0.0709 | 0.0049 | 0.0714 | 0.9841 |

SC–ANFIS | 0.1015 | 0.0103 | 0.1015 | 0.9748 | 0.0729 | 0.0055 | 0.0727 | 0.9870 |

FCM–ANFIS | 0.0880 | 0.0077 | 0.0881 | 0.9811 | 0.0707 | 0.0047 | 0.0708 | 0.9896 |

ARIMA–FCM–ANFIS–GA | 0.1017 | 0.0103 | 0.1017 | 0.9766 | 0.0666 | 0.0044 | 0.0668 | 0.9914 |

ARIMA–FCM–ANFIS–PSO | 0.1018 | 0.0104 | 0.1019 | 0.9761 | 0.0698 | 0.0049 | 0.0699 | 0.9898 |

ARIMA–FCM–ANFIS–WOA | 0.1021 | 0.0104 | 0.1022 | 0.9765 | 0.0665 | 0.0044 | 0.0667 | 0.9915 |

**Table 4.**Evaluation criterion of the vibration responses of pumping station based on D-S evidence theory.

Level | Displacement (µm) | Velocity (mm/s) | Acceleration (m/s^{2}) | Stress (MPa) | |
---|---|---|---|---|---|

Concrete Structure | Metal Structure | ||||

Level I | <140 | <3.5 | <0.7 | <12.3 | <122.5 |

Level II | 140~160 | 3.5~4.0 | 0.7~0.8 | 12.3~14.0 | 122.5~140.0 |

Level III | 160~180 | 4.0~4.5 | 0.8~0.9 | 14.0~15.8 | 140.0~157.5 |

Level IV | >180 | >4.5 | 0.9 | 15.8 | 157.5 |

m | A_{1} | A_{2} | A_{3} | A_{4} | A_{s} | A_{u} |
---|---|---|---|---|---|---|

m_{1} | 0.40 | 0.05 | 0.05 | 0.05 | 0.40 | 0.05 |

m_{2} | 0.40 | 0.05 | 0.05 | 0.05 | 0.40 | 0.05 |

m_{3} | 0.40 | 0.05 | 0.05 | 0.05 | 0.40 | 0.05 |

m_{41} | 0.40 | 0.05 | 0.05 | 0.05 | 0.40 | 0.05 |

m_{42} | 0.41 | 0.05 | 0.05 | 0.05 | 0.39 | 0.05 |

Proposition | A_{1} | A_{2} | A_{3} | A_{4} | A_{s} | A_{u} |
---|---|---|---|---|---|---|

Basic probability M | 0.9462 | 0.0240 | 0.0000 | 0.0000 | 0.0298 | 0.0000 |

Belief measure Bel | 0.9462 | 0.0240 | 0.0000 | 0.0000 | 1.0000 | 0.0000 |

Plausibility measure Pl | 0.9759 | 0.0537 | 0.0000 | 0.0000 | 1.0000 | 0.0000 |

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## Share and Cite

**MDPI and ACS Style**

Wang, S.; Zhang, L.; Yin, G.
Vibration Prediction and Evaluation System of the Pumping Station Based on ARIMA–ANFIS–WOA Hybrid Model and D-S Evidence Theory. *Water* **2023**, *15*, 2656.
https://doi.org/10.3390/w15142656

**AMA Style**

Wang S, Zhang L, Yin G.
Vibration Prediction and Evaluation System of the Pumping Station Based on ARIMA–ANFIS–WOA Hybrid Model and D-S Evidence Theory. *Water*. 2023; 15(14):2656.
https://doi.org/10.3390/w15142656

**Chicago/Turabian Style**

Wang, Shuo, Liaojun Zhang, and Guojiang Yin.
2023. "Vibration Prediction and Evaluation System of the Pumping Station Based on ARIMA–ANFIS–WOA Hybrid Model and D-S Evidence Theory" *Water* 15, no. 14: 2656.
https://doi.org/10.3390/w15142656