# Short-Term Interval Prediction of Wind Power Based on KELM and a Universal Tabu Search Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Kernel Extreme Learning Machine

_{h}, and its activation function is g(x), the relationship between input and output can be expressed as Equation (1):

^{+}is the Moore–Penrose generalized inverse matrix of the matrix H. After $\widehat{\beta}$ is determined, if the new sample is used as the input of the ELM, the output can be calculated from Equation (5):

## 3. Interval Prediction Methodology

_{i}) is assigned to be 1; otherwise, it is set to be 0. The value of the reliability should be as close as possible to the predefined confidence level $100\left(1-\alpha \right)\%$. Under the same reliability, a narrower interval is more helpful and preferred in applications to power system scheduling. Although randomly increasing or decreasing the distance between the upper and lower boundaries of the interval can easily meet the reliability requirements, it will cause a sharpness deterioration. In this respect, one defines the width of the predicted interval as Equation (11):

_{i}is the step size of the neighbor i, r is a random number that is uniformly distributed in [−1, 1], and p

_{j}is half of the dimension size of the jth variable.

_{d}, N

_{i}, and N

_{g}are three stop criteria to terminate the algorithm.

_{d};

_{i};

_{g}, go to step 1;

^{5}local optima is selected as a case study. This benchmark mathematical function is given by

_{i}= 1, (I = 1, 2, …, 5) and f

_{global}= 0. To compare the performances of different algorithms, this case study is solved by the proposed tabu search method and the original tabu search algorithm [12]. In the numerical study, the parameters of the proposed tabu search method are set as: N

_{d}= 10, N

_{i}= 5, and N

_{g}= 3. The other parameters for the two tabu search algorithms are the same for a fair comparison. To obtain the stochastic information on the averaged performances of a method, each algorithm is run 100 times by starting from different randomly initialized points. Table 1 gives the averaged performance comparison of the aforementioned two methods for solving this test mathematical function.

^{−6}in absolute values. From the numerical results in Table 1, it is observed that all the 200 independent and random runs of the two algorithms converge to the exact global optimal solution of this test function. In other words, the proposed algorithm and the original one can escape from the nearly infinite local optima of this extremely multimodal test function and converge on the global optimum with 100% probability. However, the iterative number used by the proposed algorithm is extremely reduced as compared with the original version of the tabu search algorithm, that is, reduced from 3281 to 2045. Consequently, the improved tabu search algorithm is computationally efficient in finding the global optimal solution of a multimodal objective function with continuous variables.

## 4. Case Study

- (1)
- The winter period interval prediction. In this phase, the historical wind power data in November to December will be used.
- (2)
- The summer period interval prediction. In this phase, the historical wind power data from June to July will be used.
- (3)
- The autumn period interval prediction. In this phase, the historical wind power data from September to October will be used.
- (4)
- The spring period interval prediction. In this phase, the historical wind power data from March to April will be used.

- (1)
- In view of the metric W, the proposed algorithm outperforms HIA since the values of the metric W of the predicted interval for December, July, and April using the proposed algorithm are smaller than the corresponding values using HIA;
- (2)
- In view of the metric P (%), the two methods behave almost similarly, since the proposed method performs better in the predicted interval for December and October, while HIA performs better for July and April;
- (3)
- It should be noted that the robustness of the proposed algorithm is stronger than that of HIA since the values of the metric P (%) for the four seasons of the former are always larger than that of the predefined 90% confidence level, while the values of the same metric for only July and April of the latter are larger than that of the predefined 90% confidence level.

## 5. Conclusions

- (1)
- In view of the sharpness of the predicted interval, the proposed algorithm outperforms the existing approach, HIA;
- (2)
- In view of the robustness on the reliability of the predicted interval, the proposed algorithm behaves extremely well over the existing approach, HIA.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Number of Runs to Find Global Optimal Solutions | Averaged No. of Iterations | |
---|---|---|

Proposed | That of [12] | |

100/100 | 2045 | 3281 |

Month | Proposed | HIA | ||
---|---|---|---|---|

P (%) | W | P (%) | W | |

December | 90.92 | 0.456 | 89.86 | 0.495 |

July | 91.71 | 0.482 | 93.98 | 0.491 |

October | 90.96 | 0.457 | 87.11 | 0.410 |

April | 90.35 | 0.410 | 91.58 | 0.434 |

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**MDPI and ACS Style**

Zhou, Q.; Ma, Y.; Lv, Q.; Zhang, R.; Wang, W.; Yang, S.
Short-Term Interval Prediction of Wind Power Based on KELM and a Universal Tabu Search Algorithm. *Sustainability* **2022**, *14*, 10779.
https://doi.org/10.3390/su141710779

**AMA Style**

Zhou Q, Ma Y, Lv Q, Zhang R, Wang W, Yang S.
Short-Term Interval Prediction of Wind Power Based on KELM and a Universal Tabu Search Algorithm. *Sustainability*. 2022; 14(17):10779.
https://doi.org/10.3390/su141710779

**Chicago/Turabian Style**

Zhou, Qiang, Yanhong Ma, Qingquan Lv, Ruixiao Zhang, Wei Wang, and Shiyou Yang.
2022. "Short-Term Interval Prediction of Wind Power Based on KELM and a Universal Tabu Search Algorithm" *Sustainability* 14, no. 17: 10779.
https://doi.org/10.3390/su141710779