# A Short-Term Photovoltaic Power Prediction Model Based on an FOS-ELM Algorithm

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## Abstract

**:**

## 1. Introduction

- (1)
- The computation complexion of ELM is much lower than many other machine learning algorithms.
- (2)
- The learning speed of ELM is much faster than most feed forward network learning algorithms.
- (3)
- The generalization performance of ELM is better than many others.
- (4)
- The amount of hidden layer nodes is small and they do not need to be tuned [16].

- We introduced an online learning model with a Forgetting Mechanism to the area of photovoltaic prediction, which can update the data in real time.
- We compared the ELM, OS-ELM and FOS-ELM prediction models in predicting PV power in different seasons.
- The simulation results showed that the FOS-ELM model can not only improve the accuracy but also reduce the training time.

## 2. Prediction Algorithm

#### 2.1. Classical Extreme Learning Machine (ELM)

#### 2.2. Online Sequential ELM (OS-ELM)

- (a)
- Randomly generate ${a}_{j}$ and ${b}_{j}$ where $j=1,2\dots ,L$.
- (b)
- Calculate the initial hidden layer output matrix ${H}_{0}$.$${H}_{0}=\left[\begin{array}{ccc}G({a}_{1},{b}_{1},{x}_{1})& ...& G({a}_{L},{b}_{L},{x}_{1})\\ \vdots & \ddots & \vdots \\ G({a}_{1},{b}_{1},{x}_{{N}_{0}})& ...& G({a}_{L},{b}_{L},{x}_{{N}_{0}})\end{array}\right],$$
- (c)
- Estimate the initial output weight vector:$$\mathsf{\beta}={H}^{T}(H{H}^{T}+\frac{1}{C}E)Y$$$${P}_{0}={({H}_{0}{}^{T}{H}_{0}+\frac{1}{C}E)}^{-1},\phantom{\rule{0ex}{0ex}}{Y}_{0}={\left[\begin{array}{ccc}{y}_{1}& \dots & {y}_{{N}_{0}}\end{array}\right]}^{T},$$
- (d)
- Set $k=0$.

- (a)
- When the ${(k+1)}^{th}$ chunk of new data ${\left\{({x}_{i},{y}_{i})\right\}}_{i=({\displaystyle \sum _{j=0}^{k}{N}_{j}})+1}^{i={\displaystyle \sum _{j=0}^{k+1}{N}_{j}}}$ is ready,
- (b)
- Calculate the partial hidden layer output matrix ${H}_{k+1}$ based on the latest data.$${H}_{k+1}=\left[\begin{array}{ccc}G({a}_{1},{b}_{1},{x}_{({\displaystyle \sum _{j=0}^{k}{N}_{j})+1}})& ...& G({a}_{L},{b}_{L},{x}_{({\displaystyle \sum _{j=0}^{k}{N}_{j})}+1})\\ \vdots & \ddots & \vdots \\ G({a}_{1},{b}_{1},{x}_{{\displaystyle \sum _{j=0}^{k+1}{N}_{j}}})& ...& G({a}_{L},{b}_{L},{x}_{{\displaystyle \sum _{j=0}^{k+1}{N}_{j}}})\end{array}\right];\phantom{\rule{0ex}{0ex}}{Y}_{k+1}={\left[\begin{array}{ccc}{y}_{({\displaystyle \sum _{j=0}^{k}{N}_{j})+1}}& \dots & {y}_{{\displaystyle \sum _{j=0}^{k+1}{N}_{j}}}\end{array}\right]}^{T};$$
- (c)
- Estimate the new ${P}_{k+1}$ and ${\mathsf{\beta}}^{(k+1)}$ based on (7) and (8).$${P}_{k+1}={P}_{k}-{P}_{k}{H}_{k+1}^{T}{(I+{H}_{k+1}{P}_{k}{H}_{k+1}^{T})}^{-1}{H}_{k+1}$$$${\mathsf{\beta}}^{(k+1)}={\mathsf{\beta}}^{(k)}+{P}_{k+1}{H}_{k+1}^{T}({Y}_{k+1}-{H}_{k+1}{\mathsf{\beta}}^{(k)})$$
- (d)
- Set $k=k+1$, and then go back to Step 2.

#### 2.3. OS-ELM with Forgetting Mechanism (FOS-ELM)

- (a)
- Calculate the partial hidden layer output matrix ${H}_{k+1}$, which corresponds to the latest data.$${H}_{k+1}=\left[\begin{array}{ccc}G({a}_{1},{b}_{1},{x}_{({\displaystyle \sum _{j=0}^{k}{N}_{j})+1}})& ...& G({a}_{L},{b}_{L},{x}_{({\displaystyle \sum _{j=0}^{k}{N}_{j})}+1})\\ \vdots & \ddots & \vdots \\ G({a}_{1},{b}_{1},{x}_{{\displaystyle \sum _{j=0}^{k+1}{N}_{j}}})& ...& G({a}_{L},{b}_{L},{x}_{{\displaystyle \sum _{j=0}^{k+1}{N}_{j}}})\end{array}\right],\phantom{\rule{0ex}{0ex}}{Y}_{k+1}={\left[\begin{array}{ccc}{y}_{({\displaystyle \sum _{j=0}^{k}{N}_{j})+1}}& \dots & {y}_{({\displaystyle \sum _{j=0}^{k}{N}_{j})+1}}\end{array}\right]}^{T};$$
- (b)
- Estimate the new ${P}_{k+1}$ and ${\mathsf{\beta}}^{(k+1)}$ based on (9) and (10).$${P}_{k+1}={P}_{k}-{P}_{k}{\left[\begin{array}{c}-{H}_{k-s-1}\\ {H}_{k+1}\end{array}\right]}^{T}\times {(I+\left[\begin{array}{c}{H}_{k-s-1}\\ {H}_{k+1}\end{array}\right]{P}_{k}{\left[\begin{array}{c}-{H}_{k-s-1}\\ {H}_{k+1}\end{array}\right]}^{T})}^{-1}\times \left[\begin{array}{c}{H}_{k-s-1}\\ {H}_{k+1}\end{array}\right]{P}_{k}$$$${\mathsf{\beta}}^{(k+1)}={\mathsf{\beta}}^{(k)}+{P}_{k+1}{\left[\begin{array}{c}-{H}_{k-s+1}\\ {H}_{k+1}\end{array}\right]}^{T}\times \left(\left[\begin{array}{c}{Y}_{k-s+1}\\ {Y}_{k+1}\end{array}\right]-\left[\begin{array}{c}{H}_{k-s+1}\\ {H}_{k+1}\end{array}\right]{\mathsf{\beta}}^{(k)}\right)$$
- (c)
- Set $k=k+1$, and then go back to Step 2.

## 3. Model Architecture

#### 3.1. Physical Model

^{2}); R is the solar radiation (kW/m

^{2}); e is the loss in efficiency of the array for every degree Celsius of cell temperature increase (always equals to 0.005); and ${t}_{0}$ is the ambient temperature (°C).

#### 3.2. Input Vector

#### 3.3. Data Pre-Processing

#### 3.4. Error Evaluation

#### 3.5. Flowchart of the Model

## 4. Examples and Simulation

#### 4.1. Accuracy Comparison in a Single Day

#### 4.2. Monthly Average Accuracy Comparison

#### 4.3. Comparison of Training Time

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Flowchart of the online sequential extreme learning machine with forgetting mechanism (FOS-ELM) prediction model.

**Figure 4.**Comparison of three models on a sunny day. (

**a**) FOS-ELM; (

**b**) online sequential extreme learning machine (OS-ELM); (

**c**) extreme learning machine (ELM).

Season | Model | $\mathit{n}\mathit{R}\mathit{M}\mathit{S}\mathit{E}$ | $\mathit{M}\mathit{A}\mathit{P}\mathit{E}$/% |
---|---|---|---|

Spring (Apri–June) | FOS-ELM | 0.0953 | 15.492 |

OS-ELM | 0.1041 | 16.730 | |

ELM | 0.1126 | 18.483 | |

Summer (July–September) | FOS-ELM | 0.0892 | 14.329 |

OS-ELM | 0.0933 | 15.883 | |

ELM | 0.1083 | 16.032 | |

Autumn (October–December) | FOS-ELM | 0.0974 | 15.289 |

OS-ELM | 0.1018 | 16.325 | |

ELM | 0.1219 | 17.933 | |

Winter (January–March) | FOS-ELM | 0.0876 | 15.245 |

OS-ELM | 0.0945 | 16.319 | |

ELM | 0.0983 | 17.703 |

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**MDPI and ACS Style**

Wang, J.; Ran, R.; Zhou, Y.
A Short-Term Photovoltaic Power Prediction Model Based on an FOS-ELM Algorithm. *Appl. Sci.* **2017**, *7*, 423.
https://doi.org/10.3390/app7040423

**AMA Style**

Wang J, Ran R, Zhou Y.
A Short-Term Photovoltaic Power Prediction Model Based on an FOS-ELM Algorithm. *Applied Sciences*. 2017; 7(4):423.
https://doi.org/10.3390/app7040423

**Chicago/Turabian Style**

Wang, Jidong, Ran Ran, and Yue Zhou.
2017. "A Short-Term Photovoltaic Power Prediction Model Based on an FOS-ELM Algorithm" *Applied Sciences* 7, no. 4: 423.
https://doi.org/10.3390/app7040423