# A Novel Probabilistic Power Flow Algorithm Based on Principal Component Analysis and High-Dimensional Model Representation Techniques

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Problem Formation

^{th}central moment, and f

**(**

_{X}**X**) represents the joint probability density function (PDF) of $\mathbf{X}$.

**(**

_{X}**X**) is complicated; moreover, the difficulty of integration is related to the non-linearity of ${G}^{l}\left(\mathbf{X}\right)$. Proper transformation and dimensionality reduction are needed to perform this integration. In this paper, PCA and HDMR are combined to perform the integration.

#### 2.2. Principal Component Analysis

**X**can be constructed:

**X**can be obtained

_{C}**X**is used to describe the correlation

_{C}**E**

**X**via

_{C}**E**, the PC matrix

**Z**is obtained

**Z**is an $n\text{\xd7}m$ matrix that has the same dimension as

**X**or

**X**. The ith row of the matrix

_{C}**Z**is the ith PC, and we use ${z}_{i}$ to express it. ${z}_{i}$ is a linear combination of IRVs that has no actual physical meaning.

**Z**has a descending order. According to PCA, the larger the variance is, the more information is contained in a PC. Therefore, in

**Z**, the importance of ${z}_{i}$ follows a descending order.

**Z**and matrix

**X**have the same dimension: if all of the components in

**Z**are retained, the scale of the PPF problem is not reduced. Thus, we must determine how many PCs should be retained, which can be achieved by calculating the PCs’ cumulative contributions.

**X**, we can obtain an uncorrelated $k\times m$ matrix ${\mathbf{Z}}_{k}$, where k<<n. Clearly, the calculation burden is not directly related to the dimension of IRVs but relates to the dimension of the retained PC matrix. Using PCA before the moment estimation procedure, the total efficiency of the algorithm is significantly improved; moreover, since there is no correlation between PCs, the impact of the correlations between IRVs is handled well.

_{i}(z

_{i}), i = 1…k of each PC z

_{i}is available, we can transform z

_{i}into a standard normal variable u

_{i}

**Y**can be expressed as a function of the independent standard Gaussian vector

**U**:

#### 2.3. High-Dimensional Model Representation

_{0}is a constant term that represents the mean response on

**Y**. ${H}_{\mathrm{i}}{(\mathrm{u}}_{\mathrm{i}})$ is a first-order term that represents the effects of a single IRV ${u}_{i}$ on

**Y**;${\text{}H}_{{i}_{1}{,i}_{2}}{(u}_{{i}_{1}}{,u}_{{i}_{2}}\mathrm{)}$ is a second-order term that represents the cooperative effects of the IRVs ${u}_{{i}_{1}}{,u}_{{i}_{2}}$ on

**Y**. Clearly, higher-order terms represent the cooperative effects of multiple IRVs on

**Y**, and ${H}_{{i}_{1}{i}_{2}\cdots {i}_{n}}{(u}_{{i}_{1}}{,u}_{{i}_{2}},\cdots {u}_{{i}_{n}})$ represents the cooperative effects of all IRVs ${u}_{{i}_{1}}{,u}_{{i}_{2}},\cdots {u}_{{i}_{n}}$ on

**Y**.

**Y**, and all cooperative effects of IRVs are ignored, while the second-order approximation considers the cooperative effects of any two IRVs. It is worth mentioning that the first-order approximation is exactly the same as that used in Zhao’s PEM scheme [15]; therefore, Zhao’s PEM scheme could be considered as a special case of HDMR.

## 3. Solution Procedure

**X**c and its corresponding covariance matrix $\mathsf{\Sigma}$;

## 4. Case Study

^{-}order moment and fourth-order moment, namely skewness and kurtosis, are important parameters to describe the shape of a PDF, we calculate the ORV’s moments up to the fourth-order moment. The performance of the proposed method is compared with that of HDMR alone, Zhao’s PEM proposed in [21], and 50000 iterations of MCS. Additionally, three contribution thresholds of PCA (${M}_{k}\text{}\ge \text{}99\%$, ${M}_{k}\text{}\ge \text{}95\%$ and ${M}_{k}\text{}\ge \text{}90\%$ respectively) are set subjectively to measure the influence of the contribution threshold on the accuracy and efficiency of the proposed method. The number of abscissas of HDMR is chosen as 5, and, for consistency, that of Zhao’s PEM is adjusted to be the same.

#### 4.1. IEEE-30 Test System

#### 4.2. Sensitivity to Correlations

#### 4.3. IEEE-118 Test System

## 5. Discussions

#### 5.1. Non-Linear Dependency

#### 5.2. Probability Distribution Approximation

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) ${\overline{\epsilon}}_{3}$ of evaluated methods and (

**b**) ${\overline{\epsilon}}_{4}$ of evaluated methods.

**Figure 2.**(

**a**) ${\overline{\epsilon}}_{3}$ of three correlation scenarios and (

**b**) ${\overline{\epsilon}}_{4}$ of three correlation scenarios.

Numbers | Abscissas | Weights |
---|---|---|

3 | 0 | 1.1816 |

±1.2247 | 2.9541 × 10^{−1} | |

5 | 0 | 9.4530 × 10^{−1} |

±0.9585 | 3.9362 × 10^{−1} | |

±2.0201 | 1.9953 × 10^{−2} | |

7 | 0 | 8.1027 × 10^{−1} |

±0.8162 | 4.2561 × 10^{−1} | |

±1.6735 | 5.4512 × 10^{−2} | |

±2.6519 | 9.7178 × 10^{−4} |

WF | Bus | Distribution Parameters | Fluctuation Range (MW) |
---|---|---|---|

1 | 15 | Beta (5.32, 7.34) | 0–25 |

2 | 16 | ||

3 | 26 | Beta (4.18, 1.80) | 0–18 |

4 | 30 |

Method | AREI | Calculation Time (s) | ||||
---|---|---|---|---|---|---|

Type of ORV | ${\overline{\epsilon}}_{1}$ | ${\overline{\epsilon}}_{2}$ | ${\overline{\epsilon}}_{3}$ | ${\overline{\epsilon}}_{4}$ | ||

Zhao’s PEM | P | 0.0944 | 1.3002 | 80.2243 | 47.4376 | 1.27 |

Q | 0.1273 | 2.4155 | 43.3680 | 48.1434 | ||

V | 4.1262 × 10^{−4} | 1.6993 | 189.0701 | 69.5160 | ||

$\theta $ | 0.1877 | 0.6081 | 125.3632 | 45.0952 | ||

HDMR | P | 0.0564 | 0.2419 | 10.4886 | 1.6998 | 52.13 |

Q | 0.0727 | 0.0922 | 6.8338 | 0.9072 | ||

V | 3.6439 × 10^{−4} | 0.1124 | 11.4566 | 0.6689 | ||

$\theta $ | 0.1417 | 0.0859 | 18.4550 | 1.6844 | ||

PCA+HDMR (${\mathrm{M}}_{\mathrm{k}}\text{}\ge \text{}99\%$) | P | 0.1132 | 1.3447 | 12.7240 | 1.9275 | 4.98 |

Q | 0.1986 | 0.4641 | 4.3147 | 1.0406 | ||

V | 5.3721 × 10^{−4} | 0.4107 | 11.4707 | 0.7315 | ||

$\theta $ | 0.0967 | 0.4550 | 21.3574 | 1.5991 | ||

PCA+HDMR (${\mathrm{M}}_{\mathrm{k}}\text{}\ge \text{}95\%$) | P | 0.0564 | 3.8661 | 16.1810 | 1.5683 | 1.79 |

Q | 0.1166 | 11.9587 | 31.1181 | 0.7725 | ||

V | 4.1600 × 10^{−4} | 9.9708 | 61.7536 | 0.7817 | ||

$\theta $ | 0.1902 | 2.6245 | 30.0443 | 1.4042 | ||

PCA+HDMR (${\mathrm{M}}_{\mathrm{k}}\text{}\ge \text{}90\%$) | P | 0.0704 | 9.2014 | 22.7468 | 1.3903 | 0.89 |

Q | 0.0544 | 20.5477 | 23.7019 | 0.9561 | ||

V | 3.4794 × 10^{−4} | 19.526 | 389.3838 | 1.0878 | ||

$\theta $ | 0.2106 | 6.1322 | 22.4920 | 1.2454 | ||

MCS | 164.86 |

Method | AREI | ||||
---|---|---|---|---|---|

Type of ORV | ${\overline{\epsilon}}_{1}$ | ${\overline{\epsilon}}_{2}$ | ${\overline{\epsilon}}_{3}$ | ${\overline{\epsilon}}_{4}$ | |

Low correlation | P | 0.1011 | 1.3796 | 9.5218 | 2.6647 |

Q | 0.0956 | 2.7176 | 7.3381 | 1.7268 | |

V | 3.9626 × 10^{−4} | 1.4616 | 12.7701 | 1.9133 | |

$\theta $ | 0.2716 | 0.1764 | 19.1871 | 0.6482 | |

Medium correlation | P | 0.0310 | 1.9153 | 11.0052 | 1.5038 |

Q | 0.0684 | 3.0162 | 8.6251 | 1.8370 | |

V | 1.8130 × 10^{−4} | 2.0106 | 10.8609 | 0.9980 | |

$\theta $ | 0.0883 | 1.2564 | 20.1425 | 1.6366 | |

High correlation | P | 0.0782 | 1.8078 | 11.9801 | 1.1834 |

Q | 0.0658 | 3.0001 | 7.5388 | 1.6983 | |

V | 4.7408 × 10^{−4} | 1.8222 | 10.9358 | 0.9288 | |

$\theta $ | 0.2106 | 6.1322 | 22.4920 | 1.2454 |

WF | Bus | Distribution Parameters | Fluctuation Range (MW) |
---|---|---|---|

1–4 | 2, 3, 44, 50 | Beta (3.76, 5.82) | 0–300 |

5–8 | 82, 88, 98, 115 | Beta (6.82, 2.44) | 0–160 |

Method | AREI | Calculation Time (s) | ||||
---|---|---|---|---|---|---|

Type of ORV | ${\overline{\epsilon}}_{1}$ | ${\overline{\epsilon}}_{2}$ | ${\overline{\epsilon}}_{3}$ | ${\overline{\epsilon}}_{4}$ | ||

Zhao’s PEM | P | 0.0379 | 3.3787 | 119.7405 | 51.6410 | 3.29 |

Q | 0.0334 | 8.2555 | 163.1540 | 52.8882 | ||

V | 2.9447 × 10^{−4} | 4.4120 | 155.3896 | 37.7793 | ||

$\theta $ | 0.0088 | 1.2012 | 78.0240 | 66.3490 | ||

HDMR | P | 0.0286 | 0.1707 | 9.3664 | 1.8081 | 406.33 |

Q | 0.0219 | 0.5212 | 12.8664 | 3.1527 | ||

V | 1.8866 × 10^{−4} | 0.4320 | 17.4510 | 3.1466 | ||

$\theta $ | 0.0058 | 0.1442 | 15.2332 | 0.5527 | ||

PCA+HDMR (${\mathrm{M}}_{\mathrm{k}}\text{}\ge \text{}99\%$) | P | 0.0379 | 2.4125 | 11.4793 | 2.0809 | 6.52 |

Q | 0.0314 | 7.9233 | 18.9019 | 3.6851 | ||

V | 3.3919 × 10^{−4} | 3.4223 | 19.6508 | 4.8801 | ||

$\theta $ | 0.0085 | 1.0298 | 17.0902 | 0.7055 | ||

MCS | 391.62 | 164.86 |

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**MDPI and ACS Style**

Li, H.; Zhang, Z.; Yin, X.
A Novel Probabilistic Power Flow Algorithm Based on Principal Component Analysis and High-Dimensional Model Representation Techniques. *Energies* **2020**, *13*, 3520.
https://doi.org/10.3390/en13143520

**AMA Style**

Li H, Zhang Z, Yin X.
A Novel Probabilistic Power Flow Algorithm Based on Principal Component Analysis and High-Dimensional Model Representation Techniques. *Energies*. 2020; 13(14):3520.
https://doi.org/10.3390/en13143520

**Chicago/Turabian Style**

Li, Hang, Zhe Zhang, and Xianggen Yin.
2020. "A Novel Probabilistic Power Flow Algorithm Based on Principal Component Analysis and High-Dimensional Model Representation Techniques" *Energies* 13, no. 14: 3520.
https://doi.org/10.3390/en13143520