# Dynamic Risk Assessment of Voltage Violation in Distribution Networks with Distributed Generation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Pdf Estimation Based on BKDE

_{1}, ..., X

_{N}, with an unknown density function f(x), the KDE is calculated as follows:

_{i}represents the ith sample.

_{i}from the density point x does not affect the estimation, and the closer the distance from x, the more weight should be assigned to X

_{i}. Therefore, a unimodal kernel function centered at 0 is typically chosen. In this paper, the Epanechnikov kernel function is used:

_{i})/h and substituting (2) into (1), we can obtain the complete formula for the KDE.

#### 2.1. pdf Estimation for Load

_{L}load nodes, the load data for the distribution network at time i, denoted as ${P}_{i}^{L}$, can be expressed as follows:

_{L}, can be described as follows:

_{d}represents the number of monitoring sample points within one day.

^{2}K(u)du, and K(u) is the kernel function as shown in (2).

^{2}. However, when the pdf has a boundary at 0, the above expression is no longer applicable. Instead, it is replaced by:

_{i}(x) = ${\int}_{-1}^{x/h}$u

^{i}K(u)du. It can be observed that when x ≥ h, a

_{0}(x) = 1 and a

_{1}(x) = 0, leading to no difference in bias from (6). However, when 0 ≤ x < h, both a

_{0}(x) and a

_{1}(x) are non-zero, implying that bias is always present at the boundary.

_{0}and b

_{1}are similar to a

_{0}and a

_{1}from (7), except that (8) is specific to L(x). By performing a linear combination of (7) and (8), we can obtain:

^{B}(x) will have a simple form:

^{B}(x) is the same as K(x), but when 0 ≤ x < h (i.e., at the boundary), it adjusts the original kernel function. When x < 0, the estimated value is taken as 0.

#### 2.2. Conditional pdf Estimation for DG

_{i}(x) are:

_{t}is the DG output at time t, y

_{t}is the predicted value of the conditional variable at time t, ${\widehat{f}}_{P}$ is the conditional pdf for DG, ${\widehat{f}}_{Y}$ is the pdf for the conditional variable, and ${\widehat{f}}_{P,Y}$ is the joint pdf.

_{i}represents historical samples of DG, Y

_{i}represents corresponding historical samples of conditional variables, h

_{1}and h

_{2}are bandwidth parameters, and ${K}_{1}^{B}$(∙) and ${K}_{2}^{B}$(∙) are kernel functions.

## 3. PLFCM Based on the Rosenblatt Inverter Transform

#### 3.1. Independent Transform Based on Rosenblatt Inverter Transformation

#### 3.1.1. Joint pdf for DG

_{1}, ..., h

_{d}are bandwidth parameters, and K

_{j}(∙) are kernel functions corresponding to the variable ${P}_{j}^{D}$ (j = 1, ..., d).

#### 3.1.2. Independent Transform

^{C}= (${U}_{1}^{C}$, ${U}_{2}^{C}$, ..., ${U}_{N}^{C}$)

^{T}into independent standard normal variables U

^{I}= (${U}_{1}^{I}$,${U}_{2}^{I}$,...,${U}_{I}^{n}$)

^{T}. According to the principle of equiprobability marginal transformation, it can be expressed as follows:

^{I}, which can be expressed as:

#### 3.2. PLFCM

#### 3.2.1. Linearized Load Flow Model

_{0}and T

_{0}are sensitivity matrices. J

_{0}is the Jacobian matrix obtained in the last iteration of the power flow calculation. G

_{0}= (∂Z/∂X)|

_{X}

_{=X0}.

#### 3.2.2. Cumulant Computation

_{v}represents the vth order origin moment.

#### 3.2.3. Cornish–Fisher Series Expansion

^{−1}(α), and g

_{v}represents the vth order normalized cumulant.

^{−1}(α), the CDF F(x) of the output random variable X can be determined, thereby providing the probability of node voltage violation.

## 4. The Voltage Violation Risk Assessment Metric Based on the Utility Function

#### 4.1. The Probability of Voltage Violation

_{i}) represents the voltage violation probability for node i, F(V

_{i}) represents the voltage CDF for node I, and ${V}_{i}^{min}$, and ${V}_{i}^{max}$ represent the lower and upper voltage limits permissible for node i, respectively.

#### 4.2. The Severity of Voltage Violation

_{e}, is represented by linear functions.

#### 4.3. The Comprehensive Assessment of Voltage Violation

_{i}, and the system-level comprehensive risk index R

_{s}, can be defined as follows:

_{i}) represents the voltage pdf of node i, which can be derived from F(V

_{i}) using numerical differentiation.

## 5. The Process of the Proposed Method

## 6. Case Study

#### 6.1. The Performance of pdf Modeling

_{c}represents the mean or variance of the modeled pdf, and Value

_{r}represents the mean or variance of the real measured data.

#### 6.2. Case 1: High DG Output

^{2}, while at PV2, it is 900 W/m

^{2}. Based on the BKDE, the pdfs and CDFs of WT, PV1, and PV2 can be obtained, as shown in Figure 8, along with the joint pdf and joint CDF between these variables, as illustrated in Figure 9, Figure 10 and Figure 11.

#### 6.3. Case 2: Low DG Output

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Ehsan, A.; Yang, Q. State-of-the-art techniques for modelling of uncertainties in active distribution network planning: A review. Appl. Energy
**2019**, 239, 1509–1523. [Google Scholar] [CrossRef] - Wang, Y.; Ma, H.; Xiao, X.; Wang, Y.; Zhang, Y.; Wang, H. Harmonic State Estimation for Distribution Networks Based on Multi-Measurement Data. IEEE Trans. Power Deliv.
**2023**, 38, 2311–2325. [Google Scholar] [CrossRef] - Mansouri, N.; Lashab, A.; Guerrero, J.M.; Cherif, A. Photovoltaic power plants in electrical distribution networks: A review on their impact and solutions. IET Renew. Power Gener.
**2020**, 14, 2114–2125. [Google Scholar] [CrossRef] - Palahalli, H.; Maffezzoni, P.; Gruosso, G. Gaussian Copula Methodology to Model Photovoltaic Generation Uncertainty Correlation in Power Distribution Networks. Energies
**2021**, 14, 2349. [Google Scholar] [CrossRef] - Toft, H.S.; Svenningsen, L.; Sørensen, J.D.; Moser, W.; Thøgersen, M.L. Uncertainty in wind climate parameters and their influence on wind turbine fatigue loads. Renew. Energy
**2016**, 90, 352–361. [Google Scholar] [CrossRef] - Ye, K.; Zhao, J.; Zhang, H.; Zhang, Y. Data-Driven Probabilistic Voltage Risk Assessment of MiniWECC System with Uncertain PVs and Wind Generations using Realistic Data. IEEE Trans. Power Syst.
**2022**, 37, 4121–4124. [Google Scholar] [CrossRef] - Tourandaz Kenari, M.; Sepasian, M.S.; Setayesh Nazar, M. Probabilistic assessment of static voltage stability in distribution systems con-sidering wind generation using catastrophe theory. IET Gener. Transm. Distrib.
**2019**, 13, 2856–2865. [Google Scholar] [CrossRef] - Deng, W.; Zhang, B.; Ding, H.; Li, H. Risk-Based Probabilistic Voltage Stability Assessment in Uncertain Power System. Energies
**2017**, 10, 180. [Google Scholar] [CrossRef] - Fang, S.; Cheng, H.; Xu, G. A Modified Nataf Transformation-based Extended Quasi-Monte Carlo Simulation Method for Solving Probabilistic Load Flow. Electr. Power Compon. Syst.
**2016**, 44, 1735–1744. [Google Scholar] [CrossRef] - Xie, Z.Q.; Ji, T.Y.; Li, M.S.; Wu, Q.H. Quasi-Monte Carlo Based Probabilistic Optimal Power Flow Considering the Correlation of Wind Speeds Using Copula Function. IEEE Trans. Power Syst.
**2018**, 33, 2239–2247. [Google Scholar] [CrossRef] - Mohammadi, M.; Shayegani, A.; Adaminejad, H. A new approach of point estimate method for probabilistic load flow. Int. J. Electr. Power Energy Syst.
**2013**, 51, 54–60. [Google Scholar] [CrossRef] - Zou, B.; Xiao, Q. Probabilistic load flow computation using univariate dimension reduction method. Int. Trans. Electr. Energy Syst.
**2014**, 24, 1700–1714. [Google Scholar] [CrossRef] - Singh, V.; Moger, T.; Jena, D. Probabilistic Load Flow for Wind Integrated Power System Considering Node Power Uncertainties and Random Branch Outages. IEEE Trans. Sustain. Energy
**2023**, 14, 482–489. [Google Scholar] [CrossRef] - Wang, Y.; Yu, M.; Ma, X.; Xiao, X.; Wang, C.; Liu, C. A General Harmonic Probability Model based on Load Operating states identification. IEEE Trans. Power Deliv.
**2022**, 38, 2247–2260. [Google Scholar] [CrossRef] - Funke, B.; Kawka, R. Nonparametric density estimation for multivariate bounded data using two non-negative multipli-cative bias correction methods. Comput. Stat. Data Anal.
**2015**, 92, 148–162. [Google Scholar] [CrossRef] - Zhang, Y.; Wang, J.; Luo, X. Probabilistic wind power forecasting based on logarithmic transformation and boundary kernel. Energy Convers. Manag.
**2015**, 96, 440–451. [Google Scholar] [CrossRef] - Chen, W.; Yan, H.; Pei, X.; Wu, B. Probabilistic load flow calculation in distribution system considering the stochastic char-acteristic of wind power and electric vehicle charging load. In Proceedings of the 2016 IEEE PES Asia-Pacific Power and Energy Engineering Conference (APPEEC), Xi’an, China, 25–28 October 2016; pp. 1861–1866. [Google Scholar]
- Zhu, Z.; Lu, S.; Peng, S. An Improved Stochastic Response Surface Method Based Probabilistic Load Flow for Studies on Correlated Wind Speeds in the AC/DC Grid. Energies
**2018**, 11, 3501. [Google Scholar] [CrossRef] - Der Kiureghian, A.; Liu, P.L. Structural Reliability under Incomplete Probability Information. J. Mech. Eng.
**1986**, 112, 85–104. [Google Scholar] [CrossRef] - Aven, T. On how to define, understand and describe risk. Reliab. Eng. Syst. Saf.
**2010**, 95, 623–631. [Google Scholar] [CrossRef] - Kiefer, M.; Heimel, M.; Breß, S.; Markl, V. Estimating join selectivities using bandwidth-optimized kernel density models. Proc. VLDB Endow.
**2017**, 10, 2085–2096. [Google Scholar] [CrossRef] - Helias, M.; Dahmen, D. Statistical Field Theory for Neural Networks; Lecture Notes in Physics; Springer: Cham, Switzerland, 2020; Volume 970. [Google Scholar]
- Guo, X.J.; Cai, D.F. Comparison of probabilistic load flow calculation based on cumulant method among different series expansions. Electr. Power Autom. Equip.
**2013**, 33, 85–90+110. [Google Scholar]

BKDE | KDE | |||
---|---|---|---|---|

RE of Mean (%) | RE of Variance (%) | RE of Mean (%) | RE of Variance (%) | |

WT | 5.35 | 10.20 | 15.55 | 27.43 |

PV1 | 0.070 | 2.42 | 0.23 | 7.96 |

PV2 | 0.051 | 2.98 | 0.18 | 8.07 |

Pearson | Kendall | Spearman | |
---|---|---|---|

WT/PV1 | −0.2913 | −0.2144 | −0.2922 |

WT/PV2 | −0.2882 | −0.2126 | −0.2896 |

PV1/PV2 | 0.9926 | 0.9511 | 0.9937 |

Pearson | Kendall | Spearman | |
---|---|---|---|

WT/PV1 | −0.0873 | −0.1193 | −0.0800 |

WT/PV2 | −0.0925 | −0.0625 | −0.0929 |

PV1/PV2 | 0.1256 | 0.0795 | 0.1004 |

**Table 4.**The relative errors of PLFCM with Rosenblatt inverse transform and PLFCM with Orthogonal inverse transform.

Node No. | PLFCM with Rosenblatt Inverse Transform | PLFCM with Orthogonal Inverse Transform | ||
---|---|---|---|---|

RE of Mean (%) | RE of Variance (%) | RE of Mean (%) | RE of Variance (%) | |

2 | 6.1626 × 10^{−5} | 0.8711 | 0.0012 | 22.0573 |

3 | 3.3372 × 10^{−4} | 1.1296 | 0.0078 | 23.3937 |

4 | 0.0016 | 0.5352 | 0.0159 | 30.5238 |

5 | 0.0027 | 0.3139 | 0.0247 | 34.2327 |

6 | 0.0047 | 0.2804 | 0.0404 | 32.7683 |

7 | 0.0043 | 0.2230 | 0.0329 | 34.4186 |

8 | 0.0042 | 0.4460 | 0.0119 | 33.8826 |

9 | 0.0039 | 0.5258 | 0.0118 | 30.8185 |

10 | 0.0039 | 0.5358 | 0.0288 | 37.4144 |

11 | 0.0038 | 0.5457 | 0.0311 | 30.5229 |

12 | 0.0038 | 0.5538 | 0.0350 | 36.3805 |

13 | 0.0040 | 0.5333 | 0.0437 | 36.2802 |

14 | 0.0045 | 0.4495 | 0.0448 | 32.3938 |

15 | 0.0052 | 0.4262 | 0.0406 | 32.8715 |

16 | 0.0058 | 0.3688 | 0.0447 | 37.9928 |

17 | 0.006 | 0.3173 | 0.0509 | 33.5184 |

18 | 0.0072 | 0.3027 | 0.0531 | 31.7283 |

19 | 1.2526 × 10^{−4} | 1.1032 | 0.0012 | 18.1156 |

20 | 8.4721 × 10^{−4} | 2.6008 | 0.0018 | 6.5230 |

21 | 8.8285 × 10^{−4} | 2.4950 | 0.0019 | 5.9387 |

22 | 8.6273 × 10^{−4} | 2.8694 | 0.0018 | 4.9922 |

23 | 6.0421 × 10^{−4} | 2.2021 | 0.0059 | 14.0596 |

24 | 0.0024 | 2.9833 | 0.0040 | 7.3964 |

25 | 0.0039 | 2.8185 | 0.0025 | 5.7725 |

26 | 0.0053 | 0.1665 | 0.0525 | 31.4057 |

27 | 0.0058 | 0.3903 | 0.0744 | 30.1177 |

28 | 0.0065 | 1.2671 | 0.1548 | 22.8691 |

29 | 0.0065 | 1.6573 | 0.2179 | 19.4942 |

30 | 0.0070 | 1.6806 | 0.2693 | 18.1796 |

31 | 0.0080 | 1.1913 | 0.4310 | 7.4600 |

32 | 0.0082 | 1.0459 | 0.4991 | 1.9288 |

33 | 0.0083 | 0.9570 | 0.5981 | 22.9691 |

Method | MCS of 20,000 Times | PLFCM with Rosenblatt Inverse Transform | PLFCM with Orthogonal Inverse Transform |
---|---|---|---|

Time (s) | 305.46 | 2.15 | 2.06 |

Node No. | Probability (%) | Indexes | Node No. | Probability (%) | Indexes |
---|---|---|---|---|---|

2 | 0 | 0 | 19 | 0 | 0 |

3 | 0 | 0 | 20 | 0 | 0 |

4 | 0 | 0 | 21 | 0 | 0 |

5 | 0 | 0 | 22 | 0 | 0 |

6 | 0 | 0 | 23 | 0 | 0 |

7 | 0 | 0 | 24 | 0 | 0 |

8 | 0 | 0 | 25 | 0 | 0 |

9 | 0.001 | 3.412 × 10^{−9} | 26 | 0 | 0 |

10 | 0.0512 | 3.342 × 10^{−7} | 27 | 0 | 0 |

11 | 0.087 | 6.445 × 10^{−7} | 28 | 0.072 | 4.723 × 10^{−7} |

12 | 0.185 | 1.705 × 10^{−6} | 29 | 10.888 | 3.293 × 10^{−4} |

13 | 2.199 | 3.892 × 10^{−5} | 30 | 25.763 | 1.094 × 10^{−3} |

14 | 4.307 | 9.117 × 10^{−5} | 31 | 39.445 | 2.087 × 10^{−3} |

15 | 4.263 | 1.236 × 10^{−4} | 32 | 40.484 | 2.195 × 10^{−3} |

16 | 9.208 | 2.405 × 10^{−4} | 33 | 35.688 | 2.451 × 10^{−3} |

17 | 22.188 | 7.548 × 10^{−4} | system | \ | 0.0104 |

18 | 27.372 | 1.010 × 10^{−3} |

Node No. | RE of Mean (%) | RE of Variance (%) | Node No. | RE of Mean (%) | RE of Variance (%) |
---|---|---|---|---|---|

2 | 4.7903 × 10^{−4} | 1.5650 | 18 | 1.2851 × 10^{−4} | 2.1548 |

3 | 0.0030 | 1.7234 | 19 | 5.5308 × 10^{−4} | 1.5532 |

4 | 0.0029 | 2.8588 | 20 | 1.3169 × 10^{−4} | 2.8045 |

5 | 0.0029 | 1.9286 | 21 | 3.2046 × 10^{−4} | 2.9817 |

6 | 0.0014 | 2.3425 | 22 | 3.7397 × 10^{−4} | 3.1784 |

7 | 0.0014 | 3.1850 | 23 | 0.0055 | 1.7064 |

8 | 0.0013 | 2.6184 | 24 | 0.0109 | 1.8469 |

9 | 8.6506 × 10^{−4} | 2.5399 | 25 | 0.0148 | 2.3036 |

10 | 5.2550 × 10^{−4} | 1.4705 | 26 | 0.0012 | 2.4474 |

11 | 5.0435 × 10^{−4} | 2.4533 | 27 | 0.0011 | 2.5679 |

12 | 4.6079 × 10^{−4} | 2.3993 | 28 | 0.0014 | 2.1355 |

13 | 7.9488 × 10^{−4} | 3.2392 | 29 | 0.0042 | 2.3226 |

14 | 0.0013 | 3.2261 | 30 | 0.0049 | 1.3542 |

15 | 9.9030 × 10^{−4} | 2.2084 | 31 | 0.0033 | 2.1923 |

16 | 6.5868 × 10^{−4} | 3.1785 | 32 | 0.0031 | 2.1712 |

17 | 3.5136 × 10^{−4} | 2.1632 | 33 | 0.0031 | 2.1663 |

Method | MCS of 20,000 Times | PLFCM |
---|---|---|

Time (s) | 133.66 | 2.07 |

Node No. | Probability (%) | Indexes | Node No. | Probability (%) | Indexes |
---|---|---|---|---|---|

2 | 0 | 0 | 19 | 0 | 0 |

3 | 0 | 0 | 20 | 0 | 0 |

4 | 0 | 0 | 21 | 0 | 0 |

5 | 0 | 0 | 22 | 0 | 0 |

6 | 0 | 0 | 23 | 0 | 0 |

7 | 0.0004 | 1.945 × 10^{−9} | 24 | 0 | 0 |

8 | 0.1 | 1.075 × 10^{−5} | 25 | 0 | 0 |

9 | 56.672 | 1.819 × 10^{−3} | 26 | 0 | 0 |

10 | 98.612 | 0.0101 | 27 | 0.009 | 4.600 × 10^{−8} |

11 | 99.3316 | 0.0103 | 28 | 68.155 | 3.263 × 10^{−3} |

12 | 99.831 | 0.0121 | 29 | 99.300 | 0.0139 |

13 | 100 | 0.0209 | 30 | 99.916 | 0.0190 |

14 | 100 | 0.0241 | 31 | 99.997 | 0.0250 |

15 | 100 | 0.0352 | 32 | 100 | 0.0263 |

16 | 100 | 0.0281 | 33 | 100 | 0.0359 |

17 | 100 | 0.0311 | system | \ | 0.329 |

18 | 100 | 0.0320 |

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## Share and Cite

**MDPI and ACS Style**

Hu, W.; Yang, F.; Shen, Y.; Yang, Z.; Chen, H.; Lei, Y.
Dynamic Risk Assessment of Voltage Violation in Distribution Networks with Distributed Generation. *Entropy* **2023**, *25*, 1662.
https://doi.org/10.3390/e25121662

**AMA Style**

Hu W, Yang F, Shen Y, Yang Z, Chen H, Lei Y.
Dynamic Risk Assessment of Voltage Violation in Distribution Networks with Distributed Generation. *Entropy*. 2023; 25(12):1662.
https://doi.org/10.3390/e25121662

**Chicago/Turabian Style**

Hu, Wei, Fan Yang, Yu Shen, Zhichun Yang, Hechong Chen, and Yang Lei.
2023. "Dynamic Risk Assessment of Voltage Violation in Distribution Networks with Distributed Generation" *Entropy* 25, no. 12: 1662.
https://doi.org/10.3390/e25121662