# Risk Assessment for Distribution Systems Using an Improved PEM-Based Method Considering Wind and Photovoltaic Power Distribution

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Distribution of Wind and Photovoltaic DGs

#### 2.1. Output Uncertainty of Wind Generators

_{w}is the active output power, P

_{N}is the rated output power of wind turbine generator, v

_{N}is rated wind speed, v

_{in}is cut-in wind speed, and v

_{out}is cut-out wind speed.

#### 2.2. Output Uncertainty of Photovoltaic Generators

_{max}is the maximum intensity during a certain interval. The two shape indexes α and β can be evaluated by the mean value and the variance of illumination intensity.

_{v}can be acquired:

_{v}, which can be calculated by ${P}_{v}^{\mathrm{max}}$ = A·η·I

_{max}. According to (5), P

_{v}also follows beta distribution with shape indexes α and β.

## 3. Risk Assessment Indices and Method

#### 3.1. Risk Assessment Indices

_{L}is the set for load levels, T

_{i}is the duration of i, and Q

_{i}is the set for system state s at load level i. p

_{T}(s) and C

_{0}(s) are, respectively, the occurrence probability and total load curtailment of system state s.

_{i}and m(s) is the set of component j at system state s. For component j, θ

_{j}represents transfer rate of component j, which is equal to repair rate ${\mu}_{j}$ in this paper.

_{0}(s) should be calculated beforehand; this can be calculated by optimal power flow (OPF), introduced in Section 3.3. For all of these four risk indices, the occurrence probability of system state s p

_{T}(s) is needed. Therefore, an enumeration method is applied for system state selection, which is introduced in Section 4.2.

#### 3.2. Improved Point Estimate Methods

^{j}) of F(X) (Z = F(X) = F(X

_{1}, X

_{2}, …, X

_{m})) when the stochastic numerical characteristics of variables X

_{1}, X

_{2}, …, X

_{m}are known [26,27]. The point estimation principle can be depicted as:

_{i,k}are, respectively, the expectation and standard location of variable X

_{i}, which represents illumination intensity I or wind speed v in this paper. ω

_{l,k}represents the weight of X

_{i}at x

_{l,k}, which can be calculated by (11):

_{i}. Standard location ξ

_{i,k}and weight ω

_{l,k}could be determined by (12):

_{i}on Z are the same. This assumption can simplify the calculation but may cause a big computational error as there are different impacts of X

_{i}on Z. In the improved PEM, $\sum _{k=1}^{K}{\omega}_{i,k}$ is set as ${\alpha}_{i}$, which can be determined by the analytic hierarchy process (AHP) described in [31] or the contribution of X

_{i}on Z. This equation appears to be more reasonable, as it considers the different impacts of X

_{i}on Z.

_{i}

_{,3}= 0, the equation of (12) can be solved:

_{i}

_{,3}= 0, 2m + 1 improved PEM can be constructed, and the solving results of Equation (12) are:

_{i}on Z which can be approximated by the capacity proportion of DG i was used to calculate the value of α

_{i}.

#### 3.3. Optimal Power Flow Algorithm in Distribution Systems

_{0}(s) in state s. Equations (16) and (17) were used in the optimal power flow algorithm to simulate the curtailment and dispatch work and ensure the security of distribution system [32]:

_{b}is the number of buses; ΔP

_{Gk}and ΔP

_{Lh}are respectively the power variations of generator k and load h; C

_{GK}and C

_{Lh}are respectively the control cost of generator k and load h; and I

_{ij}is the current through overload line ij.

_{0}(s), can be calculated by Equation (18):

_{d}is the value of load curtailment caused by direct structure change of the distribution system. In line with the values of C

_{0}(s) and p

_{T}(s), the risk assessment indices in Equations (6)–(9) can be calculated.

## 4. Risk Assessment Procedure for Distribution Systems

#### 4.1. Structure for Risk Assessment

#### 4.2. Procedure for Calculating Risk Indices

_{0}(s), which can be referred from Equations (16) and (17). The detailed procedure of the improved PEM-based method for computing risk indices G is summarized as follows.

_{T}(s) in system state s can be calculated by Equation (19):

_{f}is the total number of failure components, N

_{n}is the total number of normal components. For component i, ${\mu}_{i}$ is the repair rate, ${\lambda}_{i}$ is the outage rate, and these two parameters can be obtained by statistics.

_{i}(i = 1, 2, …, m). For wind generating unit i, X

_{i}represents the stochastic wind speed v; for photovoltaic generating units i, X

_{i}represents the stochastic illumination intensity I. m is the total number of wind generators and photovoltaic generators. Then, the output power of DG i is calculated by Equation (2) or (4).

_{i,k}and the weights ω

_{l,k}for random variables are computed by Equation (12) or (13). Then, the concentrations x

_{i,k}can be calculated by Equation (10). Consequently, the point (μ

_{x}

_{1}, μ

_{x}

_{2}, …, x

_{i,k}, …, μ

_{xm}) is constructed.

_{0}(s) according to Equations (15) and (16).

## 5. Case Studies

_{N}= 15 m/s, cut-in wind speed v

_{in}= 4m/s, cut-out wind speed v

_{out}= 20 m/s, shape index k = 6.23, and scale index c = 10.43.

#### 5.1. Calculation of Risk Indices Based on PEM

_{d}varies from 0 to 1. With the known capacity proportion of DGs, the value of α

_{i}can be obtained: α

_{1}= 0.2, α

_{2}= 0.3, α

_{3}= 0.2, α

_{4}= 0.3.

_{d}, all of these risk indices decrease gradually. When the generation capacity K

_{d}= 0, which means that DGs are not permeated, the value of EENS is about 3156.7 MWh/year. However, EENS decreases by nearly a half and drops to about 1582.5 MWh/year when the generation capacity K

_{d}rises to 1. Analogously, the value of PLC decreases from about 0.0062293 to 0.0045285 when generation capacity K

_{d}rises from 0 to 1.

_{d}. In addition, the slope of EENS and SI decreases with the growth of generation capacity K

_{d}. Therefore, the decreasing trend of EENS and SI is not as tangible as before, decreasingly when generation capacity K

_{d}increases.

_{d}increases to about 0.6. Namely, a large value of K

_{d}that promotes the decrease of risk indices of distribution systems is not significantly remarkable. However, the consumption of wind and photovoltaic energy becomes an increasing problem with the increasing generation capacity K

_{d}. It is with reluctance that too much wind and photovoltaic power is abandoned in distribution systems. Therefore, it is not necessary to increase the value of K

_{d}to a certain large degree.

#### 5.2. Deviation and Computational Cost Comparison

_{1}and δ

_{2}are respectively calculated by comparing the results of EENS and PLC in Table 1 and Table 2 with Table 3. As can be seen from Figure 4 and Figure 5, the maximum value of computational error δ

_{1}is about 0.99% and the maximum value of computational error δ

_{2}is about 0.37%. The computational errors δ

_{1}and δ

_{2}are all less than 1%, which greatly verifies the effectiveness of this improved PEM-based method for risk assessment of distribution systems. Also, the computational error δ

_{2}of PLC is less than the computational error δ

_{1}of EENS. This is because optimal power flow algorithm (based on Equations (15) and (16)) is used in the calculation procedure for EENS.

_{d}= 0.5, the computational costs of risk indices are as listed in Table 4.

#### 5.3. Influence of DGs on System Risk

_{d}= 0.2, which can be seen in Table 5.

_{d}= 0.2, are outlined in Table 6.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Diagram of the improved point estimate method (PEM)-based method for risk assessment of distribution lines in a distribution system.

K_{d} | 2m PEM | |||
---|---|---|---|---|

EENS (MWh/y) | PLC | EFLC (Times/y) | SI (min/y) | |

0 | 3151.2 | 62,247 × 10^{−7} | 2.4899 | 66.716 |

0.2 | 2382.5 | 59,981 × 10^{−7} | 2.3992 | 50.441 |

0.4 | 1945.7 | 53,176 × 10^{−7} | 2.1285 | 41.193 |

0.6 | 1744.4 | 48,635 × 10^{−7} | 1.9454 | 36.932 |

0.8 | 1619.1 | 47,516 × 10^{−7} | 1.9006 | 34.279 |

1 | 1580.3 | 45,243 × 10^{−7} | 1.8097 | 33.457 |

_{d}: generation capacity; EENS: expected energy not supplied; PLC: probability of load curtailment; EFLC: expected frequency of load curtailment; SI: severity index.

K_{d} | 2m + 1 PEM | |||
---|---|---|---|---|

EENS (MWh/y) | PLC | EFLC (Times/y) | SI (min/y) | |

0 | 3156.7 | 62,293 × 10^{−7} | 2.4917 | 66.832 |

0.2 | 2385.4 | 60,033 × 10^{−7} | 2.4013 | 50.502 |

0.4 | 1948.2 | 53,212 × 10^{−7} | 2.1285 | 41.246 |

0.6 | 1747.3 | 48,687 × 10^{−7} | 1.9475 | 36.993 |

0.8 | 1621.8 | 47,551 × 10^{−7} | 1.9020 | 34.336 |

1 | 1582.5 | 45,285 × 10^{−7} | 1.8114 | 33.504 |

K_{d} | Hierarchical Method | |||
---|---|---|---|---|

EENS (MWh/y) | PLC | EFLC (Times/y) | SI (min/y) | |

0 | 3178.5 | 62,434 × 10^{−7} | 2.4974 | 67.293 |

0.2 | 2406.6 | 60,164 × 10^{−7} | 2.4066 | 50.95 |

0.4 | 1963.8 | 53,353 × 10^{−7} | 2.1341 | 41.577 |

0.6 | 1759.5 | 48,812 × 10^{−7} | 1.9525 | 37.251 |

0.8 | 1634.6 | 47,677 × 10^{−7} | 1.9071 | 34.608 |

1 | 1589.2 | 45,407 × 10^{−7} | 1.8163 | 33.646 |

Risk Indices | The Computational Costs of Risk Indices(s) | ||
---|---|---|---|

2m Improved PEM | 2m + 1 Improved PEM | Hierarchical Method | |

EENS | 25.845 | 27.742 | 67.293 |

PLC | 1.985 | 2.031 | 5.950 |

EFLC | 2.157 | 2.192 | 6.107 |

SI | 25.845 | 27.742 | 67.293 |

K_{d} | 2m + 1 Improved PEM | |||
---|---|---|---|---|

EENS (MWh/y) | PLC | EFLC (Times/y) | SI (min/y) | |

0.2 | 2131.7 | 56,823 × 10^{−7} | 2.2729 | 45.131 |

K_{d} | 2m + 1 Improved PEM | |||
---|---|---|---|---|

EENS (MWh/y) | PLC | EFLC (Times/y) | SI (min/y) | |

0.2 | 2193.6 | 57,232 × 10^{−7} | 2.2893 | 46.442 |

K_{d} | 2m + 1 Improved PEM | |||
---|---|---|---|---|

EENS (MWh/y) | PLC | EFLC (Times/y) | SI (min/y) | |

0.2 | 2058.4 | 55,785 × 10^{−7} | 2.2314 | 43.579 |

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## Share and Cite

**MDPI and ACS Style**

Gong, Q.; Lei, J.; Qiao, H.; Qiu, J. Risk Assessment for Distribution Systems Using an Improved PEM-Based Method Considering Wind and Photovoltaic Power Distribution. *Sustainability* **2017**, *9*, 491.
https://doi.org/10.3390/su9040491

**AMA Style**

Gong Q, Lei J, Qiao H, Qiu J. Risk Assessment for Distribution Systems Using an Improved PEM-Based Method Considering Wind and Photovoltaic Power Distribution. *Sustainability*. 2017; 9(4):491.
https://doi.org/10.3390/su9040491

**Chicago/Turabian Style**

Gong, Qingwu, Jiazhi Lei, Hui Qiao, and Jingjing Qiu. 2017. "Risk Assessment for Distribution Systems Using an Improved PEM-Based Method Considering Wind and Photovoltaic Power Distribution" *Sustainability* 9, no. 4: 491.
https://doi.org/10.3390/su9040491