# Distributionally Robust Distributed Generation Hosting Capacity Assessment in Distribution Systems

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## Abstract

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## 1. Introduction

- We propose a DRO-based method to evaluate the HC of distribution networks considering uncertainties associated with loads and DGs’ output powers. As the statistic information (i.e., the first- and second-order moments) and the empirical distribution of uncertain variables are exploited in the assessment process, the results of the proposed method are practical and less conservative than those of the RO-based method. To the best of our knowledge, there is no such DRO application in power system that exploit both the statistic information and the empirical distribution simultaneously.
- In the proposed DRO-based method, a risk level is defined to control the robustness, enabling a trade-off between robustness and conservativeness. Unlike the robust optimization, tuning the risk level in the proposed method has a precise physical meaning, i.e., guaranteeing that the probability of operational constraints violation does not exceed a given risk threshold.
- The aggregated EVs and charging stations’ loads are modeled in the HC problem and their effects on the HC have been assessed. To the best of our knowledge, there is no such study that assesses the effect of aggregated EVs and charging stations’ loads on the HC of various types of DGs, i.e., PV, wind, and biomass.

## 2. Mathematical Modeling

#### 2.1. Distribution System Model

#### 2.2. Technical Constraints

#### 2.2.1. Steady State Voltage Constraint

#### 2.2.2. Thermal Capacity Constraints

#### 2.2.3. Short Circuit Level (SCL) Constraint

#### 2.3. Deterministic Problem Formulation Summary

#### 2.4. Distributionally Robust Optimization (DRO) HC Model

## 3. Uncertainty Modeling

#### 3.1. Uncertainty Modeling of PV Generation

#### 3.2. Uncertainty Modeling of Wind Generation

#### 3.3. Uncertainty Modeling of Biomass Generation

#### 3.4. Uncertainty Modeling of Load

#### 3.5. Uncertainty Modeling of EV Demand

#### 3.5.1. Overall Charging Demand of EVs in a Local Residential Community

#### 3.5.2. Overall Charging Demand of an EV Charging Station

#### 3.6. Modeling the Confidence Set

## 4. Solution Methodology

**Definition**

**1.**

#### 4.1. Equivalent JCC

Algorithm 1: Bisection line search algorithm for ${\tau}_{+}^{{}^{\prime}}$. |

#### 4.2. Sample Average Approximation (SAA)

**Theory**

**1.**

## 5. Numerical Results

#### 5.1. Test System

#### 5.2. Simulation Results on the Modified IEEE 33-Bus System

**PV-HC:**All DGs are PV systems.**Wind-HC:**All DGs are wind generators.**Biomass-HC:**All DGs are biomass generators.**Combined-HC:**The DGs at buses 15 and 32 are PV units, and the DGs at buses 22 and 25 are wind and biomass generators, respectively.

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DG | Distributed Generation |

HC | Hosting Capacity |

DRO | Distributionally Robust Optimization |

JCC | Joint Chance Constrained |

DSO | Distribution System Operator |

EVs | Electric Vehicles |

PV | Photovoltaic |

RO | Robust Optimization |

SO | Stochastic Optimization |

PDFs | Probability Density Functions |

SCL | Short Circuit Level |

SOC | State of Charge |

CDFs | Cumulative Density Functions |

SAA | Sample Average Approximation |

NHTS | National Household Travel Survey |

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**Figure 4.**The HC obtained via the proposed DRO-HC method with different risk levels for different technologies including: (

**a**) PV; (

**b**) wind; (

**c**) biomass; and (

**d**) combination of PV, wind and biomass.

**Figure 5.**The value of data for the HC obtained via the proposed DRO-HC method with $\tau =10\%$ for different technologies including: (

**a**) PV; (

**b**) wind; (

**c**) biomass; and (

**d**) combination of PV, wind and biomass.

**Figure 6.**The effect of aggregate demand of residential EVs on the HC obtained via the proposed DRO-HC method with $\tau =1\%$ for different technologies including: (

**a**) PV; (

**b**) wind; (

**c**) biomass; and (

**d**) combination of PV, wind and biomass.

**Figure 7.**The effect of charging stations’ demands on the HC obtained via the proposed DRO-HC method with $\tau =1\%$ for different technologies including: (

**a**) PV; (

**b**) wind; (

**c**) biomass; and (

**d**) combination of PV, wind and biomass.

**Table 1.**Value of ${\ell}_{\varphi}$, $\underline{m}\left({\varphi}^{*}\right)$, and $\overline{m}\left({\varphi}^{*}\right)$ for variation distance $\varphi $-divergence.

Divergences | ${\mathit{\ell}}_{\mathit{\varphi}}$ | $\underline{\mathit{m}}\left({\mathit{\varphi}}^{*}\right)$ | $\overline{\mathit{m}}\left({\mathit{\varphi}}^{*}\right)$ |
---|---|---|---|

Variation distance | 1 | -1 | 1 |

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**MDPI and ACS Style**

Seydali Seyf Abad, M.; Ma, J.; Ahmadyar, A.S.; Marzooghi, H.
Distributionally Robust Distributed Generation Hosting Capacity Assessment in Distribution Systems. *Energies* **2018**, *11*, 2981.
https://doi.org/10.3390/en11112981

**AMA Style**

Seydali Seyf Abad M, Ma J, Ahmadyar AS, Marzooghi H.
Distributionally Robust Distributed Generation Hosting Capacity Assessment in Distribution Systems. *Energies*. 2018; 11(11):2981.
https://doi.org/10.3390/en11112981

**Chicago/Turabian Style**

Seydali Seyf Abad, Mohammad, Jin Ma, Ahmad Shabir Ahmadyar, and Hesamoddin Marzooghi.
2018. "Distributionally Robust Distributed Generation Hosting Capacity Assessment in Distribution Systems" *Energies* 11, no. 11: 2981.
https://doi.org/10.3390/en11112981