# General Dynamic Equivalent Modeling of Microgrid Based on Physical Background

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## Abstract

**:**

## 1. Introduction

## 2. Microgrid Equivalent Model

#### 2.1. Equivalent Machine Component

#### 2.2. Equivalent Static Component

#### 2.3. Parameters of the Equivalent Model

## 3. Equivalent Model Parameter Sensitivity Analysis

#### 3.1. Trajectory Sensitivity

#### 3.2. Trajectory Sensitivity Phase

## 4. Microgrid Parameter Estimation

#### 4.1. Rotor Voltage Equivalence

#### 4.2. Parameter Estimation Based on PSO

#### 4.2.1. Review of Particle Swarm Optimization

#### 4.2.2. Particle Swarm Optimization (PSO) with Chaos Neighborhood Searching

- (1)
- Generating the initial variable ${u}_{0j}$ randomly based on the global optimal position of the particles.
- (2)
- Getting the chaotic sequences ${u}_{1j}$ using the logistic mapping ${u}_{1j}=4{u}_{0j}(1-{u}_{0j})$.
- (3)
- Generating a local neighborhood mutation variable $\text{\Delta}{x}_{j}$: $\text{\Delta}{x}_{j}=-\text{\beta}+2\text{\beta}{u}_{1j}$, where $\text{\beta}$ is the radius of the local neighborhood and is updated by Equation (24).$$\text{\beta}=({x}_{j\text{max}}-{x}_{j\text{min}})\text{cos}(\frac{\text{\pi}(t-1)}{2({t}_{\text{max}}-1)})$$
- (4)
- Local neighborhood searching: A temporary global optimal position is defined as $gbes{t}^{\prime}=gbest+\text{\Delta}X$, where $\text{\Delta}X$ is the chaos mutation variables vector $[\text{\Delta}{x}_{1}\text{\hspace{0.17em}\Delta}{x}_{j}\text{\hspace{0.17em}}\cdots \text{\Delta}{x}_{N}]$ and $gbest=[{x}_{g1}\cdots {x}_{gN}]$ is the current global optimal position. Comparing $gbes{t}^{\prime}$ with $gbest$, the current global optimal position is updated by the larger one.

#### 4.2.3. Improved Particle Swarm Optimization(IPSO) Based Parameter Estimation

## 5. Microgrid Equivalent Modeling and Discussion

#### 5.1. Microgrid System

**Figure 5.**(

**a**) The active power dynamics of the detailed model and the equivalent model of the Microgrid; (

**b**) The reactive power dynamics of the detailed model and the equivalent model of the Microgrid.

Parameter | K_{mp} | s_{0} | r_{s} | L_{sl} | L_{ad} | L_{ap} | r_{dr} |
---|---|---|---|---|---|---|---|

Value | −8.70 | −0.206 | 0.031 | 0.340 | 0.848 | 3.14 | 0.00023 |

Parameter | ${L}_{drl}$ | ${r}_{qr}$ | ${L}_{qrl}$ | ${T}_{j}$ | ${p}_{u}$ | ${q}_{u}$ | Error |

Value | 2.18 | 0.171 | 0.756 | 9.66 | 7.06 | 0.731 | 2.401 |

#### 5.2. Parameter Sensitivity Analysis and Error Analysis

Parameter | K_{mp} | s_{0} | r_{s} | L_{sl} | L_{ad} | L_{ap} | r_{dr} |
---|---|---|---|---|---|---|---|

Sensitivity | 0.0160 | 0.0352 | 0.0016 | 0.0040 | 0.1367 | 0.0377 | 0.0149 |

Parameter | ${L}_{drl}$ | ${r}_{qr}$ | ${L}_{qrl}$ | ${T}_{j}$ | ${p}_{u}$ | ${q}_{u}$ | |

Sensitivity | 0.1543 | 0.00043 | 0.0583 | 0.0015 | 0.0120 | 0.0011 |

#### 5.3. Key Parameters Estimation and Error Analysis

Parameter | K_{mp} | s_{0} | L_{ad} | L_{ap} | L_{drl} | L_{qrl} | p_{u} | Error |
---|---|---|---|---|---|---|---|---|

Value | −8.637 | −0.2161 | 0.8042 | 3.417 | 2.120 | 0.7008 | 7.039 | 2.389 |

#### 5.4. Comparing to Other Models

**Figure 9.**(

**a**) Dynamics of Active Power of different models; (

**b**) Dynamics of Reactive Power of different models.

**Figure 10.**(

**a**) Dynamics of Active Power of different models; (

**b**) Dynamics of Reactive Power of different models.

#### 5.5. Parameters Estimation under Different Operational Conditions

Operational Condition | Description |
---|---|

Condition A | The output power of gas turbine reduces 40 percent with the output active power is 3 MW. |

Condition B | The output power of gas turbine reduces 40 percent with the output active power is 3 MW, 40 percent of Static Load 2, which is 2 MW active power and 0.4 Mvar reactive power, is removed. |

Condition C | The output power of gas turbine reduces 40 percent with the active power is 3 MW. The output active power of Wind Generation reduces a half, which become 0.75 MW. PV is removed. 40 percent of Static Load 1, which is 0.4 MW and 0.2 Mvar, is removed. |

**,**while the microgrid absorbs power from the distribution network in condition A. This means that the consume power of equivalent static component comes from distribution network and equivalent machine component. In conditions B and C, the equivalent machine component absorbs power as an asynchronous induction motor with ${K}_{mp}>0$ and ${s}_{0}>0$, while the microgrid absorbs power from the distribution network.

Parameter | K_{mp} | s_{0} | r_{s} | L_{sl} | L_{ad} | L_{aq} | r_{dr} | L_{drl} | r_{qr} | L_{qrl} | T_{j} | p_{u} | q_{u} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Condition A | −4.69 | −0.17 | 0.0590 | 0.36 | 1.94 | 2.50 | 0.0003 | 3.04 | 0.2560 | 2.24 | 5.62 | 4.87 | 0.94 |

Condition B | 1.64 | 0.07 | 0.0610 | 0.10 | 4.05 | 7.54 | 0.1186 | 1.49 | 0.0024 | 4.00 | 5.18 | 8.95 | 0.61 |

Condition C | 0.99 | 0.12 | 0.0492 | 0.31 | 3.70 | 4.76 | 0.1450 | 0.65 | 0.0057 | 4.00 | 6.55 | 3.58 | 0.70 |

**Figure 11.**(

**a**) The active power dynamics of the detailed model and the equivalent model under condition A; (

**b**) The reactive power dynamics of the detailed model and the equivalent model under condition A.

**Figure 12.**The active power dynamics of the detailed model and the equivalent model under condition B.

**Figure 13.**The active power dynamics of the detailed model and the equivalent model under condition C.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Detail Data of Microgrid

U_{N}/kV | S_{N}/MVA | cosφ | R_{s}/pu | X_{s}/pu | R_{r}/pu | X_{r}/pu | X_{μ}/pu | T_{j}/s |
---|---|---|---|---|---|---|---|---|

0.96 | 2.4 | 0.8756 | 0.0100 | 0.1000 | 0.0100 | 0.1000 | 3.0000 | 1.188 |

U_{N}/kV | S_{N}/MVA | r_{s}/pu | X_{d}/pu | X_{q}/pu | X′_{d}/pu | X″_{d}/pu | X′_{q}/pu | X″_{q}/pu | T′_{d0}/s | T′_{q0}/s | T″_{d0}/s | T″_{q0}/s |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.44 | 8 | 0 | 1.5 | 1.5 | 0.256 | 0.07 | 0.3 | 0.07 | 0.0171 | 0 | 0.0125 | 0.0057 |

Load | U_{N}/kV | P_{N}/kVA | s | r_{s}/pu | X_{s}/pu | R_{r}/pu | X_{r}/pu | X_{μ}/pu | T_{j}/s |
---|---|---|---|---|---|---|---|---|---|

Load 1 | 0.4150 | 315.0 × 2 | 0.02756 | 0 | 0.0200 | 0.0347 | 0.2022 | 2.390 | 0.6230 |

Load 2 | 0.4150 | 315.0 × 5 | 0.02756 | 0 | 0.0200 | 0.0347 | 0.2022 | 2.390 | 0.6230 |

Load | P + Q |
---|---|

Load 1 | 1 MW + 0.5 MVar |

Load 2 | 5 MW + 1 MVar |

## Appendix B. Parameters Used in the PSO

x_{min} | −10 | −0.3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −1 | −1 |

x_{max} | 10 | 0.3 | 0.2 | 0.5 | 5 | 10 | 0.2 | 5 | 0.5 | 5 | 10 | 10 | 5 |

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**MDPI and ACS Style**

Cai, C.; Jiang, B.; Deng, L.
General Dynamic Equivalent Modeling of Microgrid Based on Physical Background. *Energies* **2015**, *8*, 12929-12948.
https://doi.org/10.3390/en81112354

**AMA Style**

Cai C, Jiang B, Deng L.
General Dynamic Equivalent Modeling of Microgrid Based on Physical Background. *Energies*. 2015; 8(11):12929-12948.
https://doi.org/10.3390/en81112354

**Chicago/Turabian Style**

Cai, Changchun, Bing Jiang, and Lihua Deng.
2015. "General Dynamic Equivalent Modeling of Microgrid Based on Physical Background" *Energies* 8, no. 11: 12929-12948.
https://doi.org/10.3390/en81112354