# Black-Box Behavioral Modeling of Voltage and Frequency Response Characteristic for Islanded Microgrid

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## Abstract

**:**

## 1. Introduction

- This method has wide adaptability and is not specific to a particular structure of microgrid. When the structure of microgrid changes randomly, this method is still applicable.
- The internal structure of the microgrid system and the types of loads are not considered at all. The method only pays attention to the response characteristics observed at PCC to ensure that the model structure is simple and easy to be applied.
- The proposed method is not confined to the analysis of relationship between power references and the voltage frequency, it is a general idea that can be applied in analyzing other port data of microgrid.
- Meanwhile, this method is also applicable to all kinds of microgrid systems, such as AC microgrids, DC microgrids, AC/DC hybrid microgrids, and microgrids with various types of electrical equipment and loads (like rotating generators based DG, inverter connected distributed energy resource (DER), controllable and uncontrollable loads, etc.).

## 2. Topology and Modeling Requirements of Microgrid

#### 2.1. Typical Topology of Microgrid System

#### 2.2. Modeling Requirements

## 3. Identification of Black-Box Model

#### 3.1. The Structure of Identification Model

_{0total}and Q

_{0total}refer to the total active and reactive power references in the microgrid respectively, and according to the power droop coefficients of each DG, MGCC assigns the total power reference values P

_{0total}and Q

_{0total}to each DG according to Equation (2):

_{i}and n

_{i}refer to the active and reactive power droop coefficients of ith inverter, P

_{i}and Q

_{i}refer to the active and reactive power references of ith inverter, and f

_{PCC}and E

_{PCC}are frequency and voltage amplitudes of PCC. G

_{fP}(s) and G

_{EP}(s) are transfer functions of the active power reference with respect to the frequency and the voltage amplitude of PCC respectively, G

_{fQ}(s) and G

_{EQ}(s) are the transfer functions of the reactive power reference with respect to the frequency and the voltage amplitude of PCC respectively. Among them, G

_{EP}(s) and G

_{fQ}(s) are added to the model specifically in order to analyze the cross-coupling effect.

#### 3.2. Identification Experiments Design

#### 3.2.1. Active Power Reference Step

_{fP}(s) and G

_{EP}(s). The Q

_{0total}is set to 0, while the total active power reference is regarded as the input of the step experiment by performing a step on P

_{0total}. At the same time, the f

_{PCC1}and E

_{PCC1}are sampled at PCC as the output of the step experiment.

#### 3.2.2. Reactive Power Reference Step

_{fQ}(s) and G

_{EQ}(s). The P

_{0total}is set to 0, while the total reactive power reference is regarded as the input of the step experiment by performing a step on Q

_{0total}. Meanwhile, the f

_{PCC2}and E

_{PCC2}are sampled at PCC as the output of the step experiment.

#### 3.3. Identification Strategy and Process

_{fP}(s) as an example and analyzes the identification process in detail, other transfer functions G

_{EP}(s), G

_{fQ}(s) and G

_{EQ}(s) can adopt the same process.

#### 3.3.1. Data Preprocessing

_{0total}and E

_{PCC1}. It mainly includes two steps:

- Offset removal: The black-box model of the microgrid system includes the steady-state component and the dynamic component. But the parameters to be identified only involve the system dynamics of the system model, thus it is necessary to remove the steady-state components, i.e., the steady-state values of the input and output signals (before the step is done) respectively.
- Measurement prefiltering: Since the sampling data contains some ripple and noise caused by the switching frequency, prefiltering input and output data should be considered before the recursive process to avoid the identification accuracy being affected.

#### 3.3.2. Model Order and Initial Value Selection

_{fP}(s) can be determined and remains constant in the process of recursion. The fitting performance can be calculated as follows:

_{Best fit}, the better the model can reproduce the output characteristics of the microgrid port. Considering that the high model order affects the computing speed significantly, so as long as the fitness meets the requirements, choosing a lower transfer function order is recommended. Then, the order of the numerator of G

_{fP}(s) is set to 1, while that of the denominator is 2, the corresponding expression of G

_{fP}(s) can be obtained as

#### 3.3.3. Online Recursive Algorithm

_{PCC}(k) and P

_{0total}(k) are discrete values of f

_{PCC}and P

_{0total}at time k.

- Under the condition of $\lambda $ < 0.3, it shows that the objective function J exhibits a linearity reduction trend during the recursive process, $\mu $ is needed to increase in the subsequent recursive process to ensure better frequency and voltage identification
- Under the condition of 0.3 < $\lambda $ < 0.7, it indicates that the objective function $J$ has a small change in linearity during the recursive process, therefore the damping factor is close to the best value and there is no need to change it.
- Under the condition of $\lambda $ > 0.7, a smaller damping factor $\mu $ should be chosen.

_{m}is added,

**P**

_{m}and $\widehat{\theta}$

_{m}are revised once, therefore new estimation values of parameters are derived and on-line identification of parameters are realized. The identification procedure is shown in Figure 6. Where ε is the allowable range of precision error, N is the total number of samples at the current moment. When the recursive algorithm compute to the sampling moment or the parameters estimation does not change, the recursive process is terminated until the new sampling data arrives. After obtaining the parameters of the transfer function G

_{fP}(z), the bilinear inverse transformation can be carried out to convert the transfer function from discrete domain to continuous domain.

## 4. Experiments and Model Validation

^{−6}and 4 × 10

^{−4}respectively. Under the normal operation of the experimental platform, the input and output data are sampled by the Yokogawa recorder. All devices are connected to the AC Bus and communication lines, the structure of the microgrid test platform is shown in Figure 7.

#### 4.1. Identification Experiments of G_{fP}(s) and G_{EP}(s)

_{fP}(s) and G

_{EP}(s), parallel inverters and the three-phase active load are integrated into the microgrid. After the system reaches a steady state, the active and reactive power references are set to 0 by MGCC at first. Then the active power reference is changed to 80 kW at 9 s, and set to 0 again at 15.7 s. During this experiment, the reactive power reference remains at 0. Figure 8 shows the identification data P

_{0total}, E

_{PCC1}, and f

_{PCC1}sampled by the waveform recorder at PCC.

_{0total}and f

_{PCC1}can be used to identify the model transfer function G

_{fP}(s).

_{Best fit}reaches 95.36.

_{0total}and E

_{PCC1}, the transfer function G

_{EP}(s) can be obtained:

#### 4.2. Identification Experiments of G_{fQ}(s) and G_{EQ}(s)

_{fQ}(s) and G

_{EQ}(s), parallel inverters and the three-phase reactive load are integrated into the microgrid. After the system reaches a steady state, the active and reactive power references are set to 0 by MGCC at first. Then the reactive power reference is changed to 80 kVar at 9 s, and set to 0 again after 5.6 s. During this experiment, the active power reference remains at 0. Figure 10 shows the identification data Q

_{0total}, E

_{PCC2}, and f

_{PCC2}sampled by the waveform recorder at PCC.

_{0total}and E

_{PCC2}can be used to identify the model transfer function G

_{EQ}(s).

_{Best fit}reaches 90.72.

_{0total}and f

_{PCC2}, the transfer function G

_{fQ}(s) can be obtained as follows:

#### 4.3. Recursive Model Validation

^{−6}and 8 × 10

^{−6}respectively; reactive power droop coefficients are changed to 2 × 10

^{−4}and 4 × 10

^{−4}respectively; the PV inverter which is supplied by Chroma solar PV array simulators is added to the microgrid structure, and the maximum output power of the PV inverter is set to 4 kW. During the power references step experiment, the number of inverters connected to the microgrid are changed, the active and reactive loads are set to different parameters, as listed in Table 1.

**P**and $\widehat{\theta}$ can be figured out according to the transfer function parameters obtained in Section 3.2 and Section 4.1. The frequency and voltage output of the simulation model can be obtained by using the recursive damped least squares method for online identification.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Microgrid Black-Box Model for Secondary Frequency and Voltage Regulation Control Structure.

**Figure 5.**Input power references step experiments (

**a**) Active power references step; (

**b**) Reactive power references step.

**Figure 8.**Active power step sample data for identification (

**a**) Active power references; (

**b**) Reactive power references; (

**c**) PCC frequency; (

**d**) PCC voltage amplitude.

**Figure 9.**Fitting curves of the transfer function model obtained from the active power step experiment.

**Figure 10.**Reactive power step sampling data for identification (

**a**) Active power references; (

**b**) Reactive power references; (

**c**) PCC frequency; (

**d**) PCC voltage amplitude.

**Figure 11.**Fitting curves of the transfer function model obtained from the reactive power step experiment.

**Figure 12.**Data waveforms sampled at input and output ports of the microgrid (

**a**) Active power references; (

**b**) Reactive power references; (

**c**) PCC frequency; (

**d**) PCC voltage amplitude.

**Figure 13.**Identification model and measured output curves (

**a**) PCC frequency; (

**b**) PCC voltage amplitude.

Time | Total Power References | Active Load | Reactive Load |
---|---|---|---|

t_{1} | 0/0 | 0 | 0 |

t_{2} | 0/0 | 12 kW | 20 kVar |

t_{3} | 60 kW/40 kVar | 12 kW | 20 kVar |

t_{4} | 100 kW/60 kVar | 12 kW | 20 kVar |

t_{5} | 60 kW/40 kVar | 12 kW | 20 kVar |

t_{6} | 60 kW/40 kVar | 8 kW | 10 kVar |

t_{7} | 60 kW/40 kVar (INV1 shutdown) | 8 kW | 10 kVar |

t_{8} | 60 kW/40 kVar (INV1 start up) | 8 kW | 10 kVar |

t_{9} | 0/0 | 8 kW | 10 kVar |

t_{10} | 0/0 | 0 | 0 |

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**MDPI and ACS Style**

Shi, Y.; Xu, D.; Su, J.; Liu, N.; Yu, H.; Xu, H.
Black-Box Behavioral Modeling of Voltage and Frequency Response Characteristic for Islanded Microgrid. *Energies* **2019**, *12*, 2049.
https://doi.org/10.3390/en12112049

**AMA Style**

Shi Y, Xu D, Su J, Liu N, Yu H, Xu H.
Black-Box Behavioral Modeling of Voltage and Frequency Response Characteristic for Islanded Microgrid. *Energies*. 2019; 12(11):2049.
https://doi.org/10.3390/en12112049

**Chicago/Turabian Style**

Shi, Yong, Dong Xu, Jianhui Su, Ning Liu, Hongru Yu, and Huadian Xu.
2019. "Black-Box Behavioral Modeling of Voltage and Frequency Response Characteristic for Islanded Microgrid" *Energies* 12, no. 11: 2049.
https://doi.org/10.3390/en12112049