# A Generalized and Mode-Adaptive Approach to the Power Flow Analysis of the Isolated Hybrid AC/DC Microgrids

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Primary Control in the Hierarchical Controlled AC/DC IHMG

#### 2.1. Hierarchical Control Level

- Level 1 (primary control): The control object of this level achieves voltage/frequency control for interface devices of the distributed energy resources (DER). Moreover, the power sharing and optimal power management of resources can be obtained;
- Level 2 (secondary control): The control level utilizes the low speed communication network to compensate for the voltage and frequency deviations caused by the primary control level;
- Level 3 (tertiary control): There is a positive response in this control level for external dispatching instructions to maintain the effectiveness, economy, and reliability of the system.

#### 2.2. The Primary Control of the DERs in the IHMG

^{th}bus, while the V–P droop of a DC-type DG connects to the m

^{th}bus. These are described by Equations (3)–(5), respectively [35,37].

#### 2.3. The Primary Control of the IC in the IHMG

- IC can be a slack bus for the AC subgrid, compensating power mismatch in the AC subgrid in the weak systems, while the DC subgrid has a higher power surplus [34]. In this case, the IC can operate in grid-forming mode to perform frequency and voltage control of the AC subgrid;
- IC can be a slack bus for the DC subgrid with lower power surplus capacity than that of the AC subgrid. Moreover, IC can perform voltage control of the DC subgrid as a grid-forming unit in the DC subgrid;
- In order to achieve the equal loadings of subgrids, both subgrids of the IHMG should have similar power when the IC controls the transfer of the active power between the neighboring AC and DC subgrids. Moreover, in order to adapt the active power transfer between the two subgrids, IC measures the AC frequency and DC voltage and equalizes them by normalizing. The corresponding control strategies are as follows:$$\hat{\mathsf{\omega}}=\frac{\mathsf{\omega}-0.5\left({\mathsf{\omega}}_{\mathrm{max}}+{\mathsf{\omega}}_{\mathrm{min}}\right)}{0.5\left({\mathsf{\omega}}_{\mathrm{max}}-{\mathsf{\omega}}_{\mathrm{min}}\right)},$$$${\hat{\mathrm{V}}}_{\mathrm{dc}}=\frac{{\mathrm{V}}_{\mathrm{dc}}-0.5\left({\mathrm{V}}_{\mathrm{c},\mathrm{dc}}^{\mathrm{max}}+{\mathrm{V}}_{\mathrm{c},\mathrm{dc}}^{\mathrm{min}}\right)}{0.5\left({\mathrm{V}}_{\mathrm{c},\mathrm{dc}}^{\mathrm{max}}-{\mathrm{V}}_{\mathrm{c},\mathrm{dc}}^{\mathrm{min}}\right)},$$$$\Delta \mathrm{e}=\hat{\mathsf{\omega}}-{\hat{\mathrm{V}}}_{\mathrm{dc}},$$$${\mathrm{P}}_{\mathrm{C}}=-\frac{1}{{\mathrm{k}}_{\mathrm{IC}}}\Delta \mathrm{e},$$

## 3. Definition of the Operating Modes in the AC/DC IHMG

#### 3.1. Classification of the AC/DC IHMG Configurations

- Disconnecting from the main network;
- Connecting the AC and DC subgrids through bidirectional AC/DC interfacing converters (ICs) to fulfill the bidirectional power flow between subgrids;
- Dividing the zones according to the DER type, such as RES, DG and ESS, and load type, such as AC or DC.

#### 3.2. Primary Control Operating Modes

## 4. Formulation of the Unified PF Model

#### 4.1. DER Model

#### 4.2. Formulation of the Unified PF Model

- Unit-type vector W ((N + M) × 1): It describes the unit type (i.e., grid-following or grid forming) connected to the relevant bus in the AC/DC subgrid, as follows:
- (i)
- When ${\mathrm{W}}_{\mathrm{i}}$ = 1, the bus connects to the grid-forming unit;
- (ii)
- When ${\mathrm{W}}_{\mathrm{i}}$ = 0, the bus does not connect to the grid-following unit.

- Judgment vector D ((N + M) × 1): It checks for the coexistence of grid-forming units in both AC and DC subgrids:
- (i)
- When ${\mathrm{D}}_{\mathrm{i}}$ = 1, the grid-forming unit is available in both AC and DC subgrids;
- (ii)
- When ${\mathrm{D}}_{\mathrm{i}}$ = 0, the grid-forming unit is only installed separately in the AC or DC subgrids.

- Judgment vector U ((N + M) × 1): It checks for the existence of the grid-forming unit in each AC or DC subgrid:
- (i)
- When ${\mathrm{U}}_{\mathrm{i}}$ = 1, there are grid-forming units in the AC subgrid;
- (ii)
- When ${\mathrm{U}}_{\mathrm{i}}$ = 0, there is not grid-forming unit in the AC subgrid.

- The AC admittance matrix Y(N $\times $ N):$${\mathrm{Y}}_{\mathrm{nk}}\left(\mathsf{\omega}\right)={\mathrm{G}}_{\mathrm{nk}}\left(\mathsf{\omega}\right)+{\mathrm{jB}}_{\mathrm{nk}}\left(\mathsf{\omega}\right)=\frac{1}{{\mathrm{R}}_{\mathrm{nk}}+{\mathrm{j}\mathsf{\omega}\mathrm{L}}_{\mathrm{nk}}},\text{}\mathrm{n},\mathrm{k}\in \mathrm{N}.$$
- DC conductance matrix ${\mathrm{G}}^{\mathrm{dc}}\left(\mathrm{M}\times \mathrm{M}\right)$: The element in the matrix reflects the value of the conductance of the DC line that connects two buses.

#### 4.3. Power Balance Equations

#### 4.4. Solution Procedure

## 5. Cases Studies

#### 5.1. Twelve-Bus Test System

#### 5.2. Multi-DC Subgrids IHMG Test System

#### 5.2.1. Algorithm Performance in the Normal Potation of the Multi-DC Subgrids System

#### 5.2.2. Algorithm Performance in the Operation of the Multi-DC Subgrids IHMG System during Load Fluctuation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Classification of the configurations of the isolated hybrid AC/DC microgrid (IHMG) system: (

**a**) Grid-forming units are in the AC subgrid, (

**b**) grid-forming units are in the DC subgrid, and (

**c**) grid-forming units are in both AC and DC subgrids.

Unit Type in Mode 1 | Source Type | Control Type | Output Impedance | PF Model |
---|---|---|---|---|

AC ESS or DG | Non-Ideal Voltage source | Droop ^{1} | Finite, nonzero | Droop |

RES | Ideal current source | MPPT | 0 | PQ |

IC | DC Ideal Voltage source | Constant DC Voltage control | 0 | DC constant V |

DC ESS or DG | - | - | - | - |

^{1}See Formulas (1)–(6).

Unit Type in Mode 3 | Source Type | Control Type | Output Impedance | PF Model |
---|---|---|---|---|

AC ESS or DG | - | - | - | - |

RES | Ideal current source | MPPT | 0 | PQ |

IC | AC Ideal Voltage source | Constant AC Voltage control | 0 | AC constant V |

DC ESS or DG | Non-Ideal Voltage source | Droop ^{2} | Finite, nonzero | Droop |

^{2}See Formulas (7)–(8).

Unit Type in Mode 5 | Source Type | Control Type | Output Impedance | PF Model |
---|---|---|---|---|

AC ESS or DG | Non-Ideal Voltage source | Droop ^{1} | Finite, nonzero | Droop |

RES | Ideal current source | MPPT | 0 | PQ |

IC | Non-Ideal Voltage source | Droop ^{3} | Finite, nonzero | Droop |

DC ESS or DG | Non-Ideal Voltage source | Droop ^{2} | Finite, nonzero | Droop |

^{1}See Formulas (1)–(6);

^{2}See Formulas (7)–(8);

^{3}See Formulas (9)–(14).

Subgrid | Bus Type | Number of Buses | Known Quantity | Unknown Quantity x = [x_{AC}, x_{DC}] | Number of Equations |
---|---|---|---|---|---|

AC | PQ | N_{R} | P_{n}, Q_{n} | V_{n}, ${\mathsf{\delta}}_{\mathrm{n}}$ | 2N_{R} |

Droop | N_{D} | - | P_{n}, Q_{n}, V_{n}, ${\mathsf{\delta}}_{\mathrm{n}}$ | 4N_{D} | |

Slack bus | 1 | V_{n}, ${\mathsf{\delta}}_{\mathrm{n}}$ | P_{n}, Q_{n} | - | |

DC | Const.P | M_{R} | P_{m} | V_{m} | M_{R} |

Droop | M_{D} | - | P_{m}, V_{m} | 2M_{D} | |

Const.V | M-M_{R}-M_{D} | V_{m} | P_{m} | M-M_{R}-M_{D} |

Mode Type | U_{i} | D_{i} | W_{i} |
---|---|---|---|

Mode 1 | 1 | 0 | 1 |

Mode 3 | 0 | 0 | 0 |

Mode 5 | 1 | 1 | 1 |

Bus | Bus Type | Unified PF Results | MATLAB Results | ||
---|---|---|---|---|---|

V_{n}(p.u.) | θ_{n}(p.u.) | V_{n}(p.u.) | θ_{n}(p.u.) | ||

1(AC ^{1}) | Droop | 0.9944 | 0.0337 | 0.9947 | 0.0330 |

2(AC ^{1}) | Droop (IC AC) | 0.9928 | 0.0289 | 0.9932 | 0.0301 |

3(AC ^{1}) | PQ | 0.9965 | 0.0234 | 0.9968 | 0.0256 |

4(AC ^{1}) | PQ (MPPT) | 0.9957 | 0.0440 | 0.9960 | 0.0443 |

5(AC ^{1}) | Droop (IC AC) | 0.9934 | 0.0956 | 0.9931 | 0.0900 |

6(AC ^{1}) | PQ (MPPT) | 0.9994 | 0.0000 | 0.9997 | 0.0000 |

${1}^{dc}$(DC ^{2}) | Droop (IC DC) | 1.0086 | - | 1.0082 | - |

${2}^{dc}$(DC ^{2}) | P | 1.0012 | - | 1.0013 | - |

${3}^{dc}$ (DC ^{2}) | P | 0.9986 | - | 0.9981 | - |

${4}^{dc}$(DC ^{2}) | Droop | 0.9979 | - | 0.9972 | - |

${5}^{dc}$(DC ^{2}) | Droop (IC DC) | 1.0079 | - | 1.0076 | - |

${6}^{dc}$(DC ^{2}) | R | 0.9970 | - | 0.9969 | - |

^{1}AC subgrid of the IHMG.

^{2}DC subgrid of the IHMG.

**Table 7.**Test results of the output power of the distributed energy resource (DER) in the 12-bus IHMG system.

Bus | Bus Type | Unified PF Results | MATLAB Results | ||
---|---|---|---|---|---|

P_{DR,n}(p.u.) | Q_{DR,n}(p.u.) | P_{DR,n}(p.u.) | Q_{DR,n}(p.u.) | ||

1(AC ^{1}) | Droop | −0.1325 | — | −0.1361 | — |

4(AC ^{1}) | PQ | 0.2668 | 0.2013 | 0.2698 | 0.2000 |

6(AC ^{1}) | Droop | 0.5967 | 0.3034 | 0.5924 | 0.3001 |

${2}^{dc}$(DC ^{2}) | P | 0.6098 | — | 0.6092 | — |

${3}^{dc}$(DC ^{2}) | Droop | 0.0198 | — | 0.2001 | — |

${4}^{dc}$(DC ^{2}) | Droop | 0.1998 | — | 0.1990 | — |

^{1}AC subgrid of the IHMG;

^{2}DC subgrid of the IHMG.

**Table 8.**Test results of the transferred power by interfacing converters (ICs) in the 12-bus IHMG system.

IC | $\mathsf{\Delta}\mathit{e}{\text{}}^{1}$ (p.u.) | ${\mathbf{V}}_{\mathbf{a}\mathbf{c}}^{\text{}2}$ (p.u.) | ${\mathbf{V}}_{\mathbf{d}\mathbf{c}}^{\text{}3}$ (p.u.) | ${\mathbf{P}}_{\mathbf{c}}^{\text{}4}$ (p.u.) | ${\mathbf{Q}}_{\mathbf{c}}^{\text{}5}$ (p.u.) |
---|---|---|---|---|---|

${1}^{\#}$ | 0.038 | 0.9928 | 1.0086 | −0.057 | - |

${2}^{\#}$ | 0.052 | 0.9934 | 1.0079 | −0.077 | - |

^{1}The difference between the normalized AC frequency and DC voltage.

^{2}The AC voltage amplitude of ICs.

^{3}The DC voltage of ICs.

^{4}The active power transferred by the IC.

^{5}The reactive power transferred by the IC.

MG | Bus No. | Bus Type | $\left|{\mathbf{V}}_{0}\right|$ (p.u.) | DR Type | ${\mathbf{P}}_{\mathbf{D}\mathbf{R}}^{\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{e}\mathbf{d}}$ MW | ${\mathbf{Q}}_{\mathbf{D}\mathbf{R}}^{\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{e}\mathbf{d}}$ Mvar | ${\mathsf{\omega}}_{0}$ (p.u.) | ${\mathbf{m}}_{\mathbf{p}}$ (p.u.) | ${\mathbf{n}}_{\mathbf{p}}$ (p.u.) |
---|---|---|---|---|---|---|---|---|---|

AC | 1 | Droop | 1.0 | DS | 0.8 | 0.6 | 1.0 | 0.0375 | 0.25 |

2 | Droop | 1.0 | - | - | - | 1.0 | 0.0625 | 0.4167 | |

3 | Z | 1.0 | - | - | - | 1.0 | - | - | |

4 | PQ | 1.0 | DG | 0.48 | 0.36 | 1.0 | - | - | |

5 | Droop | 1.0 | - | - | - | 1.0 | 0.0167 | - | |

6 | PQ | 1.0 | DG | 1.8 | 1.35 | 1.0 | - | - | |

DC | 1 | Droop | 1.0 | - | - | - | - | 0.0781 | - |

2 | P | 1.0 | DG | 1.92 | - | - | - | - | |

3 | P | 1.0 | DG | 0.48 | - | - | - | - | |

4 | Droop | 1.0 | DS | 0.6 | - | - | 0.3125 | - | |

5 | R | 1.0 | - | - | - | - | - | - | |

6 | Droop | 1.0 | - | - | - | - | 0.25 | - |

IC No. | AC Bus | DC Bus | ${\mathbf{P}}_{\mathbf{i}\mathbf{c}}$ (MW) | ${\mathbf{Q}}_{\mathbf{i}\mathbf{c}}$ (Mvar) | ${\mathsf{\omega}}_{0}$ | ${\mathbf{V}}_{\mathbf{d}\mathbf{c},0}$ (p.u.) | ${\mathsf{\gamma}}_{\mathbf{p}}$ | ${\mathsf{\gamma}}_{\mathbf{q}}$ |
---|---|---|---|---|---|---|---|---|

1 | 2 | 1 | 3.0 | 2.25 | 1.0 | 1.0 | 1.0 | 0.0667 |

2 | 5 | 5 | 3.0 | 2.25 | 1.0 | 1.0 | 1.0 | 0.0667 |

MG | From | To | $\mathbf{R}\text{}\left(\mathsf{\Omega}\right)$ | $\mathbf{X}\text{}\left(\mathsf{\Omega}\right)$ | Load Connected to from Bus P(MW) Q(Mvar) | |
---|---|---|---|---|---|---|

AC | 1 | 2 | 0.02646 | 0.01323 | - | |

2 | 3 | 0.04032 | 0.02016 | 0.4 | 0.3 | |

3 | 5 | 0.04032 | 0.02016 | 1.0 | 0.6 | |

5 | 6 | 0.02646 | 0.01323 | 0.8 | 0.6 | |

4 | 3 | 0.02646 | 0.01323 | - | ||

DC | 1 | 2 | 0.4340 | - | 0.6 | |

2 | 3 | 0.2279 | - | - | ||

3 | 4 | 0.4100 | - | - | ||

4 | 5 | 0.4340 | - | - | ||

5 | 6 | 0.4100 | - | 1.4 | ||

6 | 1 | 0.2279 | - | 0.5 |

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**MDPI and ACS Style**

Xiao, Y.; Ren, C.; Han, X.; Wang, P. A Generalized and Mode-Adaptive Approach to the Power Flow Analysis of the Isolated Hybrid AC/DC Microgrids. *Energies* **2019**, *12*, 2253.
https://doi.org/10.3390/en12122253

**AMA Style**

Xiao Y, Ren C, Han X, Wang P. A Generalized and Mode-Adaptive Approach to the Power Flow Analysis of the Isolated Hybrid AC/DC Microgrids. *Energies*. 2019; 12(12):2253.
https://doi.org/10.3390/en12122253

**Chicago/Turabian Style**

Xiao, Yu, Chunguang Ren, Xiaoqing Han, and Peng Wang. 2019. "A Generalized and Mode-Adaptive Approach to the Power Flow Analysis of the Isolated Hybrid AC/DC Microgrids" *Energies* 12, no. 12: 2253.
https://doi.org/10.3390/en12122253