# Admittance Criterion of Medium-Voltage DC Distribution Power System and Corresponding Small Signal Stability Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Structure of MVDC DPS

_{mmc1}is the current-limiting inductor at the output side of the voltage-controlled MMC (VCMMC). L

_{mmc2}is the current-limiting inductor at the output side of the power-controlled MMC (PCMMC). L

_{pvk}is the current-limiting inductor at the output side of the kth PVDCT. L

_{LTm}is the current-limiting inductor at the input side of the mth LTDCT. By analogy, Z

_{line_mmc1}, Z

_{line_mmc1}, Z

_{line_pvk}, Z

_{line_ltm}are the line impedances of the corresponding converters, and i

_{mmc1}, i

_{mmc1}, i

_{PVk}, i

_{ltm}are the currents of the corresponding converters.

## 3. Admittance Criterion for MVDC DPS

_{1}, the output impedance of the 1st to the nth PVDCT is Z

_{2}to Z

_{n}

_{+1}, the input impedance of the 1st to the mth LTDCT is Z

_{n}

_{+2}to Z

_{m}

_{+n+1}, the corresponding current source and the equivalent impedance are also numbered according to this law, and the current flowing into the converter is positive. So, the impedance and current of the converters can be expressed as below:

_{out_k}is the impedance of the kth converter at the MVDC side.

_{o_mmc1}can be simplified as (5), and $\widehat{i}$

_{o_mmc1}can be rewritten as (6). At last, $\widehat{v}$

_{dc_mmc1}can be simplified as (7).

_{mmc}(s) and ${Z}_{\mathrm{mmc}}^{\prime}$ are stable. Other converters in this system control their output current, so they are stable loaded by an ideal voltage source, which means that i

_{k}(s) and Z

_{k}/(Z

_{Lk}+ Z

_{k}) are stable. Then, the stability of the system can be evaluated by applying the Nyquist criterion to T

_{m}as shown below:

## 4. DC Impedance Modelling of System Converters

_{ga}, v

_{gb}, and v

_{gc}are the AC voltages of MMC, and v

_{dc}is the DC voltage of MMC. Each arm of MMC contains K series sub-modules (SMs) and an arm inductor. In Figure 2, i

_{au}is the current of the upper arm in phase a, and i

_{al}is the current of the lower arm in phase A, and so on.

_{o}is the output voltage of DCT, i

_{in}is the input voltage of DCT, i

_{1k}is the input current of the kth SM, i

_{2k}is the output current of the kth SM, k = 1, 2,…, N, and N is the number of SMs. Z

_{L}is the load impedance, and C

_{o}is the output capacitance. C

_{in}, L, n are the input capacitance, transfer inductor, and transformer ratio of the SM, respectively.

_{b}is the capacitance of the boost converter, and L

_{b}is the inductor. C

_{in_PV}is the capacitance of the DCT at the LVDC side, and C

_{o_PV}is the capacitance of the DCT at the MVDC side. L

_{PV}is the inductor of each DAB SM, and n

_{PV}is the transform ratio of each DAB SM.

_{s}is the switching frequency of the DAB modules, and D, V

_{in}, and V

_{o}are the phase shift, input voltage, and output voltage of the DAB SM under steady state, respectively. For PVDCT, f

_{s2}is the switching frequency of the DAB SM, and D

_{2}, V

_{in_PV}, and V

_{o_PV}are the phase shift, input voltage, and output voltage of the DAB SM under steady state, respectively.

## 5. Stability Analysis of the DC System

#### 5.1. Active Damping Control Strategies for DCTs

_{sh1}(s), G

_{sh2}(s) satisfy the relationship in (13) and (14) respectively, where BF

_{l}(s), BF

_{PV}(s) are the band-pass filters (BFs) for LT DCTs and PV DCTs. The center frequency of the filter is its corresponding DCT’s LC resonant frequency. The expression of the BF is written as (15); Q is the quality factor, which is 1 in this article, and f

_{p}is the center frequency of the filter. In (13), R

_{vd}is the virtual resistance of the LTDCT. In (14), R

_{vPV}is the virtual resistance of the PVDCT.

_{vd}and R

_{vPV}as 25 Ω. For LTDCT#1/#2/#3 and PVDCT#1, the central frequency for their BFs is 130 Hz. For LTDCT#4/#5/#6 and PVDCT#2, the central frequency for their BFs is 178 Hz. With the active damping loop, the bode plot of system admittance is shown in Figure 7b. The admittance overlapping frequencies are 15 Hz, 30 Hz, and 53 Hz. The phase difference is 151.4° at 11 Hz and 4.5° at 30 Hz, which all meet the stability requirements. At 53 Hz, the phase difference is decreased to 161.8°, which means that the phase margin increases to 18.2° and the system stability is effectively improved.

#### 5.2. PLECS Simulation Verification

#### 5.3. RT Box Hardware-in-Loop Simulation Verification

## 6. Conclusions

- A new admittance stability criterion is proposed in this paper. The overall stability of the system can be determined only by the equivalent admittance ratio T
_{m}in Equation (3). This criterion has clear physical meaning and concise evaluation expression. In practical engineering, the corresponding impedance sum can be obtained with frequency-sweeping impedance measurement, and the impedance stability can be determined. - The output impedance of the PVDCT shows positive resistance characteristics in the bandwidth range, which can provide damping for the system and be conducive to system stability, while the input impedance of the LTDCT shows negative resistance characteristics in the bandwidth range, which is not conducive to system stability.
- The current-limiting inductors are equipped in DCTs and have resonance with the capacitors of DCTs. Due to the negative input impedance characteristic of the LTDCT, resonance between the inductor and capacitor easily causes the instability of the system. The active damping control methods adopted in this paper can provide virtual resistance for DCTs to suppress resonance. The active damping methods can be generally configured in DCTs connected with the DC DPS to improve the stability of the DC system.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Rodriguez, E.; Vasquez, J.C.; Guerrero, J.M. Intelligent DC Homes in Future Sustainable Energy Systems: When efficiency and intelligence work together. IEEE Consum. Electron. Mag.
**2016**, 5, 74–80. [Google Scholar] [CrossRef] - Dragičević, T.; Lu, X.; Vasquez, J.C.; Guerrero, J. DC Microgrids—Part II: A Review of Power Architectures, Applications, and Standardization Issues. IEEE Trans. Power Electron.
**2016**, 31, 3528–3549. [Google Scholar] [CrossRef] - Rao, H. Architecture of Nan’ao multi-terminal VSC-HVDC system and its multi-functional control. CSEE J. Power Energy Syst.
**2015**, 1, 9–18. [Google Scholar] [CrossRef] - Gabderakhmanova, T.; Engelhardt, J.; Zepter, J.M.; Sørensen, T.M.; Boesgaard, K.; Ipsen, H.H.; Marinelli, M. Demonstrations of DC Microgrid and Virtual Power Plant Technologies on the Danish Island of Bornholm. In Proceedings of the 2020 55th International Universities Power Engineering Conference (UPEC), Torino, Italy, 1–4 September 2020. [Google Scholar]
- Stieneker, M.; Butz, J.; Rabiee, S.; Doncker, R.W.D. Medium-Voltage DC Research Grid Aachen. In Proceedings of the International ETG Congress 2015, Die Energiewende—Blueprints for the New Energy Age, Bonn, Germany, 17–18 September 2015. [Google Scholar]
- Li, W.; Yu, S.; Wei, T.; Li, Y.; Chen, J.; Liu, Y.; Xu, S. State of the Art of Researches and Applications of MVDC Distribution Systems in Power Grid. In Proceedings of the IECON 2019—45th Annual Conference of the IEEE Industrial Electronics Society, Lisbon, Portugal, 14–17 October 2019. [Google Scholar]
- Zheng, J.; Chen, J.; Liu, Y.; Qu, L.; Yu, Z.; Song, Q.; Yuan, Z.; Zhao, B.; Zeng, R. Urban Energy Internet Based on Flexible DC Distribution Network. South. Power Syst. Technol.
**2021**, 15, 25–32. [Google Scholar] [CrossRef] - Middlebrook, R.D. Input-filter considerations in design and application of switching regulators. In Proceedings of the 1976 IEEE Industy Applications Society Annual Conference, Chicago, IL, USA, 11–14 October 1976. [Google Scholar]
- Wildrick, C.M.; Lee, F.C.; Cho, B.H.; Choi, B. A method of defining the load impedance specification for a stable distributed power system. IEEE Trans. Power Electron.
**1995**, 10, 280–285. [Google Scholar] [CrossRef] - Feng, X.; Ye, Z.; Xing, K.; Lee, F.C.; Borojevic, D. Impedance specification and impedance improvement for dc distributed power system. In Proceedings of the 30th Annual IEEE Power Electronics Specialists Conference, Charleston, SC, USA, 1 July 1999. [Google Scholar]
- Sudhoff, S.D.; Glover, S.F.; Lamm, P.T.; Schmucker, D.H.; Delisle, D.E. Admittance space stability analysis of power electronic systems. IEEE Trans. Aerosp. Electron. Syst.
**2000**, 36, 965–973. [Google Scholar] [CrossRef] - Fang, J.; Shuai, Z.; Li, Y.; Wang, Z.; Wu, X.; Shen, X.; Shen, Z.J. Stability investigation and improvement for DC cascade systems with simplified impedance-based stability criterion. CSEE J. Power Energy Syst.
**2023**, 1–9. Available online: https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9770518 (accessed on 16 April 2023). - Sun, J. Impedance-Based Stability Criterion for Grid-Connected Inverters. IEEE Trans. Power Electron.
**2011**, 26, 3075–3078. [Google Scholar] [CrossRef] - Zhang, X.; Ruan, X.; Chi, K.T. Impedance-based local stability criterion for DC distributed power systems. IEEE Trans. Circuits Syst. I Regul. Pap.
**2015**, 62, 916–925. [Google Scholar] [CrossRef] - Pan, P.; Chen, W.; Shu, L.; Mu, h.; Zhang, K.; Zhu, M.; Deng, F. An Impedance-Based Stability Assessment Methodology for DC Distribution Power System With Multivoltage Levels. IEEE Trans. Power Electron.
**2020**, 35, 4033–4047. [Google Scholar] [CrossRef] - Mu, H.; Zhang, Y.; Tong, X.; Chen, W.; He, B.; Shu, L.; Ruan, X.; Zhang, X.; Gao, F.; Cao, W.; et al. Impedance-Based Stability Analysis Methods for DC Distribution Power System With Multivoltage Levels. IEEE Trans. Power Electron.
**2021**, 36, 9193–9208. [Google Scholar] [CrossRef] - Li, B.; He, J.; Li, Y.; Li, B.; Wen, W. High-speed directional pilot protection for MVDC distribution systems. Int. J. Electr. Power Energy Syst.
**2020**, 121, 106141. [Google Scholar] [CrossRef] - Liu, R.; Yang, J.; Jia, Y.; Liu, Y.; Yuan, Y.; Xiao, X. Engineering Applications of DC Transformer in Medium-voltage DC Distribution Network. Autom. Electr. Power Syst.
**2019**, 43, 131–140. [Google Scholar] [CrossRef] - Li, Z.; Wang, Z.; Wang, Y.; Yin, T.; Mei, N.; Yue, B.; Lei, W. Accurate Impedance Modeling and Control Strategy for Improving the Stability of DC System in Multiterminal MMC-Based DC Grid. IEEE Trans. Power Electron.
**2020**, 35, 10026–10049. [Google Scholar] [CrossRef] - He, B.; Chen, W.; Ruan, X.; Zhang, X.; Zou, Z.; Cao, W. A Generic Small-Signal Stability Criterion of DC Distribution Power System: Bus Node Impedance Criterion (BNIC). IEEE Trans. Power Electron.
**2022**, 37, 6116–6131. [Google Scholar] [CrossRef] - Zhang, Q.; Liu, X.; Li, M.; Yu, F.; Yu, D. Input-series-output-parallel DC transformer impedance modeling and phase reshaping for rapid stabilization of MVDC distribution systems. Electronics
**2022**, 10, 3163. [Google Scholar] [CrossRef] - Li, S.; Wang, J.; Li, X.; Xiao, X. Impedance Characteristics and Stability Analysis of Input-series Output-parallel Medium-voltage DC Transformer. High Volt. Eng.
**2022**, 48, 210–219. [Google Scholar] [CrossRef]

**Figure 7.**Stability analysis bode plots of the system under different conditions. (

**a**) Without the active damping loop. (

**b**) With the active damping loop.

**Figure 9.**Voltage waveforms of the DC system. (

**a**) With or without the active damping loop; (

**b**) Frequency spectrum of the oscillation.

**Figure 11.**Voltage waveforms of the DC system with differently rated power ratios of LTDCT and PVDCT. (

**a**) LTDCT:PVDCT = 1:2; (

**b**) LTDCT:PVDCT = 1:5.

**Figure 12.**RT Box hardware-in-loop simulation. (

**a**) Voltage waveforms of the oscilloscope; (

**b**) Back-to-back experiment platform.

LTDCT#1/#2/#3 | LTDCT#4/#5/#6 | ||
---|---|---|---|

Parameter | Value | Parameter | Value |

Number of submodules | 10 | Number of submodules | 10 |

Nominal power (MW) | 2 | Nominal power (MW) | 1 |

Switching frequency (Hz) | 1000 | Switching frequency (Hz) | 1000 |

Input voltage (kV) | 20 | Input voltage (kV) | 20 |

Output voltage (V) | 750 | Output voltage (V) | 750 |

Transformer ratio | 200/75 | Transformer ratio | 200/75 |

Energy transfer inductor of each module (mH) | 1.6 | Energy transfer inductor of each module (mH) | 3.2 |

Input capacitor of each module (mF) | 1 | Input capacitor of each module (mF) | 0.8 |

Output capacitor (mF) | 20 | Output capacitor (mF) | 15 |

Input current-limiting inductor (mH) | 15 | Input current-limiting inductor (mH) | 10 |

Low-pass filter of current control | (200π)2/ [s ^{2} + 1.414 × 200πs + (200π)^{2}] | Low-pass filter of current control | (200π)2/ [s ^{2} + 1.414 × 200πs + (200π)^{2}] |

Output current controller | (s + 100π)/(10,000s) | Output current controller | (s + 100π)/(5000s) |

Voltage controller | (s + 40) × 2850/[s(s + 400)] | Voltage controller | (s + 40) × 2140/[s(s + 400)] |

Line distance (km) | 4 | Line distance (km) | 1 |

PVDCT#1 | PVDCT#2 | ||
---|---|---|---|

Parameter | Value | Parameter | Value |

Number of submodules | 10 | Number of submodules | 10 |

Nominal power (MW) | 2 | Nominal power (MW) | 1 |

Switching frequency (Hz) | 1000 | Switching frequency (Hz) | 1000 |

Output voltage (kV) | 20 | Output voltage (kV) | 20 |

Input voltage (V) | 750 | Input voltage (V) | 750 |

Transformer ratio | 75/200 | Transformer ratio | 75/200 |

Energy transfer inductor of each module (mH) | 0.225 | Energy transfer inductor of each module (mH) | 0.45 |

Input capacitor (mF) | 20 | Input capacitor (mF) | 15 |

Output capacitor of each module (mF) | 1 | Output capacitor of each module (mF) | 0.8 |

Output current-limiting inductor (mH) | 15 | Output current-limiting inductor (mH) | 10 |

Output voltage controller | 0.157(s + 180)/[s(s + 100π)] | Output voltage controller | 0.157(s + 180)/[s(s + 100π)] |

Line distance (km) | 2.5 | Line distance (km) | 2 |

VCMMC | PCMMC | ||
---|---|---|---|

Parameter | Value | Parameter | Value |

DC Voltage (kV) | 20 | DC Voltage (kV) | 20 |

AC Voltage (kV) | 10 | AC Voltage (kV) | 10 |

Nominal power (MW) | 10 | Nominal power (MW) | 10 |

Arm inductor (mH) | 8 | Arm inductor (mH) | 8 |

Equivalent capacitor of each arm (mF) | 0.4 | Equivalent capacitor of each arm (mF) | 0.4 |

Current-limiting inductor (mH) | 10 | Current-limiting inductor (mH) | 10 |

Current controller | 3 + 300/s | Current controller | 3 + 300/s |

Low-pass filter of the DC voltage | 100π/(s + 100π) | Circulating current controller | 3 + 500/s |

Voltage controller | 3 + 300/s | Line distance (km) | 2 |

Circulating current controller | 20 + 500/s |

Line Impedance (per km) |
---|

0.0599 + j2π × 2.714 × 10^{−4} Ω |

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**MDPI and ACS Style**

Yang, J.; Wang, J.; Jin, X.; Li, S.; Xiao, X.; Wu, Z.
Admittance Criterion of Medium-Voltage DC Distribution Power System and Corresponding Small Signal Stability Analysis. *World Electr. Veh. J.* **2023**, *14*, 235.
https://doi.org/10.3390/wevj14090235

**AMA Style**

Yang J, Wang J, Jin X, Li S, Xiao X, Wu Z.
Admittance Criterion of Medium-Voltage DC Distribution Power System and Corresponding Small Signal Stability Analysis. *World Electric Vehicle Journal*. 2023; 14(9):235.
https://doi.org/10.3390/wevj14090235

**Chicago/Turabian Style**

Yang, Jinggang, Jianhua Wang, Xiaokuan Jin, Shuo Li, Xiaolong Xiao, and Zaijun Wu.
2023. "Admittance Criterion of Medium-Voltage DC Distribution Power System and Corresponding Small Signal Stability Analysis" *World Electric Vehicle Journal* 14, no. 9: 235.
https://doi.org/10.3390/wevj14090235