# Investigation of Oscillation and Resonance in the Renewable Integrated DC-Microgrid

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## Abstract

**:**

## 1. Introduction

- A comprehensive analytical model for investigating high-frequency oscillations and resonance has been developed. The impedance analysis and eigenvalue-based method are used simultaneously to identify the source of oscillation/instability in the DC microgrid.
- Most of the prior studies have considered a constant power load for the high-frequency and resonance assessment of DC microgrids. This work has considered the various types of loads in the DC microgrid and their impact on the overall high-frequency oscillations and resonance.
- A semi-global sensitivity analysis technique has been used to rank the most critical parameters of the DC microgrid, considering the cross-coupling of various parameters and uncertainties.
- Three control synthesis methods are described and compared in their suitability to the conventional PI controller in terms of high-frequency oscillations and resonance.

## 2. Oscillation Challenges in DCMG

## 3. Methodology and Modelling

#### 3.1. State Space Modelling

#### 3.2. Analytical Expression for Impedance Scanning

#### 3.3. Sensitivity Analysis

#### 3.4. Overview of Control Tuning

## 4. Numerical Analysis and Discussion

#### 4.1. Impedance Analysis

#### 4.2. Eigenvalue Analysis

_{1}) was found with 153 Hz and a damping ratio of 0.09, as shown in Figure 4. Hence, it is assumed that the critical mode might move further right in the s-plane and reduce the stability margin of the system with the variation in the system parameters. Subsequently, a participation factor analysis has been conducted to see the modes and associated devices that contribute to the modes, as presented in Table 5.

#### 4.2.1. Case Study 1 (Parameter Variation in 12-kW Boost Converter)

#### 4.2.2. Case Study 2 (Parameter Variation in Buck Converter)

#### 4.2.3. Case Study 3 (Variation in DC-Link Capacitance)

#### 4.2.4. Case Study 4 (Variation in Load Power)

#### 4.2.5. Case Study 5 (Variation in Inductance Parameter)

#### 4.3. Time-Domain Simulations and Experimental Studies

#### 4.4. Critical Parameter Ranking

#### 4.5. Control Performance Assessment

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Variables | |

BESS | Battery energy storage system |

DCMG | DC microgrid |

EMI | Electromagnetic interference |

EV | Electric vehicle |

GM | Gain margin |

HF | High frequency |

HVDC | High-voltage DC |

LF | Low frequency |

LQR | Linear Quadratic Regulator |

MVDC | Medium voltage DC |

PI | Proportional integral |

PID | Proportional integral derivative |

PM | Phase margin |

PV | Photovoltaic |

PWM | Pulse-width modulated |

RESs | Renewable energy sources |

SSS | Small-signal stability |

## Appendix A

Parameter | Symbol | Value |
---|---|---|

Boost converter source voltage | ${V}_{s}$ | $20V$ |

Boost converter inductor and capacitor | ${L}_{1}\text{}{C}_{1}$ | $147e-6H$ & $470e-6\mu F$ |

Parasitic inductor resistor | ${r}_{L1}$ | $70e-3\mathrm{\Omega}$ |

Boost converter output voltage | ${V}_{o1}$ | $40V$ |

DC link voltage | ${V}_{dc}$ | $40V$ |

Switching frequency | ${f}_{sw}$ | $20KHz$ |

Proportional and integral gain of PI controller in boost converter | ${K}_{p1}\text{}{K}_{i1}$ | $\begin{array}{l}0.0011262\text{}\\ 0.05\end{array}$ |

Input voltage of buck converter | ${V}_{in}$ | $40V$ |

Buck converter output voltage | ${V}_{02}$ | $20V$ |

Boost converter inductor and capacitor | ${L}_{2}\text{}{C}_{2}$ | $\begin{array}{l}1.5e-3H\\ 470e-6\mu F\end{array}$ |

Proportional and integral gain of PI controller in buck converter | ${K}_{p2}\text{}{K}_{i2}$ | $\begin{array}{l}0.9\\ 0.5\end{array}$ |

Switching frequency | ${f}_{sw}$ | $20KHz$ |

Load | $R$ | $75\mathrm{\Omega}$ |

Rated output power | ${P}_{o}$ | $500W$ |

Converter | PI | LQR | IP | PI+clegg |
---|---|---|---|---|

1 | Proportional: 0.9 Integral: 0.005 | Proportional: 0.0011 Integral: 1.73 | Proportional: 0.0013 Integral: 0.707 | Proportional: 0.0015 Integral: 1.25 |

2 | Proportional: 0.9 Integral: 0.005 | Proportional: 0.0011 Integral: 1.73 | Proportional: 0.0012 Integral: 0.707 | Proportional: 0.0012 Integral: 1.23 |

3 | Proportional: 0.9 Integral: 0.005 | Proportional: 0.0011 Integral: 1.73 | Proportional: 0.0011 Integral: 0.707 | Proportional: 0.0012 Integral: 1.22 |

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**Figure 3.**Frequency response of the system: (

**a**) Output and input impedance; (

**b**) total impedance seen from DC link.

**Figure 5.**Impact of variation in relation to control parameters (boost converter). (

**a**) Variation in K

_{p}in 12-kW boost converter; (

**b**) variation in K

_{i}in 8-kW boost converter; (

**c**) variation in K

_{p}in 8-kW boost converter on DC grid; (

**d**) variation in K

_{i}in 8-kW boost converter on DC grid.

**Figure 6.**Impact of variation in relation to control parameters (buck converter). (

**a**) Variation in K

_{p}in 10-kW buck converter; (

**b**) variation in K

_{i}in 10-kW buck converter; (

**c**) variation in both K

_{p}and K

_{i}in 10-kW buck converters; (

**d**) incremental K

_{p}value for multiple units of converters; (

**e**) arbitrarily selected controller parameter.

**Figure 10.**Impact of the controller and DC link dynamics on DC bus. (

**a**) DC bus voltage influenced by integral controller dynamics; (

**b**) FFT analysis on DC bus; (

**c**) parameter variations of the controller (buck converters); (

**d**) DC bus voltage influenced by the proportional controller dynamics; (

**e**) impact of simultaneous disturbances (50% voltage disturbances in source 1 and 50% load disturbances); (

**f**) impact of simultaneous disturbances (50% voltage disturbances in source 1 and source 2, and 50% load disturbances).

**Figure 11.**Experimental investigation. (

**a**) A prototype of DC microgrid setup; (

**b**) output voltage of boost converter; (

**c**) output voltage of buck converter.

Reference | Detailed Load Model | Sensitivity Analysis | Key Parameter Identification | Control Method |
---|---|---|---|---|

[13] | ✓ | ✓ | ✕ | ✓ |

[14] | ✓ | ✓ | ✕ | ✕ |

[15] | ✕ | ✓ | ✕ | ✕ |

[16] | ✕ | ✓ | ✕ | ✕ |

[17] | ✓ | ✓ | ✕ | ✕ |

[20] | ✕ | ✓ | ✕ | ✕ |

[21] | ✕ | ✕ | ✕ | ✓ |

This work | ✓ | ✓ | ✓ | ✓ |

Parameter | Symbol | Value |
---|---|---|

Rated power of the simulated system | ${P}_{o}$ | 20 kW |

Rated power of boost converter 1 | ${P}_{o-boost1}$ | 12 kW |

Rated power of boost converter 2 | ${P}_{0-boost2}$ | 8 kW |

Rated power of buck converter 1 | ${P}_{o-bucik1}$ | 10 kW |

Rated power of buck converter 2 | ${P}_{o-bucik2}$ | 5 kW |

Rated power of buck converter 3 | ${P}_{o-bucik3}$ | 1 kW |

Boost converter source voltage | ${V}_{s1}$ | 375 V |

Boost converter output voltage | ${V}_{o1}$ | 750 V |

DC link voltage | ${V}_{dc}$ | 750 V |

Switching frequency of boost converters | ${f}_{sw-boost}$ | 20 kHz |

Switching frequency of buck converter | ${f}_{sw-buck1}$ | 10 kHz |

${f}_{sw-buck2}$ | 20 kHz | |

${f}_{sw-buck3}$ | 20 kHz | |

Input voltage of buck converter | ${V}_{in}$ | 750 V |

Buck converter output voltage | ${V}_{02}$ | 400 V |

Boost Converter 1 | Boost Converter 2 | ||
---|---|---|---|

V_{s} = 375 V | L = 157 µH | V_{s} = 400 V | L = 147 µH |

V_{0} = 750 V | R = 47 Ω | V_{0} = 750 V | R = 70 Ω |

D = 0.5 (no unit) | C = 1000 µF | D = 0.5 (no unit) | C = 1000 µF |

K_{p} = 0.005 pu | K_{i} = 0.003 pu | K_{p} = 0.005 pu | K_{i} = 0.003 pu |

Parameters | Converter 1 | Converter 2 | Converter 3 |
---|---|---|---|

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

Output voltage (V) | 400 | 400 | 400 |

Inductance (H) | 0.00428 | 0.0044 | 0.0033 |

Resistance (ohm) | 16 | 145 | 147 |

Duty cycle | 0.5 | 0.5 | 0.5 |

Proportional gain (pu) | 0.9 | 0.9 | 0.9 |

Integral gain (pu) | 0.005 | 0.005 | 0.005 |

Critical Mode | Damping (%) | f (Hz) | Associated Mode | Remarks |
---|---|---|---|---|

−0.83 ± j953 | 0.09 | 151 | Controller | Boost converter |

−7.91 ± j1443 | 0.55 | 230 | Controller | Boost converter |

−7.33 ± j18,080 | 0.04 | 2878 | Controller | Buck converter |

−66.48 ± j57,931 | 0.11 | 9224 | Controller | Buck converter |

−17.36 ± j14,184 | 0.12 | 2258 | Controller | Buck converter |

Load Type | Damping of Mode 1 | Damping of Mode 2 | Damping of Mode 3 | Damping of Mode 4 | Damping of Mode 5 |
---|---|---|---|---|---|

Constant P | 0.091 | 0.56 | 0.041 | 0.112 | 0.121 |

Constant I | 0.092 | 0.55 | 0.038 | 0.111 | 0.122 |

Constant Z | 0.093 | 0.55 | 0.039 | 0.113 | 0.122 |

ZIP | 0.091 | 0.54 | 0.041 | 0.114 | 0.121 |

**Table 7.**Influential parameters identified using the different sensitivity method (oscillation stability).

Ranking | OAT | Morris | Pearson |
---|---|---|---|

1 | ${k}_{iboost1}$ | ${k}_{pboost1}$ | ${k}_{pboost1}$ |

2 | ${k}_{pboost1}$ | ${k}_{iboost1}$ | ${k}_{iboost1}$ |

3 | ${k}_{pboost2}$ | ${k}_{pboost2}$ | ${k}_{pboost2}$ |

4 | DC-link | ${k}_{iboost2}$ | ${k}_{pbuck2}$ |

Mode | Damping with PI | Damping with PI+clegg | Damping with IP | Damping with LQR |
---|---|---|---|---|

1 | 0.091 | 0.122 | 0.105 | 0.11 |

2 | 0.551 | 0.593 | 0.573 | 0.57 |

3 | 0.041 | 0.071 | 0.055 | 0.054 |

4 | 0.112 | 0.134 | 0.123 | 0.115 |

5 | 0.122 | 0.151 | 0.134 | 0.125 |

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## Share and Cite

**MDPI and ACS Style**

Habibullah, M.; Mithulananthan, N.; Shah, R.; Islam, M.R.; Muyeen, S.M.
Investigation of Oscillation and Resonance in the Renewable Integrated DC-Microgrid. *Electronics* **2023**, *12*, 1574.
https://doi.org/10.3390/electronics12071574

**AMA Style**

Habibullah M, Mithulananthan N, Shah R, Islam MR, Muyeen SM.
Investigation of Oscillation and Resonance in the Renewable Integrated DC-Microgrid. *Electronics*. 2023; 12(7):1574.
https://doi.org/10.3390/electronics12071574

**Chicago/Turabian Style**

Habibullah, Mohammad, Nadarajah Mithulananthan, Rakibuzzaman Shah, Md Rabiul Islam, and S. M. Muyeen.
2023. "Investigation of Oscillation and Resonance in the Renewable Integrated DC-Microgrid" *Electronics* 12, no. 7: 1574.
https://doi.org/10.3390/electronics12071574