# Control Design and Parameter Tuning for Islanded Microgrids by Combining Different Optimization Algorithms

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## Abstract

**:**

^{©}software.

## 1. Introduction

^{©}software. A comparative analysis among optimization algorithms is carried out for the different case studies. The main contributions of the paper are listed below.

- Proposing a new simple design approach and tuning method for the optimal setting of power, voltage, and current controllers’ coefficients.
- Proposing novel objective function evaluating optimized parameters for the controllers while ensuring the VSI stability in the whole range of operation.
- Proposing a combination of PSO and GA for parameter tuning for a VSI in an MG application.

## 2. State-Space Modelling of the Inverter

## 3. Proposed Design Approach

#### 3.1. Formulation of Optimization Problem

#### 3.2. Proposed PSO-GA

#### 3.3. Designing Controllers’ Coefficients for a Case Study

#### 3.4. Effect of Operating Point Changes

- High-frequency modes which consist of seven poles;
- Low-frequency modes which consist of three poles;
- Very low-frequency modes which consist of three poles.

#### 3.5. Effect of Output Impedance Changes

#### 3.6. Plug-and-Play Capability of the Design Approach

## 4. Simulation Results

^{©}software. The simulation is carried out for three different case studies to show the robustness of the designed controllers against various disturbances and in different MG configurations. The first one is a one-inverter system with a linear load and the second one is a one-inverter system with a nonlinear load. The last case study is a higher-order system with two inverters and a linear load. A comparative analysis among different optimization algorithms is performed in each case study.

#### 4.1. Case Study I: One Inverter with Linear Load

#### 4.2. Case Study II: One Inverter with Nonlinear Load

#### 4.3. Case Study III: Two Inverter with Linear Load

## 5. Conclusions

^{©}software. The main advantages of the proposed method are summarized below.

- Proposing a simple guideline for engineers to design controllers’ parameters in an islanded MG regardless of the number of inverters, MG configuration, output impedances, and loads types which significantly reduces the effort and complexity of the design issue.
- Improvement in the steady-state frequency, the $dq$ currents, and the three-phase voltages response under linear load changes, nonlinear load changes, and linear load changes in the islanded MG with two grid-forming inverters.
- Needless of coefficient readjustment for the whole range of operating points.
- Providing a plug-and-play design approach when a new DG wants to be added to the MG.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Artificial linear characteristics with the slope of $-{m}_{p}$ for the inverter frequency and the slope of $-{n}_{q}$ for the inverter voltage.

**Figure 5.**The validation of the small-signal model of the islanded MG with one inverter: (

**a**) d-axis current; (

**b**) q-axis current.

**Figure 10.**The eigenvalues loci for 0$$$\le {r}_{{L}_{c}}\le $10$$ and $0.35$$$$\le {L}_{c}\le $3$$.

**Figure 11.**The design of MG controllers for three different inverters with different output impedances.

**Figure 15.**The $dq$-axis inverter output current with different designed parameters: (

**a**) d-axis current; (

**b**) d-axis current zoom version; (

**c**) q-axis current; (

**d**) q-axis current zoom version.

**Figure 16.**The $dq$-axis inverter output voltage with different designed parameters: (

**a**) d-axis voltage; (

**b**) d-axis voltage zoom version; (

**c**) q-axis voltage; (

**d**) q-axis voltage zoom version.

**Figure 17.**Active and reactive power of the MG under linear load changes for different designed parameters: (

**a**) active power; (

**b**) reactive power.

**Figure 18.**Three-phase voltages and phase a voltage of the MG common bus with different designed parameters.

**Figure 22.**The $dq$-axis output current with different designed parameters under nonlinear load changes: (

**a**) d-axis current; (

**b**) d-axis current zoom version; (

**c**) q-axis current; (

**d**) q-axis current zoom version.

**Figure 23.**The $dq$-axis output voltage with different designed parameters under nonlinear load changes: (

**a**) d-axis voltage; (

**b**) d-axis voltage zoom version; (

**c**) q-axis voltage; (

**d**) q-axis voltage zoom version.

**Figure 24.**MG active and reactive power under nonlinear load changes: (

**a**) active power; (

**b**) reactive power.

**Figure 25.**MG three-phase voltages under nonlinear load changes with different designed parameters: (

**a**) conv.; (

**b**) GA; (

**c**) PSO; (

**d**) PSO-GA.

**Figure 27.**The $dq$-axis output currents and voltages in two-inverter MG: (

**A**) conv.; (

**B**) GA; (

**C**) PSO; (

**D**) PSO-GA. (

**a**) d-axis current; (

**b**) q-axis current; (

**c**) d-axis voltage; (

**d**) q-axis voltage.

**Figure 28.**The frequency, the active, and the reactive power in two-inverter MG: (

**A**) conv.; (

**B**) GA; (

**C**) PSO; (

**D**) PSO-GA. (

**a**) frequency; (

**b**) active power; (

**c**) reactive power.

**Figure 29.**The inverters three-phase output voltages with different designed parameters in two-inverter MG: (

**a**) inv. 1 with conv.; (

**b**) inv. 2 with conv.; (

**c**) inv. 1 with GA; (

**d**) inv. 2 with GA; (

**e**) inv. 1 with PSO; (

**f**) inv. 2 with PSO; (

**g**) inv. 1 with PSO-GA; (

**h**) inv. 2 with PSO-GA.

Parameters | Symbol | Value |
---|---|---|

Filter inductance | ${L}_{f}$ | $1.35$ $\mathrm{m}$$\mathrm{H}$ |

Filter capacitance | ${C}_{f}$ | 50 $\mathsf{\mu}\mathrm{F}$ |

Grid coupling inductance | ${L}_{c}$ | $0.35$ $\mathrm{m}$$\mathrm{H}$ |

Power controller bandwidth | ${\omega}_{c}$ | $31.41$ $\mathrm{rad}$/$\mathrm{s}$ |

Filter inductor resistance | ${r}_{{L}_{f}}$ | $0.1$ $\mathrm{\Omega}$ |

Filter capacitor resistance | ${r}_{{L}_{c}}$ | $0.03$ $\mathrm{\Omega}$ |

Switching frequency | ${f}_{s}$ | 8$\mathrm{k}\mathrm{Hz}$ |

Nominal frequency | ${f}_{n}$ | 50$\mathrm{Hz}$ |

Dynamic improvement loop | F | $0.75$ |

MG nominal power | S | 50 kVA |

Parameters | Value | Parameters | Value | Parameters | Value |
---|---|---|---|---|---|

${V}_{od}$ | 380 V | ${V}_{oq}$ | 0 V | ${V}_{bd}$ | 380 V |

${I}_{od}$ | 0 A | ${I}_{oq}$ | −50 A | ${V}_{bq}$ | 0 V |

Parameters | Value | Variable | Search Interval |
---|---|---|---|

Population size | 100 | ${m}_{p}$ | [0, $5\times {10}^{-3}$] |

PSO acceleration coefficients | 2 | ${n}_{q}$ | [0, $5\times {10}^{-3}$] |

PSO inertia weight | 1 | ${K}_{iv}$ | [0, 100,000] |

GA crossover rate | $0.7$ | ${K}_{pv}$ | [0, 100,000] |

GA mutation rate | $0.2$ | ${K}_{ic}$ | [0, 100,000] |

Arithmetic crossover parameter | $0.4$ | ${K}_{pc}$ | [0, 100,000] |

Parameters | Conventional | GA | PSO | PSO-GA |
---|---|---|---|---|

${m}_{p}$ | $9.4\times {10}^{-5}$ | $3.54\times {10}^{-5}$ | $4.34\times {10}^{-7}$ | $3.91\times {10}^{-7}$ |

${n}_{q}$ | $1.3\times {10}^{-3}$ | $1.87\times {10}^{-4}$ | $3\times {10}^{-4}$ | $1.42\times {10}^{-5}$ |

${K}_{iv}$ | 390 | $29.85$ | $406.09$ | $980.89$ |

${K}_{pv}$ | $0.05$ | $0.2$ | $1.386$ | $1.386$ |

${K}_{ic}$ | 16,000 | 37,469.11 | 43,762.88 | $1564.94$ |

${K}_{pc}$ | $10.5$ | $19.40$ | $3740.72$ | 10 |

Parameters | Conv. | GA | PSO | PSO-GA |
---|---|---|---|---|

${i}_{od}$ control ${t}_{s}$ (s) | 0.0084 | 0.013 | 0.0039 | 0.0029 |

${i}_{od}$ control ${t}_{r}$ (s) | 0.0017 | 0.0008 | 0.00091 | 0.0008 |

${i}_{oq}$ control ${t}_{s}$ (s) | 0.0076 | 0.0259 | 0.0077 | 0.0044 |

${i}_{oq}$ control ${t}_{r}$ (s) | 0.00082 | 0.0011 | 0.0011 | 0.00087 |

${v}_{od}$ control ${t}_{s}$ (s) | 0.0074 | 0.019 | 0.0058 | 0.0038 |

${v}_{od}$ control ${t}_{r}$ (s) | 0.0017 | 0.00073 | 0.0007 | 0.00069 |

${v}_{oq}$ control ${t}_{s}$ (s) | 0.0128 | 0.033 | 0.0106 | 0.0051 |

${v}_{oq}$ control ${t}_{r}$ (s) | 0.00082 | 0.00016 | 0.00073 | 0.00072 |

**Table 6.**The comparative analysis of designed controllers’ steady-state values in different case studies.

Variables | Case Study I | Case Study II | Case Study III ${}^{1}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Conv. | GA | PSO | PSO-GA | Conv. | GA | PSO | PSO-GA | Conv. | GA | PSO | PSO-GA | |

f | 49.46 | 49.76 | 49.971 | 49.972 | 49.44 | 49.74 | 49.971 | 49.972 | Unstable | 49.85 | 49.97 | 49.97 |

${f}_{n}-f$ | 0.54 | 0.24 | 0.029 | 0.028 | 0.56 | 0.26 | 0.029 | 0.028 | Unstable | 0.15 | 40.03 | 0.03 |

${V}_{o}$ | 351.44 | 375.65 | 372.7 | 380.28 | 354.63 | 369.22 | 371.75 | 379.38 | Unstable | 377.37 | 376.70 | 380.32 |

${V}_{n}-{V}_{o}$ | 28.56 | 4.35 | 7.3 | −0.28 | 25.37 | 10.78 | 8.25 | 0.62 | Unstable | 2.63 | 3.3 | −0.32 |

${I}_{od}$ | 68.17 | 72.98 | 72.41 | 73.85 | 67.66 | 73.33 | 72.69 | 74.17 | Unstable | 37.26 | 37.12 | 37.48 |

${I}_{oq}$ | 43.41 | 46.50 | 46.16 | 47.08 | 45.28 | 47.73 | 47.70 | 48.69 | Unstable | 22.66 | 22.54 | 22.75 |

${V}_{od}$ | 350.33 | 375.03 | 372.26 | 379.61 | 348.94 | 374.96 | 372 | 379.6 | Unstable | 377.60 | 376.18 | 379.81 |

${V}_{oq}$ | 0.0076 | 0.0011 | 0.0004 | 0.0004 | 0.082 | 0.037 | −0.0018 | −0.001 | Unstable | 0.008 | 0.00008 | 0.0003 |

^{1}The values of two inverters are the same.

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## Share and Cite

**MDPI and ACS Style**

Valedsaravi, S.; El Aroudi, A.; Barrado-Rodrigo, J.A.; Issa, W.; Martínez-Salamero, L.
Control Design and Parameter Tuning for Islanded Microgrids by Combining Different Optimization Algorithms. *Energies* **2022**, *15*, 3756.
https://doi.org/10.3390/en15103756

**AMA Style**

Valedsaravi S, El Aroudi A, Barrado-Rodrigo JA, Issa W, Martínez-Salamero L.
Control Design and Parameter Tuning for Islanded Microgrids by Combining Different Optimization Algorithms. *Energies*. 2022; 15(10):3756.
https://doi.org/10.3390/en15103756

**Chicago/Turabian Style**

Valedsaravi, Seyedamin, Abdelali El Aroudi, Jose A. Barrado-Rodrigo, Walid Issa, and Luis Martínez-Salamero.
2022. "Control Design and Parameter Tuning for Islanded Microgrids by Combining Different Optimization Algorithms" *Energies* 15, no. 10: 3756.
https://doi.org/10.3390/en15103756