# Reaction Curve-Assisted Rule-Based PID Control Design for Islanded Microgrid

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## Abstract

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## 1. Introduction

- To represent the islanded microgrid model mathematically, an equivalent transfer function is derived with the help of the first-order transfer function of all DGUs and ESDs.
- The derived transfer function is approximated into FOPDT form for ascertainment of suitable controller gain parameters.
- PID controller tuning is processed with rule-based tuning methods such as Ziegler–Nichols (ZN) step response method [29], Chien–Hrones–Reswick (CHR) method [30], approximate m-constrained integral gain optimization (AMIGO) method [31], Wang and Cluett (WC) method [32], Wang–Chan–Juang (WCJ) method [24], and Cohen–Coon (CC) method [33].
- Frequency regulation analysis for all rule-based controllers is conducted.
- The utility of the PID control design and employed rule-based controller tuning methods used to mitigate frequency deviation is analyzed by presenting step response, impulse response, Bode plot, and frequency deviation plot.

## 2. Islanded-Microgrid: Architecture and Description

#### 2.1. Mathematical Models of Microgrid Components

#### 2.1.1. Diesel Engine Generator

#### 2.1.2. Solar Photovoltaic Panel

#### 2.1.3. Wind Turbine Generator

#### 2.1.4. Biogas Turbine Generator

#### 2.1.5. Biodiesel Engine Generator

#### 2.1.6. Micro-Turbine Generator

#### 2.1.7. Aqua-Electrolyzer Fuel Cell

#### 2.1.8. Battery Energy Storage

#### 2.1.9. Flywheel Energy Storage

#### 2.1.10. Electric Vehicle

#### 2.1.11. Generator Dynamics

#### 2.2. Block Diagram of Islanded Microgrid

#### 2.3. Relation between Frequency Deviation and Net Generated Power

#### 2.4. Proportional Integral Derivative Controller

**Proportional term**reduces the rise time. However, it does not eliminate steady-state error. A large proportional gain value can cause system instability. While, a small gain results in a smaller output response.**Integral term**eliminates steady-state errors. But it may have large values of transient response. Overshoot may be caused by a high integral gain. While sluggishness may be caused by a low integral gain.**Derivative term**increases the system’s stability, reduces overshoot, and improves transient response. A large derivative gain may make the system unstable.

## 3. Controller Tuning: Rule-Based Methods

#### 3.1. Ziegler–Nichols Method

#### 3.2. Cohen–Coon Method

#### 3.3. Wang–Cluett Method

#### 3.4. Wang–Chan–Juang Method

#### 3.5. Chien–Hrones–Reswick Method

#### 3.6. Approximate M-Constrained Integral Gain Optimization Method

## 4. Results and Discussion

#### 4.1. Overall Model of Islanded Microgrid

#### 4.2. Validation of Approximated Model

#### 4.3. Frequency Regulation of Islanded Microgrid

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Equivalent block diagram of islanded microgrid depicted in Figure 2.

DGUs/ESDs | Transfer Function and Parameters of DGUs and ESDs | ||
---|---|---|---|

Equation Number | Parameters | Nominal Transfer Function | |

DEG | (1) | ${T}_{deg}=2$, | $T{F}_{deg}\left(s\right)=\frac{0.003}{1+2s}$ |

${K}_{deg}=0.003$ | |||

SPV | (2) | ${T}_{spv}=1.8$, | $T{F}_{spv}\left(s\right)=\frac{1}{1+1.8s}$ |

${K}_{spv}=1$ | |||

WTG | (3) | ${T}_{wt}=1.5$, | $T{F}_{wt}\left(s\right)=\frac{1}{1+1.5s}$ |

${K}_{wt}=1$ | |||

BGTG | (4) | ${T}_{bgtg}=0.55$, | $T{F}_{bdeg}\left(s\right)=\frac{1}{1+0.550s}$ |

${K}_{bgtg}=1$ | |||

BDEG | (6) | ${T}_{bdeg}=0.148$, | $T{F}_{bgtg}\left(s\right)=\frac{1}{1+0.148s}$ |

${K}_{bdeg}=1$ | |||

MT | (7) | ${T}_{mt}=1.5$, | $T{F}_{mt}\left(s\right)=\frac{1}{1+1.5s}$ |

${K}_{mt}=1$ | |||

AE-FC | (8) | ${T}_{ae}=0.5$, | $T{F}_{ae-fc}\left(s\right)=\frac{0.002}{1+0.5s}$ |

${K}_{ae}=0.002$ | |||

BESS | (9) | ${T}_{bess}=0.1$, | $T{F}_{bess}\left(s\right)=\frac{-0.003}{1+0.1s}$ |

${K}_{bess}=-0.003$ | |||

FESS | (10) | ${T}_{fess}=0.1$, | $T{F}_{fess}\left(s\right)=\frac{-0.01}{1+0.1s}$ |

${K}_{fess}=-0.01$ | |||

EV | (11) | ${T}_{ev}=0.9$, | $T{F}_{ev}\left(s\right)=\frac{-0.7}{1+0.9s}$ |

${K}_{ev}=-0.7$ | |||

Generator dynamics | (12) | $D=0.3$, | $T{F}_{gd}\left(s\right)=\frac{1}{0.3+0.4s}$ |

$M=0.4$ |

Method | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{t}}_{\mathit{i}}$ | ${\mathit{t}}_{\mathit{d}}$ |
---|---|---|---|

ZN | $1.2{T}_{m}/{T}_{d}$ | $2{T}_{d}$ | ${T}_{d}/2$ |

Method | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{t}}_{\mathit{i}}$ | ${\mathit{t}}_{\mathit{d}}$ |
---|---|---|---|

CC | $\frac{1.35}{a}\left(1+\frac{0.18{T}_{m}}{1-{T}_{m}}\right)$ | $\frac{2.5-2{T}_{m}}{1-0.39{T}_{m}}{T}_{d}$ | $\frac{0.37-0.37{T}_{m}}{1-0.81{T}_{m}}{T}_{d}$ |

Method | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{t}}_{\mathit{i}}$ | ${\mathit{t}}_{\mathit{d}}$ |
---|---|---|---|

WC | $\frac{0.13+0.51{T}_{d}}{K}$ | $\frac{\alpha (0.25+0.96{T}_{d})}{0.93+0.03{T}_{d}}$ | $\frac{\alpha (-0.03+0.28{T}_{d})}{0.25+{T}_{d}}$ |

Method | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{t}}_{\mathit{i}}$ | ${\mathit{t}}_{\mathit{d}}$ |
---|---|---|---|

WCJ | $\frac{\left(0.7303+\frac{0.5307{T}_{m}}{{T}_{d}}\right)\left({T}_{m}+0.5{T}_{d}\right)}{K\left({T}_{m}+{T}_{d}\right)}$ | ${T}_{m}+0.5{T}_{d}$ | $\frac{0.5{T}_{m}{T}_{d}}{{T}_{m}+0.5{T}_{d}}$ |

Method | With 0% Overshoot | With 20% Overshoot | ||||
---|---|---|---|---|---|---|

${\mathit{K}}_{\mathit{p}}$ | ${\mathit{t}}_{\mathit{i}}$ | ${\mathit{t}}_{\mathit{d}}$ | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{t}}_{\mathit{i}}$ | ${\mathit{t}}_{\mathit{d}}$ | |

$CH{R}_{SP}$ | $\frac{0.6}{a}$ | ${T}_{m}$ | $0.5{T}_{d}$ | $\frac{0.95}{a}$ | $1.4{T}_{m}$ | $0.47{T}_{d}$ |

$CH{R}_{LR}$ | $\frac{0.95}{a}$ | $2.4{T}_{m}$ | $0.42\ast {T}_{d}$ | $\frac{1.2}{a}$ | $2{T}_{m}$ | $0.42{T}_{d}$ |

Method | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{t}}_{\mathit{i}}$ | ${\mathit{t}}_{\mathit{d}}$ |
---|---|---|---|

AMIGO | $\frac{1}{K}\left(0.2+0.4({T}_{m}/{T}_{d})\right)$ | $\frac{\left(0.4{T}_{d}+0.8{T}_{m}\right)}{{T}_{d}+0.1{T}_{m}}$ | $\frac{0.5{T}_{d}{T}_{m}}{0.3{T}_{d}+{T}_{m}}$ |

Numerator’s Coefficients | Denominator’s Coefficients | ||
---|---|---|---|

${N}_{0}$ | 3.003 | ${D}_{0}$ | 0.0587 |

${N}_{1}$ | 16.67 | ${D}_{1}$ | 4.882 |

${N}_{2}$ | 34.58 | ${D}_{2}$ | 16.19 |

${N}_{3}$ | 33.32 | ${D}_{3}$ | 27.32 |

${N}_{4}$ | 15.03 | ${D}_{4}$ | 24.86 |

${N}_{5}$ | 2.832 | ${D}_{5}$ | 12.13 |

${N}_{6}$ | 0.2324 | ${D}_{6}$ | 3.147 |

${N}_{7}$ | 0.00694 | ${D}_{7}$ | 0.44 |

- | - | ${D}_{8}$ | 0.0312 |

- | - | ${D}_{9}$ | 0.00087 |

Steady-State Gain (K) | Time Constant (${\mathit{T}}_{\mathit{m}}$) | Time Delay (${\mathit{T}}_{\mathit{d}}$) |
---|---|---|

5.1877 | 3.1265 | 0.1471 |

Time Domain Specifications | ||||||
---|---|---|---|---|---|---|

Rise Time (s) | Settling Time (s) | Peak | Peak Time (s) | Overshoot | Undershoot | |

Islanded microgrid | 5.6099 | 9.8795 | 5.1123 | 17.7748 | 0 | 0 |

Approximated model | 6.8690 | 12.3781 | 5.1877 | 24.3327 | 0 | 0 |

Error Indices | IAE | ITAE | ISE | ITSE |
---|---|---|---|---|

Values | 2.03053 | 14.12585 | 0.33672 | 1.72525 |

Tuning Method | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{t}}_{\mathit{i}}$ | ${\mathit{t}}_{\mathit{d}}$ |
---|---|---|---|

ZN | 25.5129 | 0.2941 | 0.0735 |

AMIGO | 1.6779 | 5.5688 | 0.0725 |

CC | 5.5795 | 0.3607 | 0.0539 |

CHR | 4.9180 | 6.2531 | 0.0618 |

WC | 0.0395 | 0.2093 | 0.0141 |

WCJ | 24.2589 | 3.2001 | 0.0718 |

Method | Time Domain Specifications | |||||
---|---|---|---|---|---|---|

Rise Time (s) | Settling Time (s) | Peak | Peak Time (s) | Overshoot | Undershoot | |

PID-ZN | 0.0698 | 0.5369 | 1.2544 | 0.1786 | 25.4408 | 0 |

PID-AMIGO | 0.5883 | 6.8357 | 0.9984 | 21.7943 | 0 | 0 |

PID-CC | 0.1758 | 2.0267 | 1.4594 | 0.4385 | 45.9374 | 0 |

PID-CHR | 0.2261 | 1.2370 | 1.1276 | 0.4558 | 12.7572 | 0 |

PID-WC | 2.2228 | 18.5825 | 1.3544 | 5.590 | 35.4425 | 0 |

PID-WCJ | 0.0760 | 0.4844 | 1.1710 | 0.1733 | 17.0960 | 0 |

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**MDPI and ACS Style**

Bashishtha, T.K.; Singh, V.P.; Yadav, U.K.; Varshney, T.
Reaction Curve-Assisted Rule-Based PID Control Design for Islanded Microgrid. *Energies* **2024**, *17*, 1110.
https://doi.org/10.3390/en17051110

**AMA Style**

Bashishtha TK, Singh VP, Yadav UK, Varshney T.
Reaction Curve-Assisted Rule-Based PID Control Design for Islanded Microgrid. *Energies*. 2024; 17(5):1110.
https://doi.org/10.3390/en17051110

**Chicago/Turabian Style**

Bashishtha, T. K., V. P. Singh, U. K. Yadav, and T. Varshney.
2024. "Reaction Curve-Assisted Rule-Based PID Control Design for Islanded Microgrid" *Energies* 17, no. 5: 1110.
https://doi.org/10.3390/en17051110