# CPSOGSA Optimization Algorithm Driven Cascaded 3DOF-FOPID-FOPI Controller for Load Frequency Control of DFIG-Containing Interconnected Power System

^{*}

## Abstract

**:**

## 1. Introduction

- A novel cascaded 3DOF-FOPID-FOPI controller is proposed for the first time to solve the load frequency control problem of a two-area interconnected electric power system.
- The application of the CPOSGSA algorithm is extended to the load frequency control of a two-area interconnected power system through the optimal selection of gains and parameters of the cascade controller by the superior performance CPSOGSA algorithm.
- The DFIG power fluctuations due to stochastic wind speed are considered in addition to load perturbations in the area interconnected power system to verify the control performance of the controller in a more realistic scenario.
- The robustness of the proposed cascaded fractional-order controller is well illustrated by performing a sensitivity analysis under various conditions.

## 2. Systems Investigated

#### DFIG and Its Additional Frequency Response Model

## 3. Cascade Fractional Order Controller Design

#### 3.1. Implementation of Fractional Calculus

#### 3.2. Cascade 3DOF-FOPID-FOPI Controller

#### 3.2.1. 3DOF-FOPID Controller

#### 3.2.2. FOPI Controller

## 4. The Proposed CPSOGSA Algorithm

#### 4.1. Particle Swarm Optimization

#### 4.2. Gravitational Search Algorithm

#### 4.3. Improved PSOGSA Algorithm under Chaotic Map Optimization (CPSOGSA)

## 5. Simulation Analysis

#### 5.1. Scenario 1 Effect of Different Load Perturbations

#### 5.2. Scenario 2 Wind Speed Fluctuations

#### 5.3. Scenario 3 Comparison of Different Optimization Algorithms

#### 5.4. Scenario 4 Load Perturbation and Internal Parameter Changes

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CC-FOC | Cascade fractional-order controller |

CPSOGSA | Improved particle swarm optimization and gravitational search algorithm under chaotic map optimization |

DE | Differential evolution |

DFIG | Doubly-fed induction generator |

FOPI | Fractional-order proportional-integral |

GSA | Gravitational search algorithm |

ITSE | Time multiplied squared error |

LFC | Load frequency control |

PSO | Particle swarm optimization |

PID | Proportional-integral-differential |

2DOF-FOPID | Two-degree-of-freedom fractional-order proportional-integral-differential |

3DOF-FOPID | Three-degree-of-freedom fractional-order proportional-integral-differential |

## Appendix A

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**Figure 6.**Dynamic performance under Area 1 2% load disturbance (

**a**) $\Delta {f}_{1}$, (

**b**) $\Delta {f}_{2}$, and (

**c**) $\Delta {P}_{tie}$.

**Figure 7.**Dynamic performance at 2% load disturbance in area 1 and 4% load disturbance in area 2 (

**a**) $\Delta {f}_{1}$, (

**b**) $\Delta {f}_{2}$, and (

**c**) $\Delta {P}_{tie}$.

**Figure 8.**The variation of (

**a**) random wind speed ${V}_{W}$, (

**b**) wind turbine output power $\Delta {P}_{W}$.

**Figure 9.**Dynamic performance of the system under random wind speed fluctuations. (

**a**) $\Delta {f}_{1}$, (

**b**) $\Delta {f}_{2}$, and (

**c**) $\Delta {P}_{tie}$.

**Figure 10.**Algorithm iteration curve and dynamic system response (

**a**) iterative process of different algorithms, (

**b**) $\Delta {f}_{1}$, (

**c**) $\Delta {f}_{2}$, and (

**d**) $\Delta {P}_{tie}$.

**Figure 11.**Comparison of robustness results under different load disturbances and system parameters (

**a**) $\Delta {f}_{1}$ with change in loading condition, (

**b**) $\Delta {f}_{1}$ with change in $B$, (

**c**) $\Delta {f}_{1}$ with change in $R$, (

**d**) $\Delta {f}_{1}$ with change in ${T}_{w}$, (

**e**) $\Delta {f}_{1}$ with change in ${T}_{12}$, and (

**f**) $\Delta {f}_{1}$ with change in ${T}_{P}$.

No. | Chaotic Map | Function | Range |
---|---|---|---|

1 | Chebyshev | ${y}_{t+1}=\mathrm{cos}(k{\mathrm{cos}}^{-1}({y}_{t}))$ | [−1, 1] |

2 | Circle | ${y}_{t+1}=\mathrm{mod}(({y}_{t}+g-(e/2\pi )\mathrm{sin}(2\pi {y}_{t})),1)\hspace{0.17em}e=0.5,g=0.2$ | [0, 1] |

3 | Gauss/Mouse | ${y}_{t+1}=\left\{\begin{array}{l}1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{y}_{t}=0\\ \frac{1}{\mathrm{mod}({y}_{t},1)}\hspace{1em}\hspace{0.17em}otherwise\end{array}\right.$ | [0, 1] |

4 | Iterative | ${y}_{t+1}=\mathrm{sin}(e\pi /{y}_{t})\hspace{0.17em}e=0.7$ | [−1, 1] |

5 | Logistic | ${y}_{t+1}=e{y}_{t}(1-{y}_{t})\hspace{0.17em}e=4$ | [0, 1] |

6 | Piecewise | ${y}_{t+1}=\left\{\begin{array}{l}\frac{{y}_{t}}{K}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}0\le {y}_{t}\le K\\ \frac{{y}_{t}-K}{0.5-K}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0\le {y}_{t}\le K\\ \frac{1-K-{y}_{t}}{0.5-K}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}0\le {y}_{t}\le K\\ \frac{1-{y}_{t}}{K}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0\le {y}_{t}\le K\end{array}\right.\mathrm{K}=0.4$ | [0, 1] |

7 | Sine | ${y}_{t+1}=\frac{e}{4}\mathrm{sin}(\pi {y}_{t})\hspace{1em}e=4$ | [0, 1] |

8 | Singer | ${y}_{t+1}=\tau (7.86{y}_{t}-23.31{{y}_{t}}^{2}+28.75{{y}_{t}}^{3}-13.302875{{y}_{t}}^{4})\hspace{1em}\tau =1.07$ | [0, 1] |

9 | Sinusoidal | ${y}_{t+1}=e{{y}_{t}}^{2}\mathrm{sin}(\pi {y}_{t})\hspace{1em}e=2.3$ | [0, 1] |

10 | Tent | ${y}_{k+1}=\left\{\begin{array}{l}\frac{{y}_{t}}{0.7}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{y}_{t}<0.7\\ \frac{10}{3}(1-{y}_{t})\hspace{0.17em}\hspace{1em}{y}_{t}\ge 0.7\end{array}\right.$ | [0, 1] |

Controller | Unit | ${\mathit{K}}_{\mathit{P}}$ | ${\mathit{K}}_{\mathit{I}}$ | ${\mathit{K}}_{\mathit{D}}$ | $\mathit{\eta}$ | $\mathit{\zeta}$ | ${\mathit{P}}_{\mathit{f}}$ | ${\mathit{D}}_{\mathit{f}}$ | ${\mathit{G}}_{\mathit{f}\mathit{f}}$ | $\mathit{N}$ | ${\mathit{K}}_{\mathit{P}1}$ | ${\mathit{K}}_{\mathit{I}1}$ | ${\mathit{\eta}}_{1}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PID | thermal | 2 | 0 | 2 | |||||||||

hydro | 2 | 0 | 2 | ||||||||||

diesel | 2 | 2 | 2 | ||||||||||

FOPID | thermal | 2 | 2 | 2 | 0 | 0.9036 | |||||||

hydro | 0.1677 | 1.8687 | 2 | 0 | 0 | ||||||||

diesel | 2 | 2 | 2 | 0 | 0 | ||||||||

2DOF-FOPID | thermal | 1.9112 | 2 | 2 | 0 | 0.2920 | 0.0402 | 2.9999 | |||||

hydro | 0.9487 | 0 | 0 | 0.0037 | 0.5265 | 0.3364 | 0 | ||||||

diesel | 2 | 2 | 2 | 0.0845 | 0.3382 | 3 | 3 | ||||||

3DOF-FOPID | thermal | 1.9999 | 2 | 2 | 0 | 1 | 2.9910 | 3 | 1 | 46.5761 | |||

hydro | 1.2494 | 1.9858 | 0.1133 | 1 | 0.5932 | 2.2241 | 2.9977 | 0.9536 | 188.0323 | ||||

diesel | 2 | 2 | 2 | 0.0857 | 0 | 3 | 2.9998 | 1 | 143.5195 | ||||

CC-FOC | thermal | 2 | 0 | 2 | 0.9913 | 0.9186 | 3 | 3 | 0.0836 | 64.6058 | 2 | 2 | 0.0386 |

hydro | 2 | 0.0646 | 0 | 0.8879 | 0.9994 | 0 | 2.993 | 0 | 108.3272 | 0.3344 | 0 | 0.6158 | |

diesel | 1.9994 | 1.9992 | 1.9996 | 0.1592 | 0 | 2.9972 | 2.9975 | 1 | 104.8120 | 2 | 2 | 0 |

**Table 3.**Overshoot/Undershoot and Settling time of state variables for 2% load disturbance in area 1.

Controller | $\mathbf{\Delta}{\mathit{f}}_{1}$ | $\mathbf{\Delta}{\mathit{f}}_{2}$ | $\mathbf{\Delta}{\mathit{P}}_{\mathit{t}\mathit{i}\mathit{e}}$ | $\mathbf{ITSE}\times {10}^{-6}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathit{U}\mathit{S}\times {10}^{-3}\left(\mathbf{Hz}\right)$ | $\mathit{O}\mathit{S}\times {10}^{-3}\left(\mathbf{Hz}\right)$ | ${\mathit{T}}_{\mathit{s}}\left(\mathbf{sec}\right)$ | $\mathit{U}\mathit{S}\times {10}^{-3}\left(\mathbf{Hz}\right)$ | $\mathit{O}\mathit{S}\times {10}^{-3}\left(\mathbf{Hz}\right)$ | ${\mathit{T}}_{\mathit{s}}\left(\mathbf{sec}\right)$ | $\mathit{U}\mathit{S}\times {10}^{-3}\left(\mathbf{Hz}\right)$ | $\mathit{O}\mathit{S}\times {10}^{-3}\left(\mathbf{Hz}\right)$ | ${\mathit{T}}_{\mathit{s}}\left(\mathbf{sec}\right)$ | ||

PID | −1.41 | 0.3 | 10.78 | −0.26 | 0.1 | 13.2 | −0.25 | 0.132 | 10.5 | 2.88 |

FOPID | −1.88 | 0.0837 | 8.66 | −0.24 | 0.0078 | 5.46 | −0.2 | 0.005 | 6.21 | 0.79 |

2DOF-FOPID | −0.91 | 0.0286 | 5.45 | −0.074 | 0.001 | 4.65 | −0.08 | 0 | 5.65 | 0.122 |

3DOF-FOPID | −0.49 | 0.0347 | 5.35 | −0.0723 | 0.0026 | 4.25 | −0.0745 | 0.00223 | 4.45 | 0.098 |

CC-FOC | −0.1452 | 0.0096 | 1.276 | −0.0230 | 0.0008 | 1.976 | −0.0209 | 0.0006 | 1.576 | 0.0078 |

Controller | % Change | $\Delta {\mathit{f}}_{1}$ | $\Delta {\mathit{f}}_{2}$ | $\Delta {\mathit{P}}_{\mathit{t}\mathit{i}\mathit{e}}$ | $\mathrm{ITSE}\times {10}^{-6}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{U}\mathit{S}\times {10}^{-3}$ (Hz) | $\mathit{O}\mathit{S}\times {10}^{-3}$ (Hz) | $\mathit{T}}_{\mathit{s}$ (sec) | $\mathit{U}\mathit{S}\times {10}^{-3}$ (Hz) | $\mathit{O}\mathit{S}\times {10}^{-3}$ (Hz) | $\mathit{T}}_{\mathit{s}$ (sec) | $\mathit{U}\mathit{S}\times {10}^{-3}$ (Hz) | $\mathit{O}\mathit{S}\times {10}^{-3}$ (Hz) | $\mathit{T}}_{\mathit{s}$ (sec) | |||

Nominal | 0 | −0.1452 | 0.0096 | 1.276 | −0.0230 | 0.0008 | 1.976 | −0.0209 | 0.0006 | 1.576 | 0.0077 |

Loading condition | +25 | −0.1816 | 0.0119 | 1.327 | −0.0290 | 0.0009 | 2.505 | −0.0261 | 0.0008 | 2.255 | 0.0122 |

−25 | −0.1089 | 0.0071 | 1.142 | −0.0017 | 0.0006 | 1.192 | −0.0156 | 0.0005 | 1.142 | 0.0044 | |

$B$ = 0.5390 | +25 | −0.1327 | 0.0087 | 1.255 | −0.0192 | 0.0006 | 1.505 | −0.0194 | 0.0006 | 1.455 | 0.0066 |

$B$ = 0.3234 | −25 | −0.1610 | 0.0105 | 1.254 | −0.0286 | 0.0010 | 2.339 | −0.0226 | 0.0071 | 1.789 | 0.0094 |

$R$ = 3 | +25 | −0.1439 | 0.0094 | 1.279 | −0.0227 | 0.0078 | 1.929 | −0.0207 | 0.0064 | 1.529 | 0.0077 |

$R$ = 1.8000 | −25 | −0.1466 | 0.0096 | 1.277 | −0.0236 | 0.0081 | 2.027 | −0.0210 | 0.0065 | 1.627 | 0.0079 |

${T}_{w}$ = 1.25 | +25 | −0.1453 | 0.0095 | 1.277 | −0.0232 | 0.0079 | 1.977 | −0.0209 | 0.0065 | 1.577 | 0.0078 |

${T}_{w}$ = 0.75 | −25 | −0.1453 | 0.0095 | 1.278 | −0.0232 | 0.0078 | 1.978 | −0.0208 | 0.0065 | 1.578 | 0.0078 |

${T}_{12}$ = 0.0541 | +25 | −0.1446 | 0.0087 | 1.177 | −0.0266 | 0.0093 | 2.133 | −0.0239 | 0.0076 | 1.813 | 0.0074 |

${T}_{12}$ = 0.0325 | −25 | −0.1459 | 0.0100 | 1.326 | −0.0192 | 0.0061 | 1.476 | −0.0173 | 0.0049 | 1.527 | 0.0083 |

${T}_{P}$ = 14.3625 | +25 | −0.1384 | 0.0966 | 1.285 | −0.0233 | 0.0080 | 2.025 | −0.0233 | 0.0804 | 1.628 | 0.0084 |

${T}_{P}$ = 8.6175 | −25 | −0.1540 | 0.0947 | 1.243 | −0.0230 | 0.0078 | 1.925 | −0.0207 | 0.0065 | 1.534 | 0.0076 |

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## Share and Cite

**MDPI and ACS Style**

Xie, S.; Zeng, Y.; Qian, J.; Yang, F.; Li, Y. CPSOGSA Optimization Algorithm Driven Cascaded 3DOF-FOPID-FOPI Controller for Load Frequency Control of DFIG-Containing Interconnected Power System. *Energies* **2023**, *16*, 1364.
https://doi.org/10.3390/en16031364

**AMA Style**

Xie S, Zeng Y, Qian J, Yang F, Li Y. CPSOGSA Optimization Algorithm Driven Cascaded 3DOF-FOPID-FOPI Controller for Load Frequency Control of DFIG-Containing Interconnected Power System. *Energies*. 2023; 16(3):1364.
https://doi.org/10.3390/en16031364

**Chicago/Turabian Style**

Xie, Shihao, Yun Zeng, Jing Qian, Fanjie Yang, and Youtao Li. 2023. "CPSOGSA Optimization Algorithm Driven Cascaded 3DOF-FOPID-FOPI Controller for Load Frequency Control of DFIG-Containing Interconnected Power System" *Energies* 16, no. 3: 1364.
https://doi.org/10.3390/en16031364