# Modified Golden Jackal Optimization Assisted Adaptive Fuzzy PIDF Controller for Virtual Inertia Control of Micro Grid with Renewable Energy

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## Abstract

**:**

## 1. Introduction

- A modified GJO (mGJO) algorithm is suggested by incorporating Sine and Cosine Adopted Scaling Factor (SCaSF) in the original GJO method.
- The dominance of them GJO method over GJO, GWO, BBO, GSA, PSO, TLBO, MVO and ALO is demonstrated for test functions as well as the controller design problem.
- An AFPIDF structure is suggested to address the frequency regulation of an islander MG based on the VIC concept.
- The dominance of AFPID over FPID and PID is demonstrated under various levels of a symmetric renewable power penetration.

## 2. Virtual Inertia Control (VIC) in Micro Grid (MG)

#### 2.1. Studied MG

#### 2.2. Structure of VIC Loop

_{VI}, T

_{VI}and Δf

_{PLL}are the control gain, VIC time constant and the frequency variation (output of PLL), respectively.

## 3. Proposed Controller Structure and the Problem Formulation

#### 3.1. Structure of AFPIDF Controller

_{1}, K

_{2}) and the PID controller parameters (K

_{P}, K

_{I}, K

_{D}) that must be selected accurately to obtain the desired frequency regulation.

#### 3.2. Objective Function

## 4. Proposed Modified GJO Algorithm

#### 4.1. Golden Jackal Optimization (GJO) Algorithm

#### 4.1.1. Search Space Design

_{max}and X

_{min}are the bound for variables and “R

_{n}” is a random value from 0 to 1.

#### 4.1.2. Exploration Phase

_{M}(t) and X

_{FM}(t) represent the positions in t &X

_{1}(t) and X

_{2}(t) are new locations of the male and female jackal, respectively.

_{1}and E

_{0}represent the declining and initial energy of the prey. E

_{0}is varied from −1 to 1 and found as:

_{1}is found as:

_{1}equals to 1.5, E

_{1}is gradually reduced from 1.5 to 0 during iterations.

#### 4.1.3. Exploitation Phase

#### 4.1.4. Moving from Exploration to Exploitation

_{0}diminishes from 0 to −1, the prey is actually weakening, and when E

_{0}increases from 0 to 1, the strength of the prey increases. If |E| > 1, the jackal pairs search diverse segments for exploring prey, and if |E| < 1, jackals assault the prey and carries out exploitation.

#### 4.2. Modified GJO (mGJO) Algorithm

_{T}

_{1}&W

_{T}

_{2}are weighting factors. For the proper determination of weights W

_{T}

_{1}&W

_{T}

_{2}, various values of weights are tested. It is observed that the best results are acquired when W

_{T}

_{1}&W

_{T}

_{2}are chosen as 10 &9, respectively.

## 5. Simulation Results and Discussion

#### 5.1. Benchmark Functions Testing

#### 5.2. Implementation of mGJO in Engineering Design Problem

_{D}), wind power (ΔP

_{WTG}), solar power (ΔP

_{PV}), and the total renewable sources (ΔP

_{RES}) are shown in Figure 7a. To authenticate the better performance of the mGJO method, firstly, PID controllers are assumed and PID parameters (K

_{P}, K

_{I}, K

_{D})optimized by the mGJO, GJO, GWO, GSA, PSO, TLBO and ALO techniques. All the techniques are run 30 times independently and the best parameters as per minimum J value given by Equation (2) obtained are chosen, shown in Table 3. It is clear from Table 3 that a smaller amount of J value is attained with GJO compared to GWO, GSA, PSO, TLBO and ALO techniques. The J value is further reduced when mGJO is used. The % decrease in J value with the proposed mGJO technique compared to GJO, GWO, GSA, PSO, TLBO and ALO techniques are 14.68%, 15.34%, 18.36%, 23.66%, 24.21%, and 24.85%, respectively.

#### 5.2.1. Condition 1: Normal RES Integration

_{Th}) and inertia (ΔP

_{Inertia}) power for the above case. In Figure 8b, negative values indicate a decrease in respective power. It can be seen from Figure 8b that when ΔP

_{RES}increases, the ΔP

_{Th}and ΔP

_{Inertia}values decrease.

#### 5.2.2. Condition 2: Reduced RES Integration

_{Th}and ΔP

_{Inertia}is shown in Figure 9c. It can be seen from Figure 9a,c that when the penetration level of ΔP

_{RES}decreases, the ΔP

_{Th}and ΔP

_{Inertia}values increase.

#### 5.2.3. Condition 3: Increased RES Integration

_{RES}is increased to 125% as presented in Figure 10a and the ∆F response is shown in Figure 10b. Improved system responses are obtained with AFPIDF is superior to FPID and PID controllers. The response of ΔP

_{Th}and ΔP

_{Inertia}are shown in in Figure 10c, from which it is clear that as the penetration level of ΔP

_{RES}increases, ΔP

_{Th}and ΔP

_{Inertia}values are suitably changed to minimize the frequency deviation.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{g}) = 0.1 s, Turbine time constant (T

_{t}) = 0.4 s, Virtual inertia time constant (T

_{VI}) = 0.4 s, Frequency bias factor (β) = 1 p.u.MW/Hz; Droop characteristic (R) = 2.4 Hz/p.u.MW; System damping (D) = 0.015 p.u.MW/Hz, System inertia constant (H) = 0.083 p.u.MW s; Wind turbine time constant (T

_{WT}) = 1.5 s, Solar system time constant (T

_{PV}) = 1.85 s, Gate valve limits (V

_{U}/V

_{L}) = ±0.5, ESS capacity (P

_{INTERTIA}) = ±0.25, Virtual inertia gain (K

_{VT}) = 0.8; Phase detector gain (K

_{PD}) = 1.0; Loop filter gain (K

_{LF}) = 1.0; Voltage oscillator gain (K

_{VCO}) = 1.0; Time constant of PLL (T

_{PLL}) = 1.5 s.

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**Figure 9.**Response for Condition-2 (

**a**) ΔP

_{D}, ΔP

_{WTG}, ΔP

_{PV}, ΔP

_{RES}(

**b**) ΔF (

**c**) ΔP

_{Th}, ΔP

_{Inertia}.

**Figure 10.**Response for Condition-2 (

**a**) ΔP

_{D}, ΔP

_{WTG}, ΔP

_{PV}, ΔP

_{RES}(

**b**) ΔF (

**c**) ΔP

_{Th}, ΔP

_{Inertia}.

Function Name | Expression | Range | D |
---|---|---|---|

Sphere | $U{F}_{1}\left(y\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}{y}_{i}^{2}$ | [−100, 100] | 30 |

Schwefel-1 | $U{F}_{2}\left(y\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left|{y}_{i}\right|+{\displaystyle {\displaystyle \prod}_{i=1}^{n}}{y}_{i}$ | [−10, 10] | 30 |

Schwefel-2 | $U{F}_{3}\left(y\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left({\displaystyle {\displaystyle \sum}_{j-1}^{i}}{y}_{j}\right)}^{2}$ | [−100, 100] | 30 |

Schwefel-3 | $U{F}_{4}\left(y\right)=ma{x}_{i}\left\{\left|{y}_{i}\right|,1\le i\le n\right\}$ | [−100, 100] | 30 |

Quartic | $U{F}_{5}\left(y\right)={\displaystyle \sum}_{i=1}^{n}i{y}_{i}^{4}+random\left[0,1\right)$ | [−1.,28, 1.28] | 30 |

Generalized Rastrigin | $M{F}_{1}\left(y\right)={\displaystyle \sum}_{i=1}^{n}\left[{y}_{i}^{2\text{}}-10\mathrm{cos}\left(2{\mathsf{\pi}y}_{i}\right)+10\right]$ | [−5.12, 5.12] | 30 |

Ackley | $M{F}_{2}\left(y\right)=-20\mathit{exp}\left(-0.2\sqrt{\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}{y}_{i}^{2}}\right)\phantom{\rule{0ex}{0ex}}\text{}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-exp\left(\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}\mathrm{cos}\left(2\pi {y}_{i}\right)\right)+20$ | [−32, 32] | 30 |

Generalized Griewank | $M{F}_{3}\left(y\right)=\frac{1}{4000}{\displaystyle \sum}_{i=1}^{n}{y}_{i}^{2}-{\displaystyle \prod}_{j=1}^{n}\mathrm{cos}\left(\frac{{y}_{i}}{\sqrt{i}}\right)+1$ | [−600, 600] | 30 |

Kowalik | $M{F}_{4}\left(y\right)={\displaystyle {\displaystyle \sum}_{i=1}^{11}}{\left[{a}_{i}-\frac{{y}_{1}\left({b}_{i}^{2}+{b}_{i}{y}_{2}\right)}{{b}_{i}^{2}+{b}_{i}{y}_{3}+{y}_{4}}\right]}^{2}$ | [−5, 5] | 4 |

Six-Hump Camel-Back | $M{F}_{5}\left(y\right)=4{y}_{1}^{2}-2.1{y}_{1}^{4}=\frac{1}{3}{y}_{1}^{6}+{y}_{1}{y}_{2}-4{y}_{2}^{2}+4{y}_{2}^{4}$ | [−5, 5] | 2 |

Function | Indices | mGJO | GJO | GWO | GSA | PSO | TLBO | ALO |
---|---|---|---|---|---|---|---|---|

${\mathit{UF}}_{\mathbf{1}}\left(\mathit{y}\right)$ (Min = 0) | Best | 5.49 × 10^{−103} | 2.83 × 10^{−46} | 1.22 × 10^{−23} | 1.57 × 10^{−9} | 1.63 × 10^{−10} | 2.59 × 10^{−43} | 1.23 × 10^{−6} |

Worst | 1.98 × 10^{−99} | 6.4 × 10^{−40} | 2.97 × 10^{−20} | 1.53 × 10^{−6} | 2.09 × 10^{−7} | 5.1 × 10^{−41} | 2.72 × 10^{−5} | |

Ave. | 2.91 × 10^{−98} | 6.3 × 10^{−41} | 3.37 × 10^{−21} | 7.21 × 10^{−9} | 3.42 × 10^{−8} | 1.03 × 10^{−41} | 9.09 × 10^{−6} | |

SD | 5.46 × 10^{−99} | 1.51 × 10^{−40} | 6.45 × 10^{−21} | 3.78 × 10^{−9} | 5.35 × 10^{−8} | 1.47 × 10^{−41} | 7.74 × 10^{−6} | |

${\mathit{UF}}_{\mathbf{2}}\left(\mathit{y}\right)$ (Min = 0) | Best | 5.89 × 10^{−55} | 2.28 × 10^{−25} | 3.75 × 10^{−14} | 1.66 × 10^{−4} | 1.5 × 10^{−5} | 1.38 × 10^{−22} | 7.28 × 10^{−4} |

Worst | 7.74 × 10^{−53} | 2.17 × 10^{−22} | 2.99 × 10^{−13} | 4.45 × 10^{−4} | 7.56 × 10^{−4} | 7.83 × 10^{−21} | 33.87 | |

Ave. | 5.01 × 10^{−52} | 2.23 × 10^{−23} | 6.12 × 10^{−13} | 2.42 × 10^{−4} | 1.18 × 10^{−4} | 1.53 × 10^{−21} | 5.08 | |

SD | 1.03 × 10^{−52} | 4.3 × 10^{−23} | 7.08 × 10^{−13} | 5.91 × 10^{−5} | 1.14 × 10^{−4} | 1.45 × 10^{−21} | 7.81 | |

${\mathit{UF}}_{\mathbf{3}}\left(\mathit{y}\right)$ (Min = 0) | Best | 1.48 × 10^{−87} | 3.69 × 10^{−26} | 8.64 × 10^{−11} | 1.85 × 10^{−2} | 2.66 × 10^{−3} | 2.13 × 10^{−19} | 0.701 |

Worst | 1.42 × 10^{−81} | 1.6 × 10^{−19} | 3.38 × 10^{−7} | 37.29 | 1.41 × 10^{−1} | 1.94 × 10^{−16} | 1551.35 | |

Ave. | 2.16 × 10^{−80} | 1.51 × 10^{−20} | 2.4 × 10^{−8} | 7.11 | 2.43 × 10^{−2} | 2.99 × 10^{−17} | 294.37 | |

SD | 4.50 × 10^{−81} | 3.53 × 10^{−20} | 6.89 × 10^{−8} | 10.01 | 2.89 × 10^{−2} | 4.9 × 10^{−17} | 353.97 | |

${\mathit{UF}}_{\mathbf{4}}\left(\mathit{y}\right)$ (Min = 0) | Best | 5.83 × 10^{−48} | 4.93 × 10^{−18} | 5.52 × 10^{−8} | 2.93 × 10^{−5} | 7.06 × 10^{−4} | 1.77 × 10^{−18} | 9.39 × 10^{−3} |

Worst | 3.37 × 10^{−46} | 4.66 × 10^{−15} | 8.25 × 10^{−6} | 8.95 × 10^{−5} | 3.74 × 10^{−2} | 3.51 × 10^{−17} | 14.68 | |

Ave. | 1.55 × 10^{−45} | 1.28 × 10^{−15} | 1.08 × 10^{−6} | 6.25 × 10^{−5} | 1.01 × 10^{−2} | 7.68 × 10^{−18} | 3.26 | |

SD | 4.46 × 10^{−46} | 1.27 × 10^{−15} | 1.64 × 10^{−6} | 1.56 × 10^{−5} | 8.42 × 10^{−3} | 7.04 × 10^{−18} | 3.62 | |

${\mathit{UF}}_{\mathbf{5}}\left(\mathit{y}\right)$ (Min = 0) | Best | 1.61 × 10^{−5} | 6.73 × 10^{−05} | 2.35 × 10^{−4} | 2.85 × 10^{−3} | 4.79 × 10^{−3} | 3.41 × 10^{−3} | 1.59 × 10^{−2} |

Worst | 2.38 × 10^{−4} | 2.85 × 10^{−3} | 4.69 × 10^{−3} | 3.67 × 10^{−2} | 3.55 × 10^{−2} | 3.162 × 10^{−3} | 1.75 × 10^{−1} | |

Ave. | 6.84 × 10^{−4} | 7.77 × 10^{−4} | 1.37 × 10^{−3} | 1.58 × 10^{−2} | 1.81 × 10^{−2} | 1.71 × 10^{−3} | 6.65 × 10^{−2} | |

SD | 1.87 × 10^{−4} | 0.000657 | 1.06 × 10^{−3} | 7.91 × 10^{−3} | 7.87 × 10^{−3} | 7.19 × 10^{−4} | 3.68 × 10^{−2} | |

${\mathit{MF}}_{\mathbf{1}}\left(\mathit{y}\right)$ (Min = 0) | Best | 0 | 0 | 0 | 0.994961 | 2.992063 | 0.013259 | 7.95967 |

Worst | 0 | 18.13774 | 9.140608 | 14.92438 | 16.24605 | 14.22896 | 49.74783 | |

Ave. | 0 | 0.604591 | 2.653841 | 7.429027 | 8.659233 | 5.500317 | 23.74631 | |

SD | 0 | 3.311483 | 2.834879 | 3.404116 | 3.173189 | 3.437944 | 11.01983 | |

${\mathit{MF}}_{\mathbf{2}}\left(\mathit{y}\right)$ (Min = 0) | Best | 8.88 × 10^{−16} | 4.44 × 10^{−15} | 2.08 × 10^{−12} | 8.09 × 10^{−5} | 1.07 × 10^{−5} | 4.44 × 10^{−15} | 5.57 × 10^{−4} |

Worst | 4.44 × 10^{−15} | 7.99 × 10^{−15} | 1 × 10^{−10} | 1.88 × 10^{−4} | 4.44 × 10^{−4} | 7.54 × 10^{−15} | 5.191245 | |

Ave. | 4.32 × 10^{−15} | 4.8 × 10^{−15} | 2 × 10^{−11} | 1.22 × 10^{−4} | 1.51 × 10^{−4} | 4.9 × 10^{−15} | 1.389097 | |

SD | 6.48 × 10^{−16} | 1.08 × 10^{−15} | 2.13 × 10^{−11} | 2.7 × 10^{−5} | 1.15 × 10^{−4} | 1.9 × 10^{−15} | 1.345589 | |

${\mathit{MF}}_{\mathbf{3}}\left(\mathit{y}\right)$ (Min = 0) | Best | 0 | 0 | 0 | 2.16691 | 6.1535 × 10^{−2} | 0 | 5.5238 × 10^{−2} |

Worst | 0 | 0.173643 | 0.101945 | 11.33721 | 3.007362 | 9.6573 × 10^{−2} | 0.324966 | |

Ave. | 0 | 1.321 × 10^{−2} | 2.985 × 10^{−2} | 5.603493 | 0.939436 | 1.6522 × 10^{−2} | 0.179803 | |

SD | 0 | 3.9472 × 10^{−2} | 0.026988 | 2.647036 | 0.760557 | 2.3714 × 10^{−2} | 7.5028 × 10^{−2} | |

$\mathit{M}{\mathit{F}}_{\mathbf{4}}\left(\mathit{y}\right)$ (Min = 3 × 10 ^{−4})
| Best | 3.08 × 10^{−4} | 3.13 × 10^{−4} | 3.38 × 10^{−4} | 9.23 × 10^{−4} | 3.43 × 10^{−4} | 3.07 × 10^{−4} | 6.27 × 10^{−4} |

Worst | 7.35 × 10^{−4} | 2.04 × 10^{−2} | 2.10 × 10^{−2} | 1.40 × 10^{−2} | 1.35 × 10^{−3} | 2.04 × 10^{−2} | 2.11 × 10^{−2} | |

Ave. | 4.24 × 10^{−4} | 1.40 × 10^{−3} | 2.65 × 10^{−3} | 3.42 × 10^{−3} | 8.74 × 10^{−4} | 1.76 × 10^{−3} | 2.66 × 10^{−3} | |

SD | 1.01 × 10^{−4} | 1.47 × 10^{−4} | 6.08 × 10^{−3} | 3.24 × 10^{−3} | 1.91 × 10^{−4} | 5.06 × 10^{−3} | 3.67 × 10^{−3} | |

$\mathit{M}{\mathit{F}}_{\mathbf{5}}\left(\mathit{y}\right)$ (Min = −1.0316) | Best | −1.0316 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 |

Worst | −1.0254 | −1.03162 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | |

Ave. | −1.0313 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | |

SD | 7.1 × 10^{−4} | 1.63 × 10^{−6} | 1.28 × 10^{−7} | 1.56 × 10^{−10} | 5.38 × 10^{−16} | 5.61 × 10^{−16} | 3.17 × 10^{−13} |

Technique/ Controller | K_{1} | K_{2} | K_{P} | K_{I} | K_{D} | J Value |
---|---|---|---|---|---|---|

ALO/PID | _ | _ | 1.1477 | 0.8386 | 0.2995 | 11.1421 |

GSA/PID | _ | _ | 1.8912 | 1.0214 | 0.2453 | 11.0467 |

PSO/PID | _ | _ | 1.5998 | 1.0559 | 0.3187 | 10.9684 |

GWO/PID | _ | _ | 1.6556 | 1.0852 | 0.2243 | 10.2568 |

TLBO/PID | _ | _ | 1.5945 | 1.1648 | 0.2187 | 9.8906 |

GJO/PID | _ | _ | 0.9331 | 1.1375 | 0.3364 | 9.8138 |

mGJO/PID | _ | _ | 0.4855 | 1.1366 | 0.2016 | 8.3729 |

mGJO/FPID | 0.2811 | 0.0393 | 1.9883 | 1.6173 | 0.1187 | 4.6162 |

mGJO/AFPIDF | 0.3386 | 0.0764 | 1.4223 | 1.4092 | 0.0931 | 2.6341 |

Technique/ Controller | Integral Errors | MO_{S}in ΔF | MU_{s} inΔF (-ve) | |||
---|---|---|---|---|---|---|

ISE | ITAE | ITSE | IAE | |||

ALO/PID | 1.0217 | 270.6073 | 49.0962 | 5.4408 | 0.4259 | 0.4819 |

GSA/PID | 0.9974 | 263.9853 | 46.8408 | 5.3467 | 0.4561 | 0.5126 |

PSO/PID | 0.9665 | 256.7279 | 45.8277 | 5.1555 | 0.4328 | 0.4949 |

GWO/PID | 0.9435 | 240.8556 | 44.3725 | 4.9532 | 0.4678 | 0.5026 |

TLBO/PID | 0.9104 | 227.2530 | 42.7798 | 4.7245 | 0.4684 | 0.4986 |

GJO/PID | 0.8849 | 224.3486 | 41.9169 | 4.4787 | 0.4187 | 0.4775 |

Technique/ Controller | Integral Errors | MO_{S}in ΔF | MU_{s} in ΔF (-ve) | J Value | |||
---|---|---|---|---|---|---|---|

ISE | ITAE | ITSE | IAE | ||||

Case-1 | |||||||

mGJO/PID | 0.7952 | 198.4558 | 38.0886 | 4.2059 | 0.4106 | 0.4707 | 8.3729 |

mGJO/FPID | 0.4417 | 130.6643 | 20.6084 | 2.7231 | 0.3504 | 0.3881 | 4.6162 |

mGJO/AFPIDF | 0.2220 | 84.3966 | 10.0556 | 1.8131 | 0.2838 | 0.2949 | 2.6341 |

Case-2 | |||||||

mGJO/PID | 0.7035 | 158.6698 | 31.3289 | 3.4652 | 0.4060 | 0.4355 | 7.6252 |

mGJO/FPID | 0.4251 | 115.8667 | 18.6158 | 2.5230 | 0.3426 | 0.3617 | 4.7673 |

mGJO/AFPIDF | 0.2212 | 76.7245 | 9.4528 | 1.6975 | 0.2813 | 0.2860 | 3.3048 |

Case-3 | |||||||

mGJO/PID | 0.8792 | 220.4062 | 43.2489 | 4.6307 | 0.4129 | 0.4888 | 9.1711 |

mGJO/FPID | 0.4653 | 139.6949 | 21.8918 | 2.9313 | 0.3540 | 0.4083 | 4.8058 |

mGJO/AFPIDF | 0.2377 | 91.3450 | 10.8443 | 1.9856 | 0.2869 | 0.3014 | 2.6139 |

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## Share and Cite

**MDPI and ACS Style**

Nanda Kumar, S.; Mohanty, N.K.
Modified Golden Jackal Optimization Assisted Adaptive Fuzzy PIDF Controller for Virtual Inertia Control of Micro Grid with Renewable Energy. *Symmetry* **2022**, *14*, 1946.
https://doi.org/10.3390/sym14091946

**AMA Style**

Nanda Kumar S, Mohanty NK.
Modified Golden Jackal Optimization Assisted Adaptive Fuzzy PIDF Controller for Virtual Inertia Control of Micro Grid with Renewable Energy. *Symmetry*. 2022; 14(9):1946.
https://doi.org/10.3390/sym14091946

**Chicago/Turabian Style**

Nanda Kumar, S., and Nalin Kant Mohanty.
2022. "Modified Golden Jackal Optimization Assisted Adaptive Fuzzy PIDF Controller for Virtual Inertia Control of Micro Grid with Renewable Energy" *Symmetry* 14, no. 9: 1946.
https://doi.org/10.3390/sym14091946