# First-of-Its-Kind Frequency Enhancement Methodology Based on an Optimized Combination of FLC and TFOIDFF Controllers Evaluated on EVs, SMES, and UPFC-Integrated Smart Grid

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## Abstract

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## 1. Introduction

#### 1.1. Motivation and Incitement

#### 1.2. Literature Review

#### 1.3. Contribution and Paper Organization

- The suggestion of a control structure that combines the benefits of tilt, fuzzy logic, FOPID, and fractional filter regulators in a single controller known as FTFOIDFF that efficiently improves frequency stability in a hybrid two-area linked power system incorporated with severe RES penetrations.
- Utilization of a nature-inspired metaheuristic optimization technique that was recently developed (i.e., the prairie dog optimizer, or PDO) for the purpose of fine-tuning the recommended controller settings as well as the input scaling factors and membership functions for both FLC inputs and outputs in an effective manner.
- Validation of the beneficial influence of the integration of SMES, UPFC, and EVs in enhancing frequency performance during load perturbations and RESs penetrations.

## 2. Modelling of the Investigated Hybrid Power System with SMES, UPFC and EVs

#### 2.1. The Power System Structure

#### 2.2. Mathematical Representation of UPFC

#### 2.3. Mathematical Representation of SMES

#### 2.4. Mathematical Representation of the Wind Farm Unit

#### 2.5. Mathematical Representation of the PV Unit

#### 2.6. Mathematical Representation of the EVs Units

## 3. Control Strategy and Problem Presentation

#### 3.1. Prairie Dog Optimizer (PDO)

#### 3.1.1. Initialization

#### 3.1.2. The Estimation of Objective Function

#### 3.1.3. Exploration Phase

#### 3.1.4. Exploitation Phase

#### 3.2. The Detailed Configuration of The Proposed FTFOIDFF Regulator

## 4. Results and Discussion

^{®}(R2022b) is used to implement all simulation results for the investigated dual-area, multi-unit hybrid power grid in order to verify the suggested controller’s efficacy in enhancing the system’s performance. The outcomes of the simulation are generated on a computer equipped with an AMD Ryzen 7 3700U-2.30 GHz processor and 20.00 GB of RAM. By computing the value of the optimal objective function, which is represented by the ITAE value across iterations, the effectiveness of the researched power grid may be assessed. Before improving the suggested FTFOIDFF controller using the recommended PDO method, a number of preliminary issues, such as the 30 populations and 100 iterations, must be resolved. Figure 14 depicts a convergence curve that illustrates the performance of the proposed PDO algorithm in comparison to other recent optimization methodologies (i.e., Seagull Optimization Algorithm (SOA), RUNge Kutta optimizer (RUN), and Chaos Game Optimizer (CGO)). The demonstrated convergence curve can be obtained by taking on a 10% SLP at 5 s in area (a) of the investigated hybrid power grid, with no penetration of RESs in both areas. Clearly, the PDO algorithm achieved the lowest objective function value (0.0875) compared to the previously mentioned approaches. Consequently, the convergence curve demonstrates the efficacy of the proposed PDO algorithm.

#### 4.1. Case I: 10% Step Load Perturbation (SLP) at t = 5 s in Area (a)

#### 4.2. Case II: Multi-Step Load Perturbation (MSLP) in Area (a)

#### 4.3. Case III: Random Load Perturbation (RLP) in Area (b)

#### 4.4. Case IV: Pulse Load Perturbation (PLP) in Area (a)

#### 4.5. Case V: Random Sinusoidal Load Perturbation (RSLP) in Area (a)

#### 4.6. Case VI: MSLP in Area (a) with 0.01 s Communication Time Delay (CTD)

#### 4.7. Case VII: Applying RESs Fluctuations in Both Areas

#### 4.8. Case VIII: Applying RESs Fluctuations with MSLP in Area (b) and RSLP in Area (a)

#### 4.9. Case IX: UPFC and SMES Effect on the Studied System with 30% SLP in Area (a)

#### 4.10. Case X: EVs Effect on the Studied System with MSLP in Area (a)

#### 4.11. Case XI: Sensitivity Analysis

## 5. Conclusions

- The proposed control structure has efficiently improved frequency stability in a multi-area power system with severe RES penetrations.
- Integrating the benefits of tilt, fuzzy logic, FOPID, and fractional filter regulators in a single controller known as FTFOIDFF, which has superior performance over the other recent control structures.
- Application of a nature-inspired metaheuristic optimization technique that was recently developed (i.e., the Prairie Dog Optimizer, or PDO) for the purpose of fine-tuning not only the recommended controller settings but also the MFs of the FLC’s inputs and outputs in an effective manner.
- Validation of the positive effect of the integration of SMES, UPFC, and EVs in enhancing frequency performance during several harsh disturbances.

- The inclusion of conventional controllers for comparison with intelligent fuzzy-based controllers is unfair, as the incorporation of intelligent controllers, such as fuzzy logic or artificial neural networks, enhances the frequency response performance excessively.
- The use of simple structured models for EV, SMES, and UPFC will not reveal the full impact of these devices or the uncertainties that may be introduced into the systems as a result of their incorporation.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

FLC | Fuzzy Logic Control |

FTFOIDFF | Fuzzy Tilted Fractional Order Integral Derivative with Fractional Filter |

PDO | Prairie Dog Optimizer |

SOA | Seagull Optimisation Algorithm |

RUN | Runge Kutta optimizer |

CGO | Chaos Game Optimizer |

LFC | Load Frequency Control |

CTD | Communication Time Delay |

GRC | Generation Rate Constraint |

UPFC | Unified Power Flow Controller |

SMES | Superconducting Magnetic Energy Storage |

EV | Electric Vehicle |

PID | Proportional Integral Derivative |

PIDA | Proportional Integral Derivative Acceleration |

SLP | Step Load Perturbation |

MSLP | Multi-Step Load Perturbation |

RLP | Random Load Perturbation |

RSLP | Random Sinusoidal Load Perturbation |

PLP | Pulse Load Perturbation |

AGC | Automatic Generation Control |

RES | Renewable Energy Sources |

PV | Photovoltaic |

FACTS | Flexible Alternating Current Transmission Systems |

FO | Fractional Order |

PCS | Power Conversion System |

PD | Prairie Dog |

CT | Coterie |

LB | Lower Boundary |

UB | Upper Boundary |

GB | Global Best |

iter | Current Iteration |

Maxiter | Maximum Iteration Number |

MFs | Membership Functions |

E | Error |

DOE | Derivative of Error |

NB | Negative Big |

NS | Negative Small |

Z | Zero |

PS | Positive Small |

PB | Positive Big |

FIS | Fuzzy Interface System |

${\mathrm{U}}_{1}$ | First Control Law |

${\mathrm{U}}_{\mathrm{t}}$ | Total Control Law |

${\mathrm{G}}_{\mathrm{C}}$ | TFOIDFF Controller’s Transfer Function |

${\mathrm{t}}_{\mathrm{s}\mathrm{i}\mathrm{m}}$ | Simulation Time |

${\mathrm{K}}_{\mathrm{t}}$ | Tilt Gain |

${\mathrm{K}}_{\mathrm{i}}$ | Integral Gain |

${\mathrm{K}}_{\mathrm{d}}$ | Derivative Gain |

$\mathrm{n}$ | Tilt Fractional Order Power |

${\mathsf{\lambda}}_{\mathrm{i}}$ | Fractional Order Integral Operator |

${\mathsf{\mu}}_{\mathrm{d}}$ | Fractional Order Derivative Operator |

${\mathsf{\lambda}}_{\mathrm{f}}$ | Fractional Order Filter Operator |

${\mathrm{N}}_{\mathrm{f}}$ | Fractional Filter Coefficient |

${\mathrm{K}}_{1}$$,{\mathrm{K}}_{2}$ | Scaling Factor of the FLC inputs |

ACE | Area Control Error |

ITAE | Integral Time Absolute Error |

MOS | Maximum Overshoot |

MUS | Maximum Undershoot |

ST | Settling Time |

ΔFa | The frequency deviation of Area (a) |

ΔFb | The frequency deviation of Area (b) |

ΔPtie | The tie-line power deviation |

## Appendix A. The Nominal Values of the Power System’s Parameters

Parameter | Nominal Value | Parameter Definition |

${\tau}_{sg}$ | 0.08 s | Governor time constant |

${k}_{r}$ | 0.3 s | Gain of reheater steam turbine |

${\tau}_{r}$ | 10.2 s | The time constant of reheater steam turbine |

${\tau}_{t}$ | 0.3 s | Steam turbine time constant |

${\tau}_{gh}$ | 0.2 s | Hydroelectric turbine speed governor time constant |

${\tau}_{rs}$ | 4.9 s | Hydro turbine speed governor reset time |

${\tau}_{rh}$ | 28.749 s | Time constant of the transient droop |

${\tau}_{w}$ | 1.1 s | Average water string time in penstock |

${b}_{g}$ | 0.049 s | Gas turbine constant of valve positioner |

${c}_{g}$ | 1 | Valves’ gas turbine positioner constant |

${x}_{c}$ | 0.6 s | Gas turbine governor’s lead time constant |

${y}_{c}$ | 1.1 s | Gas turbine governor’s lag time constant |

${\tau}_{cr}$ | 0.01 s | Combustion response time delay in a gas turbine |

${\tau}_{fc}$ | 0.239 s | Gas turbine fuel time constant |

${\tau}_{cd}$ | 0.2 s | Volume-time constant for gas turbine compressor discharge |

${k}_{ps1}$, ${k}_{ps2}$ | 68.965, 68.965 | Power system gains |

${\tau}_{ps1}$, ${\tau}_{ps2}$ | 11.49, 11.49 s | Power system time constants |

${T}_{12}$ | 0.0433 MW | Coefficient of synchronizing |

${k}_{SMES\left(a\right)}$, ${k}_{SMES\left(b\right)}$ | 1, 1 | Gains of SMES |

${\tau}_{SMES\left(a\right)}$, ${\tau}_{SMES\left(b\right)}$ | 0.07 s | Time constants of SMES units |

${\tau}_{UPFC}$ | 0.003 s | Time constant of UPFC unit |

${k}_{EV\left(a\right)}$, ${k}_{EV\left(b\right)}$ | 1 | Gains of EVs |

${\tau}_{EV\left(a\right)}$, ${\tau}_{EV\left(b\right)}$ | 0.28 s | Time constants of EVs |

${B}_{a},{B}_{b}$ | 0.431, 0.431 MW/Hz | Frequency bias coefficients |

$R$ | 2.4 Hz/MW | Governor speed regulation constant for thermal, hydro, and gas units |

$C{F}_{T},C{F}_{H},C{F}_{G}$ | 0.5435, 0.3261, 0.1304 | Contribution factors of thermal, hydro, and gas units |

GRC with Hydro | -------- | (0.045 pu.MW/s) and (0.06 pu.MW/s. For both rising and decreasing rates), respectively |

GRC with Thermal | -------- | The GRC (generation rate constraint) for the thermal unit is set (0.0017 pu.MW/s) For rising and decreasing rates |

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**Figure 3.**UPFC integration in a dual area linked power system [67].

**Figure 4.**The circuit diagram representation of SMES [67].

**Figure 5.**The wind system representation in MATLAB/Simulink program (R2022b) [69].

**Figure 7.**The PV system representation in MATLAB/Simulink program (R2022b) [69].

**Figure 11.**The configuration of TFOIDFF controller [65].

**Figure 13.**The MFs of the proposed FTFOIDFF for both areas. (

**a**) MFs of the controllers in area 1; (

**b**) MFs of the controllers in area 2.

**Table 1.**The examined power grid’s transfer functions [69].

Power Planet | Model | Transfer Function |
---|---|---|

Thermal | Governor | $\frac{1}{{\tau}_{sg}s+1}$ |

Reheat | $\frac{{k}_{r}{\tau}_{r}s+1}{{\tau}_{r}s+1}$ | |

Turbine | $\frac{1}{{\tau}_{t}s+1}$ | |

Hydraulic | Governor | $\frac{1}{{\tau}_{gh}s+1}$ |

Transient droop compensation | $\frac{{\tau}_{rs}s+1}{{\tau}_{rh}s+1}$ | |

turbine | $\frac{-{\tau}_{w}s+1}{0.5{\tau}_{w}s+1}$ | |

Gas | Valve positioner | $\frac{1}{{b}_{g}s+{c}_{g}}$ |

Speed governor | $\frac{{x}_{c}s+1}{{y}_{c}s+1}$ | |

Fuel system and combustor | $\frac{-{\tau}_{cr}s+1}{{\tau}_{fc}s+1}$ | |

Compressor discharge | $\frac{1}{{\tau}_{cd}s+1}$ | |

Others | Power system (a) | $\frac{{k}_{ps1}}{{\tau}_{ps1}s+1}$ |

Power system (b) | $\frac{{k}_{ps2}}{{\tau}_{ps2}s+1}$ | |

T-line | $\frac{2\pi {T}_{12}}{s}$ | |

SMES (a) | $\frac{{k}_{\mathrm{S}\mathrm{M}\mathrm{E}\mathrm{S}\left(a\right)}}{{\tau}_{\mathrm{S}\mathrm{M}\mathrm{E}\mathrm{S}\left(a\right)}s+1}$ | |

SMES (b) | $\frac{{k}_{\mathrm{S}\mathrm{M}\mathrm{E}\mathrm{S}\left(b\right)}}{{\tau}_{\mathrm{S}\mathrm{M}\mathrm{E}\mathrm{S}\left(b\right)}s+1}$ | |

UPFC | $\frac{1}{{\tau}_{\mathrm{U}\mathrm{P}\mathrm{F}\mathrm{C}}s+1}$ | |

EV (a) | $\frac{{k}_{\mathrm{E}\mathrm{V}\left(a\right)}}{{\tau}_{\mathrm{E}\mathrm{V}\left(a\right)}s+1}$ | |

EV (b) | $\frac{{k}_{\mathrm{E}\mathrm{V}\left(b\right)}}{{\tau}_{\mathrm{E}\mathrm{V}\left(b\right)}s+1}$ |

E | DOE | ||||
---|---|---|---|---|---|

NB | SN | Z | SP | LP | |

NB | NB | NB | NS | NS | Z |

NS | NB | NS | NS | Z | PS |

Z | NS | NS | Z | PS | PS |

PS | NS | Z | PS | PS | PB |

PB | Z | PS | PS | PB | PB |

Controller | Thermal | Hydro | Gas | |||
---|---|---|---|---|---|---|

PID | Area (a) | ${K}_{p}=0.024,{K}_{i}=0.486,$ ${K}_{d}=0.293,{N}_{f}=102$ | Area (a) | ${K}_{p}=0.025,{K}_{i}=3.573,$ ${K}_{d}=2.025,{N}_{f}=151$ | Area (a) | ${K}_{p}=4.993,{K}_{i}=4.992,$ ${K}_{d}=4.104,{N}_{f}=192$ |

Area (b) | ${K}_{p}=0.002,{K}_{i}=0.011,$ ${K}_{d}=0.218,{N}_{f}=290$ | Area (b) | ${K}_{p}=0.368,{K}_{i}=0.176,$ ${K}_{d}=0.133,{N}_{f}=218$ | Area (b) | ${K}_{p}=0.521,{K}_{i}=1.209,$ ${K}_{d}=4.804,{N}_{f}=146$ | |

PIDA | Area (a) | ${K}_{p}=3.365,{K}_{i}=0.775,$ ${K}_{d1}=1.666,{K}_{d2}=0.001,$ ${N}_{f1}=265,{N}_{f2}=174$ | Area (a) | ${K}_{p}=0.245,{K}_{i}=0.654,$ ${K}_{d1}=9.895,{K}_{d2}=0.008,$ ${N}_{f1}=245,{N}_{f2}=194$ | Area (a) | ${K}_{p}=9.996,{K}_{i}=9.999,$ ${K}_{d1}=0.607,{K}_{d2}=0.031,$ ${N}_{f1}=191,{N}_{f2}=345$ |

Area (b) | ${K}_{p}=6.599,{K}_{i}=0.487,$ ${K}_{d1}=1.309,{K}_{d2}=0.001,$ ${N}_{f1}=249,{N}_{f2}=179$ | Area (b) | ${K}_{p}=0.714,{K}_{i}=0.451,$ ${K}_{d1}=0.217,{K}_{d2}=0.005,$ ${N}_{f1}=214,{N}_{f2}=201$ | Area (b) | ${K}_{p}=5.969,{K}_{i}=0.207,$ ${K}_{d1}=9.246,{K}_{d2}=0.003,$ ${N}_{f1}=164,{N}_{f2}=312$ | |

TFOIDFF | Area (a) | ${K}_{t}=11.173,{K}_{i}=0.011,$ ${K}_{d}=0.024,n=5.077,$ ${\lambda}_{i}=0.16,{\mu}_{d}=0.063,$ ${\lambda}_{f}=0.011,{N}_{f}=166$ | Area (a) | ${K}_{t}=0,{K}_{i}=4.453,$ ${K}_{d}=9.991,n=2.127,$ ${\lambda}_{i}=1,{\mu}_{d}=0.922,$ ${\lambda}_{f}=0.015,{N}_{f}=177$ | Area (a) | ${K}_{t}=19.988,{K}_{i}=6.801,$ ${K}_{d}=0.091,n=1.5,$ ${\lambda}_{i}=0.004,{\mu}_{d}=0.074,$ ${\lambda}_{f}=0.694,{N}_{f}=113$ |

Area (b) | ${K}_{t}=0.408,{K}_{i}=0,$ ${K}_{d}=8.042,n=5.75,$ ${\lambda}_{i}=0.29,{\mu}_{d}=0,$ ${\lambda}_{f}=0.485,{N}_{f}=161$ | Area (b) | ${K}_{t}=0.656,{K}_{i}=0.242,$ ${K}_{d}=7.655,n=3.454,$ ${\lambda}_{i}=0.184,{\mu}_{d}=0.904,$ ${\lambda}_{f}=0.581,{N}_{f}=138$ | Area (b) | ${K}_{t}=5.632,{K}_{i}=11.175,$ ${K}_{d}=2.741,n=4.476,$ ${\lambda}_{i}=0.003,{\mu}_{d}=0.008,$ ${\lambda}_{f}=0.095,{N}_{f}=396$ | |

FPID | Area (a) | ${K}_{p}=0.078,{K}_{i}=0.181,$ ${K}_{d}=0.839,{N}_{f}=100,$ ${K}_{1}=4.56,{K}_{2}=0.897$ | Area (a) | ${K}_{p}=9.889,{K}_{i}=0.662,$ ${K}_{d}=4.567,{N}_{f}=101,$ ${K}_{1}=0.65,{K}_{2}=2.073$ | Area (a) | ${K}_{p}=5.426,{K}_{i}=9.781,$ ${K}_{d}=1.289,{N}_{f}=400,$ ${K}_{1}=4.996,{K}_{2}=0.899$ |

Area (b) | ${K}_{p}=2.094,{K}_{i}=3.213,$ ${K}_{d}=0.071,{N}_{f}=400,$ ${K}_{1}=4.993,{K}_{2}=3.534$ | Area (b) | ${K}_{p}=0,{K}_{i}=0.206,$ ${K}_{d}=5.046,{N}_{f}=364,$ ${K}_{1}=0.001,{K}_{2}=0.448$ | Area (b) | ${K}_{p}=0.607,{K}_{i}=8.76,$ ${K}_{d}=0.055,{N}_{f}=156,$ ${K}_{1}=4.793,{K}_{2}=3.077$ | |

FPIDA | Area (a) | ${K}_{p}=4.444,{K}_{i}=7.397,$ ${K}_{d1}=2.01,{K}_{d2}=0.027,$ ${N}_{f1}=145,{N}_{f2}=289,$ ${K}_{1}=0.373,{K}_{2}=4.179$ | Area (a) | ${K}_{p}=0.91,{K}_{i}=0.558,$ ${K}_{d1}=3.872,{K}_{d2}=0.036,$ ${N}_{f1}=193,{N}_{f2}=279,$ ${K}_{1}=0.352,{K}_{2}=0.674$ | Area (a) | ${K}_{p}=4.151,{K}_{i}=4.41,$ ${K}_{d1}=1.015,{K}_{d2}=0.05,$ ${N}_{f1}=279,{N}_{f2}=241,$ ${K}_{1}=4.94,{K}_{2}=4.045$ |

Area (b) | ${K}_{p}=0,{K}_{i}=5.903,$ ${K}_{d1}=9.239,{K}_{d2}=0.01,$ ${N}_{f1}=354,{N}_{f2}=293,$ ${K}_{1}=1.636,{K}_{2}=0.735$ | Area (b) | ${K}_{p}=0.852,{K}_{i}=1.285,$ ${K}_{d1}=1.843,{K}_{d2}=0.028,$ ${N}_{f1}=172,{N}_{f2}=128,$ ${K}_{1}=0.006,{K}_{2}=4.879$ | Area (b) | ${K}_{p}=0.731,{K}_{i}=2.373,$ ${K}_{d1}=0.387,{K}_{d2}=0.024,$ ${N}_{f1}=399,{N}_{f2}=400,$ ${K}_{1}=2.324,{K}_{2}=3.654$ | |

FTFOIDFF | Area (a) | ${K}_{t}=16.45,{K}_{i}=0.025,$ ${K}_{d}=0.014,n=6.17,$ ${\lambda}_{i}=0.423,{\mu}_{d}=0.046,$ ${\lambda}_{f}=0.513,{N}_{f}=246,$ ${K}_{1}=4.79,{K}_{2}=2.756$ | Area (a) | ${K}_{t}=0.459,{K}_{i}=2.413,$ ${K}_{d}=7.82,n=4.261,$ ${\lambda}_{i}=0.876,{\mu}_{d}=0.452,$ ${\lambda}_{f}=0.094,{N}_{f}=284,$ ${K}_{1}=5,{K}_{2}=2.871$ | Area (a) | ${K}_{t}=12.762,{K}_{i}=4.189,$ ${K}_{d}=1.891,n=8.69,$ ${\lambda}_{i}=0.02,{\mu}_{d}=0.061,$ ${\lambda}_{f}=0.815,{N}_{f}=189,$ ${K}_{1}=3.62,{K}_{2}=3.112$ |

Area (b) | ${K}_{t}=0.783,{K}_{i}=0.874,$ ${K}_{d}=3.24,n=2.49,$ ${\lambda}_{i}=0.481,{\mu}_{d}=0.006,$ ${\lambda}_{f}=0.147,{N}_{f}=188,$ ${K}_{1}=0.782,{K}_{2}=1.023$ | Area (b) | ${K}_{t}=8.159,{K}_{i}=2.47,$ ${K}_{d}=0.489,n=8.421,$ ${\lambda}_{i}=0.452,{\mu}_{d}=0.394,$ ${\lambda}_{f}=0.023,{N}_{f}=108,$ ${K}_{1}=0.05,{K}_{2}=0.723$ | Area (b) | ${K}_{t}=2.168,{K}_{i}=10.631,$ ${K}_{d}=1.014,n=6.21,$ ${\lambda}_{i}=0.126,{\mu}_{d}=0.04,$ ${\lambda}_{f}=0.113,{N}_{f}=322,$ ${K}_{1}=1.06,{K}_{2}=1.62$ |

Controller | ΔFa (Hz) | ΔFb (Hz) | ΔPtie (pu) | ITAE | ||||||
---|---|---|---|---|---|---|---|---|---|---|

MOS | MUS | ST | MOS | MUS | ST | MOS | MUS | ST | ||

PID | 0.0025 | −0.034 | 30 | 0 | −0.024 | 13 | 0.0047 | −0.0346 | 30 | 3.312 |

PIDA | 0.0026 | −0.0424 | 20 | 0.001 | −0.0475 | 17 | 0.0035 | −0.0427 | 28 | 2.103 |

TFOIDFF | 0 | −0.063 | 13 | 0.008 | −0.0587 | 13 | 0.0016 | −0.0578 | 17 | 1.159 |

FPID | 0 | −0.0228 | 8 | 0.002 | −0.0107 | 3.3 | 0 | −0.0228 | 3 | 0.5282 |

FPIDA | 0 | −0.0158 | 0.4 | 0 | −0.008 | 1.5 | 0 | −0.0067 | 1 | 0.1409 |

FTFOIDFF | 0 | −0.0158 | 0.18 | 0 | −0.0026 | 1 | 0 | −0.0056 | 0.7 | 0.0875 |

**Table 5.**The transient response specifications of the studied system represented as ITAE value using different controllers for Case II.

Controller | ITAE | ITAE_{tot} | ||
---|---|---|---|---|

ΔFa (Hz) | ΔFb (Hz) | ΔPtie (pu) | ||

PID | 50.32 | 49.36 | 71.27 | 170.9 |

PIDA | 35.35 | 48.48 | 39.47 | 123.3 |

TFOIDFF | 28.73 | 34.73 | 24.71 | 88.17 |

FPID | 8.039 | 6.663 | 8.06 | 22.76 |

FPIDA | 1.804 | 2.866 | 1.491 | 6.162 |

FTFOIDFF | 1.451 | 2.231 | 0.923 | 4.605 |

**Table 6.**The transient response specifications of the studied system represented as ITAE value using different controllers for Case III.

Controller | ITAE | ITAE_{tot} | ||
---|---|---|---|---|

ΔFa (Hz) | ΔFb (Hz) | ΔPtie (pu) | ||

PID | 98.32 | 147.3 | 59.77 | 305.4 |

PIDA | 94.31 | 155.4 | 47.36 | 297 |

TFOIDFF | 90.72 | 117.2 | 39.85 | 247.8 |

FPID | 11.62 | 27.77 | 5.077 | 44.46 |

FPIDA | 6.283 | 20.09 | 2.724 | 29.09 |

FTFOIDFF | 1.608 | 5.676 | 0.6926 | 7.976 |

**Table 7.**The transient response specifications of the studied system represented as ITAE value using different controllers for Case IV.

Controller | ITAE | ITAE_{tot} | ||
---|---|---|---|---|

ΔFa (Hz) | ΔFb (Hz) | ΔPtie (pu) | ||

PID | 45.48 | 30.13 | 58.21 | 133.8 |

PIDA | 25.53 | 25.08 | 27.02 | 77.64 |

TFOIDFF | 20.16 | 25.54 | 17.45 | 63.15 |

FPID | 17.7 | 20.84 | 18.45 | 56.98 |

FPIDA | 6.826 | 12.11 | 5.803 | 24.74 |

FTFOIDFF | 2.776 | 1.101 | 0.5113 | 4.388 |

**Table 8.**The transient response specifications of the studied system represented as ITAE value using different controllers for Case V.

Controller | ITAE | ITAE_{tot} | ||
---|---|---|---|---|

ΔFa (Hz) | ΔFb (Hz) | ΔPtie (pu) | ||

PID | 45.53 | 53.08 | 45.56 | 144.2 |

PIDA | 26.19 | 31.95 | 26.89 | 85.03 |

TFOIDFF | 16.82 | 13.75 | 17.14 | 47.71 |

FPID | 11.75 | 5.644 | 12.08 | 29.48 |

FPIDA | 2.485 | 2.734 | 2.406 | 7.626 |

FTFOIDFF | 1.08 | 2.537 | 1.076 | 4.694 |

**Table 9.**The transient response specifications of the studied system represented as ITAE value using different controllers for Case VI.

Controller | ITAE | ITAE_{tot} | ||
---|---|---|---|---|

ΔFa (Hz) | ΔFb (Hz) | ΔPtie (pu) | ||

PID | 50.36 | 49.53 | 71.43 | 171.32 |

PIDA | 35.55 | 49.72 | 39.79 | 125.1 |

TFOIDFF | 28.93 | 35.9 | 25.22 | 90.05 |

FPID | 8.065 | 6.751 | 8.225 | 23.04 |

FPIDA | 6.222 | 4.129 | 2.413 | 12.76 |

FTFOIDFF | 2.132 | 2.619 | 1.348 | 6.099 |

**Table 10.**The transient response specifications of the studied system represented as ITAE value using different controllers for Case VII.

Controller | ITAE | ITAE_{tot} | ||
---|---|---|---|---|

ΔFa (Hz) | ΔFb (Hz) | ΔPtie (pu) | ||

PID | 129.3 | 230.4 | 170.5 | 530.1 |

PIDA | 47.43 | 245.6 | 25.69 | 318.7 |

TFOIDFF | 56.51 | 258.4 | 30.04 | 344.9 |

FPID | 5.713 | 6.861 | 5.619 | 18.19 |

FPIDA | 2.554 | 7.118 | 0.7173 | 10.39 |

FTFOIDFF | 0.777 | 3.823 | 0.6413 | 5.241 |

**Table 11.**The transient response specifications of the studied system represented as ITAE value using different controllers for Case VIII.

Controller | ITAE | ITAE_{tot} | ||
---|---|---|---|---|

ΔFa (Hz) | ΔFb (Hz) | ΔPtie (pu) | ||

PID | 4517 | 5261 | 4501 | 14,280 |

PIDA | 2626 | 3214 | 2671 | 8511 |

TFOIDFF | 1704 | 1437 | 1708 | 4849 |

FPID | 1164 | 583.7 | 1198 | 2945 |

FPIDA | 249.6 | 281.5 | 240.2 | 771.3 |

FTFOIDFF | 108.6 | 257.9 | 108.1 | 474.6 |

Controller | Conditions | ΔFa (Hz) | ΔFb (Hz) | ΔPtie (pu) | ITAE | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

MOS | MUS | ST | MOS | MUS | ST | MOS | MUS | ST | |||

FTFOIDFF | Without UPFC and SMES | 0.057 | −0.11 | 1.3 | 0.008 | −0.119 | 1 | 0.001 | −0.061 | 0.5 | 0.614 |

With UPFC only | 0.007 | −0.125 | 0.5 | 0 | −0.026 | 2.3 | 0 | −0.011 | 3 | 0.377 | |

With SMES only | 0.048 | −0.106 | 1.28 | 0.005 | −0.076 | 1 | 0 | −0.036 | 0.5 | 0.534 | |

With both UPFC and SMES | 0 | −0.06 | 0.2 | 0 | −0.01 | 2 | 0 | −0.004 | 2.5 | 0.201 |

**Table 13.**The transient response specifications of the studied system represented as ITAE value for Case X.

FTFOIDFF Optimized by PDO (Proposed) | ITAE | ITAE_{tot} | ||
---|---|---|---|---|

ΔFa (Hz) | ΔFb (Hz) | ΔPtie (pu) | ||

Without EVs | 1.946 | 3.349 | 1.367 | 6.662 |

With EVs | 1.451 | 2.231 | 0.923 | 4.605 |

Controller | Parameters Variation | % Variation | ΔFa (Hz) | ΔFb (Hz) | ΔPtie (pu) | ITAE | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MOS | MUS | ST | MOS | MUS | ST | MOS | MUS | ST | ||||

FTFOIDFF tuned by PDO (proposed) | Nominal | $0$ | 0 | −0.0106 | 0.18 | 0 | −0.003 | 1 | 0 | −0.0056 | 0.7 | 0.0875 |

${\tau}_{gh}$ | $+25\%$ | 0 | −0.0107 | 0.19 | 0 | −0.003 | 1.1 | 0 | −0.0057 | 0.7 | 0.0878 | |

$-25\%$ | 0 | −0.0106 | 0.17 | 0 | −0.003 | 1 | 0 | −0.0055 | 0.7 | 0.0873 | ||

${\tau}_{cd}$ | $+25\%$ | 0 | −0.0106 | 0.18 | 0 | −0.0032 | 1 | 0 | −0.0057 | 0.7 | 0.0875 | |

$-25\%$ | 0 | −0.0106 | 0.18 | 0 | −0.0029 | 1 | 0 | −0.0055 | 0.7 | 0.0875 | ||

${y}_{c}$ | $+25\%$ | 0 | −0.0107 | 0.17 | 0 | −0.004 | 0.9 | 0 | −0.0057 | 0.5 | 0.0871 | |

$-25\%$ | 0.001 | −0.0105 | 0.2 | 0.0004 | −0.002 | 1.5 | 0.0002 | −0.0053 | 1 | 0.0889 | ||

$B$ | $+25\%$ | 0 | −0.0106 | 0.17 | 0 | −0.003 | 0.9 | 0 | −0.0056 | 0.6 | 0.0873 | |

$-25\%$ | 0 | −0.0106 | 0.2 | 0 | −0.003 | 1.2 | 0 | −0.0056 | 0.8 | 0.0881 | ||

${k}_{EV}$ | $+25\%$ | 0 | −0.0105 | 0.18 | 0 | −0.003 | 1 | 0 | −0.0055 | 0.7 | 0.0874 | |

$-25\%$ | 0 | −0.0107 | 0.18 | 0 | −0.003 | 1 | 0 | −0.0057 | 0.7 | 0.0876 | ||

${T}_{12}$ | $+25\%$ | 0 | −0.0107 | 0.18 | 0 | −0.003 | 1 | 0 | −0.0057 | 0.7 | 0.0876 | |

$-25\%$ | 0 | −0.0105 | 0.18 | 0 | −0.003 | 1 | 0 | −0.0055 | 0.7 | 0.0874 |

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## Share and Cite

**MDPI and ACS Style**

Alghamdi, S.; Alqarni, M.; Hammad, M.R.; AboRas, K.M.
First-of-Its-Kind Frequency Enhancement Methodology Based on an Optimized Combination of FLC and TFOIDFF Controllers Evaluated on EVs, SMES, and UPFC-Integrated Smart Grid. *Fractal Fract.* **2023**, *7*, 807.
https://doi.org/10.3390/fractalfract7110807

**AMA Style**

Alghamdi S, Alqarni M, Hammad MR, AboRas KM.
First-of-Its-Kind Frequency Enhancement Methodology Based on an Optimized Combination of FLC and TFOIDFF Controllers Evaluated on EVs, SMES, and UPFC-Integrated Smart Grid. *Fractal and Fractional*. 2023; 7(11):807.
https://doi.org/10.3390/fractalfract7110807

**Chicago/Turabian Style**

Alghamdi, Sultan, Mohammed Alqarni, Muhammad R. Hammad, and Kareem M. AboRas.
2023. "First-of-Its-Kind Frequency Enhancement Methodology Based on an Optimized Combination of FLC and TFOIDFF Controllers Evaluated on EVs, SMES, and UPFC-Integrated Smart Grid" *Fractal and Fractional* 7, no. 11: 807.
https://doi.org/10.3390/fractalfract7110807