# Load Frequency Control and Automatic Voltage Regulation in Four-Area Interconnected Power Systems Using a Gradient-Based Optimizer

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- The mathematical modeling of combined AVR-LFC loops in the proposed four-area IPS, with five generation units, including two renewable energy sources in each area.
- The mathematical modeling of a proposed PID controller with four-area IPS.
- The formulations of GBO-based fitness functions for the optimal tuning of the PID controller.
- To evaluate the performance of proposed control methodology, an extensive comparison was made between GBO-PID and other controllers such as GBO-I-PD, GBO-TID, and GBO-I-P in a four-area IPS.
- The robustness of the proposed GBO-PID control methodology was validated by performing a sensitivity analysis by changing the parameters of a four-area IPS over a range of approximately ±25%.

## 2. Power System Modeling

_{1}, K

_{2}, K

_{3}, K

_{4}, and P

_{s}. T

_{ij}denotes the coefficient of synchronization between the ith and ith areas. The transfer function of the gas $({G}_{G}(s))$, reheat thermal $({G}_{T}(s))$, hydro $({G}_{H}(s))$, wind $({G}_{W}(s))$, and solar photovoltaic $({G}_{S}(s))$ systems are provided in Equations (1)–(5), respectively. Table 2 defines the terms used in the LFC and AVR systems.

## 3. Proposed Control Methodology

_{p}, K

_{i}, and K

_{d}. The control signal generated by PID controller (U

_{PID}(s)) can be written as:

_{t}(s)

^{−1/n}, where n is a real number and K

_{t}represents the gain. TID combines integer and fractional order controllers. It quickly eliminates disturbances due to its superior dynamic features. Figure 2b shows the block diagram of the TID controller. The TID controller has three gain coefficients: K

_{t}, K

_{i}, and K

_{d}. The control signal generated by the TID controller (U

_{TID}(s)) can be written as:

_{i}, K

_{p}, and K

_{d}. The control signal generated by the I-PD controller (U

_{I-PD}(s)) can be written as:

_{i}and K

_{p}. The control signal generated by the I-P controller (U

_{I-P}(s)) can be written as:

## 4. Gradient-Based Optimizer (GBO)

- A.
- GBO INITIALIZATION

_{min}and X

_{max}are the limits of the decision variable.

- B.
- GRADIENT SEARCH RULE (GSR)

_{max}and β

_{min}have values of 1.2 and 0.2, respectively. The sine function representing the change from exploration to exploitation is represented by $\alpha $. The difference ∆x between the best candidate solution (${x}_{best}$) and a position chosen at random (${x}_{r1}^{m}$) can be written as:

_{2}is a random number computed as:

_{a}and r

_{b}are random numbers between 0 and 1.

- C.
- LOCAL ESCAPING OPERATOR (LEO)

_{2}is a random number with a standard deviation of 1 and mean of 0 whereas f

_{1}is a random number between −1 and 1; pr represents the probability; and u

_{1}, u

_{2}, and u

_{3}can be obtained as:

_{1}has a value of 1 if parameter u

_{1}is less than 0.5, otherwise it has a value of 0; ${x}_{k}^{m}$can be written as:

_{2}has a value of 1 if parameter u

_{2}is less than 0.5, otherwise it has a value of 0. By choosing values for the parameters u

_{1}, u

_{2}, and u

_{3}randomly, the population becomes more diverse and is able to avoid local optimal solutions. The flow chart of GBO is given in Figure 3.

## 5. Simulations and Discussion of Results

- A.
- FOUR-AREA IPS WITH COMBINED AVR-LFC

- B.
- SENSITIVITY ANALYSIS

_{tr}) were changed to Â ± 25% of their nominal values. In this investigation, the GBO-PID controller’s optimal parameters are taken from Part A. Figure 7, Figure 8 and Figure 9 illustrate the LFC, AVR, and tie-lie power responses of the GBO-PID control methodology with variations in T

_{tr}and R, while Table 7, Table 8 and Table 9 show the numerical results of LFC’s dynamic performance specifications, respectively. Despite the ±25% variance in system parameters, it is clear from the results that all terminal voltage, frequency deviation, and tie-line power deviation responses are nearly identical to one another. The fact that values of all performance specifications including settling time, % overshoot, % undershoot, and steady-state error have barely changed with variation in system parameters is proof that the suggested technique can function well under dynamic circumstances. The results obtained categorically demonstrate that the recommended GBO-PID controller is quite robust and does not require retuning for Â ± 25% variations in T

_{tr}and R.

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Parameter | Value |

B | 0.045 |

R_{t} | 2.4 |

R_{h} | 2.4 |

R_{g} | 2.4 |

R_{w} | 2.4 |

R_{g} | 2.4 |

T_{gr} | 0.08 |

T_{re} | 10 |

K_{re} | 0.3 |

T_{tr} | 0.3 |

T_{h} | 0.3 |

T_{rs} | 5 |

T_{rh} | 28.75 |

T_{w} | 0.025 |

X | 0.6 |

Y | 1 |

a | 1 |

b | 0.05 |

c | 1 |

T_{CR} | 0.01 |

T_{f} | 0.23 |

T_{CD} | 0.2 |

D | 0.0145 |

H | 5 |

f | 60 |

K_{ps} = 1/D | 68.97 |

T_{ps} = 2*H/f*D | 11.49 |

K_{1} | 0.2 |

K_{2} | 0.1 |

K_{3} | 0.5 |

K_{4} | 1.4 |

P_{s} | 1.5 |

K_{a} | 10 |

T_{a} | 0.1 |

K_{e} | 1 |

T_{e} | 0.4 |

K_{g} | 0.8 |

T_{g} | 1.4 |

K_{s} | 1 |

T_{s} | 0.05 |

T_{w1} | 0.6 |

T_{w2} | 0.041 |

K_{w1} | 1.25 |

K_{w2} | 1.4 |

T_{pv} | 1.8 |

K_{pv} | 1 |

T_{12} | 0.545 |

T_{13} | 0.545 |

T_{14} | 0.545 |

T_{21} | 0.545 |

T_{23} | 0.545 |

T_{24} | 0.545 |

T_{31} | 0.545 |

T_{32} | 0.545 |

T_{34} | 0.545 |

T_{41} | 0.545 |

T_{42} | 0.545 |

T_{43} | 0.545 |

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**Figure 2.**(

**a**) Proposed control methodology. (

**b**) TID controller. (

**c**) I-PD controller. (

**d**) I-P controller.

**Figure 6.**Tie-line power deviation responses: (

**a**) ∆P

_{tie1}; (

**b**) ∆P

_{tie2}; (

**c**) ∆P

_{tie3}; (

**d**) ∆P

_{tie4}.

**Figure 7.**LFC responses with ±25% variations in system parameters (

**a**) ∆f

_{1}; (

**b**) ∆f

_{2}; (

**c**) ∆f

_{3}; (

**d**) ∆f

_{4}.

**Figure 8.**AVR responses with ±25% variations in system parameters: (

**a**) V

_{t1}; (

**b**) V

_{t2}; (

**c**) V

_{t3}; (

**d**) V

_{t4}.

**Figure 9.**Tie-line power deviation responses with ±25% variations in system parameters: (

**a**) ∆P

_{tie1}; (

**b**) ∆P

_{tie2}; (

**c**) ∆P

_{tie3}; (

**d**) ∆P

_{tie4}.

Reference of Paper | Area of Research | Tuning Method | Suggested Controller | Generation Sources | Year | Generation Sources in All Areas | Covered Area | Additional Incorporation for Improvements | Nonlinearities |
---|---|---|---|---|---|---|---|---|---|

[1] | LFC and AVR | AOA, LPBO, MPSO | PI-PD | - | 2022 | 2, 3 | 2, 3 | - | - |

[2] | LFC and AVR | DO | PI-PD | Reheat thermal, Hydro, and Gas | 2022 | 6 | 2, 3 | - | GDB, BD, GRC |

[3] | LFC and AVR | SA, ZN | PID | Hydro and Non-reheat thermal | 2016 | 4 | 2 | - | GDB |

[4] | LFC and AVR | NN-FTF | Hybrid NN and FTF | - | 2016 | 1 | 1 | - | - |

[5] | LFC and AVR | ZN, FLC | PID, Fuzzy | - | 2018 | 1 | 1 | - | - |

[6] | LFC and AVR | LSA | PIDF, PID ^{u}F | Reheat thermal, Wind, and Diesel | 2018 | 4 | 2 | IPFC, SMES | GDB, GRC |

[7] | LFC and AVR | MFO | FOPID | Hydro and Non-reheat thermal | 2019 | 4 | 2 | - | GDB, BD |

[8] | LFC and AVR | FA | PID | Hydro and Non-reheat thermal | 2019 | 4 | 2 | - | - |

[9] | LFC and AVR | IPSO | CPSS | Gas, Reheat thermal, and Hydro | 2020 | 1 | 1 | - | GDB, GRC |

[10] | LFC and AVR | DE-AEFA | PID | Wind , Hydro, Thermal, Gas, Solar, and Diesel | 2020 | 6 | 2 | HVDC link | GRC |

[11] | LFC and AVR | DE-AEFA | PID | Gas, Diesel, Hydro, Solar photovoltaic, Reheat thermal, and Wind | 2020 | 6 | 2 | IPFC, RFBs | GRC |

[12] | LFC and AVR | SCA | PIDF, PI | Reheat thermal and Non-reheat thermal | 2020 | 2 | 2 | UPFC, RFBs | - |

[13] | LFC and AVR | NLTA | PID | - | 2021 | 2 | 2 | - | - |

[14] | LFC and AVR | GWO | PIDD | Reheat thermal, Hydro, and Nuclear | 2021 | 6 | 2 | SMES, UPFC | GRC, GDB |

[15] | LFC and AVR | FA | PID | Reheat thermal and Hydro | 2021 | 4 | 2 | - | TD, GRC, GDB |

[16] | LFC and AVR | hFPAPFA | PIDA | Thermal | 2021 | 1 | 1 | - | - |

[17] | LFC and AVR | HHO | TIDF | Reheat thermal and Combined cycle gas turbine (CCGT) | 2021 | 6 | 3 | - | GDB, GRC, BD |

[18] | LFC and AVR | 2nd order error-driven control law | ADRC | Solar, Geothermal, Wind, and EVs | 2022 | 6 | 3 | - | - |

[19] | LFC and AVR | HHO | 2DOF I-TDF | Reheat thermal, Wind, Solar thermal, and Dish-stirling, | 2022 | 6 | 3 | - | GDB, GRC |

[20] | LFC and AVR | AFA | CFPD-TID | Hydro, Thermal, and Geothermal | 2022 | 6 | 3 | RFBs, HVDC link | GRC, DB |

[21] | LFC and AVR | AFA | CFOTDN-FOPDN | Hydro, Dish-stirling, Solar thermal, and Reheat thermal | 2022 | 4 | 2 | - | GDB, CTD, GRC |

[22] | LFC and AVR | AFA | CPDN-FOPIDN | Reheat thermal, Hydro, Gas, and Geothermal | 2022 | 6 | 3 | FESS, CES, RFBs, SMES HVDC link | GRC, GDB |

[23] | LFC and AVR | DPO | PIDA | Three Bioenergy technologies and two Solar energy sources | 2022 | 10 | 2 | - | - |

[24] | LFC and AVR | HAEFA | Fuzzy PID | Reheat thermal, Hydro, and Gas | 2022 | 6 | 2 | UCs, SMES, RFBs | - |

Proposed Method | LFC and AVR | GBO | PID | Thermal, Gas, Hydro, Wind, and Solar | 2022 | 20 | 4 | - | - |

Acronym | Definition | Acronym | Definition |
---|---|---|---|

T_{rh} | Transient droop time constant | GBO | Gradient-Based Optimizer |

SLP | Step load perturbation | K_{1}, K_{2}, K_{3}, K_{4}, | Cross-coupling coefficients for AVR and LFC loops |

PID | Proportional integral derivative | T_{CR} | Combustion reaction time delay |

V_{t} | Terminal voltage | Y | Speed governor lag time constant |

I-PD | Integral–proportional derivative | ∆f | Frequency deviation |

R_{t}, R_{h}, R_{g}, R_{w} | Speed regulation of thermal reheat, hydro, gas, and wind power plants | T_{12}, T_{13}, T_{14},T _{21}, T_{23}, T_{24},T _{31}, T_{32}, T_{34},T _{41}, T_{42}, T_{43} | Tie-line synchronizing time constants |

TID | Tilt integral derivative | K_{p} | Gain of power system |

I-P | Integral–proportional | T_{CD} | Compressor discharge volume time constant |

T_{p} | Time constant of power system | X | Speed governor lead time constant |

LFC | Load frequency control | T_{w} | Water time constant |

∆P_{tie} | Tie-line power deviation | K_{w1}, K_{w2} | Wind plant gain constants |

AVR | Automatic voltage regulator | T_{w1}, T_{w2} | Wind turbine time constants |

∆P_{D} | Load deviation | T_{PV} | Solar PV time constant |

IPS | Interconnected power system | K_{PV} | Solar PV gain constant |

T_{tr} | Time constant of thermal turbine | a,b,c | Valve positional time constant |

B | Area biasing factor | T_{h} | Main servo time constant |

T_{re} | Time constant of reheat steam turbine | K_{a} | Gain of amplifier |

T_{a} | Time constant of amplifier | K_{e} | Gain of exciter |

K_{g} | Gain of generator field | T_{e} | Time constant of exciter |

T_{gr} | Time constant of speed governor | T_{s} | Time constant of voltage sensor |

K_{re} | Gain of reheat steam turbine | T_{g} | Time constant of generator field |

K_{s} | Gain of voltage sensor | T_{f} | Fuel time constant |

D | Frequency sensitive load coefficient | V_{S} | Sensor voltage |

P_{S} | Synchronizing power coefficient | V_{e} | Error voltage |

H | Inertia constant | GDB | Governor dead band |

T_{rs} | Speed governor rest time | FTF | Fast traversal filter |

GRC | Generation rate constraints | MFO | Moth Flame Optimization |

SMES | Superconducting magnetic energy storage | DE | Differential Evolution |

NN | Neural network | GWO | Grey Wolf Optimizer |

FO | Fractional order | PFA | Pathfinder Algorithm |

AOA | Archimedes Optimization Algorithm | NLTA | Nonlinear Threshold Accepting Algorithm |

FA | Firefly Algorithm | CFPD | Cascaded Fuzzy PD |

IPSO | Improved Particle Swarm Optimization | MPSO | Modified Particle Swarm Optimization |

AEFA | Artificial Electric Field Algorithm | AFA | Artificial Flora Algorithm |

ADRC | Active disturbance rejection control | UCs | Ultra capacitors |

UPFC | Unified Power Flow Controller | CPSS | Conventional power system stabilizer |

SCA | Sine Cosine Algorithm | LPBO | Learner Performance-Based Behavior Optimization |

FPA | Flower Pollinated Algorithm | IPFC | Interline Power Flow Controller |

HHO | Harris Hawks Optimization | DO | Dandelion Optimizer |

CTD | Communication time delay | 2DOF | Two degrees of freedom |

DPO | Doctor and Patient Optimization Technique | ITSE | Integral of time multiplied by squared value of error |

Area | GBO−I-P | GBO−TID | GBO−I−PD | GBO−PID | ||||
---|---|---|---|---|---|---|---|---|

Controller Parameter | Value | Controller Parameter | Value | Controller Parameter | Value | Controller Parameter | Value | |

Area−1 | K_{p1} | 0.17 | K_{t1} | 0.76 | K_{i1} | 0.0001 | K_{p1} | 1.43 |

K_{i1} | 2.42 | K_{i1} | 1.29 | K_{p1} | 1.59 | K_{i1} | 1.27 | |

- | - | K_{d1} | 0.20 | K_{d1} | 1.97 | K_{d1} | 1.93 | |

K_{p2} | 1.37 | K_{t2} | 0.97 | K_{i2} | 1.09 | K_{p2} | 1.08 | |

K_{i2} | 1.38 | K_{i2} | 0.29 | K_{p2} | 1.15 | K_{i2} | 1.11 | |

- | - | K_{d2} | 1.18 | K_{d2} | 0.15 | K_{d2} | 0.67 | |

Area−2 | K_{p3} | 1.43 | K_{t3} | 1.54 | K_{i3} | 0.74 | K_{p3} | 1.11 |

K_{i3} | 0.82 | K_{i3} | 1.27 | K_{p3} | 0.25 | K_{i3} | 1 | |

- | - | K_{d3} | 0.30 | K_{d3} | 0.54 | K_{d3} | 1.24 | |

K_{p4} | 0.92 | K_{t4} | 0.44 | K_{i4} | 0.35 | K_{p4} | 1.37 | |

K_{i4} | 0.83 | K_{i4} | 0.18 | K_{p4} | 0.41 | K_{i4} | 1.20 | |

- | - | K_{d4} | 0.20 | K_{d4} | 0.14 | K_{d4} | 0.96 | |

Area−3 | K_{p5} | 1.15 | K_{t5} | 1.82 | K_{i5} | 0.01 | K_{p5} | 1.74 |

K_{i5} | 1.67 | K_{i5} | 0.42 | K_{p5} | 0.65 | K_{i5} | 1.76 | |

- | - | K_{d5} | 1.73 | K_{d5} | 1.28 | K_{d5} | 0.99 | |

K_{p6} | 1.11 | K_{t6} | 0.87 | K_{i6} | 1.06 | K_{p6} | 1.33 | |

K_{i6} | 1.09 | K_{i6} | 0.31 | K_{p6} | 0.85 | K_{i6} | 1.27 | |

- | - | K_{t6} | 0.24 | K_{d6} | 0.047 | K_{d6} | 1.13 | |

Area−4 | K_{p7} | 0.12 | K_{t7} | 1.78 | K_{i7} | 0.99 | K_{p7} | 0.98 |

K_{i7} | 2.78 | K_{i7} | 0.28 | K_{p7} | 2 | K_{i7} | 1.94 | |

- | - | K_{d7} | 0.62 | K_{d7} | 0.14 | K_{d7} | 1.39 | |

K_{p8} | 1.10 | K_{t8} | 1.30 | K_{i8} | 1.15 | K_{p8} | 1.24 | |

K_{i8} | 1.13 | K_{i8} | 0.069 | K_{p8} | 1.28 | K_{i8} | 0.61 | |

- | - | K_{d8} | 1.50 | K_{d8} | 0.58 | K_{d8} | 1.26 | |

ITSE | 2.18 | ITSE | 2.62 | ITSE | 3.43 | ITSE | 0.71 |

Control Strategy | Area-1 | Area-2 | ||||||

Settling Time | % Overshoot | % Undershoot | % s-sError | Settling Time | % Overshoot | % Undershoot | % s-sError | |

GBO-PID | 5.37 | 0.23 | −0.58 | 0 | 5.38 | 0.23 | −0.54 | 0 |

GBO-I-PD | 16.3 | 0 | −0.049 | 0 | 19.54 | 0.009 | −0.068 | 0 |

GBO-TID | 9.66 | 0.022 | −0.13 | 0 | 9.51 | 0.018 | −0.14 | 0 |

GBO-I-P | 16.10 | 0.0086 | −0.07 | 0 | 16.49 | 0.036 | −0.12 | 0 |

Control Strategy | Area-3 | Area-4 | ||||||

Settling Time | % Overshoot | % Undershoot | % s-sError | Settling Time | % Overshoot | % Undershoot | % s-sError | |

GBO-PID | 5.38 | 0.22 | −0.52 | 0 | 5.98 | 0.26 | −0.49 | 0 |

GBO-I-PD | 11.0 | 0 | −0.08 | 0 | 17.08 | 0.0039 | −0.046 | 0 |

GBO-TID | 9.62 | 0.011 | −0.15 | 0 | 9.68 | 0.015 | −0.13 | 0 |

GBO-I-P | 17.66 | 0.009 | −0.087 | 0 | 16.95 | 0.013 | −0.057 | 0 |

Control Strategy | Area-1 | Area-2 | ||||

Settling Time | % Overshoot | % s-sError | Settling Time | % Overshoot | % s-sError | |

GBO-PID | 3.96 | 7.52 | 0 | 4.09 | 5.45 | 0 |

GBO-I-PD | 3.72 | 0.48 | 0 | 6.75 | 6.66 | 0 |

GBO-TID | 6.60 | 21.91 | 0 | 7.68 | 19.71 | 1.8 |

GBO-I-P | 7.52 | 7.43 | 0 | 5.68 | 11.9 | 0 |

Control Strategy | Area-3 | Area-4 | ||||

Settling Time | % Overshoot | % s-s | Settling Time | % Overshoot | % s-s | |

Error | Error | |||||

GBO-PID | 4.36 | 9.0 | 0 | 2.92 | 10.13 | 0 |

GBO-I-PD | 5.72 | 18.04 | 0 | 5.27 | 5.90 | 0 |

GBO-TID | 5.24 | 33.18 | 0 | 6.72 | 11.90 | 0 |

GBO-I-P | 7.01 | 5.52 | 0 | 5.92 | 4.36 | 0 |

Control Strategy | Area-1 | Area-2 | ||||||

Settling Time | % Overshoot | % Undershoot | % s-sError | Settling Time | % Overshoot | % Undershoot | % s-sError | |

GBO-PID | 9.36 | 0.023 | −0.016 | 0 | 9.69 | 0.0025 | −0.0093 | 0 |

GBO-I-PD | 16.65 | 0.052 | −0.0035 | 0.004 | 19.48 | 0.047 | −0.034 | 0 |

GBO-TID | 11.18 | 0.012 | −0.0112 | 0 | 11.02 | 0.011 | −0.0053 | 0 |

GBO-I-P | 19.50 | 0.019 | −0.023 | 0 | 16.12 | 0.039 | −0.040 | 0 |

Control Strategy | Area-3 | Area-4 | ||||||

Settling Time | % Overshoot | % Undershoot | % s-sError | Settling Time | % Overshoot | % Undershoot | % s-sError | |

GBO-PID | 8.45 | 0.013 | −0.012 | 0 | 10.23 | 0.015 | −0.013 | 0 |

GBO-I-PD | 18.71 | 0.0038 | −0.066 | 0.004 | 19.47 | 0.043 | −0.035 | 0 |

GBO-TID | 9.7 | 0.015 | −0.017 | 0 | 9.63 | 0.0098 | −0.0081 | 0 |

GBO-I-P | 19.17 | 0.022 | −0.015 | 0 | 18.23 | 0.031 | −0.007 | 0 |

**Table 7.**Numerical results of LFC with ±25% variations in system parameters using GBO-PID control methodology.

Case | Area-1 | Area-2 | ||||||

Settling Time | % Overshoot | % Undershoot | % s-sError | Settling Time | % Overshoot | % Undershoot | % s-sError | |

−25% of T_{tr1}, T_{tr2}, T_{tr3}, T_{tr4} | 5.23 | 0.21 | −0.57 | 0 | 5.26 | 0.21 | −0.53 | 0 |

−25% of R_{t}, R_{h}, R_{g}, R_{w} | 6.38 | 0.26 | −0.55 | 0 | 6.39 | 0.24 | −0.51 | 0 |

Nominal Values | 5.37 | 0.23 | −0.58 | 0 | 5.38 | 0.23 | −0.54 | 0 |

+25% of T_{tr1}, T_{tr2}, T_{tr3}, T_{tr4} | 5.44 | 0.24 | −0.58 | 0 | 6.05 | 0.25 | −0.55 | 0 |

+25% of R_{t}, R_{h}, R_{g}, R_{w} | 4.89 | 0.21 | −0.60 | 0 | 4.90 | 0.22 | −0.56 | 0 |

Case | Area-3 | Area-4 | ||||||

Settling Time | % Overshoot | % Undershoot | % s-sError | Settling Time | % Overshoot | % Undershoot | % s-sError | |

−25% of T_{tr1}, T_{tr2}, T_{tr3}, T_{tr4} | 5.27 | 0.21 | −0.51 | 0 | 5.28 | 0.24 | −0.49 | 0 |

−25% of R_{t}, R_{h}, R_{g}, R_{w} | 6.39 | 0.21 | −0.50 | 0 | 6.38 | 0.25 | −0.47 | 0 |

Nominal Values | 5.38 | 0.22 | −0.52 | 0 | 5.98 | 0.26 | −0.49 | 0 |

+25% of T_{tr1}, T_{tr2}, T_{tr3}, T_{tr4} | 6.07 | 0.23 | −0.52 | 0 | 6.17 | 0.27 | −0.50 | 0 |

+25% of R_{t}, R_{h}, R_{g}, R_{w} | 4.90 | 0.22 | −0.53 | 0 | 4.92 | 0.25 | −0.51 | 0 |

**Table 8.**Numerical results of AVR with ±25% variations in system parameters using GBO-PID control methodology.

Case | Area-1 | Area-2 | ||||

Settling Time | % Overshoot | % s-sError | Settling Time | % Overshoot | % s-sError | |

−25% of T_{tr1}, T_{tr2}, T_{tr3}, T_{tr4} | 3.95 | 7.46 | 0 | 4.08 | 5.44 | 0 |

−25% of R_{t}, R_{h}, R_{g}, R_{w} | 3.96 | 7.57 | 0 | 4.07 | 5.43 | 0 |

Nominal Values | 3.96 | 7.52 | 0 | 4.09 | 5.45 | 0 |

+25% of T_{tr1}, T_{tr2}, T_{tr3}, T_{tr4} | 3.97 | 7.55 | 0 | 4.1 | 5.46 | 0 |

+25% of R_{t}, R_{h}, R_{g}, R_{w} | 3.92 | 7.40 | 0 | 4.07 | 5.46 | 0 |

Case | Area-3 | Area-4 | ||||

Settling Time | % Overshoot | % s-sError | Settling Time | % Overshoot | % s-sError | |

−25% of T_{tr1}, T_{tr2}, T_{tr3}, T_{tr4} | 4.36 | 9.0 | 0 | 2.91 | 10.13 | 0 |

−25% of R_{t}, R_{h}, R_{g}, R_{w} | 4.36 | 9.0 | 0 | 2.95 | 10.13 | 0 |

Nominal Values | 4.36 | 9.0 | 0 | 2.92 | 10.13 | 0 |

+25% of T_{tr1}, T_{tr2}, T_{tr3}, T_{tr4} | 4.36 | 9.0 | 0 | 2.93 | 10.14 | 0 |

+25% of R_{t}, R_{h}, R_{g}, R_{w} | 4.36 | 9.0 | 0 | 2.88 | 10.14 | 0 |

**Table 9.**Numerical results of tie-line power deviation responses with ±25% variations in system parameters using GBO-PID control methodology.

Case | Area-1 | Area-2 | ||||||

Settling Time | % Overshoot | % Undershoot | % s-sError | Settling Time | % Overshoot | % Undershoot | % s-sError | |

−25% of T_{tr1}, T_{tr2}, T_{tr3}, T_{tr4} | 9.39 | 0.022 | −0.015 | 0 | 9.71 | 0.0022 | −0.0091 | 0 |

−25% of R_{t}, R_{h}, R_{g}, R_{w} | 9.26 | 0.023 | −0.017 | 0 | 9.85 | 0.0029 | −0.0089 | 0 |

Nominal Values | 9.36 | 0.023 | −0.016 | 0 | 9.69 | 0.0025 | −0.0093 | 0 |

+25% of T_{tr1}, T_{tr2}, T_{tr3}, T_{tr4} | 9.35 | 0.023 | −0.017 | 0 | 9.67 | 0.0026 | −0.0094 | 0 |

+25% of R_{t}, R_{h}, R_{g}, R_{w} | 9.28 | 0.023 | −0.016 | 0 | 9.61 | 0.0021 | −0.0096 | 0 |

Case | Area-3 | Area-4 | ||||||

Settling Time | % Overshoot | Undershoot | % s-sError | Settling Time | % Overshoot | Undershoot | % s-sError | |

−25% of T_{tr1}, T_{tr2}, T_{tr3}, T_{tr4} | 8.48 | 0.012 | −0.011 | 0 | 10.26 | 0.014 | −0.012 | 0 |

−25% of R_{t}, R_{h}, R_{g}, R_{w} | 8.13 | 0.013 | −0.011 | 0 | 10.32 | 0.015 | −0.012 | 0 |

Nominal Values | 8.45 | 0.013 | −0.012 | 0 | 10.23 | 0.015 | −0.013 | 0 |

+25% of T_{tr1}, T_{tr2}, T_{tr3}, T_{tr4} | 8.47 | 0.013 | −0.012 | 0 | 10.21 | 0.016 | −0.013 | 0 |

+25% of R_{t}, R_{h}, R_{g}, R_{w} | 7.74 | 0.012 | −0.012 | 0 | 10.18 | 0.015 | −0.0013 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Ali, T.; Malik, S.A.; Daraz, A.; Adeel, M.; Aslam, S.; Herodotou, H.
Load Frequency Control and Automatic Voltage Regulation in Four-Area Interconnected Power Systems Using a Gradient-Based Optimizer. *Energies* **2023**, *16*, 2086.
https://doi.org/10.3390/en16052086

**AMA Style**

Ali T, Malik SA, Daraz A, Adeel M, Aslam S, Herodotou H.
Load Frequency Control and Automatic Voltage Regulation in Four-Area Interconnected Power Systems Using a Gradient-Based Optimizer. *Energies*. 2023; 16(5):2086.
https://doi.org/10.3390/en16052086

**Chicago/Turabian Style**

Ali, Tayyab, Suheel Abdullah Malik, Amil Daraz, Muhammad Adeel, Sheraz Aslam, and Herodotos Herodotou.
2023. "Load Frequency Control and Automatic Voltage Regulation in Four-Area Interconnected Power Systems Using a Gradient-Based Optimizer" *Energies* 16, no. 5: 2086.
https://doi.org/10.3390/en16052086