#
Robust Control of Frequency Variations for a Multi-Area Power System in Smart Grid Using a Newly Wild Horse Optimized Combination of PIDD^{2} and PD Controllers

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}-PD), to improve the frequency response of a multi-area interconnected power system with multiple generating units linked to it. The optimum gains of the presented controller are well-tuned using a wild horse optimizer (WHO), a modern metaheuristic optimization approach. The main study is a two-area-linked power system with varied conventional and renewable generating units. The physical constraints of the speed turbines and governors are considered. The WHO optimization algorithm is proven to outperform various other optimization approaches, such as the whale optimization algorithms (WOA) and chimp optimization algorithms (ChOA). The efficacy of the proposed WHO-based PIDD

^{2}-PD controller is evaluated by comparing its performance to other controllers in the literature (cascaded proportional integral derivative-tilted integral derivative (PID-TID), integral derivative-tilted (ID-T) controller). Multiple and varied scenarios are applied in this work to test the proposed controller’s sturdiness to various load perturbations (step, random, and multi-step), renewable energy source penetration, and system parameter variations. The results are provided as time-domain simulations run using MATLAB/SIMULINK. The simulation results reveal that the suggested controller outperforms other structural controllers in the dynamic response of the system in terms of settling time, maximum overshoot, and undershoot values, with an improvement percentage of 70%, 73%, and 67%, respectively.

## 1. Introduction

#### 1.1. Literature Review

^{2}-PD surpasses both the ID-T and PID-TID controllers in [31,32], respectively. According to prior research, picking the controller settings is just as important as choosing the controller type. The frequency stability issue has benefited greatly from the evolutionary optimization methodologies used to improve the controller parameters. As a consequence, choosing an appropriate optimization technique in the design procedure of the controller is a basic and crucial challenge. Classical optimization procedures were previously utilized to find the best frequency controller settings [18,33]. Additionally, ref. [34] presents a fuzzy gain scheduling (FGS) controller for parameter selection. These algorithms, however, face several difficulties, including slumps, deathtraps in local minimums, the demand for several iterations, and reliance on initial conditions for selecting the optimal settings. As a result, scholars overcame these obstacles by improving meta-heuristic optimization methods, such as the grey wolf optimizer [33], particle swarm optimization [35], ant lion optimization [36], chimp optimization algorithm [5], teaching-learning-based optimization [37], moth-flame optimization [11], equilibrium optimization [38], and atom search optimization [39]. Substantial emphasis has been placed on the use of various optimization techniques to assist them in tackling technical difficulties, particularly the load frequency control issue. Therefore, the author chose to use the wild horse optimizer (WHO) [40] in this work to identify the best settings for the proposed PIDD

^{2}-PD controller. The primary result from prior research is that LFC techniques that depend on the controller designer’s talents, such as fuzzy logic control, H-infinite approaches, and MPC, meet the needed performance requirements but have many design problems and take considerable time to choose the control settings. Additionally, traditional PD, PI, and PID controllers struggle to cope with system uncertainties. Numerous previous articles paid insufficient attention to robustness evaluations, such as system nonlinearities and system parameter variations. Additionally, most previous assessments failed to account for considerable renewable energy integration in the absence of system parameter changes by including system uncertainties, nonlinearities, and simultaneous load variations.

#### 1.2. Contribution of Paper

^{2}-PD controller that improves system frequency stability considering renewable power perturbations. Additionally, the suggested PIDD2-PD controller’s settings have been developed in line with the WHO to preserve both frequency and system stability under abnormal situations. In contrast to other research on related issues, the following is a summary of the paper’s key contribution:

- Using a reliable PIDD2-PD controller to enhance the frequency stability of a two-area interconnected power system considering RESs;
- Using the WHO algorithm to optimize the parameters of the presented PIDD2-PD controller, a novel and effective optimization approach for LFC design;
- Testing the effectiveness and stability of the proposed controller when the studied two-area interconnected power system is subjected to various disturbances, such as different step load disturbances (SLD), multi-step load disturbances (MSLD), random load disturbances (RLD), RESs fluctuations, and communication time delay.

## 2. The Proposed Power System Modeling

#### 2.1. Models of Dynamic Subsystems

#### 2.1.1. Thermal Power Plant Supplies 1000 MW and Includes

- Governor dead band (GDB): The GDB non-linearity formulas could be simplified as a function of changes and change rates in speeds [21]. With the aid of the Fourier series, the transfer function of a GDB with 0.5% backlash is derived as:

_{1}= 0.8 and N

_{2}= −0.2/π [19], and the time constant of the steam turbine T

_{sg}is 0.06 s.

- Reheat is modeled using the first-order transfer function:

_{r}of 0.3 and a steam turbine reheating time constant T

_{r}of 10.2 s.

- Turbine with GRC

_{t}of 0.3 s.

#### 2.1.2. Hydraulic Power Plant Supplies 500 MW and Includes

- A Governor is modeled using the first-order transfer function, with a time constant for a hydro turbine governor T
_{gh}= 0.2 s.

- Transient droop compensation is modeled using a first-order transfer function, with hydro turbine speed governor reset time T
_{rs}and a time constant of transient droop T_{rh}of 4.9 and 28.749 s, respectively.

- Penstock hydraulic turbine with GRC

_{W}= 1.1 s.

#### 2.1.3. Gas Power Station Supplies 240 MW and Includes

- The valve positioner is modeled using the first-order transfer function with a time constant of the valve positioner B
_{g}and the gas turbine valve positioner C_{g}of 0.049 and 1 s, respectively.

- The speed governor is modeled using the first-order transfer function with lead and a lag time constant of the gas turbine governor X
_{g}, Y_{g}of 0.6 and 1.1 s, respectively.

- Fuel and combustion reactions are modeled using the first-order transfer function with a gas turbine combustion reaction time delay Tcr and gas turbine fuel time constant T
_{f}of 0.01 and 0.239 s, respectively.

- Compressor discharge is modeled using the first-order transfer function with compressor discharge volume time constant T
_{cd}of 0.2 s.

_{hyd}, R

_{g}, and R

_{Th}) of 2.4 and the participation factors for each unit (PF

_{hyd}, PF

_{g}, and PF

_{Th}) are 0.2873, 0.138, and 0.5747, respectively. Table 1 shows the transfer function and parameters for power systems 1 and 2, as well as the T-line.

#### 2.2. Wind Generation Model

_{WT}= 0.025. Figure 4 depicts the wind turbine’s fluctuating power. The wind generation unit’s output power may be calculated using the following equation [44]:

^{3}, ${A}_{T}$ is the swept area by the rotor in m

^{2}, ${V}_{W}$ is the wind’s nominal speed in m/s, and ${C}_{P}$ denotes the rotor’s blade parameter. ${C}_{P}$ is calculated from the Equation (11) and ${C}_{1}$ to ${C}_{7}$ are the parameters of the turbine.

#### 2.3. PV Generation Model

_{WT}= 0.015.

## 3. Wild Horse Optimization Algorithm

#### 3.1. Population Initialization

#### 3.2. Grazing Behavior

_{2}is a uniform random value that has a range of [0, 1]. TDR is an adaptive parameter that begins at 1 and declines during the enforcement of the algorithm and reaches 0 at the end of the execution of the algorithm, according to the following Equation [40]:

#### 3.3. Behavior of Horse Mating

#### 3.4. Group Leadership

#### 3.5. Leaders Exchange and Selection

## 4. Structure of the Controller and Problem Formulation

^{2}-PD suggested controller is to manage and improve the frequency response of a power system consisting of a multi-source when it is confronted with sudden load variations and renewable energy source fluctuations. The controller is proposed in both areas to decrease frequency deviations (ΔF

_{1}, ΔF

_{2}) and the tie-line power deviation between both areas (ΔP

_{tie}−line) for different load perturbations and renewable energy sources.

^{2}-PD controller. Researchers have commonly employed the traditional PID controller because of its ease of design and operating efficiency. The PIDD

^{2}structure is identical to that of standard PID but also with the addition of second-order derivative gain [46]. The transfer functions of the PIDD

^{2}and PD controller can be represented using Equation (24), and Equation (25), respectively, as follows:

^{2}controller, in addition, ((kp) is the proportional gain, (kd) is the derivative gain, and (nf) is the filter coefficient) of the PD controller. The structure of a PIDD

^{2}-PD is depicted in Figure 9.

^{2}-PD controller parameters will be determined by reducing the fitness function (FF). The integral of time multiplied by the squared error (ITSE) is selected as the fitness function since it can minimize the settling time and quickly suppress the high oscillation [31]:

## 5. Results of Simulation and Discussions

^{2}-PD controller optimized by the WHO algorithm is compared against the ID-T and PID-TID controllers optimized by the WHO and the ID-T controller optimized by the AOA under different operating conditions.

#### 5.1. Performance Analysis of the WHO

^{2}-PD-constructed controller as determined by the three optimization strategies employed in this study: WOA, ChOA, and WHO.

_{1}), (b) frequency deviation in Area-2 (ΔF

_{2}), and (c) tie-line power deviation (ΔP

_{tie}) for load disturbances in both areas using different optimization algorithms. The PIDD

^{2}-PD controller based on the WHO response has the lowest undershoot and overshoot than the other techniques, which are (9.2 × 10

^{−3}) Hz and (2.4 × 10

^{−3}) Hz for ΔF

_{1}, respectively (see Figure 11a), and 1.49 × 10

^{−3}Hz undershoot for ΔF

_{2}with no overshoot (see Figure 11b). Additionally, the WHO application has a lowest undershoot and overshoot than the other cases when considering the tie-line power deviation (ΔP

_{tie}), which equals 6.5 × 10

^{−4}p.u for ΔP

_{tie}(see Figure 11c). Moreover, the WHO response has the lowest settling time than the other two techniques, which is 3.3 s for ΔF

_{1}and 7.2 s for ΔF

_{2}. Additionally, the WHO application has the lowest settling time than the other cases when considering the tie-line power deviation (ΔP

_{tie}), which equals 8.8 s for ΔP

_{tie}.

#### 5.2. Simulation Results and Discussions

- Scenario I: Evaluation of system dynamic response under load variation types;
- Scenario II: Evaluation of system dynamic response using RESs disturbances;
- Scenario III: Evaluation of system dynamic response with RESs disturbances, taking into consideration the communication time delay (CTD), applied to the proposed controller output;
- Scenario IV: Evaluation of system dynamic response based on RESs disturbances and changes in system settings.

#### 5.2.1. Scenario I: Evaluation of System Dynamic Response under Load Variation Types

^{2}-PD controller optimized by the WHO algorithm is compared to the performance efficiency of other controllers, such as WHO-optimized ID-T [31], PID-TID [32] controllers and ID-T controller optimized by the AOA algorithm [31]. Table 6 displays the settings of the controllers considered in this section.

^{2}-PD controller optimized by the WHO algorithm has the lowest overshoot, undershoot, and settling time of the other three controllers and provides the best objective function based on ITSE, which is 1.202 × 10

^{−4}.

^{2}-PD controller based on the WHO algorithm has been tested and assessed by applying a series of load changes in the first area and comparison with using several control strategies (i.e., PID-TID and ID-T controllers based on the WHO). Figure 13 depicts the dynamic system response. Table 8 also shows the dynamic response of the power system in this part. As a result, the proposed PIDD

^{2}-PD controller has the lowest undershoot, overshoot, settling time, and ITSE. The superiority of the proposed suggested PIDD

^{2}-PD controller based on WHO over the other controllers optimized using the WHO algorithm is that with the proposed PIDD

^{2}-PD controller it is possible to get a greater decrease in system frequency variations and power flow in the tie line compared to other controllers in this case. Therefore, the developed PIDD

^{2}-PD enhances the system’s reliability.

^{2}-PD controller in the two prior situations, RLD is a varied collection of series disturbances that may be represented by industrial loads linked to a power system network. Random load disturbances are applied to the first area shown in Figure 14a. Additionally, Figure 14 depicts the system reaction for this section using several control strategies (i.e., PIDD

^{2}-PD, PID-TID, and ID-T controllers based on the WHO). Table 9 summarizes the dynamic performance of the system in this part. In comparison to the ID-T and PID-TID controllers, the suggested PIDD

^{2}-PD controller based on the WHO has high performance in dealing with rapid and gradual load fluctuations, and the suggested controller shows better performance. It is evident that it dampens the oscillations very fast, with the lowest undershoot and overshoot, in addition to better control quality. This shows that the PIDD

^{2}-PD based on the WHO technique is a robust controller used to load frequency control LFC.

#### 5.2.2. Scenario II: Performance Evaluation Based on RESs Penetration

^{2}-PD, PID-TID, and ID-T controllers based on the WHO). Figure 15 shows the convergence characteristics of the three controllers. The evaluation by applying a series load of disturbances is shown in Figure 16a to the first area, the PV solar unit with 50 MW is linked to the first area at 250 s, and the wind farm unit with 70 MW rated power is linked to the second area at 100 s, with the note wind farm and the PV solar unit are illustrated in Figure 3 and Figure 5, respectively. Furthermore, Table 10 shows the PIDD

^{2}-PD, PID-TID, and ID-T parameters. The maximum number of iterations and the number of the population are specified to be 50 and 30, respectively. Figure 16 illustrates the dynamic power system response and the frequency deviation of the power system network (ΔF

_{1}, ΔF

_{2}), the tie-line power deviation because of the series load disturbances and the RESs penetration in this scenario. The severe fluctuations in frequency and flow power in the tie-line power occur throughout the period the RESs are connected, as shown in Figure 16. Table 11 summarizes the dynamic performance of the power system. The proposed PIDD

^{2}-PD controller can effectively dampen fluctuations in frequency and the flow power in the tie-line power. Furthermore, it obtains the lowest values for both overshoot, undershoot, settling time, and ITSE compared to the PID-TID and ID-T controllers and it has the best convergence characteristics. Additionally, this can be concluded that the ID-T controller is the least effective at controlling RESs variations with the series load disturbances.

#### 5.2.3. Scenario III: Evaluation of Performance Using RESs Disturbances and Communication Time Delay (CTD) on the Signal Output of the Controller

^{2}-PD, PID-TID, and ID-T) optimized by the WHO algorithm, and Table 13 depicts the dynamic performance of the system in this scenario. Figure 17 clarifies the convergence curve of the controllers. Figure 18 depicts the frequency fluctuation of both areas of the power system network studied and flow power in the tie-line power. Due to RESs sources disturbances and applying a communication time delay, the system’s response has severely oscillated. The suggested PIDD

^{2}-PD controller, on the other hand, can achieve adequate stability of the system power network and significantly reduce the impact of system fluctuation, and obtained the lowest overshoot, undershoot, settling time, and ITSE values than the PID-TID and ID-T controllers shown in Table 13 and Figure 18.

#### 5.2.4. Scenario IV: Performance Evaluation for RESs and Changes in System Parameters

^{2}-PD performance being investigated when system parameters, such as T

_{sg}, T

_{t}, T

_{gh}, X

_{g}, and Y

_{g}, are changed by 50%. Step load penetration occurred in the first area at 10 s and the second area at 150 s, with values of 0.01 p.u and 0.03 p.u, respectively. The PV solar system and wind turbine are linked at 80 s and 220 s, respectively. Table 10 shows the settings of the suggested PIDD

^{2}-PD controller that are employed in this scenario. Table 14 shows the dynamic performance of the power system. Figure 19 and Figure 20 clarify the frequency fluctuation of both areas of the power system network studied and flow power in the tie-line power when changing system settings by 50%. It can be concluded that changes of 50% in system settings, as well as step load penetration, applied to both areas, have a negligible effect on the functioning of the PIDD

^{2}-PD controller.

## 6. Conclusions

^{2}-PD controller is developed for improving the frequency stability in the power system network understudied. In a two-area hybrid power system, each area consists of multiple conventional power stations and renewable energy sources. The suggested controller is implemented with WOA, ChOA, and WHO algorithms. The WHO algorithm provides better performance, with a fast response. The effectiveness of the PIDD

^{2}-PD based on the WHO controller was compared to the PID-TID and ID-T controllers based on the WHO and ID-T controllers optimized for the AOA algorithm. A variety of different scenarios have been proposed to study the effectiveness of performance for the combined PIDD

^{2}-PD controller in addressing the issue of the two areas—load frequency control—by using different load patterns, RES disturbances, communication time delay, and system settings variations. From the above, it can be concluded the suggested controller achieves outstanding results in resolving all obstacles, increasing system stability, and enhancing the frequency dynamic response of the power system network. The PIDD2-PD controller has supremacy over the other controllers’ performance. The suggested PIDD2-PD controller structure has been shown to be an excellent solution to the LFC issue.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

AOA | Archimedes Optimization Algorithm |

A_{T} | The rotor swept area (m^{2}) |

B_{1}, B_{2} | Frequency bias coefficients |

ChOA | Chimp Optimization Algorithm |

C_{P} | The power coefficient of the rotor blades |

CTD | Communication time delay |

FF | Fitness function |

FO | Fractional order |

FOC | FO calculus |

FOPID | Fractional order proportional derivative |

GDB | Governor dead band |

GRC | Generation rate constraint, % (p.u) |

H | Total number of groups |

ID-T | Integral derivative—tilted |

I-PD | Integral-proportional derivative |

it | Iteration |

I-TD | Integral-tilted derivative |

ITSE | Integral time squared error |

kd | Derivative gain of PD |

KD, KDD | Derivative gains of PIDD^{2} |

KI | Integral gain of PIDD^{2} |

KP | Proportional gain of PIDD^{2} |

kp | The proportional gain of PD |

LFC | Load frequency control |

maxit | Maximum number of iterations |

Max.OS | Maximum overshoot |

MSLD | Multi-step load disturbances |

Max.US | Maximum undershoot |

N_{d}, N_{dd} | Filters’ coefficients of the PIDD^{2} |

nf | Filters’ coefficients of the PD |

PD | Proportional derivative |

PID | Proportional integral derivative |

PIDD^{2} | Proportional integral derivative—second derivative |

PV | Photovoltaics |

Q | Population size |

RESs | Renewable energy sources |

RLD | Random load disturbances |

r_{T} | The rotor radius |

SLD | Step load disturbances |

SR | Number of stallions in the population |

Set-Time | Settling time |

TDC | Transient droop compensation |

TID | Tilted integral derivative |

T_{s} | Simulation time |

V_{W} | The rated wind speed (m/s) |

WHO | Wild Horse Optimization |

WOA | Whale Optimization Algorithm |

Z | Randomly selected adaptive mechanism |

β | The pitch angle |

ΔF_{1} | The frequency deviation in Area 1 (Hz) |

ΔF_{2} | The frequency deviation in Area 2 (Hz) |

ΔP_{tie} | The tie-line power deviation (p.u) |

λ | The tip-speed ratio (TSR) |

λ_{I} | The intermittent TSR |

ρ | Air density (Kg/m^{3}) |

## References

- Magdy, G.; Mohamed, E.A.; Shabib, G.; Elbaset, A.A.; Mitani, Y. Microgrid dynamic security considering high penetration of renewable energy. Prot. Control. Mod. Power Syst.
**2018**, 3, 23. [Google Scholar] [CrossRef] - Yang, Z.; Ghadamyari, M.; Khorramdel, H.; Alizadeh, S.M.S.; Pirouzi, S.; Milani, M.; Banihashemi, F.; Ghadimi, N. Robust multi-objective optimal design of islanded hybrid system with renewable and diesel sources/stationary and mobile energy storage systems. Renew. Sustain. Energy Rev.
**2021**, 148, 111295. [Google Scholar] [CrossRef] - Dehghani, M.; Ghiasi, M.; Niknam, T.; Kavousi-Fard, A.; Shasadeghi, M.; Ghadimi, N.; Taghizadeh-Hesary, F. Blockchain-Based Securing of Data Exchange in a Power Transmission System Considering Congestion Management and Social Welfare. Sustainability
**2021**, 13, 90. [Google Scholar] [CrossRef] - Magdy, G.; Shabib, G.; Elbaset, A.A.; Mitani, Y. Optimized coordinated control of LFC and SMES to enhance frequency stability of a real multi-source power system considering high renewable energy penetration. Prot. Control. Mod. Power Syst.
**2018**, 3, 39. [Google Scholar] [CrossRef] [Green Version] - Khamies, M.; Magdy, G.; Kamel, S.; Khan, B. Optimal Model Predictive and Linear Quadratic Gaussian Control for Frequency Stability of Power Systems Considering Wind Energy. IEEE Access
**2021**, 9, 116453–116474. [Google Scholar] [CrossRef] - Jagatheesan, K.; Anand, B.; Dey, N.; Ashour, A.S.; Balas, V.E. Load Frequency Control of Hydro-Hydro System with Fuzzy Logic Controller Considering Non-linearity. In Recent Developments and the New Direction in Soft-Computing Foundations and Applications; Springer: Cham, Switzerland, 2018; pp. 307–318. [Google Scholar]
- Nguyen, G.N.; Jagatheesan, K.; Ashour, A.S.; Anand, B.; Dey, N. Ant Colony Optimization Based Load Frequency Control of Multi-area Interconnected Thermal Power System with Governor Dead-Band Nonlinearity. In Smart Trends in Systems, Security and Sustainability; Springer: Singapore, 2018; pp. 157–167. [Google Scholar]
- Dritsas, L.; Kontouras, E.; Vlahakis, E.; Kitsios, I.; Halikias, G.; Tzes, A. Modelling issues and aggressive robust load frequency control of interconnected electric power systems. Int. J. Control
**2022**, 95, 753–767. [Google Scholar] [CrossRef] - Tasnin, W.; Saikia, L.C.; Raju, M. Deregulated AGC of multi-area system incorporating dish-Stirling solar thermal and geothermal power plants using fractional order cascade controller. Int. J. Electr. Power Energy Syst.
**2018**, 101, 60–74. [Google Scholar] [CrossRef] - Sharma, M.; Dhundhara, S.; Arya, Y.; Prakash, S. Frequency excursion mitigation strategy using a novel COA optimised fuzzy controller in wind integrated power systems. IET Renew. Power Gener.
**2020**, 14, 4071–4085. [Google Scholar] [CrossRef] - Nandi, M.; Shiva, C.K.; Mukherjee, V. Moth-Flame Algorithm for TCSC- and SMES-Based Controller Design in Automatic Generation Control of a Two-Area Multi-unit Hydro-power System. Iran. J. Sci. Technol.
**2020**, 44, 1173–1196. [Google Scholar] [CrossRef] - Jagatheesan, K.; Baskaran, A.; Dey, N.; Ashour, A.S.; Balas, V.E. Load frequency control of multi-area interconnected thermal power system: Artificial intelligence-based approach. Int. J. Autom. Control
**2018**, 12, 126–152. [Google Scholar] [CrossRef] - Eltamaly, A.M.; Zaki Diab, A.A.; Abo-Khalil, A.G. Robust Control Based on H∞ and Linear Quadratic Gaussian of Load Frequency Control of Power Systems Integrated with Wind Energy System. In Control and Operation of Grid-Connected Wind Energy Systems; Springer: Cham, Switzerland, 2021; pp. 73–86. [Google Scholar]
- Habib, D. Optimal Control of PID-FUZZY based on Gravitational Search Algorithm for Load Frequency Control. Int. J. Eng. Res.
**2019**, 8, 50013–50022. [Google Scholar] - Yakout, A.H.; Kotb, H.; Hasanien, H.M.; Aboras, K.M. Optimal Fuzzy PIDF Load Frequency Controller for Hybrid Microgrid System Using Marine Predator Algorithm. IEEE Access
**2021**, 9, 54220–54232. [Google Scholar] [CrossRef] - Magdy, G.; Mohamed, E.A.; Shabib, G.; Elbaset, A.A.; Mitani, Y. SMES based a new PID controller for frequency stability of a real hybrid power system considering high wind power penetration. IET Renew. Power Gener.
**2018**, 12, 1304–1313. [Google Scholar] [CrossRef] - Sharma, J.; Hote, Y.V.; Prasad, R. Robust PID Load Frequency Controller Design with Specific Gain and Phase Margin for Multi-area Power Systems. IFAC-Pap.
**2018**, 51, 627–632. [Google Scholar] [CrossRef] - Topno, P.N.; Chanana, S. Differential evolution algorithm based tilt integral derivative control for LFC problem of an interconnected hydro-thermal power system. J. Vib. Control
**2017**, 24, 3952–3973. [Google Scholar] [CrossRef] - Khokhar, B.; Dahiya, S.; Parmar, K.P.S. Load Frequency Control of a Multi-Microgrid System Incorporating Electric Vehicles. Electr. Power Compon. Syst.
**2021**, 49, 867–883. [Google Scholar] [CrossRef] - Elmelegi, A.; Mohamed, E.A.; Aly, M.; Ahmed, E.M.; Mohamed, A.A.A.; Elbaksawi, O. Optimized Tilt Fractional Order Cooperative Controllers for Preserving Frequency Stability in Renewable Energy-Based Power Systems. IEEE Access
**2021**, 9, 8261–8277. [Google Scholar] [CrossRef] - Morsali, J.; Zare, K.; Tarafdar Hagh, M. Comparative performance evaluation of fractional order controllers in LFC of two-area diverse-unit power system with considering GDB and GRC effects. J. Electr. Syst. Inf. Technol.
**2018**, 5, 708–722. [Google Scholar] [CrossRef] - Mohamed, E.A.; Ahmed, E.M.; Elmelegi, A.; Aly, M.; Elbaksawi, O.; Mohamed, A.A.A. An Optimized Hybrid Fractional Order Controller for Frequency Regulation in Multi-Area Power Systems. IEEE Access
**2020**, 8, 213899–213915. [Google Scholar] [CrossRef] - Ali, M.; Kotb, H.; Aboras, K.M.; Abbasy, N.H. Design of Cascaded PI-Fractional Order PID Controller for Improving the Frequency Response of Hybrid Microgrid System Using Gorilla Troops Optimizer. IEEE Access
**2021**, 9, 150715–150732. [Google Scholar] [CrossRef] - Prakash, A.; Murali, S.; Shankar, R.; Bhushan, R. HVDC tie-link modeling for restructured AGC using a novel fractional order cascade controller. Electr. Power Syst. Res.
**2019**, 170, 244–258. [Google Scholar] [CrossRef] - Saha, A.; Saikia, L.C. Load frequency control of a wind-thermal-split shaft gas turbine-based restructured power system integrating FACTS and energy storage devices. Int. Trans. Electr. Energy Syst.
**2019**, 29, e2756. [Google Scholar] [CrossRef] - Mohamed, T.H.; Shabib, G.; Abdelhameed, E.H.; Khamies, M.; Qudaih, Y. Load Frequency Control in Single Area System Using Model Predictive Control and Linear Quadratic Gaussian Techniques. Int. J. Electr. Energy
**2015**, 3, 141–143. [Google Scholar] [CrossRef] - Elkasem, A.H.A.; Khamies, M.; Hassan, M.H.; Agwa, A.M.; Kamel, S. Optimal Design of TD-TI Controller for LFC Considering Renewables Penetration by an Improved Chaos Game Optimizer. Fractal Fract.
**2022**, 6, 220. [Google Scholar] [CrossRef] - Daraz, A.; Malik, S.A.; Mokhlis, H.; Haq, I.U.; Laghari, G.F.; Mansor, N.N. Fitness Dependent Optimizer-Based Automatic Generation Control of Multi-Source Interconnected Power System with Non-Linearities. IEEE Access
**2020**, 8, 100989–101003. [Google Scholar] [CrossRef] - Singh, K.; Amir, M.; Ahmad, F.; Khan, M.A. An Integral Tilt Derivative Control Strategy for Frequency Control in Multimicrogrid System. IEEE Syst. J.
**2021**, 15, 1477–1488. [Google Scholar] [CrossRef] - Kumari, S.; Shankar, G.; Das, B. Integral-Tilt-Derivative Controller Based Performance Evaluation of Load Frequency Control of Deregulated Power System. In Modeling, Simulation and Optimization; Springer: Singapore, 2021; pp. 189–200. [Google Scholar]
- Ahmed, M.; Magdy, G.; Khamies, M.; Kamel, S. Modified TID controller for load frequency control of a two-area interconnected diverse-unit power system. Int. J. Electr. Power Energy Syst.
**2022**, 135, 107528. [Google Scholar] [CrossRef] - Pahadasingh, S. TLBO Based CC-PID-TID Controller for Load Frequency Control of Multi Area Power System. In Proceedings of the 2021 1st Odisha International Conference on Electrical Power Engineering, Communication and Computing Technology (ODICON), Bhubaneswar, India, 8–9 January 2021. [Google Scholar]
- Paliwal, N.; Srivastava, L.; Pandit, M. Application of grey wolf optimization algorithm for load frequency control in multi-source single area power system. Evol. Intell.
**2020**, 15, 563–584. [Google Scholar] [CrossRef] - Revathi, D.; Mohan Kumar, G. Analysis of LFC in PV-thermal-thermal interconnected power system using fuzzy gain scheduling. Int. Trans. Electr. Energy Syst.
**2020**, 30, e12336. [Google Scholar] [CrossRef] - Jagatheesan, K.; Anand, B.; Dey, N.; Gaber, T.; Hassanien, A.E.; Kim, T.H. A Design of PI Controller using Stochastic Particle Swarm Optimization in Load Frequency Control of Thermal Power Systems. In Proceedings of the 2015 Fourth International Conference on Information Science and Industrial Applications (ISI), Busan, Korea, 20–22 September 2015. [Google Scholar]
- Ah, G.H.; Li, Y.Y. Ant lion optimized hybrid intelligent PID-based sliding mode controller for frequency regulation of interconnected multi-area power systems. Trans. Inst. Meas. Control
**2020**, 42, 1594–1617. [Google Scholar] [CrossRef] - Patel, N.C.; Sahu, B.K.; Khamari, R.C. TLBO Designed 2-DOFPIDF Controller for LFC of Multi-area Multi-source Power System. In Innovation in Electrical Power Engineering, Communication, and Computing Technology; Springer: Singapore, 2022; pp. 269–282. [Google Scholar]
- Aryan, P.; Ranjan, M.; Shankar, R. Deregulated LFC scheme using equilibrium optimized Type-2 fuzzy controller. In Proceedings of the International conference on Innovative Development and Engineering Applications, Gaya, India, 8–10 February 2021. [Google Scholar]
- Khokhar, B.; Dahiya, S.S.; Singh Parmar, K.P. Atom search optimization based study of frequency deviation response of a hybrid power system. In Proceedings of the 2020 IEEE 9th Power India International Conference (PIICON), Murthal, India, 28 February–1 March 2020; pp. 1–5. [Google Scholar]
- Naruei, I.; Keynia, F. Wild horse optimizer: A new meta-heuristic algorithm for solving engineering optimization problems. Eng. Comput.
**2021**, 1–32. [Google Scholar] [CrossRef] - Khishe, M.; Mosavi, M.R. Chimp optimization algorithm. Expert Syst. Appl.
**2020**, 149, 113338. [Google Scholar] [CrossRef] - Mirjalili, S.; Lewis, A. The Whale Optimization Algorithm. Adv. Eng. Softw.
**2016**, 95, 51–67. [Google Scholar] [CrossRef] - Parmar, K.P.S.; Majhi, S.; Kothari, D.P. LFC of an Interconnected Power System with Thyristor Controlled Phase Shifter in the Tie Line. Int. J. Comput. Appl.
**2012**, 41, 27–30. [Google Scholar] - Magdy, G.; Shabib, G.; Elbaset, A.A.; Kerdphol, T.; Qudaih, Y.; Mitani, Y. Decentralized optimal LFC for a real hybrid power system considering renewable energy sources. J. Eng. Sci. Technol.
**2019**, 14, 682–697. [Google Scholar] - Ali, M.; Kotb, H.; AboRas, M.K.; Abbasy, H.N. Frequency regulation of hybrid multi-area power system using wild horse optimizer based new combined Fuzzy Fractional-Order PI and TID controllers. Alex. Eng. J.
**2022**, 61, 12187–12210. [Google Scholar] [CrossRef] - Kalyan, C.N.S.; Suresh, C.V. PIDD controller for AGC of nonlinear system with PEV integration and AC-DC links. In Proceedings of the 2021 International Conference on Sustainable Energy and Future Electric Transportation (SEFET), Hyderabad, India, 21–23 January 2021. [Google Scholar]

**Figure 7.**Flowchart of the WHO algorithm [40].

**Figure 12.**Dynamic power system response under scenario I, Section A: (

**a**) ΔF

_{1}, (

**b**) ΔF

_{2}, and (

**c**) ΔP

_{tie}.

**Figure 13.**Dynamic power system response under Scenario I, Section B: (

**a**) MSLD, (

**b**) ΔF

_{1}, (

**c**) ΔF

_{2}, and (

**d**) ΔP

_{tie}.

**Figure 14.**Dynamic power system response under scenario I, Section C: (

**a**) RLD, (

**b**) ΔF

_{1}, (

**c**) ΔF

_{2}, and (

**d**) ΔP

_{tie}.

**Figure 16.**Dynamic power system response under Scenario II: (

**a**) Series load disturbances, (

**b**) ΔF

_{1}, (

**c**) ΔF

_{2}, and (

**d**) ΔP

_{tie}.

**Figure 18.**Dynamic power system response under Scenario III: (

**a**) ΔF

_{1}, (

**b**) ΔF

_{2}, and (

**c**) ΔP

_{tie}.

**Figure 19.**Dynamic power system response under Scenario IV with a −50 change in the system settings: (

**a**) ΔF

_{1}, (

**b**) ΔF

_{2}, and (

**c**) ΔP

_{tie}.

**Figure 20.**Dynamic power system response under Scenario IV with a +50 change in the system settings: (

**a**) ΔF

_{1}, (

**b**) ΔF

_{2}, and (

**c**) ΔP

_{tie}.

Model | Transfer Function | Parameter | Value | Description |
---|---|---|---|---|

Power system 1 | $\frac{{K}_{ps1}}{{T}_{ps1}\xb7s+1}$ | ${T}_{ps1}$$={T}_{ps2}$ ${K}_{ps1}$$={K}_{ps2}$ | 11.49 s 68.9655 | Power system time constants Power system gains |

Power system 2 | $\frac{{K}_{ps2}}{{T}_{ps2}\xb7\mathrm{s}+1}$ | |||

T-line | $\frac{2\pi {T}_{12}}{s}$ | ${T}_{12}$ | 0.0433 | Synchronization factor |

${B}_{1}$$,{B}_{2}$ | 0.4312 | Coefficient values of frequency bias |

Parameter | Value | Parameter | Value |
---|---|---|---|

${P}_{W}$ | 750 kW | ${C}_{2}$ | 116 |

${V}_{W}$ | 15 m/s | ${C}_{3}$ | 0.4 |

${r}_{T}$ | 22.9 m | ${C}_{4}$ | 0 |

$\rho $ | 1.225 kg/m^{3} | ${C}_{5}$ | 5 |

${A}_{T}$ | 1684 m^{2} | ${C}_{6}$ | 21 |

${\lambda}_{T}$ | 22.5 r.p.m | ${C}_{7}$ | 0.1405 |

${C}_{1}$ | −0.6175 |

**Table 3.**Setting values for the WHO parameters [45].

WHO Parameter | Value |
---|---|

SR | 0.2 |

H | 6 |

Q | 30 |

Number of foals | 24 |

R | 0.2372 |

WP | [2, 1.83, 0, 2, 2, 2, 0, 0, 0, 1, 0, 0, 0, 4.7, 20, 20, 3.19, 20, 20, 12.7] |

**Table 4.**Optimum settings of the suggested PIDD

^{2}-PD controller are optimized by three optimization algorithms (WOA, ChOA, WHO).

AREA 1 | |||||||||
---|---|---|---|---|---|---|---|---|---|

Algorithm | PD_{1} | PIDD^{2}_{1} | |||||||

kp_{1} | kd_{1} | nf_{1} | KP_{1} | KI_{1} | KD_{1} | KDD_{1} | Nd_{1} | Ndd_{1} | |

WOA | 14.253 | 4.4785 | 500 | 50 | 50 | 1.7171 | 0.1 | 500 | 500 |

ChOA | 14.564 | 0 | 500 | 50 | 0 | 6.3209 | 0.1228 | 401.6571 | 309.896 |

WHO | 38.475 | 0.0144 | 431.882 | 41.1532 | 0.3835 | 5.6677 | 0.1 | 100 | 478.5245 |

AREA 2 | |||||||||

Algorithm | PD_{2} | PIDD^{2}_{2} | |||||||

kp_{2} | kd_{2} | nf_{2} | KP_{2} | KI_{2} | KD_{2} | KDD_{2} | Nd_{2} | Ndd_{2} | |

WOA | 50 | 50 | 500 | 50 | 12.451 | 4.1011 | 0.8 | 500 | 500 |

ChOA | 0.008 | 0 | 496.631 | 0.0975 | 0.1436 | 0 | 0.2725 | 323.829 | 304.0149 |

WHO | 0 | 17.32 | 334.76 | 50 | 7.8044 | 0.5505 | 0.1501 | 251.83 | 498.7457 |

Optimization Techniques | ΔF_{1} (Hz) | ΔF_{2} (Hz) | ΔP_{tie} (p.u) | ITSE | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Max. OS | Max. US | Set-Time | Max. OS | Max. US | Set-Time | Max. OS | Max. US | Set- Time | ||

WOA | 0.0055 | 0.0124 | 3.6 | 0.001 | 0.00137 | 15.1 | 0.00023 | 0.00106 | 12.8 | 0.0003082 |

ChOA | 0.0042 | 0.0105 | 10.7 | 0.001 | 0.00261 | 20.8 | 0.00019 | 0.00085 | 28 | 0.0002264 |

WHO (proposed) | 0.0024 | 0.0092 | 3.3 | 0 | 0.00149 | 7.2 | 0 | 0.00065 | 8.8 | 0.0001202 |

AREA 1 | |||||||||
---|---|---|---|---|---|---|---|---|---|

Algorithm | PD_{1} | PIDD^{2}_{1} | |||||||

kp_{1} | kd_{1} | nf_{1} | KP_{1} | KI_{1} | KD_{1} | KDD_{1} | Nd_{1} | Ndd_{1} | |

PIDD^{2}-PD (WHO)(suggested) | 38.475 | 0.0144 | 431.882 | 41.1532 | 0.3835 | 5.6677 | 0.1 | 100 | 478.5245 |

AREA 2 | |||||||||

Algorithm | PD_{2} | PIDD^{2}_{2} | |||||||

kp_{2} | kd_{2} | nf_{2} | KP_{2} | KI_{2} | KD_{2} | KDD_{2} | Nd_{2} | Ndd_{2} | |

PIDD^{2}-PD (WHO)(suggested) | 0 | 17.32 | 334.76 | 50 | 7.8044 | 0.5505 | 0.1501 | 251.83 | 498.7457 |

AREA 1 | |||||||||

Algorithm | PID | TID | |||||||

kp_{1} | ki_{1} | kd_{1} | nf_{1} | KT_{1} | n_{1} | KI_{1} | KD_{1} | ||

PID-TID (WHO) | 6.1095 | 0 | 34.0678 | 489.8079 | 49.9998 | 2.5167 | 50 | 2.5459 | |

AREA 2 | |||||||||

Algorithm | PID | TID | |||||||

kp_{2} | ki_{2} | kd_{2} | nf_{2} | KT_{2} | n_{2} | KI_{2} | KD_{2} | ||

PID-TID (WHO) | 25.6245 | 12.8848 | 3.2186 | 499.8395 | 49.2407 | 2.4475 | 14.9894 | 3.8772 | |

AREA 1 | |||||||||

Algorithm | T | ID | |||||||

KT_{1} | n_{1} | KI_{1} | KD_{1} | NC_{1} | |||||

ID-T (WHO) | −31.4909 | 1.7755 | 39.3266 | 25.3455 | 499.3504 | ||||

AREA 2 | |||||||||

Algorithm | T | ID | |||||||

KT_{2} | n_{2} | KI_{2} | KD_{2} | NC_{2} | |||||

ID-T (WHO) | −15.2490 | 2.8479 | 38.8390 | 12.0328 | 336.9504 | ||||

AREA 1 | |||||||||

Algorithm | T | ID | |||||||

KT_{1} | n_{1} | KI_{1} | KD_{1} | NC_{1} | |||||

ID-T (AOA) | −4.9 | 2.17 | −3.4 | −3.6 | 496.9 | ||||

AREA 2 | |||||||||

Algorithm | T | ID | |||||||

KT_{2} | n_{2} | KI_{2} | KD_{2} | NC_{2} | |||||

ID-T (AOA) | −0.002 | 6.07 | −0.010 | −2.390 | 469.2 |

Controller | ΔF_{1} (Hz) | ΔF_{2} (Hz) | ΔP_{tie} (p.u) | ITSE | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Max. OS | Max. US | Set-Time | Max. OS | Max. US | Set-Time | Max. OS | Max. US | Set- Time | ||

PIDD^{2}-PD (WHO)(suggested) | 0.0024 | 0.0092 | 3.3 | 0 | 0.00149 | 7.2 | 0 | 0.00065 | 8.7 | 0.0001202 |

PID-TID (WHO) | 0.0013 | 0.0112 | 16.1 | 0.00021 | 0.00235 | 18.6 | 0.00009 | 0.00096 | 20.4 | 0.0002403 |

ID-T (AOA) | 0.009 | 0.028 | 11 | 0.005 | 0.024 | 12 | 0.001 | 0.004 | 11 | 0.001 |

ID-T (WHO) | 0.0042 | 0.0103 | 11.8 | 0.00098 | 0.00272 | 13.3 | 0.00017 | 0.00091 | 16 | 0.0002689 |

Controller | ΔF_{1} (Hz) | ΔF_{2} (Hz) | ΔP_{tie} (p.u) | ITSE | |||
---|---|---|---|---|---|---|---|

Max. OS | Max. US | Max. OS | Max. US | Max. OS | Max. US | ||

PIDD^{2}-PD (WHO)(suggested) | 0.0060 | 0.0123 | 0.00076 | 0.00154 | 0.00033 | 0.00067 | 0.003442 |

PID-TID (WHO) | 0.0095 | 0.0197 | 0.00146 | 0.00324 | 0.00059 | 0.00130 | 0.01051 |

ID-T (WHO) | 0.0126 | 0.0210 | 0.0029 | 0.0049 | 0.00096 | 0.00161 | 0.02664 |

Controller | ΔF_{1} (Hz) | ΔF_{2} (Hz) | ΔP_{tie} (p.u) | ITSE | |||
---|---|---|---|---|---|---|---|

Max. OS | Max. US | Max. OS | Max. US | Max. OS | Max. US | ||

PIDD^{2}-PD (WHO)(suggested) | 0.0090 | 0.0091 | 0.00113 | 0.00115 | 0.00049 | 0.00050 | 0.01756 |

PID-TID (WHO) | 0.0105 | 0.0105 | 0.00164 | 0.00163 | 0.00066 | 0.00065 | 0.02848 |

ID-T (WHO) | 0.0168 | 0.0168 | 0.0040 | 0.0039 | 0.00131 | 0.00129 | 0.1063 |

AREA 1 | |||||||||
---|---|---|---|---|---|---|---|---|---|

Algorithm | PD_{1} | PIDD^{2}_{1} | |||||||

kp_{1} | kd_{1} | nf_{1} | KP_{1} | KI_{1} | KD_{1} | KDD_{1} | Nd_{1} | Ndd_{1} | |

PIDD^{2}-PD (WHO)(suggested) | 48.2812 | 6.328 | 343.1941 | 40.63 | 2.5682 | 2.1527 | 0.0084 | 140.5213 | 421.9369 |

AREA 2 | |||||||||

Algorithm | PD_{2} | PIDD^{2}_{2} | |||||||

kp_{2} | kd_{2} | nf_{2} | KP_{2} | KI_{2} | KD_{2} | KDD_{2} | Nd_{2} | Ndd_{2} | |

PIDD^{2}-PD (WHO)(suggested) | 49.9496 | 2.0937 | 301.5396 | 44.5887 | 9.9736 | 9.8294 | 0 | 420.2037 | 118.7231 |

AREA 1 | |||||||||

Algorithm | PID | TID | |||||||

kp_{1} | ki_{1} | kd_{1} | nf_{1} | KT_{1} | n_{1} | KI_{1} | KD_{1} | ||

PID-TID (WHO) | 23.4461 | 0 | 26.4650 | 300 | 45.3144 | 2.5099 | 14.3536 | 0.9136 | |

AREA 2 | |||||||||

Algorithm | PID | TID | |||||||

kp_{2} | ki_{2} | kd_{2} | nf_{2} | KT_{2} | n_{2} | KI_{2} | KD_{2} | ||

PID-TID (WHO) | 33.2919 | 0.0947 | 3.9765 | 483.1653 | 40.6655 | 6.9517 | 0 | 2.4108 | |

AREA 1 | |||||||||

Algorithm | T | ID | |||||||

KT_{1} | n_{1} | KI_{1} | KD_{1} | NC_{1} | |||||

ID-T (WHO) | $-$39.9993 | 1.8675 | 39.9995 | 40 | 500 | ||||

AREA 2 | |||||||||

Algorithm | T | ID | |||||||

KT_{2} | n_{2} | KI_{2} | KD_{2} | NC_{2} | |||||

ID-T (WHO) | $-$25.3571 | 9.9994 | 39.9404 | 28.3108 | 495.5133 |

Controller | ΔF_{1} (Hz) | ΔF_{2} (Hz) | ΔP_{tie} (p.u) | ITSE | |||
---|---|---|---|---|---|---|---|

Max. OS | Max. US | Max. OS | Max. US | Max. OS | Max. US | ||

PIDD^{2}-PD (WHO)(suggested) | 0.0157 | 0.0157 | 0.0178 | 0.0024 | 0.00064 | 0.0015 | 0.01959 |

PID-TID (WHO) | 0.0191 | 0.0192 | 0.0218 | 0.0105 | 0.00090 | 0.00217 | 0.04153 |

ID-T (WHO) | 0.0205 | 0.0200 | 0.0249 | 0.0077 | 0.00152 | 0.00349 | 0.08325 |

AREA 1 | |||||||||
---|---|---|---|---|---|---|---|---|---|

Algorithm | PD_{1} | PIDD^{2}_{1} | |||||||

kp_{1} | kd_{1} | nf_{1} | KP_{1} | KI_{1} | KD_{1} | KDD_{1} | Nd_{1} | Ndd_{1} | |

PIDD^{2}-PD (WHO)(suggested) | 2.7227 | 0.3027 | 195.6591 | 19.6364 | 6.3706 | 7.6158 | 0.0541 | 190.8568 | 130.8450 |

AREA 2 | |||||||||

Algorithm | PD_{2} | PIDD^{2}_{2} | |||||||

kp_{2} | kd_{2} | nf_{2} | KP_{2} | KI_{2} | KD_{2} | KDD_{2} | Nd_{2} | Ndd_{2} | |

PIDD^{2}-PD (WHO)(suggested) | 7.4519 | 1.5539 | 148.7560 | 6.4190 | 12.1799 | 1.8329 | 0.0269 | 141.0946 | 100.4050 |

AREA 1 | |||||||||

Algorithm | PID | TID | |||||||

kp_{1} | ki_{1} | kd_{1} | nf_{1} | KT_{1} | n_{1} | KI_{1} | KD_{1} | ||

PID-TID (WHO) | 1.9950 | 0.0027 | 4.3505 | 375.0716 | 16.9318 | 1.738 | 3.4679 | 1.4928 | |

AREA 2 | |||||||||

Algorithm | PID | TID | |||||||

kp_{2} | ki_{2} | kd_{2} | nf_{2} | KT_{2} | n_{2} | KI_{2} | KD_{2} | ||

PID-TID (WHO) | 7.3223 | 0.0140 | 3.2081 | 312.8155 | 7.5334 | 5.1325 | 3.9343 | 1.1741 | |

AREA 1 | |||||||||

Algorithm | T | ID | |||||||

KT_{1} | n_{1} | KI_{1} | KD_{1} | NC_{1} | |||||

ID-T (WHO) | $-$5.4107 | 9.5456 | 5.0845 | 6.6880 | 389.4373 | ||||

AREA 2 | |||||||||

Algorithm | T | ID | |||||||

KT_{2} | n_{2} | KI_{2} | KD_{2} | NC_{2} | |||||

ID-T (WHO) | $-$17.4930 | 1.4958 | 4.5823 | 13.7478 | 480.3091 |

Controller | ΔF_{1} (Hz) | ΔF_{2} (Hz) | ΔP_{tie} (p.u) | ITSE | |||
---|---|---|---|---|---|---|---|

Max. OS | Max. US | Max. OS | Max. US | Max. OS | Max. US | ||

PIDD^{2}-PD (WHO)(suggested) | 0.0148 | 0.0116 | 0.030 | 0.048 | 0.00308 | 0.00176 | 0.1488 |

PID-TID (WHO) | 0.0222 | 0.0246 | 0.044 | 0.078 | 0.0075 | 0.0035 | 0.5603 |

ID-T (WHO) | 0.0323 | 0.0307 | 0.052 | 0.080 | 0.0088 | 0.0055 | 0.9892 |

Controller | ΔF_{1} (Hz) | ΔF_{2} (Hz) | ΔP_{tie} (p.u) | ITSE | |||
---|---|---|---|---|---|---|---|

Max. OS | Max. US | Max. OS | Max. US | Max. OS | Max. US | ||

PIDD^{2}-PD (suggested) | 0.0068 | 0.0033 | 0.0177 | 0.0158 | 0.00070 | 0.00078 | 0.0167 |

PIDD^{2}-PD (suggested)with +50% | 0.0068 | 0.0033 | 0.0178 | 0.0159 | 0.00072 | 0.00081 | 0.0172 |

PIDD^{2}-PD (suggested) with −50% | 0.0068 | 0.0033 | 0.0176 | 0.0157 | 0.00065 | 0.00074 | 0.01583 |

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## Share and Cite

**MDPI and ACS Style**

Khudhair, M.; Ragab, M.; AboRas, K.M.; Abbasy, N.H.
Robust Control of Frequency Variations for a Multi-Area Power System in Smart Grid Using a Newly Wild Horse Optimized Combination of PIDD^{2} and PD Controllers. *Sustainability* **2022**, *14*, 8223.
https://doi.org/10.3390/su14138223

**AMA Style**

Khudhair M, Ragab M, AboRas KM, Abbasy NH.
Robust Control of Frequency Variations for a Multi-Area Power System in Smart Grid Using a Newly Wild Horse Optimized Combination of PIDD^{2} and PD Controllers. *Sustainability*. 2022; 14(13):8223.
https://doi.org/10.3390/su14138223

**Chicago/Turabian Style**

Khudhair, Mohammed, Muhammad Ragab, Kareem M. AboRas, and Nabil H. Abbasy.
2022. "Robust Control of Frequency Variations for a Multi-Area Power System in Smart Grid Using a Newly Wild Horse Optimized Combination of PIDD^{2} and PD Controllers" *Sustainability* 14, no. 13: 8223.
https://doi.org/10.3390/su14138223