Wind turbines are typically used to harness kinetic energy derived from wind and convert it into mechanical energy (Yammani and Maheswarapu, 2019). This mechanical energy is subsequently transmitted to the generator’s rotor through a shaft, and the generator converts the mechanical energy into electrical energy. The mathematical representation of the mechanical output of a wind turbine is shown in Equation 1. P T = 1 2 ρ A C P λ , β V ω 3 (1) where, P_m represents mechanical power derived from the rotor shaft, P_w is the power of wind content in the virtual stream tube consisting Wind turbine (Karimi et al., 2016). Figure 1A represents the WTG first-order transfer function block.

A SPV system is a power generation system that converts sunlight directly into electricity, and can be used for both grid-connected mode and standalone. The increasing popularity of SPV systems in current years can be attributed to government policies that encourage their use, the increased demand for power, concerns regarding the environment, cheap operational costs, and the lack of fuel expenses. The amount of power output from a SPV system is dependent on the solar irradiance, ambient temperature, and the conversion efficiency of the SPV panel (Obaid et al., 2019). SPV power output (W) can be given as, P p v s t = η c e l l s * F F * V * I (2)

Similarly, the terminal voltage can be given as, V = V o c + K v t * T c t (3)

Also, the fill factor is given as, F F = V m p * I m p V o c * I s c (4) Where, V_mp is the voltage at maximum power in V, I_mp is the current at maximum power in A, V_oc is the open-circuit voltage in V, K_vt is the voltage temperature coefficient in mV/°C, where, η_cells is the number of SPV cells.

Similarly, the current (A) is given as, I = s t * I s c + K c t * T c t − 25 (5) Where, I_sc is the short-circuit current in A, K_ct is the current temperature coefficient in mA/°C, T_at is the ambient temperature in °C, s(t) is the random irradiance, NOCT is the nominal cell operating temperature in °C.

The cell temperature in °C is given as, T c t = T a t + s t * N O C T − 20 0.8 (6)

The SPV power’s first-order transfer function model is given in Figure 1B.

Diesel generators are a common component in power systems that are used to meet the electricity needs of consumers. These generators offer reliable and long-lasting power solutions for applications ranging from prime power to standby power. Diesel generators are used as backup power sources, emergency and reserve, etc. due to its unique features such as availability, rapid start-up, consistency, robustness, and black start ability (Zhang et al., 2017). Despite some drawbacks, these features make diesel generators a popular choice in certain locations. The rate of fuel consumption of a diesel generator at any given time t is directly related to its output power, and can be represented mathematically. F = F 0 . Y d g + F 1 . P d g (7)

Where, F₀ is the fuel curve intercept coefficient (units/h/kW), F₁ is the fuel curve slope (units/h/kW), P_dg is the electrical output of the generator (kW) and Y_dg is the rated capacity of the generator (kW).

The DEG model of the first-order transfer function block is represented in Figure 2A shown below. Using the governor control action DEG maintains the equilibrium between demand for electricity and production in an independent microgrid owing to variations in solar and wind power.

BESS are frequently employed for enhancing autonomous MG’s primary frequency control. DEG is often in charge of the principal frequency control duty. DEGs, on the other hand, have a prolonged time constant and gradual respond to frequency fluctuations, which can result in significant overshoot (Adefarati et al., 2017). BESS can be added into the system to solve this issue and improve the primary frequency response. The state of charge of BESS at any time t, as well as the limits state of charge of the BESS as a function of optimal BESS capacity E S S b e s s c a p , are calculated as follows: S O C t = S O C t − 1 1 − S b e s s + P b e s s c h t * η b e s s c − P b e s s d i s t η b e s s d (8) S O C b e s s min = 0.2 * E S S b e s s c a p (9) S O C p t e s max = 0.8 * E S S b e s s c a p (10)

Where, the present and earlier energy capacity of the BESS are represented as SOC (t) and SOC (t-1) over one-time step and the rate of self-discharge is represented as S b e s s . The BESS charging and discharging of power are represented as P b e s s c h and P b e s s d i s which are the characteristics of the electric power supply system’s surpluses and shortfalls, correspondingly. η b e s s c and η b e s s d are the BESS discharging and charging efficiencies.

The BESS transfer function first-order block is given in Figure 2B. Also, negative of ΔF represents Discharging and positive of ΔF represents Charging of the BESS with respect to the system frequency.

Kennedy and Eberhart introduced the PSO algorithm as an evolutionary optimization technique, inspired by animal social behaviors, such as the movement of schools of fish or flocks of birds, in finding food and avoiding predators. These creatures possess remarkable traits that are functional and have been optimized through many iterations of a vast optimization algorithm in the DNA of living beings (Nayak et al., 2023). Consequently, the PSO Algorithm, like other intelligent optimization algorithms, is influenced by nature. The algorithm employs random numbers and is structured in the following way:

The PSO algorithm can be broken down into six steps. The first step is the initialization of the population, which creates a random number to serve as the starting location for each particle or sequence’s motif candidate. Step 2 calculates the fitness value for each particle, and parameter pBest stores the particle with the highest fitness value. In Step 3, the global maximum fitness value (gBest) is updated. Step 4 involves calculating velocities using a randomization approach. In Step 5, each particle’s new location is updated using a velocity value. Finally, Step 6 pertains to the termination condition, where the process flow is terminated if the condition is satisfied. Otherwise, the process flow will be repeated from Step 2.

Grey wolves are known to be highly skilled predators that excel at locating prey. Their social hierarchy, as depicted in Figure 3A, is a fascinating characteristic of these animals. The group is organized into a rigid dominating structure, with the most powerful member of the pack being the Alpha, which can be either male or female. The Alpha wolf makes critical decisions about hunting, migration, sleeping locations, and eating (Ulutas et al., 2020).

The Alpha wolf is not necessarily the strongest member of the group but must be the best at controlling the pack, emphasizing that dedication and society are more significant than power. The beta wolf is the next level in the hierarchy, and they assist the Alpha wolf in making decisions. When the Alpha wolf becomes ill or dies, the beta wolf takes over as the leader. The beta wolf also serves as the disciplinarian and counsellor to the Alpha wolf. Delta, the third level in the hierarchy, is dominant in omega but must report to the Alpha and beta wolves. Scouts, carers, and hunters fall within the delta category. Omega is at the bottom of the hierarchy and is made the scapegoat, allowing them to eat last. These wolves’ absence can lead to internal strife and issues within the group. The GWO flow chart, shown in Figure 3B, follows this hierarchical structure (Gheisarnejad and Khooban, 2019).

In nature, Harris’s hawks use their powerful eyes to locate and identify their prey, but sometimes, the prey may not be easily visible. Therefore, the hawks perch, observe, and monitor the desert area for several hours to detect a potential prey. In HHO, the candidate solutions are considered as the hawks, and the finest results in each iteration is measured as the target or the ideal prey. The hawks randomly perch on various places and pauses to sight a target using two approaches. Firstly, they perched along other family members of various locations (to attack the prey while being close to each other), and the prey, which is represented by Eq. 13 when j < 0.5. Alternatively, they perch on tall trees randomly located within the group’s home range, as shown in Equation (6 and (14) when j ≥ 0.5. P s + 1 = p r a n d s − x 1 P r a n d S − 2 x 2 P S j ≥ 0.5 P r a b b i t − P m s − x 3 l b + x 4 u b − l b j 0.5 (13)

From the above, P r a n d s is a randomly selected hawk, P s + 1 is the vector of hawks position in the consecutive iteration s, P r a b b i t s represents rabbit (prey) location, P s is the vector of present hawks position, x₁, x₂, x₃, x₄ and j (random numbers) are updated in each iteration and are within (0,1), and P m is the position of the present hawks, u_b is the upper bounds and l_b is the lower bounds variables,. The normal position of the hawks is shown in (14). P m s = 1 n ∑ i = 1 n P i s (14)

Where, s and n represent the total hawks and P i s represents the iteration location of each hawk and Eq. 11 is used to calculate the fitness function.

The energy of the prey reduces significantly during its escape behavior, and this is reflected in the prey’s energy equation. as below: E = E 0 1 − t S (15)

The energy of a prey is given by equation E, where E_o is the initial energy, t is the iteration, S is the iteration maximum number. As the prey’s energy decreases during the escaping behavior, it becomes less likely to escape. Hence, the hawks adopt different strategies depending on the prey’s energy and distance. Exploitation involves besiege of hard and soft with landing dives, while exploration involves searching particles average locations.

The hawks done a rapid dive on the prey previously detected in the iteration phase and the prey normally try to avoid danger, and the chasing behavior of hawks varies in response to different escaping behaviors. To model the attacking stage in HHO, four possible strategies are proposed as shown in the below Table 1.

The HHO algorithm has two strategies to attack prey: soft besiege and hard besiege. The hard besiege has similar characteristics to the soft besiege, but its Q and R conditions are different. To visualize the HHO tracking, the vector addition structure is used and shown in Figure 4A. Additionally, the flow chart of the HHO algorithm is illustrated in Figure 4B.

In order to improve the efficiency of the conventional HHO algorithm, an enhancement is proposed by integrating the velocity updating equation of the PSO algorithm into the updating process of HHO. V i d = V i d + C 1 * r a n d   * P i d − X i d + c 2 * r a n d   * P g d − X i d (16)

Where, V_id is the velocity of the hawk; X_id is the present hawk position. P_id and P_gd shows pbest and gbest. rand () is a random number inside (0,1). cL, c2 are learning factors. Normally used as c₁ = c₂ = 2.

In this sub section, PSO based controller performance is discussed separately for each area of the microgrid sections considering WTP. The values are derived from PSO based multi-microgrid system. The performance of PSO based controller for SPV, WTG, BESS, DEG, load deviation and frequency deviation for each unit as well as tie line within each area is shown in the following figures separately as shown below. The PSO based controller has attained the average output power of SPV and WTG as 0.25 p. u. and 0.36 p. u. respectively. Similarly, the response of BESS and DEG are shown in the figure. The PSO based controller has attained the load deviation between 1.25 p. u to almost 2.1 p. u.Also, the PSO based controller performance for frequency deviation has been attained almost zero value with high deviation.

The performance of the tie line values such as frequency deviation and power of each of the Microgrid area −1, −2 and −3 under PSO is shown in the following figures. Figure 8 shows the tie line power deviation of each Microgrid such as (a) area-1 and area-2 (b) area 2 and 3 (c) area-1 and area-3. Similarly, the iteration for fitness function convergence of PSO is shown in Figure 8D showing that at 50 iteration the PSO shows best convergence result.

In this section the performance of the proposed controller and existing controllers such as PSO, GWO and EHHO based controller is compared and evaluated in this sub section. The comparative performance of the proposed and existing controller for each microgrid areas are shown and explained in the following sub sections. The following Table 2 shows the optimized controller parameters with different algorithms.

The tie line of a transmission system is defined as the connecting point or parts of different transmission line or different sub-systems, here in microgrid system a tie line represents the connecting points of two or more area or branch for exchanging power for the whole microgrid system. the performance of each three Microgrid area under comparison of different optimization technique is discussed and shown in the previous subsections. However, comparison of each optimization technique in the tie line between each microgrid area is discussed in this section and the following figure shows tie line power and frequency. Figure 11 shows the tie line output power of (a) area-1 and area-2 (b) area-2 and area-3 and (c) area-1 and area-3. From the figure it can be stated that the performance of EHHO shows better results compare to other optimization techniques in area-3. Figure 11D shows tie line power deviation of different area under EHHO. Moreover, Figure 12 shows convergence plot comparison of PSO, GWO and EHHO optimization technique. The convergence curve shows that the fitness function for EHHO is low compared to other optimization technique and also converge faster. Also, Table 3 shows the parameters of each area of the microgrid system.

The PID controller is widely recognized as the foundational and primitive control technique that the field of control has ever developed. However, it has been observed that this controller may not perform well when the system is exposed to parameter changes, uneven disturbances, or non-linearities. To address these complications, the FO based-PID controller has been integrated with the concept of higher DOF. This integration provides additional features for tuning the controller and enhances its operating capabilities (Yousri et al., 2020).

To incorporate extra inertia and other important control, a fractional order controller (specifically, the double-stage FOPID (1 + PI) controller) has been employed. The proportional derivative (PD) component of the controller delivers the extra inertia and damping, while the integral (I) component offers additional regulator to the system. As a result, the controller derivative portion enhances the transient responses and improves the response of steady-state by minimizing the error (Khokhar et al., 2021).

A 3DOF (3 Degree of Freedom) controller consists of three independent feedback loops where three inputs, R(s), Y(s), and D(s), are obtained from each of the loops. The 3DOF structure is combined with a FOPIDN (Fractional Order Proportional Integral Derivative with Notch) controller to form a 3DOF-FOPIDN controller as shown in Figure 13. This new controller has an extra independent loop, which helps in achieving a better response compared to a 2DOF-FOPIDN controller.

The fractional calculus involves using differentiation and integration with fractional-order or complex-order. Fractional derivatives have an advantage in that they can inherit the characteristics of the processes being modeled. The fractional-order PID controller (FO-PID) is a transfer function written as in Eq. 17. C o n t r o l l e r r e s p o n s e   y = D a Δ F + K S − λ Δ F + M a S μ Δ F 0 < λ ≤ 1 , 0 < μ ≤ 0 (17) with gain constants, including a proportional part, D_a, an integral part, K, and a derivative part, Ma, with two fractional operators, λ and µ. The actuating signal of the 3DOF-FOPIDN controller is described by an equation, which takes into account the three inputs from the independent loops. The transfer functions of the FOPIDN controller is shown below. U S = y x k p x + k i x S η + x x k d x s ξ N f N f + S ξ R S = − k p x + k i x S η − k d x s ξ N f N f + S ξ Y S = − g x k p x − k i x S η − g x k d x s ξ N f N f + S ξ D S (18)

In control systems, the degree of freedom (DOF) refers to the number of independent adjustments that can be made to the closed-loop transfer functions. By utilizing a 3DOF controller, closed-loop stability and dynamic response of the system can be improved while minimizing the effect of disturbances. The proposed 3DOF-TIDN controller is illustrated in Figure 14 which includes input reference R(S), system disturbance D(s), 3DOF output Y(s), and system output U(S). This controller is designed to enhance the dynamic response of the system by reducing the number of oscillations, minimizing deviation of frequency and tripping of power, and maintaining system stability. It is also intended to improve the damping ratio of the system in response to sudden loading changes (Ahmed et al., 2022). The output of the controller in a closed-loop configuration is mathematically expressed in an equation. Y s = G C O s G P s 1 + G C O s G P s G R C s R s + G P s − G C O s G P s G F F C S 1 + G C O s G P s D s (19)

The GCO(s) is designed to limit certain parameters in the control system, such as tilt, integral, derivative gain (K_Ts, K_Is, K_Ds), tilt parameter (Ns), and low-pass filter (N_Tf). The proportional (B) and derivative (C) set point weights for R(s) are also represented by the GCO(s), while the gain parameter of the GFFC Controller is represented by G_x. To determine the optimal gain values for the controllers as well as parameters, an optimization algorithm is utilized, specifically the Enhanced Harris Hawks optimization algorithm, which minimizes the ITAE (Integral absolute error, Time, and Absolute Error), while adhering to certain constraints. These constraints include minimum and maximum bounds of 0 and 2 for the controller, 0 to 200 for the filters, and 2 to 3 for N. K T s Min ≤ K T s ≤ K T s Max , K I s Min ≤ K I s ≤ K I s Max , K D s Min ≤ K D s ≤ K D s Max N T f Min ≤ N T f ≤ N T f Max , N s Min ≤ N s ≤ N s Max , c ⁡ min Max b min Max f f min f f f f Max (20)

When choosing between the 3DOF FOPIDN and 3DOF TIDN controllers for controlling three-degree-of-freedom mechanical systems, several differences should be considered. Firstly, the FOPIDN controller employs fractional calculus in its control approach, whereas the TIDN controller uses integer calculus. While the FOPIDN controller offers greater flexibility in adjusting control parameters, the TIDN controller may be simpler to implement. In terms of precision, the FOPIDN controller is generally regarded as more accurate than the TIDN controller due to its ability to utilize fractional calculus for control. Additionally, the FOPIDN controller is more robust, allowing it to effectively handle changes in the system or external disturbances. However, its implementation may be more complex than the TIDN controller, which relies on integer calculus and does not use notch filters (Guha et al., 2020).

Ultimately, the performance of each controller will depend on the specific application and the mechanical system being controlled. In some situations, the TIDN controller may be sufficient, while in others, the FOPIDN controller may be required for optimal performance. Therefore, the selection between the two controllers will depend on the specific requirements of the application and the mechanical system being controlled. Analysis of the controller in area-1, area-2 and area-3 are shown in the following figures.

Based on the analysis, the obtained results from Figures 15A–D demonstrate that the 3DOF-DOPIDN controller proposed in the study exhibits superior dynamic responses compared to other controllers. It effectively reduces oscillation and settling time in each area. Additionally, the controller helps to maintain the generated incremental power in the area. As there is no additional load demand, the total generated power in both areas remains at 0.04 p. u.MW. It also provides further insight into the controller’s performance by showing the power generation by various sources in area-2, which amounts to 2.5 p. u.MW peak in total.

Table 4 lists the gain values obtained for all three controllers studied in this section.

To evaluate the stability of the system, an eigenvalue analysis is conducted with and without various secondary controllers while violating the contract. The eigenvalue plot is displayed in Figure 16 and listed in Table 5. The damping factor of the system is an indicator of how quickly the system is damped. The damping factors for the system are also provided in Table 5. The study examines several secondary controllers, including PID, 3DOF-TIDN, and 3DOF-FOPIDN.

Upon analyzing the obtained eigenvalues, it was observed that the uncontrolled system had some eigenvalues in the right half of the s-plane, indicating instability. For the PID controllers, some of the eigenvalues were zero along with negative real parts, indicating marginal stability. However, it was observed from Figure 16 that the PID controllers resulted in initial system oscillations, but the system reached its steady-state value after some time. On the other hand, for the 3DOF-TIDN and 3DOF-FOPIDN controllers, all eigenvalues had negative real parts, indicating system stability. The damping ratio of the 3DOF-TIDN controller was found to be higher than others, indicating that the oscillation of the system may sustain for a smaller duration with this controller. Based on this analysis, it can be concluded that the 3DOF-FOPIDN controller provides more stable performance compared to the other controllers considered in the study.

Robustness analysis is a method used to evaluate the ability of a system, process, or product to maintain its performance even in the presence of uncertainties or changes in its operating conditions or environment. This analysis involves subjecting the system to different scenarios or inputs and analyzing its response to assess its performance. The objective of robustness analysis is to determine the system’s resilience and adaptability to various conditions and to identify the sources of variability and uncertainty that affect its performance.

The given passage describes a sensitivity analysis conducted to assess the robustness of a proposed 3DOF-FOPIDN controller by considering various scenarios and uncertainties. The analysis involves varying the parameters value in the system, such as varying the output of Wind, changing the system loading from the nominal scenario, and disconnecting one of the microgrid area. The results of the analysis are presented in Table 6, which shows the gains as well as other parameters under different scenarios. The system dynamics are then compared with the corresponding gain values of the 3DOF-FOPIDN controller equivalent to different conditions and their responses under nominal conditions.

Table 6 presents the ideal gains and other constraints for the 3DOF-FOPIDN controller based on the results of the sensitivity analysis. To compare the system dynamics, the gain values of the 3DOF-FOPIDN controller obtained under nominal conditions are used to generate responses for different scenarios. These responses are then compared to those obtained under changed conditions, as shown in Figures 17A, B. Overall, the sensitivity analysis involves six dynamic responses to evaluate the robustness of the 3DOF-FOPIDN controller.

The system encountered a variation in its output, which increased from 0.006 p. u.MW to 0.012 p. u.MW. The gains as well as other constraints of the anticipated 3DOF-FOPIDN controller, taking into account various uncertainties, are listed in Table 6. To assess the controller’s robustness, its performance under the changed conditions is compared to its response under nominal conditions, using the gain values obtained from both scenarios. The resulting dynamic responses are presented in the below Figures.