Enhancing the Conventional Controllers for Load Frequency Control of Isolated Microgrids Using Proposed Multi-Objective Formulation via Artificial Rabbits Optimization Algorithm

Isolated microgrids (IMGs) power remote areas. However, IMG may lower the frequency stability and increase frequency excursions with low system inertia. Load frequency management ensures system stability. Thus, the paper proposes a novel multi-objective tuning strategy to improve IMG’s load frequency control (LFC) and take the microgrid controller’s control signals into account. Diesel engine generator, fuel cell, battery energy storage system, and renewable energy sources (RESs) like photovoltaic and wind systems make up the IMG. Conventional controllers such as proportional-integral (PI) and proportional integral derivative (PID) are classically tuned based on the standard error criteria as a traditional single-objective tuning approach. Due to the low inertia of the system and the stochastic nature of RES, they cannot act as required under different operating scenarios. Therefore, the PI and PID controllers are tuned using the proposed multi-objective-based tuning approach to reduce the frequency deviations. In addition, anti-windup is applied to the enhanced classic controllers to keep them distant from the nonlinear zone and beyond the source’s physical constraints. The proposed tuning process also considers the maximum practical generation rates for different sources. The recent Artificial Rabbits Optimization (ARO) algorithm is applied to simultaneously adjust the controller parameters for several controlled sources in IMG. Extensive simulations in MATLAB and Simulink confirm the effectiveness of the proposed approach to keep the system stable even when facing high levels of disturbances. In addition, accomplishing sensitivity analysis, severe ±25% changes to the system’s parameters guarantee that the proposed tuning strategy keeps the system stable.


I. INTRODUCTION
Microgrids (MGs) have two operating modes; isolated mode and grid-tied mode. The grid-tied operation promises the balance between generation and load and has a supportive role in mitigating load disruption. Isolated microgrids (IMGs) require inventive structures and adaptable control mechanisms for performance and stability [1]. Two types of IMGs are often in low and medium-voltage networks. The first type is the single-area [2], whereas the second is the multiarea [3]. IMG is extended by adding energy storage systems (ESSs), like fuel cells (FCs) and battery energy storage systems (BESSs) [4], [5], to ensure the robustness and reliability of IMG against load disturbance with the shortage in conventional distributed generating units (DGUs). Recently, renewable energy sources (RES), such as photovoltaic (PV) modules and wind turbines (WTs) [6], controlled loads, and ESSs are combined with conventional diesel engine generators (DEGs) [7] to create the current IMG structure [2], [3], [8], [9]. In addition to IMG's inertia restrictions, inverterbased devices with modern IMG structures reduce the overall system inertia [10]. RES uncertainty also jeopardizes frequency stability, especially in single-area IMG [11]. As a vital supplemental service in IMGs, the load frequency control (LFC) scheme improves frequency stability [12], [13].
Numerous traditional and new controllers, adjusted using different tuning methodologies, have been employed to explore the LFC schemes of the single and multi-area IMGs. As reported in the literature, still the LFC can be achieved by using conventional controllers with straightforward designs and high robustness, like proportional-integral (PI) and proportional-integral-derivative (PID). However, improper PI/PID controller tuning may cause poor dynamic behavior and instability throughout the system [14]. As a result, the tuning and design approaches are significant factors in providing reliable controllers and guaranteeing overall system stability [15], [16].
Generation units have physical limitations that result in intrinsic generation rate constraints (GRC), such as ramp rate constraints and upper-lower bound constraints. Lack of effective GRC in LFC design can lead to poor controller performance and limited closed-loop system stability in the presence of disturbances [17], [18], [19]. Moreover, the antiwindup property keeps traditional controllers out of the nonlinear zone [20] and away from the source's physical restrictions, especially in the LFC scheme for nonlinear systems [21].
There have been significant efforts in the last decade to tune classical controllers properly in a variety of engineering problems, including those corresponding to the automatic generation control of IMGs, whether in the field of automatic voltage regulation [22], [23] or LFC schemes [24], [25], [26], [27], [28], [29], [30], [31], [32], [33]. The slime mould optimization (SMA) algorithm is proposed in [24] to improve the frequency stability of a two-area IMG with thermal and photovoltaic units using a single objective formulation to determine the PI controller's optimal parameters. The LFC problem and tuning procedure were made by the thermal unit's high inertia and the preset PV system profile, which made the IMG less uncertain, and the GRC of thermal unit is not considered. The LFC of two-area IMG is also investigated in [25], whereas PI controller parameters are tuned using a single objective approach based on the artificial sheep algorithm. Such IMG has reheated thermal units with a PV unit besides hydro-pumped storage. Reheated thermal units' GRC is considered, but the controller's effectiveness is tested in a high-inertia IMG with a high-power density supply, like pumped storage, to minimize traditional controllers' uncertainty constraints. In [26], the firefly and particle swarm optimization, as a hybrid optimization technique, is presented to tune the parameters of a standard PID controller of a single-area IMG that contains a flywheel. For such low inertia IMG, the authors overcome ignoring the GRC of DEG by employing the flywheel to mitigate the uncertainty due to the WT used. The mine blast algorithm (MBA) is described in [27] and [28] for tuning PID controllers and enhancing LFC of low inertia single-area IMG that includes super magnetic energy storage (SMES) and also a multiarea IMG has ultra-capacitor, respectively. Single objective tuning is applied without considering GRC to adjust the controllers' optimal gains where ultra-capacitors and SMES provide a durable network and make it easier to tune multiple controllers in the LFC manner. Without sources of a highpower density and considering sources' GRC, these MGs cannot assess controller effectiveness. Thus, the authors suggested a multi-objective-based tuning strategy to maximize controllers' significance with no high-power density sources. In [29], [30], and [31], a method for LFC for both single and multi-area IMGs is proposed using conventional PI, PID, and fuzzy-based PID controllers. The optimum traditional controller tuning for the LFC is presented based on numerous single objective error criteria that various optimization techniques have optimized. In [29], the moth flame optimizer (MFO) tunes controller parameters without considering sources' GRC of reheat-thermal units of two-area IMG. In [30] and [31], the grey wolf optimization algorithm (GWO) and whale optimization algorithm (WOA) took the sources' GRC into account. Despite the authors' consideration of GRC during tuning in [30] and [31], the simple single objective tuning is utilized with high inertia sources, and the efficiency of proposed LFC techniques is not verified for low inertia IMG.
On the other hand, simplifying tuning methods and addressing the nonlinear system limits of conventional controllers can be achieved with some advanced controllers, such as fuzzy-based PID and fractional order (FO) controllers, which may replace traditional ones. For example, the type-II fuzzy PID controller was provided by [32] to give LFC of two-area IMGs comprising microturbine and flywheels. That work uses the improved-slap swarm optimization to find the optimal gains for the PID controller using a single-objective tuning without considering the DEGs unit's GRC. In low inertia MGs with the absence of high-power density sources, many sophisticated controllers based on single-objective tuning processes may fail if the sources' GRC is ignored.
Some research studies did not address GRC during controller tuning [24], [26], [27], [28], [29]. Thus, these approaches have no theoretical basis for efficacy. On the other hand, [25], [30], [31] have addressed GRC in LFC. Unfortunately, neither of these studies has overcome MG's lack of inertia. These LFC schemes work best with structures that use high-power density or high-inertia sources, while they are not successful in demonstrating their efficacy when used with weak structures such as the low inertia single-area IMG.
The authors believe there is room to create viable strategies for adapting traditional controllers for low-inertia IMGs. A multi-objective tuning method based on the Artificial Rabbits Optimization (ARO) algorithm is proposed for PI and PID controllers. The research's main findings are: 1. Offering a novel multi-objective-based tuning technique that incorporates the integral square error (ISE) as one of the most often used error criteria, the control signals from several sources, and finally, the settling time as a vital issue in the LFC scheme of IMGs. 2. Using the ARO algorithm, a recent optimization algorithm introduced in 2022, to simultaneously tune the controller settings for a range of controlled sources to enhance IMG frequency stability. 3. Compared to the structures in the literature that use either high power density or high inertia sources, using the proposed formulation for low-inertia IMG can be seen as a real challenge. 4. Discussing the role of GRC as a significant system issue encompasses physical factors from multiple sources, which are not addressed sufficiently in the literature. When employed to control nonlinear systems, linear controllers' anti-windup property also helps improve their stability. The paper will be structured as follows: Section II introduces the planned IMG's architecture with several sources and the controllers' structures. Section III presents the proposed problem formulation. In Section IV, the optimization algorithm is explained. Section V compares the outcomes of employing the suggested multi-objective formulation to some reported criteria, and Section VI introduces the sensitivity analysis. Section VII highlights the conclusions and future work.

II. MODELING OF THE TESTED ISOLATED MICROGRID
Numerous MG models are investigated in [1], where the first-order dynamic models are adequate and simple for LFC representations over higher-order dynamic models. The firstorder dynamic models for DGUs are straightforward to avoid the aggregate complication of MGs and are also suitable for the LFC scheme [34], [35], [36], [37]. Consistent with recent research, the authors have applied uncomplicated first-order transfer functions to represent DGUs in the considered IMG.
The studied IMG is depicted in Fig. 1 with the contributing elements. Storage elements are modeled using the fuel cell (FC) and the battery energy storage system (BESS) models to provide enough energy to support load frequency stability. Wind and PV systems present renewable energy sources (RES) in this IMG, which are uncontrolled sources to facilitate other sources in different load disturbances. Three different conventional controllers are used for the LFC scheme by the reminder sources of the IMG, which are the controlled sources of diesel engine generator (DEG), FC, and BESS.
The following subsections present the model of DEG as a fast-active power supply for the IMG. Dynamic models of BESS and FC are also demonstrated. Due to the widespread collaboration of wind turbines and PV systems in the LFC scheme, a dynamic model is presented for each RES. Then, the overall MG dynamic model is discussed. Lastly, the structure of the studied controllers is briefly described.

A. DIESEL ENGINE GENERATOR MODEL
DEG is a reliable and quick source of durable electric power generation through load perturbation [35]. The assigned transfer function used for the DEG model via frequency control scheme is presented in Fig.1. A governor and a generator specified by the first-order plant model often make up this model. The system also includes a turbine to represent the inertia delay. The DEG output power variation can be represented as follows: In equation (1), the terms T 1, T 2, and T 3 stand for the Inertia delay time constant, governor time constant, and generator time constant, respectively. The symbol F refers to frequency fluctuation. The DEG speed regulation coefficient is displayed by R, while P DEG denotes the amount of DEG power variation during disturbances and U DEG is the control signal of DEG. The secondary loop adopts the DEG  power mismatches with the addition of a conventional PID controller.

B. FUEL CELL MODEL
As discussed in [38], the FC model can be established with three first-order transfer functions. This model provides sufficient small signal stability for the LFC analysis. It consists of three basic blocks: an inverter block that converts DC voltage to AC voltage; an interconnection device with two primary blocks, one for modeling the measurement delay and the other for command delay as revealed in Fig. 1. The output power of FC employed in the LFC scheme can be modeled as follows: where P FC stands for FC active power variation during system disturbances, U FC for the FC control signal, T 4 for the measurement time delay, T 5 for the command device delay, and T 6 for the inverter model time constant. In the current frequency control scheme, FC is controlled by a conventional PID controller. VOLUME 11, 2023 C. BATTERY ENERGY STORAGE MODEL For the proposed LFC, BESS performs as an ancillary source that contributes to power generation during disturbance and steady-state operations. According to [39], BESS is used in conjunction with a battery energy management system. The authors assume that the battery is functioning within a normal range. The load, which consumes a range of permissible currents at a fixed rated output voltage, is the main characteristic of the normal zone. As depicted in Fig. 1, it is critical to consider all time delays; measurement delay, command delay, and converter delay.
For the current LFC model, BESS output power can be modeled as follows: In equation (3), T 7 is the measurement time constant, T 8 is the command device delay, T 9 is the DC to AC converter model time constant, U BESS is the control signal of BESS and P BESS characterizes the change in BESS shared active power during system variations. It is worth mentioning that in the present LFC scheme, BESS is controlled by a conventional PI controller.

D. MODELS OF RENEWABLE ENERGY SOURCES
In the following subsections, the models presenting PV and WT will be briefly discussed.

1) PV MODEL
Basically, the output power of a PV system depends on solar radiation, PV panel efficiency, panel size, and ambient temperature. The extracted output power of the PV system can be presented as follows [39]: where η pv defines the efficiency of the PV system, S is the solar radiation, A pv is the area of the PV system, and T 0 is the cell temperature.
The white noise block in Fig. 2 represents the stochastic character of PV generation, whereas T 10 is the delay time constant and P PV is the amount of PV power variation throughout the simulation.

2) WIND MODEL
The following equation defines how the wind turbine generates power by harnessing the kinetic energy of the air [40]: where P w describes the wind turbine output power, v r is the rated speed of the wind, A w is the rotor swept, and C p denotes the performance coefficient of rotor blades. The approximate equation of the performance coefficient can be in terms of  turbine coefficients C 1 -C 7 as follows [40]: Referring to equation (6), β designates the pitch angle, γ i is the intermittent tip speed ratio, and γ t is the optimum tip speed. The rotor blade radius is described by r, and w is the rotational speed of the blades.
The equation that describes the intermittent tip speed ratio is expressed as shown: Figure 3 illustrates the details of the simple block shown in Fig.1 for wind turbine power generation in the LFC scheme. The stochastic behavior of wind energy generation is offered by a white noise block while T 11 describes the delay time constant, and P W defines the amount of wind turbine power variation along simulation time.

E. MICROGRID DYNAMIC MODEL
The DEG, FC, BESS, RES, and MG load are the five subsystems that make up the IMG described in Fig. 1. The following formula can be used to determine the frequency variation of that IMG: Referring to equation (8), F defines the MG frequency variation, H is the overall inertia constant of the system, D is the load damping coefficient, P L is the variation in load, P DEG is the variation in DEG power generation, P FC is the variation in output power of FC, P BESS is the variation in output power of BESS, and P RES is the variation in RES 3476 VOLUME 11, 2023 output power. The variation in RES output power is the sum of the variation of PV and WT output power i.e. P RES = P PV + P W .

F. APPLIED CONTROLLERS 1) PI-CONTROLLER
PI controllers are still frequently used in industry due to their straightforward design and adequate performance for most industrial applications. The proportional and integral gains are the two primary tuning parameters for this controller. Equation (9) shows the Laplace transform to express the transfer function (TF) of the PI-controller: where K p and K i are the proportional and integral gains, respectively, which can be optimally tuned using numerous optimization algorithms to enhance the LFC problem.
2) PID-CONTROLLER PID controllers are PI controllers modified to incorporate a derivative term to improve the dynamic response of the controlled systems. PID has excellent dynamics and robust properties, which is regarded as one of the most flexible feedback controllers [41]. PID controllers rely on proportional, integral, and derivative gains (K p , K i , and K d ). So, to keep the stability of system frequency in MG, these gains should be optimally tuned using an efficient computing technique. In the Laplace transform, the TF of a PID controller is introduced as follows:

III. PROPOSED MULTI-OBJECTIVE TUNING APPROACH A. FORMULATION
The appropriate choice of the employed objective function in the LFC scheme has a crucial impact on the controller's performance [8], [41]. Numerous objective functions, either single objective [25], [42], [43], or multi-objective [44], [45], [46] have been developed in earlier studies to reduce the frequency deviation of an IMG. Most of these studies use one of the widely accepted error criteria to enhance frequency stability. Such as the integral absolute error (IAE), the integral time absolute error (ITAE), the integral square error (ISE), or the integral time square error (ITSE). Although these criteria produced good performance indices, they do not always ensure that all performance indices will be improved, e.g., the ITAE criterion provides better settling time more sensitive than other criteria such as ISE. However, the optimal tuning-based ITAE can result in a large control signal that may be hazardous in the sudden imbalance [47]. On the contrary, the ISE criterion punishes errors with large values against lesser errors [48]. The control error's large values usually occur instantly after a step disturbance and may be represented as the system overshoot [48]. Due to the benefits of aggressive control mitigation via weak IMG, the ISE criterion is still often utilized in LFC difficulties even though it has long settling periods and is typically used to measure direct overshoot and violent control.
Researchers have also used the common error criteria to optimize several controller types for isolated networks and concluded that ISE has the potential to offer overall superior performance to other controller types, especially for low inertia IMG [33], [49], [50]. As a result, the ISE error criterion will be selected in this research to tune the various controllers of the studied IMG's LFC scheme. Additionally, in this research, the tuning procedure is carried out by utilizing a multi-objective function that aggregates many objectives to provide an effective LFC scheme.
The first objective function J 1 is ISE, it might be easier to be traced mathematically as follows: where F is the frequency deviation to be minimized and t sim is the simulation time.
More improvement in ISE necessitates high control signals from different controllers for the LFC scheme to moderate various system disturbances effectively. However, this raises the settling time, which is considered a significant performance metric. Directly including settling time as a term in a multi-objective function has a large positive impact on the settling time but a negative impact on other performance indicators [38], [51]. Thus, this research will also use an indirect novel technique by reducing the square of the control signal based on the system's output feedback, or F, besides the addition of the settling time and ISE in the overall multiobjective function.
Reduction control signals will direct the various sources' outputs when there are large disturbances, preventing different controllers from reacting aggressively to disturbances, which indirectly helps to improve the settling time. So, the second objective in the suggested multi-objective function will be expressed as follows, which is the integral square of control signals from various controlled sources.
In equation (12), U DEG , U BESS , and U FC describe the control signals of DEG, BESS, and FC, respectively.
Finally, since the settling time is a key index in the IMG LFC scheme, it is directly used as the third objective in the proposed multi-objective function as follows, where Ts is the time taken by the system to settle at F = 0: The combined multi-objective function J in equation (14) consists of three individual functions; J 1 , J 2 , and J 3 which can be formulated as follows: The weighting factors representing each function's effectiveness and contribution in the proposed multi-objective function J are described by w 1 , w 2 , and w 3 for ISE, the control signals, and settling time, respectively. Based on the weighted sum multi-objective approach [52], the sum of all weighting factors can be signified as follows: The optimal weighting values vary depending on system dynamics, and accordingly, the network operators select the best method to adjust each controller in the LFC scheme. Equal weighting is the simplest method for determining the weighting factors. So, in this study, it is assumed that all weighting factors are the same, i.e., w 1 = w 2 = w 3 = 1/3.
The inequality constraints of the proposed multi-objective function are represented by: Regarding equation (16) symbolize the minimum and maximum gains limits for the proportional, integral, and derivative parameters of various source controllers, respectively.
Zeigler-Nicholas's ultimate tuning approach [41] is used for setting the minimum and maximum limits of the PI and PID parameters in this study.

B. NORMALIZING AND SCALING
It is essential to normalize and scale different objectives since the range of each target should be changed in the same order to apply the optimization issue correctly.

1) NORMALIZING
Comparing several cost functions is one of many multiobjective optimization strategies. Comparing values from various objectives can be challenging since they may have different units and magnitudes. As a result, it is usually crucial to transform the objective functions into similar orders of magnitude. Although there are many normalization methods, the min-max strategy is the most dependable [53], allowing all normalized targets to fall between 0 and 1. The formula is outlined as follows: (17) where F in is the normalized value of i th objective function , F i min and F i max are the minima and maximum values of i th objective. Using equation (17), the proposed multi-objective function in equation (14) is reformulated to be: where J 1n , J 2n , and J 3n are the normalized values of J 1 , J 2 , and J 3 , respectively. These normalized values can be estimated by the following formulas to become in the range of (0, 1): where J 1min , J 2min , J 3min and J 1max , J 2max , J 3max are the minimum and maximum values of J 1 , J 2 , and J 3 , respectively. Through three runs, each with 50 search agents and a total of 500 iterations, the authors have determined the minimum values of J 1 , J 2 , and J 3 . The maximum value of J 1 is determined by the open loop response, the maximum value of J 2 is determined at the maximum spinning reserve by equation (12), and finally, the maximum stable value of J 3 is taken by the simulation time.

2) SCALING
Although the normalizing method guarantees that all objective values will fall within the range (0,1), it cannot ensure that all objectives will change in the same order while still falling within this range.
For example, when J 1n = 1.5 × 10 −5 , J 2n = 2.2 × 10 −2 , and J 3n = 3.5 × 10 −2 and by normalizing them, all values are less than 1, but J 1n changes in the order of 10 −5 while J 2n and J 3n change in order 10 −2 , so J 1n must be scaled by multiplying 10 3 or dividing by 10 3 for J 2n and J 3n to be scaled. Right scaling makes it possible for optimization processes to fairly compare several objective functions, as in the previous example, when the optimization process does not disregard J 1n relative to others.
Consequently, the final formulation of the suggested objective function, derived from equation (18), will be as follows: where α, β, and γ designate the scaling factors for J 1n , J 2n , and J 3n respectively. In our study we make 10 different runs, each one with 200 iterations to trace the average change order of all objective functions, then the scaling factors are decided to be as follows α = γ = 1000, and β = 1.

IV. ARTIFICIAL RABBITS OPTIMIZATION (ARO) A. SELECTION ARO AS THE OPTIMIZATION TECHNIQUE
Traditional optimization algorithms only solve convex problems. It can transform non-convex problems into convex ones within certain approximations [54]. Metaheuristic techniques are better than approximations to convert the existing non-linear and non-convex problem to a convex one before optimizing it using conventional methods. Numerous metaheuristic methods are utilized in literature to evaluate the LFC problem in recent studies [24], [25], [26], [27], [28], [29], [30], [31], [55], [56]. Artificial rabbits optimization (ARO) is one of the most recent metaheuristic methods in 2022. According to [57], the ARO approach demonstrates the superiority of optimization over a variety of well-known metaheuristic techniques, including particle swarm optimization, differential evolution, gravitational search algorithm, cuckoo search, teaching-learning-based optimization, artificial bee colony, and atom search optimization. ARO meets the Global Search Convergence theorem. The theorem declares ARO cannot miss the global optimum. As a result, the optimization problem in this study is solved effectively via ARO. It is worth highlighting here that up to the authors' knowledge, the ARO algorithm is not utilized before in literature for tuning classical controllers.

B. ARO CONCEPT
The algorithm for ARO is developed from the nature of the survival strategies of rabbits. Rabbits have two main strategies to save their life, the first strategy is detour foraging, and the second is random hiding from enemies when they are attacked [57]. The detour foraging depends on the food types of rabbits that eat forbs, grass, and leafy weeds. To avoid the nest from being attacked by predators, rabbits do not feed the grass close to the nest holes but continually pursue foods far from their nests. Rabbits have a curiously widespread vision field, which depends on overhead scanning so they can discover foods over a wide area. The detour foraging strategy is considered exploration. This survival plan of rabbits is well known as the Chinese idiom says: ''rabbits do not eat the grass near its own nest''.
The second survival strategy of rabbits relies on a random hiding idea. As illustrated from the photo of Fig. 4, taken by Amethyst in [58], rabbits will grave many burrows around their nests to cheat enemies and to make the escape implementation easier, especially if they randomly choose one of the burrows as a shelter from predators. The bodybuilding of rabbits enables them to run fast as they have short forelegs and long back legs, strong muscles, and tendons. Moreover, rabbits can confuse their enemies by escaping in a zigzag motion, suddenly stopping while running, and sharp turning. The second strategy is considered exploitation.
The penalty of fast running is energy shrink, so rabbits need to adaptively switch between detour foraging and random hiding according to the energy.

C. ARO MATHEMATICAL MODEL
ARO hires the strategies of actual rabbits in foraging and hiding, along with their energy shrink foremost to transit between both strategies. The mathematical model of the ARO algorithm is divided into three main parts. Which are: detour foraging or exploration, random hiding or exploitation, and energy shrink constrain. The mathematical model is discussed in detail in [57].

1) DETOUR FORAGING (EXPLORATION)
In ARO, suppose that each rabbit in the group has its area with some grass and (d) burrows, and the rabbits randomly visit the position of each other for feeding. With foraging, rabbits are likely to confuse around a food source to eat sufficient food. Then, the ARO detour foraging behavior implies that each search individual tends to update its position towards the other search individual chosen randomly in the group and adds a perturbation.
The detour foraging mathematical model of the rabbits is presented by: + round (0.5 · (0.05 + r 1 )) · n 1 , i, j = 1, . . . , n and j = i (23) where v i (t + 1) is the candidate position of the i th rabbit at the time t+1, x j (t) is the position of the i th rabbit at the time t, n is the size of a rabbit population, d is the dimension of the problem, T is the maximal number of iterations, · is the ceiling function, round indicates rounding to the nearest integer, rand perm returns a random permutation of the integers from 1 to d, r 1 , r 2 and r 3 are three random numbers in the range of (0,1), l is the running length which represents the movement pace when performing the detour foraging, and n 1 is subject to the standard normal distribution. Equation (23) shows that search individuals do a random search for foraging according to the position of others. This action permits a rabbit to transfer far away from its area to the area of the other rabbits. Especially a rabbit visiting others' nests instead of their nests significantly contributes to VOLUME 11, 2023 exploration and guarantees the capability of the ARO algorithm to detect global search.

2) RANDOM HIDING (EXPLOITATION)
In ARO, at every iteration, a rabbit produces d burrows around it along each dimension of the search space randomly. It selects one of the burrows for hiding to decrease the probability of being attacked. The j th burrow of the i th rabbit is generated by: From equation (29), the d burrows are created in the neighborhood of a rabbit along every dimension. H is the hiding parameter which is linearly decreased from 1 to 1 / T with a random perturbation throughout iterations. According to this parameter, initially, these burrows are produced in a bigger neighborhood of a rabbit. As the iterations rise, this neighborhood gets reduced, too. Rabbits randomly select a burrow from their burrows for shelter. To mathematically model this random hiding strategy, the following equations are applied: where b i,r (t) represents a randomly selected burrow for hiding from its d burrows, r 4 and r 5 are two random numbers in (0, 1). From equation (33), the i th search individual will try to update its position towards the randomly selected burrow from its d burrows. After one of both detour foraging and random hiding is achieved, the position update of the i th rabbit is: This equation signifies that if the fitness of the candidate position of the i th rabbit is better than that of the current one, the rabbit will wildness the existing position and stay at the candidate position generated by either equation (23) or (33).

3) ENERGY SHRINK (SWITCH FROM EXPLORATION TO EXPLOITATION)
An energy factor is calculated to model the switch from exploration to exploitation. The energy factor in ARO is defined by the following formula, where r is the random number in (0,1): In ARO, when the energy factor A(t) > 1, a rabbit is disposed to randomly discover the areas of foraging for different rabbits in the exploration phase.
Then, the detour foraging occurs; when the energy factor A(t) ≤ 1, a rabbit is motivated to randomly exploit its burrows in the exploitation phase, therefore, the random hiding occurs.
Depending on the value of the energy factor A, ARO can change between either the detour foraging or random hiding. The energy factor A is decreased with the increase of iterations, which can strengthen each individual in the population to change between the detour foraging behavior and random hiding behavior. All the updates are efficiently realized until the termination criterion happens, and then the best solution is reimbursed. The pseudocode of ARO is described in Fig. 5.

V. RESULTS AND DISCUSSION
The studied IMG is simulated using the MATLAB environment, and the LFC system is thoroughly examined under various operational scenarios. The typical IMG shown in Fig. 1 is used to evaluate the suggested LFC scheme. The nominal values of the generating units and the system demand are listed in Table 1 [33]. Moreover, this IMG operates with  parameters listed in Table 2 [38], a significant idea is to consider a specific amount of active power generated by the several DERs as a spinning reserve and a possible supply of active power for the LFC scheme.
In this section, the proposed multi-objective tuning method for different controllers of the LFC scheme is tested under different operational cases of IMG to prove its effectiveness. The optimization is accomplished for the case of 0.1 p.u step disturbance on the IMG, at 0 seconds, without the existence of RES. Simulations are carried out to study the frequency response considering t sim = 60 s. The maximum iterations are chosen by 500, and the population size is 50 for this study.
The following suggested case studies will be extensively tested, where each case study has different load scenarios: First case study: load variations without RES availability. Second case study: load variations with the availability of RES.

A. THE EFFECTIVENESS OF ARO
At first, the performance of ARO is evaluated and compared with numerous recent optimization techniques applied in the literature for the LFC of IMG. The comparative study will include the optimization algorithms: genetic algorithm (GA), particle swarm optimization (PSO), GWO [30], WOA [31], MFO [29], MBA [27], and SMA [24]. The controllers of all controlled sources (DEG, BESS, and FC) are tuned based on the four popular error criteria: ISE, IAE, ITAE, and ITSE. Each distinct optimization algorithm has four runs that represent the four error criteria. For implementing a fair comparison among different optimization techniques, the cost function of all algorithms is equally evaluated with 100 iterations and a population size of 50. The IMG is subsequently subjected to a step load of 0.1 p.u at 0 sec, the optimal output gains of each algorithm are then applied to the controllers, and the objective functions are computed. Table 3 displays the optimized values of these common error criteria for individual algorithms. As revealed, the ARO achieves better-optimized values than others in all single objective functions. So, as ARO has approved its trusted performance, the ARO algorithm will be applied to the proposed multi-objective optimization issue in this work in all the following scenarios for tested case studies.

B. TEST CASES AND TUNING OBJECTIVES
To distinguish the proficiency of the proposed multi-objective function, described in Section III, which consists of three objectives (J 1 , J 2 , & J 3 ), the authors will evaluate tuning the controller parameters of different controlled sources using the following formulations: 1. The first formulation is the single objective J 1 (optimal controller parameters are only based on ISE).  Table 4 shows the optimally tuned gains of the four tested formulations for different sources' controllers. These optimal parameters will be applied to the system's controllers for investigating LFC. For providing a complete LFC analysis of the tested IMG, two case studies are proposed to test the system under different scenarios. The first case is with no availability of RES, while the second is with the availability of RES.

C. FIRST CASE STUDY WITH NO AVAILABILITY OF RES
In this investigated case study, the IMG is tested without the availability of RES, it is exposed to four different disturbances as follows: 1. The well-known performance indices, ISE, the overshoot (OSH), the undershoot (USH), and Ts (the settling time) are assessed to evaluate and compare various tuning formulations. It is necessary to identify the priorities of each unique performance metric in the LFC problem to judge the various performances. The system operator determines the indices' significances based on the different MG operating conditions. The system's settling time presents the priest index in the LFC of IMG. Peak deviation at the time of load disturbance (USH in step-up load) is given as the second priority. Last but not least, OSH is taken into account to ensure system performance following the moment of disturbance.      Table 5 offers the performance indices for various tuning formulations. Since all controllers consider sources' GRC, Table 5 indicates that all objectives have the same USH; hence, the formulation with the lowest OSH will settle faster. According to Fig. 6-a and Table 5, the proposed multi-objective formulation J delivers the fastest reaction with 0.0133 p.u overshoot. Using the multi-objective formulation J , Ts and OSH are improved by 45.16 and 21.2%, respectively. The system settles in 2.96 s, compared to 5.4 s of the single objective J 1 .
It is vital to notice that including Ts in the second formulation of the multi-objective (J 1 & J 3 ) directly improves Ts, but it has the penalty of high control action. It results in a larger OSH of 0.0226 p.u and a slower response than the proposed formulation J with Ts = 4.03 s. The formulation of J 1 and J 2 may adopt the system OSH after disturbance's instant to be lower than the formulation of J 1 & J 3 with OSH of 0.0145 p.u, and the settling is faster at 3.37 s by adding the control signal in the objective function. On the other hand, the proposed multi-objective formulation J loses disregarded undershoot percent and tiny ISE percent w.r.t. the single objective J 1 by −0.14 % and −4.43 %, respectively.
When the IMG undergoes 0.15 p.u step disturbance at t=1 s, the proposed multi-objective formulation J was superior to the others as shown in Fig. 6-b. Table 6 displays the proposed tuning objective performance indices w.r.t others. It settled in 3.8 s with a 15.73% improvement over J 1 . Despite tuning the controller's parameters based on J 1 (ISE), they failed to achieve minimum ISE in this scenario. The proposed multi-objective formulation J has an ISE of 0.01941, while J 1 's was 0.02623. The proposed multi-objective formulation J reduces the system's OSH to 0.1054 p.u and improves it by 25.39 %. Scenario 1, under the two different step load disturbances, reveals how a faster system's settlement with less overshoot is achieved by using the proposed multi-objective formulation J and the formulation (J 1 & J 2 ) that take into account control signals. On the other hand, a slower system settlement and more overshoots would result from using the formulations that neglect the control signals during the tuning, such as the single objective J 1 or the multi-objective formulation of J 1 & J 3 .

2) SCENARIO 2 WITH A SEVERE STEP CHANGE OF 0.2 P.U
The key purpose of this scenario is to examine the robustness of the proposed multi-objective formulation J when the system is exposed to a severe step disturbance of 0.2 p.u w.r.t other formulations. Whether the system responds faster and keeps improving or not is the main issue here.
According to Fig. 7, even after subjecting the system to a severe step of 0.2 p.u at t = 1 s, the proposed multi-objective formulation J and (J 1 & J 2 ) maintained system stability. They successfully managed the sudden and severe load by limiting the control signals to specific controllers, delivering the right amount of power, and guaranteeing system stability. In contrast, the other tuning objectives J 1 and (J 1 & J 3 ) failed to preserve the system's stability. They will subject the tuned controllers to high control signals for the controlled sources, making them unable to react sensibly enough to keep the system stable. Moreover, the objective (J 1 & J 3 ) generates more control actions than objective J 1 , so undershoots and overshoots occur later for objective (J 1 & J 3 ) compared with objective J 1 .

3) SCENARIO 3 WITH A MULTI-STEP LOAD
In this scenario, the IMG is exposed to multi-step disturbances have maxima and minima values of 0.075 p.u and −0.075 p.u, respectively, while the time difference between two successive steps is 5 s as illustrated in Fig. 8. The IMG will have two repetitive states. The first state is overfrequency when the load steps down from 0.075 p.u to −0.075 p.u (e.g., at t=5 s). The second state is underfrequency when the load steps up from −0.075 p.u to 0.075 p.u (e.g., at t=10 s). Thus, it is essential to ensure that the controllers' tuned parameters have enabled the system to settle before the next disturbance.
The performance analysis is then complete. Consequently, various tuning objectives' system responses are evaluated in Fig. 9 indicating the system responses following the overfrequency and under-frequency states, respectively. As previously discussed in Scenario 1, the peaks during the instants of disturbances initiation will be excluded from comparison since they are nearly identical. As displayed in Fig. 9, the proposed multi-objective formulation J for parameters tuning has a superior response to other formulations.  Table 7 summarizes the performance indices for such multi-step disturbance (two states of over-frequency at t=5 s, and under-frequency at t=10 s), which are the settling time, the first peak following the instant of disturbance, and the decay ratio (DR) index. It is worth noting that the DR index is a good measure of how rapidly the oscillations are decreasing. The lower DR, the rapid decay oscillation until the system settles.
The proposed multi-objective formulation J has superiority for all estimated indices over others. It could enable the system to settle with minimum times either in over or underfrequency states. The system has a minimum 1st peak in two states (1st USH and 1st OSH of over and under-frequency states after step instant) of −0.1056 p.u and 0.1048 p.u, respectively. So, it can provide the minimum setting times of 3.75 s for the over-frequency state and 3.8 s for the underfrequency state.
On the other hand, the single objective J 1 and the multiobjective formulation of (J 1 & J 3 ) provide higher control signals to different sources which significantly help the system to settle close to the time difference limit (5 seconds). VOLUME 11, 2023

TABLE 7. Performance indices in Scenario-3 (a multi-step load).
In the over-frequency condition, the system settles in 4.82 s and 4.61 s, respectively, while in the under-frequency state, it takes 4.72 s and 4.59 s, respectively.
The other multi-objective formulation of (J 1 & J 2 ) has achieved very well performances and enabled the system to settle in times almost equal to the proposed multi-objective formulation J of 3.85 s and 3.86 s, respectively.
DR index is estimated in Table 7 to indicate how quickly the oscillations are diminishing. As illustrated, the proposed multi-objective formulation J has the lowest DR values in the over-frequency and under-frequency states of 17.8 % and 18.45 %, respectively. Besides, the formulation (J 1 & J 2 ) could provide a very satisfactory value for DR of 19.87 % in the over-frequency condition and 20.83 % in the underfrequency state when the tuning process considers the control signals. Recalling the penalty of higher control actions will be for objectives that do not consider control signals in the tuning procedure, such as the single objective J 1 and multiobjective formulation (J 1 & J 3 ). They offered a terrible DR (high values) of 59.24 % and 39.39 % for the over-frequency state and 54.15 % and 38.76 % for the under-frequency state.

4) SCENARIO 4 WITH A RANDOM LOAD
In this scenario, the tested IMG is examined for severe random load conditions, as given in Fig. 10, using the block of band-limited white noise that generates normally distributed random numbers. Figure 11 displays the variations in frequency for the different investigated four objectives for parameters' tuning in this scenario of a random load. Due to the reduced inertia in FIGURE 9. Frequency deviations of the tested IMG for different tuning objectives in Scenario 3 with multi-step load changes every 5 seconds.
the system, the frequency oscillates with a larger amplitude of frequency. As the system is very sensitive to random load, it may not have any fast-responding standby to suppress the variations in frequency, which may result in system instability.
As shown, all objectives respond well till 30 seconds. The zoom region in Fig. 11 ensures that the tuned parameters based on the proposed multi-objective formulation J have the lowest frequency variations at high spikes of the random load (dashed circles in the zoom region in Fig. 11).
As shown, both the proposed multi-objective formulation J and the formulation (J 1 & J 2 ), considering the control signals in the tuning process, could maintain the system stability when the system is subjected to severe value random load after t=30 s, in contrast to the other tuning objectives which fail in preserving the system stability. Indeed, they behaved sensibly to fast oscillations of the random load by limiting the control signals for different controllers to supply sufficient power and keep the system stable.
On the contrary, the other objectives (single objective J 1 and multi-objective formulation (J 1 & J 3 ) will be subject to large control signals for various controlled sources and thus fail to respond fast during random load oscillations. Therefore, when the system is exposed to that severe random load, it cannot maintain system stability after t=30 s. VOLUME 11, 2023 A. E. Khalil et al.: Enhancing the Conventional Controllers for LFC of IMGs

D. SECOND CASE STUDY WITH AVAILABILITY OF RES
For this case study, the IMG is tested under different scenarios with RES availability (PV and wind). As shown in Fig. 12, the RESs freely supply all their available power to the IMG, while the DEG, BESS, and FC sources change their power output based on system frequency fluctuations.
RES has two distinctive impacts on the IMG's frequency. The first effect is the over-frequency when RESs supply excessive power to the IMG (e.g., at t=30 s in Fig. 12), and the second effect is the under-frequency which occurs when their available power decreases according to the weather conditions (e.g., at t=50 s in Fig. 12).
The tested scenarios will be as follows: 1. PV and WT systems are added to the tested IMG without load disturbances. 2. A multi-step load is subject to the system in addition to the availability of RES. 3. A step load is applied with the critical excessive power of RES. 4. A step load is applied with the critical shortage power of RES.

1) SCENARIO 1 WITH RESs INCLUDED WITHOUT LOAD VARIATIONS
The variations in frequency are studied in this scenario as demonstrated in Fig. 13 when adding RESs without any load variation to check the robustness of different tuning objectives to enhance the system stability.
As illustrated in Fig. 13, all objectives can enhance the system stability along different variations of RES. But for the high variations shown in the zoom areas of Fig. 13 (upper, down), the proposed multi-objective tuning formulation J is preferred over other alternatives because it settles more quickly and makes sense to assign control actions to various controllers.
Additionally, the formulation (J 1 & J 2 ) can perform better than the single objective and the formulation (J 1 & J 3 ) in larger RES swings. It is clear that the formulations depend on limiting control actions, like the proposed multi-objective tuning formulation J , and the formulation (J 1 & J 2 ) can perform better than the others in critically high RES variations and deliver acceptable responses in small RES fluctuations.

2) SCENARIO 2 WITH RESs INCLUDED AND A MULTI-STEP LOAD
A periodic multi-step load is applied in this scenario every 15 s. It has maxima and minima of 0.075 p.u and −0.075 p.u, respectively. The system's frequency responses are displayed in Fig. 14. The system exhibits repeated critical over-frequency (e.g., at t=15 and 45 s) because the load is reduced by 0.15 p.u with only a little increase in RES power.
Critical under-frequency (e.g., at t=30 s) occurs because the load is increased by 0.15 p.u with the shortage in RES power.
The multi-objective tuning formulation J that was proposed for tuning parameters has the ability to sensibly limit the control signals of multiple controllers, thereby ensuring low overshoots and undershoots after the instant of the disturbance. As a consequence, it has the benefit of being able to settle any violations in the system frequency faster than other formulations, regardless of whether the fluctuations were significant or minor.

3) SCENARIO 3 WITH A STEP-DOWN LOAD AND EXCESSIVE RESs' POWER
As revealed in Fig. 13, the first critical variation in the system frequency due to adding RESs sources arises at t=30 s, which presents the critical over-frequency state as the available RESs power is excessive. In this scenario, it is supposed to apply a step-down load at that critical instant to maximize the over-frequency problem. A step-down load of −0.1 p.u is applied at t=30 s to study the system stability.
As seen in Fig. 15, a decrease in load and the availability of surplus power from RES may cause the system's critical over-frequency to occur, which may cause instability. The proposed multi-objective formulation J and the formulations (J 1 & J 2 ) are superior as they preserve system stability. On the contrary, the single-objective  formulation J 1 and (J 1 & J 3 ) that neglect control signals during the tuning process are unable to respond appropriately because of their enlarged control actions, which causes system instability.
Moreover, the proposed objective has provided little more improvement in the overall performance than the formulation (J 1 & J 2 ), which does not consider the settling time as shown in the lower zoom area of Fig. 15.

4) SCENARIO 4 WITH A STEP-UP LOAD AND SHORTAGE OF RESs' POWER
This scenario is investigated to evaluate the stability of IMG has RES and a step-up load of 0.1 p.u is applied at t=50 s. It is assumed to maximize the critical under-frequency proposed by RESs due to their shortage in available power at the instant of t=50 s, as demonstrated previously in Fig. 13.
Again, as shown in Fig. 16, the tuned parameters based on the single objective J 1 and the formulation of (J 1 & J 3 ) can not preserve the frequency stability at the instant for applying the step-up disturbance at t=50 s. In contrast, the tuned parameters based on the proposed multi-objective formulation can keep the system stable after the step-up disturbance at t=50s, besides providing better performance than the formulation (J 1 & J 2 ).

VI. SENSITIVITY ANALYSIS (SYSTEM PARAMETERS' UNCERTAINTY)
A comprehensive sensitivity analysis is performed to evaluate how the system responds to a step change while the system parameters vary by a margin of ±25% from their nominal values. There are two distinct ways to alter the system's parameters. The first includes varying one of the following separately: the system's inertia, damping coefficient, speed regulation constant, or sources' time constants. Second, all system parameters are changed at once.
For the first way, Fig. 17 illustrates the system frequency response of controllers tuned based on the proposed multi-objectives with the individual variation of system parameters such as inertia constant (H), damping coefficient (D), speed regulation constant (R), and sources time constants. The suggested LFC system responds well to the variations in H, as seen in Fig. 17-b, and stabilizes rather closely to the scenario with nominal parameters. The suggested LFC technique would successfully stabilize the system for H fluctuations of −25% in less than 1 s after the case of nominal parameters has stabilized. As demonstrated in Fig. 17-b and Fig. 17-c, peak OSH, and USH are quite similar. It reveals how well the system adapts to variations in R and D and how it is capable of settling simultaneously. No matter what adjustments are made to the time constants of the sources as shown in Fig.17-d, the system response does not cause any considerable differences in the settling time compared to the nominal case. The peak OSH and USH also closely approximate one another.
For the second way, Fig. 18 displays how the suggested LFC scheme behaves when the system's parameters are altered by ±25% at once. As can be observed, changing the system's parameters by a value of −25% would result in a noticeable change in how it responds. Even yet, the established LFC technique has the capacity to stabilize the system and settle it in less than two seconds after the nominal case has been settled. Moreover, the system's behavior differs slightly depending on a +25% change in system parameters related to the situation of nominal parameters. VOLUME 11, 2023 FIGURE 17. Frequency deviations of the tested IMG for different tuning objectives in case of changing the system parameters separately.

FIGURE 18
. Frequency deviations of the tested IMG for different tuning objectives in case of the system parameters are changed at once.

VII. CONCLUSION
A novel multi-objective tuning approach is introduced to optimize IMG's LFC's traditional PI and PID controller parameters. The proposed multi-objective tuning uses controller control signals, settling time, and the integral square of frequency deviation to determine the optimal gains of various controllers used in the LFC scheme of a low inertia IMG. It considers anti-windup and practical generation rate limits for different sources.
At first, the recent ARO algorithm has approved its superiority among several optimization algorithms based on the conventional single-criterion tuning formulations: ISE, IAE, ITAE, and ITSE. Consequently, the proposed multi-objective formulation is applied based on ARO to tune the controllers.
For distinct IMG operating scenarios, the proposed multi-objective formulation for controller tuning is extensively compared to three others. The three formulations were ISE-based single tuning, ISE with control signals coming from different controllers, and ISE with settling time. The frequency stability of the IMG is evaluated in two extensive case studies, one without RES and one with RES.
In the absence of RES, the IMG is tested under four load scenarios: a step, a severe step disturbance, a multi-step, and a severe random disturbance. ISE, OSH, USH, and settling time performance indicators are assessed to examine frequency responses.
− The proposed tuning approach outperformed other tuning formulas in OSH and the settling time under a 0.1 p.u step disturbance.
− It enhanced all performance metrics with a high step disturbance of 0.15 p.u.
− It provided the best settling time and fastest decay oscillations for periodic multi-step disturbances.
− It could maintain system stability for the severe step of 0.2 p.u and random disturbances, unlike those that do not consider control signals. In the presence of RES, IMG stability is tested under four scenarios: no load disturbance, multi-step, step-up, and stepdown disturbances.
-In the first and second scenarios, the proposed multiobjective tuning was better than other tuning objectives for both small and large fluctuations. -During critical RES excess and shortage, the third and fourth scenarios use step-up and step-down disturbances. The proposed multi-objective formulation preserved system stability. On the contrary, the formulations that neglect control signals during tuning cannot respond effectively, resulting in system instability Finally, sensitivity analyses of the proposed LFC technique are carried out for ±25% system parameter change. The suggested multi-objective formulation keeps the system stable when system parameters change.
Notably, future research will investigate the viability of integrating the proposed multi-objective tuning in more sophisticated controllers like fractional-order PID and cascaded loop controllers.