# An FRLQG Controller-Based Small-Signal Stability Enhancement of Hybrid Microgrid Using the BCSSO Algorithm

Authors: Ginbar Ensermu; M. Vijayashanthi; Merugu Suresh; Abdul Subhani Shaik; B. Premalatha; G. Devadasu
Journal: Journal of Electrical and Computer Engineering
 (2023)

---

## Abstract

The development of a network termed microgrid (MG) has been motivated owing to augmentation in renewable energy source (RES) infiltration along with the utilization of enhanced power electronic technologies. Recently, more popularity has been gained by the hybrid MG (HMG). Maintaining the power system’s (PS) small-signal stability (SSS) is highly complicated during the energy enhancement of RES. The enhancement of the SSS has been focused on by numerous existing methodologies; however, the optimal solution was not obtained by those methodologies. A new controller with the assistance of bell-curved squirrel search optimization (BCSSO) is proposed to address the aforementioned issue. Initially, for PSs such as photovoltaic (PV), wind turbines, along with fuel cells, a mathematical model is ascertained. Then, in this, the converter design has been developed. The PV’s maximum power flow is recognized by maximum power point tracking (MPPT) in the bidirectional switched buck-boost converter (BSBBC), which is utilized in this research, and by utilizing the fuzzy ruled linear quadratic Gaussian (FRLQG), the converters are controlled to assure safe operation along with soft dynamics. By employing the BCSSO, the parameters are modified in this controller which in turn ameliorates the SSS. The experiential evaluation of the proposed system’s performance is analogized with the existing methodologies. Consequently, the outcomes confirmed that a better performance was attained by the proposed methodology than the prevailing works.


---

## Body

## 1. Introduction

Owing to the rise in the utilization of DC sources along with loads, the survey on DC MGs has attracted more attention in the past few years. A group of distributed generators (DG), energy storage systems (ESS), local controllable along with noncontrollable loads are together combined into local distribution systems called MG [1]. Through various kinds of power electronics converters (PEC), all of the above said are joined to a common DC bus [2]. AC-MG, DC MG, along with AC/DC MGs are the “3” types of MGs. It assists in the integration along with the application of large-scale RES and is considered a small PS [3]. The prevailing grid infrastructure, safety, and technologies are incorporated into the AC-MG, and it is a highly adapted configuration. The disadvantages are the necessity for the synchronization of DG and losses due to reactive power circulation [4]. The DC MG has the features of excessive conversion efficiency along with smaller line loss when correlated with the AC-MG [5]. AC along with DC in the same distribution grid is combined into a hybrid AC/DC MG in which the direct amalgamation of AC- and DC-centric renewable energy is provided. The conversion stages along with the energy losses are reduced by HMG. A minimal quantity of interface elements is used devoid of the synchronization of generation along with storage units, as they are directly linked to the AC/DC network [6]. Normally, regarding the technical, economic, and environment is unclear. Please rephrase it for clarity., DG is beneficial [7]. The small-signal oscillations are caused in the distribution systems (DS) owing to suboptimal sizing, inappropriate type, and the improper location of renewable energy hybrid DGs (REHDG) such as wind as well as PV [8]. Irregular clean energies such as wind along with solar energy are extensively employed, and regarding the distribution of time and resources, these energies are more supportive [9]. The main sources of energy for the traditional power grid are fossil fuels, and it causes the release of carbon dioxide along with dust, alterations in the climates, and the ceaseless exhaustion of fossil fuels [10]. The stability along with control of MG has been a significant topic owing to the augmentation in the exploitation of RES [11, 12].An essential tool for analyzing the PS’s performance when they are subjected to minor disturbances is known as small-signal analysis. Individual subgrids and interlinking converters (ILC) play a big role in the stability of an HMG. Consequently, the significance of converter control methods should not be ignored [13]. Subsequently, the solution for instability is sought by the control scheme [14]. Various forms of stability analysis have received a lot of attention. The influences of control system architectures on stability analyses are unquestionable [15]. Either eigenvalue techniques or frequency domain procedures are utilized to appraise the system’s stability. As a result, a guarantee can be given to SSS only, and owing to the existence of large-signal disruptions, system stability cannot be assured [16]. In several MG studies, owing to the significance of stability, the power converter controllers’ design that are concentrated on preserving and/or increasing the system stability has turned into a focal point [17]. There are numerous literature works on frequency control that are centered on the heuristic optimization method, and they predominantly examine the traditional thermal, hydro-PSs which contain load variation disturbance [18]. The ideas and practices employed for the traditional PSs are insufficient for the MGs along with PSs’ stability evaluation with a higher level of linked DGs [19]. So, this paper provides new ways to enrich the MHG’s stability and to unravel the inconvenience and challenges regarding the stability.The paper’s remaining part is structured as follows: the prevailing work pertinent to the SSS is examined in Section2, the proposed system is discussed in Section 3, with the prevailing methodologies, the proposed methodology’s performance is analyzed in Section 4, and the paper’s future work is listed in Section 5.


## 2. Related Work

Haseena Kuttomparambil Abdul khader [20] used a metaheuristic hybrid algorithm to present a Type-2 fuzzy fractional-order PID (T2FFOPID) centered PSS, which promoted the power device’s electromechanical oscillation damping performance to enrich the dynamic steadiness. In opposition to a huge variety of device fluctuations found in the MG, the massive bandwidth, reminiscence impact, along with flat phase contribution in the frequency reaction of FOPID were formulated to craft the controller and were executed nicely. From a hybrid algorithm, the amalgamation of a dynamic genetic with a bacteria foraging approach, initially, and the T2FFOPID’s parameter tuning trouble were changed  [21]. Via nonlinear time domain simulations along with performance indices underneath variant disturbances along with working criteria, the controller’s functionality in the lower-frequency oscillations within the device having numerous MG penetration ratios was examined. The goal feature of the system was that the integral of the time-weighted absolute of the error (ITAE) had a superior convergence time, and hence, it could have an impact on the device’s performance.Yi Zhang et al. [22] facilitated a network of AC/DC MGs in an adaptively coordinated control strategy (CS) to enrich the system’s frequency along with DC voltage, and at the same time, the power-sharing is maintained properly. Initially, the converters linking the AC and DC MGs were produced with the help of a CS on the grounds of a synchronverter along with a virtual DC machine. Except for the power-sharing controller, it contains an adaptive virtual governor along with a virtual inertia regulator. Then, the AC/DC MGs’ SSS’s evaluation with the offered CS was made to instruct the structure and, in addition, to select the control parameters. Lastly, the technique improved the frequency/DC voltage nadir along with the device’s powerful performance, which could be seen through the simulation and investigational outcomes. The optimal values of those parameters were not considered by the adaptive virtual inertia along with a virtual governor gain CS, and so, good performance was not assured [23].Pankaj Negi et al. [24] analyzed an approach for constructing the control parameters to a voltage source converter and to amplify the steadiness of PV generation in diverse grid capacities, and this strategy is termed as model reference adaptive control (MRAC). A proportional resonant (PR) controller was employed to construct a better damping approach called MRAC, whose gain parameters were adjusted using the ant lion optimization (ALO) algorithm. The LCL filter, PR current, and grid-connected PV, controllers along with their parameters, particularly proportional as well as resonant gains, were primarily concentrated. Additionally, to appraise the controller’s functional situation for diverse gain parameters, the phase margin along with the controller transfer functions’ gain was developed using an abode plot. Moreover, for optimizing the controller’s parameters and stabilizing the device with zero steady-state error, a vast variety of control approaches were made compulsory. With the prevailing controllers, the acquired outcomes and the consequences of the enhanced MRAC approach were compared [25, 26]. The enhanced MRAC strategy’s outcomes were provided, and its efficacy was correlated with certain prevailing controllers. Due to its long run time, local optima stagnation, together with premature convergence, and the ALO algorithm could damage the system’s performance.Fan Feng et al. [3] targeted to augment the steadiness of the dual active bridge (DAB) converter-centered ESSs by developing two impedance compensation techniques. Initially, by considering the steadiness of DAB-based ESSs, the layout of the DAB converter’s standard feedback controllers was revisited. To resolve the unsteadiness deficiency of the DAB-centric ESS, the optimized design procedures for the DAB feedback controller were used, which did not need any extra passive components or active control loops [27, 28]. Next, to alter the DAB converter’s input impedance, “2” types of impedance shaping regulators (ISRs), such as the bus voltage-centric ISR (BV-ISR) and the bus current-centered ISR (BC-ISR), were utilized. Ultimately, the investigational outcomes confirmed the competence of the optimized DAB controller along with ISRs. But, under all load circumstances, the performance was not verified.Hossein Shahsavari and AlirezaNategh [29] developed an optimal fractional-order PID controller to expand the probabilistic SSS of the power networks by contemplating the variability of system operating situations while analyzing the consequences of higher infiltration of PV power plants using mathematical techniques. This paper exhibited the principle modules as well as the mathematical illustration of the larger-scale PV generation included with the single-machine infinite-bus PS. Next, an efficient methodology was introduced which revamps the PS stabilizers (PSS) and was robust enough to restrain electromechanical oscillations in power devices that integrate a random PV power [30, 31]. So, this led to the construction of a vigorous PSS centered on the controller’s hybridization along with a nondominated sorting genetic algorithm (NSGAII). The NSGAII-based controller’s performance was appraised underneath several solar irradiances, temperature settings, together with disturbances. As of the simulation outcomes, it was evident that the system was utilized for developing indispensable controllers for larger scale PV power plants. Only for certain circumstances, the system stabilized the signals [32].Hassan Pourvalisouraki et al. [33] enforced a DC- AC HMG system to achieve stable responses in steady-state and dynamic operations, by devising a diverse control approach. To construct the controller, the small-signal linearization (SSL) technique and the direct Lyapunov method were merged. Different from other existing Lyapunov-centered methods, the controller’s the steady-state components were derived using SSL-based mathematical terms [34, 35]. The various contributions to the control method’s structure are (a) controlling the PCC voltage frequency along with DC-link voltage errors directly; (b) more efficient DC/AC converter current decoupling features; and (c) the bidirectional DC/DC converter can be supervised directly by the input voltage error [36]. Lastly, relative MATLAB/simulink simulation outcomes were offered to publish the superiority of the prevailing controller united with SSL underneath AC/DC load variations, storage unit’s discharging/charging, and the sharing of AC loads amongst grid sides together with DC sources [37]. In the Lyapunov theory, the initial polynomial functions were very intricate, which could impact the system’s running time.


## 3. Small Signal Stability Enhancement Using the Novel Controller with the Evolutionary Algorithm

For distributed renewable energies linking to the utility grid, the MG is considered an effectual conception. Nevertheless, the study of MG has faced numerous challenges. The instability is the major issue in the HMG. A novel evolutionary algorithm centered on SSS advancement in DC-AC HMG is proposed here to conquer this complication. Firstly, the hybrid energy resources’ mathematical model is procured. Then, to control the converter’s operation, the BSBBC is designed; in addition, the FRLQG controller is utilized in this methodology. The BCSSO, which ameliorates the system’s stability, is utilized to tune the controller’s parameters. Figure1 shows the proposed methodology’s block diagram.Figure 1
Block diagram for the proposed research methodology.


### 3.1. Mathematical Model of Distributed Renewable Energies

Initially, for wind turbines, PV arrays, batteries, along with fuel cells, the mathematical model is recognized.


#### 3.1.1. Mathematical Model for PV Arrays

Sunlight is transmuted into electricity by the PV system. The photovoltaic cell is the PV system’s fundamental component. The PV cell’s characteristics are identical to the normal diode in the dark. The electrons are set free, whilst the cell is hit by sunlight with energy higher than the semiconductor energy gap; thus, an extensive current flows in the external circuit. PV cells are grouped into modules, encapsulated as of the front, and backed by a metallic panel for protection since they are fragile along with lower voltage. PV’s mathematical formulation is as follows:(1)CU,S,A=Cph−CoeA+CRs/nAth−1−A+C⋅RsRsh=Cph−CD−Csh,Cph=Cpho⋅SSnom,CphU=Cph+KoU−Umeas,Ko=CphU2−CphU1U2−U,Co=CscU1.UU13/n⋅e−Eg/Ps1/U1U−1,CoU1=CSCU1eqAOCU1/nkU1−1,RsU=−dAdCAOC−1CoU1⋅q/nkU1⋅eqAOCU1/nkU1,Rsh=AOCCph−CoeqAOC/nkUmeas−1,RshU=Rsh⋅UUmeasα,where the photo-generated current in amperes is specified as Cph, the photo-generated current at the nominal radiation is signified as Cpho, the diode dark saturation current is denoted as Co, the diode dark current is represented as CD, the shunt current is illustrated as Csh, the series resistance is indicated as Rs, the shunt resistance is defined as Rsh, the solar radiation in w/m2 is proffered as S, the Snom is the radiation, the photo voltaic module standardized at the ideality factor is denoted as e, the electron charge is indicated as k, Boltzmann’s constant is specified as Pg, the semiconductor energy gap is denoted as K0, the short-circuit current temperature coefficient is represented as q, the electron change is defined as Es, the number of cells in series is signified as Es, the short-circuit current is illustrated as Csc, the open-circuit voltage is defined as AOC, and the thermal voltage is proffered as Ath, which is expressed as follows:(2)Ath=nkUce.


#### 3.1.2. Mathematical Model of Wind Turbine

The wind turbine-generated power relies on wind speed. The algebraic sum of gust wind speed (WS), base WS, and noise WS, along with ramp WS is termed the wind. The wind turbine’s mechanical power is as follows:(3)Mm=0.5ρWaPCp¯vsw3,where the mechanical power in watts is specified as Mm, the air density kg/m3 is signified as p, the swept area m2 is denoted as Wa, the turbine power coefficient is represented as PCp¯, and the WS m/s is indicated as vsw. The turbine conversion efficiency is represented by the power coefficient. If the pitch angle B=0, then the tip speed function is denoted as PCp¯, and the turbine’s conversion efficiency is indicated as δ and is expressed as follows:(4)PCp¯δ=c1c2δ−c4e−c5/δ+c6δ,where(5)δ=ωBrvsw,where the coefficients are given as c1,c2,c4,c5, and c6, the rotational speed rad/s is indicated as w, and the blades’ radius is specified as Br.


#### 3.1.3. Mathematical Model for Fuel Cells

The fuel cell is a device in which electricity is generated as of the chemical reaction of hydrogen along with oxygen. The fuel cells’ mathematical model is as follows:(6)FOC=kcFn,CT0=zHLIH2+IO2Lhe−ΔG/LU,Vol=LuzγH,where the Nernst voltage is defined as Fn, the voltage constant at nominal conditions is indicated as kc, the number of moving electrons is represented as z, the energy values are symbolized as H,L, the partial pressure of hydrogen is explained as IH2, the partial pressure of oxygen is represented as Io2, Plank’s constant is defined as h, the size of the activation barrier is signified as ΔG, the temperature is represented as U, and the charge transfer coefficient is specified as γ.


#### 3.1.4. Battery System Modelling

When the system has power deficiency, batteries that store surplus power in the HPS are utilized. The number of times the battery is charged/discharged in a day specifies the battery’s lifetime.(7)BTT=C+Mm+FOC−CCl¯,where the battery is indicated as BTT and the hybrid DC and AC dynamic load is denoted as CC1¯. The remaining power is stored in the battery following the AC and DC load.Figure2 shows the HMG structure. In this, AC loads are linked in the AC bus; similarly, in the DC bus, the battery PV, DC loads, along with fuel cells are connected.Figure 2
Structure of a hybrid microgrid.


### 3.2. Proposed Algorithm

In this section, the electrical energy flow is processed along with a controlled converter by providing voltages and currents in a way, which optimally suits the loads. The BSBBC (bidirectional converter) is utilized here. It has the potency to obtain the required output voltage in both directions of power flow by boosting the input voltage in a single stage. For the intermediate network betwixt lower voltage DC and the converter leg, “1” capacitor, “2” relays, “1” inductor, “3” active switches, and “1” diode are utilized here. On the AC along with the DC bus energy status, the direction of power flow is dependent upon, and also, it is centered on the direction of power flow switches that are being operated. In the AC bus as well as the DC bus, the frequency of deviation in energy status is utilized to be lesser; thus, the switches’ one part is employed as relays. Higher-frequency control switches are considered as the other part of the switches. These switches are functional regarding the shoot through duty ratio to control the voltage gain in the inverting mode along with the rectifying mode. To discover the maximum power as of the PV module under variable temperatures along with shading conditions, there is a necessity for the development of the MPPT. Furthermore, the FRLQG controls all the converters to assure safe operation along with soft dynamics. It is an amalgamation of fuzzy rules and linear quadratic control. In the linear quadratic control, the problem of poor tracking performance may be caused owing to the consumption of extra power; thus, the fuzzy rules are utilized in this methodology regarding which the controller is executed. The main objective is to minimize the operating cost of the consumed energy by a microgrid. The algorithm was developed to find the optimal operating set-points for which a power system control is required to maintain a continuous balance between power generation and load demand. The load frequency controller and automatic voltage regulator play an important role in maintaining constant frequency and voltage in order to ensure the reliability of electric power. In order to improve the performance and stability of these control loops, proportional-integral-derivative (PID) controllers are normally used. But these fixed gain controllers fail to perform under varying load conditions and hence provide poor dynamic characteristics with a large settling time, overshoot, and oscillations. In order to achieve better dynamic performance, system stability, and sustainable utilization of generating systems, controller parameters must be well-tuned. In this paper, evolutionary algorithms are proposed to find the optimum parameters of the controller to control the voltage and frequency of the generating system within the permissible limit. These algorithms offer a more stable and faster convergence towards the best parameters and also with minimum computational time. These algorithms are appropriate to model the uncertainties found in the power demand and improve the flexible nature of the controllers.The plant’s state-space model is(8)Xt=OmXt+Imυt+ξ1at,(9)Yt=OmXt+ξ2a¯t.Here,X communicates to the state vector, the contribution aimed at the framework is signified as υ, and the deliberate output as of the ideal plant is defined as Y, together with a specifies the Gaussian repetitive disturbance following up on the framework; the system matrix, the input matrix, and the output matrix are represented as Sm, Im, and Om; the noise matrices are denoted as ζ1 and ζ1. In this, to control the battery charging/discharging current, the fuzzy logic is implemented. Xt and Yt are the parameters estimated by a set of fuzzy rules of the form:IfΔOm is ωi and ΔΔOm is ωi, then(10)Xt=Ji,Yt=Qi,where ϖi, ωi, Ji, and Qi are fuzzy sets on the respective supporting sets.The fuzzy membership function(MF) sets are offered as triangular partitions in the FRLQG; positive small (PS), positive big (PB), negative big (NB), zero (ZO), and negative small (NS) are the partitions. To ameliorate the SSS and to augment the methodology’s performance, the controller’s parameters are tuned before controlling the power flow. The BCSSO algorithm is utilized here. Southern flying squirrels’ dynamic foraging behavior through gliding is imitated by the squirrel search optimization (SSO) algorithm. It is also an effective approach utilized by small mammals for traveling long distances in the deciduous forests of Europe as well as Asia. Acorn nuts are effortlessly found by them to meet their everyday energy requirements. The following phases illustrate the SSO mathematically regarding the flying squirrels’ food foraging strategy. In general, SSO, regarding the uniform distribution, the random value is chosen which devalues the performance because numerous chances are there to choose the worst optimal resolution at this random selection. Thus, a bell-shaped constant centered on the Cauchy distribution function is utilized in this research. Parameter initialization, random initialization of flying squirrel locations, sorting, declaration, random selection, generating new locations, and inspecting seasonal conditions are the steps included in this algorithm. In this, the squirrels are regarded as the controller parameters.


#### 3.2.1. Initialization of the Parameters

The SSA major parameters are the population sizePs, the maximum number of iteration mai, the predator presence probability pyi, the number of decision variables X, the gliding constant Φ, the scaling factor sf, along with the upper and lower bounds for the decision variable are ψu as well as ψ1. At the beginning of the SSA procedure, these parameters are set.


#### 3.2.2. Random Initialization of Flying Squirrel’s Locations

In a forest, there arenn flying squirrels (η); thus, the squirrels are regarded as the input parameters. The ith flying squirrel’s location on the ith tree is specified by a vector,(11)ηi,j=ηi1,ηi2,ηi3,..........,ηid,i=1,2,3,......,nn,where the ith flying squirrels of the jth dimension are denoted as ηij and the number of dimensions is indicated as d. After that, in the search space, the flying squirrels’ locations are initialized randomly as follows:(12)ψi,j=ψl+Bs∗ψu−ψl,i=1,2,.....,Ps,j=1,2,......,nn,wherein a bell-curved number is specified as Bs and this is derived by utilizing the Cauchy distribution function. Next, for the optimized controller, the fitness function (FF) is analyzed. The FF is regarded as the maximization of the SSS along with the minimization of the quadratic cost functions (QCFs). The enhancement of SSS’s objective function is as follows:(13)maxκi,where the SSS term is denoted as κi. The optimal LQG design starts with a model in the form of equations (18) and (19). In this, the process noise is described as at, and the measurement noise is illustrated as a¯t. In the optimal LQG, the feedback control is built to mitigate the following QCF,(14)QCF=limT⟶∞EV1T∫0TXtTFXt+vtTRRvt.dt.Here, symmetric weighting matrices are specified asF≥0 and RR≥0, and the expected value is denoted as EV⋅. To limit the control inputs’ states along with size, the terms XTFX and vTRRv are analogized to a prerequisite in (14). In the cost function, the selection of the matrices F and RR relies on the system’s required performance. The robustness problems against the plant uncertainty are not addressed directly by the controller’s design. The LQG’s robustness is leaned by the noise’s considerable decision regarding the plant model equations (18) and (19), which reflects the required bandwidth along with robustness characteristics. In this, mitigating the tracking error among the command signal along with the gauged output is the control objective. To execute this, an IC is related to an SN system owing to its phenomenal lower-frequency tracking operation. Tracking errors are reduced by the higher gain of the IC even though it displays a lower bandwidth, which proves the robustness insufficiency. Therefore, f=f1,f2,…,fPs denotes the fitness value (FV) derived. An individual flying squirrel’s location’s FV is gauged by substituting the decision variables’ value into a FF:(15)fi=fiψi,1,ψi,2,…,ψi,nn,i=1,2,…,Ps.


#### 3.2.3. Sorting, Declaration, and Random Selection

All the FS position FVs are sorted in ascending order. It is observed that the squirrel with the minimal FV is on the hickory nut tree (optimal food source), the next “3” best flying squirrels are on the acorn tree (normal food source), and the remaining squirrels are on the normal trees (no food source). Moreover, by executing randomized selections, some FSs are believed to fly toward the direction of the hickory nut tree concerning that the squirrels have got satisfied with their daily energy requirements. The remaining FSs are glided towards the acorn nut tree.


#### 3.2.4. Generation of New Locations

There occurs “3” situations in the nonexistence of a predator at the time of the dynamic forging of flying squirrels. In this, the flying squirrel glides together with searches all over the forest for its favorite food. However, the flying squirrel is watchful; in addition, it is forced to take a small random walk to search for the closest hiding location. The dynamic foraging behavior “3” mathematically modeled cases are given as follows:(a)
Flying squirrels on acorn nut trees (ψat) may move towards the hickory nut tree. Here, the squirrels’ new location is acquired as follows:(16)ηctt+1=ηctt+gld.lsc.ηktt−ηott,if,r1≥ppp,ranl,otherwise.Here, the flying squirrel location on the acorn tree att and t+1 iteration is specified as ηctt and ηctt+1, the gliding distance is indicated as gld, the sliding constant is represented as lsc, the flying squirrel on the hickory tree is indicated as ηkt, the flying squirrel on a normal tree is described as ηott, the random number in the range of [0, 1] is denotes as r1, the predator presence probability is signified as ppp, and the random location is represented as rani.(b)
To accomplish their daily energy requirements,ψnt may move towards acorn nut trees. The squirrels’ new location is updated as follows:(17)ηott+1=ηott+gld.lsc.ηctt−ηott,if,r2≥ppp,ranl,otherwise.Here, the flying squirrels’ location on the normal tree att and t+1 iteration is represented as ηott+1, and the random number 2 is indicated as r2, which existed in the range of [0, 1].(c)
To store the hickory nuts for consuming them when there is a shortage of food, few FS as of the normal trees, which have already consumed acorn nuts, might glide in the hickory nut tree’s direction. Here, the FS updated positions are obtained utilizing the following equation:(18)ηott+1=ηott+gld.lsc.ηktt−ηott,if,r3≥ppp,ranl,otherwise.Here, the random number 3 presented in the range of [0, 1] is represented asr3.


#### 3.2.5. Check Seasonal Condition Monitoring

The flying squirrel foraging activity has got affected significantly by seasonal changes. Therefore, the changes in weather affect the flying squirrels movement, and a highly realistic approach towards optimization is provided by the addition of such behavior. Consequently, the solution is prevented as of the local optimal solutions by monitoring the seasonal condition. Initially, the seasonal constantseac is computed as follows:(19)seact=∑j=1dηct,jt−ηkt,jt2,where the seasonal constant at iteration t is specified as seact, the squirrel on the hickory and acorn trees at jth dimension is signified as ηkt,jt and ηct,jt, and the total dimension is represented as d. Subsequently, in the 2nd step, the seasonal monitoring criterion is checked by seact<seamin in which the seasonal constant minimum value is represented as seamin, and it is calculated as follows:(20)sea10e−6365t/tmax/2.5min,where the maximum number of iterations is indicated as tmax. The end of the winter season is found if the seasonal condition is true; thus, the flying squirrels, which lose their potency to travel around the forest, will randomly transfer their searching positions for food sources over again.(21)ηott+1=η↓+evy×η↑−η↓,where the levy distribution function is denoted as evy. It promotes better search space exploration and is computed as follows:(22)evy=0.01×τa×στb1/ε,where the “2” normally distributed random numbers in the range [0, 1] are represented as τa and τb, a positive constant is defined as ∂, which is lesser than 2, and σ is computed as follows:(23)σ=Γ1+ε×sinπε/2Γ1+ε/2×ε×2ε−1/21/ε,where the factorial function is denoted as Γ (Algorithm1).


#### 3.2.6. Pseudocode for the Proposed BCSSO Algorithm

Algorithm 1: Bell-curved squirrel search optimization (BCSSO) algorithm.
Input: parameters (t) and Y (t)Output: optimal parametersBeginInitialize location ηij = (ηi1, ηi2, ηi3……….ηid) seasonal constants seact and maximumIteration MaxuDefine initial location of each flying squirrel by,ψij = ψl + Bs ∗ (ψu − ψl)Calculate fitness functionSet iteration IIt = 1While (IIt ≤ Maxit) doGenerate locations using 3 locationsScenario 1:If (r1 ≥ ppp); {Update the squirrel locationηctt+gld.lscηktt−ηott} else {ηctt+1=ranl}end ifScenario 2If (r2 ≥ ppp); {Update the squirrel locationηott+gld.lscηctt−ηott} else {ηott+1=ranl}end ifScenario 3If (r3 ≥ ppp); {Update the squirrel locationηott+gld.lscηktt−ηott} else {ηott+1=ranl}end ifPerform seasonal monitoring by using a seasonal constantCalculate fitness functionSet IIt = IIt + 1End whileReturn optimal parametersThe BCSSO’s pseudocode is demonstrated. The parameter initialization, fitness analysis along with the updating process for the squirrel locations are explicated.


## 3.1. Mathematical Model of Distributed Renewable Energies

Initially, for wind turbines, PV arrays, batteries, along with fuel cells, the mathematical model is recognized.


### 3.1.1. Mathematical Model for PV Arrays

Sunlight is transmuted into electricity by the PV system. The photovoltaic cell is the PV system’s fundamental component. The PV cell’s characteristics are identical to the normal diode in the dark. The electrons are set free, whilst the cell is hit by sunlight with energy higher than the semiconductor energy gap; thus, an extensive current flows in the external circuit. PV cells are grouped into modules, encapsulated as of the front, and backed by a metallic panel for protection since they are fragile along with lower voltage. PV’s mathematical formulation is as follows:(1)CU,S,A=Cph−CoeA+CRs/nAth−1−A+C⋅RsRsh=Cph−CD−Csh,Cph=Cpho⋅SSnom,CphU=Cph+KoU−Umeas,Ko=CphU2−CphU1U2−U,Co=CscU1.UU13/n⋅e−Eg/Ps1/U1U−1,CoU1=CSCU1eqAOCU1/nkU1−1,RsU=−dAdCAOC−1CoU1⋅q/nkU1⋅eqAOCU1/nkU1,Rsh=AOCCph−CoeqAOC/nkUmeas−1,RshU=Rsh⋅UUmeasα,where the photo-generated current in amperes is specified as Cph, the photo-generated current at the nominal radiation is signified as Cpho, the diode dark saturation current is denoted as Co, the diode dark current is represented as CD, the shunt current is illustrated as Csh, the series resistance is indicated as Rs, the shunt resistance is defined as Rsh, the solar radiation in w/m2 is proffered as S, the Snom is the radiation, the photo voltaic module standardized at the ideality factor is denoted as e, the electron charge is indicated as k, Boltzmann’s constant is specified as Pg, the semiconductor energy gap is denoted as K0, the short-circuit current temperature coefficient is represented as q, the electron change is defined as Es, the number of cells in series is signified as Es, the short-circuit current is illustrated as Csc, the open-circuit voltage is defined as AOC, and the thermal voltage is proffered as Ath, which is expressed as follows:(2)Ath=nkUce.


### 3.1.2. Mathematical Model of Wind Turbine

The wind turbine-generated power relies on wind speed. The algebraic sum of gust wind speed (WS), base WS, and noise WS, along with ramp WS is termed the wind. The wind turbine’s mechanical power is as follows:(3)Mm=0.5ρWaPCp¯vsw3,where the mechanical power in watts is specified as Mm, the air density kg/m3 is signified as p, the swept area m2 is denoted as Wa, the turbine power coefficient is represented as PCp¯, and the WS m/s is indicated as vsw. The turbine conversion efficiency is represented by the power coefficient. If the pitch angle B=0, then the tip speed function is denoted as PCp¯, and the turbine’s conversion efficiency is indicated as δ and is expressed as follows:(4)PCp¯δ=c1c2δ−c4e−c5/δ+c6δ,where(5)δ=ωBrvsw,where the coefficients are given as c1,c2,c4,c5, and c6, the rotational speed rad/s is indicated as w, and the blades’ radius is specified as Br.


### 3.1.3. Mathematical Model for Fuel Cells

The fuel cell is a device in which electricity is generated as of the chemical reaction of hydrogen along with oxygen. The fuel cells’ mathematical model is as follows:(6)FOC=kcFn,CT0=zHLIH2+IO2Lhe−ΔG/LU,Vol=LuzγH,where the Nernst voltage is defined as Fn, the voltage constant at nominal conditions is indicated as kc, the number of moving electrons is represented as z, the energy values are symbolized as H,L, the partial pressure of hydrogen is explained as IH2, the partial pressure of oxygen is represented as Io2, Plank’s constant is defined as h, the size of the activation barrier is signified as ΔG, the temperature is represented as U, and the charge transfer coefficient is specified as γ.


### 3.1.4. Battery System Modelling

When the system has power deficiency, batteries that store surplus power in the HPS are utilized. The number of times the battery is charged/discharged in a day specifies the battery’s lifetime.(7)BTT=C+Mm+FOC−CCl¯,where the battery is indicated as BTT and the hybrid DC and AC dynamic load is denoted as CC1¯. The remaining power is stored in the battery following the AC and DC load.Figure2 shows the HMG structure. In this, AC loads are linked in the AC bus; similarly, in the DC bus, the battery PV, DC loads, along with fuel cells are connected.Figure 2
Structure of a hybrid microgrid.


## 3.1.1. Mathematical Model for PV Arrays

Sunlight is transmuted into electricity by the PV system. The photovoltaic cell is the PV system’s fundamental component. The PV cell’s characteristics are identical to the normal diode in the dark. The electrons are set free, whilst the cell is hit by sunlight with energy higher than the semiconductor energy gap; thus, an extensive current flows in the external circuit. PV cells are grouped into modules, encapsulated as of the front, and backed by a metallic panel for protection since they are fragile along with lower voltage. PV’s mathematical formulation is as follows:(1)CU,S,A=Cph−CoeA+CRs/nAth−1−A+C⋅RsRsh=Cph−CD−Csh,Cph=Cpho⋅SSnom,CphU=Cph+KoU−Umeas,Ko=CphU2−CphU1U2−U,Co=CscU1.UU13/n⋅e−Eg/Ps1/U1U−1,CoU1=CSCU1eqAOCU1/nkU1−1,RsU=−dAdCAOC−1CoU1⋅q/nkU1⋅eqAOCU1/nkU1,Rsh=AOCCph−CoeqAOC/nkUmeas−1,RshU=Rsh⋅UUmeasα,where the photo-generated current in amperes is specified as Cph, the photo-generated current at the nominal radiation is signified as Cpho, the diode dark saturation current is denoted as Co, the diode dark current is represented as CD, the shunt current is illustrated as Csh, the series resistance is indicated as Rs, the shunt resistance is defined as Rsh, the solar radiation in w/m2 is proffered as S, the Snom is the radiation, the photo voltaic module standardized at the ideality factor is denoted as e, the electron charge is indicated as k, Boltzmann’s constant is specified as Pg, the semiconductor energy gap is denoted as K0, the short-circuit current temperature coefficient is represented as q, the electron change is defined as Es, the number of cells in series is signified as Es, the short-circuit current is illustrated as Csc, the open-circuit voltage is defined as AOC, and the thermal voltage is proffered as Ath, which is expressed as follows:(2)Ath=nkUce.


## 3.1.2. Mathematical Model of Wind Turbine

The wind turbine-generated power relies on wind speed. The algebraic sum of gust wind speed (WS), base WS, and noise WS, along with ramp WS is termed the wind. The wind turbine’s mechanical power is as follows:(3)Mm=0.5ρWaPCp¯vsw3,where the mechanical power in watts is specified as Mm, the air density kg/m3 is signified as p, the swept area m2 is denoted as Wa, the turbine power coefficient is represented as PCp¯, and the WS m/s is indicated as vsw. The turbine conversion efficiency is represented by the power coefficient. If the pitch angle B=0, then the tip speed function is denoted as PCp¯, and the turbine’s conversion efficiency is indicated as δ and is expressed as follows:(4)PCp¯δ=c1c2δ−c4e−c5/δ+c6δ,where(5)δ=ωBrvsw,where the coefficients are given as c1,c2,c4,c5, and c6, the rotational speed rad/s is indicated as w, and the blades’ radius is specified as Br.


## 3.1.3. Mathematical Model for Fuel Cells

The fuel cell is a device in which electricity is generated as of the chemical reaction of hydrogen along with oxygen. The fuel cells’ mathematical model is as follows:(6)FOC=kcFn,CT0=zHLIH2+IO2Lhe−ΔG/LU,Vol=LuzγH,where the Nernst voltage is defined as Fn, the voltage constant at nominal conditions is indicated as kc, the number of moving electrons is represented as z, the energy values are symbolized as H,L, the partial pressure of hydrogen is explained as IH2, the partial pressure of oxygen is represented as Io2, Plank’s constant is defined as h, the size of the activation barrier is signified as ΔG, the temperature is represented as U, and the charge transfer coefficient is specified as γ.


## 3.1.4. Battery System Modelling

When the system has power deficiency, batteries that store surplus power in the HPS are utilized. The number of times the battery is charged/discharged in a day specifies the battery’s lifetime.(7)BTT=C+Mm+FOC−CCl¯,where the battery is indicated as BTT and the hybrid DC and AC dynamic load is denoted as CC1¯. The remaining power is stored in the battery following the AC and DC load.Figure2 shows the HMG structure. In this, AC loads are linked in the AC bus; similarly, in the DC bus, the battery PV, DC loads, along with fuel cells are connected.Figure 2
Structure of a hybrid microgrid.


## 3.2. Proposed Algorithm

In this section, the electrical energy flow is processed along with a controlled converter by providing voltages and currents in a way, which optimally suits the loads. The BSBBC (bidirectional converter) is utilized here. It has the potency to obtain the required output voltage in both directions of power flow by boosting the input voltage in a single stage. For the intermediate network betwixt lower voltage DC and the converter leg, “1” capacitor, “2” relays, “1” inductor, “3” active switches, and “1” diode are utilized here. On the AC along with the DC bus energy status, the direction of power flow is dependent upon, and also, it is centered on the direction of power flow switches that are being operated. In the AC bus as well as the DC bus, the frequency of deviation in energy status is utilized to be lesser; thus, the switches’ one part is employed as relays. Higher-frequency control switches are considered as the other part of the switches. These switches are functional regarding the shoot through duty ratio to control the voltage gain in the inverting mode along with the rectifying mode. To discover the maximum power as of the PV module under variable temperatures along with shading conditions, there is a necessity for the development of the MPPT. Furthermore, the FRLQG controls all the converters to assure safe operation along with soft dynamics. It is an amalgamation of fuzzy rules and linear quadratic control. In the linear quadratic control, the problem of poor tracking performance may be caused owing to the consumption of extra power; thus, the fuzzy rules are utilized in this methodology regarding which the controller is executed. The main objective is to minimize the operating cost of the consumed energy by a microgrid. The algorithm was developed to find the optimal operating set-points for which a power system control is required to maintain a continuous balance between power generation and load demand. The load frequency controller and automatic voltage regulator play an important role in maintaining constant frequency and voltage in order to ensure the reliability of electric power. In order to improve the performance and stability of these control loops, proportional-integral-derivative (PID) controllers are normally used. But these fixed gain controllers fail to perform under varying load conditions and hence provide poor dynamic characteristics with a large settling time, overshoot, and oscillations. In order to achieve better dynamic performance, system stability, and sustainable utilization of generating systems, controller parameters must be well-tuned. In this paper, evolutionary algorithms are proposed to find the optimum parameters of the controller to control the voltage and frequency of the generating system within the permissible limit. These algorithms offer a more stable and faster convergence towards the best parameters and also with minimum computational time. These algorithms are appropriate to model the uncertainties found in the power demand and improve the flexible nature of the controllers.The plant’s state-space model is(8)Xt=OmXt+Imυt+ξ1at,(9)Yt=OmXt+ξ2a¯t.Here,X communicates to the state vector, the contribution aimed at the framework is signified as υ, and the deliberate output as of the ideal plant is defined as Y, together with a specifies the Gaussian repetitive disturbance following up on the framework; the system matrix, the input matrix, and the output matrix are represented as Sm, Im, and Om; the noise matrices are denoted as ζ1 and ζ1. In this, to control the battery charging/discharging current, the fuzzy logic is implemented. Xt and Yt are the parameters estimated by a set of fuzzy rules of the form:IfΔOm is ωi and ΔΔOm is ωi, then(10)Xt=Ji,Yt=Qi,where ϖi, ωi, Ji, and Qi are fuzzy sets on the respective supporting sets.The fuzzy membership function(MF) sets are offered as triangular partitions in the FRLQG; positive small (PS), positive big (PB), negative big (NB), zero (ZO), and negative small (NS) are the partitions. To ameliorate the SSS and to augment the methodology’s performance, the controller’s parameters are tuned before controlling the power flow. The BCSSO algorithm is utilized here. Southern flying squirrels’ dynamic foraging behavior through gliding is imitated by the squirrel search optimization (SSO) algorithm. It is also an effective approach utilized by small mammals for traveling long distances in the deciduous forests of Europe as well as Asia. Acorn nuts are effortlessly found by them to meet their everyday energy requirements. The following phases illustrate the SSO mathematically regarding the flying squirrels’ food foraging strategy. In general, SSO, regarding the uniform distribution, the random value is chosen which devalues the performance because numerous chances are there to choose the worst optimal resolution at this random selection. Thus, a bell-shaped constant centered on the Cauchy distribution function is utilized in this research. Parameter initialization, random initialization of flying squirrel locations, sorting, declaration, random selection, generating new locations, and inspecting seasonal conditions are the steps included in this algorithm. In this, the squirrels are regarded as the controller parameters.


### 3.2.1. Initialization of the Parameters

The SSA major parameters are the population sizePs, the maximum number of iteration mai, the predator presence probability pyi, the number of decision variables X, the gliding constant Φ, the scaling factor sf, along with the upper and lower bounds for the decision variable are ψu as well as ψ1. At the beginning of the SSA procedure, these parameters are set.


### 3.2.2. Random Initialization of Flying Squirrel’s Locations

In a forest, there arenn flying squirrels (η); thus, the squirrels are regarded as the input parameters. The ith flying squirrel’s location on the ith tree is specified by a vector,(11)ηi,j=ηi1,ηi2,ηi3,..........,ηid,i=1,2,3,......,nn,where the ith flying squirrels of the jth dimension are denoted as ηij and the number of dimensions is indicated as d. After that, in the search space, the flying squirrels’ locations are initialized randomly as follows:(12)ψi,j=ψl+Bs∗ψu−ψl,i=1,2,.....,Ps,j=1,2,......,nn,wherein a bell-curved number is specified as Bs and this is derived by utilizing the Cauchy distribution function. Next, for the optimized controller, the fitness function (FF) is analyzed. The FF is regarded as the maximization of the SSS along with the minimization of the quadratic cost functions (QCFs). The enhancement of SSS’s objective function is as follows:(13)maxκi,where the SSS term is denoted as κi. The optimal LQG design starts with a model in the form of equations (18) and (19). In this, the process noise is described as at, and the measurement noise is illustrated as a¯t. In the optimal LQG, the feedback control is built to mitigate the following QCF,(14)QCF=limT⟶∞EV1T∫0TXtTFXt+vtTRRvt.dt.Here, symmetric weighting matrices are specified asF≥0 and RR≥0, and the expected value is denoted as EV⋅. To limit the control inputs’ states along with size, the terms XTFX and vTRRv are analogized to a prerequisite in (14). In the cost function, the selection of the matrices F and RR relies on the system’s required performance. The robustness problems against the plant uncertainty are not addressed directly by the controller’s design. The LQG’s robustness is leaned by the noise’s considerable decision regarding the plant model equations (18) and (19), which reflects the required bandwidth along with robustness characteristics. In this, mitigating the tracking error among the command signal along with the gauged output is the control objective. To execute this, an IC is related to an SN system owing to its phenomenal lower-frequency tracking operation. Tracking errors are reduced by the higher gain of the IC even though it displays a lower bandwidth, which proves the robustness insufficiency. Therefore, f=f1,f2,…,fPs denotes the fitness value (FV) derived. An individual flying squirrel’s location’s FV is gauged by substituting the decision variables’ value into a FF:(15)fi=fiψi,1,ψi,2,…,ψi,nn,i=1,2,…,Ps.


### 3.2.3. Sorting, Declaration, and Random Selection

All the FS position FVs are sorted in ascending order. It is observed that the squirrel with the minimal FV is on the hickory nut tree (optimal food source), the next “3” best flying squirrels are on the acorn tree (normal food source), and the remaining squirrels are on the normal trees (no food source). Moreover, by executing randomized selections, some FSs are believed to fly toward the direction of the hickory nut tree concerning that the squirrels have got satisfied with their daily energy requirements. The remaining FSs are glided towards the acorn nut tree.


### 3.2.4. Generation of New Locations

There occurs “3” situations in the nonexistence of a predator at the time of the dynamic forging of flying squirrels. In this, the flying squirrel glides together with searches all over the forest for its favorite food. However, the flying squirrel is watchful; in addition, it is forced to take a small random walk to search for the closest hiding location. The dynamic foraging behavior “3” mathematically modeled cases are given as follows:(a)
Flying squirrels on acorn nut trees (ψat) may move towards the hickory nut tree. Here, the squirrels’ new location is acquired as follows:(16)ηctt+1=ηctt+gld.lsc.ηktt−ηott,if,r1≥ppp,ranl,otherwise.Here, the flying squirrel location on the acorn tree att and t+1 iteration is specified as ηctt and ηctt+1, the gliding distance is indicated as gld, the sliding constant is represented as lsc, the flying squirrel on the hickory tree is indicated as ηkt, the flying squirrel on a normal tree is described as ηott, the random number in the range of [0, 1] is denotes as r1, the predator presence probability is signified as ppp, and the random location is represented as rani.(b)
To accomplish their daily energy requirements,ψnt may move towards acorn nut trees. The squirrels’ new location is updated as follows:(17)ηott+1=ηott+gld.lsc.ηctt−ηott,if,r2≥ppp,ranl,otherwise.Here, the flying squirrels’ location on the normal tree att and t+1 iteration is represented as ηott+1, and the random number 2 is indicated as r2, which existed in the range of [0, 1].(c)
To store the hickory nuts for consuming them when there is a shortage of food, few FS as of the normal trees, which have already consumed acorn nuts, might glide in the hickory nut tree’s direction. Here, the FS updated positions are obtained utilizing the following equation:(18)ηott+1=ηott+gld.lsc.ηktt−ηott,if,r3≥ppp,ranl,otherwise.Here, the random number 3 presented in the range of [0, 1] is represented asr3.


### 3.2.5. Check Seasonal Condition Monitoring

The flying squirrel foraging activity has got affected significantly by seasonal changes. Therefore, the changes in weather affect the flying squirrels movement, and a highly realistic approach towards optimization is provided by the addition of such behavior. Consequently, the solution is prevented as of the local optimal solutions by monitoring the seasonal condition. Initially, the seasonal constantseac is computed as follows:(19)seact=∑j=1dηct,jt−ηkt,jt2,where the seasonal constant at iteration t is specified as seact, the squirrel on the hickory and acorn trees at jth dimension is signified as ηkt,jt and ηct,jt, and the total dimension is represented as d. Subsequently, in the 2nd step, the seasonal monitoring criterion is checked by seact<seamin in which the seasonal constant minimum value is represented as seamin, and it is calculated as follows:(20)sea10e−6365t/tmax/2.5min,where the maximum number of iterations is indicated as tmax. The end of the winter season is found if the seasonal condition is true; thus, the flying squirrels, which lose their potency to travel around the forest, will randomly transfer their searching positions for food sources over again.(21)ηott+1=η↓+evy×η↑−η↓,where the levy distribution function is denoted as evy. It promotes better search space exploration and is computed as follows:(22)evy=0.01×τa×στb1/ε,where the “2” normally distributed random numbers in the range [0, 1] are represented as τa and τb, a positive constant is defined as ∂, which is lesser than 2, and σ is computed as follows:(23)σ=Γ1+ε×sinπε/2Γ1+ε/2×ε×2ε−1/21/ε,where the factorial function is denoted as Γ (Algorithm1).


### 3.2.6. Pseudocode for the Proposed BCSSO Algorithm

Algorithm 1: Bell-curved squirrel search optimization (BCSSO) algorithm.
Input: parameters (t) and Y (t)Output: optimal parametersBeginInitialize location ηij = (ηi1, ηi2, ηi3……….ηid) seasonal constants seact and maximumIteration MaxuDefine initial location of each flying squirrel by,ψij = ψl + Bs ∗ (ψu − ψl)Calculate fitness functionSet iteration IIt = 1While (IIt ≤ Maxit) doGenerate locations using 3 locationsScenario 1:If (r1 ≥ ppp); {Update the squirrel locationηctt+gld.lscηktt−ηott} else {ηctt+1=ranl}end ifScenario 2If (r2 ≥ ppp); {Update the squirrel locationηott+gld.lscηctt−ηott} else {ηott+1=ranl}end ifScenario 3If (r3 ≥ ppp); {Update the squirrel locationηott+gld.lscηktt−ηott} else {ηott+1=ranl}end ifPerform seasonal monitoring by using a seasonal constantCalculate fitness functionSet IIt = IIt + 1End whileReturn optimal parametersThe BCSSO’s pseudocode is demonstrated. The parameter initialization, fitness analysis along with the updating process for the squirrel locations are explicated.


## 3.2.1. Initialization of the Parameters

The SSA major parameters are the population sizePs, the maximum number of iteration mai, the predator presence probability pyi, the number of decision variables X, the gliding constant Φ, the scaling factor sf, along with the upper and lower bounds for the decision variable are ψu as well as ψ1. At the beginning of the SSA procedure, these parameters are set.


## 3.2.2. Random Initialization of Flying Squirrel’s Locations

In a forest, there arenn flying squirrels (η); thus, the squirrels are regarded as the input parameters. The ith flying squirrel’s location on the ith tree is specified by a vector,(11)ηi,j=ηi1,ηi2,ηi3,..........,ηid,i=1,2,3,......,nn,where the ith flying squirrels of the jth dimension are denoted as ηij and the number of dimensions is indicated as d. After that, in the search space, the flying squirrels’ locations are initialized randomly as follows:(12)ψi,j=ψl+Bs∗ψu−ψl,i=1,2,.....,Ps,j=1,2,......,nn,wherein a bell-curved number is specified as Bs and this is derived by utilizing the Cauchy distribution function. Next, for the optimized controller, the fitness function (FF) is analyzed. The FF is regarded as the maximization of the SSS along with the minimization of the quadratic cost functions (QCFs). The enhancement of SSS’s objective function is as follows:(13)maxκi,where the SSS term is denoted as κi. The optimal LQG design starts with a model in the form of equations (18) and (19). In this, the process noise is described as at, and the measurement noise is illustrated as a¯t. In the optimal LQG, the feedback control is built to mitigate the following QCF,(14)QCF=limT⟶∞EV1T∫0TXtTFXt+vtTRRvt.dt.Here, symmetric weighting matrices are specified asF≥0 and RR≥0, and the expected value is denoted as EV⋅. To limit the control inputs’ states along with size, the terms XTFX and vTRRv are analogized to a prerequisite in (14). In the cost function, the selection of the matrices F and RR relies on the system’s required performance. The robustness problems against the plant uncertainty are not addressed directly by the controller’s design. The LQG’s robustness is leaned by the noise’s considerable decision regarding the plant model equations (18) and (19), which reflects the required bandwidth along with robustness characteristics. In this, mitigating the tracking error among the command signal along with the gauged output is the control objective. To execute this, an IC is related to an SN system owing to its phenomenal lower-frequency tracking operation. Tracking errors are reduced by the higher gain of the IC even though it displays a lower bandwidth, which proves the robustness insufficiency. Therefore, f=f1,f2,…,fPs denotes the fitness value (FV) derived. An individual flying squirrel’s location’s FV is gauged by substituting the decision variables’ value into a FF:(15)fi=fiψi,1,ψi,2,…,ψi,nn,i=1,2,…,Ps.


## 3.2.3. Sorting, Declaration, and Random Selection

All the FS position FVs are sorted in ascending order. It is observed that the squirrel with the minimal FV is on the hickory nut tree (optimal food source), the next “3” best flying squirrels are on the acorn tree (normal food source), and the remaining squirrels are on the normal trees (no food source). Moreover, by executing randomized selections, some FSs are believed to fly toward the direction of the hickory nut tree concerning that the squirrels have got satisfied with their daily energy requirements. The remaining FSs are glided towards the acorn nut tree.


## 3.2.4. Generation of New Locations

There occurs “3” situations in the nonexistence of a predator at the time of the dynamic forging of flying squirrels. In this, the flying squirrel glides together with searches all over the forest for its favorite food. However, the flying squirrel is watchful; in addition, it is forced to take a small random walk to search for the closest hiding location. The dynamic foraging behavior “3” mathematically modeled cases are given as follows:(a)
Flying squirrels on acorn nut trees (ψat) may move towards the hickory nut tree. Here, the squirrels’ new location is acquired as follows:(16)ηctt+1=ηctt+gld.lsc.ηktt−ηott,if,r1≥ppp,ranl,otherwise.Here, the flying squirrel location on the acorn tree att and t+1 iteration is specified as ηctt and ηctt+1, the gliding distance is indicated as gld, the sliding constant is represented as lsc, the flying squirrel on the hickory tree is indicated as ηkt, the flying squirrel on a normal tree is described as ηott, the random number in the range of [0, 1] is denotes as r1, the predator presence probability is signified as ppp, and the random location is represented as rani.(b)
To accomplish their daily energy requirements,ψnt may move towards acorn nut trees. The squirrels’ new location is updated as follows:(17)ηott+1=ηott+gld.lsc.ηctt−ηott,if,r2≥ppp,ranl,otherwise.Here, the flying squirrels’ location on the normal tree att and t+1 iteration is represented as ηott+1, and the random number 2 is indicated as r2, which existed in the range of [0, 1].(c)
To store the hickory nuts for consuming them when there is a shortage of food, few FS as of the normal trees, which have already consumed acorn nuts, might glide in the hickory nut tree’s direction. Here, the FS updated positions are obtained utilizing the following equation:(18)ηott+1=ηott+gld.lsc.ηktt−ηott,if,r3≥ppp,ranl,otherwise.Here, the random number 3 presented in the range of [0, 1] is represented asr3.


## 3.2.5. Check Seasonal Condition Monitoring

The flying squirrel foraging activity has got affected significantly by seasonal changes. Therefore, the changes in weather affect the flying squirrels movement, and a highly realistic approach towards optimization is provided by the addition of such behavior. Consequently, the solution is prevented as of the local optimal solutions by monitoring the seasonal condition. Initially, the seasonal constantseac is computed as follows:(19)seact=∑j=1dηct,jt−ηkt,jt2,where the seasonal constant at iteration t is specified as seact, the squirrel on the hickory and acorn trees at jth dimension is signified as ηkt,jt and ηct,jt, and the total dimension is represented as d. Subsequently, in the 2nd step, the seasonal monitoring criterion is checked by seact<seamin in which the seasonal constant minimum value is represented as seamin, and it is calculated as follows:(20)sea10e−6365t/tmax/2.5min,where the maximum number of iterations is indicated as tmax. The end of the winter season is found if the seasonal condition is true; thus, the flying squirrels, which lose their potency to travel around the forest, will randomly transfer their searching positions for food sources over again.(21)ηott+1=η↓+evy×η↑−η↓,where the levy distribution function is denoted as evy. It promotes better search space exploration and is computed as follows:(22)evy=0.01×τa×στb1/ε,where the “2” normally distributed random numbers in the range [0, 1] are represented as τa and τb, a positive constant is defined as ∂, which is lesser than 2, and σ is computed as follows:(23)σ=Γ1+ε×sinπε/2Γ1+ε/2×ε×2ε−1/21/ε,where the factorial function is denoted as Γ (Algorithm1).


## 3.2.6. Pseudocode for the Proposed BCSSO Algorithm

Algorithm 1: Bell-curved squirrel search optimization (BCSSO) algorithm.
Input: parameters (t) and Y (t)Output: optimal parametersBeginInitialize location ηij = (ηi1, ηi2, ηi3……….ηid) seasonal constants seact and maximumIteration MaxuDefine initial location of each flying squirrel by,ψij = ψl + Bs ∗ (ψu − ψl)Calculate fitness functionSet iteration IIt = 1While (IIt ≤ Maxit) doGenerate locations using 3 locationsScenario 1:If (r1 ≥ ppp); {Update the squirrel locationηctt+gld.lscηktt−ηott} else {ηctt+1=ranl}end ifScenario 2If (r2 ≥ ppp); {Update the squirrel locationηott+gld.lscηctt−ηott} else {ηott+1=ranl}end ifScenario 3If (r3 ≥ ppp); {Update the squirrel locationηott+gld.lscηktt−ηott} else {ηott+1=ranl}end ifPerform seasonal monitoring by using a seasonal constantCalculate fitness functionSet IIt = IIt + 1End whileReturn optimal parametersThe BCSSO’s pseudocode is demonstrated. The parameter initialization, fitness analysis along with the updating process for the squirrel locations are explicated.


## 4. Results and Discussion

The proposed SSS enhancement of the HMG’s performance is assessed here. The proposed system is executed in MATLAB/simulink shown in Figure3.Figure 3
The MATLAB/simulink model of HMG with the PV, battery, and fuel cell.


### 4.1. Performance Analysis

In this, the phase of the proposed system along with its magnitude is evaluated; subsequently, the frequency of the system with and without tuning the controller’s parameters is appraised.The present controller phase behavior evaluation is depicted in Figure4. The controller phase is altered regarding the frequency. In this, the frequency level is regarded as of 10−3 to 103 rad/s. The controller is presented at −90° at the 10−3 frequency, and then, after 10° frequencies, the phase is completely altered.Figure 4
Analysis of the phase behavior of the controller.The controller’s magnitude is displayed in Figure5. Regarding the frequency, the magnitude is altered. The controller’s magnitude is different for varied frequencies. The controller has the 30 dB, 60 dB, and 90 dB magnitude for frequencies above 10°; similarly, the controller has −30 dB, −60 dB, and −90 dB magnitudes for frequencies below 10°.Figure 5
Analysis of the magnitude of the controller.The frequency evaluation with and without tuning the controller’s parameter is exhibited in Figure6. Regarding the time variation, the frequency is assessed. When the time is 0.1 sec, the light deviation occurred whilst tuning the controller’s parameters, whereas a higher level frequency variation occurred without tuning the controller’s parameters. Consequently, it is confirmed that better outcomes are obtained by tuning the controller’s parameters than without the tuning of the controller’s parameters.Figure 6
Frequency analysis.


### 4.2. Comparative Analysis

The performance of the proposed controller and the prevailing PID controller and the proportional integral (PI) controller is assessed regarding the voltage evaluation. Next, by tuning the controller’s parameter, the BCSSO-centered FRLQG controller is analogized with the prevailing evolutionary algorithm-centered controllers such as particle swarm optimization-FRLQG (PSO-FRLQG), SSO-FRLQG, and genetic algorithm-FRLQG (GA-FRLQG) regarding the QCF, stability, along with speed deviation. Finally, regarding the fitness evaluation, the performance of the proposed evolutionary algorithm and the existing PSO, SSO, and GA is analogized.Figure7 illustrates the controlled switching signal voltage comparison meant for the FRLQG and the prevailing LQG, PID, and PI controllers. Regarding the time in seconds, the voltage gain is assessed. The voltage gain at time 0.2 sec is 355 V for the proposed controller and 212 V and 185 V for the prevailing controllers. Likewise, at time 0.3 sec, the proposed system has the 303 V, and the prevailing LQG, PID, and PI controllers have obtained 202 V, 196 V, and 174 V, respectively. For the remaining time seconds also, a better outcome is acquired by the proposed than the existent controller. Therefore, the proposed systems’ better performance is proved.Figure 7
Analysis of voltage.The speed deviation of the proposed and the existent controller is evaluated in Figure8. The frequency regulation is centered on the speed control of all generators joined to the grid; thus, the speed deviation evaluation is a significant procedure. Regarding time seconds, the speed deviation is varied. As of the evaluation, it is perceived that a higher-speed deviation is experienced by the previous GA-FRLQG than the other existent along with the proposed methodologies all the time. For instance, at time 6 sec, a speed deviation of 1 p.u is possessed by the BCSSO-FRLQG; similarly, speed deviations of 1.2 p.u, 1.5 p.u, and 1.7 p.u are possessed by the prevailing SSO-FRLQG, PSO-FRLQG, and GA-FRLQG, respectively. Thus, a better outcome is attained by the proposed BCSSO-centered FRLQG, and the worst outcome is attained by the prevailing GA-centric FRLQG. Consequently, it is confirmed that a lesser speed deviation is accomplished by the proposed system than by the prevailing controller.Figure 8
Speed deviation analysis.Figure9 exhibits the QCF evaluation for the tuned controllers. The controller with minimal QCF shows better performance. The proposed controller’s QCF value for 0sec is 0, 43%, 49%, 56%, 55%, and 61%, and for 0.5 sec, it is 0.1 sec, 0.2 sec, 0.3 sec, and 0.4 sec. The prevailing controllers have a higher QCF than the proposed system. For example, the QCF for the SSO-FRLQG, PSO-FRLQG, and GA-FRLQG at time 0.5 s is 75%, 77%, and 83%. A higher QCF is attained by the prevailing controllers for the remaining time also. As a consequence, it is confirmed that better performance was attained by the proposed algorithm centric grid design than by the other methodologies.Figure 9
Comparison analysis of tuned controllers based on the QCF metric.The system stability enhancement is exhibited in Figure10. The stability gauge for any dynamic system exposed to smaller perturbations is mentioned as SSS. This research model’s major purpose is stability evaluation. The higher stability of 93% is attained by BCSSO-FRLQG at 0.5 sec than the other prevailing SSO-FRLQG, PSO-FRLQG, and GA-FRLQG controllers, which obtained stability of 74%, 67%, and 56%, respectively. For the remaining time seconds also, higher stability is attained by the proposed than the prevailing controller-centered HMG. Hence, it is proved that better performance was acquired by the proposed methodology than by the prevailing methodologies.Figure 10
Stability enhancement analysis.Fitness vs. iteration of the BCSSO with the previous SSO, PSO, and GA are depicted in Table1. A fitness of 0.978 is attained by the BCSSO, and a fitness of 0.906, 0.864, and 0.796 is attained by the prevailing SSO, PSO, and GA for the 25th iteration. A higher fitness is attained by the proposed than the prevailing algorithms. The proposed model attains better fitness than the existent algorithms for the remaining iterations as well. Accordingly, it is proved that a better outcome was attained by the proposed evolutionary algorithm. Figure 11 exhibits the graphical plot of fitness vs. iteration analysis.Table 1
Fitness vs. iteration analysis.
IterationsProposed BCSSOSSOPSOGA50.7210.6840.6530.613100.7960.7230.6910.634150.8310.7860.7490.696200.9010.8420.8130.753250.9780.9060.8640.796Figure 11
Graphical plot for fitness vs. iteration analysis.


## 4.1. Performance Analysis

In this, the phase of the proposed system along with its magnitude is evaluated; subsequently, the frequency of the system with and without tuning the controller’s parameters is appraised.The present controller phase behavior evaluation is depicted in Figure4. The controller phase is altered regarding the frequency. In this, the frequency level is regarded as of 10−3 to 103 rad/s. The controller is presented at −90° at the 10−3 frequency, and then, after 10° frequencies, the phase is completely altered.Figure 4
Analysis of the phase behavior of the controller.The controller’s magnitude is displayed in Figure5. Regarding the frequency, the magnitude is altered. The controller’s magnitude is different for varied frequencies. The controller has the 30 dB, 60 dB, and 90 dB magnitude for frequencies above 10°; similarly, the controller has −30 dB, −60 dB, and −90 dB magnitudes for frequencies below 10°.Figure 5
Analysis of the magnitude of the controller.The frequency evaluation with and without tuning the controller’s parameter is exhibited in Figure6. Regarding the time variation, the frequency is assessed. When the time is 0.1 sec, the light deviation occurred whilst tuning the controller’s parameters, whereas a higher level frequency variation occurred without tuning the controller’s parameters. Consequently, it is confirmed that better outcomes are obtained by tuning the controller’s parameters than without the tuning of the controller’s parameters.Figure 6
Frequency analysis.


## 4.2. Comparative Analysis

The performance of the proposed controller and the prevailing PID controller and the proportional integral (PI) controller is assessed regarding the voltage evaluation. Next, by tuning the controller’s parameter, the BCSSO-centered FRLQG controller is analogized with the prevailing evolutionary algorithm-centered controllers such as particle swarm optimization-FRLQG (PSO-FRLQG), SSO-FRLQG, and genetic algorithm-FRLQG (GA-FRLQG) regarding the QCF, stability, along with speed deviation. Finally, regarding the fitness evaluation, the performance of the proposed evolutionary algorithm and the existing PSO, SSO, and GA is analogized.Figure7 illustrates the controlled switching signal voltage comparison meant for the FRLQG and the prevailing LQG, PID, and PI controllers. Regarding the time in seconds, the voltage gain is assessed. The voltage gain at time 0.2 sec is 355 V for the proposed controller and 212 V and 185 V for the prevailing controllers. Likewise, at time 0.3 sec, the proposed system has the 303 V, and the prevailing LQG, PID, and PI controllers have obtained 202 V, 196 V, and 174 V, respectively. For the remaining time seconds also, a better outcome is acquired by the proposed than the existent controller. Therefore, the proposed systems’ better performance is proved.Figure 7
Analysis of voltage.The speed deviation of the proposed and the existent controller is evaluated in Figure8. The frequency regulation is centered on the speed control of all generators joined to the grid; thus, the speed deviation evaluation is a significant procedure. Regarding time seconds, the speed deviation is varied. As of the evaluation, it is perceived that a higher-speed deviation is experienced by the previous GA-FRLQG than the other existent along with the proposed methodologies all the time. For instance, at time 6 sec, a speed deviation of 1 p.u is possessed by the BCSSO-FRLQG; similarly, speed deviations of 1.2 p.u, 1.5 p.u, and 1.7 p.u are possessed by the prevailing SSO-FRLQG, PSO-FRLQG, and GA-FRLQG, respectively. Thus, a better outcome is attained by the proposed BCSSO-centered FRLQG, and the worst outcome is attained by the prevailing GA-centric FRLQG. Consequently, it is confirmed that a lesser speed deviation is accomplished by the proposed system than by the prevailing controller.Figure 8
Speed deviation analysis.Figure9 exhibits the QCF evaluation for the tuned controllers. The controller with minimal QCF shows better performance. The proposed controller’s QCF value for 0sec is 0, 43%, 49%, 56%, 55%, and 61%, and for 0.5 sec, it is 0.1 sec, 0.2 sec, 0.3 sec, and 0.4 sec. The prevailing controllers have a higher QCF than the proposed system. For example, the QCF for the SSO-FRLQG, PSO-FRLQG, and GA-FRLQG at time 0.5 s is 75%, 77%, and 83%. A higher QCF is attained by the prevailing controllers for the remaining time also. As a consequence, it is confirmed that better performance was attained by the proposed algorithm centric grid design than by the other methodologies.Figure 9
Comparison analysis of tuned controllers based on the QCF metric.The system stability enhancement is exhibited in Figure10. The stability gauge for any dynamic system exposed to smaller perturbations is mentioned as SSS. This research model’s major purpose is stability evaluation. The higher stability of 93% is attained by BCSSO-FRLQG at 0.5 sec than the other prevailing SSO-FRLQG, PSO-FRLQG, and GA-FRLQG controllers, which obtained stability of 74%, 67%, and 56%, respectively. For the remaining time seconds also, higher stability is attained by the proposed than the prevailing controller-centered HMG. Hence, it is proved that better performance was acquired by the proposed methodology than by the prevailing methodologies.Figure 10
Stability enhancement analysis.Fitness vs. iteration of the BCSSO with the previous SSO, PSO, and GA are depicted in Table1. A fitness of 0.978 is attained by the BCSSO, and a fitness of 0.906, 0.864, and 0.796 is attained by the prevailing SSO, PSO, and GA for the 25th iteration. A higher fitness is attained by the proposed than the prevailing algorithms. The proposed model attains better fitness than the existent algorithms for the remaining iterations as well. Accordingly, it is proved that a better outcome was attained by the proposed evolutionary algorithm. Figure 11 exhibits the graphical plot of fitness vs. iteration analysis.Table 1
Fitness vs. iteration analysis.
IterationsProposed BCSSOSSOPSOGA50.7210.6840.6530.613100.7960.7230.6910.634150.8310.7860.7490.696200.9010.8420.8130.753250.9780.9060.8640.796Figure 11
Graphical plot for fitness vs. iteration analysis.


## 5. Conclusion

This paper presents a framework for stability analysis and performance evaluation of microgrids. In addition, various parameters and their impacts on the microgrid dynamic performance are investigated. To modify the parameter, a new controller with the BCSSO was proposed here; thus, it meliorated the SSS. Firstly, for hybrid renewable energies, mathematical models are recognized. After that, the converter is designed; in addition, the evolutionary algorithm is utilized to control the power flow. Regarding the phase behavior along with magnitude, the proposed methodologies’ performance is assessed. Next, the frequency is evaluated with and without tuning the controller’s parameter. In accordance with the evaluation, the best outcomes are obtained by the tuning parameter-centered controller. Then, regarding QCF, speed deviation, along with stability, the performance of the BCSSO- FRLQG controller, and the prevailing PSO-FRLQG, SSO-FRLQG, and GA-FRLQG are analogized. Subsequently, for the proposed work and for the existing converters such as PID, LQG, along with PI, the voltage is evaluated. The comparison outcomes proved that a better outcome along with higher stability was attained by the proposed methodology. A stability of 93% was obtained by the BCSSO-FRLQG at time 0.5 sec, which is higher than the other methodologies. After that, the BCSSO’s fitness is assessed with the prevailing evolutionary algorithms; subsequently, it is established that the proposed one attained the best outcome. Therefore, it is recognized that the current converter and controller-centric HMG are highly efficient for PSs to ameliorate the SSS. In the upcoming future, to promote the system performance, the proposed methodology can be advanced with highly renewable energies and enhanced algorithms.



---

*Source: 8404457-2023-02-04.xml*