Among all the types of RESs, SPG is the most common and committed eco-friendly energy resource. In the past few years, rigorous research and development related to SPG units has reduced the per kilowatt/hour generation cost. Furthermore, the extensive capability of SPG units makes them a deserving alternative to traditional power generators. The power generation from SPG highly depends on solar irradiance and ambient temperature (Sanki and Basu, 2018; Sahu et al., 2022). The basic equation of SPG-generated power can be presented as Eq. 1: P S P G = η ϕ A 1 − 0.005 T a + 25 . (1)

According to the small-signal stability analysis, control system engineers have the freedom to investigate any system dynamics using the transfer function model. The considered linearized transfer function of SPGs is depicted in Figure 2.

WPGs are evolving as one of the most promising clean power producers, along with SPGs. The progressive development of WPGs has already crossed the global utilization threshold of 21%. These DERs convert the kinetic energy from the wind into electrical energy (Tah and Das, 2016; Latif et al., 2019b). The power expression of WPGs can be expressed as shown in Eq. 2. P W P G = 0.5 ρ A r C p λ w , β V w i n d 3 . (2)

This work utilizes a linearized transfer function model of the WPG, as shown in Figure 3.

The last few decades have utilized the extensive applications of DEG units in microgrid scenarios. In addition to renewable energy sources, non-renewable generators like DEGs are very effective in AGC operations (Tah and Das, 2016). The primary objective of the DEG unit is to ensure the frequency deviation is zero at steady-state conditions under any load perturbations. In this regard, a linearized model of DEGs is utilized along with the renewable energy sources, which is expressed in Eq. 3. Accordingly, a schematic diagram is presented in Figure 4. Moreover, a linearized model of a delay unit is utilized along with the DEG unit to represent the overall system performance more realistically. ∆ P DEG = K DEG 1 + s T DEG 1 1 + s T D e l a y ∆ u DEG . (3)

Renewable source-based power plants frequently suffer from a mismatch between the generated power and load demand due to their intermittent nature (Siti et al., 2022). In order to achieve a seamless power balance between the generating units and loads, ESSs can be used as backup power units. Based on the detailed literature reviews, it is observed that in most cases, BESS and FESS units are utilized as ESSs. Therefore, in this work, the BESS and FESS models are taken into consideration. The considered linearized models of BESS and FESS are presented in Figures 5, 6, respectively.

The power fluctuations between generation and power demand directly affect the system’s operating frequency and cause oscillations. Thus, the total generated power should be controlled and matched with the load power demand to avoid such an unwanted situation (Sanki et al., 2021). Furthermore, the mismatch between the generated and demanded power can be expressed as shown in Eq. 4: ∆ P = P G − P L . (4)

Now, the relation between power mismatch and frequency perturbation is presented in Eq. 5. ∆ f = P G − P L κ . (5)

In Eq. 5, for the P G < P L condition, − ∆ f will be considered and for the P L < P G condition, + ∆ f will be considered. Furthermore, a time delay operator is incorporated to represent the relationship between the change in power and frequency. Therefore, the change in frequency with the change in power is presented in Eq. 6. G P S = ∆ f ∆ P = 1 κ 1 + s T P S = 1 D + s M . (6)

The linearized power system model is depicted in Figure 7.

The tie-lines between the interconnected areas under any power system play a significant role in sustaining synchronism by exchanging power between the control areas during load disturbances. To date, the majority of AGC studies have considered a multi-area power system model without including the effect of line resistance R _Ln (Tah and Das, 2016). Thus, the power flow equations only consist of line reactance X _Ln . Considering the effect of X _Ln , the conventional tie-line power flow P ₁₂ is presented in Eqs 7, 8. A typical two-area power system schematic diagram is depicted in Figure 8A. P 12 = E A 1 E A 2 X Ln sin ∠ δ 1 − ∠ δ 2 = E A 1 E A 2 X Ln sin ⁡ ∠ δ , (7)

where, ∠ δ = ∠ δ 1 − ∠ δ 2 . (8)

This article focuses only on the frequency regulation issue. Therefore, a change in active power causes a change in frequency.

Under small load perturbations, the change in δ can be considered ∆ δ . Hence, the power flow equation under small load perturbations becomes (Eq. 9) ∆ P 12 = T 12 ∠ δ 1 − ∠ δ 2 , (9)

where T ₁₂ is the tie-line synchronizing co-efficient, which is presented in Eq. 10. T 12 = E A 1 E A 2 X Ln cos ∠ δ 1 − ∠ δ 2 = E A 1 E A 2 X Ln cos ⁡ ∠ δ . (10)

Finally, according to the conventional AGC studies, the frequency deviation due to load perturbation is presented in Eq. 11. ∆ P 12 s = 2 π T 12 ∫ ∆ f 1 − ∆ f 2 d t = 2 π T 12 s ∆ f 1 s − ∆ f 2 s . (11)

In modern decentralized power systems like microgrids, the exchange of power between the interconnected areas is very frequent, as they consist of intermittent energy resources. In such a scenario, the losses due to R _Ln are essential for power flow calculations. Therefore, modification of traditional power flow equations for such decentralized interconnected systems is highly required. Accordingly, the line impedance Z _Ln is considered to appear in the tie-line, connecting different areas under the microgrid scenario. For example, a schematic diagram of a two-area power system model is considered in Figure 8B. According to Figure 8B, the improved power flow equation P 12 becomes P 12 = E A 1 E A 2 Z Ln cos ⁡ ∠ θ Z − E A 1 E A 2 Z Ln cos   ∠ δ + ∠ θ Z . (12)

Here, current expression from bus 1 to 2 is as follows: I → 12 = E A 1 ∠ δ 1 Z Ln ∠ θ Z − E A 2 ∠ δ 2 Z Ln ∠ θ Z . (13)

Simplifying Eq. 12, 13 by substituting Z Ln = R Ln 2 + X Ln 2 , cos ⁡ ∠ θ Z = R Ln Z Ln , and sin ⁡ ∠ θ Z = X Ln Z Ln , we get Eq. 14 P 12 = E A 1 2 R Ln Z Ln 2 − E A 1 E A 2 R Ln Z Ln 2 cos ⁡ ∠ δ + E A 1 E A 2 X Ln Z Ln 2 sin ∠ δ . (14)

Under a small load perturbation situation, the governing power flow equation from bus 1 to 2 becomes ∆ P 12 = E A 1 E A 2 R Ln 2 + X Ln 2 R Ln ⁡ sin   ∠ δ 1 − ∠ δ 2 + X Ln ⁡ cos   ∠ δ 1 − ∠ δ 2   × ∆ ∠ δ 1 − ∆ ∠ δ 2 . (15)

Similar to Eq. 15, the power flow equation from bus 2 to 1 becomes as follows (Eq. 16). ∆ P 21 = E A 2 E A 1 R Ln 2 + X Ln 2 X Ln ⁡ cos   ∠ δ 1 − ∠ δ 2   − R Ln ⁡ sin   ∠ δ 1 − ∠ δ 2   × ∆ ∠ δ 1 − ∆ ∠ δ 2 . (16)

Furthermore, ▽ 1 = R Ln ⁡ sin   ∠ δ 1 − ∠ δ 2 + X Ln ⁡ cos   ∠ δ 1 − ∠ δ 2 and ▽ 2 = X Ln ⁡ cos   ∠ δ 1 − ∠ δ 2   − R Ln ⁡ sin   ∠ δ 1 − ∠ δ 2 , as presented in Figure 1. The power loss during small load perturbation can be presented using Eq. 17. ∆ P L o s s = ∆ P 12   + ∆ P 21 . (17)

The contribution from the renewable generating components, i.e., SPGs and WPGs, is always non-deterministic in nature. As a result, frequency fluctuations with load perturbations are very frequent phenomena in microgrids. It is essential to incorporate suitable control schemes in order to ensure proper system stability and smooth frequency regulation by maintaining a power balance between generation and load demand. To date, PID or various combinations of PID controllers are extensively used in AGC due to their faster and more flexible operations. Therefore, this paper proposes a CPIPD controller (as depicted in Figure 9) for a two-area microgrid to achieve dynamic frequency regulation by ensuring proper power balance. The PD feedback controller is effective in filtering out unwanted noises due to the available system disturbances, and the forward path PI controller is efficient in providing zero steady-state error of the frequency and tie-line power deviations under load perturbation (Sanki et al., 2021). The proposed CPIPD controller parameters can be represented using Eqs 18, 19, where suffixes 1 and 2 present the forward and feedback paths of the proposed controller, respectively. Suffix “n” presents the specific area of the microgrid. C 1 s = K p i 1 , n + K i 1 , n s . (18) C 2 s = K p d 2 , n + s K d 2 , n . (19)

According to Figure 9, the frequency error due to any load perturbation is corrected by the PI controller, and noise filtration due to uncertainties is accomplished by the PD controller.

To date, numerous metaheuristic algorithms have been proposed in order to evaluate the optimum controller gain parameters. It is always desired that these metaheuristic algorithms exhibit a faster convergence rate with less computational time to tune the gain parameters. However, the majority of them struggle to satisfy such basic criteria. Recently, the SPBO technique has been proposed to overcome such hardships (Das et al., 2020). In this regard, a detailed discussion of SPBO is presented in the upcoming section.

The controller gain parameter evaluation plays an important role in achieving the desired responses. Generally, metaheuristic algorithms are utilized to obtain suitable controller gain parameters with minimum objective functions. In this scenario, various performance indices like ISE, IAE, ITSE, and ITAE have so far been incorporated in the presence of different optimization techniques to derive the objective functions (Antonopoulos et al., 2020). In this work, different performance indices are adopted to evaluate the suitable controller parameters. The detailed mathematical expressions of the different performance indices are presented in Eqs 20–23. J I S E = ∫ 0 T ∆ f i 2 + ∆ P t i e , i j 2 d t . (20) J I A E = ∫ 0 T ∆ f i + ∆ P t i e , i j d t . (21) J I T S E = ∫ 0 T t { ∆ f i 2 + ∆ P t i e , i j 2 d t . (22) J I T A E = ∫ 0 T t ∆ f i + ∆ P t i e , i j d t . (23)

The considered objective functions of this work can be represented as Eqs 24, 25. J min = ∫ 0 T ∆ f 1 2 + ∆ f 2 2 + ∆ P t i e , 12 2 + ∆ P t i e , 21 2 , (24) where, K p i , ⁡ min < K p i < K p i , ⁡ max ,   K i , ⁡ min < K i < K i , ⁡ max K p d , ⁡ min < K p d < K p d , ⁡ max ,   K d , ⁡ min < K d < K d , ⁡ max . (25)

Performance analysis of any student in a particular class during an examination is mostly carried out based on the secured marks. The student who secures the overall highest mark is generally considered the class topper and is appreciated with some reward. Therefore, all students in any class attempt to provide maximum effort for each subject to secure the overall highest mark. The SPBO algorithm works with the same student psychology, where each student in a class tries to score the overall highest mark by improving their performance.

In order to sustain the position of class topper, gradual performance improvement for each subject is required. Therefore, students need to provide equal effort in all subjects to improve their overall scores. However, in reality, the effort of each student does vary based on their merit, efficiency, and subject interest. Additionally, the physiology of each student does differ, and as a result, performance during examinations discriminates against individuals. Some of the students consider the class topper as their reference and accordingly manage their efforts to become a topper. On the other hand, some of them consider both the class topper and the average student to manage their efforts to become a topper. Based on this analysis, the students of a particular class can be categorized into the following sections:

Topper: in an examination, the student who scores the maximum overall marks is generally considered a class topper. In order to sustain the position of the topper, a student must put forth equal effort in all the subjects to secure the overall maximum marks. Now, the improvement in the performance of a topper can be evaluated using Eq. 26. X t o p − n e w = X t o p + − 1 k × r n d × X t o p − X i . (26)

Good student: the choice of this section of students is a random process as the psychology of individual students differs. The students who manage their effort, considering the class topper as their reference, can be assessed as Eq. 27: X n e w , i = X i + r n d × X t o p − X i . (27)

However, the students who consider both the class topper and the average student to manage their effort regarding the study can be assessed as Eq. 28: X n e w , i = X i + r n d × X t o p − X i +   r n d × X i − X m e a n . (28)

Average student: this section of students can be considered subject-wise average candidates. Effort on any particular subject depends greatly on interest in its topics. Therefore, a student who has less interest in a particular subject tries to focus more on the subjects of their interest for overall improvement. The performance analysis of this category of student can be carried out as Eq. 29: X n e w , i = X i + r n d × X m e a n − X i . (29)

Students who improve randomly: this category of students randomly puts extra effort into a particular subject of their interest to improve their overall score during examinations. The performance analysis of this category of students can be carried out as Eq. 30: X n e w , i = X min + r n d × X max − X min . (30)

The detailed flowchart of the SPBO algorithm is presented in Figure 10.

This case study is performed in a two-fold process. First, a stability margin analysis of the proposed controller gain parameters is presented in order to ensure proper and stable system operation. Based on the stable gain parameter values, the ranges of the controller gain parameters are decided during the optimization procedure. However, in the earlier research works, the controller gain parameters were randomly selected to find out the minimum objective function. Furthermore, a detailed comparative analysis is presented between the traditional controllers, state-of-the-art control techniques, and the proposed control scheme. The extensive performance analysis is presented in the following sections.

This section presents a comparative performance analysis of the proposed two-area microgrid with and without considering the effect of R _Ln during SLP. Conventional power systems are extensively utilized for high-voltage and low-current applications. Therefore, the power angle value (δ) is taken considerably high (within the range of 45°), and the tie-line is considered lossless (Tah and Das, 2016). Therefore, in conventional power systems, the effect of R _Ln becomes negligible during system performance evaluation. However, in decentralized multi-area power systems, like microgrids during interconnected mode operation, the above-mentioned parametric values are completely different. In this scenario, the system operates with comparatively low-voltage and high-current ratings. Accordingly, the value of δ is found to be considerably low. Thus, the inclusion of tie-line R _Ln in decentralized power systems becomes essential for power flow calculation. In the previous section, the superior performance of the proposed CPIPD controller over the traditional PID for an improved two-area microgrid model was already established. Thus, in this test case, the proposed CPIPD controller with SPBO is considered for the two-area microgrid to analyze the significance of R _Ln during performance evaluation.

In this section, 1% SLP is considered at area-1 of the proposed microgrid scenario. Like in the previous case, the frequency regulation of area-1, area-2, and the power flow through the tie-line is examined with and without considering the effect of R _Ln . As depicted in Figures 14A–C, the oscillations in frequency and tie-line power signals have reduced much after considering the effect of R _Ln . It should be noted that the power tie-line also acts as a feedback signal in each area. Therefore, the oscillation in the power tie-line is responsible for the oscillations in the frequency signals. Based on the dynamic responses, it is clear that the inclusion of R _Ln has reduced the system oscillations significantly. Hence, the effectiveness of the suitable tie-line design can be established.

In real-life power systems, the load is always random in nature. Therefore, the performance of the proposed CPIPD controller using SPBO in an improved two-area microgrid is further examined under random load perturbation (RLP) and in the presence of uncertain renewable energy resource generation. The RLP is applied in area-1 to investigate the system performance. The nature of the load (small or large change in magnitude) with its switching instants is presented in Table 3. Furthermore, the available power contributions from SPG and WPG based on their intermittent nature are presented in Table 4. Additionally, the patterns of the RLP and the SPG–WPG units are depicted in Figures 15A, B, respectively.

The detailed dynamic responses are presented in Figure 16. According to the obtained results, it can be stated that the system performance outperforms the other available control schemes in the presence of the proposed control topology in terms of the peak of over/undershoots, oscillations, and setting time. The effectiveness of the proposed control scheme under the improved microgrid scenario can also be validated with the available magnified responses. Therefore, it can be stated that the proposed controller with SPBO is capable of providing improved performance in a multi-area microgrid model during RLP with uncertain power output from SPG–WPG.

The capacity and size of the microgrid will increase in the future with increasing load demand. Accordingly, the integration of renewable source-based energy will increase. Therefore, there will be a requirement for design modifications where the microgrid will increase in size, capacity, and interconnections. This case study presents a detailed performance analysis of a three-area microgrid scenario with an SPBO-optimized CPIPD controller and an improved power tie-line model. In this section, the RLP is considered as the load disturbance. In addition, to examining the controller performance under RLP, the intermittent nature of the SPG and WPG is also considered to create a realistic situation. The nature of RLP and the intermittent nature of SPG–WPG are considered the same, as described in Section 5.3.

The frequency fluctuation of the considered three-area interconnected microgrid scenario and the overall tie-line power deviation are depicted in Figure 17. Based on the obtained results, it can be concluded that the frequency and the tie-line power deviations are efficiently handled by the proposed SPBO-tuned CPIPD controller. Furthermore, the overall power sharing between the areas has taken place seamlessly without abrupt oscillations.

In this section, the performance analysis is presented by considering the improved two-area microgrid integrated with a 12-node radial distribution network, as shown in Figure 18A. The proposed system is connected by bus numbers 9 and 11, respectively. The substation (SS), which is situated far away from the connecting nodes, is considered as the swing bus in the network. In this test case, prior to integration, the voltages of nodes 9 and 11 of the considered distribution network were 0.9895 pu and 0.986 pu, respectively. The corresponding loss of the distribution network was 19.02 kW. However, after integrating the proposed two-area microgrid, the node voltages are improved significantly from 0.9895 pu to 0.9980 pu (for node 9) and from 0.9860 pu to 0.9954 pu (for node 11). In addition to that, a notable power loss reduction takes place (from 19.02 kW to 15.34 kW) accordingly. The simulated responses of the corresponding bus voltages and power losses are presented in Figure 18B. The improved voltage profile and reduction of power loss clearly show the effectiveness of the improved two-area microgrid interconnection in the 12-node distribution network. A critical performance analysis of the improved two-area microgrid under 12-node distribution networks in terms of pu voltage (nodes 9 and 11) and power loss reduction is presented in Table 5. Based on the obtained results, the efficacy of the proposed microgrid model is justified.