## 1 Introduction
As RFBs have so many advantages, their application in a hydrothermal- biodiesel-GPPt system with time delay needs elaborate assessments. The proper selection of a secondary controller is of high importance in AGC learning regarding control. Numerous forms of subsidiary controllers, such as integer-order (InOr), fractional-order (FrOr), and cascade controllers (CaC), are stated in literature to be linked to AGC. The usage of various InOr controllers, such as integral (I) [26], proportional-integral (PI) [27], and proportional-integral-derivative with filter (PIDN) [28], is reflected in the literature. Two [29] or three [30] higher degrees of freedom controllers have been inspected in this ground of training, which were also of the InOr type. Limited FrOr controllers that have found to be of use are FOPI [31], FOPIDN [32; 33; 34], and proportional with TIDN [35; 36; 37; 38]. The literature on AGC systems provides an argument about the training of InOr amalgamated controllers (PD-PID) [39], FrOr amalgamated controllers (FOPI-FOPD) [20], and the assemblage of InOr and FrOr controllers (PIDN-FOPD). [PERSON] et al. [40] analyzed a system with a cascade IADuN controller. In addition to all these controllers, various authors reported on fuzzy and intelligent controllers. [PERSON] et al. [41] worked on an AGC system incorporating ANN controllers. [PERSON] et al. [42] have reported on the usage of fuzzy PIDF controllers. However, the series cascade combination of TIDN with FrOr integral-derivative termed TIDN-FOID has not yet been quantified in AGC works. Additionally, the function of the TIDN-FOID subsidiary controller in this trinity-arena hydrothermal-biodiesel-GPPt arrangement has not yet been researched. The enactment of any subordinate controllers is dominant solitary if the finest values of attributes are suitably selected. These could be prepared with the aid of traditional/optimization practices [43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60]. However, the traditional practice has moderate difficulty to deliver suboptimum consequences. A few algorithms used for AGC are bacterial foraging optimization (BFO) [10], whale optimization algorithm (WOA) [19], sine-cosine algorithm (SCA) [20], cuckoo search (CS) [29], particle swarm optimization (PSO) [33], coyote optimizer algorithm (COA) [37], bat algorithm (BA) [39], salp swarm algorithm [50], grey wolf [51], firefly algorithm (FA) [53], flower pollination algorithm (FPA) [54], differential evolution (DE) [55], and Jaya algorithm [56]. A newly established bioinspired meta-heuristic procedure labeled as SHO has also been developed [57]. SHO was recognized from spotted hyenas' behavioral nature, which depicts the communal promise amidst spotted hyena and their cooperative performances. To a great wonder, SHO has not identified its implementation in AGC learning for obtaining premium values of controller attributes, thus demanding advance examination. With concern to overhead deliberations, the major persistence of the current artifact is as follows:  
1. Construction of a trio-arena structure with thermal-thermal-biodiesel energy in arena-1, thermal-thermal-GPPt energy in arena-2, and hydrothermal energy in arena-3. 2. The attributes of PIDN/TIDN/TIDN-FOID are concurrently augmented separately via the SHO algorithm so as to attain an outstanding controller. 3. The impact of biodiesel and GPPt energy on system dynamics is studied with the best obtained controller from (b). 4. The impact of time delay on system dynamics is studied with the best obtained controller from (b). 5. The impact of RFB on system dynamics is studied with the best obtained controller from (b). 6. Eigenvalue assessment is performed to comment on the stability of the system. 7. Sensitivity analysis is performed to examine the robustness of the best controller's gains with a higher value of step load disturbance.