## 2 Scheme Representation  
### Overall Representation of the Scheme  
A trinity-arena structure of a non-identical kind is engaged with the interpretation of analyses with an arena size ratio of 2:3:4. The structure includes thermal, thermal, and biodiesel (T-T-Bd) resources as generating elements in arena-1. Similarly, thermal, thermal, and geothermal power plants (GPPt) (T-T-GPPt) energy is used in arena-2 and thermal and hydro (T-H) energy in arena-3. The traditional diesel plant is currently replaced with biodiesel elements, since they are not contaminated as well recyclable and of somewhat denser, lesser viscosity; furthermore, it results in a lower amount of carbon monoxide. Consequently, it can be employed as a reserve for a power source. It entails a combustion engine and valve controller. The primary order transfer function (Trfn) of the valve controller and ignition appliance of biodiesel element is specified by Equations (1) and (2), respectively [22]:  
\[Tf_{\mathit{value~{}regulator_{Bio-diel}}}=\frac{K_{VR}}{1+sT_{VR}} \tag{1}\]  
\[Tf_{\mathit{Combustion~{}engine_{Bio-diel}}}=\frac{K_{CE}}{1+sT_{CE}} \tag{2}\]  
Geothermal energy is a possible inexhaustible source of energy where underground thermal energy is transmuted into electrical energy. The Trfn modeling of GPPt is comparable to thermal elements but it does not have the cistern for warming condensation. The initial order Trfn of the governor and turbine of GPPt are specified by Equations (3) and (4), respectively [20]:  
\[Tf_{\mathit{GCPP}}=\frac{1}{1+s\mathit{GCPP}} \tag{3}\]  
\[Tf_{\mathit{TCPP}}=\frac{1}{1+sT_{GPP}} \tag{4}\]  
where \(\mathit{GCPP}\) and \(T_{GPP}\) are the time constants of the governor and turbine of GPPt, respectively. These standards are attained by augmentation procedure SHO, given the restrictions.  
The systems that are associated with a generation rate constraint and governor dead band display non-linearity. The contributing factors (_cf_) of every generating unit of individual arenas are _cf_\({}_{11}\) = 0.4, _cf_\({}_{12}\) = 0.4, and _cf_\({}_{13}\) = 0.2 in arena-1, _cf_\({}_{21}\) = 0.4, _cf_\({}_{22}\) = 0.39, and _cf_\({}_{23}\) = 0.21 in arena-2, and _cf_\({}_{31}\) = 0.655 and _cf_\({}_{32}\) = 0.345 in arena-3. This makes up structure-1. The system with only thermal energy in area-1 and 2 and thermal and hydro energy in area-3 is considered structure-2. Consequently, structure-1 is integrated with time delay in all areas in order to make the system more practical. This is treated as scheme-3. The energy storage section, specifically RFB, is included in area-1. This is treated as structure-4. The representation and _Trfn_ prototypical of schemes are shown in Figure 1. The basic standards of the scheme constraints are specified in Appendix A. The finest standards of controller attributes are attained by the assistance of spotted hyena optimizer procedure using the integral squared error as the performance index (\(PIx_{ISE}\)) specified by Equation (5) [19]:  
\[PIx_{ISE}=\int_{0}^{T}\Bigl{\{}(\Delta f_{1})^{2}+(\Delta f_{2})^{2}+(\Delta f_{3 })^{2}+\left(\Delta P_{tie_{1-2}}\right)^{2}+\left(\Delta P_{tie_{2-3}}\right)^{2 }+\left(\Delta P_{tie_{1-3}}\right)^{2}\Bigr{\}}\;dt \tag{5}\]