Table 7 Specifications of the analyzed system's transient response for scenario A.2.

From: Frequency regulation in a hybrid renewable power grid: an effective strategy utilizing load frequency control and redox flow batteries

Controller structures

Considered dynamic response of (\({\Delta {\text{f}}}_{{\text{a}} })\)

Considered dynamic response of (\({\Delta {\text{f}}}_{{\text{b}} })\)

Considered dynamic response of (\({\Delta {\text{P}}}_{{\text{tie}}}\))

PD-PI-optimized via HHO 3

\({O}_{sh}\)= 10.121

\({O}_{sh}\)= 1.52

\({O}_{sh}\)= 0.5469

\({U}_{sh}\) =  − 15.181

\({U}_{sh}\) =  − 2.28

\({U}_{sh}\) =  − 0.82035

FO-(PD-PI)-optimized via MRFO3

\({O}_{sh}\)= 7.45

\({O}_{sh}\)= 0.94914

\({O}_{sh}\)= 0.31477

\({U}_{sh}\) =  − 11.203

\({U}_{sh}\) =  − 1.451

\({U}_{sh}\) =  − 0.48686

Fuzzy-PID + (T \({I}^{\lambda }{D}^{\mu }\))-optimized via COA

\({O}_{sh}\)= 1.12

\({O}_{sh}\)= 0.3051

\({O}_{sh}\)= 0.1670

\({U}_{sh}\) =  − 1.817

\({U}_{sh}\) =  − 0.5570

\({U}_{sh}\) =  − 0.2364

  1. All values are multiplied by (\({10}^{-3}\)).