Table 4 Optimized controller parameters and ST and JITAE results of Test-system-3
Algorithm | MGWO: PID [27] | DSA: FOPID [23] | DSA: FOPI–FOPD [23] | CGO: FOPID–FOPI |
|---|---|---|---|---|
Controller parameters | Thermal | Thermal | Thermal | Thermal |
\({K}_{P}=1.7502\) | \({K}_{P}=1.9998\) | \({K}_{P1}=-2.0000\) | \({K}_{P1}=1.9999\) | |
\({K}_{I}=-0.0087\) | \({K}_{I}=2.0000\) | \({K}_{P2}=-2.0000\) | \({K}_{P2}=1.9787\) | |
\({K}_{D}=0.7499\) | \({K}_{D}=1.9999\) | \({K}_{I}=-2.0000\) | \({K}_{I1}=1.9878\) | |
Hydro | \(\lambda =0.5021\) | \({K}_{D}=-2.0000\) | \({K}_{I2}=2.0000\) | |
\({K}_{P}=0.3110\) | \(\mu =0.2386\) | \(\lambda =0.5040\) | \({K}_{D}=1.9715\) | |
\({K}_{I}=0.3102\) | Hydro | \(\mu =0.0100\) | \({\lambda }_{1}=0.3023\) | |
\({K}_{D}=0.0034\) | \({K}_{P}=1.2029\) | Hydro | \({\lambda }_{2}=0.3013\) | |
Gas | \({K}_{I}=1.1908\) | \({K}_{P1}=1.7433\) | \(\mu =1e{-}9\) | |
\({K}_{P}=0.0091\) | \({K}_{D}=-0.4375\) | \({K}_{P2}=-2.0000\) | Hydro | |
\({K}_{I}=1.2409\) | \(\lambda =0.5072\) | \({K}_{I}=-2.0000\) | \({K}_{P1}=2e{-}7\) | |
\({K}_{D}=0.6901\) | \(\mu =0.3748\) | \({K}_{D}=-1.3626\) | \({K}_{P2}=0.1269\) | |
Gas | \(\lambda =0.9506\) | \({K}_{I1}=2.1e{-}8\) | ||
\({K}_{P}=0.9272\) | \(\mu =0.0877\) | \({K}_{I2}=1.9642\) | ||
\({K}_{I}=2.0000\) | Gas | \({K}_{D}=1.8441\) | ||
\({K}_{D}=1.9999\) | \({K}_{P1}=-2.0000\) | \({\lambda }_{1}=0.0061\) | ||
\(\lambda =1.2105\) | \({K}_{P2}=-1.7815\) | \({\lambda }_{2}=1.0000\) | ||
\(\mu =1.2682\) | \({K}_{I}=-2.0000\) | \(\mu =1.5e{-}6\) | ||
\({K}_{D}=-2.0000\) | Gas | |||
\(\lambda =0.8692\) | \({K}_{P1}=1.5012\) | |||
\(\mu =1.2000\) | \({K}_{P2}=2.0000\) | |||
\({K}_{I1}=2.0000\) | ||||
\({K}_{I2}=2.0000\) | ||||
\({K}_{D}=1.6228\) | ||||
\({\lambda }_{1}=1.1267\) | ||||
\({\lambda }_{2}=1.1e{-}4\) | ||||
\(\mu =1.3075\) | ||||
Ts (s) (2% band) | ||||
\(\Delta {F}_{1}\) | 15.69 | 8.31 | 1.81 | 1.74 |
\(\Delta {F}_{2}\) | 20.05 | 6.25 | 4.51 | 3.53 |
\(\Delta {P}_{tie}\) | 17.82 | 6.51 | 3.98 | 2.86 |
\({J}_{ITAE}\) | 0.9197 | 1.557 | 0.0461 | 0.0259 |