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Table 6 Performance parameters of LFC with ISCA, SSA, ALO, SCA and PSO in the case of SLD of 2% in each area

From: Design of a 2-DOF-PID controller using an improved sine–cosine algorithm for load frequency control of a three-area system with nonlinearities

Algorithm

\(\Delta F_{1}\)

\(\Delta F_{2}\)

\(\Delta F_{3}\)

\(\Delta P_{1 - 2}\)

\(\Delta P_{2 - 3}\)

\(\Delta P_{3 - 1}\)

ISCA

Settling time

10.7728

12.9398

13.0042

7.253

6.2267

10.965

Maximum deviation

− 0.00606

− 0.005649

− 0.00630

− 5.744 × 10−5

− 1.9 × 10−5

− 1.972 × 10−5

SCA

Settling time

12.539

12.6376

14.288

16.717

13.6541

21.91

Maximum deviation

− 0.006370

− 0.01117

− 0.00630

− 14.2 × 10−5

− 44.19 × 10−5

− 10.95 × 10−5

ALO

Settling time

19.1768

16.2105

15.315

17.4759

17.58

22.56

Maximum deviation

− 0.007476

− 0.009389

− 0.002249

− 18.9 × 10−5

− 113.35 × 10−5

− 56.62 × 10−5

SSA

Settling time

15.2973

19.9389

19.6692

18.9614

16.5634

25.91

Maximum deviation

− 0.009796

− 0.008697

− 0.008903

49.62 × 10−5

− 73.22 × 10−5

− 44.47 × 10−5

PSO

Settling time

19.3201

19.8754

15.6837

20

20

29.759

Maximum deviation

− 0.013279

− 0.01183

− 0.01385

46.81 × 10−5

− 170.89 × 10−5

− 57.945 × 10−5