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Table 3 Details of benchmark-test-functions [39]

From: Hybrid intelligence approach for multi-load level reactive power planning using VAR compensator in power transmission network

Benchmarks Range Functions Type GOV
Sphere [− 100, 100] \( f(u)=\sum \limits_{i=1}^d{u}_i^2 \) UM 0
Step [−100, 100] \( f(u)=\sum \limits_{i=1}^d{\left({u}_i+0.5\right)}^2 \) UM 0
Dejong’s [−1.28, 1.28] \( f(u)=\sum \limits_{i=1}^d\left[{iu}_i^4+\mathit{\operatorname{rand}}\left(0,1\right)\right] \) UM 0
Ackley’s [−32, 32] \( f(u)=-20\exp \left(-0.2\sqrt{\frac{1}{d}\sum \limits_{i=1}^d{u}_i^2}\right)-\exp \left(\frac{1}{d}\sum \limits_{i=1}^d\cos \left(2\pi {u}_i\right)\right)+20+e \) MM 0
Griwank [−600, 600] \( f(u)=\sum \limits_{i=1}^d\frac{u_i^2}{4000}-\prod \limits_{i=1}^d\cos \left(\frac{u_i}{\sqrt{i}}\right)+1 \) MM 0
Schwefel’s [− 500, 500] \( f(u)=\sum \limits_{i=1}^d\left[100{\left({x}_{i+1}-{x}_i^2\right)}^2+{\left({x}_i-1\right)}^2\right] \) MM 0
  1. UM uni-modal, MM multi-modal, GOV global optimum value