Appendix
1.1 SG, IG, STATCOM-PIDF and load parameters
The terms K1, K2, K3, K4, K5 and TG related to the eqs. (1–3) are further described as
$$ {K}_1=\frac{2\ast V\ast {X}_{eq}}{\left({R}_{IG}^2+{X}_{eq}^2\right)} $$
(16)
$$ {K}_2=\frac{V\ast \cos \delta }{X_d^{\hbox{'}}} $$
(17)
$$ {K}_3=\frac{\left(E\ast \cos \delta -2\ast V\right)}{X_d^{\hbox{'}}} $$
(18)
$$ {K}_4=\frac{X_d^{\hbox{'}}}{X_d} $$
(19)
$$ {K}_5=\frac{\left({X}_d^{\hbox{'}}-{X}_d\right)\ast \cos \delta }{X_d^{\hbox{'}}} $$
(20)
$$ {T}_G={T}_{d0}^{\hbox{'}}\frac{X_d^{\hbox{'}}}{X_d} $$
(21)
In (16), the denominator terms i.e., RIG and Xeq are again related to the other IG parameters and may written as
$$ {R}_{IG}=\frac{r_2^{\hbox{'}}}{s}\ast \left(1-s\right)+\left({r}_1+{r}_2^{\hbox{'}}\right) $$
(22)
$$ {X}_{eq}=\left({x}_1+{x}_2^{\hbox{'}}\right) $$
(23)
The different IG and SG parameters value in (16)-(23) are given as [16].
For IG: r1 = \( {r}_2^{\hbox{'}} \) = 0.19 p.u., x1 = \( {x}_2^{\hbox{'}} \) =0.56 p.u., s = − 3.5%.
For SG: V = 1.0 p.u., δ = 17.24830, \( {T}_{d0}^{\hbox{'}} \) = 5.0 s, Xd =1.0 p.u., and \( {X}_d^{\hbox{'}} \) =0.15 p.u.
The other considered data of the proposed STATCOM-PIDF based wind-diesel HPSM are as in [14, 16].
PIG=0.6 p.u. kW, QIG = 0.291 p.u. kVAr, Pin = 0.667 p.u. kW, power factor in IG = 0.9, η = 90%, PSG = 0.4 p.u. kW, QSG = 0.2 p.u. kVAr, Eq = 1.12418 p.u., \( {E}_q^{\hbox{'}} \) = 0.9804 p.u., Pload = 1.0 p.u. kW, Qload = 0.75 p.u. kVAr, power factor of load = 0.8, Qcom = 0.841 p.u. kVAr and α = 53.320.
1.2 Simulink model based power system data
The different constant data used in the simulation of STATCOM – PIDF based wind-diesel HPSM (Fig. 1) are G1 = 1.478, G2 = 3.8347, KV = 0.667, TV = 0.0007855, KA = 200, TA = 0.05, TR = 0.02, H = 1.0, D = 0.8, ω0 = 314.
1.3 Different system matrix [A] components
Different system matrix [A] components are as follow:
\( a0101=-\frac{D}{2H} \); \( a0102=-\frac{K_{11}}{2H} \); \( a0105=-\frac{K_{22}}{2H} \).
\( a0303=-\frac{1}{T_a} \); \( a0304=-\frac{K_a}{T_a} \).
\( a0404=-\frac{1}{T_r} \); \( a0406=\frac{1}{T_r} \).
\( a0503=\frac{E_2}{T_g} \); \( a0505=-\frac{1}{T_g} \); \( a0506=\frac{E_1}{T_g} \).
\( a0605=\frac{K_1{K}_v}{T_v} \); \( a0606=\frac{L}{T_v} \); \( a0610=\frac{G_1{K}_v}{T_v} \).
a0706 = − m × Q; a0707 = − M × Ki × Q; a0708 = − M × Q.
\( a0804=-\frac{MK_v{K}_1}{mT_v} \); \( a0806=-\left(M\times {K}_i\times Q\right)-\frac{ML}{mT_v}-\frac{N_1}{M} \)
$$ a0807=\left(m\times {K}_i^2\times Q\right)+\frac{N_1{K}_i}{M}+\frac{K_i}{mMQ}; $$
\( a0808=-\left(m\times {K}_i\times Q\right)-\frac{1}{mMQ}-\frac{N_1}{M} \); \( a0810=-\frac{MK_v{G}_1}{mT_v} \).
\( a0908=\frac{1}{T_c} \); \( a0909=-\frac{1}{T_c} \).
\( a1009=\frac{1}{T_d} \); \( a1010=-\frac{1}{T_d} \).
Here, L = − 1 − (Kv × E2) − (Kv × G2) + (Kv × K2);
\( m=\frac{T}{N} \); M = (m × Kp) + Kd; N1 = (m × Ki) + Kp; \( Q=\frac{1}{M-\left(m\times {N}_1\right)} \)
1.4 Algorithms parameters
For GSA [25]: jumping rate, Jr = 0.3; G0 = 100; τ = 20. rNorm =2, rPower =1 and ξ =0.0001.
For BGA [15]: number of bits = (number of parameters) × 8 (as considered in the present work), mutation probability = 0.001, crossover rate = 80%.