### Appendix

### 1.1 SG, IG, STATCOM-PIDF and load parameters

The terms *K*_{1}*, K*_{2}*, K*_{3}*, K*_{4}*, K*_{5} and *T*_{G} 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.*, R*_{IG} and *X*_{eq} 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: *r*_{1} = \( {r}_2^{\hbox{'}} \) = 0.19 p.u., *x*_{1} = \( {x}_2^{\hbox{'}} \) =0.56 p.u., *s* = − 3.5%.

For SG: *V* = 1.0 p.u., *δ* = 17.2483^{0}, \( {T}_{d0}^{\hbox{'}} \) = 5.0 s, *X*_{d} =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].

*P*_{IG}=0.6 p.u. kW, *Q*_{IG} = 0.291 p.u. kVAr, *P*_{in} = 0.667 p.u. kW, power factor in IG = 0.9, *η* = 90%, *P*_{SG} = 0.4 p.u. kW, *Q*_{SG} = 0.2 p.u. kVAr, *E*_{q} = 1.12418 p.u., \( {E}_q^{\hbox{'}} \) = 0.9804 p.u., *P*_{load} = 1.0 p.u. kW, *Q*_{load} = 0.75 p.u. kVAr, power factor of load = 0.8, *Q*_{com} = 0.841 p.u. kVAr and *α* = 53.32^{0}.

### 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 *G*_{1} = 1.478, *G*_{2} = 3.8347, *K*_{V} = 0.667, *T*_{V} = 0.0007855, *K*_{A} = 200, *T*_{A} = 0.05, *T*_{R} = 0.02, *H* = 1.0, *D* = 0.8, *ω*_{0} = 314.

### 1.3 Different system matrix [*A*] components

Different system matrix [*A*] components are as follow:

\( a0101=-\frac{D}{2H} \); \( a0102=-\frac{K_{11}}{2H} \); \( a0105=-\frac{K_{22}}{2H} \).

\( a0303=-\frac{1}{T_a} \); \( a0304=-\frac{K_a}{T_a} \).

\( a0404=-\frac{1}{T_r} \); \( a0406=\frac{1}{T_r} \).

\( a0503=\frac{E_2}{T_g} \); \( a0505=-\frac{1}{T_g} \); \( a0506=\frac{E_1}{T_g} \).

\( a0605=\frac{K_1{K}_v}{T_v} \); \( a0606=\frac{L}{T_v} \); \( a0610=\frac{G_1{K}_v}{T_v} \).

*a*0706 = − *m* × *Q*; *a*0707 = − *M* × *K*_{i} × *Q*; *a*0708 = − *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 − (*K*_{v} × *E*_{2}) − (*K*_{v} × *G*_{2}) + (*K*_{v} × *K*_{2});

\( m=\frac{T}{N} \); *M* = (*m* × *K*_{p}) + *K*_{d}; *N*_{1} = (*m* × *K*_{i}) + *K*_{p}; \( Q=\frac{1}{M-\left(m\times {N}_1\right)} \)

### 1.4 Algorithms parameters

For GSA [25]: jumping rate, *J*_{r} = 0.3; *G*_{0} = 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%.