## 2 FPSO and FPSO-NTE  
[PERSON] and coworkers [34; 35] first proposed FPSO, which introduced a fractional-order derivative in PSO for rearranging and modifying the order of the velocity derivative. In a fractional-order derivative, common definitions include the Riemann-Liouville definition, the Caputo definition, and the Grunwald-Letnikov definition [55; 56; 57]. In the FPSO proposed by [PERSON] and coworkers [34; 35], the order of the velocity derivative is derived from the Grunwald-Letnikov definition, as shown below:  
\[D^{\lambda}[x(t)]=\lim_{h\to 0}\left[\frac{1}{h^{\lambda}}\sum_{k=0}^{+ \infty}\frac{(-1)^{k}\Gamma(\lambda+1)x(t-kh)}{\Gamma(k+1)\Gamma(\lambda-k+1)}\right], \tag{1}\]  
where \(D\) is the derivative operator, \(\lambda\) is the fractional order of the derivative, \(h\) is the difference operator, \(\Gamma\) is the Gamma function, and \(x(t)\) is the object for the derivative. For an application in discrete time, Expression (1) can be approximated using:  
\[D^{\lambda}[x(t)]=\frac{1}{T^{\lambda}}\sum_{k=0}^{r}\frac{(-1)^{k}\Gamma( \lambda+1)x(t-kT)}{\Gamma(k+1)\Gamma(\lambda-k+1)}, \tag{2}\]  
where \(T\) and \(r\) are the sampling period and the truncation order, respectively. The update expression of the velocity for PSO is:  
\[V_{i}(t+1)=V_{i}(t)+c_{1}r_{1}\Big{(}P_{i}^{l}(t)-p_{i}(t)\Big{)}+c_{2}r_{2}( P^{S}(t)-p_{i}(t)). \tag{3}\]  
The update expression of the position for PSO is:  
\[P_{i}(t+1)=p_{i}(t)+V_{i}(t+1), \tag{4}\]  
where \(i=1,2,\,\ldots,\,S;S\) is the number of particles; \(t\) is the current iteration; \(\lambda\) is the fractional order of the derivative; \(V_{i}(t)\) is the velocity; \(c_{1}\) and \(c_{2}\) are the cognitive and social coefficients, respectively; \(p_{i}(t)\) is the current position; \(P_{i}^{l}(t)\) and \(P^{S}(t)\) are the position of the self-best solution in the current iteration and the position of the global solution in the current iteration, respectively; and \(r_{1}\) and \(r_{2}\) are random constants between 0 and 1. In Expression (3), the position term \(p_{i}\) of \((P_{i}^{l}-p_{i})\) is rewritten as follows using fractional-order derivation:  
\[ \begin{array}{ll}D^{\lambda}[p_{i}(t+1)]=&\sum\limits_{k=0}^{r}\frac{(-1)^{k }\Gamma(\lambda+1)x(t-k)}{\Gamma(k+1)\Gamma(\lambda-k+1)}\\ &=\lambda p_{i}(t)+\frac{\lambda}{2}(1-\lambda)p_{i}(t-1)+\frac{\lambda}{6}(1 -\lambda)(2-\lambda)p_{i}(t-2)\\ &+\frac{\lambda}{24}(1-\lambda)(2-\lambda)(3-\lambda)p_{i}(t-3).\end{array}  \tag{5}\]  
According to [PERSON] et al. [34] and [PERSON] et al. [36], \(r\) is set to 4 to obtain the best balance of convergence rate and accuracy. Additionally, tests show that an \(r\) larger than 4 obtains the same results. Therefore, the updated expression of the velocity of the FPSO is:  
\[ \begin{split} V_{i}(t+1)=V_{i}(t)+c_{1}\cdot r_{1}\left[P_{i}^{l}(t) -\lambda p_{i}(t)-\frac{\lambda}{2}(1-\lambda)p_{i}(t-1)-\frac{\lambda}{6}(1- \lambda)(2-\lambda)\right.\\ p_{i}(t-2)-\frac{\lambda}{24}(1-\lambda)(2-\lambda)(3-\lambda)p_{ i}(t-3)\left]+c_{2}\cdot r_{2}(P^{g}(t)-p_{i}(t)).\end{split}  \tag{6}\]  
To improve the PSO performance and effectiveness, [PERSON] et al. [16] and [PERSON] and [PERSON] [58] introduced a non-linear NTE in PSO. [PERSON] et al. [36] introduced a non-linear time-varying inertia weight in FO-DPSO. In the current study, NTE is integrated in FPSO to enhance the performance and effectiveness. The resulting FPSO-NTE is expressed as follows:  
\[ \begin{split} V_{i}(t+1)=\omega(t)\cdot V_{i}(t)\\ +c_{1}(t)\cdot r_{1}\left[P_{i}^{l}(t)-\lambda p_{i}(t)-\frac{ \lambda}{2}(1-\lambda)p_{i}(t-1)-\frac{\lambda}{6}(1-\lambda)(2-\lambda)p_{i }(t-2)\right.\\ \left.-\frac{\lambda}{24}(1-\lambda)(2-\lambda)(3-\lambda)p_{i}(t -3)\right]+c_{2}(t)\cdot r_{2}(P^{g}(t)-p_{i}(t))\end{split}