Fractional calculus entered the control field in the late 1990s, though the concept of fractional order calculus (FOC) dates back to 1695 with [PERSON] and [PERSON]. FOC employs non-integer orders, including complex values, as can be seen in the non-integer compilation provided by [10], offering additional degrees of freedom for various objectives such as better modelling of complex systems or new proposal of controllers [11; 12; 13]. FOC was first applied to a PI controller in 1999 [14]. A summary table of the real order FOC formulae in different controller type adoption can be seen in Table 1[10; 15; 16; 17]. In Table 1, the orders \(n\), \(\lambda\), \(\mu\) and \(\alpha\) of controller actions are real numbers in the range of 0 and 2 (to ensure causality), the terms \(k_{x}\) represent the controller gains, and \(\omega_{x}\) are angular pulsations used for tuning purposes. The promising advancements of FOC in control have led to efforts in robust tuning criteria [18; 19] and dynamic enhancement [20; 21]. In 2018, a formal translation from PR to fractional PR (FPR) controller was established [22], following an initial presentation in 2016 [23]. The FPR was later renamed as fractional ideal PR controller in 2021 [24]. Research [16; 24] shows that FPR controller increases control bandwidth with a lower computational burden than PRHC controllers, though these studies primarily focus on current regulation. The study in [24] presents a stability analysis by examining the trajectory of eigenvalues, but it is still under the current control framework. From [22] to [24], it can be concluded that the fractional component helps to modify the frequency trajectory, mainly in terms of phase lead-lag manipulation. According to the aforementioned studies, when using FPR controllers, the frequency domain response has a potential manipulation capability. This idea can be extended for applications other than controlling the current loop of VSCs. Thus, regarding voltage regulation in VSCs, several studies have explored the application of FOC to PI controllers [25; 26]. For instance, the fractional use of PID in [27] demonstrates improved time response and superior disturbance rejection. However, applying FOC to PR controllers is less widespread in the voltage control framework. The study presented in [28] is one of the few examples of applying FPR to voltage regulation. Although the tuning criterion employed is based on an \begin{table}
\begin{tabular}{l l l} \hline \hline
**Author** & **Controller** & **Transfer Function** \\ \hline A. [PERSON] [10] & CRONE derivative & \(k_{d}s^{\alpha}\) \\ \hline A. [PERSON] [10] & CRONE integrator & \(k_{i}\frac{1}{s^{\alpha}}\) \\ \hline I. [PERSON][14] & \(\text{PI}^{\lambda}\) & \(k_{p}+k_{i}/s^{\lambda}\) \\  & \(\text{PID}^{\mu}\) & \(k_{p}+k_{i}/s+k_{d}s^{\mu}\) \\  & \(\text{PI}^{\lambda}\text{D}^{\mu}\) & \(k_{p}+k_{i}/s^{\lambda}+k_{d}s^{\mu}\) \\ \hline [PERSON] [15] & (\(\text{PI}\))\({}^{n}\) & \(s^{-n}(k_{p}+k_{i}/s^{\lambda})\) \\  & (\(\text{PD}\))\({}^{n}\) & \(s^{-n}(k_{p}+k_{i}g^{\lambda})\) \\  & (PID)\({}^{n}\) & \(s^{-n}(k_{p}+k_{i}/s^{\lambda}+k_{d}s^{\mu})\) \\ \hline [PERSON] [15] & IMC-FOID & \(k_{p}+k_{i}/s^{\alpha}+\frac{k_{d}s^{\alpha}}{s^{\alpha}+1}\) \\ \hline [PERSON] [15] & (\(\text{PI}\))\({}^{\lambda}\) & \((k_{p}+k_{i}/s)^{\lambda}\) \\  & (\(\text{PD}\))\({}^{\mu}\) & \((k_{p}+k_{d}s)^{\lambda}\) \\ \hline [PERSON] [15] & Mod-FOPID & \(k_{c}\frac{(1+\frac{s^{\alpha}}{s^{\alpha}})^{2}}{s^{\alpha}}\) \\ ine [PERSON]. [PERSON] [15] & Non-Linear FOPID & \(k_{d}|D^{\mu}e(t)|^{\delta}sign(D^{\mu}e(t))\) \\  & & \(+k_{i}|D^{\lambda}e(t)|^{\gamma}sign(D^{\lambda}e(t))\) \\ \hline [PERSON] [16] & FPR & \(k_{p}+k_{i}\frac{s^{\alpha}}{s^{\alpha}+\omega_{i}^{2}}\) \\ \hline [PERSON] [15] & Fractional Lead-Lag & \(\left(\frac{1+s/\omega_{i}}{1+s/\omega_{i}}\right)^{r}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: State of the art of the main research on non-integer controllers.