## 3 Materials and Methods
### IRLA-SPM Control System  
The overview of the IRLA-SPM control system is shown in Figure 6; its main task consists of tracking the reference power profile \(P_{m}^{*}\) at best, by complying with all M/G constraints simultaneously. In this regard, it is worth noting that \(P_{m}^{*}\) already complies with torque and power limitations, thus just current limitation (\(C_{i}\)) and voltage saturation (\(C_{v}\)) have to be considered; these can be expressed as  
\[C_{i}:\ \left|i_{dq}\right|\leq\hat{I}_{m}\,\ C_{v}:\ \left|v_{dq}\right|\leq \frac{V_{DC}}{\sqrt{3}}\,\ x_{dq}=x_{d}+j\ x_{q}\,\ x\in\{v,i\} \tag{6}\]  
where \(v_{dq}\) and \(i_{dq}\) denote voltage and current space vectors in the \(dq\) synchronous reference frame, while \(\hat{I}_{m}\) is the maximum M/G current magnitude. Considering now M/G voltage equation, this can be expressed as  
\[v_{dq}=(r+j\ p\omega_{m}L)i_{dq}+L\frac{di_{dq}}{dt}+j\ p\omega_{m}\Lambda \tag{7}\]  
in which \(r\) and \(L\) are the phase resistance and synchronous inductance, respectively, \(p\) is the number of pole pairs, and \(\Lambda\) is the equivalent magnetic flux linkage due to PMs. Electromagnetic torque and power equations can be instead expressed as  
\[T_{e}=\frac{3}{2}p\Lambda\ i_{q}\,\ P_{m}=-T_{e}\ \omega_{m} \tag{8}\]  
in which the minus sign appearing in the torque-power relationship is due to the fact that positive \(P_{m}\) values mean FESS discharging and, thus, negative torque values applied to the M/G shaft. Consequently, since IRLA-SPM torque depends on q current component only, \(i_{d}^{*}\) can be set in accordance with DC-link voltage capability and M/G voltage demand (flux-weakening). In particular, assuming to be at steady-state operation, substituting (7) into (6) yields  
\[C_{i}:\ \left|i_{dq}\right|\leq\hat{I}_{m}\,\ C_{v}:\ \left|i_{dq}-\varsigma \right|\leq\rho \tag{9}\]  
\[\varsigma=-\frac{p\omega_{m}\Lambda}{r^{2}+\ (p\omega_{m}L)^{2}}(p\omega_{m}L+j \ r)\,\ \rho=\frac{V_{DC}}{\sqrt{3}}\frac{1}{\sqrt{r^{2}+\ (p\omega_{m}L)^{2}}}. \tag{10}\]  
Figure 5: M/G power limitation occurring at _ready_ operating mode. The graphical representation of (9) onto the (\(i_{d}\),\(i_{q}\)) plane is shown in Figure 7, together with the torque locus defined by (8). Since \(\varsigma\) moves to the left-side of the plane and \(\rho\) significantly decreases with the rotor speed, it becomes apparent that negative \(i_{d}^{*}\) values are required at high-speed operation because both \(C_{i}\) and \(C_{v}\) must be satisfied. Among the several flux-weakening methods proposed in the literature for determining \(i_{d}^{*}\)[34], the voltage-follower PI regulator is here considered [35]; particularly, as soon as M/G voltage demand exceeds \(V_{DC}\), this deficit is processed by a PI regulator in order to inject a suitable demagnetizing d current component, leading to flux-weakening operation, as highlighted in Figure 6. This approach is simple and effective, although it suffers from relatively low dynamic performance; this is because increasing PI gains excessively would lead to d current component injections at dynamic operations, in correspondence of which M/G voltage demand is high but just transiently. Consequently, a trade-off between dynamic and steady-state performance has to be achieved. Once \(i_{d}^{*}\) has been determined by limiting it in accordance with \(\hat{I}_{m}\), both \(i_{d}^{*}\) and \(\hat{I}_{m}\) concur to limit \(i_{q}^{*}\), which is preliminary determined in accordance with \(P_{m}^{*}\). Both \(i_{d}^{*}\) and \(i_{q}^{*}\) are then tracked by means of two PI-based current loops, which can be designed in accordance with (7) but expressed as  
\[\widetilde{v}_{dq}=r\,i_{dq}+L\frac{di_{dq}}{dt}\,,\,\,\widetilde{v}_{dq}=v_{ dq}-j\,\,p\omega_{m}\Big{(}\Lambda+Li_{dq}\Big{)}. \tag{11}\]  
In particular, M/G phase currents (\(i_{uvw}\)) are converted into the corresponding dq space vector through the Clarke and Park transformations.