## 3 FSMPSCC Assessment with Performance Maps
### Optimal Tuning  
As previously indicated, tuning in FSMPSCC consists of selecting the WF for the CF (6). In [22], the authors propose the use of a different tuning for each operating point of the system. In this way, the compromise between conflicting criteria is tackled independently for each operating point. This is in contrast with the traditional FSMPSCC setup, where the WF is computed considering the range of operation of the system as a whole [9]. This method can be improved as it is realized that for different regions of operation, the terms  
Figure 3: Example of trajectories gathered in the experimental setup for model tuning. present in (6) take different relative values. In the case at hand, it will be assumed that the VSI limitations impose a limit \(U\) on \(f_{sw}\). Then, the WF tuning should provide a \(f_{sw}\) below the limit for all possible operating points, that is, \(f_{sw}<U\) for all \((I^{*},f_{e})\in\Omega\). On the other hand, the tuning should provide the minimum tracking error in \(\alpha-\beta\) for each operating point. This can be expressed as  
\[e_{\alpha-\beta}^{o}=\min_{\lambda_{sc}}e_{\alpha-\beta} \tag{9}\]  
As an example, the performance maps of Figure 4 are presented. They correspond to controller C1 with \(\lambda_{sc}=0.014\) (left) and with an optimized \(\lambda_{sc}\) (right). The first tuning (\(\lambda_{sc}=0.014\)) provides a value of \(f_{sw}\leq 11\) kHz for all operating points using a single value of the WF for the whole operating space; this represents the standard method of tuning the FSMPS. It is interesting to compare this case with the optimized \(\lambda_{sc}\) (right), where a value of \(f_{sw}\leq 11\) kHz is also achieved. The values of \(e_{\alpha-\beta}\) are clearly better for the optimized case. This kind of analysis is allowed by using performance maps since all operating points are easily visualized. In the following, optimized tuning will always be considered. The criterion is that of Equation (8). The method is the independent tuning of \(\lambda_{sc}\) for each VVV set and for each operating point. Since there is only one tuning parameter, almost any optimization method can be used [30]. In our case, a first-order Newton-like method has been used. In order to manage the continuous operating space it is discretized by taking 12 sub-divisions of the load and 12 subdivisions of speed. It must be stressed that in FSMPSCC, the controller configuration (selection of \(\lambda_{sc}\)) is not a trivial task, since the various performance indices (\(e_{\alpha-\beta}\), \(f_{sw}\)) are affected by tuning in a way that depends on the particular operating point. As a result, some operating points might get a reduction in either index at the cost of an increase for other operating points. As an example, consider Figure 5 where the optimal values of \(\lambda_{sc}\) are given for controller C1 for two scenarios: to attain \(f_{sw}\leq 11\) kHz (left) and to attain \(f_{sw}\leq 8\) kHz (right). A non-linear relationship between the optimal WF and the operating points is clearly visible. Please notice that for each operating point, there is a unique value for the WF. The existence of one or several peaks must not be interpreted as a choosing problem. A similar procedure can be followed to assess the capabilities of C2. This controller uses a set of VVV composed of the S + ML and ML + L coronas, plus zero VV. Figure 6 shows the performance map for the C2 controller, \(e_{\alpha-\beta}\) and each operating point, with an optimal value for \(\lambda_{sc}\). These results are quite interesting and remarkable. First, it should be noted that most articles in the literature dealing with VVV focus on C1, and the C2 controller clearly reduces \(e_{\alpha-\beta}\) at many points in the operating space. Second, one might argue that the C2 controller should always outperform C1, given the fact that it incorporates all the VVV that C1 uses plus the extra S + ML combinations. In other words, C2 has more degrees of freedom than C1.