## 3 \(\mathcal{H}_{\infty}\)-Based Control Strategies
### Design Alternative 3: Voltage Control Based on \(\mathcal{H}_{\infty}\) Synthesis  
The main purpose of this design is to obtain a voltage controller based on \(\mathcal{H}_{\infty}\) synthesis using the same sensors (voltage \(V_{\text{C}}\) and current \(I_{\text{W}}\)) as the calculated controllers in the previous subsections. Considering that the controller is digitally implemented with a 0.25 ms sampling rate, the plant described in (1) and (2) has been discretised by assuming a zero-order hold (ZOH) input. Then, a time delay of one period \(z^{-1}\) has been introduced, to include the PWM effect and the computational delay [25].  
The following step consists of obtaining the generalised plant. Figure 7 shows the input weights \(W_{du}\), \(W_{j}\) and \(W_{p}\) and the output weights \(W_{u}\), \(W_{i}\) and \(W_{y}\) in the generalised plant.  
\begin{table}
\begin{tabular}{c} \hline \hline
**PR Tuned Using \(\mathcal{H}_{\infty}\) Synthesis** \\ \hline Voltage control: \(G_{PR,V}\): \(K_{P,V}=0.881\), \(K_{R,V}=1180.0\) \\ \hline Current control: \(G_{PR,I}\): \(K_{P,I}=0.709\), \(K_{R,I}=924.0\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Proportional–resonant control parameters.  
Figure 6: Weights for cascaded \(\mathcal{H}_{\infty}\) design: Current (**left**); Voltage (**right**).  
As commented before, the uncertainty must be \"plugged in\" the weights to obtain the optimal \(\mathcal{H}_{\infty}\) controller:  
* \(W_{p}\) is the reference weight, considering that the system is scaled to pu values. The weight is a constant, and it is fixed to 1 to ensure reference shaping, as in the prior \(\mathcal{H}_{\infty}\) designs;
* \(W_{du}\) is the weight of a perturbation at the control action. This weight is fixed to 0.01, considering a 1% perturbation at the input. In this design, \(W_{du}\) is lower than the previous designs because now the complete model (LC filter plus WTG transformer) has been considered, not just the L transfer function \(G_{1 ru}\);
* \(W_{f}\) is a weight of perturbation at the current \(I_{W}\). This weight is fixed to 0.02, considering a 2% perturbation at the current;
* \(W_{u}\) is the control action weight. This weight is fixed to 0.2 to allow the controller to apply an increment of 5 at the control action (this means 1.1 volts per each required volt, in absolute value);
* \(W_{i}\) is the current weight. This weight is unnecessary but is introduced to limit the current dynamics. This weight is fixed to 0.02;
* \(W_{err}\) is the voltage error weight. This weight assumes the same consideration as the weights \(W_{errV}\) and \(W_{errt}\) for the previous controllers. Hence, this weight is also tuned to achieve a closed-loop \(\mathcal{H}_{\infty}\) norm less than 1. The resulting values of \(K_{Low}\), \(W_{err-res}\) and \(W_{High}\) from \(W_{err}\) for this weight are listed in Table 5.  
The selected weights used for the controller synthesis are shown in Figure 8.  
Equation (20) shows the obtained \(\mathcal{H}_{\infty}\) controller. Table 5 shows the used weights in the generalised plant that allow the controller synthesis using Matlab's hinfsyn function. The selected weights achieve closed-loop \(\mathcal{H}_{\infty}\) norm less than 1. This implies that the frequency responses of all outputs from the generalised plant remain below the inverse of their corresponding weights for a disturbance input spectrum below the specified weights in the generalised plant.  
Figure 8: Weights for \(\mathcal{H}_{\infty}\) design of the single-loop voltage control.  
Figure 7: Generalised plant of the single-loop voltage control for the grid-side converter of a grid-forming WTG.  
The obtained cascade controller \(K(z)\) is composed of a voltage and a current controller:  
\[K(s)=\big{[}\ K_{\mathcal{H}_{\infty},V}\quad K_{\mathcal{H}_{\infty},I}\ \big{]}. \tag{20}\]