# Enhanced Control Strategy for Three-Level T-Type Converters in Hybrid Power-to-X Systems
## 4 Proposed Control Design
### Inverter Side Inductor Inner Control Loop  
Since the disturbance of the current loop, i.e., the output voltage, is eliminated by the FF action, which makes the system dynamics decoupled from external disturbances, the controller \(C_{1}\)(s) goal is to track the reference inductor current \(i_{L1}(t)\), given by the following:  
\[i_{L1 ref}(t)=K_{I}\sin(\omega_{0}t)u(t) \tag{4}\]  
where \(K_{I}\) is the current amplitude and \(\omega_{0}\) is the grid frequency. Since the current amplitude is produced from the external loop of output voltage control, it is unknown or a requirement for executing a computationally intensive real-time Fourier transform. However, in order  
Figure 5: Single-phase _LCL_-filter-based converter representation. Figure 6: Block diagram of the proposed control structure. to decouple the controller design from the current amplitude information requirement, a control strategy based on time domain transient response of single resonant controllers is proposed. The design procedure is based on setting a closed-loop system response to a sinusoidal reference with a unit step magnitude (Equation (4)) or in the Laplace domain ([35; 36]) as in the following:  
\[I_{L1 ref}(s)=\frac{K_{l}\omega_{0}}{s^{2}+\omega_{0}^{2}}. \tag{5}\]  
the time- and Laplace-domains output current are defined by  
\[i_{L1}(t)=K_{l}\sin(\omega_{0}t)u(t)\big{(}1-e^{-\omega_{C}t}\big{)}, \tag{6}\]  
and  
\[I_{L1}(s)=\frac{K_{l}\omega_{0}}{s^{2}+\omega_{0}^{2}}\cdot\frac{2\omega_{CI}s +\omega_{CI}^{2}}{\left(s+\omega_{CI}\right)^{2}+\omega_{0}^{2}} \tag{7}\]  
where \(\omega_{CI}\)\({}^{-1}\) is the transient time constant of the current loop. Dividing (5) and (7) leads to the following closed-loop transfer function:  
\[\frac{I_{L1 ref}(s)}{I_{L1}(s)}=\frac{2\omega_{CI}s+\omega_{CI}^{2}}{\left(s+ \omega_{CI}\right)^{2}+\omega_{0}^{2}}. \tag{8}\]  
On the other hand, the closed loop transfer function is defined by the following:  
\[\frac{I_{L1 ref}(s)}{I_{L1}(s)}=\frac{LG(s)}{1+LG(s)} \tag{9}\]  
hence, comparing (8) and (9), the corresponding open-loop transfer function can be obtained as the following:  
\[LG(s)=C(s)P(s)=\frac{2\omega_{CI}s+\omega_{CI}^{2}}{s^{2}+\omega_{0}^{2}}, \tag{10}\]  
where the controller can then be derived accordingly as in the following:  
\[C(s)=\frac{2\omega_{CI}s+\omega_{CI}^{2}}{s^{2}+\omega_{0}^{2}}P(s)^{-1}, \tag{11}\]  
where \(P(s)\) is defined in (3). To enhance robustness against frequency variations, a damping term is incorporated into the resonant controller. While adding damping reduces the resonant peak gain, it helps mitigate frequency shifts caused by digital implementation constraints such as finite word length and quantization effects. Without damping, these factors can lead to deviations in the resonant frequency, reducing system performance. The trade-off involved is that increasing damping improves stability but slightly compromises steady-state accuracy. To counteract this, the crossover frequency should be set as high as feasible to maintain adequate control performance. This approach ensures reliable tracking and rejection of disturbances while preventing instability issues commonly associated with ideal resonators. The current controller is then defined with practical modification according to [33] as follows:  
Figure 5:0  
the controller coefficients \(L_{1}\) and \(R_{1}\) are known from the system plant, and the remaining controllers coefficients, i.e., resonant term bandwidth \(\xi\) and transient response frequency \(\omega_{CI}\), have yet to be determined. In order to design the controller, loop-gain shaping, according to [29], is proposed.