# Comparative Analysis of Current Control Techniques to Support Virtual Inertia Applications +
## 2 Virtual Inertia and Current Control Techniques
### Proportional-Integral Controller  
The current control of grid-connected inverters using PI controllers is a mature control strategy. The design method is fairly simple and straightforward, but has limitations especially in terms of the transient performance. The main distinct advantages are that the switching signals have a constant frequency and the harmonics are well-defined. The PI control method used for virtual inertia emulation uses a nested loop structure with two loops. An outer loop generates the reference power based on Equation (1), and the reference current is then calculated. The faster inner loop is a current control loop used to control the output current of the inverter to this reference value. An inherent drawback of the PI controller is that it cannot track sinusoidal current references, thus exhibiting a steady-state error [19]. To overcome this limitation, a transformation into the direct-quadrature (\(dq\)) synchronous reference frame is used, which transforms the three-phase \(abc\) currents into constant \(dq\) currents, which can be tracked by PI controllers. Furthermore, this allows the application of linear control techniques for the design of the current controller. This approach is classified as synchronous \(dq\) frame control in the literature [20].  
Figure 2 shows the control structure of a PI Type-2-based current control of the grid-connected inverter along with the benchmark that will be used throughout this paper. The PI Type-2 controller is a variation of the traditional PI controller with an extra pole at the high frequency range to compensate for the high frequency switching noises. A PLL is used to track the phase \(\omega t\) of the grid voltage [20; 21]. The three-phase inverter output current is continuously measured using current sensors and first transformed into the \(dq\) domain using the Park transformation (\(abc\to dq\)). The cross-coupling terms (\(\omega LI_{d}\) and \(\omega LI_{q}\)), which are usually neglected during the design of the PI controller, can also be compensated in the control structure, which improves the dynamics of the controller [20]. Moreover, the grid voltage \(v_{abc}\) is also transformed into the \(dq\) domain and fed forward, which helps in stabilizing the output voltage of the inverter faster and also aids in the synchronization process [22]. The output of the PI controllers is then transformed back into the \(abc\) domain using the inverse Park transformation (\(dq\to abc\)) and used as modulating signals for the pulse width modulation (PWM) modulator. The transfer function of the plant (i.e., a grid-connected inverter with an LC filter) for the design of the PI controller can be modeled as [22]:  
Figure 1: Operating principle of a virtual inertia emulation system.  
\[G_{plant}(s)=\frac{\frac{1}{L}}{s+\frac{r}{L}} \tag{2}\]  
where \(r\) represents the internal resistance of the filter inductor and \(L\) is the inductance of the filter inductor. The effect of the capacitance \(C\) can also be neglected from the point-of-view of the current control design. A modified form of a PI controller, referred to as a PI Type-2 controller, which includes a high frequency pole in the transfer function compared to a classical PI controller [23], is used in this paper. The transfer function is given by:  
\[C_{PI}(s)=K_{PI}\frac{1+s\tau}{s\tau}\frac{1}{1+sT_{p}} \tag{3}\]  
where \(K_{PI}\) is the controller gain, \(\tau\) is the time constant, and \(T_{p}\) is the time constant of an additional high frequency pole introduced to reduce the effects of higher frequency switching noises.