# An Optimal Power Control Strategy for Grid-Following Inverters in a Synchronous Frame
## 1 Introduction  
Due to the inherent low inertia present in voltage-source inverters (VSI), microgrid control is becoming an important object of study nowadays in order to improve penetration of renewable energy. Microgrid control is generally defined by levels. In [1], the authors present a hierarchical scheme that divides VSI controllers in different levels. First, voltage-current (V-I) control level is used to regulate the VSI current and voltage waveforms according to a specific reference without having significant harmonic distortion [2]. The power control level, or primary control level, regulates VSI power sharing to the microgrid. This power sharing could occur in grid-connected or islanded mode. For the grid-connected mode, the power grid imposes the voltage amplitude and frequency in the point of common coupling (PCC). In this mode, the VSI behaves as grid-follower. In islanded mode, the voltage and frequency in the PCC is determined by each of the generators in the microgrid. Typically, droop control is used in this control level because it allows to distribute power generation among VSI proportionally without communication. Higher control levels are in charge of maintaining microgrid's power quality and regulating power sharing from a microgrid to the power grid or even another microgrid. Droop control is used with proportional-resonant (PR) or proportional-integral (PI) controllers to regulate power sharing as shown in Figure 1[3, 4]. To share power from the inverter to the main grid, the droop Equations (1) and (2) are used:  
\[\omega = \omega_{0}-\mathrm{m}(\mathrm{P}\,-\,\mathrm{P}_{0}) \tag{1}\] \[\mathrm{E} = \mathrm{E}_{0}-\mathrm{n}(\mathrm{Q}\,-\,\mathrm{Q}_{0}) \tag{2}\]  
where \(\omega_{0}\), \(\mathrm{E}_{0}\), \(\mathrm{P}_{0}\), and \(\mathrm{Q}_{0}\) are the operating frequency, amplitude, and reference values for the active/reactive power, respectively. Coefficients \(\mathrm{m}\) and \(\mathrm{n}\) are known as the droop gains. As analyzed in [2, 5], the stability of the grid-connected IBG is mainly determined by the primary control level parameters. These parameters are the cut-off frequency of the power calculation low-pass filter and the coefficients \(\mathrm{m}\) and \(\mathrm{n}\). Due to the fact that droop controllers do not require a communication link between generators, this control method has become popular for VSI primary control. However, droop controllers are based on the assumption that active and reactive power are directly related to VSI output voltage amplitude and frequency. In addition, droop controllers rely on a low-pass filter that is used to separate the dynamics from the V-I and primary control levels. Thus, some common issues have been widely reported for droop controllers such as performance fulfilment [6, 7, 8], dependency on line impedance values [1, 3, 9], and frequency and voltage deviations from the nominal values [3]. Many approaches for improving droop control performance have been developed [2, 10, 11, 12, 13, 14, 15, 16, 17]. Most of them are based on adding supplementary loops to improve transient response, power quality, harmonic power sharing, or robustness. The gain selection of the supplementary loops is commonly based on heuristic methods and do not rely on an open-loop IBG model to use modern control design methods. Closing several loops of feedback control and designing with heuristic methods does not guarantee robustness of the closed-loop system [18]. For grid-following IBG control, it is common to implement output current controllers. Some of the most relevant publications in this area may be found in [19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. The open-loop state-space models for these controllers allow the use of modern control methods such as LQR, Kalman filters, or robust controllers based on \(\mathrm{H}_{\infty}\) theory. However, these models do not include power sharing dynamics and rely on a higher control level to regulate active and reactive power. Figure 1: Typical droop control structure.