## 3 Hierarchical State-of-the-Art Control
### Overview of the Inner-Loop Control Methods  
As previously mentioned, the use of power electronic interfaces is able to make the microgrid operate in both islanded and grid-connected modes. Thus, VSI realizes the flexibility of microgrid operation modes by relying upon VCM and CCM behaviors. For instance, VCM is used in islanded mode to maintain the stability of voltage. In contrast, in grid-connected mode, CCM is commonly used to inject current into the primary grid. The inner current loop is mandatory to provide the current reference associated with the external voltage control loop. * _Proportional Integral (PI)_: The PI method is the most applied over the years to serve different purposes and is practically employed with the synchronous reference frame associated with _dc_ variables. It delivers numerous benefits, such as less complicated controller designs, applicability in linear time-invariant systems, practicability for the single-input-single-output (SISO) system, etc. For instance, [40] discusses the effects of the fractional order of the PI method for load-sharing revamping, and the same method is proposed in [41] and [42] in their respective applications. The PI controller does not solely correspond on the _dc_ variables but also on the sinusoidal variables. As mentioned in [43], the PI current controller integrators associate with the stationary frame under undesirable and distorted operating conditions. It offers low computational burden with zero steady-state error for current harmonics concerns. * _Proportional resonant (PR)_: A viable alternative to the synchronous frame PI controllers is the stationary frame PR compensators [44]. The PR controller is often interpreted as a sinusoidal controller as it is widely associated with the stationary reference frame (\(\alpha\beta\)) as well as the natural reference frame (\(abc\)) due to its capability of forcing errors to be zero and for its harmonics compensation [45; 46]. Similar methods are proposed in [47] for the distributed power generation systems (DPGS). Studies demonstrate that resonant controllers are an adequate option for inner current control loops to show anticipated performances. It has a high gain around the natural resonant frequency, \(\omega\)[48], as it is able to introduce a low-order harmonics compensator to improve the current controller. It offers no excessive computational resources, fast transient performance, and the capability of zero steady-state error at the AC frequency, among others. * _Dead-beat control (DB)_: Among discrete-time linear controls, dead-beat is the most commonly employed due to its fast transient response in different applications, which makes it a suitable approach for current regulation. The DB control is able to place all the closed-loop poles at zero [49]; thus, the tracking error settles to zero within a few sampling steps equivalent to the SISO systems. Worth mentioning, the DB control is in the predictive family, which forecasts the effect of the control action by tracking the current reference accurately without any error. Some in literature have proposed methods to improve the conventional DB controller's limitations, such as being vulnerable to model uncertainties, sensitive to model and parameter mismatch, etc. For instance, [50; 51] employ an adaptive self-tuning for the pulse-width modulated (PWM) converter by adopting a delay compensation caused by voltage calculation, synchronous frame rotation, and the PWM converter itself in order to be homogeneously relocated outside the closed-loop control system, thus compensating for their effects on the closed loop's stability. The implementation of the current controller with a higher degree of disturbance rejection enables swift current tracking with higher bandwidth qualities. In [52], the study uses a combination of a DB controller with neural-network identification to present nonlinear estimation problems and to operate as a grid voltage estimator to realize a grid voltage sensorless scheme to guarantee high-quality power injection.