# An Adaptive Model Predictive Voltage Control for LC-Filtered Voltage Source Inverters
## 1 Introduction  
Nowadays, three-phase inductor and capacitor (LC)-filtered voltage source inverters (VSIs) have found widespread applications in renewable energy resources, such as alternating current (AC) microgrids, distributed generation systems, energy storage systems, and uninterruptible power supplies, to produce a stable sinusoidal output voltage of the desired magnitude and frequency under different load conditions [1, 2, 3, 4, 5, 6]. Several high-performance control schemes have been reported to meet this goal, including proportional-integral (PI) and proportional-resonant (PR) control methods [5, 6, 7, 8, 9, 10], model predictive control (MPC) method [11, 2, 12, 13, 14], sliding mode control (SMC) method [15, 16, 17, 18], and adaptive control methods [4, 5, 19, 20, 18]. PI and PR control methods usually use a cascaded or dual loop structure. An inner-current control loop is used for better disturbance rejection and to improve system stability, and an outer voltage control loop to achieve zero steady-state error and generate the reference signal for the inner-control loop [8, 9, 10]. They are well-known to obtain zero steady-state errors and provide a smooth control input. However, their dynamic is limited due to the cascaded control loops and they employ many integrators and resonators. Also, under nonlinear load conditions, they need harmonic compensators (HCs), which lead to extra complexity and computational burden [7]. Moreover, they are usually designed basedon a continuous-time model. This is contrary to their practical implementation, which is generally based on the powerful digital processors. Therefore, this conflict may lead to some discretization errors. Unlike the PI and PR control methods, MPC is known for coinciding with the discrete nature of VSIs and simplicity in digital implementation. Also, an excellent dynamic performance due to the inverter output voltage's direct regulation is another essential advantage of MPC. Despite these advantages, their performance is sensitive to control system delays and parameter mismatches, which may lead to system instability [1; 2; 11; 12; 13; 21; 22]. Moreover, they need extra load-current sensors to provide good disturbance rejection, increasing system size, cost, and measurement losses of VSIs. To reduce the number of sensors, many sensorless MPCs using different observers and estimators have been proposed so far [1; 11; 12; 23; 24]. However, high sensitivity to model and parameter mismatches has remained a significant drawback. SMC is a robust control method against parameter uncertainties and unknown disturbances. In SMCs, the tracking problem of the n-dimensional system is converted to a stabilizing problem of one scalar variable (called a sliding variable). Therefore, it dramatically simplifies the tracking problem for high-order systems. High chattering in the control input and the infinite time convergence of tracking variables to zero are mentioned as the major drawbacks of this method [14; 15; 16; 17; 25; 26; 27]. The previous control methods are designed for the worst-case scenario and employ a fixed structure with constant control gains, which increases system conservatism. Unlike them, adaptive control is a control method with adjustable control gains. The control gains can be updated based on the adaption mechanism concerning the variable or initially uncertain parameters in the controlled system. Therefore, this control method keeps the system performance and stability at the desired level under different conditions. Generally, adaptive control methods can be classified into two categories: indirect adaptive control (IAC) and direct adaptive control (DAC) [28]. In IAC, at first, system parameters and disturbances are estimated based on a proper online identification technique; afterward, control gains will be updated based on the estimated values. It is worth mentioning that the IAC also reveals the plant information used for other control and protection goals [5; 18; 29; 30].