## 1 Introduction  
The behavior of the power system in the modern age is highly complicated due to the presence of non-linear loads, devices, and green energy resources. The presence of such systems or devices raises power quality issues. The flow and reliability of the power include the use of robust optimization and artificial intelligence control. A comprehensive review and details related to this area are reported in [1]. The application area of the AC-AC converters includes grid voltage compensation during the voltage sags or swells and speed control systems of industrial loads such as grinding mills employed in mining and steel rolling processes. Currently, the power quality of the grid voltage is highly affected due to the use of non-linear devices and loads in modern power systems. The disturbance of the power system includes the fluctuations of load voltage or unequal voltage problems in a three-phase system. The short duration voltage variation may be categorized as voltage sag once line rms voltage is reduced. It is known as voltage swell if line rms voltage increases beyond a certain level or value. The correction of such voltage variation can be achieved with a series injection of voltage. These voltage compensators may include flexible AC transmission devices, also known as FACTS devices or controllers [2; 3]. The line voltage can be corrected by adding and subtracting injected voltage to increase and decrease the overall line voltage once there is a reduction or an increase in the line voltage [4]. These features of the injected voltage can be practiced with the inverting and non-inverting output voltage forms of the converters. The traditional voltage injection approach converts the DC voltage of a DC source to AC with the help of a voltage source inverter. A transformer at the final stage facilitates the injection of inverter voltage to the line voltage [5; 6]. The limited capacity of the DC sources may avoid the solution for longer-duration problems. This drawback is tackled by obtaining the DC voltage from the AC line by employing an AC-to-DC converter. In both previous approaches for voltage compensation, a DC link capacitor is always present between the DC source and inverter, and its use leads to the problem of low reliability [7; 8; 9]. In these converters, improving the power quality for input and output is a big challenge. This issue is addressed and tackled in the converters implemented with the multilevel approach, as reported in [10; 11; 12]. This power conversion approach may not be effectively employed for the application where voltage regulation or frequency regulation at a discrete level is required. The use of direct AC-AC converters avoids the use of a DC-link capacitor to reduce the overall size and increase its reliability, as well as the life of a power converting system. The initially developed direct AC-AC converters have only unipolar voltage gain [13], meaning that their output can only be a non-inverted form of the input voltage with voltage buck and boost operation and the inverted version of the input for voltage buck-boost operation. These converters cannot solve all power systems' issues and cannot be used to govern the output frequencies. Bipolar voltage gain is the mandatory requirement to provide the solution for the preceding problems. The bipolar voltage gain capabilities of the output in the direct AC-AC converters can be achieved through various power converting topologies such as Z-source converters [14; 15], PWM control with unified power converting approach [16], and switching cell structure [17]. The z-source structure is initially introduced in high switching frequency inverters to avoid the possibility of the short-circuiting of the input DC source as two high-frequency operating transistors in an inverter leg may remain on at the same time. This may happen as there may be some internal defect in the manufacturing process of the switching transistor, or it may be due to a fault in the generation of the control inputs of the operating transistors.