Design of super-twisting algorithm control and observer for three-phase inverter in standalone operation

Received May 5, 2021 Revised Jan 29, 2022 Accepted Feb 5, 2022 This paper develops a new control algorithm of a distributed generation system in the standalone operation. Behavior of three-phase voltage source inverter is investigated and the guidelines for tuning the control parameters are presented. Based on super-twisting algorithm, the proposed controller guarantees the load voltage performance under different types of loads. The proposed controller is established for an inner-loop current controller and an outer-loop voltage controller in a dual control scheme. The proposed scheme is very simple, thus tuning control parameters is easy and the computational burden of the controllers is low. In order to validate the load current of the proposed system feasibility, a reduced-order observer is adopted. The simulation results indicate a more reliable and efficient performance compared to the standard sliding control and the adaptive control.


INTRODUCTION
Last century, there has been a rapid increase in the number of distributed generation systems (DGSs) because of their merits compared to the conventional central generation systems, such as decreasing systems and fuel cost, complying with stricter environmental regulations, and technological advances in small generators, power electronics, and energy storage devices [1]− [6]. The DGSs could be implemented in two different modes that are in stand-alone application (Islanded mode) [7] or grid-connected applications [8]. Especially, the stand-alone mode is applied as an optimal option in some cases, such as: expensive connection, and impractical location rural areas. During the stand-alone applications process, the DGSs implement in independent mode or in a parallel mode by connecting the DG units together. The most remarkable advantage of the parallel mode is good load sharing (harmonic power sharing, reactive power sharing, and real power sharing) by regulating real power and reactive power independently [9]. On the other hand, the independent mode has to maintain good performance (small steady-state error, low harmonic distortion, fast transient response) of the load voltage among complicated scenarios. In this article, the authors concentrate more on each DG behaviors during the independent operation in the Islanded mode.
Recently, several studies have been investigated on designing voltage controllers for three-phase voltage inverters in order to improve the load voltage performance. First, the conventional proportion-integral (PI) technique was presented for a single voltage control loop in [10]. Zhang et al. [11] proposed dual-loop scheme (the PI inner-loop current and the PI outer-loop voltage controllers). Even they maintain a good quality  [13], an adaptive lyapunov-based control strategy for a single-phase uninterruptible power supply (UPS) inverter was proposed. Although this controller showed a good performance in the case of the nonlinear loads, it still consumes more time to validate and evaluate the output response of the micro-grid system. A discrete-time repetitive controller is designed to maintain the performance of the UPS inverters load voltage [14]− [16]. Nevertheless, this method suffers from a series of pitfalls such as reduced conversion efficiency of the inverter or distorted output voltage in the UPS. Lucena et al. [17], a single-phase AC/DC/AC five-leg multilevel converters are proposed to ensure the output voltage quality in case of the unbalanced loads and the nonlinear loads. However, these controllers are too complicated to implement. The (proportional-derivative) PD voltage controller is equipped for DC/DC boost converters via current sensorless approach [18], but this approach could only process in the areas which install the active damping. Besides, one serious weakness of this method is the ability to stabilize the error dynamic. In order to deal with these complex issues, the sliding-mode control (SMC) has been proposed for UPS inverters [19], [21]. Komurgucil [19], precented the UPS inverters are designed by using the rotating-sliding-line-based slidingmode controller. Despite of smaller total harmonic distortion in the load voltage, the designed controller is only for the single-phase inverter. The SMC strategy is approached for doubly fed induction generator using extended active power [20]. Unfortunately, this method neglects to supervise the variable switch frequency of the converter effectively. Therefore, it causes the high chattering problem. The SMC strategy is also proposed to control the four-leg inverter [21]. This control strategy proves the robustness of the SMC system. However, total harmonic distortion (THD) of the load voltage is relatively high in the case of the nonlinear loads. The disadvantage of classical SMC above is characterized by a discontinuous control action which causes the chattering problem increasement. To overcome this problem, a super-twist sliding-mode control (STSMC) is proposed in many literature sources [22]− [28]. The advantage of this technique is characterized by continuous time of the control action. Thus, the chattering phenomenon is eliminated significantly.
In this paper, the STSMC strategy is involved in the dual-scheme with a STSMC inner-loop current controller and a STSMC outer-loop voltage controller under various load types. The method for tuning parameters of the proposed controller is designed. The most remarkable point to emerge from the STSMC controller is the continuous control actions which reduce the load voltage chattering. Hence, the quality of the load voltages is significantly improved. The STSMC controller is sufficiently simple, thus tuning controller parameters is easy and the computational burden of the controller is low. Moreover, a reduced-order observer of the load current is established to reduce the number of current sensors. As a result, the reliability and the cost-effectiveness of the designed system are enhanced. Finally, the simulation results show the good performance of the load voltages under various load types when compared to the standard sliding controller and adaptive controller.
The rest of this novel is divided as follows: The next section indicates a DGS description and mathematical model of a three-phase inverter. The STSMC controller and reduce-order observer designs are described in section 3. The simulation results are reported in section 4. The final section illustrates the brief conclusion.

STANDALONE DGS DESCRIPTION AND MATHEMATICAL MODEL
A brief overview of the DGS based on renewable energy sources is depicted in Figure 1. There are three main parts in standalone DGS: Input powers, local loads, and the power conversion unit. To begin with, the input powers consist of different kinds of renewable energy sources, such as wind, fuel, and solar energy. Secondly, the power conversion unit is established by coupling the input converter (AC/DC converter or DC/DC boost converter), a three-phase inverter with an LC output filter. Finally, the power converter transmits the output voltage and the electric power to the local loads. The quality and reliability of them are regulated by the controllers. In this paper, the input powers and the input converter are replaced by a stiff DC voltage source.
The structure diagram of a three phase DC/AC converter applying the LC filter is presented in Figure 2. The mathematical model of the designed converter can be described as (1). V is defined as a vector output voltage 1 I is defined as a vector current 11 , LC denotes the filter inductance and filter capacitor, respectively. The (1) which places in the stationary frame convert into the following equations in the synchronously rotating d-q frame: where as w is defined as the frequency of the angular, 22 In (2) could be rewritten as:

STSMC TECHNIQUE AND REDUCE-ORDER OBSERVER DESIGN
The proposed controller is designed based on the STSMC technique. To be more specific, a dual structure is established by combining the inner loop of current controller and the outer loop of voltage controller. Both of them are involved the STSMC technique to calibrate the load voltage. In addition, an observer of the load current is established for calculating the voltage loop controller. The proposed STSMC and the observer in stand-alone mode is depicted in Figure 3.  Figure 3. Schematic diagram of the STSMC and observer

Controller design
From in (3), the equation of inner loop current controller can be described as: It reveals that the system in (6) is applied one-order sliding mode control (I-SMC) strategy or twoorder sliding mode control (II-SMC) strategy. As mentioned above, the I-SMC controller causes the variable frequency of the DC/AC inverter switch, therefore increasing the chattering phenomenon. To handle this issue, the 2-STSMC controller known as STSMC is adopted. Consequently, the action control variables consist of two terms [28].
Where ̃ (i=1,2) denote the control variables terms which is designed by using the STSMC technical, denote the equivalent control terms for the nominal and unperturbed model derived by letting ̇= 0, ̃= 0. As a result, the equivalent control terms eqi u are written as, substitution of (7) and (8) into (6) produces: The equation of the control terms i u are derived from [28].
Where the values The STSMC terms ̃3,̃4 are similar to the control terms which described in (10). The block diagram of the STSMC current and voltage controller is shown in Figure 4. This block diagram is composed of the equation blocks. The inputs of the block are the state variables, observer variables, and given constant parameters, while the outputs of the block diagram are the control signals i u . it is obvious that ( ) = | | , taking the time derivative of (15) produces, from (5) and (12), the sliding variables are generalized as, because of the effect of the STSMC, the sliding regime would happen and the sliding variable i s close to zero, so| | ≤ with are the positive constants which close to zero. Considering the most disadvantageous case, in which | | = , and substitution in (17) to (16) produces.
Taking the time derivative of (18), the reflecting the i e error dynamics is obtained, where s is the Laplace operator, the design of the STSMC parameters 12 ,, i i i c k k is made to ensure the poles of the characteristic in (20) equal the poles of the desired characteristic determined as, where s is the Laplace operator, the design of the STSMC parameters where the pole of desired in (21) 1 = − is the dominant poles, so the choice of i  is enough high, is the damping coefficient, is the natural frequency. The performance of the error dynamics in (19) is achieved by turning the controller parameters , 1 , 2 to guarantee 2 = 2 , 1 = 1 , 0 = 0 simultaneous fulfillment. From simultaneous implementation 2 = 2 , 1 = 1 , 0 = 0 the controller parameters , 1 , 2 are calculated as,

Load current observer
It is obvious that the STSMC requires the load current information. The information can be obtained by using sensors, but the sensors make the system more expensive and less reliable. Liu et al. [29], the authors applied the full-order observer, which causes a high computational burden in the controller. Hence, the reduce-order observer is utilized as a better option to estimate the current load. The reduced-order observer of load current is represented as,  The design of a reduced-order observer rests on coordinate transformation [30]- [32].
The coordinate transformation is obviously nonsingular for any 1 M . The equation with respect to x is obtained from in (25), (26).
In this section,  (26), the load current observer is obtained as: The block diagram of the load current observer is expressed in Figure 5.

RESULTS AND DISCUSSION
In this section, we utilized the MATLAB/Simulink to evaluate the efficiency of proposed observer and controller. The simulation model of the DGS is defined by the stiff DC voltage source, DC-AC inverter with an LC filter connected to the various loads, the STSMC, and the reduce-order observer. The nominal parameters employed in these simulations are listed in Table 1. During the simulation process, the discrete model is designed with a time step of 2 s  . The switching frequency converters are set at 5 kHz. The parameters of STSMC for the current loop control and the voltage loop control can be turned as: 1 = 500, 1 = 1, 1 = 100, 1 = 0.05, 1 = 100; 2 = 500, 2 = 1, 2 = 100, 2 = 0.05, 2 = 100 3 = 2500, 3 = 1, 3 = 100, 3 = 0.05, 3 = 100; 4 = 2500, 4 = 1, 4 = 100, 4 = 0.05, 4

= 100
The variables of the matrix 1 M can be chosen as follows: 1 = 0.005, 2 = 0.01.   Tables 2 and 3 respectively.  Table 3 compares the THD results between the proposed method and the other controllers among different types of loads. Generally, it is clear that the THD rate of the proposed method in the case of the balanced resistive load is the lowest, at about 0.05%. By contrast, the figures for the standard sliding controller and the adaptive voltage controller are much higher, nearly 2.15% and 0.094% respectively. In addition, for the nonlinear load, the THD proportion of the proposed control is noticeably low, at about

CONCLUSION
In this paper, the STSMC of a three-phase inverter is designed to ensure the good performance of the load voltage among various load conditions. The proposed STSMC is developed for the outer-loop voltage controller and inner-loop current controller. Hence, it is not only easy to implement but also requires little computational effort. Furthermore, we have found an innovative reduced-order observer to estimate the load currents precisely. Hence, the parameter calculations are more robust to system disturbances. However, one minor limitation of our research is the harmonic distortion of the inverter current in the case of the nonlinear loads. As is well known, the high THD of the inverter current is extremely hard to control and could not be eliminated completely. We propose that further research should be undertaken in this mentioned area. In general, the paper points out the advantages of the proposed controller and observer based on the MATLAB/Simulink. The simulation results have indicated the feasibility and effectiveness of the proposed approach.