Model Predictive Control for Three-phase Grid-Connected Inverters

Demands of renewable energy are increasing due to its effectiveness and sustainability. However, this energy source depends much on the weather and is unstable. Therefore, it needs to be connected to the power network via grid-connected inverters using power electronics devices. The power quality of inverter outputs depends much on the control strategy and modulation. The conventional control methods such as the proportional-integral (PI) and proportional resonance (PR) use the control loops and depend on the controller coefficients. The hysteresis current control method offers the best dynamic response. However, its switching frequency is very difficult to control. This paper presents a method basing on the model predictive control. In the proposed method, the inverter switching states are optimally chosen to minimize the cost function. This helps inverters reduce the switching counts while ensuring the low output harmonics. Thus, this can help inverters decrease the switching loss. The simulation results on Matlab/Simulink have validated the effectiveness of the proposed control method compared with that of the hysteresis current one. Keywords— Grid-connected inverters, current harmonics, PR control, hysteresis control, model predictive control.


I. INTRODUCTION
The electric systems using renewable energy through the three-phase grid-connected inverters are increasing [1]. The power quality of inverter outputs depends much on the control strategies. There are many types of current controllers used for the three-phase grid-connected inverters such as PI, PR, and hysteresis current (HC). The PI and PR controllers are often used very popular in the control of grid-connected inverters due to their simplicity. However, the quality of these controllers depends much on the controller coefficients. In addition, the controller coefficients adjusted to increasing the dynamic response of these controllers make the overshoot increase. This can cause overcurrent and damage power electronic devices. Meanwhile, the HC controller offers the fast response and low overshoot [2]- [7]. However, this HC controller has the switching frequency to vary in a wide range and difficult to control. Therefore, in order to keep the switching frequency constant, the HC controller needs to apply the adaptive hysteresis band as proposed in [6], [8]. This leads to the complex calculation of digital signal processors. In addition, the use of the three independent HC controllers for the three phases makes the switching states difficult to be optimal. This leads the number of switching commutations of the HC controller to increase high. When increasing the hysteresis bandwidth to reduce the switching frequency, the inverter output harmonics increase significantly. The current harmonics of the inverters cause negative effects for the power quality of the power network [9]. In order to ensure the electric energy operation and transmission are safe and stable, the grid codes are promulgated by the electric system operators such as IEEE-929 (2000) [ [12] are also applied for grid-connected inverters. Then, the conventional control methods can cause overcurrent for IGBTs due to the high overshoot.
Moreover, the digital control platforms basing on DSPs become very popular due to the semiconductor technology development and suitable for the discrete control. This helps the digital control methods increase advantages. In which, a method basing on the model prediction will promote these benefits. The method of model predictive control (MPC) can completely reject the control loops. However, the MPC in [13] has not been used popularly in the field of grid-connected inverters because of the dependence of the system parameters [14]- [16]. Therefore, this paper proposes a control method of three-phase grid-connected inverters using the model predictive control. Due to its good dynamic response, the HC control method will be described in Section II to make the fundamentals compare with the MPC method. The MPC method is also presented in detail in Section III. The results and discussion in Section IV will show the effectiveness of the MPC method compared with that of the HC one. The harmonics and switching counts are also considered quantitively. In addition, a strategy for decreasing the number of switching commutations is also proposed in this section. This strategy will help inverters reduce the switching loss. Section V will include the advantages of the MPC method.

II.
HYSTERESIS CURRENT CONTROL A common grid-connected inverter has a structure as Fig.  1. The required active and reactive powers of the system needed to inject into the grid will be calculated according to the reference currents Id-ref and Iq-ref respectively. A phase-locked loop (PLL) is used to extract the grid voltage angle . This angle is used to convert the currents Id-ref and Iq-ref in the synchronous frame dq into the three phase currents as Fig. 2. In the HC control method, three reference phase currents are compared with three output phase currents of inverter [6], [8] respectively. Thus, there are three HC controllers used in this method.  The switching principle of each phase current is showed in Fig. 3. Then, the pulse-width modulation (PWM) depends on the hysteresis bandwidth (HB). The switching frequency is difficult to control. Especially, in the regions of small current value, the switching frequency can increase highly. To solve this issue, an adaptive hysteresis bandwidth can be used [8], [17]. However, the calculation will be more complex. So, in reality, the fixed HB is often used [7]. A principle model on Matlab/Simulink in Fig. 4, using the HC control method, has a 2-level 3-phase inverter with 6 IGBTs as Fig. 5. The outputs consist eight switching states of 3 phases, Sa, Sb, and Sc. The space vector modulation and development of DSP help the MPC be popularly applied [18]. Moreover, the control concepts in the MPC are also very intuitive. The principle diagram of the MPC method is showed in Fig. 6 and its operational principle is also showed in Fig. 7. The state space model is described as (1).

Cost function minimization IGBT inverter
x k Ax k Bu k y k Cx k Du k (1) Where k is the sample instant. Then, the cost function is described as (2) and represents the expected response of the system. In the 2-level 3-phase inverter, the number of states is defined as (3) and showed in Fig. 8.
Then, the cost function of the current control using the MPC will be defined as follows.  The phase voltage equations will be defined as  (8) Then, the voltage vector V can be inferred as  (12) Then, the inverter output voltage will be as follows.
Ri L e dt (13) The derivative of the current in the discrete domain with the sampling cycle Ts according to the forward Euler method will be as follows. (14) The predictive current at the time k+1 will be as. (15) Where ê(k) is the estimated grid voltage. The algorithm of the MPC is showed in Fig. 9. Where the weight factor Lamda () is used to consider the reduction of the number of switching commutations. S is calculated as (16) and is the sum of switching commutations and x is the phases A, B, and C respectively.
A simulation model on Matlab/Simulink using the MPC is showed in Fig. 9. The dynamic circuit parameters are the same as those of the Fig. 4. In the MPC model, the controller uses the block Matlab function in Simulink.   The system parameters are showed in Table 1   However, the current steady state error of the HC method is significantly high. This is showed clearly in Fig. 14(a), in which the squares of the current errors of the HC method in the red are higher than 5 A. While those of the MPC one in the black are always lower than 4 A. The phase A current harmonics of the two methods are also measured at the final fundamental period of each interval and showed in Table 2. In addition, the number of switching commutations in each fundamental period in Fig. 15, with =0, has showed that the switching counts of the MPC in the black are lower than 80. While those of the HC method in the red are always higher than 80. When considering the reduction of switching count with the weight factor =0.01, the number of switching commutations of the MPC, in the blue in Fig. 16, is always lower than 65. Although the number of switching commutations of the MPC is lower than those of the HC, the current THD of the MPC is still lower than that of the HC in Fig. 17. CONCLUSION This paper has presented a method using the model predictive control for three-phase grid-connected inverters. An algorithm for reducing the number of switching commutations is also proposed by including the weight factor in the cost function of the MPC. The simulation models of the MPC and HC methods built on Matlab/Simulink are used for verifying the proposed algorithm. The performance of the proposed MPC method has also been validated when comparing the simulation results of the MPC with those of the HC one. The current harmonics, the number of switching commutations, overshoot, and settling time are also considered quantitively.