An efficient predictive current controller with adaptive parameter estimation in 3-Φ inverter

In this paper, a detail design and description of a predictive current control scheme are adopted for three-phase grid-connected two-level inverter and its application in wind energy conversion systems. Despite its advantages, the predictive current controller is very sensitive to parameter variations which could eventually affected on system stability. To solve this problem, an estimation technique proposed to identify the value of harmonic filter parameter based on Model reference adaptive system (MRAS). Lyapunov stability theory is selected to guarantee a robust adaptation and stable response over large system parameter variation. The simulation results shows the efficiency of the proposed techniques to improve the current tracking performance.


INTRODUCTION
In recent years, the grid-connected inverters have made a giant strides in the various applications of industries and renewable energies , that embody by introduce a new concepts of advanced control in detriment to other conventional controls. The grid-connected inverters is widely used in wind energy systems. The simplified structure of the grid-connected inverters is illustrated in Figure 1. Where, it's possible to replace the maximum amount of power that can be withdrawn from the wind turbine, generator, and rectifier flowed through the inverter by adequate variable DC-current source without causing any damage in system caracteristic [1]- [6].

SYSTEM STRUCTURE AND MODELING
The conventional power circuit of a three-phase grid-connected two-level inverter applied to wind energy is modeled as shown in Figure 2. Where, a harmonic filter (r g ,L g ) takes place between the grid and the inverter. The models of the three-phase inverter and grid current dynamic are briefly described in the following subsections..

Inverter model
The two-level inverter is composed of six bidirectional switches. Since there are three phases with two operating modes in each phase, the inverter is able to generate only 2 3 = 8 states as different possible output voltages. The operating modes of the inverter are summarized as follows [2]: Figure 2. Topology of grid-connected two-level inverter in wind energy conversion systems By means of the space vector tool, The output voltage generated by the inverter can expressed in terms of the different switching states by Where V dc is DC link voltage, v αi and v βi are α and β component of the inverter output voltage vectors. Figure 3 shows the eight output inverter voltage vectors in space vectors corresponding the all possible switching states which including six different active voltage vectors (V 1 to V 6 ) and two other zero vectors (V 0 and V 7 ).
Where v di and v qi are dq-axis components of the inverter output voltage.
Where v dg and v qg are dq-axis components of the grid voltage and ω g is the angular fraquency of the grid, r g filter resistance and L g is filter inductance.
The grid active and reactive powers are expressed by [19]:

VOLTAGE ORIENTED VECTOR (VOC)
In the literature, the phase-locked loop (PLL) is needed not only to find the grid voltage angle θ g for the grid synchronization, but also to realize the voltage oriented control (VOC). Where, the dq-axis components of the grid voltage is oriented to be align only with its d-axis component and eliminate the second component. That means, the active power equation and reactive power equation are simplified to [20] As described previously, It is noted that the d-axis current reference is i * dg related to the grid active power. It can be adjusted through PI controller which maintains the measured DC-link voltage v dc at its given reference value v * dc , we can write The q-axis current reference i * qg can be calculated from reference reactive power of the grid Q * g as

PREDICTIVE CURRENT CONTROL (PCC)
According to the 8 switching possible states for two-level inverter, the predictive current controller exploit the discrete-time model with one-step for grid-connected inverter to predict at the next instant the future behavior of the controlled grid current. It consists of three major subsystems: extrapolation of reference currents, predictive model, and cost function minimization. At each sampling time, cost function values are calculated for all of possible commutation states, based on predefined criterion. Then, the smallest value of the cost function will be used to determine the optimal switching state applied to the inverter at the next period [2]. Approximating the grid current derivatives by [ ❒ ISSN: 2088-8694 The discrete-time model on dq-frame of grid currents at (k + 1) state is expressed as follows: However, the future reference of the grid current i * g,dq (k + 1) can be estimated by means of current and previous value of the reference current with the help of second-order Lagrange extrapolation as follows [2], [3].
In this paper, the first target of the predictive current controller is to achieve the smallest current error between the extrapolated reference currents and the predicted current. The second target is to reduce significantly the switching frequency by adjusting the commutation number between two successive sampling instants. In situations where the switching loses are important. In brief, These all objectives can be expressed in the form of a cost function g to be minimized. The cost function summarizes the desired behavior of the inverter can be obtained using the error squared such as : Where λ sw is weighting factors for switching frequency reduction, S(k) and S(k − 1) are the present switching state and past applied switching state respectively. Figure 4 shows the proposed predictive current controller for the grid-connected two-level inverter. In this scheme, getting the angle θ g from the measured grid voltage is done via PLL method to make v dg equal to V g and also to transforme the variables measured ( v g and i g ) from abc-frame to dq -frame. Once reference grid currents i * g,dq (k) is calculated, the PCC procedure is established to select the optimal switching signal applied to the inverter. Moreover, the scheme involves the MRAS observer to improve the robusrness of PCC by identifying the filter parameter.
The adaptive model is given as follows: i g andî g are the output current of reference model and adaptive model respectively. The measured grid current is tuning as the reference model witch is strongly reliant on the rated filter parameters (r g and L g ), and the estimated grid current is tuning as the adaptive model witch uses a value of estimated filter parameters (r g andL g ), and then, the output current error of two models is used by adaptive controller to adjust the parameter in adaptive controller until the identification becomes asymptotically stable and the current error approaches zero as small as possible.
The current error is defined as e = i g −î g = [ e d e q ] T , the error state equation can be written: Letting The (15) can be written asė Considering a Lyapunov function candidate as follow: V (e, t) = 1 2 (e T P e + ∆ T Z∆) Where P = I 2×2 and A = λ 1 0 0 λ 2 −1 λ 1 and λ 2 are arithmetic number.
Starting from Lyapunov stability concept, the Lyapunov function must respond the following three necessary conditions to ensure that the identification is asymptotically stable [22]- [25]: 1) V (e, t) > 0, . It is evident that the first and third conditions are met. The second condition can be discussed as follows:V (e, t) = 1 2 (ė T P e + e T Pė +∆ T Z∆ + ∆ T Z∆) = 1 2 (e T (P A + P A T )e + a(î d e d +î q e q +ȧ λ 1 ) For P A + P A T = 2αI < 0 it is clear that e T (P A + P A T ) is negative definite. The termV (e, t) must be negative definite, this yieldṡ The adaptive law can be easily expressed as following:

SIMULATION RESULTS
To evaluate the efficiency of the considered predictive current control model with MRAS oberver for grid connected two-level inverter under various values of parameter filter, the whole simulation studies are implemented by means of MATLAB/simulink tools. The main parameters are indicated in Table 1.

Ideal case operation
The performance of the proposed predictive current controller scheme for three-phase grid-connected inverters considering the rated value of filter parameter is tested by applying a DC-link voltage reference V * dc = 1220 V, and a zero reference power reactive Q * g = 0 to unitary the power factor. Figure 6 (a) shows the weighting factor impact λ sw on the grid current quality via Total Harmonic Distortion (THD) and the average switching frequency f sw . It's clearly noted when the weighting factor is increasing, the of the grid current is changed progressively from 1.94 to 3.57 %, while the average switching frequency is reduced considerably from 8.86 to 3.56 [KHz]. For this reason, the weighting factor can be adjusting at 1700 for operating the inverter at f sw = 3.703 [KHz] and for having a suitable value of THD at 3.30 %. Figure 6 (b) shows the obtained simulation result where the DC-link voltage is controlled to maintain at its reference value 1200 V and it is stable during steady state operation. This simulated response is obtained with a PI controller having a k p = 6.90 and a k i = 1.952 10 3 . The dq-axis grid current components either the reference i * g,dq or the measured i g,dq are tracking each other perfectly as shown in Figure 6 (c). Consequently, The amplitude of the reference d-axis grid current component i * dg created from the voltage PI controller is equal to 887.5 A (peak). In addition, the grid-connected inverter is controlled to supply zero reactive power corresponding the zero reference q-axis grid current. Figure 6 (d) illustrates the waveform of a-phase grid current, where the peak value of i ag is equal to i dg .

Non-ideal case operation without MRAS
It's the case where the variation in filter parameter has been considered as follows: • Filter resistance and inductance are 50 % lower than its nominal value • Filter resistance and inductance is also 100 % of its nominal value • Filter resistance and inductance are 150 % higher than its nominal value These values are applying at the following moments 0.1 and 0.2 (s) respectively. Figure 7 shows the effect of parameters variation of both r g and L g on the overall performance of the controller as well as its stability. Under these conditions, it is clear to note the tracking errors between the measured values of i dg , i qg , and V dc and their references, which leads to unbalanced value of THD especially when both the resistance and the inductance are higher than the nominal values as presented in Figure 8.  Figure 8. Simulation results explains the influence of filter parameters variation on the performance of PCC r g and L g are: (a) 50 % lower than its nominal value , (b) aquals to nominal value, (c) 150 % higher than its nominal value. Figure 9 shows the dynamics performance of PCC scheme when the MRAS observer based on lyapunov function is enable under parameters variation. The adaptive gains are tuned using a tuning method (trial and error) to give suitable performance. After some tuning, the adaptive gains are setting as following: λ 1 = 0.625 and λ 2 = 6.795. the simulation results consists of two parts: in the first one, the filter parameters are increasing and switching from 50 % of its nominal value to nominal value, and in the second one the filter parameters are decreasing and switching from 150 % of its nominal value to nominal value. The switching is applied at the moment 0.15 (s). In the both simulation, the MRAS observer track and make the estimated values converges to parameter variation. Moreover, the estimated currentî g and the measured current i g tracks each other. Despite large error at the same time where the switching is applied. the error current e converges to zero. It is worth noting that the estimate value of the filter resistance is more accurate than the estimated value of the filter inductance with steady-state estimation error in range of 1.763 10 −4 and 0.1 % respectively.

Non-ideal case operation with MRAS
To show the robustness of PCC controller with a MRAS observer for a three-phase grid-connected inverters, a comparison of performance can be done through two cases: The first is in agreement with the second scenario, where the filter parameters are changed in the absence of a MRAS observer, while PCC controller is operating on the normal values. As for the second case, which corresponds to the third scenario, the change of the filter parameters are in the presence of the MRAS observer, which estimates the filter parameters so that the PCC controller uses them. The comparison was done by measuring THD and f sw , as shown in Figure 10. Although the filter parameters are changed, the MRAS observer helps the PCC controller to select a switching states that corresponds to the minimal values of f sw with slight difference in THD. 7. CONCLUSION Sensitiveness to system parameter variations and its influences on three-phase grid-connected inverters for wind energy conversion systems has been treating in this paper. A MRAS estimator based on lyapunov function is developed to identify the RL filter parameter. the performance is verified via computer simulation, and various scenario has been studied. the simulation results show that the proposed MRAS estimator gives a good performance and the stability is guaranteed against parameter variations. In the future, a practical implementation of predictive current control model with MRAS observer for grid connected two-level inverter is planned. Moreover, a comparison between the adopted MRAS observer and others techniques such as Neural Network (NN) and Extended Kalman Filter (EKF) will be carried out.