Research of Active Power Filter Modeling with Grid Impedance in Feedback Linearization and Quasi-Sliding Mode Control

Active power filter (APF) is the most popular device in regulating power quality issues. Currently, most literatures ignored the impact of grid impedance and assumed the load voltage is ideal, which had not described the system accurately. In addition, the controllers applied PI control; thus it is hard to improve the compensation quality. This paper establishes a precise model which consists of APF, load, and grid impedance. The Bode diagram of traditional simplified model is obviously different with complete model, which means the descriptions of the system based on the traditional simplified model are inaccurate and incomplete. And then design exact feedback linearization and quasi-sliding mode control (FBL-QSMC) is based on precise model in inner current loop.The system performances in different parameters are analyzed and dynamic performance of proposed algorithm is compared with traditional PI control algorithm. At last, simulations are taken in three cases to verify the performance of proposed control algorithm.The results proved that the proposed feedback linearization and quasi-sliding mode control algorithm has fast response and robustness; the compensation performance is superior to PI control obviously, which also means the complete modeling and proposed control algorithm are correct.


Introduction
Active power filter (APF) is the most popular device in harmonics compensation; it commonly consists of current filter, electronic converter, and DC-link capacitor.The APF connected to grid and compensated power quality problems [1].
Recently, the interactions between grid connected devices and the grid have drawn greater attention [2].However, most literatures established the model of isolated APF and rarely consider the impact of grid impedance [3].Thus the descriptions of the system based on the traditional simplified model are inaccurate and incomplete.If the mathematical models of the grid impedance and the loads were contained in the APF control system, then the specific effects of each of them could be studied more accurately and completely [4].
In recent decades, nonlinear control theory has made a great progress, especially feedback linearization (FBL) theory based on differential geometry.Feedback linearization control can achieve global linearization by using a certain nonlinear state transformation or feedback transformation [5].All these methods can solve the problem of nonlinear system and obviously improve static and dynamic performance.Yet, this control method depends on an accurate mathematical model and is sensitive to system parameters [6].There are many literatures that adopted feedback linearization in three-phase APF [7] and single-phase APF [8].However, none of them consider the impact of grid impedance.Thus, the optimum performance of APF has not been achieved.
Sliding mode control shows great robustness and fast convergence when the system is running in the sliding surface [9].Thus, slide model control (SMC) is adopted to increase i La i Lb i Lc u sa u sb u sc N Figure 1: Structure of  APF with grid impedance.the robustness in model uncertainties and reduce disturbance responses [10].In [11], sliding mode control has been applied to the three-phase APF and achieved a good result.This paper establishes a precise model consisting of APF, load, and grid impedance.And then it applies exact feedback linearization and quasi-sliding mode control method (FBL-QSMC) in inner current loop.At last, it analyzes the performances of proposed algorithm and compares with traditional PI control algorithm in dynamic respond and simulation.

Mathematical Model of APF with Grid Impedance
The structure diagram of LC APF with grid impedance is shown in Figure 1.Because a three-phase APF system is symmetrical, the system is analyzed with its single-phase circuit [12].The single-phase circuit diagram of an APF with current controller is shown in Figure 2.
In Figure 2,  dc is voltage of DC-link,  dc is capacitor of DC-link,   is output voltage of converter,  1 is output current of converter,  1 / 1 is filter impedance,  is filter capacitor,   is current of APF compensating,   is voltage in load side,   is load current,   /  is equivalent impedance of load,   /  is grid impedance, and   is grid voltage.
From Figure 2, there are the following equations: where   =   −   .The role of inner current loop is keeping source current   tracking with reference current  *  and realizing system unity power factor operation.This paper applies feedback linearization in inner current loop, designed state variables are is  =   , and the output variable is  = ℎ(X) =  1 .The following affine nonlinear equations are obtained. where ] .

Control System Design
Taking Lie derivative for (2), the following is obtained: From (4), the relative degree  = 3 is less than the system dimension  = 4, which means the nonlinear systems are designed to be minimum phase with its zero dynamics.Because the zero dynamics are assumed to be globally stable, the state variables converge and so does the closed-loop system [13].
After taken Lie derivative obtained the variables transformation, the state equations of APF with grid impedance control system altered to Thus, There are Since the state variables converge to zero due to the linear characteristics, the closed-loop trajectories can be well controlled.Furthermore, some assumption must be applied to the fourth equation of ( 7) [13]: That is, Equation ( 9) shows zero dynamics is bounded and exponentially stable.
In (7), (, ) =    2  ℎ() and (, ) =  3  ℎ(), assuming the new input is the original nonlinear state equations can be written as standard linear equation: The input transformation can be expressed as The equivalent control input of feedback linearization is In Lie derivative where ) , Thus

Quasi-Sliding Mode Controller Design.
Quasi-sliding mode control (QSMC) can eliminate the chattering of sliding mode and keep stable and thus is used in widespread manner recently [14].
From the above analysis it is known that this is a 3-order system, and  =  1 −  1ref , thus defining the sliding surface as In QSMC, reaching law is designed as where > 0.
(18) Consider a positive augmented Lyapunov candidate as where  > 0 is a design parameter.
The time derivative of  is Substituting ( 16) and ( 17) into (20) obtains By the Lyapunov theorem of stability and LaSalle's invariance principle, the control system is proved to be stable.

Analysis of System Performance
The system parameters are listed in Table 1.

Analysis of Transfer Functions.
In traditional method [15], the LC filter is designed based on the transfer function of converter output voltage   to load voltage   without load and grid impedance.The transfer function is In traditional model, it is unable to obtain the transfer relationship of converter output voltage   to source current   .Only the relationship of converter output voltage   to converter output current  1 is obtained, in assuming the load voltage   is constant, such that The resonant frequency   is Substitute the system parameters  1 and  into (29); the resonant frequency   = 2055 Hz.
From the above equations, applying Matlab obtained the transfer functions and the Bode diagrams are shown in Figure 4.
In Figure 4 From Figure 4 it is seen that the frequency characteristics of traditional simplified model are obviously different with proposed precise model, which means the traditional model cannot describe the system correctly.Thus it is necessary to establish a precise model for analyzing parameters and designing controller.

Impact of Grid Impedance.
The Bode diagram of converter output voltage   to source current   "  2  " in different grid inductive impedance   is shown in Figure 5.
From Figure 5 it is seen that with the grid inductive impedance increasing the high frequency magnitude of   decreases, and the resonant frequency also decreases.
The Bode diagram of converter output voltage   to source current   "  2  " in different grid resistance   is shown in Figure 6.
From Figure 6 it is seen that with the grid resistance increasing the steady magnitude of   decreases, and the resonant quality factor decreases.

Comparison of Control Systems.
In PI controller, the parameters are inner loop proportionality coefficient   = 10, integral coefficient   = 2000, outer loop proportionality coefficient   = 1, and integral coefficient   = 1000.
In FBL-QSMC controller, the inner loop is based on equivalent input  eq and QSMC input   , the outer loop is based on PI controller, the proportionality coefficient   = 1, and integral coefficient   = 1000.
Comparison of the step response performances is shown in Figure 7.
From Figure 7 it is known that in PI controller converges slowly.However, FBL controller responds quickly and converges fast.
In cos signal input, the responses of PI controller and FBL controller are shown in Figure 8.
From Figure 8 it is known that in PI controller the convergence time is more than a period.However, FBL controller responds quickly and converges fast.

Simulation Analysis
The simulations are in Matlab/Simulink; the simulation parameters are listed in Table 1.The load consists of both linear and nonlinear type.The system is analyzed for different operating conditions as discussed in the following cases.

Case 1: Load Change.
Cutting down nonlinear load at  = 0.105 s, the results in PI controller are shown in Figure 9, and the results in FBL-QSMC controller are shown in Figure 10.
In Figure 9(a), the upper part is load current and source current, the total harmonic distortion (THD) of large load current is 16.25%, the THD of source current is 4.98%, the  THD of small load current is 12.65%, and the THD of source current is 4.96%.When cutting down the nonlinear load, the THD of load current decreases obviously; however, the ratio frequency harmonic increases; thus the THD decreases a little.The load change means the active power change; thus there is an impulse to direct current error   .In Figure 9(b), the load voltage increases when load power change is smaller, which is because the load current decreases and the voltage loss decreases.Also there is an impulse to DC-link voltage.
In Figure 10(a), the upper part is load current and source current, the THD of load current is 16.37%, the THD of source current is 2.58%, the THD of load current is 12.74%, and 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0. the THD of source current is 2.55%.When cutting down the nonlinear load, the THD of load current decreases obviously; however, the ratio frequency harmonic increases; thus the THD decreases a little.There is an impulse to direct current error   ; however, convergence is fast; in addition, the direct current error   and quadrature current error   are very small.In Figure 10(b), the load voltage increases when load power change is smaller.The DC-link voltage is similar to PI controller, which is because the outer loops are both applying PI control.

Case 2: DC-Link Voltage
Change.The reference DC-link voltage changes from 700 V to 680 V at  = 0.105 s; the results in PI controller are shown in Figure 11; the results in FBL-QSMC controller are shown in Figure 12.
In Figure 11(a), the upper part is load current and source current, the THD of load current is 16.24%, and the THD of source current is 5.01%.With DC-link voltage decrease, the compensating current track ability decreases.The compensation harmonic increases more than frequency harmonic decrease; thus the THD increases.The DC-link voltage decrease means exporting active power; thus the source current decreases; there is an impulse to direct current error   .In Figure 11  In Figure 12(a), the upper part is load current and source current, the THD of load current is 16.37%, and the THD of source current is 2.55%.With DC-link voltage decrease, the compensating current track ability decreases, and the frequency harmonic decreases.The compensation harmonic increases less than frequency harmonic decrease; thus the THD decreases.There is an impulse to direct current error   and converge fast.In addition, the direct current error   and quadrature current error   are very little.In Figure 12(b), the load voltage increases when DC-link voltage change is smaller.The DC-link voltage response is similar to PI controller which is because the outer loops are both applying PI control.

Case 3: Reactive Power Change.
At  = 0.105 s without reactive power compensation, the results in PI controller are shown in Figure 13; the results in FBL-QSMC controller are shown in Figure 14.
In Figure 13(a), the upper part is load current and source current, the THD of load current is 16.24%, and the THD of source current is 4.95%.Without reactive power compensation the compensating current decreases, which can improve the harmonic current tracking ability; thus the THD decreases.Without reactive power compensation the source current's power angle is equal to the load current's, and there is an impulse to quadrature current error   .In Figure 13(b), the change of reactive power produces a little impact on the load voltage and no impact on DC-link voltage.In Figure 14(a), the upper part is load current and source current, the THD of load current is 16.37%, and the THD of source current is 2.46%.Without reactive power compensation, there is an impulse to quadrature current.The tracking current error is obviously smaller than in PI controller.In Figure 14(b), with the reactive power change there is little impact on load voltage and no impact on DClink voltage.

Conclusion
The Bode diagram of traditional simplified model is obviously different with complete model, which means the descriptions of the system based on the traditional simplified model are inaccurate and incomplete.Thus, it is necessary to popularize complete model of APF.The proposed feedback linearization and quasi-sliding mode control algorithm has fast response and robustness; the compensation performance is superior to PI control obviously.The simulation results proved that the complete modeling and proposed control algorithm are correct.

Figure 2 :
Figure 2: Single-phase circuit diagram of  APF with grid impedance.

Figure 9 :
Figure 9: Results in load change with PI controller.
(b), the load voltage increases when DClink voltage decreases, which is because the source current decreases and the voltage loss decreases.

Figure 14 :
Figure 14: Results in reactive power change with FBL-QSMC controller.

Table 1 :
The control system consists of outer voltage loop and inner current loop.The outer voltage loop applied PI control, and the inner current loop applied FBL-QSMC control.It is necessary to remind that there are three identical inner current loops in three phases, respectively.However, there is only drawn one phase in the diagram.As shown in Figure3, the source reference active current   is equal to the sum of load's active current  *  and DC-link voltage regulating current  * dc ; the reference reactive current   and zero sequence current are equal to zero.After Park's transformation, the reference current from 0 coordinate transformed to  coordinate.The reference current   subtracts the source current   which obtains the source error current   in  phase, respectively.The source error current   applied sliding mode control obtains switching input   and state variables in feedback linearization control obtain equivalent control input  eq and then sum up them equal to the control input signal .System parameters.
Feedback Linearization Sliding Mode Control.The control diagram of feedback linearization quasi-sliding mode control (FBL-QSMC) is shown in Figure 3.
, the line "  2  " is represented transfer function of converter output voltage   to source current   in proposed model; the line "  2  " is represented transfer function of converter output voltage   to load voltage   in proposed model; the line "  2 1 " is represented transfer function of converter output voltage   to converter output current  1 in proposed model; the line "  2  " is represented transfer function of converter output voltage   to load voltage   in traditional model; the line "  2 1 " is represented transfer function of converter output voltage   to converter output current  1 in traditional model.