Internal-Model-Principle-Based Specific Harmonics Repetitive Controller for Grid-Connected PWM Inverters

This paper analyzes the general properties of IMP-based controller and presents an internal-model-principle-based (IMP-based) specific harmonics repetitive control (SHRC) scheme. The proposed SHRC is effective for specific nk ± m order harmonics, with n > m ≥ 0 and k = 0, 1, 2, . . .. Using the properties of exponential function, SHRC can also be rewritten into the format of multiple resonant controllers in parallel, where the control gain of SHRC is n/2 multiple of that of conventional RC (CRC). Therefore, including SHRC in a stable closed-loop feedback control system, asymptotic disturbance eliminating, or reference tracking for any periodic signal only including these specific harmonic components at n/2 times faster error convergence rate compared with CRC can be achieved. Application examples of SHRC controlled three-phase/single-phase grid-connected PWM inverters demonstrate the effectiveness and advantages of the proposed SHRC scheme.


Introduction
Repetitive control (RC), as an internal model principle (IMP) [1,2] based controller, can track or eliminate periodic signals in an effective way if it is included in a stable closed-loop system.Conventional RC (CRC) [3], presented since the early 1980 and being the most widely used RC format nowadays, can achieve zero-error tracking or disturbance rejection for any periodic signal whose fundamental period time is known at first.Through fast-Fourier transformation (FFT) analysis, any known periodic signal can be decomposed into dc, fundamental component, and all harmonic components.That is, CRC is effective to track or eliminate any periodic signal including any order harmonic.Due to the time delay line being included in the RC structure, CRC is slower than any instantaneous feedback controller although it has better tracking accuracy.RC has been studied in several aspects [4][5][6][7][8][9][10][11][12], including its properties and structures.Moreover, its applications can be found in many fields, such as disc drives [13], robots [14], satellites [15], and PWM converters [8,9,[16][17][18][19][20][21][22].
In order to solve this problem, this paper proposes an IMP-based specific harmonics RC (SHRC) for threephase/single-phase grid-connected PWM inverters.To better understand the proposed SHRC, the general properties of IMP-based controller and the performance of IMP-based CRC are analyzed at first.Then, the performance of IMPbased SHRC and its error convergence rate analysis are also given.Finally, two application cases of SHRC controlled three-phase/single-phase grid-connected PWM inverters are provided to demonstrate the effectiveness and advantages of the proposed SHRC scheme.initial conditions.Thus, a general disturbance signal can be described as the format of a differential equation as follows:

IMP-Based Controller
Using Laplace transform, (1) can be derived into where Γ  () is the disturbance generating polynomial (D-GP), which can be defined into Notice that this D-GP is only related with the denominator of (2).Similarly, reference signal can be described in the same way, as long as replacing the symbol "" with "."Reference generating polynomial (R-GP) Γ  () can be defined as follows: Then, internal model principle (IMP) [2,23] can be expressed as follows: If disturbance () or reference input   () can be expressed as its D-GP or R-GP just like Γ  () or Γ  (), asymptotic error elimination or reference tracking can be achieved for the control system using the following controller: or Therefore, only D-GP or R-GP, that is, Γ  () or Γ  (), is needed when using IMP, while the amplitude information of disturbance or reference input is not needed.

The General Properties of IMP-Based Controller.
Figure 1 shows the diagram of a closed-loop feedback control system.Let the model of plant be and assume that Γ  () or Γ  () is not the factor of   ().
Then, the output sensitivity function of closed-loop feedback system shown in Figure 1 is or and the output complementary sensitivity function is or Assume the selected () and () can make the closedloop characteristic equation or have the negative realistic roots.Thus, the output response of disturbance () can be obtained as Because  cl () has stable roots, the inverse Laplace transform of () can asymptotically converge to zero; that is, ( → ∞) = 0.
The system error response of disturbance () is Because  cl () has stable roots, the inverse Laplace transform of () can asymptotically converge to zero; that is, ( → ∞) = 0.
The system error response of reference  ref () is Because  cl () has stable roots, the inverse Laplace transform of () can asymptotically converge to zero; that is, ( → ∞) = 0.
The system output response of reference  ref () is Because  cl () has stable roots, the inverse Laplace transform of () can asymptotically track its reference input; that is, () →  ref ().
Therefore, if D-GP or R-GP is included in the denominator of a controller, asymptotic disturbance eliminating or reference tracking can be achieved for the control system.
Then, we will see why conventional RC (CRC) is an IMPbased controller and why it can track/eliminate all harmonics.

IMP-Based Conventional Repetitive Controller (CRC).
According to the properties of exponential function [24], CRC can be rewritten into where  rc is the control gain and   = 2/  = 1/  is the fundamental period of signals with   being the fundamental frequency,   being the fundamental angular frequency.
If the disturbance () or reference input  ref () of closedloop feedback control system is a periodic signal, which can be decomposed into the summation of all harmonic components, that is, dc, fundamental component and any order harmonic component using FFT, its D-GP or R-GP, that is, Γ  () or Γ  (), can be derived as follows: or Then, CRC can be derived into or From ( 21) or (22), it can be seen that D-GP Γ  () or R-GP Γ  () is included in the denominator of CRC controller.Thus, asymptotic disturbance eliminating or reference tracking for any periodic signal can be achieved using IMP indicated in Section 2.1, if CRC is used as the controller in a closedloop feedback control system.Therefore, CRC is an IMPbased controller and is effective for any periodic signal that is composed with all harmonic components.

Specific Harmonics RC (SHRC).
A specific harmonics RC (SHRC) shown in Figure 2 for efficient removing/tracking specific  ±  order harmonics is proposed as follows: where  and  are integers with  >  ≥ 0.
Using the similar analysis with CRC in Section 2.3, it can be achieved that SHRC is also an IMP-based controller in the following subsection.

IMP-Based SHRC.
According to the properties of exponential function [24], ( 23) can be rewritten into where  ̸ = 0.If  = 0, (23) can be rewritten into If the disturbance signal () of closed-loop feedback control system is a periodic signal only including specific  ±  order harmonic components, its D-GP, that is, Γ  (), can be derived as follows: Then, SHRC can be derived into Similarly, if the reference input  ref () is a periodic signal only including specific  ±  order harmonic components, its R-GP, that is, Γ  (), and corresponding SHRC can also be derived.
So, it can be seen that D-GP Γ  () or R-GP Γ  () is included in the denominator of SHRC controller.Thus, asymptotic disturbance eliminating or reference tracking for any periodic signal only including these specific  ±  order harmonic components can be achieved using IMP indicated in Section 2.1, if SHRC is used as the controller in a closedloop feedback control system.Therefore, SHRC is also an IMP-based controller and is effective for any periodic signal only including these specific harmonics.

Error Convergence Rate of SHRC.
According to the properties of exponential function [24], SHRC can also be rewritten into the format of multiple resonant controllers in parallel as follows: From (28), it can be achieved that, for specific ± order harmonics, the control gain of SHRC is (/2) rc /  .From [25], the corresponding control gain of CRC is  rc /  .So,  the control gain of SHRC is /2 multiple of that of CRC.The error convergence rate of SHRC is also /2 times faster than that of CRC.Particularly, for the three-phase PWM inverters application, the error convergence rate of SHRC with  = 6 and  = 1 is three times (i.e., /2 = 3) faster than that of CRC.For the single-phase PWM inverters application, the error convergence rate of SHRC with  = 4 and  = 1 is two times (i.e., /2 = 2) faster than that of CRC.Therefore, SHRC can be used to enhance the error convergence rate for tracking/removing specific harmonics and thus improve the performance of three-phase/single-phase PWM inverter control system.
In the next two sections, SHRC will be used in threephase and single-phase grid-connected PWM inverter systems to verify the effectiveness of SHRC and to improve the control performance of both inverter systems.

Case 1: Three-Phase Grid-Connected PWM Inverters
4.1.Modeling.Figure 3 shows a three-phase grid-connected PWM inverter system, where  dc is the dc-side voltage;  and  are inductance and resistor, respectively;   ,   , and   are the inductance currents; V  , V  , and V  are the , , and  three-phase grid voltages.The control objective is to achieve high current tracking accuracy through forcing   ,   , and   to exactly track their reference  aref ,  bref , and  cref , which are generated by using PQ control method and instantaneous power theory [26].System parameter values are shown in Table 1.
Table 1: Parameters for three-phase grid-connected inverter system. Inverter The mathematical model [18] can be described as ) . (29)
A dead-beat (DB) current controller is chosen as follows: where  = , , .Letting  jref () =   ( + 1), the closed-loop transfer function without RC is () =  −1 .Therefore, the control delay for DB current controller is only one step, that is, one sample period, in theory, which can achieve fast dynamic response.

Experimental Results
. Figure 4(a) shows the experimental result with single DB controller.It can be seen that current distortion is relatively serious; that is, the current waveform is not a smooth sine wave.Inverter Figures 4(b), 4(c), and 5 show the steady-state response and the current error convergence histories with CRC controller and the proposed SHRC controller being plugged into the DB controlled three-phase grid-connected inverter, respectively.For comparability, both RC gains are  rc = 0.2.Figures 4(b) and 4(c) show the current distortions can be greatly improved with both CRC and SHRC controllers.Moreover, it can be clearly seen from Figure 5 that the current error convergence times for CRC and SHRC are 0.32 s and 0.12 s, respectively.Therefore, the current error convergence rate of the proposed SHRC controller is nearly three times faster than that of CRC controller (as indicated in Section 3.3).
Therefore, the effectiveness of SHRC for three-phase gridconnected PWM inverter system and its advantage compared with CRC are verified.

Case 2: Single-Phase Grid-Connected PWM Inverters
5.1.Modeling.Figure 6 shows a single-phase grid-connected PWM inverter system, where  dc is the dc-side voltage;  and  are inductance and resistor, respectively;   is the inductance current; and V  is the grid voltage.The control objective is to achieve high current tracking accuracy through forcing   to exactly track its reference  sref which is generated by using PQ control method and instantaneous power theory [26].System parameter values are shown in Table 2.
The mathematical model can be described as The data-sample format of (35) can be gotten: where   () is the single-phase duty cycles,  1 = /, and  2 = .
A DB current controller is chosen as follows: Letting  jref () =   ( + 1), the closed-loop transfer function without RC is () =  −1 .Therefore, the control delay for DB  current controller is only one step, that is, one sample period, in theory, which can achieve fast dynamic response.
(39) 5.2.Experimental Results. Figure 7(a) shows the experimental result with single DB controller.It can be seen that current distortion is relatively serious; that is, the current waveform is not a smooth sine wave.current error convergence times for CRC and SHRC are 0.32 s and 0.16 s, respectively.Therefore, the current error convergence rate of the proposed SHRC controller is nearly two times faster than that of CRC controller (as indicated in Section 3.3).
Therefore, the effectiveness of SHRC for single-phase PWM grid-connected inverter system and its advantage compared with CRC are verified.

Conclusions
This paper proposed an internal-model-principle-based (IMP-based) specific harmonics RC (SHRC).Using the general properties of IMP-based controller and the properties of exponential function, it can be concluded that SHRC can achieve zero-error tracking or perfect disturbance rejection of specific periodic signal only including these specific  ±  order harmonic components at /2 times faster error convergence rate compared with conventional RC (CRC).
Two application examples of SHRC controlled threephase/single-phase grid-connected PWM inverters have been given to show the effectives and promising advantages of SHRC.The experimental results show that, compared with CRC, SHRC offers three times (for three-phase inverter) or two times (for single-phase inverter) faster current error convergence rate and yields nearly the same current tracking accuracy.The proposed IMP-based SHRC provides a high performance control solution to grid-connected PWM inverters.

Figure 5 :
Figure 5: Current tracking error histories with two RCs plugged into for three-phase grid-connected inverter system.

Figures 7 (
b), 7(c), and 8 show the steady-state response and the current error convergence histories with CRC controller and the proposed SHRC controller being plugged into the DB controlled single-phase grid-connected inverter, respectively.For comparability, both RC gains are  rc = 0.2.Figures7(b) and 7(c) show the current distortions for both CRC and SHRC controllers can be greatly improved.Moreover, it can be clearly seen from Figure8that the

Figure 8 :
Figure 8: Current tracking error histories with two RCs plugged into for single-phase grid-connected inverter system.

Table 2 :
Parameters for single-phase grid-connected inverter system.