2. Modeling of the PV inverter for harmonic analysis

In this section, PWM inverter framework with current feedback control for single-phase fullbridge PV inverter, which is generally used in the commercial products, conventional model of the current regulation scheme for that kind of inverter, and general inverter model proposed for the harmonic analysis are presented.

to control the output current Iout to track a reference output current Iref . A phase-locked loop (PLL) has been used to obtain the phase angle of Iref from the grid voltage Vg. The amplitude

MPPT process. The design of the voltage control loop may vary according to different

 

Figure 3 shows the conventional control structure diagram of the current-controlled inverter. This model can be analyzed by using conventional linear analysis methods. It can help the designer to tune the controller [21] and investigate the control performance and stability [22].

where GPI, GPWM, Ginv, and Gf are the transfer functions for the PI controller, PWM, inverter, and filter, respectively. In this model, only the fundamental waveforms are considered, and

Figure 4 shows the generalized model which is derived from the conventional current control structure diagram for a PWM inverter with harmonic information. The location and types of harmonic sources, which need to be added, are shown in this figure. The output current S5 is generated based on a reference current by the full-bridge inverter with current control, as shown in the first trace. The model of current control scheme, which includes the harmonics information, is shown in the second trace. Compared with Figure 3, the switch harmonic source Vswitch harmonics in the PWM section is added to generate a pulse waveform on top of the sinusoidal signal. This harmonic source contains the characteristic of the PWM, including the type of PWM method and the switching frequency. The voltage difference between the grid voltage and the inverter output voltage will cause the changes in the output current. Therefore, in Figure 4, the grid voltage harmonic source is added at the inverter

Iref � Gf

can be determined by the voltage control loop according to the

1 þ GPIGPWMGinvGf

Vg (1)

can be found in [10, 20].

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of the reference current Iref

 

inverter topologies. The detailed derivation of Iref

The closed-loop transfer function is given by

2.2. Conventional model of current regulation scheme

Iout <sup>¼</sup> GPIGPWMGinvGf 1 þ GPIGPWMGinvGf

2.3. The general inverter model for the harmonic analysis

Figure 3. Conventional control structure diagram of the current-controlled inverter.

harmonic information is required for the harmonic distortion analysis.

#### 2.1. Single-phase full-bridge PV inverter with current control

An example of PWM inverter framework with current feedback control is shown in Figure 2. It is the most common structure which used by the commercial products. The inverter is formed by one output inductor, a DC-link capacitor CDC, and four power switches. The DClink voltage VDC presents two different scenarios: one is with voltage ripple and another is without voltage ripple. The following sections analyze these two different cases separately. Vinv is the full-bridge inverter output voltage and Vg is the grid voltage, Iout is the inverter output current. A fixed grid voltage has been applied to the grid-connected inverter output terminals, and the inverter input voltage is controlled to provide MPP tracking. A current control scheme is used, since only the AC output current can be controlled. A filter has been used to connect between the inverter and the grid. In this chapter, a single inductor is used to simplify the analysis. A feedback control with the PI controller is used for the PWM inverter

Figure 2. PWM inverter framework with current-controlled feedback loop.

to control the output current Iout to track a reference output current Iref . A phase-locked loop (PLL) has been used to obtain the phase angle of Iref from the grid voltage Vg. The amplitude of the reference current Iref can be determined by the voltage control loop according to the MPPT process. The design of the voltage control loop may vary according to different inverter topologies. The detailed derivation of Iref can be found in [10, 20].

### 2.2. Conventional model of current regulation scheme

2. Modeling of the PV inverter for harmonic analysis

54 Power System Harmonics - Analysis, Effects and Mitigation Solutions for Power Quality Improvement

2.1. Single-phase full-bridge PV inverter with current control

Figure 2. PWM inverter framework with current-controlled feedback loop.

for the harmonic analysis are presented.

In this section, PWM inverter framework with current feedback control for single-phase fullbridge PV inverter, which is generally used in the commercial products, conventional model of the current regulation scheme for that kind of inverter, and general inverter model proposed

An example of PWM inverter framework with current feedback control is shown in Figure 2. It is the most common structure which used by the commercial products. The inverter is formed by one output inductor, a DC-link capacitor CDC, and four power switches. The DClink voltage VDC presents two different scenarios: one is with voltage ripple and another is without voltage ripple. The following sections analyze these two different cases separately. Vinv is the full-bridge inverter output voltage and Vg is the grid voltage, Iout is the inverter output current. A fixed grid voltage has been applied to the grid-connected inverter output terminals, and the inverter input voltage is controlled to provide MPP tracking. A current control scheme is used, since only the AC output current can be controlled. A filter has been used to connect between the inverter and the grid. In this chapter, a single inductor is used to simplify the analysis. A feedback control with the PI controller is used for the PWM inverter Figure 3 shows the conventional control structure diagram of the current-controlled inverter. This model can be analyzed by using conventional linear analysis methods. It can help the designer to tune the controller [21] and investigate the control performance and stability [22]. The closed-loop transfer function is given by

$$I\_{out} = \frac{G\_{Pl}G\_{PWM}G\_{inv}G\_f}{1 + G\_{Pl}G\_{PWM}G\_{inv}G\_f}I\_{ref} - \frac{G\_f}{1 + G\_{Pl}G\_{PWM}G\_{inv}G\_f}V\_g \tag{1}$$

where GPI, GPWM, Ginv, and Gf are the transfer functions for the PI controller, PWM, inverter, and filter, respectively. In this model, only the fundamental waveforms are considered, and harmonic information is required for the harmonic distortion analysis.

#### 2.3. The general inverter model for the harmonic analysis

Figure 4 shows the generalized model which is derived from the conventional current control structure diagram for a PWM inverter with harmonic information. The location and types of harmonic sources, which need to be added, are shown in this figure. The output current S5 is generated based on a reference current by the full-bridge inverter with current control, as shown in the first trace. The model of current control scheme, which includes the harmonics information, is shown in the second trace. Compared with Figure 3, the switch harmonic source Vswitch harmonics in the PWM section is added to generate a pulse waveform on top of the sinusoidal signal. This harmonic source contains the characteristic of the PWM, including the type of PWM method and the switching frequency. The voltage difference between the grid voltage and the inverter output voltage will cause the changes in the output current. Therefore, in Figure 4, the grid voltage harmonic source is added at the inverter

Figure 3. Conventional control structure diagram of the current-controlled inverter.

section. The lowest trace in Figure 4 sketches the waveform of each stage, and the details are described as follows:

3. Current harmonic caused by DC-link voltage ripple

can be evaluated for any particular operating condition by using this model.

y n½ �¼ ½ � cos ð Þ� ωon y n½ � � 1 ½ � cos 2ð Þþ ωon VDC

y n½ � � 1 so that it can be used in the computation of y n½ �. The y n½ � � 1 is

Figure 5. Model of inverter with the DC-link voltage ripple.

solution for the current harmonics has been provided.

In this section, the current harmonics caused by DC-link voltage ripple has been analyzed. The model for considering the double-line frequency voltage ripple has been built. The closed-form

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Figure 5 shows the model of the inverter based on Figure 4, and the DC-link voltage ripple has been taken into account. The inverter transfer function Ginv shown in Figure 4 is replaced by the section under the triangle shading, which is a sinusoidal signal Vrip at double-line frequency on top of the DC component VDC. Since the voltage ripple is time-varying, the transfer function for this section cannot be derived. In [23], the authors point out that closed-form solutions cannot be derived when the harmonic ripple components are not neglected. However, numeric solutions

The harmonic characteristics of the output current shown in Figure 5 can be identified by qualitatively analyzing the simplified loop model. The section under the triangle shading is also known as the amplitude modulation; the feedback loop with unit delay is shown in Figure 6, where Z�<sup>1</sup> denotes the delay of a unit sample period. Compared with Figure 6, in this simplified model, several linear blocks are left out. Due to the system linearity, the signal frequency characteristics will remain the same. A similar analysis method, which has been used in sound processing research [24], is adopted in this chapter to analyze this time-varying system. Two discrete-time sinusoidal example signals Iref ½ �¼ n cos ð Þ ωon and Vrip½ �¼ n cos 2ð Þ ωon are used. The output signal y n½ � can be illustrated as the result of subtraction between the reference signal cos ð Þ ωon and the delayed output signal y n½ � � 1 then timed with the AM section, which is cos 2ð Þþ ωon VDC

¼ cos ð Þ ωon ½ cos 2ð Þþ ωon VDC� � y n½ � � 1 ½ � cos 2ð Þþ ωon VDC

For n ≤ 0, ω<sup>o</sup> is the angular velocity of a signal in the fundamental frequency, VDCis constant, at the initial condition, y n½ �¼ 0,. The delay exists at any point in time n, and we need to store

(2)

1. S1 is the error between the current reference and the output current of the inverter, S1 ¼ Iref � Iout ¼ Iref � S5.

2: S2 is the amplitude modulation (AM) ratio, S2 ¼ S1GPI, where the PI controller's transfer function is GPI ¼ kp þ ki=s. kp and ki are the proportional and the integral gain.

3: S3 is the gate drive signal. S3 ¼ S2GPWM þ Vswitch harmonics, where GPWM ¼ 1=Cpk and Cpk is the carrier signal's peak value.

4: S4 is the output voltage of the inverter Vinv, S4 ¼ S3Ginv, where Ginv ¼ VDC. The VDC can be either a time-varying or a constant signal; these two cases need to be treated separately.

5: S5 is the output current of the inverter Iout. S5 ¼ S4 � Vg Gf . The grid voltage Vg may contain the voltage harmonics Vg harmonics:S4 � Vg is the voltage difference between the output filter. The transfer function of the filter is Gf ¼ 1=ð Þ Ls , where L is the filter's inductance.

The main causes of harmonic in PV inverter can be summarized into several categories: grid background voltage distortion, switch harmonics (high frequency), DC-link voltage variation due to MPPT, and some other causes (PLL blocks, etc.). Harmonic distortion for both cases, with or without voltage ripple on the DC link, can be analyzed by using this generalized model.

Figure 4. Model of current-controlled PWM inverter with harmonic information.
