High-Performance Voltage Controller Design Based on Capacitor Current Control Model for Stand-alone Inverters

Abstract

This study proposes high-performance voltage controller design that employs a capacitor current control model for single-phase stand-alone inverters. The single-phase stand-alone inverter is analyzed via modeling, which is then used to design the controller. A design methodology is proposed to maximize the bandwidth of the feedback controller. Subsequently, to compensate for the problems caused by the bandwidth limitations of the controller, an error transfer function that includes the feedback controller is derived, and the stability of the repetitive control scheme is evaluated using the error transfer function. The digital repetitive controller is then implemented. The simulation and experimental results show that the performance of the proposed controller is high in a 1.5 kW single-phase stand-alone inverter prototype.

1. Introduction

Stand-alone type inverters are widely used to supply electrical power at places with no grid connections. Moreover, they can be used for emergency power supply under grid power fault conditions. Therefore, the output waveforms of these inverters must be sinusoidal, similar to those of the ac supply of commercial utility lines.

Recently the nonlinear loads of residential electrical loads are increasing dramatically because of the introduction of electronic loads, thus leading to problems such as voltage variations and waveform distortion. These problems can result in steady-state errors and distortions in supply voltage waveform in stand-alone type inverters. Therefore, control strategies for stand-alone type inverters have been proposed to overcome these problems [1-31].

Typically, the structure of an inverter output voltage controller comprises multiple feedback loops rather than a single feedback loop. Although the response of an inverter is faster with a single feedback loop, a multiple feedback loop is more advantageous in shutting off overcurrent and improving the transient response of inductor [1-8]. The multiple feedback loop control method is widely used by researchers because of the ease of implementation and good performance [9, 10]. A multiple feedback loop controller is typically composed of voltage control in the outer loop and current control in the inner loop. Numerous control strategies such as PI [9, 10], predictive control [11, 12], deadbeat control [13, 14], and repetitive control [15-33] are used for controlling the outer and inner loops. However, these controllers have a problem to make a design of controller optimally, which should be considered on both outer loop and inner loop.

Generally, a repetitive control scheme is used to compensate the periodic components at multiples of the electrical fundamental frequency. Studies on the repetitive control scheme can be classified into two categories: those focusing on compensating periodic components using the repetitive control scheme [15-26] and those focusing on improving the repetitive control scheme [27-33]. This study belongs to the first category. Numerous studies exist on controlling the output voltage of stand-alone inverters. In order to reduce steady-state errors, this controller can be used directly with an open-loop SPWM inverter [15]. A zero-phase odd-harmonic repetitive controller has been proposed to reduce data memory usage and ensure faster convergence of the tracking error [16, 17]. Moreover, this controller can be employed in uninterruptible power supply (UPS) inverters to supply sinusoidal voltage under nonlinear loads [18-23]. Further, by connecting the repetitive controller in parallel, steady state and rapid response can be achieved and harmonic components can be reduced. In [24], both repetitive control and resonance control schemes were used to achieve rapid response. Also, control structure of combining with PI controller and repetitive controller is researched by [25, 26] for reducing the output voltage distortion. However, those studies are just focused on repetitive controller’s performance with feedforward controller for diminishing nonlinear characteristics without considering on feedback controller’s performance.

In this study, analytical design method is proposed for performance improvement of output voltage controller in the stationary reference frame. The controller consists of both multiple feedback controller and feedforward controller. The feedforward controller is chosen as repetitive controller, and the multiple feedback controller is selected as PI controller at the outer loop and Type-III compensator at the inner loop. Prior to designing the controller, the inverter is modeled for comparing capacitor and inductor current control models, and a Type III compensator-based feedback controller is designed. Thereafter, a plug-in repetitive controller is designed to compensate the frequency component, which cannot be handled with a feedback controller alone. In order to verify the effectiveness of the designed controllers, simulations were performed using the PSIM simulation software package. Experiments are conducted with a 1.5 kW single-phase stand-alone inverter prototype to verify the proposed control strategy.

 

2. Single Phase Inverter Modeling and Controller Design

2.1 Modeling of single-phase inverter

Single-phase inverters with two types of controllers are shown in Fig. 1. Input voltage is usually supplied by lithium-ion or lead batteries. They are composed of four switches (S1, S2, S3, S4 ), an output filter inductor (L), an output filter capacitor (C), an inductor parasitic resistance (RL ), and a capacitor parasitic resistance (Rc ). The output filter plays a role in rejecting switching noise. The single-phase inverter uses the multiloop control method in which the outer loop is the capacitor voltage control loop and the inner loop is either capacitor current or inductor current control loop, as shown in Fig. 1. In this study, the implemented controller is a digital controller. Therefore, the voltage and current controllers can be adjusted. Because digital controllers are different from analog controllers, they must consider digital delay effects. These effects are generated owing to sampling of the control variables, computation of the control algorithm, updating of the pulse-width modulation (PWM) signal in single PWM switching mode [25].

Fig. 1.Schematic of single-phase stand-alone inverter.

The inverter must be first modeled for designing the controller. The two types of inverters can be modeled from Fig. 1. The inductor current control model can be given by (1), and the capacitor current control model can be obtained as (2).

where, Vdc is inverter dc-link voltage, and uc represents the inverter output voltage before the LC filter is scaled by dc-link voltage.

In order to analyze the characteristics of these models, the bode plots of both duty ratio to voltage and duty ratio to current of each model are needed. There is no difference of the bode plot of duty ratio to voltage between these models. But the bode plot of duty ratio to current has a different characteristics of frequency response as like Fig. 2. The Fig. 2 shows the bode plot of duty ratio to current. The magnitude and phase vary between 10 Hz and 1000 Hz in the inductor current control model whereas they are nearly the same in the capacitor current control model according to load power. Therefore, the capacitor current control model is more independent of load than the inductor current control model and thus is more suitable for controller design.

Fig. 2.Frequency response of duty ration to current: (a) Capacitor current control model; (b) Inductor current control model

2.2 Design of controller for single-phase inverter

As mentioned above, the capacitor current control model is more appropriate for the controller of this study; thus, the inverter control block diagram shown in Fig. 1(b) is selected. The purpose of a single-phase inverter is to supply a stable sinusoidal voltage. The multiple feedback loop control strategy is selected for controlling the single-phase inverter. The outer loop Gdcv (z) is controlled through a PI controller that is easy to implement and exhibits good performance. The inner loop Gdci(z) is implemented with a Type III compensator. The Type III compensator can boost the phase and gain of the controller and enables analytic controller design. The conventional PI controller, namely, s-domain expression, is given by Eq. (3). The voltage controller Gdcv (z) used in this study is given by Eq. (4) by converting the s-domain to the z-domain.

where, v*ref is the reference output voltage, i*ref is the reference inductor current. Further, ki and kp are the PI controller gains, and Ts is the control period.

The switching frequency is denoted as fs, and the maximum bandwidth of the outer loop controller is limited to approximately fs/50 [34-36]. However, the bandwidth of the outer loop voltage controller is dependent on the bandwidth of the inner loop current controller. Therefore, the design of the inner loop controller must be optimized, and that affects control performance. The inner loop controller Gci (z) must be optimized for tracking the voltage controller output. Moreover, the maximum bandwidth of the inner loop controller is limited to approximately fs/10 in the digital controller [34-36]. Therefore, the inner loop controller must be analytic control design to ensure optimal performance. In this case, the Type III compensator is more suitable than other types of compensators because the phase boost can range from 0º to 180º in the highfrequency region. Therefore, the phase near the crossover frequency can primarily be determined by the phase of Gci(z). In this study, the current controller is designed such that the targeted phase margin is 80º at the crossover frequency. The results of the design are shown in Fig. 3. As shown in Fig. 3, the crossover frequency and the phase margin, which are the bandwidth and the phase margin of the system, satisfy the design specifications.

Fig. 3.Compensated frequency response of current controller

 

3. Repetitive Controller Design

As mentioned above, the capacitor current control model is more independent of loads than the inductor current control model. Accordingly, the capacitor current controller achieves high quality sinusoidal voltage regulation. However, both residential and industrial applications primarily have nonlinear loads. The nonlinear characteristics distort the output voltage thus resulting in quality degradation because a conventional controller cannot completely compensate the distortion. In order to compensate the harmonics, a plug-in type repetitive controller is used in this study for supplying high-quality voltage. Conventional repetitive controllers are compared in terms of frequency response, and the most suitable repetitive controller is selected.

3.1 Repetitive controller components selection

The repetitive control method, which is based on the internal model principle, is widely used to obtain very accurate reference tracking for closed-loop control systems in which the tracking error is repeated periodically [25]. A repetitive controller can be divided into three parts, as shown in Fig. 4. First, the data saving section is used for storing data during the sampling period of fundamental frequency for eliminating repetitive errors to obtain the internal model. Second, the data filtering part rejects harmonics on the basis of the stability factor q(z). Third, the output scaling part is obtained by evaluating the system model gain of the repetitive controller Gf(z). Moreover, according to previous research, there are two types of repetitive controllers, as shown in Fig. 4(a) and Fig. 4(b). The two types of repetitive controllers are represented as follows:

Fig. 4.Types of repetitive controllers: (a) Plug-in repetitive structure; (b) Cascade repetitive structure.

where L is the constant that compensates the time delay effect.

The time-delay effects are caused by the digital PWM update and the computation delay. Further, L leads the actual error [25]. In addition, as can be seen from (5) and (6) and Fig. 8, it differs with the type of repetitive controller used. Stability factor q(z) such as a low pass filter (LPF) typically rejects harmonic components. When q(z) is an

LPF, Fig. 5 shows the frequency response of Grp(z). For (6), we can see a deviation from the original source in the magnitude and phase in the high frequency region. The harmonic rejection performance of (5) is better than that of (6). Therefore, in this study, the repetitive controller given by (5) is chosen. There are many types of stability factors q(z) in a repetitive controller. The stability factors are given by

Fig. 5.Comparison of frequency responses of repetitive controllers.

Fig. 6 compares the frequency responses of (7), (8), and (9) up to the Nyquist frequency. Further, qcons(z) is the magnitude of the repetitive controller output amplified at all-frequency range, and the repetitive controller has zero phase delay whereas qlpf (z) is the magnitude of the repetitive controller output in the low-frequency range and reduces from the cutoff frequency range, and the phase of the repetitive controller is delayed in the high-frequency range. Further, qnt(z) is the magnitude of the repetitive controller output equaling that of an LPF, and the repetitive controller has zero phase delay. In this study, the stability factor is chosen to be qnt(z).

Fig. 6.Comparison of frequency responses of repetitive controllers: (a) Constant type stability factor; (b) Low pass filter type stability factor; (c) Zero phase delay low pass filter type stability factor.

3.2 Design of repetitive controller

The proposed repetitive control algorithm is shown in Fig. . ★The control algorithm includes a plug-in repetitive control scheme such as a feedforward structure. In Fig. 7, Tv(z) is defined as follows:

Fig. 7.Plug-in repetitive control scheme.

In order to design a stable controller, the relationship between the reference input voltage (v*) and the voltage error (verr) can be obtained from Fig. 7, which is similar to the form derived in [25], as follows:

If all the magnitudes of the roots for Ge(z) are less than unity, all the poles of Ge(z) are located in the unit circle in the z-domain so that Ge (z) remains stable. In that situation, the stability of Ge (z) depends on arp (z) and brp(z). The roots of the denominator of arp (z) are identical to the roots of Tv (z). Hence, the roots of the denominator of arp (z) are located within the unit circle as long as Tv (z) is stable. Accordingly, the stability of Ge (z) is determined by the roots of brp (z). Let the function of the roots of H(z) be defined as follows:

In order to design a stable repetitive controller, the design parameters L, q(z), and Gf (z) must be selected to satisfy (14).

Fig. 8 shows a comparison of the trajectories of the roots up to the Nyquist frequency (10 kHz) at different values of Gf (z). In Fig. 8, if Gf (z) is less than 0.2, the trajectories of the roots converge into the unit circle. Accordingly, the repetitive control system is considered to be stable because condition (14) is satisfied. Fig. 8 shows the relationship between v* and the output voltage vo of the repetitive controller. Fig. 9 shows the depths of each notch at multiples of 60 Hz. This indicates that the repetitive controller can effectively block the repetitive error.

Fig. 8.Trajectories for the root of H(z).

Fig. 9.Frequency response of Ge (z).

 

4. Simulation Results

In order to investigate the effectiveness of the designed controllers, simulations were performed using the PSIM simulation software package.

All the parameters used in the simulation are identical to the values given in Table 1. Fig. 10 shows the simulation results of the repetitive controller with the inductor current control model. It shows the results for a load of 1500 W, and RMS of the output voltage and peak-to-peak magnitude of the voltage error are 220 V and approximately 110 V, respectively. Fig. 11. shows the simulation results with the capacitor current control model as load.

Table 1.Single-phase stand-alone inverter parameter

Fig. 10.Simulation results with inductor current control model under 1.5 kW loads.

Fig. 11.Simulation results with capacitor current control model under 1.5 kW loads.

It shows the results for a load of 1.5 kW, and the RMS of the output voltage is 220 V. Further, the peak-to-peak magnitude of the voltage error is approximately 30 V. Figs. 12 and 13 show the simulation results under different nonlinear load conditions. Without the repetitive controller, the peak-to-peak magnitude of the current error is approximately 38 V, as can be seen in Fig. 12(a). However, with the designed repetitive controller, the peak-to-peak magnitude of the voltage error is limited to 1.5 V, as can be seen in Fig 12(b). Further, the total harmonic distortion (THD) varies from 0.08% to 0.04%.

Fig. 12.Simulation results under nonlinear loads: (a) without and (b) with repetitive controller.

Fig. 13.Simulation results under step nonlinear loads: (a) without and (b) with repetitive controller.

Before using the repetitive controller, the peak-to-peak magnitude of the voltage error is approximately 38 V, as shown in Fig. 13(a). However, with the designed repetitive controller, the peak-to-peak current errors are limited to 4 V. Moreover, the THD varies from 3.8% to 1.3%, and the RMS value varies from 216 V to 219 V.

 

5. Experimental Results

The proposed control scheme has been tested with a 1.5-kW single-phase inverter whose parameters are the same as the values in Table 1. The SKM75GM128D IGBT module manufactured by SEMIKRON was chosen as the switch, and the switching frequency is 20 kHz. The TMS320F28335 32-bit floating-point digital signal processor (DSP) manufactured by Texas Instruments was used as the digital controller. Numerous experiments were conducted to analyze the closed-loop performance of the proposed controller. The closed-loop performance of the controller investigated under the following conditions:

Controller based on both inductor and capacitor current control models (linear load) Nonlinear load

1) Controllers Based on Inductor and Capacitor current control Models: The closed-loop performances of the controllers based on the capacitor and inductor current control models are compared. Each test was conducted using linear loads from no-load to full-load conditions for analyzing characteristics of the output voltage. Fig. 14 shows the output waveforms of the controller based on the inductor current control model under different loads. The controller is optimized to full-load condition, and Figs. 14(a) - (b) show the output waveforms corresponding to 0 W and 1500 W loads. As the load power increases, the output voltage approaches 311 V; however, the error of the output voltage also increases.

Fig. 14.Experimental results with the inductor current control model: (a) no load; (b) 1.5 kW load

Fig. 15 shows the output waveforms of the controller based on the capacitor current control model under different loads. This controller is also optimized to full-load condition, and Figs. 15(a) - (b) show the output waveforms at 0 W, 500 W, 1000 W, and 1500 W loads. As opposed to the inductor current control model, the output voltage is controlled and nearly constant at 311V and the magnitude of voltage error also varies minimally regardless of load.

Fig. 15.Experimental results with the capacitor current control model: (a) no load; (b) 1.5 kW load.

Figs. 16, 17, and 18 compare the peak output voltage values, output voltage errors, and THDs of the output voltage waveforms, respectively, of the controllers based on the inductor and capacitor current control models. As can be seen in Fig. 16, the peak output voltage value of the inductor current control model varies from 0% to 8% whereas that of the capacitor current control model varies slightly from 0.3% to 0.7% regardless of load. The output voltage error of the inductor current control model is approximately twice as that of the capacitor current control model, as shown in Fig. 17. Additionally, the THD of the inductor current control model is evaluated to be approximately 2.5% on average, and that the capacitor current control model is evaluated to be approximately 1.5%, as shown in Fig. 18. Therefore, the experimental results show that the capacitor current control model is superior to the inductor current control model in all the cases.

Fig. 16.Comparison of output voltage peak values.

Fig. 17.Comparison of output voltage errors between reference and output voltage.

Fig. 18.Comparison of THDs.

2) Nonlinear Loads: The Fig. 19 shows two types of nonlinear loads and nonlinear thyristor loads. These load configuration is followed by IEC 6240-3. Former consists of a diode rectifier, Rserise, RL and C, where (Rserise = 9 Ω, RL = 48 Ω /C= 3300 μF) and latter consists of a thyristor rectifier and RL where (RL = 23.8 Ω). Figs. 20 and 21 show the output voltage, output current and voltage error without and with the proposed repetitive controller under the nonlinear loads. The analysis of the experimental results is given in Table 2. From Fig. 20, the output voltage, voltage error, and THD are determined to be 310.6 V, 42 V, and 2% without the repetitive controller and 311 V, 0 V, and 1.37% with the repetitive controller, respectively.

Table 2.Comparison controller performance

Fig. 19.Nonlinear load structure: (a) nonlinear load; (b) nonlinear thyristor load.

Fig. 20.Experimental results of a diode rectifier load: (a) without repetitive controller; (b) with repetitive controller.

In Fig. 21, the output voltage, voltage error, and THD are 304.5 V, 28.2 V, and 4.07% and 310.6 V, 0.42 V, and 4.76% without and with the repetitive controller, respectively. In the case of Fig. 21, the output voltage THD is degraded; however, its level is within a permissible range of grid code. This problem can be overcome by increasing the capacitance in the output filter. The results clearly show that the proposed repetitive controller improves the quality of the output voltage under nonlinear loads.

Fig. 21.Experimental results of a thyristor rectifier load: (a) without repetitive controller; (b) with repetitive controller.

 

6. Conclusion

A high-performance voltage controller based on capacitor current control model is proposed for single-phase stand-alone inverters in this study. The proposed controller was designed using the repetitive control scheme to handle voltage distortions under nonlinear loads in 1.5kW single phase stand-alone inverter prototype. The simulation and experimental results verify the effectiveness of the proposed control scheme under nonlinear loads. The output voltage error is decreased by 39.8% in rate power, and the THD is increased by 36.8% under stable linear load condition. The output voltage error is decreased by over 98% in step response. This high-performance voltage controller can be employed in ESS and UPS applications.