Simplified Controller Design Method for Digitally Controlled LCL-Type PWM Converter with Multi-resonant Quasi-PR Controller and Capacitor-Current-Feedback Active Damping

Abstract

To track the sinusoidal current under stationary frame and suppress the effects of low-order grid harmonics, the multi-resonant quasi-proportional plus resonant (PR) controller has been extensively used for digitally controlled LCL-type pulse-width modulation (PWM) converters with capacitor-current-feedback active damping. However, designing the controller is difficult because of its high order and large number of parameters. Moreover, the computation and PWM delays of the digitally controlled system significantly affect damping performance. In this study, the delay effect is analyzed by using the Nyquist diagrams and the system stability constraint condition can be obtained based on the Nyquist stability criterion. Moreover, impact analysis of the control parameters on the current loop performance, that is, steady-state error and stability margin, identifies that different control parameters play different decisive roles in current loop performance. Based on the analysis, a simplified controller design method based on the system specifications is proposed. Following the method, two design examples are given, and the experimental results verify the practicability and feasibility of the proposed design method.

I. INTRODUCTION

With the development of renewable energy and smart grid, energy storage systems (ESSs) have become increasingly interesting. ESSs could smooth the output power and decouple energy generation from demand [1]. As an interface between storage elements and the power grid, a voltage source pulse-width modulation (PWM) converter plays an important role in the single-stage and multistage power conversion systems (PCS) for ESSs [1], [2]. To smooth the injected currents, the conventional L filter is replaced by the LCL filter because of its better harmonic attenuation ability [3]-[10]. However, given the resonance hazard of the LCL filter, damping solutions are required to stabilize the system.

Two main methods are used to dampen resonance, namely, passive damping and active damping. However, active damping is more well known than passive damping because no additional power loss occurs [3]-[10]. Among the various active damping solutions, capacitor–current–feedback active damping is selected in this study because of its effectiveness, simple implementation, and extensive application [6]-[10]. Capacitor–current–feedback active damping is equivalent to a virtual resistor connected in parallel with the filter capacitor [5]. This conclusion is drawn by excluding the delay effect.

However, computation and PWM delays occur in the digitally controlled system. The computation delay is the interval between the sampling instant and duty ratio update instant. The PWM delay is caused by the zero-order hold effect, which keeps the duty ratio constant after it has been updated [10]-[13]. Given the delay effect, the capacitor–current–feedback active damping is equivalent to a variable virtual impedance, which consists of a resistor connected in parallel with a reactor, rather than a virtual resistor. When the virtual resistor is negative, two unstable poles will be generated in the grid current loop [10]. As a result, the resonance peak should not be dampened to less than 0 dB to ensure system stability [8]. Thus, the capacitor–current–feedback gain should be selected with extreme caution.

In addition to system stability, high-quality injected power is another essential object in the control of the LCL-type PWM converter. Thus, the selection and design of the current controller is crucial. The stationary α–β frame is selected in this study to prevent the inconvenient decoupling in the synchronous d–q frame [6]. To track the sinusoidal current reference and suppress the selected low-order current harmonics, the proportional plus multifrequency resonant (multi-resonant proportional plus resonant [PR]) controller has been used extensively [14]-[19]. An ideal resonant controller can provide infinite gain to eliminate the steady-state error, but it occurs at the target frequency only. Any perturbation, such as frequency deviation, will lead to a significant reduction in the generated gain [18]. However, in fact, the grid frequency is allowed to deviate by ±0.5 Hz. Hence, the performance of the controller will be reduced, especially when applied to weak grids and microgrids where the frequency deviates even worse [20]. Moreover, attaining an ideal resonant controller is sometimes impossible because of finite precision in digital systems. To address these issues, the quasi-resonant controller is proposed [12], [19], [21]-[24]. The quasi-resonant controller can provide a sufficiently large gain around the target frequency to reduce its sensitivity to the grid frequency fluctuation and can be attained in digital platforms with a higher accuracy.

However, the design of the quasi-resonant controller, especially the multifrequency quasi-resonant controller (multi-resonant quasi-PR controller), is more difficult than that of the ideal resonant controller because the steady-state error should be taken into account in addition to stability and the stability margin. The design of a single quasi-PR controller is relatively easy and has been presented in [9], [12]. In general, the single quasi-PR controller can be designed based on steady-state error, crossover frequency (fcs), phase margin (PM), and gain margin (GM) of the system, which have a significant effect on system performance and stability margin. However, these design methods are not applicable for the multi-resonant quasi-PR controller because calculating the PM of the controller is impossible because of its high order and large number of parameters. In [21], a guideline on designing the multifrequency quasi-resonant controller (without the proportional controller) is presented, which considered grid frequency deviation, grid synchronization, grid impedance variation, and transient response. In [24], pole placement is used to determine the controller parameters by properly selecting the poles to guarantee system stability and acceptable performance of the current loop. In [23], the controller parameters are designed separately mainly based on the requirements of the steady-state errors and PM. However, these design methods are inconvenient for engineers, and previous studies do not focus considerable attention on capacitor–current–feedback active damping.

The effect of computation and PWM delays on the active damping performance is analyzed in detail by using Nyquist diagrams. The effect of the controller parameters on the current loop performance with the application of frequency response theory in the continuous domain is also investigated. Then, a simplified practical design method of the multi-resonant quasi-PR controller and capacitor–current–feedback coefficient is proposed in this study. The paper is organized as follows: In Section II, the average switching model (ASM) of the internal current loop considering the control delay is derived. Based on the derived ASM, the effect of the delay on the active damping performance, which influences the stability constraint condition of the current loop, is investigated by using the Nyquist stability criterion in Section III. In Section IV, the effect of the control parameters on system performance, that is, system stability constraint condition, steady-state error, and stability margin, are investigated. Based on the analysis, a simplified design method is proposed in Section V, and two design examples are conducted step by step by using the proposed method. In Section VI, the effectiveness of the proposed design method is verified by using the experimental results from a prototype of a three-phase LCL-type PWM converter. The conclusion is given in Section VII.

 

II. CONTROL STRATEGY AND MODEL OF THE LCL-TYPE PWM CONVERTER

Fig. 1 shows the configuration of a three-phase LCL-type grid-connected PWM converter in the stationary α–β frame. The LCL filter is composed of L1, C, and L2. Cdc is the direct current (DC) link capacitor. As the equivalent series resistors (ESRs) of L1, C, and L2 can provide a certain degree of damping and help stabilize the system, the ESRs are omitted in this study to obtain the worst case.

Fig. 1.Topology and control strategy of a three-phase LCL-type PWM converter in the stationary α–β frame.

As an interface between storage elements and the power grid in two-stage PCS, the primary objective of the PWM converter is to exchange power with the grid by controlling the grid current i2 directly. To directly control the battery charge–discharge and prolong its service time, the DC link voltage udc is also controlled by the PWM converter. As such, the d-axis current reference is generated by the outer DC voltage loop. Thus, the α-axis and β-axis current references are obtained by using the reverse Park transformation to the d-axis and q-axis current references To synchronize with the grid voltage ug, the phase angle of ug is detected through a decoupled double synchronous reference frame phase-locked loop [25]. The capacitor current iC serves as feedback to damp the LCL filter resonance actively, and K is the feedback coefficient. The capacitor–current–feedback signal is subtracted from the output of the current controller. Then, the capacitor–current–feedback signal is normalized with respect to udc/2 to obtain the modulation reference, which is fed to a digital PWM modulator.

As previously mentioned, computation and PWM delays occur in the digitally controlled system. The computation delay is one sampling period Ts in the synchronous sampling case when sampling is conducted at the beginning of a switching period. The calculated duty ratio is not updated until the next sampling instant. The PWM delay is definitely a half sampling period. Thus, the total delay is one and a half sampling periods (1.5Ts) [10]–[13]. The single-phase equivalent ASM of the current loop for the converter in inverter mode is shown in Fig. 2. We noted that the antialiasing filter could be removed in the synchronous sampling case [11]. Therefore, the grid current i2 can be derived as follows:

Fig. 2.Single-phase equivalent ASM of the current loop for the digitally controlled LCL-type PWM converter in inverter mode.

where T(s) is the loop gain of the system and is expressed as follows:

and Gg(s) is expressed as follows:

where is the resonance angular frequency of the LCL filter.

As shown in Eq. (1), the grid voltage low-order harmonics have a significant effect on the grid current i2. To suppress the effect of the low-order harmonics, the multi-resonant quasi-PR controller is employed. The transfer function is expressed as follows:

where h can take the values 1, 3, 5, 7, …, m, with m being the highest current harmonic to be attenuated.

 

III. EFFECT OF THE COMPUTATION AND PWM DELAYS ON THE ACTIVE DAMPING PERFORMANCE

As shown in Fig. 2, considering ug as the disturbance, the block diagram can be transformed into the standard dual-loop structure shown in Fig. 3. The loop gain of the active damping loop Tic(s) can be derived as follows:

Fig. 3.Standard dual-loop structure of the grid current loop with capacitor current active damping.

As shown in Fig. 3, the grid current loop T(s) has no open-loop poles that lie in the right half plane (RHP), except for the active damping loop. That is to say, the number of RHP closed-loop poles of the active damping loop determines the number of RHP open-loop poles of T(s). In this study, the Nyquist diagram of Tic(s) is used to determine the number of RHP closed-loop poles by examining the magnitude at the negative real axis crossing frequency ωpc. Notably, the crossing points at high frequencies caused by the delay effect affect stability only slightly. As such, the conclusions obtained on the delay effect are summarized as follows. Similar results can be found in [10].

Fig. 4.Nyquist diagrams for the positive frequency of the active damping loop with different ωres. (a) Analog control (no delay). (b) ωs/6 > ωres. (c) ωs/6 ≤ ωres ≤ ωs/2. (d) ωs/2 < ωres.

Notably, when ωres > ωs/2, the Nyquist curve may encircle the critical point [see Fig. 4(d)]. However, this case will never occur because ωres < ωs/2 is required to ensure system controllability [26].

 

IV. SYSTEM PERFORMANCE ANALYSIS

As shown in Eqs. (2) and (4), the system is of high order and contains many control parameters. Thus, analyzing system performance is difficult. As such, the controller and system model have to be simplified first.

A. Simplified Controller and System Model

The quasi-PR controller shown in Eq. (4) can be rewritten as follows:

where = nKrh/Kp is the relative resonant gain of the PR controller and n is the number of resonant controllers.

Fig. 5 shows the Bode diagram of the PR controller derived using Eq. (7) with different parameters. The following conclusions can be drawn: (1) determines the relative gain at the target frequency ωh. The gain gradually increases with the increase in . However, the phase lag introduced by the controller is also increased. (2) ωc mainly influences the resonant bandwidth at the target frequency to improve its robustness against the frequency fluctuation. (3) Kp shifts the magnitude plot up and down and has only a slight effect on the phase plot.

Fig. 5.Bode diagram of the multi-resonant quasi-PR controller (ωc = 3) with different parameters.

Based on Eq. (7), the gain at ωh can be obtained as follows:

As shown in Fig. 5, the quasi-PR controller can be approximated to Kp at frequencies greater than ωm, that is,

Typically, the crossover angular frequency ωcs is restricted to a value lesser than ωres. Therefore, the LCL filter can be simplified as an L filter when calculating the magnitude at ωcs and the frequencies lesser than ωcs, which is also applicable for digitally controlled systems [8]. As such, the magnitude of T(s) and Gg(s) at ωcs and the frequencies lesser than ωcs can be simplified as follows:

Moreover, |T(jωcs)| = 1, combining Eqs. (9) and (10) produces the following equation:

B. System Stability Constraint Condition Analysis

As analyzed previously, two RHP open-loop poles might be generated in T(s) because of the delay effect. Thus, to ensure system stability, the Nyquist curve for positive frequency has to make one counterclockwise encirclement of the point (−1, j0). In the Nyquist diagrams of T(s) shown in Fig. 6, the negative real axis crossings might occur at ωres or at ωres and ωs/6. Combining Eqs. (2), (9), and (12), the loop gain T(s) at ωres and ωs/6 can be obtained as follows:

Fig. 6.Nyquist diagrams of the loop gain T(s) with GPR(s) = Kp. (a) ωres < ωs/6. (b) ωres ≥ ωs/6.

From Eq. (13), we observed that the Nyquist curve of T(s) always crosses over the negative real axis at ωres for K > 0. As shown in Eq. (6), Kc > 0 for ωres < ωs/6 and Kc ≤ 0 for ωres ≥ ωs/6. Thus, if ωres < ωs/6, the Nyquist curve crosses over the negative real axis one more time at ωs/6 for K > Kc [see Fig. 6(a)] and, if ωres ≥ ωs/6, the Nyquist curve certainly crosses over the negative real axis at ωs/6 for K > 0 [see Fig. 6(b)]. We noted that, if ωres = ωs/6, the crossing points at ωres and ωs/6 coincide with each other.

We assume the magnitude requirements of T(s) at ωres and ωs/6 are M1 and M2, respectively. Based on the previous analysis, the stability constraint condition on the grid current loop can be derived as follows:

Fig. 7.Curves of fcs with the increase in fres for M1 = 0.707.

C. Steady-state Error Analysis

As shown in Eq. (1), the grid current i2 comprises two parts. One part is the command–current component generated by the current reference . The other part is the voltage–current component generated by the grid voltage ug. Based on Eq. (1), the grid current error can be derived as follows:

Considering the role of the controller, if the magnitude of T(s) at the fundamental angular frequency ω1 is sufficiently large, then 1 + T(jω1) ≈ T(jω1). Moreover, as the influence of the filter capacitor is negligible at ω1, considering Eqs. (1), (10), and (11), the fundamental component of i2(s) can be approximated as follows:

As the quasi-PR controller can provide sufficiently large gain at ω1, the voltage–current component could be attenuated to decrease its value, that is, −ug(jω1)/(Kp + Kr1) ≈ 0. Thus, simplification of the steady-state error involves the amplitude error only, not the phase error. Moreover, as shown in Eq. (1), the harmonic currents are generated only by the grid harmonics. Therefore, the steady-state error requirement can be converted to the amplitude error requirements of the current components at ω1 and ωh which are denoted by εi and εuh, respectively. Based on Eq. (19), εi and εuh are defined as follows:

To ensure that the system is stable, ωh should be lesser than ωcs. Accordingly, substituting Eqs. (10) and (11) into Eq. (21), the relationship between the gain of the PR controller and steady-state amplitude errors can be approximated as follows:

Considering Eqs. (8) and (12), the relationship between and the steady-state errors can be calculated as follows:

D. Stability Margin Analysis

Based on the stability constraint condition analysis discussed previously, we noted that the magnitude requirements M1 and M2 determine the GM of the system. Therefore, we focus on the PM only. As shown in Eq. (2), the PM is codetermined by the phases of the control object GLCL(s) and PR controller at ωcs. The phase of GLCL(s) at ωcs decreases with the increase in K [see Fig. 8(a)]. With respect to the PR controller, Fig. 5 shows that the phase lag caused by the PR controller increases with the increase in ih . ωc is relatively small. Thus, the effect of ωc on the PM is disregarded. Kp has no effect on the phase response, but Kp affects ωcs. Thus, the PM of the system is related to K, , and Kp.

Fig. 8.Bode diagrams of the control objective GLCL(s) with different K (a) and the loop gain T(s) with different Kp (b).

As analyzed previously, K regulates system stability and influences steady-state error. Therefore, when system stability and steady-state error have been ascertained, the system phase response could be derived by using Eq. (2) with Kp/n = 1, and PM is related to Kp only. As shown in Fig. 8(b), the PM is changed with different values of ωcs, which is approximately proportional to the value of Kp. As |T(jωcs)| = 1, substituting Eq. (9) into Eq. (2), the accurate relationship between Kp and ωcs can be derived as follows:

 

V. DESIGN OF THE CURRENT CONTROLLER AND CAPACITOR–CURRENT–FEEDBACK COEFFICIENT

A. Design Procedure of the Control Parameters

As analyzed previously, the damping gain K mainly influences the system stability, the relative resonant gain ih mainly regulates the steady-state error, and the proportional gain Kp mainly affects the PM of the system. Thus, a simplified controller design method based on the specifications of the current loop is proposed as follows:

Step 1. The specifications of the grid current loop are determined, specifically εi and εuh by the requirements of the steady-state errors at the target frequencies, the PM by the requirements of the dynamic response and robustness, and ωcs by the requirement of the dynamic response speed.

Step 2. K is designed based on the stability requirement.

Step 3. ωc is designed based on the deviation range of the grid fundamental frequency.

Step 4. is designed based on the steady-state error requirements.

Step 5. Kp is designed based on the PM requirement.

B. Design Example

Based on a 5 kW prototype in the laboratory, two different filter capacitor values are considered to range the filter resonance frequency. The parameters of the LCL-type PWM converter are given in TABLE I. We consider four resonant controllers (h = 1, 5, 7, 11). Given that fres in Case I is close to fs/6, K > Kc is preferred rather than K < Kc. From [20], the maximum deviation of the grid fundamental frequency is approximately 0.5 Hz. As such, Δf is equal to 0.5 Hz in Step 3. The design procedures and results are shown in Table II, where GM1 and GM2 denote the GM around ωres and ωs/6, respectively. The parameters of the quasi-PR controller corresponding to Eq. (4) are Kp = 9.6, Kr1 = 180, and Krh = 84 for Case I and Kp = 7.8, Kr1 = 146.25, and Krh = 68.25 for Case II.

TABLE ILCL FILTER SYSTEM PARAMETERS

TABLE IIDESIGN PROCEDURE AND RESULTS

Fig. 9 shows the Bode diagrams of the grid current loop before and after compensation with different controllers. By comparison, we observed that the phase lag introduced by the controller will shift the −180° crossing point, which could improve system robustness to a certain extent, especially for fres close to fs/6 when the active damping loop is unstable. Specifically, for Case I, the GM increases from GM1 = 0.898 dB and GM2 = −0.782 dB with one resonant controller (n = 1) to GM1 = 1.27 dB and GM2 = −1.27 dB with four resonant controllers (n = 4), but the PM decreases from 33.7° to 31.2°.

Fig. 9.Bode diagrams of the grid current loop before and after compensation with different controllers: (a) Case I and (b) Case II.

For Case II, the GM increases slightly from 2.21 dB to 2.27 dB and the PM decreases from 34.1° to 29.3°. Moreover, considering the delay effect, we observed that the LCL resonant frequency deviates from ωres. The actual resonant angular frequency and the actual damping ratio ξ' are derived in the Appendix and expressed as follows:

In Eq. (27), by letting , obtaining sin(1.5Tsω) > 0, f(ω) > 1 for ω < ωs/3 and sin(1.5Tsω) < 0, f(ω) < 1 for ω > ωs/3 becomes relatively easy. As a result, with the increase in K, is greater than ωres for ωres < ωs/3 and lesser than ωres for ωres > ωs/3, but never exceeds ωs/3. This finding means that will be close to ωs/3 with the increase in K. We noted that, for K = Kc, the active damping loop is marginally stable with ωpc = ωs/6 and has no contribution to the resonance damping, = ωs/6.

In the actual condition, L1 and C do not significantly change, except for L2 (considering the impact of grid impedance). Fig. 10 shows the Nyquist plots of the grid current loop around the critical point when L2 is decreased by 50% or increased by 100%. This finding indicates that the grid current loop remains stable for both cases when L2 is decreased by 50% or increased by 100%. Nevertheless, the PM of Case I changes from 23.9° to 33.3° [see Fig. 10(a)] and the PM of Case II changes from 26.6° to 2.07° [see Fig. 10(b)]. We noted that the GM decreases significantly for Cases I and II, and the active damping loop of Case II becomes unstable when L2 is decreased by 50%. Thus, to improve system robustness, K should have a larger value in Step 2 when the active damping loop is stable.

Fig. 10.Nyquist plots of the grid current loop around the critical point when L2 changes from −50% to 50%: (a) Case I and (b) Case II.

 

VI. EXPERIMENTAL VERIFICATION

A 5 kW prototype has been constructed in the laboratory to verify the effectiveness of the proposed design method. The key parameters of the prototype are listed in TABLE I. A Yy-type galvanic isolation transformer is placed between the LCL-type PWM converter and the grid. The grid voltages and currents are sensed by voltage/current halls. The control algorithm is implemented in a 32-bit float-point digital signal processor (TMS320F28335). The quasi-PR controller is discretized by Tustin transformation. In this study, the capacitor current is indirectly sensed through the difference between i1 and i2, and the current waveforms are inverted on the oscilloscope.

Fig. 11 shows the experimental waveforms at no load for Cases I and II, where the current tracking error of α-axis eiα is measured through the analog-to-digital conversion interface on the control board. From the spectra of uga, we observed that low-order harmonics exist in the real power grid, and the total harmonic distortion (THD) is 2.293%. The measured steady-state errors normalized with respect to ug are listed in TABLE III. Considering the current distortion and the effect of dead time, the errors are slightly larger than the actual values of εu1, which are calculated from Eq. (21) using the designed parameters.

Fig. 11.Experimental waveforms at no load for (a) Case I and (b) Case II. From top to bottom: the DC link voltage udc, the grid voltage uga, the grid current i2a, the spectrum of uga, and the current tracking error of α-axis eiα.

TABLE IIIMEASURED RESULTS

Fig. 12 shows the experimental waveforms when the DC resistant load is 40 Ω, with power factor (PF) set to 1.0. We observed that the grid current is sinusoidal and the harmonics at the target frequencies (5th, 7th, and 11th) have been well suppressed. Moreover, the resonant peak is not dampened to less than 0 dB, which coincided with the design results. However, considering the leakage inductance of the isolation transformer, the actual resonant frequency is slightly lesser than the theoretical value shown in Fig. 9. The actual resonant frequencies are approximately 1,900 Hz for Case I and 1,650 Hz for Case II. The measured error eiα, PF, and harmonic contents at the target frequencies are listed in TABLE III. The measured errors are 0.524% for Case I and 0.636% for Case II, which are slightly smaller than that at no load because the command–current error εi at rectifier mode can cancel some of the voltage–current error εu1, which can be observed in Eq. (19) when the direction of current i2 at inverter mode is positive in the actual system. Given that the gain at ω1 for Case I is larger than that for Case II, the measured error and PF for Case I are slightly smaller than that for Case II.

Fig. 12.Experimental waveforms with a DC resistant load R = 40 Ω and PF set to 1.0 for (a) Case I and (b) Case II. From top to bottom: the DC link voltage udc, the grid voltage uga, the grid current i2a, the spectrum of i2a, and the current tracking error of α-axis eiα.

To evaluate dynamic performance, the LCL-type PWM converter operating without the outer voltage loop is used. The DC link voltage is provided by a three-phase noncontrolled rectifier. Fig. 13 shows the transient experimental results when the grid current reference ranges between 1 A and 5 A for Cases I and II with PF set to 1.0. We observed that the inverters rapidly responded to the reference change and the current tracking error of α-axis eiα is sustained at approximately zero all the time. Nevertheless, oscillation occurs during the current step change because the resonant peaks are not dampened to less than 0 dB, which implies that the actual damping ratios are small because of the delay effect. Based on Eq. (28), the actual damping ratio ξ' can be calculated at approximately 0.07 for Case I and 0.01 for Case II.

Fig. 13.Transient experimental results at inverter mode when the grid current reference ranges between 1 A and 5 A: (a) Case I and (b) Case II.

 

VII. CONCLUSIONS

In this study, we analyzed the characteristics and controller design method for the digitally controlled LCL-type PWM converter based on the multi-resonant quasi-PR controller and capacitor–current–feedback active damping. The effect of the delay on the active damping performance is investigated by using the Nyquist diagrams. If the damping loop is unstable, two RHP open-loop poles are generated in the grid current loop, which is codetermined by the LCL resonant frequency (fres) and the active damping gain (K). Then, the system stability constraint condition can be obtained based on the Nyquist stability criterion. Moreover, impact analysis of the control parameters on the current loop performance identifies that different control parameters play different decisive roles in the current loop performance: K mainly influences the system stability, the relative resonant gain mainly regulates the steady-state error, and the proportional gain mainly affects the PM of the system. Based on the analysis, a simplified controller design method based on the system specifications is proposed. The proposed method can obtain the optimum controller, which ensures system stability with high robustness and strong ability to suppress the effect of the grid voltage low-order harmonics. Following the method, two design examples are given and the design results are directly used on a laboratory prototype. The experimental results are consistent with the design specifications. These findings confirm the practicability and operability of the proposed design method.