# Stability analysis of high power factor Vienna rectifier based on reduced order model in *d*-*q* domain

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## Abstract

For a DC distributed power system, system stability can be predicted by dividing it into source and load subsystems, and then applying the Nyquist criterion to the impedance interaction between the source and load model. However, the generalized Nyquist criterion is extremely complicated and cannot directly reveal effective control strategies to reduce interaction problems of cascade three-phase AC systems. Specifically, as a current force rectifier, this characteristic makes it difficult to judge the stability of a cascade three-phase Vienna AC system. To deal with the aforementioned problems, a simplified small signal stability criterion is presented for an AC distributed power system. Based on the criterion, the small signal model and impedance based on the reduced order model in the *d*-*q* domain are studied theoretically. For the instability issue, an impedance regulator design method is presented. The correctness of the simplified stability criterion and the effectiveness of the proposed impedance regulator method are validated by extensive simulation and experiment.

## Keywords

Three-level Vienna rectifier Small-signal stability Nyquist criterion Grid impedance Impedance regulator## 1 Introduction

In recent years, with the fast development of communications technology, the electric vehicle (EV) charging station and the computer industry, the safe and reliable operation of high voltage direct current (HVDC) power supply systems has become an important research topic all over the world [1, 2]. Specifically, the three-phase three-level boost-type Vienna rectifier [3] has proven to be a cost-effective and very efficient solution, maximizing the power density of industrial motor drives, active filters, the EV charging station and data center HVDC power supply system [4]. However, power electronic converter systems with regulated output voltage feature negative incremental input impedance, which translates into a constant power load (CPL) behavior. This characteristic may makes the distributed power system (DPS) suffer from small-signal instability issues because of the dynamic interactions between the converters and passive components in the systems [5, 6]. This issue exists in both DC and AC DPSs. For the DC DPS, the impedance criterion was first established by Middlebrook, who states that the system stability can be predicted by dividing it into the source and load subsystems, and then applying the Nyquist criterion to the ratio between the source output impedance and the load input impedance [7, 8, 9]. For the AC DPS, however, there has been much less research.

It should be noted that the definition of the small-signal impedance in three-phase AC DPS is complicated compared to the DC DPS. This makes it extremely complicated and cannot directly suggest effective control strategies to improve interaction problems of the source and load subsystems. Most of the work on the AC DPS can be categorized into two approaches. One is the analysis of the system in the *d*-*q* domain, whereas the other applies harmonic linearization in the phase domain through symmetric components [10]. The impedance-based stability criteria developed by Macfarlane and Postlethwaite in the 1970’s has also been proposed for the three-phase AC DPS. This method employs the generalized Nyquist criterion (GNC) in the *d*-*q* domain to study the stability of a cascaded AC system [11]. However, calculating the singular value of the matrix was complicated and inconvenient for the design of AC DPSs. A significantly improved modeling method was presented by Mao in 1996 [12, 13]. By developing a reduced-order (RO) model for active front-end (AFE) converters, Mao showed that the GNC could be used to study the stability of the converter using the RO model [14, 15, 16]. In return, the stability of an AC DPS can be predicted by examining the locus described by the impedance models based on analytical calculations or numerical simulation. As a result, the RO model and impedance analytical calculations are critical for the design of three-phase power conversion systems. With the rapid development of digital controllers, various digital control strategies have been proposed for the control of the Vienna rectifier [4, 17, 18, 19]. However, as a current force rectifier, detailed analysis on the RO averaged model and small signal stability analysis of the Vienna rectifier have rarely been reported.

In order to further simplify the theoretical analysis, this paper proposes a simplified stability criterion and an input impedance regulator design method to analyze the interaction problems between three-phase Vienna AC systems and the variable grid impedance. First, averaging and local linearization techniques are used to derive the dynamic RO model expressed in the *d*-*q* domain. Then the impedance model is computed neglecting interactions between the *d*-*q* components of control inputs and currents, respectively. For unstable cascade AC systems, an input impedance regulator design method is proposed to improve the interaction problem of the system. Most importantly, the derived small-signal stability criterion and impedance regulator provide a simple and useful theoretical basis for the design process. Instead of trial and error, the proposed criterion can be used to predict and guarantee the stability operation of an AC DPS during the design process, and though this paper is mainly concerned with the three-phase boost-type rectifier, the approaches can be applied to other topologies.

The contribution is organized as follows. First, a simplified small signal stability criterion for AC DPS is presented in Section 2. Then the RO small signal and impedance model are discussed, neglecting the interactions between the *d*-*q* components, in Sections 3 and 4, respectively. The stability issue between the Vienna rectifier and grid impedance is proposed in Section 5. Then, for unstable systems, an input impedance regulator design method is discussed. Finally, the analysis of previous sections is verified in Section 6.

## 2 Simplified small signal stability criterion for AC system in synchronous *d*-*q* frame

*Q*

_{a},

*Q*

_{b}, and

*Q*

_{c}. Here,

*u*

_{sx}(subscript

*x*denotes a, b, c) is the electromotive force of the power grid,

*u*

_{gx}is the grid voltage at the point of common coupling (PCC),

*i*

_{gx}is the grid current,

*L*is the inductance which is being used to suppress the high-frequency harmonics,

*L*

_{sg}denotes the equivalent inductance of the grid power,

*C*

_{g}denotes the distributed capacitance of the grid power,

*R*

_{L},

*R*

_{sg}and

*R*

_{cg}denote the parasitic resistances,

*C*

_{1}and

*C*

_{2}are the DC-link capacitors,

*i*

_{op}and

*i*

_{on}are the output currents, and

*u*

_{0}is the output voltage of the DC bus.

**u**_{sdq}(

*s*) is the disturbance of the source subsystem.

**Z**_{sdq}(

*s*) is the output impedance of the source subsystem seen from the PCC.

**Y**_{Ldq}(

*s*) is the input admittance of the Vienna rectifier seen from the PCC.

**u**_{bdq}(

*s*) is the grid voltage at the PCC in the

*d*-

*q*domain.

**G**_{sdq}(

*s*) is the voltage gain of the input filter.

**G**_{ldq}(

*s*) is the voltage gain of the Vienna rectifier.

**Z**_{sdq}(

*s*) and

**Y**_{Ldq}(

*s*). As a result, the stability of the AC DPS is mainly described by the transfer matrix as:

**Y**_{ldq}(

*s*) and

**Z**_{sdq}(

*s*). Providing that the three-phase Vienna rectifier is fully decoupled by unity-power factor control, as compared with

**Y**_{Lqq}(

*s*) or

**Y**_{Ldd}(

*s*), the value of the cross-coupled input admittances

**Y**_{Ldq}(

*s*) and

**Y**_{Ldq}(

*s*) can be ignored [20, 21]. By neglecting the cross-coupling between the input and output variables of the rectifier, the three-phase Vienna rectifier can be divided into two independent DC-DC channels. As a result, the small signal stability is derived as:

Based on the assumption above, the three-phase Vienna rectifier can be independent of the *d*-*d* and *q*-*q* channel. As presented in [9, 13, 20, 21, 22], since the phase of **Z**_{sqq}(*s*)**Y**_{lqq}(*s*) was always leading the phase of **Z**_{sdd}(*s*)**Y**_{Ldd}(*s*) by 180°, then the stability of the converter and filter system could be studied by analyzing **Z**_{sdd}(s)**Y**_{ldd}(s) alone. This validated the use of the RO model of the three-phase boost rectifier, which modeled the converter as an equivalent dc-dc converter. The stability of the converter and filter system can then be evaluated using the standard Nyquist stability theorem for a single input single output (SISO) system proposed in Middle-Brook’s criterion for DC-DC converters. Moreover, modeling is of great importance since it is a first step toward control design, and the control scheme analysis can be effectively simplified using the RO model. Nevertheless, very little research about the RO model and stability analysis of Vienna rectifier has been reported. To deal with those aforementioned issues, using the two-port network theory, the corresponding impedance transfer functions based on the RO small-signal model in the *d*-*q* domain must be derived.

## 3 Reduced order averaged model of Vienna rectifier

*Q*

_{a},

*Q*

_{b},

*Q*

_{c}).

- 1)
The switching frequency is much higher than the grid frequency and the output voltage and grid current are all in steady state.

- 2)
The DC-link capacitor is sufficiently large that, as a result, the output DC voltage is constant during every switching period, and the neutral voltage is zero.

*S*

_{x}is the switching function of switch

*Q*

_{x}; \(\text{sgn} \left( {i_{x} } \right)\) is the threshold function.

Since the Vienna rectifier is a current force converter, the pole voltage (*u*_{AO}, *u*_{BO}, *u*_{CO}) is determined by the switching state and the polarity of the AC phase current [24].

The expression of *u*_{AO} is shown as:

*Q*

_{x}is

*T*

_{x}and the switching period is

*T*

_{s}, the duty ratio of switch

*Q*

_{x}is defined as:

*d*-

*q*equations of the system are obtained as:

*is the abc/*

**K***dq*0 transform matrix.

*U*

_{s}is the root mean square (RMS) value of the source line-to-neutral voltage;

*I*

_{s}is the RMS value of the line current;

*i*

_{0}is the output current of the system (

*i*

_{0 }=

*i*

_{op }+

*i*

_{on}).

*d*-

*d*channel in the

*d*-

*q*domain is written as:

*(*

**Y***s*) denotes output vector;

*(*

**U***s*) denotes the control input vector.

## 4 Impendence analysis of Vienna rectifier in synchronous *d*-*q* frame

*d*-

*q*domain is shown in Fig. 4, where SVPWM stands for space vector pulse width modulation. It is compatible with any

*d*-

*q*frame alignment of the input AC voltages. With this control scheme, the unity power factor is achieved by the proportional-integral (PI) controller and the internal phase-locked loop (PLL). In addition, the PLL aligns the converter with the existent

*d*-

*q*domain. The DC-link voltage

*u*

_{0}must be assumed as constant, because the rectifier is regulated as an AFE. Compared with the previous modeling method, the proposed method has several special features. It ensures

*d*-

*q*frame compatibility. On the other hand, it achieves all control dynamics.

To simplify the analysis, the equivalent carrier-based PWM modulation method is adopted. The corresponding modulation waveforms without and within zero-sequence injection are shown in Fig. 4. The modulation index (*M*_{r}) is set to \(\sqrt 3 /2\), *u*_{ap} is the waveform before the injection of common-mode component, *u*_{z} is the common-mode component, *u*_{a} is the waveform after the injection of the common-mode component.

*d*-

*d*channel. The control block diagram of the Vienna rectifier based on the RO model is shown in Fig. 5. Here,

*G*

_{PIu}(

*s*) and

*G*

_{PIi}(

*s*) are the PI controller transfer functions of outer voltage loop and inner current loop, respectively.

*G*

_{SVM}(

*s*) is the transfer function of the carried-SVM, which consisted of the modulation and the delay link. The modulation transfer function of carried-SVM [25, 26] is obtained as:

*G*

_{delay}(

*s*) is obtained as:

## 5 Small-signal stability analysis and virtual impedance design

In order to validate the theoretical analysis proposed, a simulation model and experimental prototype of the Vienna rectifier are built. The test parameters are kept consistent with the theoretical analysis. Unless otherwise noted, in all the simulation cases the grid phase voltage is 380 V/50 Hz, the grid voltage frequency is 50 Hz, the DC bus voltage is 750 V, the DC bus capacitor is 1080 μF, and the switching frequency adopted in this paper is 50 kHz. The parameters of grid LC filter I is as below, *L*_{sg }= 0.3 mH, *R*_{sg }= 0.02 \(\varOmega\), *C*_{g }= 20 μF, *R*_{cg }= 0.03 \(\varOmega\). For LC filter I, the value of *C*_{g} is 5 μF. The parameters of PI regulator involved is as list, the *K*_{p} of outer loop is 0.1 and *K*_{i} is 10,the *K*_{p} of inner loop is 0.01 and *K*_{i} is 20.

### 5.1 Small signal stability analysis of Vienna rectifier with LC filter

*d*-

*q*domain is derived as:

*C*

_{g}. The impedance ratio curve of phase difference is nearly 180° as the filter capacitor decreases to 20 μF. However, it should be pointed out that there must be a certain absolute error because the disturbance term of (36) was ignored. Considering many simulation and experiment results, the derived error of phase between the theoretical calculation and actual system is about 5°. In order to maintain the system in a stable condition, the phase difference between the impedance of LC filter and Vienna rectifier should be kept smaller than 175°.

### 5.2 Stability analysis of system with increasing number of CPLs

*G*

_{11c1}(s),

*G*

_{11c2}(s), …,

*G*

_{11cn}(s) are the input impedances of the Vienna rectifiers;

*Z*

_{eq}(

*s*) is the equivalent input impedance;

*n*is the number of multi Vienna rectifiers in parallel operation.

### 5.3 Design method of virtual impedance

*d*-

*d*channel of the Vienna rectifier as an example, the proposed method is shown in Fig. 9.

*f*

_{1},

*f*

_{2}] must be kept between [− 90°, 90°]. This means that the input impedance of CPL must be in the quadrant I or IV in the complex plane. The input impedance in the complex plane is shown in Fig. 10.

The input impedance with the impedance regulator introduced is adjusted from the left half plane of the complex plane to the right half plane [27]. The real part (*R*_{e}) of input impedance \(Z_{\text{id,vi,c}}^{{}} (s)\) with the impedance regulator introduced in the cross section [*f*_{1}, *f*_{2}] should satisfy:

The coefficient of impedance regulation *k* can be calculated by (46).

## 6 Digital simulation and experiment verification

### 6.1 Simulation results

To verify the proposed stability criterion and impedance regulator design method, a whole digital simulation model based on the PSIM is built. The value of the filter inductance is set at 0.35 mH. Note that in all simulation cases, the simulation and experimental parameters are consistent with the theoretical analysis.

### 6.2 Experimental results

To avoid the influence of unknown disturbance on parallel operation,a model of the parallel system is built based on the Real Time-Laboratory (RT-LAB). The grid current of a #1 and #2 Vienna rectifier is shown as Fig. 12. It is clear that the grid current is unstable. All the above results are consistent with the theoretical analysis.

## 7 Conclusion

A simplified small signal stability criterion is presented for the three-phase unity-power factor Vienna cascaded AC systems. Based on this criterion, an input impedance regulator design method to stabilize the unstable cascaded AC systems is proposed. In order to verify the correctness and superiority of the proposed methods, a complete simulation is carried out and an RT-LAB experimental model is built. The results demonstrate that the stability of a cascaded AC system is fully determined by the SISO return-ratio of *d*-*d* channel impedances, and with the proposed impedance regulator the system stability is greatly improved and is not sensitive to the grid impedance. In addition, the proposed criterion can predict and guarantee the stable operation of a cascaded system during the design process, and the proposed method could also be used for other three-phase topologies.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Fund of China (No. 51677151), National Natural Science Youth Fund of China (No. 51507138), the Major Scientific and Technological Innovation Projects of Shaanxi Province, China (No. 2015ZKC02-01) and International Exchange And Cooperation Project of Key R&D Program in Shaanxi (No. 2017KW-035).

## References

- [1]Dang CL, Tong XQ, Huang JJ et al (2017) The neutral point-potential and current model predictive control method for Vienna rectifier. J Frankl Inst 354(17):7605–7623MathSciNetCrossRefGoogle Scholar
- [2]Dang CL, Tong XQ, Huang JJ et al (2017) QPR and duty ratio feed-forward control for Vienna rectifier of HVDC supply system. IEEJ Trans Electr Electron Eng 12(4):501–509CrossRefGoogle Scholar
- [3]Kolar JW, Zach FC (1997) A novel three-phase utility interfaces minimizing line current harmonics of high-power tele communications rectifier modules. IEEE Trans Ind Electron 44(4):456–466CrossRefGoogle Scholar
- [4]Ma H, Xie YX, Shi ZY (2016) Improved direct power control for Vienna-type rectifiers based on sliding mode control. IET Power Electron 9(3):427–434CrossRefGoogle Scholar
- [5]Wen B, Boroyevich D, Mattavelli P et al (2012) Experimental verification of the generalized Nyquist stability criterion for balanced three-phase AC systems in the presence of constant power loads. In: Proceedings of IEEE energy conversion congress and exposition (ECCE), Raleigh, USA, 15–20 September 2012, pp 3926–3933Google Scholar
- [6]Liu Z, Liu JJ, Bao WH et al (2015) Infinity-norm of impedance-based stability criterion for three-phase AC distributed power systems with constant power loads. IEEE Trans Power Electron 30(6):3030–3043CrossRefGoogle Scholar
- [7]Cespedes M, Sun J (2014) Impedance modeling and analysis of grid connected voltage-source converters. IEEE Trans Power Electron 29(3):1254–1261CrossRefGoogle Scholar
- [8]Middlebrook RD (1976) Input filter consideration in design and application of switching regulators. In: Proceedings of IEEE industry applications society annual meeting, Chicago, USA, 10–11 October 1976, pp 94–107Google Scholar
- [9]Rygg A, Molinas M, Zhang C et al (2016) A modified sequence-domain impedance definition and its equivalence to the
*d*-*q*domain impedance definition for the stability analysis of AC power electronic systems. IEEE J Emerg Sel Top Power Electron 4(4):1383–1396CrossRefGoogle Scholar - [10]Vesti S, Suntio T, Oliver JA et al (2013) Impedance based stability and transient-performance assessment applying maximum peak criteria. IEEE Trans Power Electron 28(5):2099–2104CrossRefGoogle Scholar
- [11]Mao HC, Boroyevich DS, Lee FCY (1998) Novel reduced-order small-signal model of a three-phase PWM rectifier and its application in control design and system analysis. IEEE Trans Power Electron 13(3):511–521CrossRefGoogle Scholar
- [12]Belkhayat M (1997) Stability criteria for AC power systems with regulated loads. Dissertation, Purdue UniversityGoogle Scholar
- [13]Feng X, Liu J, Lee FC (2002) Impedance specifications for stable DC distributed power systems. IEEE Trans Power Electron 17(2):157–162CrossRefGoogle Scholar
- [14]Francis G, Burgos R, Boroyevich D et al (2011) An algorithm and implementation system for measuring impedance in the
*D*-*Q*domain. In: Proceedings of 2011 IEEE energy conversion congress and exposition (ECCE), Phoenix, USA, 17–22 September 2011, pp 3221–3228Google Scholar - [15]Phattanasak M, Ghoachani RG, Martin JP (2016) Lyapunov-based control and observer of a boost converter with LC input filter and stability analysis. In: Proceedings of 2016 international conference on electrical systems for aircraft, railway, ship propulsion and road vehicles & international transportation electrification conference (ESARS-ITEC), Toulouse, France, 2–4 November 2016, pp 1–6Google Scholar
- [16]Cao WC, Ma YW, Liu Y (2017)
*D*-*Q*impedance based stability analysis and parameter design of three-phase inverter-based AC power systems. IEEE Trans Ind Electron 99(3):1–11Google Scholar - [17]Wen B, Boroyevich D, Burgos R (2015) Small-signal stability analysis of three-phase AC systems in the presence of constant power loads based on measured
*d*-*q*frame impedances. IEEE Trans Power Electron 30(10):5952–5963CrossRefGoogle Scholar - [18]Debranjan M, Debaprasad K (2015) Voltage sensor less control of the three-level three-switch Vienna rectifier with programmable input power factor. IET Power Electron 8(8):1349–1357CrossRefGoogle Scholar
- [19]Freddy FB, Hugo VB, Luis MS et al (2014) Control of a three-phase AC/DC Vienna converter based on the sliding mode loss-free resistor approach. IET Power Electron 7(5):1073–1082CrossRefGoogle Scholar
- [20]Lin HW, Jia CX, Vasquez JC (2017) Angle stability analysis for voltage-controlled converter. IEEE Trans Ind Electron 99(3):1–11Google Scholar
- [21]Burgos R, Boroyevich D, Wang F et al (2010) On the AC stability of high power factor three-phase rectifiers. In: Proceedings of 2010 IEEE energy conversion congress and exposition (ECCE), Atlanta, USA, 12–16 September 2010, pp 2047–2054Google Scholar
- [22]Youssef NBH, Al-Haddad K, Kanaan HY (2008) Implementation of a new linear control technique based on experimentally validated small-signal model of three-phase three-level boost-type Vienna rectifier. IEEE Trans Ind Electron 55(3):1666–1676CrossRefGoogle Scholar
- [23]Gabriel HPO, Maswood AI, Lim ZY (2016) Grid connected three-phase multiple-pole multilevel unity power factor rectifier with reduce components count. IET Power Electron 9(7):1437–1444CrossRefGoogle Scholar
- [24]Thandapani T, Paramasivam S, Karpagam R (2015) Modelling and control of Vienna rectifier a single phase approach. IET Power Electron 8(12):2471–2482CrossRefGoogle Scholar
- [25]Lee JS, Lee KB (2015) Carrier-based discontinuous PWM method for Vienna rectifiers. IEEE Trans Power Electron 30(6):2896–2900CrossRefGoogle Scholar
- [26]Hang LJ, Li B, Zhang M et al (2013) Space vector modulation strategy for Vienna rectifier and load unbalanced ability. IET Power Electron 6(7):1399–1405CrossRefGoogle Scholar
- [27]Yu H, Ruan XB, Wang XH et al (2014) Stability analysis of cascade AC system based on three-phase voltage source PWM rectifier. In: Proceedings of 2014 international power electronics and application conference and exposition, Shanghai, China, 5–8 November 2015, pp 847–852Google Scholar
- [28]Xue MY, Zhang Y, Kang Y (2012) Full feedforward of grid voltage for discrete state feedback controlled grid-connected inverter with LCL filter. IEEE Trans Power Electron 27(10):4234–4247CrossRefGoogle Scholar

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