An Inductance Voltage Vector Control Strategy and Stability Study Based on Proportional Resonant Regulators under the Stationary αβ Frame for PWM Converters

Abstract

The mathematical model of a three phase PWM converter under the stationary αβ reference frame is deduced and constructed based on a Proportional-Resonant (PR) regulator, which can replace trigonometric function calculation, Park transformation, real-time detection of a Phase Locked Loop and feed-forward decoupling with the proposed accurate calculation of the inductance voltage vector. To avoid the parallel resonance of the LCL topology, the active damping method of the proportional capacitor-current feedback is employed. As to current vector error elimination, an optimized PR controller of the inner current loop is proposed with the zero-pole matching (ZPM) and cancellation method to configure the regulator. The impacts on system's characteristics and stability margin caused by the PR controller and control parameter variations in the inner-current loop are analyzed, and the correlations among active damping feedback coefficient, sampling and transport delay, and system robustness have been established. An equivalent model of the inner current loop is studied via the pole-zero locus along with the pole placement method and frequency response characteristics. Then, the parameter values of the control system are chosen according to their decisive roles and performance indicators. Finally, simulation and experimental results obtained while adopting the proposed method illustrated its feasibility and effectiveness, and the inner current loop achieved zero static error tracking with a good dynamic response and steady-state performance.

I. INTRODUCTION

Sinusoidal current regulation with low total harmonic distortion (THD) of a 3-phase bidirectional power flowing PWM voltage source converter (VSC) is an aspect of paramount importance to obtain good performance in many different applications, such as active power filters, motor drives, variable-speed wind turbines and photovoltaic inverters [1]-[8]. As a result, the research and development of various methods in the field of current regulators have been put forward to realize good power quality for VSCs. Over the past few years, pulse width modulation (PWM) and hysteresis controls with their optimization have been popularly adopted due to their advanced features [5]-[9]. Hysteresis modulation introduces a minor error to the average current and offers much better dynamic current tracking than the PWM method. However, the inherent disadvantage is its variable switching frequency, which makes it rather difficult to design a power filter for harmonics attenuation and to reduce the switching loss for efficiency improvement [5]. Some researchers have proposed a nonlinear sliding mode variable structure approach based on the adaptive band hysteresis theory, and a lot of effort has been done to achieve a fixed switching frequency. However, obtaining a high-frequency chatter free switching frequency on the sliding surface requires prior knowledge of the upper bounds of the system uncertainties [6]. Space vector modulation (SVM) is a widely used PWM strategy due to its constant switching frequency, and because its chosen switching sequence can be easily implemented with Park transformation [7], [8]. Meanwhile, this voltage oriented control divides alternating current into two parts, active and reactive segments, and then controls them separately with linear PI regulators, which achieves a good dynamic response by the inner current control loop with accurate decoupling calculations and proportional-integral regulating [9]. Some alternative control algorithms have been proposed such as instance predictive control [10], direct power control (DPC) [11], [12], deadbeat control [13]-[15], etc. The main principle of the DPC method is to directly regulate the instantaneous power instead of using the inner current loop, which will produce serious power ripples and a variable switching frequency along with pre-defined on-off forms and hysteresis comparators. In addition, the main shortcoming of the deadbeat strategy is its request for a small sampling period so as to achieve satisfactory performance. In terms of the inner current-loop controllers, there are some approaches to improve system characteristics of the PWM converter. A great deal of effort has been expended to tackle the above problems for tracking error cancellation. PI and PR approaches have been widely adopted in linear tuning strategies. The control of the VSC can be implemented under different reference frames, such as the abc, αβ or dq reference frames. Detailed information on the reference frames for the VSC is introduced in [4]. There is strong coupling between the d and q-axes for a PI regulator, and sinusoidal values lead difficulty in precisely tracking the reference AC signals without a static error [16], [17]. For solving this issue, a PR controller is introduced into the inner current loop [18]-[22]. Thus, the steady state error of a specific frequency can be completely removed since this ideal PR controller provides an infinite gain and a 180° phase shift at the points where the stability problem is produced. Nevertheless, unlike the PI controller, the PR controller is generally complicated to design for whole control systems with stability [22].

In addition, the harmonics generated by converters having an impact on other grid connected utilities and devices are limited by IEEE 519-2014, which presents limits for the THD of currents. Therefore, it is important to do filter studies for eliminating current harmonics [23]-[25]. A third order LCL-filter is quite suitable for grid-tied VSCs with good performance in current ripple attenuation even with small inductances. In addition, an LCL filter exhibits an undesired resonance effect that causes stability problems. A passive damping method has good reliability and simplicity, but it leads to a lot of power dissipation [26]. These problems can be addressed by adopting the capacitor current feedback strategy of active damping, since it has better robustness to external disturbances than passive damping when the line current includes lots of harmonics [27], [28]. However, the damping regulator introduces a voltage component into the VSC modulation index reference that is difficult to achieve for the traditional DPC. This is due to the fact that the PWM modulation is not directly controlled via the inner current loop. Thus, it can be tackled through a collaboration between the deadbeat predictive power control approach and an optimized PR regulator under the stationary αβ reference frame with the SVM. This is done to obtain a high dynamic instantaneous active and reactive power track with a variable on-off frequency prevention and a relatively satisfactory performance. Normally, the system’s stability margin and characteristics can be illustrated by the means of the steady-state error, crossover frequency, phase margin (PM), and gain margin (GM) [17]. As a result, the above mentioned methods can be employed for the system parameter and robustness design of the current controller and the capacitor-current-feedback active-damping.

First, this paper analyzes the mathematical model of a three-phase grid connected VSC under the stationary αβ reference frame, and then presents an accurate calculation of the inductance voltage vector approach with a predictive power control outer loop. The impact of parameter variations on an inner-current loop adopting an active damping PR regulator on the system dynamic and static performance is mechanismly studied. Furthermore, the stability problem and optimal parameter configuration of the control block diagram correctly addressed by considering the pole-zero assignment method of the frequency domain response are also reviewed and discussed. A specific and effective approach for the control of an active damping LCL-topology-based VSC with a PR controller is proposed. Finally, simulation and experimental results are presented in a systematic way to verify the theoretical concepts and implementation is discussed throughout the whole presentation.

 

II. CALCULATION METHOD OF THE REFERENCE VOLTAGE

A grid-connected 3-phase VSC adopting an LCL-filter is illustrated in Fig. 1, where Lg, Lr and Cf are the grid-side inductor, converter-side inductor and branch capacitor respectively, and the VSC typically consists of a 3-phase IGBT bridge and a DC side electrolytic capacitor.

Fig. 1.Topology diagram of three-phase PWM converter.

The following equation could be defined by Kirchhoff Law when system is under the rectifier mode.

Where Vdc and idc are the DC-link voltage and current, RL is the equivalent resistance considering the line impedance and series resistance of the inductors Lg and Lr, VL is inductance voltage vector, and Vk is the converter average voltage vector. Thus, the corresponding inductance voltage vector can be represented as:

To achieve the real parameter values of the series resistance, capacitor and reactor, a Precision Impedance Analyzer WK6500B is employed. Thus, the following expression can be written, considering (1) and (2), as:

To realize the actual current tracking its reference, the vector direction of di/dt should be approximately the same as –ΔI. This is equivalent to seeking the corresponding Vk with the smallest included angle between -ΔI and -VLk, as shown in Fig. 2.

Fig. 2.Distribution of inductance voltage vectors for different switching states.

The equivalent control model for the VSC is generally achieved by neglecting the influence of the series resistance R [8], [25], [27]. However, through the observation of (3) and Fig. 2 of the inductance voltage vectors distribution, the series resistance directly impacts the calculation accuracy for the reference inductance voltage vector selection, and the resistance value for underestimating or overestimating has large effects on the integral absolute error although it is relatively small, as described Dr. Ana Vidal and Dr. Alejandro G. Yepes at the University of Vigo [29]. The function IAE should be defined to identify the underestimated and overestimated cases. The error ε(k, t) between both the curves and the integral absolute error IAE(k) are calculated. IAE is decided through the area covered between the two curves iR=R’ and iR≠R’. Moreover, the sign of the error area is calculated via the integral error IE(k).

By this means, the weighted IAE (WIAE) can be defined as:

As in the above analysis, the cost function IAE illustrates considerable influence on accurate reference vector selection, and the area between iR=R’ and iR≠R’ is relatively less when overestimating than when underestimating. Thus, a LCR analyzer WK6500B for precise vector computation in the VSC control strategy, which characterizes components up to 120 MHz with 0.05% basic measurement accuracy, can generally satisfy the requirement. And equation (4) and (5) along with Fig. 3 theoretically verified whole analysis.

Fig. 3.Effects of WIAE for R’ = R, R’ > R and R’ < R.

Define the inductance voltage vector VLc, which is composed by the adjacent vectors VLi and VLj, is in the same direction, as ΔI in Fig. 4. Assume the parameter Γ as follows.

Fig. 4.Distribution of voltage vector VLc and reference axis YII.

Define Tdes as the vector composition period of VLi and VLj for ΔI elimination.

The reference inductance voltage vector can be expressed as follows.

In order to calculate the parameter Γ, the reference axes YS are needed. They are located in six sectors and are perpendicular to axes a, b and c, as shown in Fig. 5. Assume that YII(ΔI), YII(VLc) and YII(U) are projections of ΔI, VLc and U on the reference axis YII. Since ΔI and VLc are in the same direction, the parameter Γ can be calculated via (7) and (8).

Fig. 5.Distribution of reference axis YS.

Therefore, if the reference frame of a certain sector is confirmed, the parameter Γ of this sector will be obtained.

Each sector corresponds to a pair of adjacent inductance voltage vectors VLi and VLj as shown in Fig. 6. Assume that ΔI is located in the region S, the reference axis YS should be used to calculate Γ.

Fig. 6.Location of ΔI and distribution of VLc1, VLc2 and VLc3.

The following vector and variables are defined.

When ΔI is located in section II, the projection value can be obtained as and which is substituted into (10) for the calculation of Γ. Thus, Γ with all of the conditions is obtained as shown in Table I by parity of reasoning.

TABLE ICOMPUTATION OF EACH SECTOR’S Γ

In addition, as to the outer control loop, according to the instantaneous power theory and the equivalent coordinate transformation, the system’s instantaneous power S can be derived by multiplication between the voltage vector and the current vector conjugate, and it is expressed as follows:

Define vαβ = [vα vβ]T, uαβ = [uα uβ]T and iαβ = [iα iβ]T as the average voltage vector of the VSC, and the line voltage and current vectors in the stationary αβ frame, respectively.

Under the α–β plane, the instantaneous active and reactive power model of the three-phase VSC is defined as follows.

Assume that the line voltage uαβ during instant Ts is ideally invariant (uαβ(k+1)=uαβ(k)). Consequently, the algebraic iterative expression of the instantaneous powers during 2 continuous sampling periods is observed as:

To achieve the aim of the predictive control, the controlled object is adjusted to track its dynamical given values by the next period as shown in the following expression.

Accordingly, the converter average voltage vector is expressed as follows:

The reference input current in the α-β plane by the transformations from (14)-(16) can be performed by means of the following relation:

Therefore, the topology of the proposed current control strategy adopting a PR regulator under the stationary αβ frame for a PWM converter is shown in Fig. 7.

Fig. 7.Deadbeat predictive instantaneous power control strategy with PR controller based on inductance voltage vector under αβ frame.

 

III. ROBUSTNESS DESIGN OF THE CURRENT REGULATOR

The transfer function under the α-β plane of a PI regulator achieved by a positive or negative-sequence is computed via the usage of the frequency shift adopting internal model control, and the optimized state transformation or frequency domain approach of a PR regulator is presented, which directly regulates the overall current, including both the positive and negative components under stationary αβ coordinates [21], [30]. An ideal PR regulator has an infinite gain and a 180° phase shift at the fundamental frequency ω0, and it has little phase shift and gain except for ω0. In order to solve the stability problems caused by an infinite gain, an optimized PR regulator can be used in practical implementations with a finite gain, but high enough to achieve little static error [18], [19]. The improved PR controller is in the following form:

Where KP and KR are the proportional and resonant coefficients, ωc is represents the bandwidth at -3 dB which can enhance the robustness and stability in the presence of line frequency variations.

A. Model and Control of the System

For the inner current control loop, the grid side inductor current ig is compared with its reference ig* and produces an error into the PR regulator. This generates a given current value ic*. The capacitor current ic is fed back to adapt ic*, and along with the proportion loop for active damping to produce the modulation reference for the converter. Therefore, the control block diagram can be derived as follows.

In terms of Mason's Rule and the system model, the input current expression can be introduced as:

By (19), the gain of GPR(s)·GD(s)·GVSC(s)·GLCL(s) at the control system’s resonant frequency is much larger than 1. Thus, the first item approximation of (19) is 1, and the grid voltage disturbance is approximately 0. The transfer function of the system current open loop is derived as:

Instability will be led to by a time delay of the calculation and the PWM loop [31], [32]. In addition, the first-order delay is on behalf of the ZOH of the VSC current. In addition, the delay Td is considered to represent a digital implementation of the computational delay. The Taylor series expansion is used for modeling approximations of the two first-order inertial elements.

Fig. 8.Control block diagram of inner current loop.

For a three phase PWM converter with synchronous sampling of the input current and with a sampling period Ts, the total time delay including the sampling and transport delay is generally given by ΣTi = 1.5Ts.

The active damping technique using capacitor current feedback in Fig. 7 is a type of control algorithm rather than physical elements. After the introduction of a second order oscillation element by the active damping gain Kw to suppress resonance peak, its Laplace transfer function becomes:

The system order is added greatly after employing a PR regulator. Thus, the equivalent expression of (20) can be written in the form of (23).

The damping ratio can be written as:

With the damping ratio ζ increasing gradually from 0 to 0.9, the resonance peak of the transfer function GLCL(s) changes from 30dB to disappearance, so as to effectively restrain the amplification effect at the resonance frequency point, which is illustrated in Fig. 9. Since a larger Kw leads to the controller’s saturation and stability, the parameters are chosen as ξ=0.707 and Kw=1.36 based on the good convergence performance of the second-order optimal theory.

Fig. 9.Bode frequency characteristics corresponding to variable damping coefficient.

The controller’s bandwidth ωc reflects the ability to track the input signal. Therefore, the system should have a larger bandwidth in order to enhance the dynamic response characteristics [20], [21]. However, high frequency interference noise such as the switching frequency affects the system’s stability when ωc increases. The change rule for this is depicted in Fig. 10. Since the VSC is required to run well while the grid fundamental frequency fluctuates between 49.5 Hz and 50.5 Hz [17], ωc = 2πΔf = π rad/s is defined to get a lot of gain in whole operating frequency scale with a related maximum frequency variation of Δf = 0.5 Hz.

Fig. 10.Frequency response of resonant term for variation in ωc.

B. Influence of System Parameters on the Root Locus

The six-order system in (23) is a little too complex to get a practical solution. It shows that there are two zeros and the Bode diagram in Fig. 9 shows that the general derivation LCL topology behaves like L at frequencies lower than the approximate resonant frequency [29]. Compared with the original system, it is easier to calculate with pole-zero cancellation from six to four. As illustrated in Fig. 11, the control target is to configure P1 and P2 as the dominant poles of a second order under-damped linear element with ζ=0.707, and the distance of P3 and P4 off the imaginary axis is 5 times bigger than that of the dominant poles. This ensures stability with enough of a margin for system parameter variations. As shown in the following analysis, parameter-corresponding pole-zero locus for the closed-loop transfer function is employed for optimal parameter configuration.

Fig. 11.Pole-zero locus of inner current loop corresponding to variable parameter value. Arrows show the variation of pole placement. (a) With variable Kw, ΣTi and L. (b) With variable KP and KR.

From Fig. 11(a), when the equivalent inductance L is increasing, the system’s poles distribution moves off the imaginary axis although its impact on the pole-zero locus of the system is relatively less. This improves the system stability. Thus, a good robustness to reactor value variations can be achieved. When the active damping coefficient Kw increases, the root locus of the dominant poles P1 and P2 moves from instability in the right half plane to a stable state with damp increasing, and then into the instable region again after the critical value is reached with Kw=1.4. This is in accordance with the optimum value mentioned above. Correspondingly, the poles P3 and P4 lean to shift far off of the imaginary axis with the system damp increasing and good resonance peak suppression. Meanwhile, the poles generated from the LCL-filter and PR controller shift quickly near the imaginary axis while the total delay ΣTi increases. It can be seen that the parameters scale that leads to system instability is extended with a time delay. This makes the controller and poles hard to configure. From Fig. 11(b), KP and KR should be determined via the predicted placement of P1 and P2 so that the damping ratio and inherent frequency are involved. When KP increases, the dominant poles shift off of the imaginary axis and then near it, when the system’s damp changes from ‘large’ to ‘small.’ This has almost as much influence as Kw. KR has less influence on the root locus than KP. This leads to the poles shifting near the imaginary axis. Thus, its damping is decreased. It is well-known that KP is directly related to the dynamic performance of regulator. However, KR mainly decides its gain at a specific frequency and adjusts its bandwidth close to the resonance. According to the pole-zero locus, a parameters scale that meets the system stability and performance requirements can be obtained.

Based on the principle discussed above, the rules of the parameters’ influence are illustrated, and their optimal ranges are derived. The frequency response of the system Bode and Nyquist diagram with selected optimal parameters is depicted in Fig. 12. In terms of the Nyquist theory, the control system is for stability since its trajectory does not encircle the critical point (-1, j0) [33]. Compared with the original continuous s-domain model, they have an amplitude margin of h = -20lg|G(jωgs)| = 9.7dB and a phase margin of γ = π + ∠G(jωcs) = 30.2° at the cross over frequency. In addition, the corresponding frequency of the dominant poles at -3dB is about 220Hz, which is far enough from the fundamental frequency. This is good for the system’s characteristics. Consequently, the phase and amplitude margin guarantees the control system’s stability.

Fig. 12.Frequency response performance of current loop. (a) Overview of Nyquist diagram. (b) Bode plot diagram.

C. Digital Implementation of the Current Regulator

Generally, the current regulator is analyzed under the continuous time domain and it is necessary to adopt an appropriate discretization approach in the digital control system, such as the Tustin (bilinear transformation), first-order hold, and impulse invariance to guarantee the discrete domain of current controller can accurately match its continuous model [14], [31], [33]. In addition, an inappropriate discretization method leads to the zeros and poles shifting in terms of the system’s transfer function, which may deteriorate controller’s stability and decrease its tracking precision.

During the discretization procedure, the impacts of aliasing distortion in the impulse invariance method are indistinguishable from the original continuous model. In addition, the time lagging response and frequency shift in the Tustin method lead to an unsatisfied frequency response as a result of the continuous s-domain transfer function [33]. In order to improve the signal tracking precision in a practical system, the ideal digital implementation approach cannot cause a gain attenuation or phase shift at the resonant frequency. To solve the above problems during the controller’s discretization, the zero-pole matching (ZPM) discipline is predicted as follows:

There are proportional and resonant parts for a PR controller. The proportional part is linear with a small constant coefficient Kp. Thus, the controller’s discretization derivation about the resonant segment is as follows.

Where, and

Define and , and according to the zero-pole matching rule shown in (25), the discretization result of the PR controller using the ZPM method is derived as follows:

Define P = e-p1Ts and Q = e-p2Ts, and then the transfer function in the z domain can be obtained.

From (28), the differential expression to be coded for the discrete-time implementation can be written as:

Equation (29) is implemented by using the digital implementation structure shown in Fig. 13.

Fig. 13.Digital implementation diagram of controller.

As shown in Fig. 14, the discretization results using the zero-pole matching method can offer reasonable positions for the system’s resonant peaks and a similar frequency response as the continuous transfer function. When the Tustin method is applied, due to its simple calculation, the results are quite unsatisfactory. The resonant frequency shifts from 50 Hz to 49.02 Hz, and phase response has a delay of −23.5°, which is harmful to the regulator stability. Define ωA as the angular frequency of the continuous time domain and ωD as the the angular frequency of discrete-time domain, so the transformation mechanism from continuous to the discrete domain via Tustin is written as following substitution.

Fig. 14.Discretization results adopting Tustin and ZPM method with 4 kHz sample frequency.

The expression is obtained, and when the sampling frequency is quite fast, the approximate equation ωA≈ωD can be reached theoretically. Thus, the ZPM method has better performance due to the accurate zero/pole transformation and mapping between the two domains without frequency aliasing distortion and phase shift.

 

IV. SIMULATION AND EXPERIMENTAL RESULTS

A. Simulation Results

In this section, the results of the described mathematical analysis and the effectiveness of the suggested strategy have been verified by means of numerical simulations. The system’s parameters are listed in Table II.

TABLE IIPARAMETERS OF ACTIVE-DAMPING LCL VSC PROTOTYPE

Fig. 15 and Fig. 16 show the AC-side current trajectory presented in this paper compared with the traditional SVM. The three marked sections of the trajectory mean: [I] the rectifier state, [II] the inverter state, and [III] the state that limits the amplitude of the current. The dynamic tracking performance of the proposed solution is much better than that of the traditional SVM.

Fig. 15.The trajectory of the current vector I under traditional DPC-SVM.

Fig. 16.The trajectory of the current vector I under proposed method.

It can be observed from Figs. 17-20, that the instantaneous power and the DC voltage both track each reference and include good stability and little static error. The input current has an almost sinusoidal waveform (THD = 2.38%) and is synchronous to the grid voltage. Thus, the unity power factor running of the VSC is successfully implemented with a reactive power that is approximately zero.

Fig. 17.Waveforms of converter side current and voltage. (a) Current and voltage of converter side. (b) Line voltage vs converter side current.

Fig. 18.Waveforms of grid side A-phase voltage and current.

Fig. 19.Current spectrum of phase A.

Fig. 20.DC-Link voltage dynamic response.

B. Experimental Verification

The considered grid-connected VSC prototype with a LCL-filter has been experimentally tested according to the design strategy proposed in this paper. The parameters of the system are reported in Table II and a PR regulator is designed as analyzed in Section III.

The voltage and current waveforms of phase A both at the VSC and the grid sides are shown in Fig. 21 and Fig. 22, respectively. The harmonics component analysis of the full load is given in Fig. 23. The current THD analysis at different loads by adopting the modified PR regulator with active damping is compared to that obtained by adopting the traditional PI controller with normal DPC-SVM, as shown in Fig. 24. Fig. 25 shows the experimental waveforms of the DC voltage and line current transient response with less over shoot of the DC bus reference voltage track. The reference for the active power has been varied in five steps with a constant zero reactive power. Each of them has a width of 100ms. The given and measured power components are plotted in Fig. 26 with good dynamic track performance. It can be concluded that the proposed algorithm can improve the quality of the grid current, and that the robustness of the controller is verified through experimental results in accordance with the simulation results.

Fig. 21.Experimental waveforms of converter side current and voltage. (a) Current and voltage of converter side. (b) Line voltage vs converter side current.

Fig. 22.Experimental waveforms of line side voltage and current.

Fig. 23.Experimental input current spectrum of phase A.

Fig. 24.THD comparison chart of grid side phase current.

Fig. 25.Experimental waveforms of DC voltage transient response. (a) DC voltage step from 93V to 106V. (b) DC voltage step from 93V to 118V. (c) Zoom-in graphic of Fig. 25(a).

Fig. 26.Dynamic response of reference and measured power value.

 

V. CONCLUSION

This paper presented a theoretical analysis of the inductance voltage vector control strategy based on a Proportional Resonant regulator under the stationary αβ frame and the stability robustness approach for a three-phase VSC with respect to an optimum parameter match. To achieve an appropriate inductance voltage vector for each sector with accurate calculations on the current error to modulate the PWM converter, the judgment rule of sector selection and the optimal switch state by the geometrical distribution model is established. This can replace the trigonometric function calculation, Park transformation, real-time detection of the PLL and feed-forward decoupling used in the traditional space vector modulation. The proposed strategy with an active damping LCL filter achieves tracking errors of the dynamic instantaneous power via the application of the required inductance voltage vector under the stationary αβ coordinate during each switching period. In addition, the PR controller is designed and discretized by means of the frequency response of the zero-pole assignment matching and cancellation with an in-depth performance influence analysis, which is frequently encountered when considering a system’s robustness in the presence of parameter fluctuations and regulator digital discretization of a current-controlled VSC. The methodology to analyze and enhance the transient response of the inner current loop, based on the study of the system’s transfer function locus through pole–zero placement, is also discussed. The optimal gain margin and phase margin are configured to get a rapid and non-oscillating transient response. Finally, the feasibility and effectiveness of the proposed design method are verified by the means of simulation and experimental results on a laboratory PWM converter prototype.