Current Decoupling Control for the Three-level PWM Rectifier with a Low Switching Frequency

Abstract

Three-level PWM rectifiers applied in medium voltage applications usually operate at low switching frequency to keep the dynamic losses under permitted level. However, low switching frequency brings a heavy cross-coupling between the current components $i_d$ and $i_q$ with a poor dynamic system performance and a harmonic distortion in the grid-connecting current. To overcome these problems, a mathematical model based on complex variables of the three-level voltage source PWM rectifier is firstly established, and the reasons of above issues resulted from low switching frequency have been analyzed using modern control theory. Then, a novel control strategy suitable for the current decoupling control based on the complex variables for $i_d$ and $i_q$ is designed here. The comparisons between this kind of control strategy and the normal PI method have been carried out. MATLAB and experimental results are given in detail.

1. Introduction

Nowadays, PWM rectifiers have been widely applied in series of industrial fields, such as Static Reactive Power Generator (SVG), Active Power Filter (APF), Unified Power Flow Controller (UPFC), High Voltage DC Transmission (HVDC) and some renewable energy generation system [1-6]. The key issues concerning the control of a threelevel voltage source PWM rectifier are about neutral potential balancing, switch losses and the cross-coupling between id and iq. In order to improve the output power of the converters, the switching frequency of the semiconductors (IGBT, IGCT or GTO) are usually kept low to restrain switch losses at permitted level [7-9]. However, low switching frequency not only brings about a heavy cross-coupling between the current components id and iq as well as a poor dynamic system performance [10, 11], also a severe harmonic distortion, especially the low order harmonics, in the grid-connecting current [12].

Considering the situation that rectifiers usually being connected to grid, a heavy harmonic distortion can’t meet the grid standards, and will be harmful to other grid loads. A novel dead beat control scheme combining with repetitive control was presented in [13], while it didn’t take consideration of the influence of low switching frequency. Literature [14] put forward a model predictive control strategy in static coordinates, which had a nice performance with a low switching frequency, however, its characteristic of frequency fluctuation brought about some other difficulties in filter design and issues of electromagnetic compatibility. Two kinds of approaches have been proposed in [15] to improve the harmonic distortion resulted from low switching frequency, in which a LCL filter was considered, whose reasonable parameters and volume could be a problem for its development.

Studies on a novel control scheme for a high-power three-level voltage source PWM rectifier with a low switching frequency have been carried out in this paper, which are focusing on current decoupling controller. Section 2 establishes a complex model with low switching frequency for a three-level PWM rectifier case considered in this paper, and Section 3 analyze the problems resulted from low switching frequency in detail. Section 4 describes a novel complex current controller to realize the decoupling between id and iq, along with some simulation verification. The whole control scheme is shown in Section 5 and the performance are evaluated based on a experimental platform whose microprocessor being DSP and FPGA, at switching frequency fs=500Hz.

 

2. Complex Model for Three-level PWM Rectifier

The topology of a three-level PWM voltage source rectifier studied in this paper is shown in Fig. 1.

Fig. 1.Topology of the three-level PWM voltage source rectifier

Where, ea, eb, ec represent the grid voltages; ia, ib and ic are grid-side currents; νaO, νbO, νcO are ac voltages; L and R are the filter inductance and resistance respectively; RL stands for load while iL is the load current; idc is the current of DC-link and Vdc represents the DC-link voltage; C1 and C2 are DC-link capacitors whose voltages are uc1 and uc2; O stands for clamped point while O’ is the midpoint of the grid.

The traditional models for PWM rectifier are described in dq coordinates as following

Where, ed and eq are the dq components of the grid voltage; νd, νq represent the dq components of the ac voltage respectively and id, iq are the grid-side currents in dq coordinates; p is differential operator; ωs stands for grid angular frequency and SdP, SqP, SdN, SqN represent the switch status also in dq coordinates.

Defining complex variables as

With the definitions in Eq. (3), the complex models can be obtained without the consideration of the influence of switching frequency

Where, τs = L/R is the time constant of the ac side; es, is, νs are the complex variables of grid voltage, grid-side current and ac voltage respectively; SP and SN are complex variables about the switch statuses.

From Eq. (4), the complex transform of the current loop can be obtained as

A low switching frequency influence the control system in terms of an effect on the PWM link and we use Eq. (6) to make an approximation, which containing a crosscoupling factor jωsτdνs.

While, ν*s is the reference voltage vector resulted from the current controller; τd stands for the delay time with the low switching frequency and the sampling delay, normally,τd=0.75/fs (fs is switching frequency) [11,16]. And the complex transform of Eq. (6) is

The complex transform of the whole current loop can written as

 

3. Cross-coupling Problem with a Low Switching Frequency

The corresponding open loop zero-pole of Eq. (8) is obtained as shown in Fig. 2, in which, complex roots p1 = −1/τd − jωs, p2 = −1/τs − jωs, whose positions have relationships with the time constants τd and τs. When fs is high, τs >> τd, the pole p2 is the dominant one while p1 has a little effect on the system performance. Reducing fs makes p1 be near to the zero shaft, and has a gradually enhanced influence on the system performance. Actually, the complex factor j in Eq. (8) is the essential reason why does the cross - coupling exists.

Fig. 2.Zero-Pole map of the unregulated open loop

In order to analyze the influence on cross-coupling in further, a coupling function in frequency domain is defined to describe the coupling degree.

The coupling degree between id and iq for this three-level PWM rectifier with a low switching frequency can be obtained by substituting Eq. (8) into Eq. (9), which is shown in Fig. 3. From Fig. 3, it can be seen that low switching frequency brings a more serious cross-coupling.

Fig. 3.Cross-coupling degree of the current loop at different switching frequency

 

4. Complex Current Controller Design

4.1 Normal PI current controller

A normal PI with a feedforward compensation can realize the decoupling control for the current components id and iq. For the complex transform of the current loop described as Eq. (5), a feedforward compensation jωsτsis is introduced firstly, and then, a PI controller is given as

By designing k0 and τi properly, the current loop can realize a nice control performance. However, with the consideration of a low switching frequency, extra crosscoupling, jωsτdis exists as shown in Eq. (8). There are two parameters need to be adjusted along with double feedforward compensations if using normal PI control.

The Bode diagram and Step response of normal PI current with different switching frequency fs are shown in Fig. 4.

From Fig. 4 (a), it can be known that the control bandwidth will reduce along with the decrease of the switching frequency, while the rising time, peak time, adjusting time and overshoot increase shown in Fig. 4 (b), which mean that normal current controller doesn’t suitable anymore with a low switching frequency.

Fig. 4.Influence on normal current controller with different switching frequency: (a) Bode analysis; (b) Step response

By tuning the gain value KiP of the PI controller can get good performance without considering the switching frequency, as shown in Figs. 5(a) and 5(b). However, when the switching frequency is low, the adjustment of gain tuning can’t improve the system performance to a great degree compared with the complex current controller. The comparisons between the normal PI controller and the complex one are shown in Fig.6 when fs = 500 Hz (both of which have achieved the best effect).

Fig. 5.Influence on normal current controller with different gain value (a) Bode analysis; (b) step response

Fig. 6.Comparisons between the normal PI controller and the complex one

4.2 Complex current controller design

From above theoretical analysis, we know that crosscoupling has closely relationship with imaginary parts existing in the complex transform. Given that a novel controller Fr(s) designed to emit two complex poles at the same time, the coupling phenomenon would be eliminated fundamentally.

In this paper, Fr(s) is re-designed as Eq. (11), which is also called a complex current controller.

Where, there is only one parameter k0 need to be designed and the complex signal graph of the current control system is shown in Fig. 7.

Fig. 7.Complex signal flow graph of the current control system with a complex current controller

Substituting (11) into (8), the open loop current transform with a complex controller can be obtained as

Both complex poles now are eliminated and the coupling issue is solved.

In order to design the parameter k0, a closed loop transform with current controller is given as

Where, 2ζωn = 1/τd , ωn2 = k0 /(τs td)

Eq. (13) shows that this kind of closed loop is a typical second order system, whose damping coefficient ζ should be designed as 0.707 to reach an optimal performance [17], and then, the theoretical k0 should be designed as

Considering the parameters of the three-level PWM rectifier are: amplitude of es is 690V, L = 5mH, R = 0.1 Ω, C1 = C2 = 6800μF, Vdc = 1800V, RL = 0.1 Ω, fs=500Hz, so the calculated τs = 0.05 and τd = 0.0015, the theoretical k0 is 16.67, and the influence for different k0 is shown in Fig. 8.

Fig. 8.Influence on the control system with different k0

From Fig. 8, it can be seen that, larger k0 can brings a more broad control bandwidth and a faster response, however, the overshoot increases. Fig. 8 gives a recommendation that k0 in this paper is 20, which also verify the robustness of the designed complex current controller.

4.3 Simulation comparisons

The whole control scheme is designed as shown in Fig. 9, in which, the PI controller is used to control the DC-link voltage Vdc. The power factor is set as Pf =1 with taking the output of the PI voltage controller as the given value of id and the given value of iq setting as zero.

Fig. 9.Whole control scheme of the proposed strategy

Simulation comparisons have been carried out in MATLAB/simulink according to the control scheme as shown in Fig. 9, between the normal PI controller and the complex one while fs = 500Hz, and a conventional PI controller is used for voltage outer loop. The simulation parameters are as same as the ones in the part 5; at t = 0s, RL = 500Ω, at t =1.2s, RL=100Ω.

The current components id and iq are shown in Fig. 10, the grid voltage ea and the enlarged grid-side current ia are shown in Fig. 11, Fig. 12 is the current trajectory of the grid-side current with the complex current controller.

Fig. 10.Current components id and iq with different current controllers

Fig. 11.Grid voltage ea and the enlarged grid-side current ia with different current controllers

Fig. 12.Current trajectory of the grid-side current with the complex current controller

From Fig. 10, it can be known that, this kind of complex current controller can realize the decoupling with a better characteristic compared with the normal PI controller. The harmonic distortion of grid-side current ia also has been improved as shown in Fig. 11. The dynamic current trajectory shown in Fig. 12 verifies that this kind of complex controller has a nice dynamic performance.

 

5. Experimental Verification

Experimental platform has been established as shown in Fig. 13, where, a TI TMS320F28335 DSP is used to realize the control algorithm and a Xilinx Spartan3E-FPGA is adopted to implement A/D,D/A, pulses generation. Type of the Power devices is Infineon 450A, 1700V IGBT, whose drivers are the type of Concept (3W, 20A). The detailed scale of the experimental platform is shown in Table 1.

Fig. 13.Experimental platform

Table 1.Detail experimental parameters

The implementation of the complex current controller is the most challenging part during the digital implementation. Firstly, Eq. (11) can be decomposed into the real and imaginary components. According to the definition in Eq. (3), that a real component corresponds the direct-axis one (d coordinate) while an imaginary part corresponds the quadrature axis one (q coordinate). Then, the complex current controller could be designed in dq coordinates.

The steady waveforms of grid voltage ea and grid-side current ia are shown in Fig. 14 with the different control strategy.

Fig. 14.The grid voltage ea and the grid-side ia with different control strategy

Fig. 14 shows that complex current controller has a better characteristic in reducing the harmonic distortion of the grid-side current.

During the experiment, the given Vdc = 300V, at t = 2s, Vdc = 450V and then reduced to 300V again. The corresponding DC-link voltage, current in dq coordinates are shown in Fig. 15. Sudden loading and unloading experimental results are shown in Fig. 16.

Fig. 15.Waveforms with a changement on the DC-link voltage, CH1: given DC-link voltage; CH2: real DC-link voltage; CH3: current of d coordinate; CH4: current of q coordinate

Fig. 16.Waveforms with a changement on the load, CH1: given DC-link voltage; CH2: real DC-link voltage; CH3: current of d coordinate; CH4: current of q coordinate

When a normal PI current controller was adopted, the dynamic waveforms of grid-side current ia, the DC-link voltage Vdc and the current in dq coordinates are shown in Fig. 17 with the same experimental processes (the given Vdc = 300V, at t=2s, Vdc=450V and then reduced to 300V again), which is consistent with the simulation result shown in Fig. 10(a) and Fig. 11(a).

Fig. 17.Waveforms by PI controller CH1: grid-side current; CH2: real DC-link voltage; CH3: current of d coordinate; CH4: current of q coordinate

When the load changed suddenly, the dynamic grid voltage ea and current ia are shown in Fig.18 as well the dynamic DC-link voltage, which verify the dynamic performance of this kind of control scheme for three-level PWM rectifier.

Fig. 18.Experimental waveforms in dynamic state, CH1: DC-link voltage ; CH2: grid-side current ia; CH3 grid voltage ea

 

6. Conclusions

Some research focusing on the decoupling control have been carried out for the three-level voltage source PWM rectifier considering a low switching frequency. Firstly, a novel complex model was established for the three-level PWM rectifier, and the influence resulted from low switching frequency was considered based on the detailed analysis on the influence on cross-coupling between in dq current components and performance of the normal current controller.

A complex current controller was proposed for the threelevel PWM rectifier along with its parameter design, and the comparisons have been conducted between the normal PI current controller and the complex one.The whole control scheme was also given based on the microprocessors DSP and FPGA, and experiments were carried out to verify the effectiveness of the designed control system.