The equivalent model of controller in synchronous frame to stationary frame

1 Introduction

The controller in synchronous frame is widely applied to the electrical engineering field[1-3], such as power delivery control, phase locked loop (PLL), power quality control, motor speed regulation, etc. For synchronous frame, the input signal whose frequency is consistent with rotating frequency can be transformed into dc signal. Then by controlling dc signal, the ac signal in stationary frame can be controlled indirectly. For example, PI controller in dq synchronous frame is one of the mainstream control strategies for fundamental frequency[4,5]. The common PI controller in stationary frame holds a finite gain for ac component, thus cannot guarantees the system to track ac reference without steady-state error[6,7]. However PI controller in synchronous frame holds an infinite gain for the input signal whose frequency is consistent with rotating frequency, and thus achieves zero steady-state error. Besides, applying a low-pass first order filter in synchronous frame can extract the input signal whose frequency is consistent with rotating frequency quickly,and get similar performance to band-pass filter in stationary frame[8].

Most research about the controller in synchronous frame just focuses on the control performance of input signal at rotating frequency[9,10]. Lacking a universal equivalent model that can transform a controller in synchronous frame to stationary frame, it is difficult to obtain frequencydomain control performance of input signal at other frequency, and cannot build an unified frequency-domain model of the closed-loop system in stationary frame. Thus,it is hard to achieve the overall performance of system.

In the early literature [11], PI controller in positive sequence synchronous frame is equivalent to proportionalresonant (PR) controller in stationary frame. The equivalent explains why the two control strategies have similar performance at rotating positive sequence fundamental frequency[12-14]. But, for input signal at other frequency,such as, negative sequence fundamental frequency, the PI controller in positive sequence synchronous frame provides finite gain at negative sequence fundamental frequency, and yet, the PR controller provides an infinite gain at negative sequence fundamental frequency. So there is pretty difference between the performance of the two controllers, and the equivalent is hard to complete.Therefore, the deduced model cannot reflect the output control performance of input signal at non-rotating frequency. Another stationary frame equivalent model of PI controller in synchronous frame can correctly describe the output control performance of input signal at different frequency in [15]. However, the paper only deduced the equivalent model of PI controller in synchronous frame.Owing to the tedious deduction, the conclusions are difficult to replicate when other controller in synchronous frame is applied. Thus, it is necessary to discuss the stationary frame universal equivalent model of arbitrarily controller in synchronous frame.

As how to deduce the universal equivalent model of arbitrarily controller in synchronous frame to stationary frame stationary is still a difficult problem, the closedloop control system containing controller in synchronous frame is difficult to carry out frequency analysis and stability analysis. In this paper, taking an arbitrarily controller in single synchronous frame as an example, the stationary frame equivalent model of arbitrarily controller in synchronous frame is deduced by comparing the input and output signal. The derived model can not only reflect the output control performance of input signal at rotating frequency, but also reflect the control performance at nonrotating frequency, and can be applicable to various types of controller. Applying the deduced model, the unified frequency-domain model of the overall system in stationary frame can be built. Therefore, the model can provide theoretical guidance for frequency analysis and stability analysis.

2 Deduction of equivalent model of controller in single synchronous frame

The block diagram of a controller in positive sequence synchronous frame is shown in Fig. 1.

Fig. 1 The controller in positive sequence synchronous frame

In Fig. 1, the I denotes input while O denotes output.Geq(s) is assumed to be the choice of controller in synchronous frame, and the rotation matrix converter is

Where ω0 is the fundamental frequency. Geq(s)αβ+is defined as the equivalent transform expression from controller in positive sequence synchronous frame to stationary frame, and we get

According to Fourier’s theorem, a periodic signal can be decomposed into the sum of difference sinusoidal signals and dc component. For an arbitrary signal in a three-phase stationary frame

Where V is the amplitude of input signal, and φ is the initial phase angle. ω represents the angle frequency,and it can be positive, negative or zero, which means the signals is in positive, negative sequence or dc component,respectively. After Clarke’s transform the input signal into stationary frame, we get

The Clarke’s matrix is

Through the transformation matrix from stationary frame to positive sequence synchronous frame, the (t)can be written as

To simplify the derivation process, the controller Geq in synchronous frame can be regarded as M∠θ. Where M and θ are the amplitude and phase of the controller Geq,separately. According to the concept of vector method, the angle frequency of the vector is greater than zero generally.In (6), when ω≥ω0, all kinds of vector turn counterclockwise at the angle frequency of ω-ω0, M and θ are given as

When ω＜ω0, the vector turn clockwise at the angle frequency of ω0-ω, which opposite to the rotation direction of the stipulated vector. In order to conform to the regulation, (6) can be transformed as

In (8), M and θ are shown as

By concerning (7) and (9), M and θ are represented as

The equivalent model Geq(s)α+ or Geq(s)β+ can be defined as T∠a similarly, where T and α are the amplitude and phase. According to the regulation of vector, and concerning (4) and (13), T and α are given as

In (14), when ω＜0, make appropriate transform to the expression of the input and output signals, ω can be converted as −ω . Thus, when ω＜0,α=θ. When 0＜ω＜ω0, the −θ can be shown as

Like the above derivation, for the controller in negative sequence synchronous frame, when Geq(s) is the choice of the controller in negative sequence synchronous frame,Geq(s)α- or Geq(s)β- which is the universal equivalent model of controller in negative sequence synchronous frame to stationary frame, can be shown as

The (17) and (18) represent the stationary frame equivalent model of controller in single synchronous frame separately.

3 Deduction of equivalent model of controller in double synchronous frame

Fig. 2 The controller in double synchronous frame

The controller in double synchronous frame is shown in Fig. 2, which structure is often used in unbalanced conditions. For controlling fundamental positive and negative sequence components at the same time, PI controller in double synchronous frame can be adopted.Geq(s) is also assumed to be the selection of controller in double synchronous frame. Geq(s)αβ, which is the equivalent model of controller in double synchronous frame can be deduced. Based on double synchronous frame, we can obtain using (17) and (18)

One thing to note: when the input signal is positive sequence, ω＞0,s=jω; when the signal is negative sequence, ω＜0 ,s= −jω. Therefore, according to (17),(18) and (19), the equivalent model of controller in double synchronous frame to stationary frame becomes

When PI controller is chosen as the controller in double synchronous frame, the expression is

Substitute (21) into (17) and (18), the equivalent model of PI controller in single synchronous frame to stationary frame can be deduced

By substituting (21) into (20), the equivalent model of PI controller in double synchronous frame to stationary frame can be obtained

Comparing (22), (23) and (24), PI controller not in single synchronous frame but in double synchronous frame is equivalent to PR controller in stationary frame.

4 Simulation results

As the controller in double synchronous frame is composed of the controller in single synchronous frame,the focus of this paper is to verify the correctness of equivalent model of controller in single synchronous frame to stationary frame.

The simulated system and closed-loop control block diagram of controller in positive sequence synchronous frame are shown in Fig. 3 and Fig. 4, respectively. In Fig. 4,Geq(s)αβ+ is the equivalent transfer function of controller in positive sequence synchronous frame to stationary frame,and Gplant(s) is the device transfer function. The Irefαβ and Ioutαβ stand for input and output signals of the closed-loop control block diagram, separately.

Fig. 3 Simulated system

Fig. 4 The closed-loop control block diagram of the simulated system

Table 1 System parameters and values

Parameter Values DC-link voltage 500V Filter inductor L= 1mH kP*kPWM 0.3 kI*kPWM 30 Cut-off frequency ωc 20Hz

The transfer function Gplant(s) of the inverter is

Where kPWM is the inverter gain, L is the filter inductor.Choosing the PI controller as the controller in positive sequence synchronous frame, the expression of Geq(s)αβ+can be obtained

Concerning Fig. 4, the open-loop and closed-loop transform functions are

Key values are given in Table 1, the open-loop and closed-loop bode diagram are shown in Fig. 5 and Fig. 6,respectively.

Fig. 5 shows the open-loop bode diagram with PI controller, when Gopen(s) = 0dB, the phase margin is 46o in positive sequence while is 81o in negative sequence.There is greatly difference between the positive and negative sequence of phase margin. Thus, the phase margin in positive and negative domain should be considered simultaneously in the design of system parameter when controller in synchronous frame is used.

Fig. 6 shows the closed-loop bode diagram with PI controller, the closed-loop control in positive sequence fundamental domain can track reference without error,cannot track negative sequence fundamental reference with steady-state error. The result is consistent with the actual condition.

Fig. 5 Open-loop bode diagram (PI controller)

Fig. 6 Closed-loop bode diagram (PI controller)

A simulated model is performed to verify the correctness of the proposed model further. The simulation parameters are derived from Table 1. The simulation waveform of negative sequence input and output signals in alpha axis are presented in Fig. 7. The amplitude and the initial phase angle of input signal are 100A and 90o,separately. According to the closed-loop bode diagram in Fig. 6, the gain of the closed-loop control system at negative sequence fundamental frequency is 0.757, and the tracking phase angle difference is -50.60. The amplitude and the phase angle of theoretical output will be 75.7A and 39.4o. And the amplitude of actual output signal is 75.6A,and the phase angle of output is 39o. Thus, the simulation results agree with theoretic analysis. Fig. 8 shows the simulation waveform of 10 Hz negative sequence input and output signals in alpha . Where the amplitude of input signal in alpha axis is 100A, and the initial phase angle is 90o. According to the closed-loop bode diagram, the gain of the closed-loop control system at 10 Hz negative sequence is 1.03, and the tracking phase angle difference is -11.60.The amplitude and the phase angle of theoretical output will be 103A and 78.4o. And the amplitude and phase angle of actual output signal are 103.2A and 78.3o, separately.The simulation results further validate the correctness of the proposed model.

Fig. 7 The simulation results of the input signal at negative sequence fundamental frequency (PI controller)

Fig. 8 The simulation results of the input signal at 10 Hz negative sequence (PI controller)

In order to verify the generality of the proposed equivalent model, the first-order low-pass filter is chosen as the controller Geq(s) in positive sequence synchronous frame. The expression of Geq(s)αβ+ can be obtained

In (29),ωc is the cut frequency of the first-order lowpass filter. According to the values of parameter in Table 1,the closed-loop bode diagram is shown in Fig. 9.

Similarly, by observing the difference of amplitude and phase angle between input and output signals to verify the correctness of the proposed model. Fig. 10 shows the simulation waveform of input and output signals at negative sequence fundamental frequency in alpha axis.The amplitude and the initial phase angle of input signal are 100A and 90o, separately. According to the closedloop bode diagram in Fig. 9, the gain of the closedloop control system at negative sequence fundamental frequency is 1.53, and the tracking phase angle difference is -1510. The amplitude and the phase angle of theoretical output will be 153A and -61o. And the amplitude of actual output signal is 153A, and the phase angle of output is -60.8o. Thus, the simulation results correspond with theoretic analysis.

Fig. 11 shows the simulation waveform of input and output signals at 10 Hz negative sequence in alpha axis.Where the amplitude of input signal is 100A, and the initial phase angle is 900. According to the frequency-domain curve of closed-loop model, the gain of the closed-loop control system at 10 Hz negative sequence is 1.228, and the tracking phase angle difference is -4.40. The amplitude and the phase angle of theoretical output will be 122.8A and 85.6o. And the amplitude and phase angle of actual output signal are 122.8A and 85.5 o, separately. The simulation results further validate the correctness and generality of the proposed model.

5 Conclusions

This paper has researched on the stationary frame equivalent model of arbitrarily controller in synchronous frame, and gets the following conclusion:

Fig. 9 Closed-loop bode diagram (low-pass filter)

Fig. 10 The simulation results of the input signal at negative sequence fundamental frequency (low-pass filter)

Fig. 11 The simulation results of the input signal at 10 Hz negative sequence (low-pass filter)

1) The stationary frame equivalent model of controller in single synchronous frame is deduced. And the model not only reflect the control performance of the input signal at different frequency, but also provide theoretical support for modeling and frequency-domain analysis when controller in synchronous frame is adopted.

2) PI controller in double synchronous frame is equivalent to PR controller in stationary frame, but PI controller in single synchronous frame is not equivalent to PR controller in stationary frame.

3) The stability of closed-loop system need to synthetically consider the stability margin in positive and negative sequence frequency-domain when controller in synchronous frame is adopted.

Acknowledgements

This work was supported by SGCC Scientific and Technological Project (5216A018000J); National Key R &D Program of China (2016YFB0900900).