Passive control of three-phase PWM rectifier and its damping characteristics of CSR

Abstract

Due to the current mode pwm rectifier model has strong coupling and nonlinear characteristics, the traditional linear control theory can’t achieve satisfactory effect based on passive control theory to realize current-mode PWM rectifier state variable linear resolve coupling, decoupling model. Based on the design of the direct current control strategy, the theory and simulation experiments prove the feasibility and the effectiveness of solutions and achieve the ideal control effect. Damping is studied through experiments of passive control of the effect on the stability of the current-mode PWM rectifier system. It provides a theoretical and technical support in the application of the high-power current type inverter.

Keywords

PBC CSI pwm rectification damping characteristics

1. Introduction

Current-mode PWM rectifier is easy to control current is more direct, internal short circuit protection, as well as a faster dynamic response. With the development of superconducting technology, it will improve the energy storage efficiency of the current-mode PWM rectifier, this will become an advantages for the application of current mode PWM rectifier in high-power. However, because of the inductance and capacitance filter, current type rectifier will suffering severe vibration and current distortion, such as current mode rectifier relatively complex structure and control [1, 2]. Current-mode PWM rectifier model has strong coupling and nonlinear characteristics, but the traditional linear control theory can’t achieve satisfactory result. Using state feedback exact linearization method is feasible, but with complex model controller and parameter setting [3, 4]. A hysteresis control method is proposed, based on the active and reactive power rectifier hysteresis comparison, sort out the switch status table, so as to realize the direct power control of the rectifier. But this method cannot effectively control switch frequency, at the same time depends on the width of the hysteresis control precision, thus not suitable for high power occasion [5, 6]. The forecast control strategy, through to the rectifier AC, DC side for testing, is to calculate the estimate parameters of the next moment, finally according to the value of the building function control power devices work. The method has good performance, but the value function parameter setting doesn’t have a clear step. This makes face different control object, still need to staff of parameters for the rich experience and repeated experiment [7, 8, 9]. In this paper, a method of active damping control using the control algorithm to join the “virtual resistance” to change the system damping is Put forward, so as to realize the purpose of improving control performance and get a certain application in industry.

Passive theory is a special case of the theory of dissipation. Based on passive and stability are closely linked and passive control has good robustness [10, 11]. Passive control was applied to current source PWM rectifier, prove that the passive of three phase current type PWM rectifier, and to strictly passive, namely system near balance on a wide range of gradual stability, passive design can be carried out. In the lyapunov theory and quadratic error storage function, based on the passive control to achieve the state variable linear resolve the coupling. And through the simulation experiments prove that increase the damping system, improve the stability of the system.

2. The topology structure and mathematical model of three-phase CSR current-mode PWM rectifier

2.1 Three-phase CSR current-mode PWM rectifier topology

Figure 1 shows the topology structure. Current mode PWM rectifier is based on the principle of Buck step-down converter, the DC voltage can adjust the regulation between zero and rating, and short circuit protection easy current-mode PWM rectifier, flexible net side current control, so in recent years, the study of the current-mode PWM rectifier received great attention.

2.2 Three-phase CSR current-mode PWM rectifier mathematical model

Shown in Fig. 1, in the main circuit of three phase current type PWM rectifier hypothesis [12].

1)
AC and DC side filtering inductance $L$ , $L_{d}$ all is linear, and saturation;
2)
switch loss has been reduced to CSR dc side, and is included in $R_{L}$ ;
3)
grid electromotive force for three-phase stable pure sine wave of electromotive force ( $u_{a}$ , $u_{b}$ , $u_{c}$ ).

By the literature [1, 2] are:

$\displaystyle\left\{\begin{array}[]{l}\displaystyle{L\frac{di_{q}}{dt}+\omega Li% _{d}=u_{q}-v_{q}-i_{q}R}\\ \displaystyle{L\frac{di_{d}}{dt}+\omega Li_{q}=u_{d}-v_{d}-i_{d}R}\\ \displaystyle{C\frac{dv_{q}}{dt}+\omega Cv_{d}=i_{q}-s_{d}i_{dc}}\\ \displaystyle{C\frac{dv_{d}}{dt}+\omega Cv_{q}=i_{d}-s_{q}i_{dc}}\\ \displaystyle{L_{d}\frac{di_{dc}}{dt}=\frac{3}{2}(s_{q}v_{q}+s_{d}v_{d})-i_{dc% }R}\\ \end{array}\right.$ (1)

By Eq. (1) as you can see, in the ( $d$ , $q$ ) system, the mathematical model of the current-mode PWM rectifier is a multiple input multiple output system, input is low frequency switching function $s_{d}$ and $s_{q}$ , the output is the net side current direct axis $i_{d}$ and quadrature axis component $i_{q}$ , Eq. (1) also suggests that current type PWM rectifier is a nonlinear system (the product of the state variables $i_{dc}$ and input variables $s_{d}$ , $s_{q}$ ), and is a coupling system. Therefore, to the current mode PWM rectifier based on the linear state equation to solve coupling.

Figure 1.
Three phase current PWM rectifier structure.

3. Three-phase CSR current-mode PWM rectifier system of passive control strategy

3.1 The passivity of three-phase PWM rectifier of CSR system analysis

Passive theory is a special case of the theory of dissipation. Passive concept first developed in circuit theory after Kalman and Popov, as well as the development of the famous scholars such as Hill, the formation of the passive theory of the system.

For nonlinear system, if there is a positive semi-definite scalar functions $V(x)$ , makes $u\in R$ 3.2 the inequality for any input, said the system is dissipative, $V(x)$ as energy storage function, the inequality is called the dissipation inequality.

.

If the energy of the system is always less than or equal to the initial, the system is the sum of the energy and external energy, these systems are becoming passive.

.

The three-phase PWM rectifier system of CSR, stored energy function to take:

$\displaystyle V=\frac{1}{2}x^{T}Mx$ (2)

Then the system strictly passive, namely system can near balance on a wide range of gradual stability, passive design can be carried out.

Proof..

By Eq. (1)

The mathematical model of CSR two phase rotating $d$ - $q$ ,

$\displaystyle\left\{\begin{array}[]{l}\displaystyle{\frac{Ldi_{d}}{dt}=-Ri_{d}% +\omega Li_{q}+u_{d}-v_{d}}\\ \displaystyle{\frac{Ldi_{q}}{dt}=-Ri_{q}-\omega Li_{d}+u_{q}-v_{q}}\\ \end{array}\right.$ (3) $\displaystyle\left\{\begin{array}[]{l}\displaystyle{\frac{Cdv_{d}}{dt}=\omega Cv% _{q}+i_{d}-S_{d}i_{dc}}\\ \displaystyle{\frac{Cdv_{q}}{dt}=-\omega Cv_{d}+i_{q}-S_{q}i_{dc}}\\ \end{array}\right.$ (4) $\displaystyle L_{dc}\frac{di_{dc}}{dt}=\frac{3}{2}(v_{d}S_{d}+v_{q}S_{q})-R_{l% }i_{dc}$ (5)

CSR Eulerian – Lagrangian (EL) mathematical model:

$\displaystyle\begin{bmatrix}\frac{2}{3}L_{dc}&0&0&0&0\\ 0&L&0&0&0\\ 0&0&L&0&0\\ 0&0&0&C&0\\ 0&0&0&0&C\\ \end{bmatrix}\begin{bmatrix}di_{dc}/dt\\ di_{q}/dt\\ di_{d}/dt\\ dv_{q}/dt\\ dv_{d}/dt\\ \end{bmatrix}+\begin{bmatrix}0&0&0&-S_{q}&-S_{d}\\ 0&0&WL&1&0\\ 0&-WL&0&0&+1\\ S_{q}&-1&0&0&WC\\ S_{d}&0&-1&-WC&0\\ \end{bmatrix}\begin{bmatrix}i_{dc}\\ i_{q}\\ i_{d}\\ v_{q}\\ v_{d}\\ \end{bmatrix}+$ (6) $\displaystyle\phantom{\quad}\begin{bmatrix}\frac{2}{3}R_{L}&0&0&0&0\\ 0&R&0&0&0\\ 0&0&R&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ \end{bmatrix}\begin{bmatrix}i_{dc}\\ i_{q}\\ i_{d}\\ v_{q}\\ v_{d}\\ \end{bmatrix}=\begin{bmatrix}0\\ u_{q}\\ u_{d}\\ 0\\ 0\\ \end{bmatrix}$

Equation (3.1) can be written as EL form:

$\displaystyle M\dot{x}+Jx+\Re x=u$ (7)

The matrix $M$ is a positive definite diagonal matrix, matrix $J$ is an antisymmetric matrix, $J$ reflects the system internal structure of the Internet; $R$ indicates the dissipation of the system; The energy exchange with external of the system is indicated by $u$ . In the matrix expression, letter $R$ is for positive definite symmetric matrix;

$\displaystyle M=\begin{bmatrix}\frac{2}{3}L_{dc}&0&0&0&0\\ 0&L&0&0&0\\ 0&0&L&0&0\\ 0&0&0&C&0\\ 0&0&0&0&C\\ \end{bmatrix},X=\begin{bmatrix}di_{dc}/dt\\ di_{q}/dt\\ di_{d}/dt\\ dv_{q}/dt\\ dv_{d}/dt\\ \end{bmatrix},$ (8) $\displaystyle J=\begin{bmatrix}0&0&0&-S_{q}&-S_{d}\\ 0&0&WL&1&0\\ 0&-WL&0&0&+1\\ S_{q}&-1&0&0&WC\\ S_{d}&0&-1&-WC&0\\ \end{bmatrix},\Re=\begin{bmatrix}\frac{2}{3}R_{L}&0&0&0&0\\ 0&R&0&0&0\\ 0&0&R&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ \end{bmatrix},u=\begin{bmatrix}0\\ u_{q}\\ u_{d}\\ 0\\ 0\\ \end{bmatrix}$ (9)

For three phase current source PWM rectifier, the stored energy of the system functions:

$\displaystyle V=\frac{1}{2}x^{T}Mx$ (10)

∎

The derivative of Eq. (10) is:

$\displaystyle\left\{\begin{array}[]{l}\dot{V}=x^{T}M\dot{x}=x^{T}u-x^{T}\Re x% \\ x^{T}Jx=0\\ x^{T}\Re x>0\\ \end{array}\right\}\to\dot{V}\leqslant x^{T}u\quad x^{T}u=i_{d}u_{d}+i_{q}u_{q}$ (11)

Equation (11) satisfied the dissipation inequality, the rectifier passive resistance. And strictly passive. The system can be near balance on a wide range of gradual stability, passive design can be carried out.

3.2 Three-phase PWM rectifier system of CSR passive decoupling control algorithm

Error energy function for the quadratic function:

$\displaystyle V=\frac{1}{2}x_{e}^{T}Mx_{e}$ (12)

Derivative of Eq. (12):

$\displaystyle\dot{V}=x_{e}^{T}M\dot{x}=x_{e}^{T}\left\{u-M\dot{x}^{\ast}-J(x^{% \ast}+x_{e})-\Re(x^{\ast}+x_{e})\right\}$ (13)

When:

$\displaystyle u=M\dot{x}^{\ast}+J(x^{\ast}+x_{e})+\Re x^{\ast}$ (14)

Equal to:

$\displaystyle\dot{V}=-x_{e}^{T}\Re x_{e}<0$ (15)

The presence of source control rule is:

$\displaystyle u=Jx+\Re x^{\ast}$ (16)

The $J$ , $R$ , $x$ , $x^{\ast}$ , and $u$ is plug in Eq. (13):

$\displaystyle\left\{\begin{array}[]{l}\displaystyle{S_{q}=\frac{i_{q}-WC(U_{m}% +WLi_{q}-I_{m}R)}{i_{dc}}}\\ \displaystyle{S_{d}=\frac{i_{d}+WC(-WLi_{d})}{i_{dc}}}\\ \end{array}\right\}$ (17)

Combine Eq. (17) with Eq. (16):

$\displaystyle\left\{\begin{array}[]{l}\displaystyle{\frac{Ldi_{d}}{dt}+Ri_{d}=% I_{m}R}\\ \displaystyle{\frac{Ld_{q}i_{q}}{dt}+Ri_{q}=0}\\ \displaystyle{\frac{L_{dc}di_{dc}}{dt}+R_{L}i_{dc}=R_{L}i_{dc}^{\ast}}\\ \end{array}\right.\quad\left\{\begin{array}[]{l}\displaystyle{\frac{Cdv_{d}}{% dt}=0}\\ \displaystyle{\frac{Cdv_{q}}{dt}=0}\\ \end{array}\right.$ (18)

CSR is obtained by Eq. (18) in the state variables of the $i_{d}$ , $i_{dc}$ , $i_{q}$ , $v_{d}$ , and $v_{q}$ static decoupling is realized. $I_{m}$ value by steady state when the ac/dc power balance equation:

$\displaystyle\frac{3}{2}U_{m}I_{m}-\frac{3}{2}RI_{m}^{2}=i_{dcR}^{2}R_{L}$ (19) $\displaystyle I_{m}=\frac{1}{2}\left(\frac{U_{m}}{R}-\sqrt{\left(\frac{U_{m}}{% R}\right)^{2}-\frac{8i_{dcR}^{2}R_{L}}{3R}}\right)$ (20)

Which makes the $x_{e}=x-x^{\ast}$ , $x_{1}^{\ast}=I_{DCR}$ , $x_{2}^{\ast}=I_{q}^{\ast}=0$ , $x_{3}^{\ast}=I_{d}^{\ast}=I_{m}$ , $u_{d}=U_{m}$ , $u_{q}=0$ . At this point the passive decoupling design is complete.

3.3 Passive damping to control the influence of three-phase PWM rectifier system of CSR

For the realization of the system dynamic response speed, high tracking precision, will be expected to inject damping, join the resistance of Ra. Error energy function for the quadratic function:

$\displaystyle V=\frac{1}{2}x_{e}^{T}Mx_{e},\quad x_{e}=x-x^{\ast}$ (21) $\displaystyle\dot{V}=-x_{e}^{T}\Re x_{e}<0$ (22) $\displaystyle\dot{V}=-x_{e}^{T}(\Re+\Re_{a})x_{e}<0$ (23)

By increasing the resistance Ra, speeds up the dissipative, system status, quickly return to equilibrium. To speed up the system stability, and quick response time.

Figure 2.

Three-phase CSR control system based on passivity principle diagram.

3.4 Three-phase PWM rectifier of CSR passive control system

Three phase current type PWM rectifier net side current control strategy can be divided into two types: indirect current control and direct current control. Direct current control has good control performance and has a fast current response, the main circuit parameters have stronger robustness. Three-phase CSR control system based on passivity principle is shown in Fig. 2. Three-phase CSR dc side is constant current source, but the constant current source load. Control objectives:

1)
the power factor is close to 1;
2)
the net side current is sinusoidal;
3)
the wide-ranging asymptotically stable equilibrium system good robustness;
4)
the balance change tracking accuracy is higher, the steady-state error to zero;
5)
the dynamic response is fast;
6)
strong anti-interference system;
7)
under non-sinusoidal voltage, prevent low voltage harmonic oscillation.

4. Three-phase PWM rectifier passive control system simulation of CSR and damping characteristics

System simulation model including sampling module, three-phase CSR filtering and synchronization module, setting calculation module, PI control module, passive decoupling control module, based on SPWM three value logic function for decoupling and modulation module, CSR three-phase rectifier bridge modules. On the ac side inductance and capacitance parameter setting, respectively according to the setting value into the module.

Figure 3.

Grid side voltage and current waveform.

4.1 Three-phase PWM rectifier passive control system simulation analysis of CSR

Grid side voltage and current waveform is shown in Fig. 3, you can see from the picture the power supply voltage at 0.01 ms, namely the current achieved synchronization, achieve the goal of the power factor of 1, short response time, and the waveform is stable. After the ac capacitor filter net side current for a smooth sine wave, the green line in the waveform is shown in Fig. 3. PI control module, the adjustment realization of the inner ring current $I_{d}$ , $I_{q}$ can be achieved by adjusting the proportion, integral parameter, make the $I_{d}$ to track the IM, achieve high accuracy and rapid adjustment. Adjust the $I_{q}$ is zero at the same time, the power factor of 1.

4.2 Three-phase PWM rectifier load changes of CSR simulation analysis

Simulation parameters, and grid voltage: 380 v, frequency: 50 hz, switch frequency: 2 mh 1800 hz, ac filter inductor, the equivalent resistance, ac filter capacitor: 0.50 500 uf, dc output rated voltage: 300 v; Load $R=$ 5 $\Omega$ ; 50 mh, inductance capacitance 50 muF.

Figure 4 for when load changes, the dc side current waveform, the current recovery time soon, 0.02 ms, ac current is sine wave at the same time.

Figure 4.

Dc side current waveform (upper) the ac current waveform (below).

Figure 5.

A phase current harmonic network side.

Figure 6.

A phase current harmonic network side.

Figure 7.

Dc side current harmonic.

Figure 8.

Dc side current harmonic.

4.3 The ac resistance change of harmonic analysis

Shown in Fig. 5 the ac resistance of 0.1 $\Omega$ net side current waveform, harmonic produced by the figure visible twin peaks, and the peak value is greater than five times the peak value, and the current waveform is not stable, low harmonic oscillation. When the damping resistance to increase 0.5 $\Omega$ , harmonic amplitude decreases, and twin peaks disappear, current waveform is close to sine wave, as shown in Fig. 6. Also conforms to the dissipation principle, resistance hours slow dissipation system is not stable, the dissipation of fast, large resistance circuit stability.

4.4 When the ac capacitance change of harmonic analysis

As shown in Fig. 7, when reduce the ac capacitor, the system damping is reduced, low harmonic oscillation caused by intensifying, low harmonic, system is not stable. As shown in Fig. 8, when adding ac capacitance, increase the system damping, harmonic is reduced, can increase the capacitance value, can increase the damping of the system stability.

5. Conclusion

Three-phase PWM rectifier of CSR passive control system, dc current and reactive current net side as control variable, implements the dc current and power factor of the independence, decoupling control and good dynamic response.

Passive control system, the three-phase PWM rectifier of CSR for the realization of the system dynamic response is fast, high tracking precision and the need to inject damping.

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