Research on the modeling and control mechanism of three-phase cascade rectification stage of non-power frequency transformer

Non-power frequency transformers are essential and important devices for new energy generation and smart grid construction. This article analyzes the working principle of voltage cascade rectifier stage, constructs a mathematical model of three-phase cascade rectifier stage system, and simplifies its model. Finally, the idea of overall control of the voltage loop in the rectifier stage system was derived, and the traditional control method of rectification was directly transplanted into the overall control system of the cascade rectifier stage. At the same time, the characteristics of voltage loop and current loop of cascade rectifier stage system are analyzed, and the relevant control structure block diagram and controller design idea are given, which lays an important theoretical foundation for in-depth study of non-power frequency transformer.


Introduction
In recent years, research on cascaded inverters, cascaded active power filters, and cascaded reactive power compensators has been quite extensive and continuously deepening in China.However, there is relatively little research on multi-level collaborative control cascaded rectifier stages that mainly transmit functional quantities.The cascade rectifier stage of the new generation of high power converters without power frequency transformers has a wide range of applications in future smart grids, high-power AC drives, and other fields, and will have a profound impact on existing high-power transmission technologies [1][2] .This paper first analyzes the circuit topology and working principle of voltage type cascaded rectifier stage, and establishes the mathematical models of single-phase and three-phase voltage type cascaded rectifier CHBR (Cascaded H-Bridge Rectifier) from the perspective of low frequency and high frequency [3] .

Three-phase cascade rectifier stage topology
The three-phase cascade rectifier stage adopts a Y-shaped connection, and the topology structure is shown in Figure 1.In the figure, point O is the midpoint of the power grid, N is the midpoint of the Yshaped connection of the cascade rectifier stage system, and u NO is the voltage between point N and point O. u cona , u conb and u conc are the voltages between a and b, c and d, and e and f in the figure, respectively [4] [5] .Assuming that in a three-phase cascade rectifier system, the internal resistance The capacitance values on the DC side are equal, and the capacitance voltage is u dcij (i=a, b, c; j=1，2.The i and j in the following modeling are the same); The power grid is an ideal voltage source with three-phase balance and symmetry [6] : sin( )

Mathematical modeling of 2-phase cascaded rectifier stage
Similar to the modeling of single-phase cascaded rectifier stage switches, each cascaded H-bridge of the three-phase cascaded rectifier stage can be represented by a switch function [7] , as shown in equation (2), with switch labels corresponding to Figure 1: According to Kirchhoff voltage theorem, the Equation of state of AC side of rectifier stage system can be established [8] : Due to And because the power grid is a three-phase balanced voltage source [9] : According to equations (3), (4), and (5): ) 3  3 According to Kirchhoff current theorem, the equation of state of DC side of rectifier stage system can be established: Linearize the switching function S ij and find the average value ξ of one switching cycle, obtained: By introducing equation ( 8) into equation (3) ( 7), the average model of the three-phase rectifier stage system in the abc coordinate system can be obtained [10] .The average model is shown in Figure 2. The average model of the three-phase cascade rectifier stage shown in Figure 2 is too complex, which is not conducive to the design of the controller [11] .Therefore, further simplification is needed for this average model.Considering that the capacitor voltage on the DC side of the H-bridge at all levels is equal in steady-state, and the fluctuation of the capacitor voltage is very small, therefore: Among above parameters, dcij u is DC component contained in the DC side voltage of the H-bridge;  dcbj u is its AC component.Therefore, the following formulas can be derived as follows: Assuming the following formula: 3 The AC side of the three-phase cascade rectifier stage can be simplified as: Similar to the simplification of single-phase cascade rectifier stages, the DC side of three-phase rectifier stages can be simplified in the form of equation ( 16): Among above parameters: By incorporating equations ( 11) and ( 12) into ( 16), it can be concluded that: Therefore, the average model of the three-phase cascade rectifier stage can be simplified in the form of Figure 3: The above mathematical model of the three-phase cascade rectifier stage is established in the abc coordinate system, which has the advantages of clear and clear physical meaning.However, due to the time-varying AC quantity on the AC side of the rectifier stage, it brings significant difficulties to the design of the system controller.Compared to traditional vector control methods, the mathematical model of the rectifier stage in the abc coordinate system can be transformed into a dq coordinate system that rotates synchronously with the fundamental frequency of the power grid voltage, so that all sinusoidal variables are simultaneously converted into relative DC variables, thereby simplifying the design of the control system.
By performing Park transformation on both sides of (15) and ( 18), the mathematical model of the rectifier stage system in the dq rotating coordinate system can be obtained: 0 In formula: ξ jd 、ξ jq --the component of the switching function in the dq coordinate system; i sd 、i sq --the component of the AC-side current in the dq coordinate system; u sd 、u sq --the component of the power supply potential in the dq coordinate system; v d 、v q --the fundamental component of the synthesized vector u con on the AC side of the rectifier stage in the dq coordinate system.Park transformation to: Thus, the mathematical model of the three-phase cascade rectifier stage in the dq rotating coordinate system can be obtained, as shown in Figure 4.

Control of the three-phase cascade rectification stage
The model structure in the dq coordinate system is shown in Figure 5. Comparing the average model of the cascaded rectifier stage with the model structure diagram in the dq coordinate system, it can be found that its overall control is roughly the same as traditional three-phase full bridge cascaded rectifier.In the dq coordinate system, the d-axis and q-axis variables of the cascaded rectifier stage are coupled with each other, which is not conducive to controller design.Traditional three-phase full bridge rectifier control methods can be referred to, and feedforward decoupling control of i d and i q can be introduced to compensate for input v q and v d .The control equations of v q and v d can be expressed as:  The active current i sd and reactive current i sq in this equation have been completely decoupled.When the active current i sd is less than the active command current i sd * , i sd * -i sd >0.Because R s i sd is very small, L(di sd /dt)>0, i sd active current gradually increases; On the contrary, the active current gradually decreases.The control principle of reactive current i sq is the same.The decoupling control structure of the inner loop of the rectifier stage system is shown in Figure 6: If the Current loop controller adopts traditional PI control, i sq and i sd can be expressed as: ) Among the above parameters, i sq and i sd1 are generated by instruction signals, while i sqg and i sdg are generated by disturbances from the power supply.It can be seen that although traditional PI controllers have good current following performance in the decoupled dq coordinate system, the system has weak anti-interference ability against grid voltage fluctuations.The usual approach is to design the system as a typical Type II system, and perform filtering control or composite compensation control for the overshoot current of the typical Type II system.Here, the proportional resonance controller adopted by the Current loop during single-phase rectification is introduced into the three-phase rectification to make up for the shortcomings of the PI controller.The formula (27) (29) can be summarized as follows: When the angular frequency of AC signal is ω R (s=jω R ).Both i sqg and i sdg are 0, which enables the system to track the command current without static error, while improving the system's anti-interference ability against grid voltage fluctuations.In actual system, it is necessary to consider the influence of digital system accuracy, analog system component parameters, etc.The quasi PR controller mentioned earlier can be considered for implementation.
From the analysis of the input and output characteristics of the rectifier stage, it can be concluded that the total active power loss of the three-phase rectifier stage, that is, the transmission of system active energy, is directly related to the capacitor voltage.Therefore, to stabilize the voltage loop of the threephase connected rectifier stage, it is necessary to start from the perspective of overall control.Firstly, it is necessary to stabilize the average value of each phase capacitor voltage to obtain the required active power flow i sd * for the average value.The control diagram of the voltage loop is shown in Figure 8.  Due to the main transmission of active energy in the rectifier stage system, as well as various losses such as parallel losses and mixed losses, load imbalance and fluctuations become the main impact on the voltage balance of the rectifier stage system.The voltage balance control of three-phase cascaded rectifier stages can be divided into two categories: one is the voltage balance of the DC side capacitors of the H-bridge in each stage, and the other is the voltage balance between the three-phase phases.Simulations were conducted to validate the accuracy and theoretical correctness of the model for a threephase cascaded rectifier with no line-frequency transformer under unbalanced three-phase loads, as depicted in Figures 10(a

Conclusions
On the basis of analyzing the topology of the three-phase cascade rectifier stage, this article establishes a mathematical model of the three-phase cascade rectifier stage system and simplifies the model.Conduct in-depth research on the control mechanism of three-phase cascaded rectifier stages, and obtain the model structure of the cascaded rectifier stages in the dq coordinate system and the decoupling control structure diagram of the inner loop of the rectifier stage system.At the same time, the characteristics of the voltage loop and the Current loop of the cascade rectifier system are analyzed, and the relevant control structure block diagram and controller design ideas are given, which lays a very important theoretical foundation for further control strategy research and analysis.

Figure 1 .
Figure 1.The topology of three phase CHBR stage

Figure 2 .
Figure 2. Average model of three phase CHBR stage

Figure 3 .
Figure 3. Simplified equivalent circuit model of CHBR stage on abc coordinate system

Figure 4 .
Figure 4. Equivalent circuit model of CHBR stage on dq coordinate system

Figure 5 .
Figure 5. Model structure of CHBR stage on dq coordinate system

Figure 6 .
Figure 6.The structure of current loop decoupling control of CHBR stageDue to the symmetry of the two current inner loops, the design of the controller can be done in the same way.Similar to the single-phase rectifier stage: considering the sampling delay, and the PWM control is equivalent to a small inertial link, the Current loop control block diagram of the rectifier stage system in the dq coordinate system is shown in Figure7:

Figure 8 .
Figure 8.Total voltage control structure diagram of three phase CHBR stage

Figure 9 .Figure 10 .
Figure 9.Total voltage loop control diagram of three phase CHBR stage )(c)(e)(g).The simulation results obtained using the zero-sequence voltage injection method are shown in Figure10(b)(d)(f)(h).These simulation results provide academic evidence for the accurate modeling and control mechanism of the three-phase cascaded rectifier with no line-frequency transformer.