Small-signal Stability Analysis of Paralleled Converter System under Power Quality Control Strategy for Microgrid

In order to enhance the output voltage stability of converter system under non-linear and unbalanced load conditions, the power quality control strategy based on fault-tolerant control architecture is adopted. By analyzing the principle of power quality control strategy, a mathematical model of parallel converter system is established. The mathematical model of the paralleled converter system is linearized at the steady-state operating point, and the corresponding small-signal mathematical model is established. The small-signal stability of the paralleled converter system under the power quality control strategy is analyzed. The results show that under the power quality control strategy, the output voltage stability of the converter system is improved, and the small interference stability of the parallel converter system is guaranteed.


INTRODUCTION
In recent years, distributed power generation and energy storage technologies have received increasing attention from scholars, and distributed energy generation systems with three-phase converters in parallel play an important role in electrical energy conversion. Distributed power generation and energy storage devices are connected to the microgrid through converters to realize plug-and-play of renewable energy supply devices [1][2] . However, the fast response speed, small inertia, and poor overload capacity of the converter affect the stable operation of the microgrid [3][4][5][6] , and it is necessary to study the improved converter control strategy and its feasibility and stability in order to improve the reliability of the distributed microsource power supply, increase the stability of the supply voltage, and improve the microgrid power quality [7][8][9][10] .
For the small-signal stability analysis of the parallel converter system, the power differential link is introduced on the basis of droop control in [11], which solves the problem of disturbance oscillation, analyzes the small-signal stability of the system, and summarizes the selection of control parameters. method. In [12], a parallel inverter system with synchronous generator characteristics is studied and a small-signal model is established, and the small-signal stability analysis is used as a reference basis for the configuration of system parameters. In [13][14], a full-order small-signal model of a three-phase AC microgrid was established to determine the effects of each control parameter on the dynamic [18] The converter system under the power quality control strategy mainly includes: the control object LC filter, the voltage and current double closed loop, the residual generator based on the observer, and the compensation matrix Q. The schematic diagram of the topological structure of the parallel connection of two converters under the power quality control strategy is shown in Figure. Among them: f L is the filter inductance, H; f R is the parasitic resistance of the filter inductance,

POWER QUALITY CONTROL STRATEGY AND ITS MATHEMATICAL MODEL
Where, is the unit matrix of 4 4, and A and B1 are:

2.2.Compensation Matrix Q
The corresponding mathematical description of the structure of the design parameter matrix Q is: unknown parameters to be designed, and T represents a time constant.
The state space mathematical model of parametric matrix Q matrix in discrete state can be obtained as follows:： , ,, r D is the coefficient matrix with unknown parameters to be designed in the parameter matrix Q.

2.3.Parameter Design
In order to obtain the appropriate parameters of the Q matrix of the compensation link, the model matching method is chosen. The schematic diagram of model matching is shown in Fig. 2. , then the theoretical compensation matrix can eliminate any disturbance signal acting on the converter system, from the purpose of eliminating the disturbance and improving the output voltage stability of the converter.
Where the load current odq I is the disturbance input, r is the residual signal output by the observer, and r u is the compensation signal obtained by matrix calculation. r G is the transfer function corresponding to the observer, and r G is the transfer function of the disturbance signal acting on the LC filter, the corresponding state space mathematical expression is.
is the transfer function of the control signal acting on the LC filter alone when there is no disturbance, and the corresponding state space mathematical expression is Then the expression for calculating the matrix parameters is:

3.1.Single Converter Small Signal Model
In operation, due to the variation of various parameters, small disturbance is common. If the disturbance is sufficiently small, the nonlinear equation describing the system is linearized at the steady-state operating point, and the stability of the actual nonlinear system can be studied according to the stability of the linearized system.
The small signal model of current loop is: The state space equation of the observer-based residual generator is linearized, and the small signal model of the observer-based residual generator is: LG , , , Linearized the state space equation of parameter matrix Q in the fault-tolerant control architecture, and obtained the small signal model of parameter matrix Q: The corresponding small signal equation is: (12) Linearized the state space equation of LC filter, which is a control object considering coupling resistance and inductance, and obtained the corresponding small signal model: The small signal model of the converter under the power quality control strategy of a single unit is: Where, the state variable is:

3.2.Parallel Small Signal Model with Two Converters
Single converter in each part of the small signal model is built on their rotation coordinate system, when two parallel inverters, need to choose a reference coordinate to global coordinates of the converter, the other a small signal model of current transformer transformation to the global coordinate system, the definition of  for the second stage converter and the phase Angle difference of the global coordinate system. Fig. 3 The parallel circuit model of two converters is shown in Fig. 4.
The small signal equation of the line obtained by linearization is: The mathematical model of resistive load is as follows: The small signal equation of load obtained by linearization is: To eliminate ac bus node voltage, virtual impedance n r is introduced between each node and ground. The parallel small signal model of two converters is as follows:

SIMULATION EXPERIMENT [18]
Select the converter parameters as shown in Table I. The same microgrid transient simulation model of the simulation is also built in MATLAB/Simulink, and the steady-state operating point obtained from the simulation is brought into the state matrix as the initial value for the small-signal analysis. The small-signal stability of the system is analyzed at this equilibrium point, and the dynamic characteristics of the system under different operating conditions can be obtained, with the dynamic characteristics of the system during the parameter change. All the characteristic roots of the system are obtained, and it can be seen from Fig. 5 that all the characteristic roots of the system are distributed in the left half plane, which proves that the system is stable. Analyze the effect of Q matrix parameter variation on system stability under the electrical energy control strategy used in this paper. When the parameter T is changed from 100 to -10, the trend of the root trajectory of the converter is shown in Fig. 6. When T changes from positive to negative, the characteristic root near the origin gradually moves to the right and the system stability is weakened. And it crosses the origin when T becomes negative and the system is unstable. Therefore, when designing Q, it should ensure that the T parameter is positive.
When the parameter k changes, the trend of the root trajectory of the system is shown in Fig. 7. When the parameter k2 changes from 0 to 20, the root trajectory trend of the system is shown in Fig.  7(a), when k2 gradually becomes larger, the left low-frequency characteristic root gradually approaches the imaginary axis; when the parameter k4 changes from 10 to -10, the root trajectory trend of the system is shown in Fig. 7(b), when k4 gradually becomes smaller, the left low-frequency characteristic root gradually moves away from the imaginary axis, and the system tends to be more stable, when the parameter k6 changes from -125 to 125, the root trajectory trend of the system is shown in Fig. 7(c), when k6 gradually becomes larger, the left low-frequency characteristic root gradually approaches the imaginary axis; when the parameter k8 changes from -1000 to 0, the root trajectory trend of the system is shown in Fig. 7(d), it can be seen that when k8 is negative to zero, the low-frequency characteristic root moves to the origin or even crosses the imaginary axis, positive root appears, and the system is easy to be unstable. From the above, the coefficient k6 and k8 changes of the residual signal have a large impact on the stability of the system.

CONCLUSION
In order to improve the stability of the output voltage of the converter, this paper proposes to adopt the power quality control strategy, and also takes the converter parallel system under this control strategy as the research object, establishes the full-order small-signal dynamic model, and uses MATLAB