Stability Analysis Method of Parallel Inverter

In order to further provide theoretical support for the stability of an auxiliary inverter parallel system, a new model which covers most of control parameters needs to be established. However, the ability of the small-signal model established by the traditional method is extremely limited, so this paper proposes a new small-signal modeling method for the parallel system. The new smallsignal model not only can analyze the influence of the droop parameters on the system performance, but also can analyze the influence of the output impedance of the inverter, the unbalanced and nonlinear loads, and the power calculation method and cut-off frequency of the low-pass filter on the system performance and stability. Based on this method, this paper carries out a comprehensive analysis on the performance of a parallel inverter system. And the correctness of the modeling method and analysis process of the system performance and stability are verified by the consistency of the simulation and experimental results.


Introduction
The stability of a parallel system is one of the important indexes to evaluate the performance of a parallel system.For a high power auxiliary parallel system, a change of the various control parameters has an extremely large impact on the stability of the system.Therefore, it is very important and necessary to analyze the stability of the parallel system.
In general, the stability analysis of a parallel system should be based on a specific parallel control strategy.At present, the wireless parallel system usually adopts the droop method to achieve current sharing automatically [1][2][3][4][5][6], the basic approach of which is to use active power and reactive power to droop the amplitude and frequency of the voltage.The paper also adopts the droop method but has been modified to improve effectively [7][8][9][10] and the whole cycle adaptive droop method (TWCADM) is proposed.It is pointed out that a new power calculation method without a large delay lowpass filter, combined with the improved droop method, can greatly enhance the dynamic response of the parallel system.However, the author does not demonstrate the reason of the improvement of dynamic response in theory.Therefore, it is necessary to establish a unified model of the parallel system, using the mathematical tools to prove the improvement of the dynamic response in theory.
For the mathematical model of a parallel inverter system, the relevant literature [11,12] is not abundant.Antonio et al. establish a small-signal model of grid-connected inverter for the first time [13], but he only considers the outer power loop and regards the inner voltage loop as the ideal state.This method is relatively simple, the established model is of 3rd-order, and, to a certain extent, it can reflect the change trend of a single inverter.However, a big drawback of this method is that the voltage of the parallel point is regarded as an independent variable, so that the parallel inverter automatically shields the influence of another inverter, and most papers adopt this method [14][15][16].Marwali et al. propose a small-signal modeling method which can consider the whole parallel system [17], and the voltage of a parallel point can be expressed as the physical variable of each inverter.All inverters in a parallel system can be coupled together by the voltage at a parallel point, and the inverters in the model can affect each other, so the established model is more accurate.But in Marwali et al. 's paper, the model established by the discrete modeling method in the  domain  is extremely complex.Actually, it is not greatly significant for the parallel system with large delay link to adopt discrete analysis; besides, this method fails to take the impact of the power calculation method into account.
Aiming at the defects of the methods above, this paper proposes a new method to establish the small-signal model of a parallel system, which can take the factors into account such as the coupling effect between the inverters in the parallel system, the influence of the line parameters, load characteristics, and the power calculation method and the cut-off frequency of the low-pass filter.The model established by this method can effectively analyze the influence of each correlation variable of the parallel inverter on the system performance, has a very wide universality, and is simpler than the model established by Marwali et al.

The Control Method of Parallel Inverters
Figure 1 is the topology of the parallel system, which indicates that the inverter adopts the three-phase four-wire system with split capacitor topology, and each inverter is controlled in two stages: parallel control [18][19][20][21][22][23] and inverter control [24][25][26][27][28], and the output voltage and output current of each phase need to be sampled.
Figure 2 is the schematic diagram of inverter control and we can obtain the output voltage of the inverter from the diagram: where In (1), () is control branch and () is inverter internal impedance.Equation (1) indicates that both () and () can be controlled by the voltage controller  v () and the current controller  i ().If the appropriate voltage controller and current controller are selected, the internal impedance of the inverter can be designed.
In this paper, the improved resonant controller is used as the voltage controller and a proportional controller is used as the current controller.
Figure 3 is the bode diagram of the internal impedance of the inverter and we can discover an interesting result that the method shown in Figure 2 leads the internal impedance to be resistive.Therefore, the resistive droop method is suitable for the parallel system: where  p is the proportion a droop coefficient of active power,  d is the differential coefficient of active power,  p is the proportion of a droop coefficient of reactive power,  i is the integral coefficient of reactive power, and  d is the differential coefficient of reactive power. and  represent the active power and reactive power, respectively, and both of them are the values after the large delay low-pass filter.
Traditionally, the power calculation method is described by the following equation: where In (5),  max is the maximum value of voltage,  u is the initial voltage phase angle,  max is the maximum value of current, and  i is the initial current phase angle.
In the power calculation method above, the obtained power contains an AC component at two-times the fundamental frequency.Therefore, a large delay low-pass filter is Fi gure 4: The model of a parallel inverter.adopted to filter the AC component, which deteriorates the dynamic performance of the parallel system.
In order to improve the dynamic performance of the parallel system, the power calculation method can be improved as the following equation: Equation (6) indicates that both the voltage and current need to be delayed by 90 degrees which leads to more storage cost payment.However, the obtained power does not contain an AC component any more, which enables the cutoff frequency to be set higher, and the dynamic performance of parallel system can be effectively enhanced, as shown in

The Traditional Small-Signal Modeling Method
According to the above derivation, we can simplify the parallel inverter model, as shown in Figure 4, where Z∠ is the output impedance of the inverter, consisting of two parts: internal impedance and line impedance.
From Figure 4, according to the concept of complex power, the output power of the inverter can be obtained: Considering the small disturbance existing near the steady-state value (,  e ,  e ), (8) can be linearised as where Based on the droop method mentioned in (3), the differential calculation result can be obtained as follows: Combining ( 9), ( 10), ( 12), (13), and ( 7), Δ and Δ can be obtained: Simplifying ( 14) the small-signal model can be obtained as follows: where Equation ( 15) is the traditional small-signal model, and it can be found that the traditional model is of 3rd-order, so the traditional small-signal modeling method is relatively simple and apt to analyze.However, this method is not accurate, mainly for the following reasons: (1) The voltage at the parallel point is regarded as an independent variable, and the coupling effect between two inverters is eliminated.However, for an actual parallel system, there must be a coupling action between the two inverters, and the change of output characteristics of one inverter will inevitably lead to that of the other inverter.Therefore, the traditional small-signal modeling method is not accurate.
(2) It is well known that the load characteristics have a great impact on the stability of the parallel system, but the traditional small-signal model cannot analyze the influence of the load characteristics.
(3) The traditional small-signal model cannot analyze the influence of different power calculation methods.

The New Small-Signal Modeling Method
Aiming at the drawbacks of the traditional small-signal modeling method, this paper proposes a new small-signal modeling method.The model established by this modeling method has a wide enough amount of information and it Fi gure 5: Simplified parallel model.
can not only analyze most of the control parameters outside the voltage loop of the control system but also analyze the influence of the circuit parameters.What is more, the model is simpler than the model established by Marwali et al. [17].
For convenience, the parallel system shown in Figure 4 can be simplified as shown in Figure 5.The simplification is just to simplify the modeling and description process; readers can restore the output impedance of inverter and the load into any characteristics; it is even possible to consider the influence of two inverters on the parallel system with inconsistent impedance characteristics.
Based on Figure 5, the output current of the inverter can be obtained as The sum of the output current of two inverters is the load current and the output voltage can be obtained by the load current as Combining ( 17) and ( 18), we can obtain From Figure 5, we can discover that the voltage phase angle at the parallel point is zero.In fact, the parallel voltage is sinusoidal and the phase angel is varied, and, for the convenience of modeling, the phase angel is set to the reference value so the phase angle  1 and  2 can be understood as the phase angle difference, rather than a time varying variable.The meaning of the equivalence is that the time varying variable can be transformed into an invariant, which is suitable for analysis at the small-signal model.
According to (19), we can get two equations as follows.
Equation (20) specifies the relationship between the parameters of the two inverters and the parallel point voltage.
Equation (21) specifies the relationship between the steadystate voltages of the inverters.
Combining (19) and ( 17), we can obtain According to the traditional power calculation method [29,30], we can find that before the power calculation we need to obtain İo ∠ − 90 ∘ .In practical operation, the current waveform can be stored in electronic memory in real time.However, in the process of stability analysis, it is known that İo ∠ − 90 ∘ is actually the result of the exchange between the real part and imaginary part of İo , so we can obtain Combining ( 20), (23), and (24), we can get the traditional power calculation method: where,  i ,  i ,  i , and  i can be expressed by the inverter parameters which are given in Appendix C. Similarly, adopting the same method, a new method of power calculation can be obtained, which is the same as (25).And only the values of  i ,  i ,  i , and  i are different, and they are given in Appendix D.
where  is the coefficient matrix: Combining ( 28) and ( 29), we can obtain the smallsignal model of the parallel system, as shown in (30).The coefficients in the equation are given in Appendix.
Similarly, we can construct the small-signal model based on the new power calculation method; the model is still as shown in (30)   The model established by ( 30) is a six-order system, and its physical meaning is aimed at the whole parallel system while the traditional small-signal model established by ( 9) is aimed at the single inverter only.Although the model established by (30) is simpler than the model by Marwali et al., it still cannot be analyzed directly and is required to be solved by mathematical tools such as MATLAB.

Small-Signal Model of Traditional Power Calculation
Method.First, the model established by the traditional power calculation method is analyzed, the circuit parameters are given in Table 1, and it is should be noted that the steady-state values should be satisfied (21).Figure 6 is a root locus of the small-signal model using the traditional power calculation method.(a) is the local root locus when  i is equal to 1 and  p increases from 0 to 1, and (b) is the local root locus when  p is equal to 1 and  i decreases from 1 − 2 to 1 − 5.
Observing Figure 6, we discover that the parallel system, using traditional power calculation method, is not stable in the root locus, for which the traditional power calculation method will introduce an AC variable of double frequency.In order to suppress the oscillation,  p should be increased or  i should be decreased.We can discover from the Table 1 that, after introducing differential control, a pair of conjugate complex roots are generated.This is the basic reason why the dynamic response of the model can be improved.

Small-Signal Model of the New Power Calculation Method.
From the small-signal model, established by the traditional power calculation method, it can be found that the robustness of the parallel inverter is not strong.In order to improve it, the author uses the new power calculation method without a large delay low-pass filter, and this paper regards the model based on the new power calculation method as the main analysis object.Figure 8 is the root locus of the small-signal model based on the new power calculation method only using proportional droop control.(a) is the root locus when  i is equal to 1 − 4 and  p increases from 0 to 1 − 2, and (b) is the root locus when  p is equal to 1 − 4 and  i increases from 0 to 5 − 3. Three important points can be obtained from Figure 8: firstly, the new power calculation method is better than the traditional power calculation method, and the poles always fall on the left side of the imaginary axis; secondly, when only the proportional droop is adopted, the poles fall on the negative real axis, which indicates that the dynamic performance is good; thirdly, with the increase of the proportional droop coefficient, the poles of the system shift to the left and the stability is improved.
Figure 9 is the root locus of the small-signal model based on the new power calculation method with differential control.In Figure 9,  p =  d = 0, and the four curves correspond to four local root locus, respectively, when  i =  p = 1 − 1, 1 − 2, 1 − 3, and 1 − 4 and  d increases from 0 to 1 − 5. From Figure 9, we can discover that the introduction of differential control generates conjugate complex roots.And the significance of the conjugate complex roots is to improve the dynamic performance of the system, but the excessive increase of the differential coefficient will lead to the instability of the system.In addition, it can be found in Figure 9 that the increase of the proportional droop coefficient will lead the poles to move to the left and increase the stability of the system.
The control parameters of Figure 10 10(a) indicates that  p mainly affects the conjugate complex roots and can be modified to fine-tune the dynamic response of the system. d mainly affects the poles on the negative real axis, but this pole is not the main pole.So its influence on the system performance is not very large.
Figure 11 is the root locus when the impedance varies, and all the droop coefficients are given as follows: (a) indicates that with the increase of resistance, that is, the load decrease, the poles on the negative real roots move to two sides of the axis, respectively, but the main poles are getting closer to the imaginary axis.So the stability of the system becomes worse when the load becomes smaller.(b) shows that, with a change of the output impedance of the inverter, the pole moves to the right but the main pole has not been changed.So a change of the output impedance has a limited effect on the parallel system.
Figure 12 is the root locus when the cut-off frequency of the low-pass filter increases from 1 − 2 to 20, and the droop coefficients used are the same as those in Figure 11. Figure 12 shows that, to a certain extent, the system gets more stable if the cut-off frequency is increased, but the cut-off frequency cannot increase without limit.When the cut-off frequency increases to a certain value, the main pole of the system turns into a pair of conjugate complex roots that move right with the increase of the cut-off frequency.Then the system stability will get worse.

The Research on Application of the Model
Based on the small-signal model introduced in the this paper and the droop control that we researched before, the problem of unbalanced current and harmonic current caused by the unbalanced and nonlinear loads can be resolved and the output current can be shared equally.Thus, this part establishes the simulated and experimented platform to verify the theory.The simulated and experimental parameters are  all within the range of previous stability analysis so that the correctness of the stability analysis can be well verified.The simulated model is built by using Simulink/sfunction.Power is expressed by per unit value and 1 means 11375 kVA.The simulated parameters are shown in Table 2.
First, the simulated results and analysis of the control strategy in the parallel inverter system with an unbalanced load are introduced as follows.Figure 13 is the unbalanced load simulated results waveform, which indicates that the unbalanced currents can be shared effectively.
Second, the simulated results and analysis of the control strategy in the parallel inverter system with a nonlinear load are illustrated, and Figure 14 is the nonlinear load simulated waveforms, which indicates that the proposed droop method can share harmonic currents effectively.
The experimental results and analysis of the parallel inverter system with an unbalanced load are described as follows.Figure 15 is the experimental unbalanced load waveforms, and we can discover that the unbalanced current can be accurately shared.The experimental results are consistent with the simulated results.Similarly, the experimental results and analysis of the nonlinear load are introduced.Figure 16  proposed droop method can share harmonic currents accurately and has good robustness.
Figure 17 is the experimental waveforms with an unbalanced load and nonlinear load at the same time.It can be seen from the figure that the unbalanced and harmonic currents can both be effectively shared.
Figure 18 is a photograph of the experimental platform.All the experiments in this paper were completed using this experimental platform.

Conclusions
This paper proposes a new small-signal modeling method for an auxiliary parallel inverter system and analyzes the influence of the parameters on the stability characteristics of the system in detail.The traditional analysis method is based on a small-signal model, but the model established by the traditional small-signal method is not accurate without considering the coupling relationship between parallel inverters, so this model cannot analyze the influence of the load and the output impedance of inverter.The small-signal modeling method proposed in this paper regards the whole parallel system as the research object, including the influences of the control system information outside the voltage loop and circuit parameters of inverter, which provide the most comprehensive analysis method by far.

A. The Coefficient Matrix 𝐷 of Traditional Method
The coefficient matrix  of the traditional power calculation method is as follows:

B. The Coefficient Matrix 𝐷 of New Method
The coefficient matrix  of the new power calculation method is as follows:

C. The Coefficients of Traditional Method
The coefficients of traditional power calculation method are given as follows:

D. The Coefficients of New Method
The coefficients of new power calculation method are given as follows:

Figure 1 :
Figure 1: The topology of auxiliary parallel inverter system.

Figure 2 :
Figure 2: The schematic diagram of inverter control.

Figure 3 :
Figure 3: The bode diagram of the internal impedance of the inverter.

Figure 6 :
Figure 6: The root locus of the small-signal model based on the traditional power calculation method.

Figure 7 :
Figure 7: The root locus of the small-signal model based on the traditional power calculation method.

Figure 7
Figure7is a root locus of the small-signal model based on the traditional power calculation method after introducing differential control.The droop coefficients are as follows:  i = 1 − 3,  p = 1 − 1,  p = 1 − 4, and  d = 1 − 4, and  d increases from 0 to 8 − 3. We can discover from the Table1that, after introducing differential control, a pair of conjugate complex roots are generated.This is the basic reason why the dynamic response of the model can be improved.

Figure 8 :Figure 9 :
Figure 8: The root locus of a small-signal model based on the new power calculation method.

5 ,
and  d = 1 − 8. (a) is the root locus when  o increases from 1 − 2 to 1e2 and (b) is the root locus when  o increases from 1−3 to 5−1.

Figure 10 :Figure 11 :Figure 12 :
Figure 10: The root locus of the small-signal model based on the traditional power calculation method.

Figure 13 :
Figure 13: The simulated results with an unbalanced load.

Figure 14 :
Figure 14: The simulated results with a nonlinear load.

Figure 18 :
Figure 18: The photograph of experiment platform.

Table 2 :
Parameters of simulation and experiment.