Wide frequency band oscillation mechanism and suppression measures for grid-interfaced PV inverters based on impedance analysis

With the rapid development of PV and wind power, the energy structure of the power grid is changing dramatically. The dominant power supply role played by conventional power sources such as hydropower and thermal power is gradually replaced by PV and wind power. All the power electronics-based equipment makes the power system stable form more complex, and one of the most significant challenges is wide frequency band oscillation (WFBO). This paper proposes an impedance analysis method for PV inverters in high proportion to new energy-integrated power systems. First, we investigate WFBO mechanism caused by the PV inverter based on its impedance model, and then a detailed impedance model of parallel PV inverters is developed to study WFBOs in PV stations. Finally, the WFBO suppression measure based on active damping control is proposed. Simulation results prove the correctness and feasibility of the active damping control strategy.


Wide frequency band oscillation
At present, with China's vigorous promotion of new energy (NE), the proportion of electronic equipment in the power system continues to increase, weakening the AC coupling between the power grid to a certain extent, weakening the electromechanical oscillation between synchronous generator sets, and the electromagnetic oscillation problem caused by electronic equipment is gradually prominent.In recent years, there have been WFBO events related to NE power and electronic equipment in various countries in the world [1-4] .This paper gives several typical oscillation events, as shown in Table 1.
Compared with the oscillation problem of the traditional power system, the WFBO problem of the power system with a high proportion of new energy is mainly caused by the power electronic equipment (PEE) and its control system.The oscillation frequency is wide, from Hz to several kHz.The scope of influence may involve multi-regional, multi-unit, and multi-electrical equipment [5] .

Wide frequency band oscillation analysis mode
At present, state-space and impedance analysis are common methods for analyzing the stability of NE power access to the power grid.The state space analysis method is to establish the time domain state space Equation of the NE power generation system first, then obtain the system characteristic Equation based on the set state space Equation, and finally determine the stability of the NE power generation system by analyzing the typical values and eigenvectors of the NE power generation system [6] .In the traditional power system, the time constant of the equipment is large, the system state space model is relatively simple, and the use of the state space analysis method to analyze the stability of the traditional power system is very effective and has a clear significance.Hence, the stability analysis of the conventional power system generally uses the state space method.Different from traditional power systems, the time constant of NE power generation equipment is small, the corresponding system state space model is more complex, the low-order model is not enough to express all the essential characteristics of the system, and the high-order system state space model will make stability analysis quite difficult [7] .In particular, when considering the influence of external factors such as phase-locked loops and grid impedance of NE power generation equipment, the difficulty of system stability analysis will be further increased [8] .

Wide frequency band oscillation suppression
In terms of wide-band oscillation suppression, cutting protection, changing the system operation mode and enhancing the grid structure are generally used in actual projects, but this will limit the gridconnected consumption of NE power generation and the economic operation of the power system, which is a technical means of last resort.Therefore, there is an urgent need to explore more costeffective wide-band oscillation suppression methods.
To improve the stability of the PV inverter grid-connected system of new energy power generation, such as increasing hardware circuits or changing control strategies [9] , are often used in theoretical research.The Reference [10] proposes using a converter in series as an impedance eliminator, which can make grid impedance influences system stability less, but a converter in series significantly increases the system cost.Reference [11]- [13] proposed that the retrofitting resistors on the output capacitor of the grid-connected inverter are proposed to increase PV inverter grid-connected system damping, the oscillation suppression method is simple and easy, and the resonance suppression effect is very significant, but the damping resistor consumes active power.Reference [14]- [15] applied of virtual impedance method to shape the output impedance of PV inverter grid-connected inverters.Reference [16]- [17] proposes to add a filter to the controller so that only the impedance characteristics of the gridtied inverter at the resonant frequency are changed.
With the increase of PEE in the power grid, WFBO has become one of the key elements restricting the further development of new energy.Based on the WFBO impedance analysis method, this paper explores the WFBO mechanism of the NE high-proportion power system.It proposes the active impedance control strategy of the filter capacitor unit of the LCL circuit to suppress the occurrence of WFBO effectively.

Output impedance model of PV inverter
To research the influence of phase-locked loop (PLL) on the stability of PV inverter system, this paper takes a single-phase PV inverter as a research object and performs modeling and analysis.According to Norton's theorem, the PV inverter impedance model, when incorporating PLL into the modeling process, is established, which provides a basis for further theoretical analysis.This paper focuses on the single-phase PV inverter, and its control block strategy is shown in Figure 1: Since the DC side is usually equipped with specific voltage regulation measures, its fluctuations can be ignored, so the voltage outer loop is omitted and replaced by a constant DC voltage source Udc.When using the impedance method to analyze PV inverter grid-connected system, firstly, establish the impedance model of the PV inverter and further establish the impedance model of the inverter incorporating PLL.Since inverter output current into the grid is controllable, the Norton equivalent circuit is generally used to simulate the PV inverter without considering the PLL, composed of a current source is(s), inverter output impedance Zout(s) in parallel.Under the condition of upcc(s)=0 and iref(s)=0, is(s) as shown in Equation ( 1) and is(s) and Expression for Zout(s): In order to study PLL on PV inverter system stability, it is necessary to introduce a phase-locked link in the above derivation.Based on the working principle of the PLL and the reference current generated block strategy shown in Figure 3, the relationship between iref(s) and upcc(s) can be obtained Equation ( 4) shows that the PV inverter, after considering the PLL, is equivalent to a parallel impedance model, and Figure 4 is its corresponding equivalent circuit diagram.The independent current source iref(s) contained in the original Norton model of the PV inverter is replaced with the negative impedance link Zpll(s) introduced by the PLL.
Figure 4 PV inverter grid-connected system when considering PLL where Zpll(s) is the PLL equivalent impedance: Equation ( 5) can obtain the output impedance of the PV inverter after considering the influence of the PLL: From Equation ( 6), it can be seen that the mathematical model Gpll(s) of the PLL needs to be established before analyzing the Zout_pll(s) characteristics.Commonly used PLLs include zero-crossing phase-locked loops and SRF-PLL, etc. SRF-PLL with strong harmonic suppression ability is selected in this paper.The harmonic linearization method is now used to model the PLL.
Figure 5 The PLL control loop circuit The PLL control strategy is shown in Figure 5, and a set of voltage quadrature components can be obtained by converting upcc through Clark, and uα and uβ.Then the uα and uβ under the stationary coordinate system are transformed by Park Tαβ/dq to the synchronous rotating coordinate system dq axis components (ud, uq).Gp(s) is the controller of the phase-locked loop, usually a PI controller, and its transfer function is kpi(s)=kp+ki/s。 Then, obtain a mathematical model of the PLL.From the control strategy shown in Figure 5, a series of time domain Equations can be obtained, and the steady-state value of each variable can be added to the small signal disturbance to obtain:

Multi-inverter grid-tied impedance model
Under the premise that the PV inverter grid-connected system is stable, this section studies the stability of the multi-PV inverter grid-connected system.The structure of the multi-PV inverter grid-connected system can be equivalent, as shown in Figure 6.From the figure, each inverter connects to a common bus in parallel through an LCL filter, and the bus voltage is Vpcc.In a multi-PV inverter system, each inverter's control strategy is independent.A closed-loop control scheme with the grid-connected current as the feedback can ultimately simplify the system control block diagram into the form shown in Figure 7.The expression for the grid-connected current that can be deduced from Figure 7 is: Because each inverter is controlled independently, it can be equivalent to the circuit shown in the figure above.Each inverter in Figure 7 is equally comparable to the Norton equivalent circuit, allowing multiple machines to be connected to the Norton equivalent model.Apply Kirchhoff's law of currents at the PCC point in Figure 6, get: Bring Equation ( 9) into Equation ( 8) to obtain: Then bring Equation (10) into Equation ( 9) to obtain the output current expression of any PV inverter as follows: From Equation ( 11), the output current of each PV inverter is composed of three parts.The first part is generated by its excitation current, but its coefficient not only includes its inverter control parameters, but also contains other inverter control parameters, which can reflect the mutual coupling between gridconnected inverters.The second part is generated by the excitation current of other inverters in parallel with it, which reflects the rest parallel inverter's control performance influence the current.The third part is the generation of grid voltage, and its coefficients reflect the grid impedance parameter's influence on the current.From Figure 8, the response sensitivity of the multi-inverter impedance to the number of inverter impedances in the high-frequency band is high, mainly concentrated in 1000 Hz ~10000 Hz.As the number of inverters connected to the grid increases, the trough of the inverter impedance model increases, and the intersection point with the grid impedance is offset in the high-frequency direction.When n = 5, 10, the phase margin corresponding to the grid intersection is negative, and the entire system will oscillate.

The influence of each parameter on the impedance model
To effectively suppress the WFBO of the system, this project proposes an improved non-beat control scheme embedded in active damping to find a suitable virtual impedance change control strategy, change L1, L2, and C respectively, obtain their respective Bode plots, and analyze the response sensitivity of the inverter impedance model to each parameter change.
First, change the parameter in the LCL circuit to obtain the corresponding inverter impedance baud diagram, as shown in Figure 9 (a) change L1 (b) change L2 (c) change C Figure 9 The effect of the LCL filtering link on the impedance model As can be seen from Figure 9, when changing L1, L2, and C is expanded to 5 times and ten times the original value, the impedance model has a poor response to the L1 change and cannot be used to change the LCL Circuit parameters, in turn, suppress system WFBO.The impedance model has a good response to L2 changes, but L2 has a small range of changes, and cannot be used to change LCL circuit parameters to suppress system WFBO.The impedance model has a good response to C changes, and the C change range is wide, and all finally, C is selected to change the LCL circuit parameters shows in figure 10 which suppress the WFBO of the system.
Figure 10 LCL filter with a virtual damping resistor

Active damping control
To suppress WFBO, the resonant current between inverters needs to be reduced.Using the inverter control strategy and according to the analysis of the bode diagram above, the broadband oscillation suppression virtual resistance of the photovoltaic inverter system is obtained.The response of the line current is expressed by the inverter output current and the grid voltage in the continuous s-domain. ) For simplified second-order systems (12), the damping ratio (ζ) is determined to be: Formula ( 13) is the damping resistance corresponding to the damping ratio of WFBO suppression.The resistance in Equations ( 12) and ( 13) is changed to virtual impedance control, guaranteeing the damping and stability of the inverter system in the z-domain.
According to the above processing method, to suppress the resonance of the output current and power, the main consideration is to connect a resistor in parallel at both ends of the capacitance of the LCL circuit so that the output current can produce a more apparent damping effect because the actual resistance in parallel will increase a large power loss, so here it is hoped that the equivalent current that appears on the resistor will be superimposed on the output current of the inverter through control, therefore Improve the virtual impedance control strategy based on the block strategy in Figure 11, and the virtual impedance is introduced in the following way:  From Figure 12(a), before the active damping control is added, when the number of system gridconnected PV inverters is large, the inverter output current and power spontaneously oscillate, that is, the corresponding Figure 8 with the increase in the number of inverters, the system phase margin is negative, and the system loses stability; from Figure 12(b), when active damping control is added, the multi-photovoltaic inverter system maintains the current power stability after the current power is stable.No broadband oscillation occurs; that is, although the system phase margin in Figure 13 is reduced due to the increase in the number of grid-connected PV inverters, it can maintain a positive phase margin under the regulation of active damping, that is, suppress the occurrence of WFBO and maintain system stability.At the same time, no resonant peak was found with the addition of virtual resistor control.Explains that the virtual resistance control strategy is valid.

Conclusion
This paper, based on the impedance analysis method, analyzed the grid-connected WFBO of the inverter, and as the number of PV inverters connected to the grid increases, the stability of the power grid and the possibility of WFBO increases.According to the analysis of the mechanism of WFBO, the active impedance control of the capacitance of the LCL filter is proposed to suppress the generation of WFBO.According to the simulation analysis, the results of this method are remarkable.

Figure 1
Figure 1 Block diagram of LCL type PV inverter system

Figure 1 Figure 2
Figure 2 Mathematical model of a PV inverter

Figure 3
Figure 3 Reference current generation block diagram

Figure 6
Figure 6 Equivalent structure diagram of PV multi-inverter grid-connected system

Figure 7
Figure 7 PV inverter current control strategy

7Figure 8
Figure 8 Bode diagram of the impedance model of the multi-inverter

Figure 11
Figure 11 Virtual impedance introduction block diagram That is, the current generated by Vc on the virtual resistor should be fed back to the reference current.Acquisition of active damping control before and after the inverter output current and output power waveform plot is shown in Figure 12. Figure (a) is the virtual impedance control before the inverter waveform diagram; figure (b) is the inverter waveform after the addition of active damping control.

Figure 13
Figure12A diagram of the system's wide-band oscillation suppression effect after adding virtual impedance Gpll(s) is a mathematical model of PLL, Equation (3) can be concluded 4where