Small signal stability analysis of paralleled inverters for multiple photovoltaic generation units connected to weak grid

: In this study, a small signal model of paralleled inverters for multiple photovoltaic (PV) generation units (MPGUs) connected to weak grid is developed. Based on the proposed small signal model, eigenvalue analysis is employed to study the stability of MPGUs within different grid strength, different control parameters in each phase-locked loop (PLL) controller and varying operating points in each PV generation unit. Furthermore, based on the transfer function of the power control loops in dq rotation frame, the coupling mechanism of operating point in each PV unit is revealed. The results show that: with the increasing of output power of PV unit among the PV generation and the decreasing of the grid strength, it may occur oscillation phenomena, the oscillation frequency is about 5 Hz; tuning gains of PLL has noticeable effect on the damping characteristic of the system, and larger PLL gains can improve system damping; each PV unit power control loop is coupled by the point of common coupling voltage, increasing the output power of PV unit, it may reduce the phase margin of the system transfer function and deteriorate the system stability. Theoretical analysis is veri ﬁ ed with simulation of MPGUs connected to weak grid.


Introduction
PV power generation is an important way to promote transformation of energy and deal with environmental challenges. By the end of 2015, the national cumulative installed PV capacity was 43.18 GW in China and 200 GW worldwide [1]. The global installed PV capacity is expected to exceed 450 GW by 2017; the trend of PV development is rapid.
Due to the restriction of resource endowment, large scale PV generation is mostly located in deserts or semi-deserts where the grid structure is relatively weak, and integrated to power systems through voltage source converter (VSCs). Ideally, the point of common coupling (PCC) voltage is stable, the weak coupling between VSCs and the PCC voltage, they have less interactions with each other. With the increasing capacity of grid-connected PV generation, the grid impedance cannot be ignored, it may lead to VSCs connected to weak grid [2,3], and the ideal condition is destroyed. The PCC voltage is unstable, it will be affected by the output power disturbance of PV array and the grid disturbance. With the strong coupling between VSCs and the PCC voltage, they have more interactions with each other. Therefore, this situation poses challenges on the control and safe operation of PV generation.
Stability of PV generation connected to weak grid has been investigated in some papers [3][4][5][6][7]. In [4], an impedance-based stability criterion for grid-connected VSCs is proposed; it reveals that the system will be unstable if the ratio between the grid impedance and the VSC output impedance cannot satisfy the Nyquist stability criterion. Yan et al. [5] proposed a small signal model to analyse the stability of DC-link voltage in VSC and discussed the effect of P-V characteristic of PV array on the VSC stability. Agorreta et al. [6] proposed an equivalent VSC that models the N-paralleled gridconnected VSCs in PV plants and the coupling effect due to grid impedance is described.
The above studies have explained the operation stability of PV generation qualitatively under the condition of weak grid; however, the control system plays an important role which has not been widely considered. In [8], it was found that the instability phenomenon is quite related with converter control loops such as AC terminal control loop and phase-locked loop (PLL). In addition, most of these studies adopt the single-machine-infinite-bus model, few literatures have adopted the paralleled model for multiple photovoltaic generation units (MPGUs), it cannot represent the PV generation stability within interactions among machines, the research results are limited. This paper establishes a small signal model of paralleled inverters for MPGUs connected to weak grid, eigenvalue analysis is employed to study the stability of MPGUs within different grid strength, different control parameters in each PLL controller and varying operating points in each PV generation unit. Based on the transfer function of the power control loops in dq rotation frame, the coupling mechanism of operating point in each PV unit is revealed.
2C o n figuration of MPGUs connected to weak grid Fig. 1 shows the configuration of MPGUs connected to AC grid. In Fig. 1, C i is the input filter capacitor; U dci is the DC voltage; L fi is the output filter inductor; C fi is the output filter capacitor; R ci and L ci are the resistor and inductor of the collector system, respectively; R g and L g are the resistor and inductor of the grid, respectively; U ti is the terminal voltage; U pcc is the PCC voltage; U g is the grid voltage.
In Fig. 1, the strength of the AC system is generally described by a short-circuit ratio (SCR), as shown in (1). The AC system is regarded as a weak grid under the condition that the SCR is < 3. From (1), it can be obtained that SCR is decreased with increase of grid impedance.
where S ac is the short-circuit capacity of the AC system; S N is the rated power of the PV generation; Z g is the grid impedance.

Modelling of MPGUs connected to weak grid
A small signal model is derived for stability analysis and some assumptions are made as follows: (1) C fi and R g are not included in the stability analysis; (2) The system is lossless.

PV array model
This part is intended to derive the mathematical model of PV array. Fig. 2 shows the configuration of PV array, N p and N s are the number of parallel and series connected cells, U dci is the DC voltage, I PVi is the DC current. The U dci -I PVi characteristic of a PV array is shown in (2), which can be obtained from the manufacturer where I sci is the short-circuit current; U oci is the open-circuit voltage; C 1 and C 2 are constants.  (3), which is transformed into dq frame. A typical grid voltage oriented-based vector control system of VSC is shown in Fig. 4. The outer loop in the d frame is the DC voltage controller which produces the current reference i dref for the current inner loop. The outer loop in the q frame is the AC-bus voltage controller which produces the current reference i qref for the current inner loop. The inner current controllers produce the modulation signals in dq frame which are then translated back to the abc frame

VSC model and control
In Fig. 4, U dcrefi and U dci are the DC voltage reference value and the actual value, respectively; U trefi and U ti are the terminal voltage reference value and the actual value, respectively; i drefi and i qrefi are the current reference values in dq frame; i di and i qi are the actual values in dq frame; U di and U qi are the modulation voltages in dq frame; θ plli is the output of PLL.
According to Fig. 4, the following equations can be obtained: where x 1i , x 2i , x 3i and x 5i are the state vectors; k p1i , k p2i , k p3i and k p5i are the proportional gains of the controllers; k i1i , k i2i , k i3i and k i5i are the integral gains of the controllers. Fig. 5 shows the working principle of PLL, which is employed to make sure that d frame is always aligned with the terminal voltage where x pll is the state vector; k p4i and k i4i are the PLL parameters.

Grid model
The equivalent circuit of the AC side of MPGUs connected to AC grid is shown in Fig. 7, the KVL equation of the AC side in the xy frame can be written as where x g is the grid impedance; x ci is the collector system impedance; U txi and U tyi are the terminal voltage in xy frame; U gx and U gy are the grid voltage in xy frame. The transformation for the terminal voltage and current from dq frame to xy frame is given as Equations (2)-(7) are the differential algebraic equations, which can describe the MPGUs connected to weak grid accurately. Taking the two photovoltaic (PV) generation units paralled as an example, from linearisation of the equations, a small signal model of the system is deduced. The small signal model of the system is expressed in state-space equations asḊ 4 Small signal stability analysis

Eigenvalues analysis
In order to obtain the eigenvalues, taking the two PV generation units paralleled as an example, the system parameters are shown in Table 1. Table 2 shows all the system modes with frequency and damping of the oscillation, under the condition that SCR is equal to 1.5. As can be seen From Table 2, the system has four oscillation modes and 10 damping modes. All the eigenvalues of the system are distributed on the left side of the complex plane, the system is stable.

Effect of grid strength
Influence factors on stability of the system involve grid strength, different control parameters in each PLL controller and varying operating points in each PV generation unit. Based on the small signal model of MPGUs connected to weak grid, we discuss the effects of above factors on the stability of the system.  The system parameters are shown in Table 1; the eigenvalues locus for SCR varies from 4 to 1.2 is shown in Fig. 8. As the grid strength decreases, it can be seen that there are eight eigenvalues changing. λ 1 , λ 7 , λ 8 and λ 14 move to left-half plane; λ 9 , λ 10 , λ 11 and λ 12 move to right-half plane. When SCR turns to be 1.2, λ 9 and λ 10 enter the unstable region, the system becomes unstable. Therefore, the stability of the MPGUs connected to weak grid becomes worse with reduction of grid strength.

Effect of PLL gains
When SCR turns to be 1.5 and the proportional gain of PLL k p42 turns to be 50, the eigenvalues locus for another proportional gain of PLL k p41 varies from 10 to 100 is shown in Fig. 9.A s k p41 increases, it can be seen that there are seven eigenvalues changing. λ 7 , λ 8 , λ 9 , λ 10 , λ 11 and λ 12 move to left-half plane; λ 14 moves to right-half plane. When k p41 turns to be 10 and k p42 turns to be 50, λ 9 and λ 10 enter the unstable region, the system becomes unstable.
The eigenvalues locus for both proportional gains of PLL k p41 and k p42 vary from 10 to 100 is shown in Fig. 10.A sk p41 and k p42 increase, it can be seen that there are also seven eigenvalues changing. λ 7 , λ 8 , λ 9 , λ 10 , λ 11 and λ 12 move to left-half plane; λ 14 moves to right-half plane. When both k p41 and k p42 turn to be 20, λ 9 and λ 10 enter the unstable region, the system becomes unstable.
The oscillation frequency and damping ratio are shown in Figs. 11 and 12. As can be seen from Fig. 12, tuning gains of PLL has noticeable effect on the damping characteristic of the system, and larger PLL gains can improve system damping, enhance the system stability.

Effect of operating point
The MPGUs change their operating point through the active power output varies. The eigenvalues locus for active power output varies from 0.45-1.0 pu is shown in Fig. 13. As the active power output increases, it can be seen that there are also seven eigenvalues changing. λ 7 , λ 8 and λ 14 move to left-half plane; λ 9 , λ 10 , λ 11 and λ 12 move to right-half plane.
In order to reveal the coupling mechanism of operating point in each PV unit, according to the method in [9], neglecting the effect of current control loop dynamic and PLL dynamic. The power control loops of paralleled inverters for MPGUs connected to weak grid is simplified as shown in Fig. 14. In Fig. 14, P PV1 and      Simplified the small signal model further, which is described in Fig. 15. The open loop transfer function G o (s) can be obtained in (10) According to (10), the bode plot of G o (s) in given in Fig. 16.I t can be seen from Fig. 16, with the output power ratio k increases, the phase margin of G o (s) decreases and the stability of the system becomes weaken.

Simulation results
In order to verify the effectiveness of the small signal analysis, a simulation model of MPGUs connected to weak AC grid is built in power systems computer aided design (PSCAD)/electromagnetic transients (EMTDC). The parameters are shown in Table 1.
When the proportional gains of PLLs k p41 and k p42 turn to be 50, the integral gains of PLLs k i41 and k i42 turn to be 1500, Figs. 17 and 18 show DC voltages (U dc1 and U dc2 ) response and terminal voltages (U t1 and U t2 ) response of a corresponding detailed model for a step change of illumination at t = 8 s with different grid strengths. Fig. 17 illustrates that the system becomes stable after a step change of illumination in SCR = 1.5. Fig. 18 illustrates that the system becomes unstable after a step change of illumination in SCR = 1.2 and the oscillation frequency is about 5 Hz. The  When SCR turns to be 1.5, Figs. 19 and 20 show DC voltages (U dc1 and U dc2 ) response and terminal voltages (U t1 and U t2 ) response of a corresponding detailed model for a step change of illumination at t = 8 s with different PLL gains. In Fig. 19, k p41 = 15, k i41 = 1500, k p42 = 50 and k i42 = 1500; in Fig. 20, k p41 = 30, k i41 = 1500, k p42 = 50, k i42 = 1500. Fig. 19 illustrates that the system becomes unstable after a step change of illumination and the oscillation frequency is about 5 Hz. Fig. 20 illustrates that the system becomes stable after a step change of illumination. The simulation results are coherent with the eigenvalue locus analysis in Fig. 9.

Conclusions
A small signal model of paralleled inverters for MPGUs connected to weak grid is presented in this paper. Eigenvalue analysis is employed to study the stability of MPGUs within different grid strength, different control parameters in each PLL controller and varying operating points in each PV generation unit. The following conclusions can be drawn: (i) With the increasing of output power of PV unit among the PV generation and the decreasing of the grid strength, it may occur oscillation phenomena (SCR < 1.2), the oscillation frequency is about 5 Hz. (ii) Tuning gains of PLL has noticeable effect on the damping characteristic of the system, and larger PLL gain can improve system damping (30-100), enhance the system stability. (iii) Each PV unit power control loop is coupled by the PCC voltage, increasing the output power of PV unit, it may reduce the phase margin of the system transfer function and deteriorate the system stability.

Acknowledgments
This work was supported in part by National Natural Science Foundation of China (Grant no. 51277024) and Research Program of State Grid Corporation of China.