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Single stationary domain equivalent inverter admittance for three‐phase grid‐inverter system considering the interaction between grid and inverter

2019; Institution of Engineering and Technology; Volume: 12; Issue: 6 Linguagem: Inglês

10.1049/iet-pel.2018.5856

1755-4543

Xiaoming Zou, Xiong Du, Guoning Wang, Heng‐Ming Tai,

Islanding Detection in Power Systems

IET Power ElectronicsVolume 12, Issue 6 p. 1593-1602 Research ArticleFree Access Single stationary domain equivalent inverter admittance for three-phase grid-inverter system considering the interaction between grid and inverter Xiaoming Zou, State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing, 400044 People's Republic of ChinaSearch for more papers by this authorXiong Du, Corresponding Author duxiong@cqu.edu.cn State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing, 400044 People's Republic of ChinaSearch for more papers by this authorGuoning Wang, State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing, 400044 People's Republic of ChinaSearch for more papers by this authorHeng-Ming Tai, Department of Electrical and Computer Engineering, University of Tulsa, Tulsa, OK, 74104 USASearch for more papers by this author Xiaoming Zou, State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing, 400044 People's Republic of ChinaSearch for more papers by this authorXiong Du, Corresponding Author duxiong@cqu.edu.cn State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing, 400044 People's Republic of ChinaSearch for more papers by this authorGuoning Wang, State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing, 400044 People's Republic of ChinaSearch for more papers by this authorHeng-Ming Tai, Department of Electrical and Computer Engineering, University of Tulsa, Tulsa, OK, 74104 USASearch for more papers by this author First published: 13 March 2019 https://doi.org/10.1049/iet-pel.2018.5856Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract Admittance matrix has been used to represent the characteristics of three-phase inverter for grid-inverter system stability evaluation in both stationary, sequence, and dq domain. For the dq domain admittance matrix, the PLL and Park transformation introduce additional dynamics during the admittance measurement that may influence the application of dq model. For the stationary and sequence domain admittance matrix, the model measurement is not affected by the above additional parts but may also encounter other difficulties in actual measurement. First, current frequency response analyser (FRA) is not able to perform two-frequency measurement required by the admittance matrix. Second, practical limitations exist in the course of solving complex-valued admittance matrix equations. Here, a single stationary domain equivalent admittance of three-phase inverter for grid-inverter system with considering the interaction between grid and inverter is defined so that the above two difficulties can be avoided. Then, an equivalent admittance model of prototype inverter is constructed for stability analysis of the grid-inverter system using Nyquist stability criterion, instead of the generalised Nyquist criterion (GNC). The effectiveness of the proposed equivalent inverter admittance model is validated through simulation and experiment verification. 1 Introduction The development of distributed generation renders many applications of three-phase grid-connected inverters in power system [1]. The interaction between grid and inverter, however, may cause the stability issue [2], thus affects the safe and reliable operation of the distributed generation systems. Impedance-based stability analysis has been a useful tool in this regard [3]. The inverter admittance can be modelled in dq domain, sequence domain, or stationary domain [4-6]. Modelling in which domain is determined by the definition of admittance. The inverter admittance/impedance matrix have been developed in dq domain for the stability analysis of grid-inverter system [4, 7]. Then, the modified sequence impedance and equivalent impedance [8, 9] are derived from dq impedance by symmetrical decomposition. For the measurement of dq and modified sequence domain impedance, the PLL and Park transformations are required [7, 10]. These parts introduce additional dynamics, which may affect the accuracy of impedance measurement [11]. In [5], the pure positive- and negative-sequence inverter impedance is developed in sequence domain by harmonic linearisation with neglecting the frequency coupling term. With the pure positive- and negative-sequence impedance, the stability of grid-inverter system can be obtained using Nyquist criterion. Moreover, [12] further pointed out that the frequency coupling must be considered otherwise the stability of grid-inverter system cannot be estimated precisely. This implies that the sequence impedance/admittance of inverter also has to be represented as a matrix similar to dq model, and the generalised Nyquist criterion (GNC) must be used. Then, a new method called multi-harmonic linearisation is used to capture the frequency coupling term in sequence domain [13]. For the measurement of sequence domain model, the frequency response analyser (FRA) has been widely used in practice because it is convenient to inject perturbation excitation and extract the corresponding response [14]. However, frequency coupling introduced by phase locked loop (PLL) generates matrix admittance representation [6, 12], which results in current FRA being not able to measure the coupling admittance. In the grid-inverter system, the interaction between grid and inverter makes the measurement of the inverter matrix admittance even difficult. Moreover, complex-valued matrix equation solving encounters practical limitations [14]. In the grid-inverter system, the perturbation voltage at frequency generates current at frequency and , where is the grid-line frequency. These currents result in voltages at the point of common coupling (PCC) containing frequency components at and . Then, the admittance matrix would be measured and calculated with at least two different sets of test at each frequency. In each test, two response currents at two different frequencies, and , need to be extracted at the same time. The current FRA does not have such two-frequency measurement capability. It only measures the response which has the same frequency as the input frequency. This paper defines the single equivalent inverter admittance for grid-inverter system to avoid two-frequency measurement and complex-matrix equation solving, which makes off-the-shelf FRA-based measurement possible. The equivalent admittance is defined in stationary domain by means of complex space vector and complex transfer function similar to [6, 15], which can cover positive and negative sequence characteristic under one expression. Moreover, the proposed equivalent admittance can be used to analyse the stability of grid-inverter system with Nyquist criterion instead of GNC. Stability concern of the multi-inverter grid-connected system has been reported in the literature [16-19]. A harmonic instability analysis method using the state-space modelling and participation analysis was proposed in [16]. An improved resonance modal analysis method is adopted for the stability analysis of multi-inverter grid-connected systems [17]. The current separation scheme was used in the modelling and analysis of a multi-inverter system [18]. A nodal admittance matrix was developed in [19]. Therefore, the proposed modelling and analysis method here is also extended for the stability analysis of a multi-inverter grid-connected system. 2 Challenges for measurement of stationary or sequence domain inverter admittance matrix The inverter admittance measurement is critical in actual grid-inverter system applications. In stationary or sequence domain, when frequency coupling is considered, the small signal three-phase inverter admittance can be defined and modelled as an admittance matrix [6, 12] (1)where (2) and are the voltage and current phasors at frequency , respectively. and are that at frequency . To validate the analytical admittance model or measure the admittance, each element of the admittance matrix can be measured by the scheme shown in Fig. 1. The measurement is performed by injecting a small signal perturbation voltage vpa to the grid via a power amplifier (PA) and an isolation transformer. The PCC voltages vpcca, vpccb and grid currents ia, ib, are measured and recorded by frequency response analyser. Fig. 1Open in figure viewerPowerPoint Inverter admittance measuring scheme The equivalent circuit of the grid-inverter system injected by grid-voltage perturbation is shown in Fig. 2. represents the fundamental components at frequency and the injected voltage perturbation of frequency . Fig. 2Open in figure viewerPowerPoint Equivalent circuit of the grid-inverter system with voltage perturbation Consider inverter itself alone that means grid impedance is zero, i.e. in Figs. 1 and 2. Thus, there is no interaction between grid and inverter. In the first test, the perturbation voltage at frequency can be seen directly by the inverter at PCC as . Due to the frequency coupling of the inverter, currents at and at are generated. Then, two elements of the admittance matrix in (2) can be calculated with the FRA output, (3) (4), , and are the phasors of , , and , respectively. In the second test, change the perturbation voltage frequency to . Then, the other two elements in (2) can be calculated as (5) (6)According to above description, if the FRA can extract the amplitude and phase information of grid currents at frequency and simultaneously, then the inverter admittance matrix defined in (2) can be measured with two sets of test. However, such FRA is not available and such equipment is difficult to construct. Commercially available FRA only can extract the response having the same frequency as the perturbation frequency. Under this circumstance, only (3) and (5) can be measured. In addition to above challenge, the complex-valued matrix equations solving is required in practical measurement. In actual grid-inverter system, the existence of grid impedance results in the mutual interaction between grid and inverter. The perturbation voltage at frequency will generate grid currents at frequency and . In turn, these two current components generate PCC voltages and in the presence of grid impedance. Under this circumstance, the inverter admittance elements in (2) can be obtained by the following complex-valued matrix equations (7)Practical limitations do exist to obtain solutions of such complex-valued matrix equations. It can be seen from (7) that if and are not linearly independent, then the determinant of the matrix formed by these two vectors will be zero, and (7) does not have solutions. Moreover, measurement errors must be carefully controlled to reduce the effects of numerical inverse on complex-valued matrix [14]. In summary, for grid-inverter system, measurement of the inverter admittance matrix faces two challenges. First, the two-frequency extraction FRA is not available. Second, practical constraints exist for finding solutions of the complex-valued matrix equations due to the interaction between grid and inverter. 3 Single stationary domain equivalent inverter admittance for grid-inverter system Here, the single stationary domain equivalent inverter admittance for grid-inverter system is defined in a way that it can be measured using the commercial FRA. With this , restrictions encountered by the admittance matrix of (2) can be lifted. Moreover, stability analysis can be performed using Nyquist stability criterion, instead of the GNC. 3.1 Single equivalent inverter admittance definition and measurement method The small signal representation of the grid-inverter system including is depicted in Fig. 3. The equivalent inverter admittance is defined as the ratio between the grid current and PCC voltage at the perturbation frequency (8)where is the grid current complex space vector at frequency and is the PCC voltage complex space vector at . The grid current at frequency can be expressed as (9) Fig. 3Open in figure viewerPowerPoint Small signal representation of the three-phase grid-inverter system at frequency To measure the equivalent admittance at frequency , a procedure using the measuring circuit in Fig. 1 is developed. First, a perturbation voltage with frequency is injected to the grid voltage. Second, the PCC voltage component and the grid current component are extracted using FRA. Third, is obtained from (8). In actual grid-inverter system, the inverter and grid interact with each other. Here, the equivalent inverter admittance model is defined considering the interaction. The admittance characterises the equivalent inverter side of the grid-inverter system, rather than the inverter itself. 3.2 Modelling of single equivalent inverter admittance In this section, the frequency coupling mechanism of three-phase grid-inverter system is analysed and the equivalent inverter admittance model is constructed. A three-phase grid-inverter system is shown in Fig. 4. As indicated in [15], an input voltage of the inverter at the frequency will result in an output current at the frequency and another output current at the frequency , due to the PLL effect. The inverter exhibits a single-frequency input double-frequency ( and ) output (SIDO) phenomenon in stationary domain. Here, similar to [20, 21], two admittance representations, self-admittance and accompanying-admittance , are derived to characterise the SIDO phenomenon by means of complex space vector and complex transfer functions. Fig. 4Open in figure viewerPowerPoint Three-phase grid-inverter system The modelling is achieved by injecting a small signal symmetrical perturbation voltage at frequency into the AC side of inverter. First, referring to [21], the small signal model of Park transformation can be expressed as (10) (11)where and can be complex space vector of the perturbation voltage, or the current, the duty ratio, and so on. and are the corresponding vectors in dq domain. is the corresponding vector at fundamental frequency. The duty ratio vector at the fundamental frequency can be expressed as (12)The PLL can be modelled by (13)where , and are the PI parameters of PLL. The small signal model of inverter power stage in stationary domain can be expressed as (14) (15)where denotes the constant DC bus voltage, and represent the small signal responses of duty radio. The small signal models of current controller in dq domain also can be derived at frequency and , and are shown in (16) and (17), respectively. In (16) and (17), denotes the amplitude of carrier signal which is equal to half of the DC bus voltage, is the current controller, and is the emulated delay effect. Combining the models of power stage circuit (14),(15) and current controller(16), (17) through Park transformationmodel (10), (11), self-admittance and accompanying-admittance for the inverter itself can be derived asshown in (18) and (19). (16) (17) In (18) and (19) at the top of the next page, are transfer functions of the voltage andcurrent sensing filters, (20) (21) (22) and (see (22)), where and are respective filter time constants. The self-admittance represents the relationship between the input voltage at frequency and output current at frequency . The accompanying-admittance describes the relationship between the input voltage at frequency and output current at frequency . These two admittances and characterise the SIDO phenomenon of the inverter itself. Further, for the grid-inverter system in Fig. 4, current generation considering the frequency coupling can be illustrated by the block diagram of Fig. 5. It can be seen from Fig. 5 that, according to the SIDO model and , a perturbation of the grid voltage generates one grid current at frequency and another grid current at frequency . Due to the existence of grid impedance , these two current components will generate two PCC voltage components via the blocks and . The corresponding PCC voltage at each frequency will generate currents at frequency and again. Thus, the currents at frequency and are coupled with each other by way of and . Fig. 5Open in figure viewerPowerPoint Block diagram of current generation considering frequency coupling The single equivalent inverter admittance , shown in (22), can be derived from Fig. 5, which is defined as the ratio between the grid current and PCC voltage . It represents the current generation mechanism due to frequency coupling and characterises the grid-inverter interaction. It is worth noting that is defined in stationary domain by means of complex transfer function and can be viewed as a positive sequence equivalent admittance if is positive and as a negative sequence equivalent admittance if is negative. Thus, it considers the positive and negative sequence admittance under one expression, which is different from the sequence domain admittance. 3.3 System stability analysis In this section, we show that the stability for grid-inverter system can be evaluated by Nyquist stability criterion using the equivalent inverter admittance . For the grid-inverter system at frequency shown in Fig. 3, its stability can be determined by . That is, if satisfies the Nyquist criterion, the grid-inverter system at frequency is stable. System stability at frequency is also discussed. It can be derived from Fig. 5 that the grid current at frequency is (23)Fig. 6 shows the small signal representation of the grid-inverter system at frequency , as described by (23). Stability of the grid-inverter system at frequency can be investigated using Fig. 6. That is, stability of the grid-inverter system at frequency can be determined by Nyquist criterion using . Fig. 6Open in figure viewerPowerPoint Equivalent small signal representation of the three-phase grid-inverter system at frequency It can be seen from (22) that the open loop gain of equivalent inverter admittance is . According to the frequency shift property, Nyquist plots of and hold the same stability property. That means the system stability at both frequencies can be obtained by Nyquist criterion using the equivalent inverter admittance and its open loop gain. 4 Equivalent admittance model validation and analysis 4.1 Admittance model validation Simulation circuit of Fig. 4 has been built in Simulink to measure and validate the equivalent inverter admittance model of (22). Key parameters of the circuit are listed in Table 1. The one-line topology of grid for three-phase grid-inverter system is depicted in Fig. 7, in which , , and . In the simulation, the admittance test method discussed in Section 3 is used. A three-phase grid-inverter system, shown in Fig. 8, was constructed to experimentally verify the equivalent admittance model. The system structure and parameters are the same as in the simulation. Table 1. System symbols and values Symbol Value DC source voltage Vdc 400V d-axis current reference idr 6A q-axis current reference iqr 0A grid voltage grid frequency f0 50 Hz inductor L 1.5 mH ESR of inductor RL 0.15 Ω proportion coefficient of current controller kp 1.77 integration coefficient of current controller ki 706 proportion coefficient of PLL kpp 6.44 integration coefficient of PLL kpi 4280 sampling period Ts 10−4s Fig. 7Open in figure viewerPowerPoint One-line topology of grid for three-phase grid-inverter system Fig. 8Open in figure viewerPowerPoint Experimental platform of three-phase grid-inverter system Bode plots from the analytical model of (22) and from the point–point simulation results are shown in Fig. 9, where the blue solid line denotes the analytical results from (22) and the red dotted line denotes the simulation result. It is observed from Fig. 9 that the magnitude and phase responses by the analytical and simulation approaches match very well both at the positive and negative frequencies. This verifies the correctness of the equivalent inverter admittance model. Fig. 9Open in figure viewerPowerPoint Bode plots of . Blue solid line: analytical results; Red dotted line: simulation (a) Magnitude response, (b) Phase response Besides, the experiment is conducted to verify the correctness of model. Fig. 10 shows the Bode plots of from the experiment result. The red dotted line denotes the measured data using a commercial FRA and the blue solid line denotes the analytical data from (22). It can be seen from Fig. 10 that the magnitude and phase responses by the analytical and experimental results are in close agreement at the positive and negative frequencies. Experimental results also verify the accuracy of the equivalent inverter admittance model. Fig. 10Open in figure viewerPowerPoint Bode plots of . Blue solid line: analytical results; Red dotted line: experimental measurement (a) Magnitude response, (b) Phase response Next, the measurement of is also discussed in this section. Since describes the relation between the input voltage at frequency and the output current at frequency for the inverter itself, it can be calculated by measuring the related current and voltage using FRA directly. To verify the correctness of , the grid impedance is set to zero in the above simulation and experimental prototype. Bode plots of obtained by analytical formula and simulation are shown in Fig. 11, and that obtained by experimental results are shown in Fig. 12. The blue solid line denotes the Bode plots of analytical model in both figures. The simulation and experimental measurement result are depicted as red dotted line in Figs. 11 and12, respectively. It is observed from Figs. 11 and 12 that simulation and experimental results match very well with the analytical model. The results verify the correctness of . Fig. 11Open in figure viewerPowerPoint Bode plots of . Blue solid line: analytical results; Red dotted line: simulation (a) Magnitude response, (b) Phase response Fig. 12Open in figure viewerPowerPoint Bode plots of . Blue solid line: analytical results; Red dotted line: experimental measurement (a) Magnitude response, (b) Phase response 4.2 Effects of parameter variation The effects of parameter variation to the equivalent inverter admittance are examined in this section. In particular, the effects of grid inductance and PI parameters and of the current controller are considered. Consider the grid inductance of values 2, 3, 4, and 5 mH. Bode plots of the equivalent inverter admittance under the above four grid inductances are shown in Fig. 13. It can be seen from Fig. 13 that no significant impact on is observed, except in the frequency interval from −200 to 300 Hz. Fig. 14 is the zoom-in plot of that interval. In the frequency ranging from-100 to 0 Hz, the magnitude and phase of increase along with the size of grid inductance. Fig. 13Open in figure viewerPowerPoint Bode plots of equivalent admittance under various . Green line: 2 mH; Pink line: 3 mH; Blue line: 4 mH; Black line: 5 mH (a) Magnitude response, (b) Phase response Fig. 14Open in figure viewerPowerPoint Zoom-in plots of Fig. 13 in the frequency interval from −200 Hz to 300 Hz (a) Magnitude response, (b) Phase response Next, we examine the effect of PI controller parameters. Consider of values 1.77, 2.665, 3.54, and 4.425, while . Fig. 15 shows the Bode plots of in the frequency up to 1000 Hz under various . It can be seen from Fig. 15 that smaller produces larger magnitude for frequencies <500 Hz. The phase response decreases from −1,000 to −100 Hz and increases from 100 1,000 Hz, along with the increase of . No significant changes occur on the magnitude and phase responses of against when the frequency is higher than 1000 Hz. Fig. 15Open in figure viewerPowerPoint Bode plots of in the frequency region −1000∼1000 Hz for different . Green line: 1.77; Pink line: 2.665; Blue line: 3.54; Black line: 4.425 (a) Magnitude response, (b) Phase response Bode plots of under various are shown in Fig. 16. The values are 706, 1059, 1412, and 1765, while is fixed at 1.77. It can be seen from Fig. 16 that, with the increase of , the phase response of decreases from −400 Hz to 50 Hz, and increases from 50 Hz to 400 Hz. The magnitude response over is complex. Fig. 16Open in figure viewerPowerPoint Bode plots of in the frequency band of −500 and 500 Hz for different . Green line: 706; Pink line: 1059; Blue line: 1412; Black line: 1765 (a) Magnitude response, (b) Phase response 4.3 Model comparison This subsection compares the proposed stationary domain equivalent admittance model with the dq model [4], the sequence models [5, 12], the -frame model [6], and the SISO modified sequence model [9]. The compared characteristics include the representation, frequency coupling, reference frame, measurement, and stability criterion. The results are summarised in Table 2. Table 2. Comparison of the proposed admittance model with other existing models Admittance/impedance model Representation Frequency coupling Reference frame FRA Stability criterion dq model [4] matrix do not exist dq no GNC sequence model [5] two elements (positive and negative sequences) no stationary yes Nyquist Criterion sequence model [12] matrix sequence coupling stationary no GNC -frame model [6] matrix yes stationary no GNC SISO modified sequence model [9] two elements (positive and negative sequences) sequence coupling dq no Nyquist Criterion proposed model single element yes stationary yes Nyquist Criterion Some observations from Table 2 are given in the following. The proposed model is represented as a single element. In contrast, the models in [4, 6, 12] are represented by a matrix and the models in [5, 9] are represented by the positive-sequence and the negative-sequence elements. The frequency coupling effect is factored into the proposed model and the -frame model in [6], while the model in [5] is not. The models in [9, 12] consider the coupling between the positive and negative sequences. On the other hand, the dq model in [4] encounters the cross-coupling between the d-axis and the q-axis, rather than the frequency coupling. Consider the model measurement by an off-the-shelf frequency response analyser (FRA). The proposed model and the model in [5] can be obtained by an FRA; all others cannot. 5 Application of the admittance model for stability analysis of grid-inverter system The proposed equivalent inverter admittance model can be applied to stability analysis of the grid-inverter system. Using the admittance model, system stability can be analysed with Nyquist criterion instead of the GNC. Moreover, the phase margin can be provided to help for the design and modification of the inverter controller. Stability of the grid-inverter system is examined for different values of grid impedance. Two cases are considered. Case I has a grid inductance and Case II has . The system prototype structure and key parameters are shown in Fig. 4 and Table 1, respectively. For Case I, Nyquist plots of and are shown in Fig. 17. In the figure, the pink dotted line represents the Nyquist plot of and the blue solid line denotes that of . The inset is the zoomed-in version of the plot around the point (−1, 0). One can see from Fig. 17 that both Nyquist plots do not encircle ( − 1,0). This implies that the system is stable. Under this case, the experimental results are shown in Fig. 18. It can be seen that PCC phase voltage and three-phase grid currents waveforms do not exhibit noticeable distortion. As such, the system is stable. This demonstrates that stability analysis result is correct. Fig. 17Open in figure viewerPowerPoint Nyquist plots of and for Case I Fig. 18Open in figure viewerPowerPoint Waveforms of PCC voltage and three phase grid currents for Case I Under Case II, the corresponding Nyquist plots are depicted in Fig. 19. It is observed from Fig. 19 that the Nyquist plot of encircles ( − 1, 0). This indicates that the system is not stable, which will result in harmonic oscillations [22]. Experimental results for Case II are shown in Fig. 20. One can see that three-phase grid currents are destabilised with the harmonic oscillations, which implies that the system is not stable. This verifies the stability analysis result by Nyquist criterion and validates the effectiveness of proposed equivalent inverter admittance model. Fig. 19Open in figure viewerPowerPoint Nyquist plots of and for Case II Fig. 20Open in figure viewerPowerPoint Waveforms of PCC voltage and three phase grid currents for Case II 6 Modelling and analysis of multi-inverter grid-connected system In this section, the modelling and analysis method proposed here is extended for the stability analysis of multi-inverter grid-connected system. The multi-inverter grid-connected system is depicted in Fig. 21. Assume that there are N inverters connected to the PCC in the system. The self-admittance and accompanying-admittance of the ith inverter can be obtained by (24) (25) Fig. 21Open in figure viewerPowerPoint Multi-inverter grid-connected system The small signal representation of the ith inverter is shown in Fig. 22. The small signal representations for other inverters can be obtained in a similar manner. Fig. 23 depicts the small signal representation of a multi-inverter system. Fig. 22Open in figure viewerPowerPoint Small signal representation of the ith inverter Fig. 23Open in figure viewerPowerPoint Small signal representation of the multi-inverter It can be seen from Fig. 23 that the terminal current can be obtained as (26). Then, the equivalent self-admittance and accompanying-admittance of the multi-inverter can be expressed as (27) and (28), respectively. Similar to the single-inverter grid-connected system, the equivalent inverter admittance for the multi-inverter grid-connected system considering the interaction between inverter and grid can be derived as (29). Then, the stability of the multi-inverter grid-connected system can be determined using the Nyquist criterion on and . (26) (27) (28) (29) For multi-inverter grid-connected system shown in Fig. 21, parameters of each inverter are assumed to be the same and are listed in Table 1. The grid inductance is set as 0.5 mH. Nyquist plots of and for a five-inverter system are shown in Fig. 24. In the figure, the pink dotted line denotes the Nyquist plot of and the blue solid line denotes that of . The inset is the zoom-in plot around the point (−1, 0). It can be seen from Fig. 24 that both Nyquist plots do not encircle (−1, 0). Thus, the system is stable. The PCC currents displayed in Fig. 25 validates that the system is stable. Fig. 24Open in figure viewerPowerPoint Nyquist plots of and for a five-inverter grid-connected system Fig. 25Open in figure viewerPowerPoint PCC current waveforms for a five-inverter grid-connected system For a six-inverter grid-connected system, the Nyquist plots are shown in Fig. 26. It can be seen from the zoom-in plot of Fig. 26 that the Nyquist plot of encircles (−1, 0). This result indicates that the system is not stable. Fig. 27 shows the PCC current waveforms, which are highly distorted. That means the system is not stable. Fig. 26Open in figure viewerPowerPoint Nyquist plots of and for a six-inverter grid-connected system Fig. 27Open in figure viewerPowerPoint Waveforms of PCC currents for a six-inverter grid-connected system In conclusion, the proposed inverter admittance model also enables accurate stability analysis for the multi-inverter grid-connected system. 7 Conclusions The single equivalent inverter admittance representation considering the interaction between grid and inverter has been presented to avoid the difficulties faced by measuring the coupling admittance matrix for the grid-inverter system. The admittance is defined in stationary domain by means of complex transfer function, which considers the positive and negative sequence admittance under one expression. Analytical equivalent inverter admittance model considering frequency coupling due to the PLL has also been constructed and validated. The effects of parameter variation to the equivalent inverter admittance have been investigated. 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