Generalized Short Circuit Ratio for Multi Power Electronic Based Devices Infeed Systems: Defi- Nition and Theoretical Analysis

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Generalized Short Circuit Ratio for Multi Power Electronic based Devices Infeed Systems: Defi- nition and Theoretical Analysis

Huanhai Xin, Member IEEE, Wei Dong, Deqiang Gan, Senior Member IEEE, Di Wu, Member IEEE, Xiaoming Yuan, Senior Member IEEE

AC grid and evaluate the small signal stability of Abstract—Short circuit ratio (SCR) is widely applied to analyze grid-connected PED system qualitatively [8-10]. Lager SCR the strength of AC system and the small signal stability for single leads to a stronger AC grid and a more stable system, and vice power electronic based devices infeed systems (SPEISs). However, versa. SCR is also applied to measure the static voltage stability there still lacking the theory of short circuit ratio applicable for multi power electronic based devices infeed systems (MPEIS), as of AC-DC systems, which involves two indices, i.e., the critical the complex coupling among multi power electronic devices SCR (CSCR) and the boundary SCR (BSCR) [11-12]. Usually, (PEDs) leads to difficulties in stability analysis. In this regard, this CSCR  2 is used to differentiate very weak systems from weak paper firstly proposes a concept named generalized short circuit systems. BSCR 3 is used to differentiate weak systems from ratio (gSCR) to measure the strength of connected AC grid in a strong systems. However, to the best knowledge of the authors, multi-infeed system from the small signal stability point of view. Generally, the gSCR is physically and mathematically extended there is no quantitative indices as well as physical mechanism from conventional SCR by decomposing the multi-infeed system for the small signal stability analysis of PED systems. Moreo- into n independent single infeed systems. Then the operation ver, the previous researches on SCR mainly focuses on the gSCR (OgSCR) is proposed based on gSCR in order to take the static voltage stability of AC-DC systems and the small signal variation of operation point into consideration. The participation stability of single power electronic based devices infeed sys- factors and sensitivity are analyzed as well. Finally, simulations tems (SPEIS). How to measure the strength and stability of are conducted to demonstrate the rationality and effectiveness of the defined gSCR and OgSCR. MPEIS from the small signal stability point of view is still Index Terms—multi power electronic based devices infeed sys- under challenge. tems; system decoupling; generalized short circuit ratio; small Compared with a single grid-connected device system, there signal stability; operation generalized short circuit ratio exists complex interaction mechanisms not only between grid-connected devices and AC grid but also among I. INTRODUCTION grid-connected devices in multi grid-connected devices system, With fast development of smart grid and urgent demand for which increase the difficulty to analyze it. Fortunately, since flexibility and controllability enhancement of power system, the grid-connected devices are highly similar with multi-infeed power-electronic-based devices (PEDs) are increasing used for system, and analytical method can be applicable for stability the integration of renewable energy generations, VAR com- analysis. Hence, the concept of multi-infeed short circuit ratio pensation devices in ac transmissions systems [1-3]. As a result, is proposed aiming at simplifying multi-infeed system into a large number of PEDs will be accessed to the AC power grid. single-infeed systems and extending the concept of SCR to These PEDs and the AC grid together constitute the multi analyze the multi-infeed systems. For example, CIGRÉ pro- power electronic based devices infeed systems (MPEIS). posed multi-infeed short circuit ratio (MISCR) for multi-infeed The high penetration of PEDs inevitably enlarges the HVDC systems (MIDC) by considering the neighboring equivalent AC grid impedance and weakens the AC grid, which HVDC’s voltage influence [13]. However, MISCR is a makes the interactions between PEDs and AC grid more com- rule-of-thumb extension of SCR and it is lacking of strict theory plex [4-5]. As a result, the risk of oscillation issues arises or basis [14]. Ref. [15] proposed generalized short circuit ratio becomes more critical in a weak AC grid [6-7]. The previous (gSCR) by eigenvalue decomposition from the voltage stability researches show that the characteristics of AC grid play an point of view, which overcome the rule-of-thumb basis of important role in the stability of grid-connected PEDs system: MISCR. Nevertheless, since the small signal stability problems the structure and parameters of AC grid affect the resonance of are more likely to occur in typical MPEISs such as wind plants PEDs filter, and the strength of AC grid affect the interactions and photovoltaic plants [8,16], the gSCR index proposed in [15] among PEDs. For this reason, to quantitatively evaluate the is still not able to evaluate the stability of MPEIS. strength of AC grid and the interactions of PEDs is in urgent In this paper, a static index named generalized short circuit demand. ratio (gSCR) is proposed for the small signal stability evalua- In theoretical research and engineering application, short tion of MPEIS. Firstly, the mathematical model of SPEIS is circuit ratio (SCR) is widely applied to measure the strength of formulated via Jacobian transfer matrix. The relationship be- > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 2

tween the SCR and small signal stability criterion is particularly 0  2 *50 is the nominal angular frequency of AC system, analyzed. Then, based on the dynamic characteristics of multi the subscript 0 denotes the initial value of steady operating infeed system, the MPEIS is decomposed into n independent point and will be omitted for convenience. single infeed systems. In this regard, the concept of gSCR is It can be observed from (1) and (2) that the closed-loop proposed to analyze the small signal stability of MPEIS, which system becomes a multi-input-multi-output (MIMO) feedback can be used to measure the strength of AC systems. Finally, control system shown as Fig. 2 and the characteristic equation derived from gSCR, OgSCR is proposed for multi VSCs infeed is 1 system. The analysis on participation factors and sensitivity to detIJJ2PED _ sss net _ s   0 (4) OgSCR and gSCR is conducted as well. The effectiveness of where I is 2-dimension identity matrix. Substituting (1) the proposed index is validated by simulations on the 2 Matlab/Simulink. and (2) in (4) yields the detailed characteristic equation  GP  s G PU  s c s  det SB   G s G s II. SINGLE-INFEED SHORT CIRCUIT RATIO  Q   QU   The typical SPEIS is shown as Fig. 1, which consists of two s BU22  Q  s BU  P   0 (5) parts, i.e., the PED and the AC grid. P and Q are the real and 22 s BU P  s BU  Q  reactive power output of PED. U and  are the magnitude and B. Relationship of SCR and small signal stability phase of PED terminal voltage. Lg is the Thévenin inductance of AC grid. In SPEIS, the SCR is defined as SU2 11 A. Characteristic equation of SPEIS SCR ac  t   (6) SZSSZBBB P  in UU PP E0    QQ where Sac is the capacity of AC short circuit, ZB1 is the U dc reactance of AC grid and the resistance is neglected to simplify L I LBg   f the analysis. C f PED AC grid Combining (5) and (6), the closed characteristic equation (5)

Fig. 1 Single power electronic based devices infeed systems can be rewritten as the determinant of Jacobian transfer matrixes (7) and the explicit function of SCR (8):

, U PQ,  1 GP  s  Q b G PU s P b J s + PED   cs( ) det  -  G s P G s Q U Q   b QU  b J 1 s net s  SCR  s  SCR  +0  (7) Fig. 2 Equivalent feedback-control diagram s  SCR  s  SCR  The linearized input-output characteristics of PED and AC  grid can be expressed via Jacobian transfer matrixes [17-18], SCR2 a s SCR  b s  0 (8) given as follows: where PPSbB and QQSbB are the power output based on P G s G s   P   PU     the rated capacity of AC system. as  and bs  are given by  =SB    (1) QUU GQ  s G QU  s    1 J s a s= s G s G s PED_ s      2 2 2   P   QU   s   s U J s J s  P P   PU      = (2)       + s GQ  s  G PU  s  QUU JQ  s J QU  s      1 Jnet_ s s b s= G s G s G s G s   2 2 4  P  QU  PU  Q   where S is the rated capacity of PED, subscript s represents  s   s U B   SPEIS. And the negative sign in (2) indicates that J s has  GsGsQ     GsGsPPQ   22  net_ s  QU  P   b  PU  Q   b b b  the opposite power positive direction to JPED_ s s . The transfer Equation (8) reflects the explicit relationship between the functions in are given as in (3), with detailed deriva- small interference stability of the system and the short circuit ratio. Since the stability of the SPEIS increases with the in- tion process shown in Appendix A. crease of SCR [8,16,19], the small signal stability of SPEIS can 2 JP  s=  s BU00 Q be quantitatively evaluated via SCR.  2 Definition1: When SPEIS has two conjugate eigenvalues lo- JPU  s=  s BU00 P  (3) cating at the imaginary axis, the SCR is defined as the critical J s=  s BU2 P  Q     00 short circuit ratio (CSCR).  2 JQU  s=  s BU00 Q If SCR is smaller than CSCR, SPEIS is unstable and vice 1 s  versa. Besides, the stability margin can be determined by the ，， 0 ， where BL 0 g  s= 2  s= 2 difference between SCR and CSCR. s 0  1 s 0  1 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 3

Note that the PEDs are supposed to have a better dynamic represents diagonal matrix diag(a1,a2...an) for convenience. The property in a stronger system in the analysis above, which is transfer functions in Jnet_ m s are given by (10), and the de- satisfied by most kinds of PEDs such as the power electronic tailed derivation process is shown in Appendix A. devices based on PLL and vector control. Since the detailed JMNPi  s  s  diag Q  model and parameters of PED is not used in the derivation (4)  -(8), the analysis above has generality. However, despite SCR JMNPU s  s  diag P i  (10) is only related to the impedance of AC grid, CSCR and the  JMNQi s  s  diag P  relationship between stability margin and SCR is determined JMN s  s  diag Q by the detail model of PED shown as (8).  QU     i  where

III. MULTI-INFEED SHORT CIRCUIT RATIO MUUBij i j ijcos ij  (11) The equivalent circuit of MPEIS is shown as Fig. 3. To NUUBij i j ijsin ij simplify the stability analysis, the resistance is neglected, and here Bij is the elements of node admittance matrix B , Bii  0 . the following assumptions hold in this paper: MPEIS can be also regarded as 2n-dimension MIMO feed- Assumption1: The PEDs are similar, which means that the back control system shown as Fig. 2 and the characteristic control strategy, the parameters and the compensation capaci- equation is tors of PEDs based on their individual rated capacity are same. 1 detIJJss   0 (12) Assumption2: The topology of AC system is connected and n PED__ m net m inductive. The nodal admittance matrix is reversible and Since assumption 3 yields sin(ij ) 1 , N  0 and symmetrical. UUUcos 2 1 . Equation (12) can be further simplified as Assumption3: The power across interconnection lines is i j ij i much less than its limitation.  G s G s diag Q  diag P  det P PU S ii  B  GQ  s G QU  s diag Pii diag Q  PP11   QQ E1 11 ss BB   UU B  detJ sim   0 (13)  11 10 ss BB        PED1  11 PP12 12 Note that the terminal voltage of PEDs U is assumed to be QQ i B 12 12 12 1p.u. in (13). However, this assumption is not necessary and the stability of system when PEDs not at rated operation point will PP22 E2 QQ22 be analyzed later in this paper via operation gSCR.

B20 UU22  B. MPEIS Decoupling    1 PED2  22 PP Multiplying (13) left by IS , (13) is equivalent to B1n 11nn 2B QQ11nn PP22nn  G s G s B P   PU   2n QQ det In 22nn  G s G s  Q QU PPnn

QQnn En sJJ diag Q  s  diag P   eq b eq b   0 (14)  sJJeqdiag P b  s eq  diag Q b  UUnn   Bn0     PEDn nn where In is n-dimension identity matrix, PPSb i Bi , PEDs AC System QQSb i Bi , Jeq is defined as the extended Jacobian matrix 1 Fig. 3 Multi power electronic based devices infeed system JSBeq= B (15) A. Characteristic equation of MPEIS According to assumption 2, all eigenvalues of the matrix Jeq are positive. And the minimum eigenvalue of Jeq is a simple Similar to SPEIS, MPEIS can be also regarded as a combi- eigenvalue, which means its geometric multiplicity and alge- nation of PEDs and AC system, with linearized input-output braic multiplicity is one, and the corresponding eigenvectors characteristics expressed via Jacobian transfer matrixes are positive as well [15]. Thus, there exists a matrix T which JPED_ m s and Jnet_ m s respectively: decomposes the matrix J eq to diagonal matrix sorted as  P GP  s G PU  s   0       by ascending order: = S 12 n  B   QUU GQ  s G QU  s   -1  TJTeq = = diag i  (16) JPED_ m s Combining (15) and (16) yields

P JJP ss PU      G s G s = (9) P   PU       det  In QUUJJQ ss QU    G s G s      Q   QU  

Jnet_ m s s diag Qbb  s   diag P   where  denotes Kronecker product, subscript “m” denotes  0 (17) sdiag P  s  diag Q     bb      MEPIS, SB=diag S Bi  . And in the following article, diag(ai) > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 4

Since all the elements in (17) are identical diagonal matrixes, 4) CgSCR of MPEIS equals the CSCR of the equivalent we can rewrite (17) as follows SPEISs numerically. Moreover, the relationship between gSCR

c12( s ) c ( s )  cn ( s )  0 (18) and stability margin of MPEIS is the same as that of SPEIS. where There are two characteristics of gSCR. One the hand, gSCR is easy to calculate since it relates only to the node admittance G s Q G s P ss    P   b PU  b   ii  matrix B and rated capacity of the PEDs. One the other hand, csi ( ) det + =0 G s P G s Q ss    Q   b QU  b   ii  although gSCR is a static index, it is able to evaluate the small signal stability of MPEIS. In other words, the dynamic part of It can be observed from (5) and (18) that the factors csi () in (18) is the same as cs() in (5), which means the closed char- AC grid (  s and  s ) and PEDs are viewed as a whole shown as the coefficient as and bs in (8), after MPEIS acteristic equation of n-infeed system can be regarded as a     decomposition. Thus, gSCR is only determined by the static part production of n independent SPEISs. The AC system rated capacity of equivalent SPEISs and MPEIS are assumed to be of AC grid. same for convenience, then the equivalent SPEISs follows the It is noticeable that the proposed gSCR index for MPEIS in following equation this paper has the same expression as that for multi-infeed MIDC in [15]. And the gSCR for MIDC is a special case of (PQUBPQSbie____ , bie , ie , ie )(,,1, b b Bii ) (19) gSCR for MPEIS from the eigenvalue point of view when s=0. where the subscript “e” donates equivalent single-infeed sys- However, these two gSCR indexes aim at different issues with tem. The SCR of each equivalent SPEIS is different physical mechanism. The gSCR for MIDC is a static SCR S1  S   (20) i Bi Bi i i index aiming at evaluate the static stability of system, which As analyzed above, the dynamic characteristics of MPEIS defined from the static voltage stability point of view. And the with n PEDs can be represented by n independent SPEISs, gSCR for MPEIS is a static index aiming at evaluate the dy- namely the n-infeed system can be decomposed into n inde- namic stability of system, which defined from the small signal pendent single infeed systems and the eigenvalues of MPEIS stability point of view. can be obtained by calculating the eigenvalues of n equivalent SPEIS. Moreover, it can be observed from (19) that the equiv- D. Verification of Simplification in MPEIS Decoupling alent SPEISs have the same parameters and operation condition In this subsection, the influence of neglecting the elements except AC grid admittance. Therefore, the stability analysis of related to N in the simplification from (12) to (13) on eigen- MPEIS can be transformed to the stability analysis of n iden- value calculation is discussed. The neglected part is given by tical PEDs connected to different ac grid with different SCR. ss NN  N neg   (22) C. Generalized short circuit ratio (gSCR) ss NN  Since the stability of MPEIS can be obtain by analyzing Since the simplified MPEIS described by (13) is decom- equivalent SPEISs, the necessary and sufficient conditions for posed into n equivalent SPEISs, the simplified MPEIS totally the MPEIS to be stable is that all the equivalent SPEISs are have n(m+2) eigenvalues, where m and m+2 is the number of stable. Therefore, the small signal stability of MPEIS is up to state variables in a PED and a SPEIS respectively. Compared the weakest equivalent SPEIS, namely the SIPES correspond- with n equivalent SPEISs, the original MPEIS described by (12) ing to the smallest eigenvalue of J eq . If the equivalent SIPES has the transmission lines between the terminal of each PED. with the smallest SCR is stable, the MPEIS is stable and vice Thus, the original MPEIS has n(m+2)+n(n-1) eigenvalues, versa. Therefore, the gSCR for MPEIS is defined as follows. where n(n-1) is the number of state variables transmission lines Definition2: The minimum eigenvalue of extended Jacobian between each PED. That is to say, the simplified MPEIS loses matrix Jeq is defined as the generalized short circuit ratio, as n(n-1) eigenvalues after neglecting N neg . However, are mainly related to the dynamic of inductances. Thus, the lost gSCR  minJeq  (21) eigenvalues are mainly determined by the dynamic of induct- It can be seen from aforementioned analysis that SPEIS is a ances, which have little influence on the stability analysis of special MPEIS which dimension is 1 and the J of SPEIS is eq MPEIS. 1 SB1 11 . Hence, the properties of SCR can be extended to On the other hand, since the diagonal elements of are gSCR: all 0 and sinij 1, the following equation is satisfied 1) The gSCR is called critical gSCR (CgSCR) when MPEIS NJ neg sim ss (23) has two conjugate eigenvalues locating at the imaginary axis, ss i F i F which is used to differentiate unstable systems from stable where si is the eigenvalues of the simplified MPEIS. It can be systems. observed from (23) that the Frobenius norm of is much

2) The strength of AC grid and the stability of MPEIS in- less than that of Jsim , which means has little influence on crease with gSCR. the remaining eigenvalues of original MPEIS. In summary, the 3) If gSCR is less than CgSCR, MPEIS is unstable, and vice influence of neglecting on stability analysis is negligible. versa. The stability margin can be evaluated by the difference between gSCR and CgSCR. IV. OPERATION GSCR FOR GRID-CONNENCTED VSC As aforementioned discussion on gSCR, the power injection and terminal voltage of PEDs are assumed to be the same based > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 5

on their individual capacity, which may not resemble the prac- Similar to the derivation (16)-(18), there exists a matrix To tical system. In this section, the operation gSCR (OgSCR) is which decomposes the Jeqo sorted as 0 o12   o    on by deduced for the grid-connected VSCs to measure the strength ascending order. Then (26) is simplified as of multi VSCs infeed power systems (MVIS) when the VSCs c( s ) c ( s )  c ( s )  0 (28) operate at an arbitrary operation point. o12 o on where

GP  s G PU  s ss oi    oi csoi ( ) : det + 0 G s G s ss    Q   QU     oi  oi It can be observed from (28) that the MVIS can be decomposed into n equivalent independent single VSC infeed power

systems (SVIS) which characteristic equations are csoi (). As- suming the equivalent single-infeed systems are all working at rated operation point for convenience, then the parameters of SVIPS satisfies the following equation:

(Pbiebie_____ , Q , U ieBieie , S , B , SCR i ) (1,0,1,1, oioi , ) (29) Similar to the analysis of gSCR, the small signal stability of MVIS is up to the weakest equivalent SVIS, namely the SVIS

corresponding to oi . And the operation gSCR (OgSCR) is defined as follows: Definition3: The minimum eigenvalue of Jeqo is defined as the operation generalized short circuit ratio, as

Fig. 4 Block diagram of the a single VSC infeed system OgSCR  minJeqo  (30)

1 A. Operation gSCR for Grid-connected VSCs It can be seen from (15) and (27) that JJeqo diag P bi U i eq , The VSC control system used in this paper can be referred to which means that gSCR is a special case of OgSCR when [20], which is designed as a PLL based dual-loop vector con- 1 diag Pbi U i   I . In terms of the physical meaning, OgSCR is troller as shown in Fig. 4. The outer control loop can be designed as either output power controller or the dc voltage con- the extension of gSCR by taking the output power and terminal troller. Besides, the power factor of the converter is assumed to voltage into consideration. The OgSCR can be usedto reflect be closed to 1. the influence of operation conditions of PEDs on the strength Based on the small signal model that developed in [20], the and stability of system. Besides, it is noticeable that OgSCR is Jacobian transfer matrixes of two different types of VSCs can no longer applicable when the Jacobian transfer matrix of PED be derived and they all have the form as (24). The derivation does not meet the form as (24). However, gSCR does not have process and the expression of J s are shown in Appendix B this restriction. VSC The proposed OgSCR has the following properties on the UGoP  s UG oPU  s 1 analogy of gSCR: JJPED_ ss  S B VSC s S B P b  (24) UGoQ  s 1 UG oQU  s 1) Critical OgSCR (COgSCR) is defined on the condition where PPS is the power injection based on the rated when MVIS has two conjugate eigenvalues locating at the b i B imaginary axis. MVIS is unstable when OgSCR is less than capacity of VSC. COgSCR, and vice versa. Besides, the difference between Combining (9), (24) and considering assumption 3, the closed characteristic equation of MVIS can be expressed as: gSCR and CgSCR represents stability margin. 2) COgSCR equals the CSCR. The relationship between  G s G s 1 oP   oPU    OgSCR and stability margin of MVIS is the same as that of det diag Ui PSPS b B    b B   G s G s   oQ   oQU   1 SVIS.

s MM  s   diag Qii  diag P    B. Participation factors and sensitivity analysis        0 (25) sMM s  diag P  diag Q        ii     Based on the aforementioned analysis of OgSCR, MVIS can where P  diag P  , PS diag P  and diag Q   0 . Multi- be decomposed into n equivalent SVIS, and each SVIS is a b bi b i B i combination of the dynamic each PED on different scales. The 11 1 plying (25) left by IPS2  bB and right by I2  diag Ui  participation factor of the nth PED to the mth equivalent SVISs yields can be indicated by pom, n v omn u onm (31) GoP  s G oPU s   sJJ eqo s eqo  det In     0 (26) G s G s sJJ s where vo and uo are left and right eigenvector of Jeqo respec- oQ   oQU     eqo  eqo  where tively. Since B is a symmetric matrix, the equation 1 T 1  1  1  1  1  1 diag Pi Ui vuoo (31) is equal to Jeqo P b S B Mdiag U i diag P bi S Bi U i B = diag P i U i  B (27)   12 pm, n v omn u onm P m U om u onm (32) > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 6

Since the first equivalent SVIS, namely the SVIS corre- teristic equation (8), the relationship between SCR and the sponding to 1 , has the weakest eigenvalue of MVIS, the par- damping ratio of SVIS is obtained and plotted in Fig. 5. ticipation factors of each VSC to SVIS1 reflects the participa- Table 1. Parameters of VSC tion degree of weakest eigenvalue of MVIS, which can be used Symbol Description Value to identify strongly correlated generators of weak damped S Base value of AC system 1500kVA modes. UB Base value of voltage 690V Lf,Cf Inductance and capacitance of convert 0.05pu,0.05pu Moreover, the sensitivity of rate capacity, output power and Hdc(s) Transfer function of the dc voltage controller 0.2+200/s system admittance to OgSCR can be written as Hi(s) Transfer function of the current controller 0.6+15/s H (s) Transfer function of the PLL 2+3020/s diag P11 S U pll OgSCR Jeqo  bi Bi i   vo u o v o B uo SSSBi  Bi  Bi (33) Pbi 2  OgSCRuoi 0 Ui

OgSCR Jeqo SBi 2 voou  OgSCRuoi  0 (34) PUbiPbi i 1 OgSCR diag Pii U  B B vuuT u BBBo  o o  o ij ij ij (35)  2 (uo11 i u o j )  0 i  j j  0   Fig. 5 The Relationship between Stability Margin and SCR uo11 i u o j 0 i  j j  0 Sensitivity indicates the direction of improving system small It can be seen from Fig. 5 that the damping ratio increases signal stability and we can draw conclusions that: increasing with SCR and the SVIS has two conjugate imaginary eigen- system admittance, decreasing the capacity and the output values when SCR=2.86, i.e., CSCR is 2.86. Therefore, the power of PED can increase OgSCR, namely, improve system SVIS is stable only if SCR is larger than 2.86. Since there is a voltage stability. positive correlation between damping ratio of the weakest Moreover, it can be seen from that (32)-(35) that participa- eigenvalue of SIVS and SCR, the difference between SCR and tion factors and sensitivity are easy to calculate, since their CSCR can evaluate the stability margin, which met the prop- expressions are only related to output power, admittance matrix, erties of SCR analyzed above. rated capacity and the right eigenvectors of Jeqo . B. MVIS decoupling and OgSCR verification Similarly, the participation factors and sensitivity expressions of gSCR can be derived as, The five VSCs infeed system is constructed with the pa- 2 rameters given in Table 2 and Table 3. p v u S u (36) m, n mn nm Bi nm Table 2. Parameters of AC grid gSCR 2 Admittance Value(p.u.) Admittance Value(p.u.) Admittance Value(p.u.)  gSCRui  0 (37) SBi B10 0.2 B20 0.15 B34 0.07 B 0.15 B 0.18 B 0.05  2 12 23 35 gSCR (u11ij u )  0 i  j j  0 B13 0.1 B24 0.2 B40 0.1  (38)  B14 0.06 B25 0.21 B45 0.11 Bij u u0 i  j j  0  11ij B15 0.09 B30 0.25 B50 0.2 where v and u are left and right eigenvector of Jeq . The con- Table 3. Parameters of VSCs clusions drawn from the sensitivity expressions is similar as that from OgSCR, namely, increasing system admittance and VSC1 VSC2 VSC3 VSC4 VSC5 decreasing capacity of PED can improve system voltage sta- Pbi 0.8 0.7 0.9 1 0.5 S 1.5 2 1 1.8 1.5 bility. Bi

Table 4. The relationship between equivalent SVISs and the eigenvalue of Jeqo V. SIMULATION VALIDATION SVIS 1 2 3 4 5

In this section, case studies are carried out on a single VSC λi 6.0944 24.9627 46.6669 56.6602 94.3062 infeed system and a five VSCs infeed system to verify the Firstly, decomposing the five VSCs infeed system into five effectiveness of multi-infeed system decoupling as well as the SIVSs, and the corresponding eigenvalue of Jeqo is given in proposed OgSCR index. Since the gSCR is a special case of Table 4. The eigenvalues directly calculated by five-infeed OgSCR, the verification of OgSCR also validates the effec- system and those of the five equivalent SVISs are shown in Fig. tiveness of gSCR. The PED model used in simulation is the 6. It can be seen that the eigenvalues calculated by MVIS SVISs VSC with dc-voltage control shown as Fig. 4 and the parame- are almost the same. Therefore, the equivalent SVISs can ex- ters are listed in Table 1. actly represent the stability of MVIS, which verifies the rationality of multi-infeed system decoupling. A. Analysis of SCR and CSCR Moreover, it can be seen from Table 4 shows that OgSCR is By changing the admittance of AC grid and solving charac- 6.0944 since the smallest eigenvalue of Jeq is 1 . From Fig. 5 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 7 it can be observed that OgSCR satisfies OgSCR> COgSCR = Moreover, Fig. 7 gives the trajectories of the OgSCR when CSCR=2.86 and the damping ratio of the weakest eigenvalue is the output power P b1 of VSC1 or line admittance B10 varies. It 0.0059, which means MPEIS is stable. On the other hand, t the can be seen clearly that OgSCR decrease when PED output weakest eigenvalue is -0.3132 +53.3411i and the damping ratio power increase or line admittance decrease, which meet the is 0.0059, which is the same as the conclusion obtained by conclusion of sensitivity analysis. OgSCR. This means the stability assessment from gSCR is effective.

Fig. 7 OgSCR variation with power and admittance

VI. CONCLUSION Based on multi-infeed system decoupling method, the concept of gSCR is proposed to analyze the small signal stability of multi infeed system, which can be further used to judge the strength of AC systems. The gSCR is defined from the point of view of small signal stability, which physically and mathematically unify the concept of SCR in single and multi-infeed system. The proposed gSCR is a static index and easy to calculate using the proposed method, which provide theoretical support for the analysis of multi identical devices infeed system. Moreover, the OgSCR are proposed as the extension and application of gSCR. Simulation results validate the effectiveness of OgSCR and gSCR. Further work will be extended to ex- ploring the interaction of nonidentical PEDs and the corresponding stability evaluation method.

Appendix A. Derivation of J s and J s Fig. 6 Eigenvalues comparison between MEPIS and equivalent SIPESs net_ s   net_ m   C. Participation factors and sensitivity analysis Since the AC grid in SPEIS is the special case of AC grid in MPEIS when n=1, we only derive here and J s To verify the effectiveness of participation factors on net_ s   strongly correlated generators identification, the participation equals to when n=1. factors of each VSC to the weakest equivalent SVIS and the The dynamic equations of inductance between node i and j is participation factors of state variables in MVIS to the weakest shown as follows: eigenvalue (-0.3132 +53.3411) are calculated and given in Uix UIjx sLij0 L ij  ijx  Table 5.     (39) U  L sL Table 5. Participation factors to the weakest eigenvalue of MVIS iy UIjy 0 ij ij  ijy  Participation factors Participation factor to VSC State Variables PUIUIij ix ix iy iy to weakest SVIS the weakest eigenvalue  (40) PLL integrator 1 QUIUI   4 1  ij ix iy iy ix PLL PI 0.9991 PLL integrator 0.8689 P 2 0.8730  ij PLL PI 0.8681 Pi j0, j i 0  =  (41) PLL integrator 0.6935 Q 0 1 0.6948 i  Qij  PLL PI 0.6928 j0, j i PLL integrator 0.4810 3 0.4810 I PLL PI 0.4806 where is the output current of PED, subscribe “x” and “y” PLL integrator 0.3028 represent the x-axis component and y-axis component in syn- 5 0.3029 PLL PI 0.3025 chronous rotating coordinate respectively. It can be seen from Table 5 that the participation factors of The transformation of voltage form rectangular coordinate each VSC to the weakest SVIS and the weakest eigenvalue are form into polar form is expressed as: almost the same, which means the participation factors of each UUix= i cos i VSC to the weakest SVIS can reflect the participation degrees  (42) UU= sin of each VSC weakest eigenvalue and are able to be used for  ix i i strongly correlated generators identification. In addition, the following equations are satisfied at steady > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 8

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