> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1 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 detIJJ2PED _ sss 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 E0 QQ where Sac is the capacity of AC short circuit, ZB1 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 ma- trixes (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).