Generalized Discrete-Time Equivalent Model for the Dynamic Simulation of Regional Power Grids

Transcription

1 eneralize Discrete-Time Euivalent Moel for the Dynamic Simulation of Regional Power ris Fu Shen Stuent Member IEEE Ping Ju Senior Member IEEE Mohamma Shahiehpour Fellow IEEE Zhiyi Li Member IEEE an ueping Pan Abstract The introuction of an euivalent moel for regional power gris in a large-scale power system with complex loas is essential for reucing the computation buren in real-time ynamic analyses. In this paper we propose a generalize iscrete-time euivalent moel (DEM) for simulating the physical characteristics of a regional power gri. DEM facilitates the interconnection of euivalent moels for representing regional power gris an improving the accuracy an spee in ynamic simulations of large-scale power systems. The paper first investigates the inherent relationships among coefficients in the iscrete-time moels of synchronous generators an composite loas so as to guie the estimation of coefficients of the DEM for regional power gris. The paper then evelops relationships among coefficients associate with DEM for a regional power gri. The DEM that is forme by combining iscrete-time moels of synchronous generators an composite loas represents specific ynamic characteristics of regional power gris. Numerical experiments are conucte by simulating groun faults in the China Electric Power Research Institute system an the accuracy of the propose DEM is verifie by analyzing the simulation results. In aition the paper has applie DEM to stuy the regional power gri of central China which valiates the use of DEM in practical power system analyses. Inex Terms Large-scale power system operation iscretetime euivalent moel for regional power gris regional moel. Inices i K Symbols Parameters E U I U I I U U E f I gr I g I gr I g U gr U g I cr I c NOMENCLATURE Inex for coefficients Inex for iscrete time steps Incremental value Subscript for steay state Transient voltage Terminal voltage Terminal current Amplitue of the bus voltage Currents of -axis an -axis Voltage of -axis an -axis The terminal voltage of excitation Real an imaginary currents of generator Real an imaginary voltages of generator Real an imaginary currents of composite loa This work was supporte in part by the proect of Renewable Energy an Smart ri (B4) the National Key Basic Research Program of China (973 Program) (3CB84) an the 6 Orinary University rauate Stuent Scientific Research Innovation Proect of Jiangsu Province (6B4434). F. Shen P. Ju an. Pan are with the School of Energy an Electrical Engineering Hohai University Naning China (shenfu@hhu.eu.cn; pu@hhu.eu.cn; xueping_pan@63.com). Z. Li an M. Shahiehpour are with the Electrical an Computer Engineering Department Illinois Institute of Technology Chicago USA (zhiyi.li@hawk.iit.eu; ms@iit.eu). I r I Real an imaginary currents flowing into the regional power gri Bus voltage phase angle T T Short-circuit transient an sub-transient time constants of -axis f x f y Variables in x-y coorinates f f Variables in - coorinates T T Open-circuit transient an sub-transient time constants of -axis T D Short-circuit transient time constant of D- branch T D Open-circuit transient time constant of D- branch T Short-circuit sub-transient time constant of - axis T Open-circuit sub-transient time constant of - axis D Damping coefficient Angle of the generator Angular velocity of rotor R s R r Resistance of stator an rotor s r Reactance of the stator an rotor Steay state reactance Transient reactance m Exciting reactance x ff x ff Euivalent excitation reactance of -axis an -axis x x Positive seuence euivalent reactance of - axis x a Reactance of armature reaction of -axis x Reactance of -axis x x Transient an sub-transient reactance of - axis R f R f Euivalent excitation resistance of -axis an -axis R R Positive seuence euivalent resistance of - axis an -axis Flux linkages of -axis an -axis u f Excitation wining voltage h Sampling time step Variables (s) (s) Inuctances of -axis an -axis gi an gi Both are the iscrete-time moel coefficient of the synchronous generator ci Discrete-time moel coefficient of composite loa i Discrete-time moel coefficient of regional power gri s Laplace transformation z Z transformation Other notations are efine in the text.

2 L I. INTRODUCTION ARE power gris are often operate through coorinate controls of a hierarchy of regional power gris. Iniviual regional power gris euippe with energy management systems (EMSs) establish euivalent moels for their external regional systems with numerous challenging tasks for ata exchanges among regional power system moels [] [3] which has consistently threatene the stability of power systems [4][5]. Recently several stuies have propose euivalent moels for regional power systems [6]-[7]. The euivalent moels that were wiely aopte for maintaining the ynamic characteristics of regional power gris represente three categories of coherent-base moels [8][9] analytical moels [][] an estimate euivalent moels [][3]. In most cases an aggregate generator was use to represent multiple generators in the euivalent moel of a regional power gri an an euivalent loa was aopte to represent a multitue of loas [4] [6]. With the aitional uncertainty introuce in power system operations an the increasing penetration of renewable energy resources into regional power gris the attainment of an euivalent moel for a regional power gri has become more cumbersome [7][8] leaing to strong nonlinearities in regional power gris [9]. In [] a seventh-orer nonlinear uasi-state space moel was erive for an active istribution network. In [] a secon-orer transfer function was use as a ynamic euivalent moel for a istribution network. In [] the artificial neural network (ANN) metho was introuce to represent a moel for a regional power gri. However the euivalent moel epene on the operating state of the regional power gri for estimating ANN weights. It is ifficult to obtain an euivalent representation of regional power gris for ynamic stability analyses as power systems are inherently nonlinear an analytical moels are generally lacking for euivalent representations of such nonlinear euations [3]. Ref. [4] propose a semi-implicit formulation of ifferential-algebraic euations (DAEs) escribing power system moels for transient stability analyses which reuce computation burens an increase the sparsity of the Jacobian matrix of the power system. Ref. [5] escribe the Power System Analysis Toolbox (PSAT) an open software package for analysis an esign of small to meium size electric power systems. A regional power gri euivalent was often represente by a static loa which was a conservative moel for emboying the power system operation in critical conitions such as the northeast blackout of 3 in the Unite States. Therefore [6] propose a large power systems euivalent represente by an aggregate generator which characterizes coherent combinations of strongly connecte machines in an area after the northeast blackout; however the aggregate generator i not emboy all typical types of loas. The power system moel represents the integration of smaller moels for regional power gris to aress the operation of the integrate power system that cannot be represente by iniviual smaller moels. The power system moel integration incluing a hierarchy of regional power gris are interconnecte with external regional moels for ynamic simulations. Ref. [7] presente the characteristic of a complex high-orer continuous system base on an all-coefficient aaptive control metho applie to the integrate power system moel. Yet regional power gris cannot strictly satisfy the reuire conition that the external euivalent moel parameters be time-inepenent. An euivalent moel that is universally applicable to regional power gris with iverse compositions is conucive to the assembly of the moels of power system regions for real-time analyses that guie the secure operation of the whole power system. Dynamic components in a regional power gri usually inclue generators an inuction motors. Accoringly the generalize iscrete-time moel of a regional power gri can be erive base on iscrete-time moels of synchronous generators an composite loas incluing various types of loas. This paper propose DEM for a regional power gri base on iscrete-time moels of synchronous generators an composite loas consiste by inuction motor loa an static loa as shown in Fig. which is regional power gri. Fig. (a) is original regional power gri an Fig. (b) is regional power gri representation where is a generator M is inuction motor loa incluing the typical type of loa. DEM consiers the terminal voltage of the bounary bus an the inecte current by the external gri as input an output state variables in each regional power system respectively which is ifferent from the previous methos for eveloping the power system ynamic euivalence (e.g. [8] [3]). Therefore the practicability of DEM is on representing regional power gris which consist of ynamic components (e.g. synchronous generators inuction motors) or static components (e.g. lights). BUS BUS BUS4 BUS4 BUS6 BUS8 BUS9 BUS5 BUS3 BUS7 BUS3 BUS8 BUS9 BUS5 BUS34 BUS33 M Inuction Static Loa Motor Loa Composite Loa enerator (a) (b) Fig.. Regional power gri (a) Original regional power gri (b) Regional power gri representation The contributions of this paper are liste as follows: ) The paper has propose DEM for a regional power gri which can be erive base on iscrete-time moels of iniviual synchronous generators (or an euivalent aggregate generator) an iniviual composite loas (or an euivalent aggregate composite loas). ) DEM investigates the inherent relationship among coefficients of the iscrete-time moels of synchronous generators an composite loas so as to guie the estimation of coefficients for representing those components in regional power gris. 3) The paper emonstrates analytically as well as via simulations that the sum of output state coefficients in the regional power gri moel is approximately eual to an the sum of input state coefficients is approximately eual to in DEM. On the one han the relationship among coefficients coul provie a theoretical basis for valiating DEM

3 3 parameters; on the other han the relationship among coefficients reuces the number parameters to be ientifie. 4)The paper stuies the regional power gri of central China using DEM which valiates the use of DEM in practical applications. The broaer applications of DEM are also iscusse in the paper. The remainer of this paper is organize as follows. First iscrete-time moels are erive for a synchronous generator an a composite loa in Sections II an III respectively. Then Section IV presents a DEM of a regional power gri base on the iscrete-time moels of generators an composite loas through the weighte sum metho. The moeling steps for DEM an case stuies are consiere in Section V to verify the accuracy of the propose moel. Finally Section VI conclues this paper. I gr ( s) a a U gr ( s) I g ( s) a a U g ( s) where a sin ( s) ( s) a a a cos ( s) ( s) ( s) cos ( s) ( s) ( s) sin. ( s) ( s) y (7) II. DISCRETE-TIME MODEL OF A SYNCHRONOUS ENERATOR Assume that there exists an euivalent wining in both -axis an -axis of a synchronous generator where flux linkages of -axis an -axis are expresse as [8][9] ( s) u f ( s) i () () s i Here (s) is the stator to fiel voltage transfer function [9]. (s) an (s) are expresse by T T s ( T D T ) s () s T T s ( T D T ) s () T s () s T s The armature resistance R is generally small enough to be ignore. The impeances are state in (3) when the automatic voltage regulator (AVR) is applie ( s) U ( s) ( s) E f ( s) I( s) I( s) (3) ( s) U ( s) I () s Accoringly (3) is rewritten as ( s) E f ( s) ( ) () s I s U ( s) () s I ( s) U ( s) (4) () s Quantities in coorinates are transforme to those in xy coorinates given in the Fig.. Using -xy coorinates in Fig. we can get ( ) f f ( f f ) e (5) x y Through the transformation we rewrite the (5) as f sin cos x f f cos sin y f When the transformation matrix applie to (4) we have sin cos (6) cos sin in (6) is f f y Fig.. -xy coorinates As the moel accuracy suffers an the number of parameters is increase when bus voltage phase angles are explicitly expresse in (7). Accoringly the bus voltage phase angle is ignore for facilitating the moel erivation an (7) is simplifie as I gr ( s) sin U ( s) ( s) ( s) cos (8) ( s) ( s) I g ( s) U ( s) ( s) ( s) ( s) The transfer functions relating the generator s current to its voltage are erive as 3 I gr () s agrs bgrs cgrs gr 3 (9) U() s e s f s g s h gr gr gr gr I () s a s b s c s U() s e s f s g s h 3 g g g g g 3 g g g g f x f x () where etaile expressions of parameters are presente in Appenix A. The bilinear transformation is a nonlinear mapping metho that compresses the infinite freuency range to a finite one to avoi the spectrum aliasing cause by the continuous-iscrete transformation [5]. Accoring to the bilinear transformation applie to (9) an () we have I ( k 3) I ( k ) I ( k ) I ( k) gr g gr g gr g3 gr U ( k 3) U ( k ) U ( k ) U ( k) g 4 g5 g6 g7 I ( k 3) I ( k ) I ( k ) I ( k) g g8 g g9 g g g U ( k 3) U ( k ) U ( k ) U( k) g g g3 g4 () ()

4 4 where etaile expressions of all parameters are presente in Appenix B. The real-part coefficients g g g3 g4 g5 g6 g7 an the imaginary-part coefficients g8 g9 g g g g3 g4 in () an () respectively are relate to time constants T T T T T D T D T T an the sampling time h (where z s h z ) in the bilinear transformation. In turn the time constants are epenent on reactance x ff x ff x x x a x x x an resistances R f R f R R. Hence the iscrete-time moel of a synchronous generator is epenent on its physical characteristics. The following relationships are observe in () an () when sampling time h is small 3 7hgr h ggrh 4 fgrh 8egr g g g3 3 (3) h h g h 4 f h 8e gr gr gr gr 8 h 3 gr g 4 g5 g6 g7 3 hgrh ggrh 4 fgrh 8egr 3 7hgh ggh 4 fgh 8eg g8 g9 g 3 hgh ggh 4 fgh 8eg 8 h 3 g g g g3 g4 3 hgh ggh 4 fgh 8eg (4) (5) (6) It is shown here that the sum of output state coefficients of a synchronous generator is approximately eual to an the sum of input state coefficients is approximately eual to. When the variation of bus voltage phase angle is consiere the transfer functions relating the generator current to its voltage magnitue an phase angle are erive as I ( s) ( s) U ( s) ( s) ( s) (7) gr gr gr I ( s) ( s) U ( s) ( s) ( s) (8) g g g where gr(s) an gr(s) are the transfer functions of I gr with respect to U an. g(s) an g(s) are transfer functions of I g with respect to U an. Accoring to the bilinear transformation applie to (7) an (8) we have I ( k 3) I ( k ) I ( k ) I ( k) gr g gr g gr g 3 gr U ( k 3) U ( k ) U ( k ) U ( k)+ g 4 g5 g 6 g 7 ( k 3) ( k ) ( k ) ( k) g g g 3 g 4 I ( k 3) I ( k ) I ( k ) I ( k) g g8 g g9 g g g U ( k 3) U ( k ) U ( k ) U ( k) g g g3 g4 ( k 3) ( k ) ( k ) ( k) g 5 g 6 g 7 g8 (9) () The etaile expressions are not state here. However when the bus voltage phase angle is ignore in () an () we fin that the number of moel parameters is 4. When the bus voltage phase angle is consiere in (9) an () we fin that the number of moel parameters increases to which can be a computation buren. Corresponingly the voltage phase angle variations are ignore in our stuy in orer to facilitate the erivation an the simulation of the propose moel. The etaile analyses are provie in Section V. III. DISCRETE-TIME MODEL OF A COMPOSITE LOAD A composite loa consists of a static loa an an inuction motor connecte in parallel [9] [3] which is expresse as E T E I ET () t where E U R I ( U E) Rs s sm ( s m) T ( ) R. r m r When the terminal voltage an the inecte current of the composite loa are consiere as input an output variables respectively () is transforme as [33] Icr U Ar Icr Br Ic CrU t t () Ic U AIcr BIc C U B t t A B T Br / T where r / C T B r / A / T / C B / T B B T Rs / ( Rs ) B / ( Rs ). The freuency omain representation of () is obtaine by applying the Laplace transformation to () sicr ( s) Ar Icr ( s) BrI c ( s) CrU ( s) su ( s) s Ic ( s) A Icr ( s) B Ic ( s) C U ( s) sb (3) U ( s) Accoringly transfer functions relating real an imaginary parts of inecte current to terminal voltage is obtaine as Icr () s BrC sbr B scr s BCr sb (4) U () s s sb sa A B B A r r r I () s A C sa sc s B A C sba c r r r () r r r U s s sa sb A B A B s m (5) The following ifference euations are obtaine by applying the bilinear transformation to (8) an (9) Icr ( k ) c I cr ( k ) cicr ( k) (6) U ( k ) U ( k ) U ( k) c3 c4 c5 I ( k ) I ( k ) I ( k) c c6 c c7 c (7) c8u ( k ) c9u ( k ) c U ( k) where etaile expressions of all parameters are presente in Appenix C. The real-part coefficients c c c3 c4 c5 an the imaginary-part coefficients c6 c7 c8 c9 c in (6) an (7) are relate to the composition of the composite loa as well as the sampling time h. Hence the iscrete-time moel of a composite loa is epenent on its physical characteristics. The following relationships among the coefficients in (6) an (7) are erive when h is small enough 3 h( Br A Ar B ) h( Ar B )+4 c c (8) h ( A B B A ) h( A B ) 4 r r r 4 h ( BrC BCr ) c3 c4 c5 h ( A B B A ) h( A B ) 4 r r r 3 h( Br A Ar B ) h( Ar B)+4 c6 c7 h ( Ar B Br A ) h( Ar B) 4 4 h ( ACr Ar C ) c8 c9 c h ( A B B A ) h( A B ) 4 r r r (9) (3) (3) We conclue that the sum of output state coefficients of a composite loa is approximately eual to an the sum of

5 5 input state coefficients is approximately eual to which coul provie a theoretical basis for valiating the DEM parameters. IV. DEM OF REIONAL POWER RID The aggregate regional power gri is illustrate in Fig. 3 in which the synchronous generator moel is expresse by () an () an the composite loa moel is expresse by (6) an (7). The iscrete-time moel of the regional power gri is euce by introucing the terminal voltage of the bounary bus an the currents inecte by the external power gri as input an output state variables respectively. İc =Icr+Ic Static Loa CP M Inuction Motor Loa Composite Loa Fig. 3. Regional power gri İ =Ir+I enerator U İg =Igr+Ig In Fig. 3 the current flow into the coupling point (CP) is expresse as I I c I g (3) where is the current flow into the composite loa an is the current flow into the synchronous generator. The incremental form of (3) is expresse as I I r I Ic Ig (33) We employ weighting factors to coorinate synchronous generator an composite loa characteristics in the iscretetime moel of a regional power gri consiering by concluing that at steay state there exists a certain correlation between real an imaginary parts of the currents flowing into the synchronous generator an the composite loa. Assume that a proportion factor K r exists between the real part of the synchronous generator current an that of the composite loa while another proportion factor K woul apply to imaginary parts which are state as Icr KrI gr (34) Ic K I g Consiering () () (6) (7) (33) an (34) the CP current in Fig. 3 is expresse as Ir ( k ) Ir ( k ) Ir( k 3) 3 I r( k) I ( k 3) 8 9 I ( k ) I ( k ) I ( k) (35) Uk ( 3) Uk ( ) 3 4 Uk ( ) Uk ( ) I c I g c Krg c Krg where K K 5 g5 c4 K 9 K c7 g9 3 g3 c r 6 g6 c5 g 4 g 4. r 7 g 7 g c8 3 g 3 c6 K g 8 8 K 4 g4 c3 g c9 Base on the relationships among coefficients in (3) (6) (8) (3) an (33)-(34) when h is small the following relationships among coefficients in (9) are obtaine: c c Kr ( g g g3) g3 3 (36) K ( c3 c4 c5) ( g4 g5 g6 g7) (37) c6 c7 Kr( g8 g9 g) g 8 9 (38) K 3 4 ( c8 c9 c ) ( g g g3 g4) (39) Accoringly we raw the following conclusions for DEM: ) The regional power gri moel can be expresse similar to that of a synchronous generator or a composite loa. ) The coefficients of euivalent components in the regional power gri moel are functions of h. 3) When h is small the sum of output state coefficients in the regional power gri moel is approximately eual to an the sum of input state coefficients is approximately eual to which coul provie a theoretical basis for valiating the estimate DEM parameters. We isregare the effect of power network in our regional moel erivation. Ref. [34] propose a ynamic power system euivalent moel using power transfer istribution factors. Accoringly the moel in Fig. 4 represents a regional power gri consiering the regional power gri network. In Fig. 4 z fm an z fg are composite loa an synchronous machine impeances respectively in the network amittance matrix. When the regional power gri network is consiere the DEM imension will be increase. M Inuction Static Loa Motor Loa Composite Loa enerator Fig. 4. Regional power gri representation consiering power gri network z fm We can estimate the DEM parameters using the least suares estimation metho [35][36]. Here (35) is represente as Y where an Y are the DEM input an output signals respectively measure in the regional power gri terminal bus an represents the DEM parameters. Accoringly r r z fg (4)

6 6 [ I ( k ) I ( k ) I ( k) I ( k ) I ( k ) I ( k) r r r U ( k 3) U ( k ) U ( k ) U( k) ] Ir ( k3) Y I ( k3) To estimate the parameter we apply the least suares metho in which the resiual J is minimize as min J = ˆ Y (4) where ˆ When values Ŷ are the estimate DEM parameters. J ˆ the estimate parameters T ˆ an output are attaine as ˆ T ( ) T Y (4) (43) Ŷ ˆ where Ŷ is the estimate DEM output. Accoring to (4) an (43) when we get the values of the an Y from the regional power gri the estimate parameters an estimate output Ŷ can be calculate. As DEM is in the incremental form when estimate output Ŷ is known Ir an I in the regional power gri will be calculate by aing steay state values of Ir an I to the estimate values. ˆ V. SIMULATION AND VERIFICATION In this section two regional power gris are simulate. The results verify the accuracy an effectiveness of the propose euivalent moel for representing the regional power gris. All simulations are conucte on a computer running a 64-bit Winows with a 3.6 Hz Intel (R) Core (TM) i7-77 CPU an 8B memory. The version 7. of the Power System Analysis Software Package (PSASP) software an the version R6a of the Matlab software are use for simulation. When DEM is applie to a simple regional power gri with composite loa for verifying the DEM moel the steps for eriving DEM parameters inclue: ) The power system moel structure with parameters for the regional power gri are presente. ) DEM parameters are obtaine relationships among DEM parameters are verifie an aaptability an ynamic characteristic of DEM are examine through erivation an estimation methos. Case : Regional power gri is represente by a composite loa For a certain regional power gri the CEPRI test system as illustrate in Fig. 5 is consiere for simulation in which a composite loa is connecte to BUS5 in the regional power gri. To verify the feasibility of the propose moel typical values are utilize for representing the composite loa: R s= R r=. s=.8 r=. m=3.5 an h=.. A single-phase-togroun fault is applie to the transmission line locate between BUS6 an BUS9 at t=s. The fault is cleare at t=.s an the original steay state is restore. Then the DEM of a composite loa is estimate base on (6) an (7) as: Icr ( k ).9976 Icr ( k ).9976 Icr ( k) (44).88 U ( k ).49 U ( k ).87 U ( k) I ( k ).9976 I ( k ).9976 I ( k) c c c U ( k ).6687 U ( k ) 5.83 U ( k) (45) In orer to verify the DEM approach the coefficients of iscrete-time moel of the regional power gri are also obtaine by the least-suare curve fitting in which the sum of suare ifferences between fitte values an CP current traectories is minimize in orer to estimate the coefficients. BUS4 BUS BUS3 BUS9 BUS BUS BUS BUS5 BUS BUS3 Fig. 5. Test system in Case BUS5 BUS4 BUS BUS4 BUS6 BUS5 BUS BUS6 BUS3 BUS7 BUS7 BUS6 BUS3 BUS8 BUS8 BUS33 BUS3 BUS9 BUS8 BUS5 BUS34 BUS9 BUS5 CLM Regional Power ri Two isturbances corresponing to the fault current are represente by voltage ips which are simulate by moifying the grouning resistance; the isturbe CP current traectories use for estimating the coefficients in the iscrete-time moel are liste in Table I. The fitte curves for real an imaginary parts of the CP current uner two isturbances are epicte in Figs. 6 an 7 respectively. Here the black curve correspons to actual values in the terminal bus 5 the green one correspons to estimate values using (44) an (45) an the re one correspons to calculate values by least suares fitting base on (6) an (7). Figs. 6 an 7 show that the estimate an fitte real an the imaginary parts of currents are approximately the same as the actual curves representing subtle ifferences in enlarge spots. TABLE I PARAMETERS IN CASE Parameters Values 3% 3% c c c c c c c c c c Relationship c+ c c3+ c4+ c5 c6+ c7 c8+ c9+ c Using Table I an Figs. 6 an 7 in this case stuy we conclue that: ) The coefficients values in the Table I are close to the true coefficients values in (44) (45) verifying the correctness of the propose iscrete-time moel. ) The coefficients are similar in the two isturbances inicating that the propose iscrete-time moel is robust against operating conitions in the external power gri. 3) The sum of output coefficients is approximately eual to an the sum of input coefficients is approximately eual to which accor with (8) (3). The relationship among

7 Ic (p.u.) Icr (p.u.) 7 coefficients provie a theoretical basis for valiating DEM parameters. In aition the relationship among coefficients reuces the number of parameters that nee to be estimate for representing DEM. In the (8) an (3) if we know one parameter in the euation we can etermine the other one. In (9) an (3) if we know two parameters in the euation we can get the last one. Ic (p.u.) Icr-act Icr-cal Icr Ic-act Ic-cal Ic Fig. 6. Dynamic response of Ir an I with isturbance.65 Icr (p.u.) Icr-act Icr-cal Icr Ic-act Ic-cal Ic Fig. 7. Dynamic response of Ir an I with isturbance In Case the DEM methos has been verifie for through the research on DEM consiste by composite loa. When DEM is applie to a more complex regional power gri consiering the following steps ) The terminal voltage of bounary bus an the inecte current by external gri are calculate as input an output state variables in each regional power system. ) DEM parameters are estimate relationships among DEM parameters are verifie an aaptability an ynamic characteristic of DEM are examine through estimation methos. Case : Regional power gri is represente by a synchronous generator together with a composite loa Consiering another certain regional power gri the CEPRI system epicte in Fig. 8 is use again for simulation. The regional power gri consists of a synchronous generator an a composite loa locate at BUS5 an BUS5 respectively. When we consier the generator as aggregate generator an the composite loa as aggregate composite loa the results will be the same. The synchronous generator is locate at BUS5 an its parameters are given in Table II. The composite loa is accesse at BUS5 an the proportion of the inuction motor is 3% the static loa consists of constant reactance an the inuction motor parameters are given in Table III. In this stuy a single-phase-to-groun fault is applie at t=s to the transmission line locate between BUS6 an BUS9 an the fault is cleare at t=.s. The 3% % an 3% voltage ips are attaine by setting appropriate grouning resistances. The DEM moel parameters with these isturbances are presente in Table IV which reveal the following observations ) The coefficients in the iscrete-time moel are similar in two isturbances. ) The sum of output state coefficients of the moel is approximately eual to an the sum of input state coefficients of the moel is approximately eual to which accor with (36) -(39). On the one han the relationship among coefficients provie a theoretical basis for valiating DEM parameters; on the other han the relationship among coefficients reuces the number of parameters that nee to be estimate for representing the euivalent system. In (36) an (38) we can etermine the last one parameter if we know the two parameters in the euation. In (37) an (39) we can etermine the last parameter if we know the three parameters in the euation. BUS4 BUS BUS3 BUS9 BUS BUS BUS BUS5 BUS BUS3 Fig. 8. Test system in Case BUS5 BUS4 BUS BUS4 BUS6 BUS5 BUS3 BUS BUS6 BUS9 BUS7 BUS5 BUS7 BUS6 BUS8 BUS3 BUS8 BUS8 CLM BUS5 Regional Power ri BUS33 TABLE II PARAMETERS IN SYNCHRONOUS ENERATOR LOCATED AT BUS T T T T T D TABLE III PARAMETERS IN INDUCTION MOTOR LOCATED AT BUS5 Rr s r m T s BUS3 BUS9 BUS34

8 8 TABLE IV PARAMETERS IN CASE Parameters Values 3% % 3% Relationship The isturbe power system curves with DEM moel an the actual curves corresponing to two isturbances are illustrate in Figs. 9-. In Figs. 9 an the black curve is an actual value in terminal bus 5 an the re correspons to estimate values by least suares fitting base on DEM. In Fig. 9 the blue line is Ir when the regional power gri euivalent is represente by the aggregate generator. In the Fig. 9 the fitte DEM is more accurate than the moel represente by the euivalent aggregate generator. It is possible that the propose DEM linearization woul limit its application to a more localize section of the power system operating point. In orer to emonstrate the impact of linearization we have mae a comparison of DEM with an original nonlinear moel an the corresponing results are shown here. In Fig. the blue line is Ir when the regional power gri euivalent is represente by the original nonlinear moel. In Fig. the fitte DEM is close to the moel represente by the original nonlinear moel an the actual value of Ir. Fig.. Dynamic response of Ir an I with isturbance Fig. 9. Dynamic response of Ir an I with isturbance Fig.. Dynamic response of Ir an I with isturbance 3 In Case DEM is use to erive the moel of the regional power gri with relatively complex components. In Fig. Ir accoring to the least suare fitting curves for representing DEM cannot fit the actual Ir well when the regional power gir results in a 3% voltage ip in the regional power gri operating point. However simulations an verifications in Case emonstrate that DEM can be use for the regional power gri representation when the isturbance results in a less than 3% voltage ip (which essentially covers the maority of regional power gri isturbances). The root mean suare error (RMSE) is applie to calculate the ifference between actual an least suare fitting curves as presente in Table V. The parameters with isturbance are use in curve fitting with isturbance an isturbance 3 an vice versa to verify the correctness of the moel for mutualfitting RMSE are liste in Table VI. In Figs. 9- the fitte curves using DEM are approximately the same as the actual curves representing subtle ifferences in enlarge spots with the small RMSE liste in Table V. The mutual-fitting RMSE of Ir

9 9 an I is small enough so that the valiity an aaptability of the propose DEM for the regional power gri are verifie. TABLE V RMSE FOR DISTURBED CURVES Voltage RMSE ip Ir I 3% 3.6e e-4 % 5.6e e-4 3%.3e e-4 TABLE VI MUTUAL-FITTIN RMSE FOR DISTURBED CURVES Voltage RMSE of Ir RMSE of I Dip 3% % 3% 3% % 3% 3% 3.6e-4 5.6e-4.5e e-4.33e-3 6.e-4 % 7.84e-4 5.6e-4.e e e e-4 3% 4.5e-4.63e-3.3e-3.e e e-4 In this case we mae a simulation to consier voltage phase angle variation. The generator locate at BUS4 is chosen as a stuy obect with parameters given in Table VII. The same single-phase-to-groun fault is applie at t=s to the transmission line locate between BUS6 an BUS9 an the fault is cleare at t=.s with a % voltage ip. The least suare fitting results are shown in Fig. for the real part of the generator current at terminal Bus 4 where the black curve Igractual is the actual values of Igr the blue curve Igr an the re curve Igr are with an without voltage phase angle variations which are estimate values by least suares fitting base on () an (9) respectively. Igr (p.u.) TABLE VII PARAMETERS OF SYNCHRONOUS ENERATOR LOCATED AT BUS T T T T T D Igr-actual Igr Igr' Fig.. Comparison of Igr-actual Igr Igr In Fig. the RMSE of Igr-actual with Igr an Igr are.5e-3 an 3.5e-3 respectively. Accoringly the fitting accuracy in the simulation is not improve when the bus voltage phase angle is consiere. Furthermore the number of parameters woul be increase when the variations in bus voltage phase angle are explicitly expresse in (7). Corresponingly the variations are ignore in our stuy in orer to facilitate the erivation an the simulation of the propose moel. Case 3: A regional power gri of Central China In Cases an we make a DEM verification in which we apply DEM to a regional power gri of central China (see Fig. 3). On a single-phase-to-groun fault occurre in the 5Kv line between buses an 9. The fault measurement (terminal voltage terminal active an reactive power) of the regional power gri are provie by phase measurement units (PMUs). Hyro Power Plant Thermal Power Plant kv Substation 5kV Substation kv Substation 5kV Switch Station kv 5kVkV Transfer Line A certain regional power gri Fig. 3. Topology of the regional power gri of central China Here are steps to verify the DEM in the case stuy in which we have mae a comparison of actual fault measurements an those obtaine through the DEM application. ) Collect the steay state PMU measurements of the regional power gri. ) Use the steay state measurement values to estimate the DEM parameters (see Table VIII) by least suare fitting metho. 3) Use the voltage ip fault values in the regional power gri measure by PMUs an apply the DEM obtaine in Step to calculate the ynamic response of Ir an I. 4) Compare the PMU fault measurements with the DEM values. In Table VIII we also fin that the sum of output coefficients is approximately eual to an the sum of input coefficients is approximately eual to in the regional power gri of Central China which provie a theoretical basis for valiating the corresponing DEM parameters. The RMSE of Ir an I in Case 3 are 4.69e-3 an 9.48e-3 respectively which emonstrate that the ynamic response of Ir an I obtaine through DEM in Step are almost the same as those of actual PMU measurements in a certain stable range. The Case of the regional power gri of central China has exhibite an verifie the application of DEM. TABLE VIII PARAMETERS IN CASE 3 Parameter Value Relationship In Step 3 voltage ip occurs in the regional power gri of central China as epicte in Fig. 4. The corresponing ynamic response of Ir an I are epicte in Figs. 5 an

10 I (p.u.) Figs. 4-6 show that actual PMU measurements inclue some ripples aroun the steay state value which are from power electronic evices an electromagnetic transient properties. The propose power gri has a self-healing capability which allows voltages an active an reactive power in the regional power gri of central China to recover its steay state uickly when the single-phase-to-groun fault occurs. Other potential applications of DEM inicate that DEM is not only conucive to the assembly of regional power system moels for real-time analyses that guie the secure operation of large scale power systems but coul also be use for the euivalent moeling of a host of istribute AC/DC microgris an the generalize loa moeling an simulation in large-scale power systems. Some of these topics will be analyze further in our future stuies. U (p.u.) /3/8 6/3/8 4:8:.96 4:8:5.96 6/3/8 6/3/8 4:8:.96 4:8:4.96 Fig. 4. Voltage ip in the regional power gri of central China Ir (p.u.) /3/8 6/3/8 4:8:.96 4:8:5.96 U-measurement 6/3/8 4:8:7.96 Ir-measurement Ir-DEM 6/3/8 4:8:3.96 6/3/8 6/3/8 6/3/8 4:8:.96 4:8:4.96 4:8:7.96 Fig. 5. Dynamic response of Ir in the regional power gri of central China /3/8 6/3/8 4:8:.96 4:8:5.96 6/3/8 6/3/8 6/3/8 4:8:.96 4:8:4.96 4:8:7.96 Fig. 6. Dynamic response of I in regional power gri of central China VI. CONCLUSION I-measurement I-DEM 6/3/8 4:8:3.96 6/3/8 4:8:3.96 In this paper a DEM is evelope for regional power gris which encompasses the funamental moels of synchronous generators an composite loas. The inherent relationships among DEM coefficients in a iscrete-time moels are introuce which are linke with coefficients associate with electric power components. As shown in case stuies using the DEM coefficients prouce current traectories that are very similar to the actual power system cases. Simulation results also valiate the relationships among coefficients in the propose DEM for the regional power gris. The case stuy pertaining to the regional power gri of central China verifie the application of DEM. REFERENCES [] P. Ju L. Q. Ni an F. Wu Dynamic euivalents of power systems with online measurements. Part IEE Proceeings- eneration Transm. Distrib. vol. 5 no. pp [] M. Shahiehpour Z. Li S. Bahraamira Z. Li an W. Tian Networke microgris: Exploring the possibilities of the IITbronzeville gri IEEE Power Energy Mag. vol. 5 no. 4 pp [3]. Liu M. Shahiehpour Z. Li. Liu Y. Cao an Z. Bie Microgris for enhancing the power gri resilience in extreme conitions IEEE Trans. Smart ri vol. 8 no. pp [4] F. Milano Small-signal stability analysis of large power systems with inclusion of multiple elays IEEE Trans. Power Syst. vol. 3 no. 4 pp [5] A. Ortega an F. Milano Moeling simulation an comparison of control techniues for energy storage systems IEEE Trans. Power Syst. vol. 3 no. 3 pp [6] N. Kishor J. M. Seppänen J. Turunen A.-J. Nikkilä L. C. Haarla an S. Purwar Low-orer controller esign using remotely measure time elaye signals for stabilisation of euivalent power gri IET ener. Transm. Distrib. vol. 9 no. 6 pp [7] Z. Liu D. Li an S. Liu Stuy on ynamic euivalence an moeling of regional power gri base on RTDS in IEEE Powertech 3 pp. 6. [8] A. Vahinia. Lewich E. Palmer an A. hosh enerator coherency an area etection in large power systems IET ener. Transm. Distrib. vol. 6 no. 9 p [9] S. Kim S. Member an T. J. Overbye Enhance measurement-base ynamic euivalence using coherency ientification in IEEE Power an Energy Conf. at Illinois 3 pp. 5. [] M. A. Mctho Moifie selective moal analysis metho an its application in the analysis of power system ynamics IEEE Trans. Power Syst. vol. 6 no. 3 pp [] R. M. Lin Development of a new an effective moal ientification metho-mathematical formulations an numerical simulations J. Vib. Control vol. 7 no. 5 pp [] B. Zhang Y. Zhang M. Liao an Y. ie Stuy on ynamic euivalent coherency-base of Hainan power gri in Power an Energy Engineering Conference no. pp. 5. [3] M. Shiroei B. Mohammai-Ivatloo an M. Parniani Loworer ynamic euivalent estimation of power systems using ata of phasor measurement units Int. J. Electr. Power Energy Syst. vol. 74 no. 7 pp [4] S. Yu S. Zhang an. Zhang Two-step metho for the online parameter ientification of a new simplifie composite loa moel IET ener. Transm. Distrib. vol. no. 6 pp [5] IEEE Task Force Loa representation for ynamic performance analysis IEEE Trans. In. Appl. vol. 8 no. pp [6] P. Ju et al. Composite loa moels base on fiel measurements an their applications in ynamic analysis ener. Transm. Distrib. IET vol. no. 5 pp [7] D. N. Kosterev C. W. Taylor an W. a. Mittelstat Moel valiation for the August 996 WSCC system outage IEEE Trans. Power Syst. vol. 4 no. 3 pp [8] C. Qin P. Ju F. Wu Y. Liu. Ye an. Wu An efficient multi-area networks-merging moel for power system online ynamic moeling CSEE J. Power Energy Syst. vol. no. 4 pp

11 [9] P. Ju E. Hanschin an D. Karlsson Nonlinear ynamic loa moelling: moel an parameter estimation IEEE Trans. Power Syst. vol. no. 4 pp [] J. V. Milanovic an S. Mat Zali Valiation of euivalent ynamic moel of active istribution network cell IEEE Trans. Power Syst. vol. 8 no. 3 pp. 3. [] S. M. Zali an J. V. Milanović Dynamic euivalent moel of istribution network cell using prony analysis an nonlinear least suare optimization in 9 IEEE Bucharest PowerTech: Innovative Ieas Towar the Electrical ri of the Future 9 pp. 6. [] H. Chen C. Deng an D. Li Recurrent neural network-base ynamic euivalencing in power system 7 IEEE Int. Conf. Control Autom. vol. pp [3] A. Chang an M. M. ADIBI Power system ynamic euivalents IEEE Trans. Power Appar. Syst. vol. PAS-89 no. 8 pp [4] F. Milano Semi-implicit formulation of ifferential-algebraic euations for transient stability analysis IEEE Trans. Power Syst. vol. 3 no. 6 pp [5] F. Milano An open source power system analysis toolbox IEEE Trans. Power Syst. vol. no. 3 pp [6] A. Chakrabortty J. H. Chow an A. Salazar A measurementbase framework for ynamic euivalencing of large power systems using wie-area phasor measurements IEEE Trans. Smart ri vol. no. pp [7] H. Wu J. Hu an Y. ie Characteristic moel-base allcoefficient aaptive control metho an its applications IEEE Trans. Syst. Man Cybern. Part C Appl. Rev. vol. 37 no. pp [8] T. Sugiyama T. Nishiwaki S. Takea an S. Abe Measurements of synchronous machine parameters uner operating conition IEEE Trans. Power Appar. Syst. no. 4 pp [9] P. Kunur Power system stability an control New York: Mcraw-hill 994 pp [3] F. Shen P. Ju L. u. Huang B. Lou an H. Huang Mechanism analysis of power loa using ifference euation approach in 6 IEEE International Conference on Power System Technology 6 pp. 6. [3] J. Ma D. Han R. M. He Z. Y. Dong an D. J. Hill Reucing ientifie parameters of measurement-base composite loa moel IEEE Trans. Power Syst. vol. 3 no. pp [3] C. Cai Y. Jin Y. Yu an P. Ju Loa moeling with consiering freuency characteristic Sustain. Power ener. Supply pp. 6. [33] P. Ju The theory an metho of power system moeling Naning: China Science Press pp [34] F. Milano an K. Srivastava Dynamic REI euivalents for short circuit an transient stability analyses Electr. Power Syst. Res. vol. 79 no. 6 pp [35] Y. Wehbe L. Fan an Z. Miao Least suares base estimation of synchronous generator states an parameters with phasor measurement units in North American Power Symposium pp. 6. [36] S. Nabavi an A. Chakrabortty Structure ientification of reuce-orer moels of power systems in a ifferentialalgebraic form IEEE Trans. Power Syst. vol. 3 no. pp APPENDICES A. Specific parameters of (7) an (8) agr sin ( T T T T T T ) T ( T D T ) T T bgr sin T T ( T D T ) T cgr sin ( TD T ) ( TD T ) T gr sin ( T ) e T T T f T ( T T ) T T gr g T ( T T ) gr D gr D h gr ag cos T T T T T T T T T 3 T T T T ( T D T ) T T bg cos T T ( T D T ) T T ( T D T ) T T 3 T T 3 ( T D T ) T cg cos ( ) ( ) T D T T D T T ( T D T ) 3 ( T D T ) 3 T g cos ( T ) ( T ) e T T T f T ( T T ) T T g g D g T ( T T ) hg. g D B. Specific parameters of (9) an () 3 3hgr h ggrh 4 f grh 4egr g 3 hgrh ggrh 4 f grh 8egr 3 3hgr h ggrh 4 f grh 4egr g 3 hgrh ggrh 4 f grh 8egr h h g h 4 f h 8e 3 gr gr gr gr g3 3 hgrh ggrh 4 fgrh 8egr h c h 4b h 8a 3 gr gr gr gr g 4 3 hgrh ggrh 4 f grh 8egr 3 3 grh cgrh 4bgrh 4agr g5 3 hgrh ggrh 4 fgrh 8egr 3 3 grh cgrh 4bgrh 4agr g6 3 hgrh ggrh 4 fgrh 8egr h c h 4b h 8a 3 gr gr gr gr g7 3 hgrh ggrh 4 f grh 8egr 3 3hg h ggh 4 fgh 4eg g8 3 hgh ggh 4 fgh 8eg 3 3hg h ggh 4 fgh 4eg g9 3 hgh ggh 4 fgh 8eg h h g h 4 f h 8e 3 g g g g g 3 hgh ggh 4 f gh 8eg

12 h c h 4b h 8a 3 g g g g g 3 hgrh ggrh 4 f grh 8egr 3 3 h c h 4b h 4a g 3 hgh ggh 4 fgh 8eg 3 3 gh cgh 4bgh 4ag g3 3 hgh ggh 4 f gh 8eg h c h 4b h 8a 3 g g g g g4 3 hgh ggh 4 fgh 8eg C. Specific parameters of () an () h( Br A Ar B )+8 c h ( A B B A ) h( A B ) 4 r r r h Br A Ar B h Ar B c h Ar B Br A h Ar B ( ) ( ) 4 ( ) ( ) 4 h ( BrC B Cr ) h( B Br B Cr ) 4 c3 h ( Ar B Br A ) h( Ar B ) 4 ( r r ) 8 h B C B C c4 h ( A B B A ) h( A B ) 4 r r r h ( BrC BCr ) h( B Br B Cr ) 4 c5 h ( Ar B Br A) h( Ar B) 4 h( ABr B Ar )+8 c6 h ( A B B A ) h( A B ) 4 r r r h A Br B Ar h B Ar c7 h Ar B Br A h Ar B ( ) ( ) 4 ( ) ( ) 4 h ( ACr Ar C ) h( BAr A C ) 4B c8 h ( Ar B Br A) h( Ar B) 4 h ( ACr Ar C ) 8B c9 h ( Ar B Br A) h( Ar B) 4 h ( ACr ArC ) h( BAr A C ) 4B c h ( Ar B Br A ) h( Ar B ) 4.. Innovation at Illinois Institute of Technology. He is a member of the US National Acaemy of Engineering an a Fellow of the American Association for the Avancement of Science (AAAS). Zhiyi Li (SM 4-M 7) receive the B.S. egree in electrical engineering from i an Jiaotong University i an China in an the M.S. egree in electrical engineering from Zheiang University China in 4. He complete his Ph.D. egree in 7 in the Electrical an Computer Engineering Department at Illinois Institute of Technology. He is a visiting faculty in the Robert W. alvin Center for Electricity Innovation at Illinois Institute of Technology. ueping Pan receive the Ph.D. egree from Zheiang University Hangzhou China in 8. She is presently a professor in the College of Energy an Electrical Engineering Hohai University Naning China. Her research interests inclue moeling of renewable power generation system power system ynamic analysis etc. BIORAPHIES Fu Shen (S'6) receive the B.S. egrees in Hohai University an he is currently an MD-PhD stuent of Electrical Engineering at Hohai University Naning China. He works as research scholar in the Illinois Institute of Technology (IIT) Chicago. His research interests inclue moeling an control of power system. Ping Ju (M'95 SM') receive the B.Sc. an M.Sc. egrees from Southeast University Naning China in 98 an 985 respectively an the Ph.D. egree from Zheiang University Hangzhou China. From 994 to 995 he was an Alexaner von Humbolt Fellow at the University of Dortmun ermany. He is currently a Professor of Electrical Engineering at Hohai University Naning China. His research interests inclue the moeling an control of power systems. He has publishe five research books an authore an coauthore over ournal papers. Prof. Ju receive the Scientific Funs for Outstaning Young Scientists of China. Mohamma Shahiehpour (F ) receive the Honorary Doctorate egree from the Polytechnic University of Bucharest Bucharest Romania. He is a University Distinguishe Professor an Boine Chair Professor an Director of the Robert W. alvin Center for Electricity