INVERTER-interfaced islanded microgrids serve the primary. Spatio-Temporal Model Reduction of Inverter-Based Islanded Microgrids

Transcription

1 Spato-Temporal Model Reduton of Inverter-Based Islanded Mrogrds Lng Luo and Saraj V. Dhople, Member, IEEE 1 Abstrat Computatonally effent and salable models that desrbe droop-ontrolled nverter dynams are key to modelng, analyss, and ontrol n slanded mrogrds. Typal models developed from frst prnples n ths doman desrbe detaled dynams of the power eletrons nverters as well as the network nteratons. Consequently, these models are very nvolved; they offer lmted analytal nsghts and are omputatonally expensve when appled to nvestgate the dynams of large mrogrds wth many nverters. Ths alls for the development of redued-order models that apture the relevant dynams of hgher-order models wth a lower dmensonal state spae whle not ompromsng modelng fdelty. To ths end, ths paper proposes model reduton methods based on sngular perturbaton and Kron reduton to redue large-sgnal dynam models of nverter-based slanded mrogrds n temporal and spatal aspets, respetvely. The redued-order models are tested n a modfed IEEE 37-bus system and verfed to aurately desrbe the orgnal dynams wth lower omputatonal burden. In addton, we demonstrate that Kron reduton solates the mutual nverter nteratons and the equvalent loads that the nverters have to support n the mrogrd ths aspet s leveraged n the systemat seleton of droop oeffents to mnmze power losses and voltage devatons. Index Terms Droop ontrol, Islanded mrogrd, Kron reduton, Model reduton, Sngular perturbaton. I. INTRODUCTION INVERTER-nterfaed slanded mrogrds serve the prmary objetve of meetng rtal loads whle mantanng frequeny and voltage wthn presrbed performane lmts. Inspred by the operaton and ontrol of synhronous mahnes n bulk power systems, droop ontrol s an effetve prmarylevel ontrol strategy that has wdely been appled n ths doman [1]. The premse of droop ontrol s to trade off voltage and frequeny based on the atve and reatve power njeted by the power eletron nverters n the mrogrd. Control synthess, performane evaluaton, stablty analyss, and relablty estmaton of nverter-nterfaed mrogrds requre salable and omputatonally effent models that aurately apture the dynams of droop-ontrolled nverters and desrbe all nteratons between the onsttuent soures, loads, and power eletron nverters. However, typal models for droop-ontrolled nverters are very detaled, and nlude myrad states from nternal ontrol loops and flters [] []. Conevably, ontrol desgn, numeral smulatons, and stablty assessment wth suh models n slanded mrogrds omprsng tens of or even hundreds of nverters s omputatonally Ths work was supported by the Insttute of Renewable Energy and the Envronment (IREE) grant no. RL , Unversty of Mnnesota. The authors are wth the Department of Eletral and Computer Engneerng at the Unversty of Mnnesota, Mnneapols, MN, (emal: luoxx7,sdhople@umn.edu). expensve and offers lmted analytal nsghts. These aspets all for the development of model-reduton methods to solate relevant spato-temporal dynams and mutual nteratons of nterest. Whle model reduton methods are well establshed for synhronous generators n bulk power systems, a systemat model-reduton proedure for droop-ontrolled slanded nverters has thus far been lakng. Ths paper proposes model reduton methods based on sngular perturbaton and Kron reduton to redue large-sgnal dynam models of nverterbased slanded mrogrds n temporal and spatal aspets, respetvely. As mentoned above, related to ths work are model reduton tehnques for dynam models n bulk power systems. Examples nlude Krylov subspae methods [7], Gramanbased methods [], aggregate slow ohereny [], and sngular perturbaton methods [] [1]. Moment mathng model reduton by projeton on Krylov subspaes s proposed n [7] to redue the power system dynamal model order. Gramanbased methods proposed n [] redue model order by omputng approxmatons to the ontrollablty and observablty gramans of large sparse power system models. Aggregate slow ohereny methods used n bulk power system analyss, smulatons and slandng strategy studes [13], [1] are mplemented wth steps of lnearzaton, alulaton of egenvalues and egenvetors, and systemat dvson nto oherent groups. Sngular perturbaton methods provde a systemat approah to multple tme sales modelng n dynamal systems wth state varables that evolve at dfferent speeds [1] [17]. In sngular perturbaton methods, redued-order models are obtaned by dentfyng fast varables and subsequently negletng ther dynams n a systemat fashon (ths amounts to assumng they reah steady state muh faster than the slow varables n the system) [], [1], [1], [1]. Related to our approah, sngular perturbaton methods were appled n [1] to obtan redued-order models negletng load dynams n parallelonneted droop-ontrolled nverters. Sne the proposed approah apples to ruts that le at the nterseton of power eletrons and power systems, another body of related work pertans to model reduton methods that have been appled n the doman of power eletrons ruts, see, e.g., [0] [3] and the referenes theren. Compared to Krylov-subspae, Graman-based, and slowohereny methods whh requre lnearzaton and alulaton of egenvalues, sngular perturbaton methods are muh more ntutve to onstrut n dynamal models omposed of dfferental equatons wth multple tme sales. Gven these advantages, redued-order models for droop-ontrolled nverters are obtaned by applyng sngular-perturbaton methods.

2 In addton, Kron reduton s utlzed to redue the network n the spatal aspet by elmnatng the algebra power-flow equatons orrespondng to the loads and other nodes n the network that are not onneted to the nverters. Orgnally proposed n 13 [], Kron reduton s a standard model reduton tool employed n power networks for applatons suh as transent stablty assessment [], []. Here, the tool elmnates exogenous non-nverter nodes and solates the mutual nverter nteratons and the equvalent loads that the nverters have to satsfy. We demonstrate the model reduton methodology wth the the large-sgnal dynam model of droop-ontrolled nverters desrbed n [] hosen as a benhmark. The nverter ontrol system s omposed of a power ontroller (wthn whh the droop ontrol laws are embedded), as well as addtonal loops for voltage and urrent ontrol. The model nludes dfferental equatons wth multple tme sales whh motvates the applaton of sngular-perturbaton methods for model reduton. The model n [] s hosen as a benhmark to ensure the results are wdely aessble to researhers and pratsng engneers; n partular, our hoe s motvated by the fat that smlar models have been utlzed n [], [7] [33]. The approah to model reduton s presented wth a broad level of generalty so t an be adopted by researhers and pratng engneers n other settngs. A partularly nterestng result of the modelreduton proedure s that the ubqutous frst-order droop law that s ommonly employed n modelng, analyss, and ontrol (see, e.g., [3]) s reovered from the full nonlnear dynamal system after suessve applatons of sngular perturbaton. The remander of ths paper s organzed as follows. Seton II ntrodues the large-sgnal dynam model of an nverter-based slanded mrogrd, nludng the model of a sngle nverter, the oordnate transformatons nvolved n norporatng multple nverters, and the eletral network. Seton III presents: ) the temporal model reduton approah based on sngular perturbaton methods, and ) the spatal model reduton approah based on Kron reduton. Seton IV nludes numeral smulaton studes to valdate the spatotemporal redued models n a modfed IEEE 37-bus network. In addton, a strategy for the systemat seleton of droop oeffents to mnmze power losses and voltage devatons s also presented. We antpate ths approah beng omplementary to reent work that has addressed the optmal desgn of droop oeffents and laws from the perspetves of energy-storage management, power sharng, and eonom optmzaton [3] [37]. Conludng remarks and dretons for future work are presented n Seton V. are modeled ether as onstant mpedanes or onstant urrent soures. Fgure 1 shows a blok dagram of the ontroller employed n eah nverter. In addton to the power ontroller (wthn whh the droop laws are embedded) there are addtonal nner ontrol loops to regulate the nverter output urrent and termnal voltage. Let ω denote the eletral frequeny of the nverter, and let ω om denote an adopted ommon eletral frequeny. (For modelng purposes, ths s typally hosen to be the eletral frequeny of some nverter n the system. In ths work, we assume that nverter = 1 sets the ommon frequeny.) Then, the evoluton of the power angle of the nverter, δ, (defned wth respet to the power angle of the referene nverter) s governed by dδ dt = ω ω om, (1) Consequently, we an transform the termnal voltage and output urrent for eah nverter from ab to dq oordnates, usng the termnal voltage angle δ + ω om t as the referene frame angle to obtan v o := v od +jv oq and o := od +j oq. The dynamal equaton that aptures the operaton of the power ontroller s gven by: 1 ds ω dt = S + v o o, () where ω s the ut-off frequeny, and S = P + jq s the apparent (low-pass fltered) power delvered by the nverter. 1 The outputs of the power ontroller are the voltage referene and the nverter frequeny governed by the voltage-reatve power droop law, and the frequeny-atve power droop law [], [], [7] [33], [37], [3]: vo ref S S = v nom n Q Q = v nom n Q, (3) S + S ω = ω nom m P P = ω nom m P, () where v nom and ω nom are the nomnal system voltage and frequeny, respetvely. The droop oeffents, n Q and m P, are the slopes of the voltage-reatve power and frequenyatve power urves, respetvely, and they are set as: m P = ω P max, n Q = V Q max, () 1 Notaton: ( ) T denotes transposton; ( ) denotes the omplex onjugate; Re{ } and Im{ } denote the real and magnary parts of a omplex number, respetvely; and j := 1 s the magnary unt. The (p, q) entry of the matrx A s denoted by [A] pq. II. SYSTEM MODEL In ths seton, the large-sgnal nonlnear dfferental algebra system model of nverter-based slanded mrogrds s ntrodued. We frst present the model of a sngle nverter, and then desrbe the mrogrd network model. A. Inverter Model Consder an slanded mrogrd omprsng N droopontrolled nverters, and suppose all loads n the mrogrd Fgure 1: Blok dagram of the ontroller for a sngle nverter.

3 3 where P max (Q max ) s the maxmum atve power (reatve power) that the nverter s expeted to delver to support frequeny (termnal voltage), and ω ( V ) s the permssble frequeny (voltage) devaton. For the nverter seleted to be the ommon referene nverter, the frequeny s gven by S + S ω om = ω nom m P P = ω nom m P. () The voltage- and urrent-ontroller state varables are denoted by φ = φ d + jφ q and γ = γ d + jγ q, respetvely. Followng [], [3], [], [3], a onventonal PI ontroller s utlzed to regulate the termnal voltage and output urrents to ther referene values, denoted by vo ref and ref o, respetvely. The voltage and urrent ontrollers generate the referenes v ref ref o = F o + K φ p = jω nom L f o + K γ p dφ dt + Kφ φ, (7) dγ dt + Kγ γ, () where Kp φ (Kp γ ) and K φ (Kγ ) are the parameters for the urrent (voltage) PI ontrol bloks, and F s the stat gan of the urrent feedforward ontrol blok. The ontroller dynams are governed by dφ dt = vref o v o, dγ dt = ref o o. () For ontrol purposes, the quadrature omponent of the referene termnal voltage voq ref s set to zero, and vo ref = v od as shown n Fg.. Fnally, the dynams of the output urrent are aptured by ( L d o R dt = 1 + jω L ) o + v o v b, () R R where v b := v bd + jv bq s the bus voltage at the pont of ommon ouplng (PCC). Equatons (1), (), (), and () desrbe the large-sgnal nnth-order dynamal model of a sngle nverter n the dq referene frame. Note that the followng dynam states are nluded n ths model: δ, P, Q, φ d, φ q, γ d γ q, od, and oq. In pratal mrogrds, a ommon referene frame has to be adopted for all the nverters sne there are multple nverters n the system. We denote the ommon referene frame as the DQ referene frame, and suppose t s spefed by the = 1 nverter. When nverters are ntegrated to the mrogrd through the PCC, the PCC voltage and output urrent should be transformed to the ommon referene frame DQ-axs by B. Network Model Gven the hgh swthng frequeny of the nverters, the dynams of the transmsson lnes and loads are negleted and network nteratons are desrbed by algebra relatons based on Ohm s and Krhoff s laws. Suppose, that n addton to the N nverter buses, the mrogrd nludes M buses, that may be onneted to loads. The mrogrd buses are olleted n the set A := {1,..., N + M}, and dstrbuton lnes are represented by the set E := {(p, q)} A A. The seres admttane of the lne (p, q) s denoted by y pq = (R pq + jω om L pq ) 1, where R pq, L pq, and ω om denote the lne resstane, ndutane, and the ommon angular frequeny. Let Y C N+M N+M denote the omplex admttane matrx of the network, v, denote the vetors of bus voltage and nodal urrent njetons (expressed n DQ oordnates). Expressng Krhoff s rut laws n matrx-vetor form, we an wrte = Y v, (13) where the entres of Y are defned as j N p y pj, f p = q [Y ] pq := y pq, f (p, q) E 0, otherwse and N p := {j A : (p, j) E} denotes the set of buses onneted to the p th bus through a dstrbuton lne. III. SPATIO-TEMPORAL MODEL REDUCTION In Seton II, we desrbed the omplete large-sgnal dynam model for droop-ontrolled nverters that nludes states from multple nternal ontrol loops and flters [], [3], []. In ths seton, we desrbe how Kron reduton allows us to smplfy the network equatons, and how sngular perturbaton methods allow us to derve redued models of dfferent orders for ndvdual nverters. A. Kron Reduton Kron reduton s a standard tool employed n modelng, analyss, and ontrol of power systems (see [] and referenes theren for a detaled overvew). In ths work, Kron reduton s utlzed to elmnate nternal, non-nverter buses, and solate the mutual nverter nteratons. Reall from Seton II, the nteratons of the nverters and loads wthn the eletral network are desrbed by (13). To v DQ o = T v o, DQ o = T o, (11) where the dq to DQ-axs transformaton matrx, T, s: e jδ 0 T = (1) 0 0 e jδ N Note that the (1, 1) entry of T, [T ] 11 = 1, sne δ 1 (t) = 0 by defnton n (1). Fgure : Angle referene frames for multple nverters.

4 elmnate the non-nverter buses, frst we dvde the buses nto two groups, and let v N C N 1 (v M C M 1 ) denote the vetor of voltages of the nodes onneted (not onneted, respetvely) to nverters. Smlarly, let N C N 1 ( M C M 1 ) denote the vetor of urrent njetons at the nodes onneted (not onneted, respetvely) to the nverters. Note that non-zero elements n the vetor M denote onstanturrent loads. Then (13) an be expressed as [ ] [ ] [ ] N YN = N Y N M vn. (1) M v M Y T N M Y MM Elmnatng v M from the equaton above allows us to wrte: N = Y eq v N + Y N M Y 1 MM M, (1) Y eq = Y N N Y N M Y 1 MM Y N T M, (1) where Y eq s referred subsequently as the Kron redued admttane matrx. From (1), t s evdent that we assume the matrx Y MM s nonsngular. Ths mples that the non-nverter node voltages have a unque soluton, gven by v M = Y 1 MM Y N T M v N. Indeed, t s mportant to qualfy the lass of networks for whh Y MM s nonsngular. For RLC networks wth shunt elements, t s possble to obtan a sngular Y MM, suh that the nteror voltages v M are not unquely determned (ths an happen, e.g., when apatve lnes exatly anel out ndutve loads). However, for RLC networks wthout shunt elements, the matrx Y MM s always nonsngular beause Y A s rredubly blok dagonally domnant [0], [1]. In ths paper, we assume that Y MM s always nonsngular. Basally, (1) ndates that the nverter urrent njetons and termnal voltages are oupled through Y eq ; furthermore, onstant-urrent loads n the network are mapped to ndvdual nverters through the matrx Y N M Y 1 MM. Note that the dagonal entres of Y eq apture the equvalent loal loads that eah nverter has to support, and off-dagonal entres ndate equvalent mpedanes that ouple nverters. The overall dynam model of the nverters n the mrogrd s omposed of two parts: ) the dfferental-algebra equatons for the nverters, and ) the algebra equatons of the network. For smplty of dsusson, here we assume there are only mpedane loads n the network (.e., M = 0 n (1)). For the purpose of smulaton, frst, we solve for the vetor of bus voltages as: v N = Yeq 1 N, wth N = T o and v N = T v b. Then, the PCC bus voltage vetor s transformed from DQ-axs to the dq-axs as nput of the nverter dfferental equatons by settng u = v b = T 1 Yeq 1 T o. Note that applyng Kron reduton restrts our attenton to loads that are ombnatons of mpedanes and urrent soures/snks. Constant power loads have to be approxmated as equvalent mpedanes as s done ommonly n transent stablty assessment studes for synhronous generators n bulk power systems []. B. Sngular Perturbaton Methods For dynamal systems whh exhbt dfferent dynam response speeds, sngular perturbaton methods an be appled to redue the model order by negletng the fast dynam state varables. The premse s to assume that the fast dynam varables nstantaneously reah a quas-steady-state soluton whh an be obtaned by the soluton of algebra equatons that result from relevant orgnatng dynamal equatons [], [1], [1], [1], [3]. To formalze the presentaton, the mrogrd dynam model wth multple tme sales s expressed n the followng standard sngular perturbaton form [], [11], []: dx dt ε dz dt = f(x, z, u, t, ε), (17) = g(x, z, u, t, ε), (1) where x C n 1, s the vetor that ollets the slow dynam varables, z C m 1, s the vetor that ollets the fast dynam varables, u s the vetor of nputs, and ε = dag {ε 1, ε,, ε m } denotes a dagonal matrx wth non-zero entres omprsed of small model parameters. From [11], [], f the Jaoban of g( ), gven by g(x, z, u, t, ε) z, (1) ε=0 s nonsngular wth egenvalues that have negatve real parts, then we an redue the n + m order system n (17)-(1) to an n-order system. To ths end, followng [11, Theorem 11.1], we set ε = 0; obtan the quas-steady-state (QSS) soluton z = h(x, u, t), (0) and substtute h(x, u, t) n (17), to get the redued-order model dx dt = f(x, h(x, u, t), u, t, 0) =: f(x, u, t). (1) In the next seton, we apply the general method desrbed above to suessvely redue the orgnal large-sgnal nverter models to obtan redued models of varyng orders. C. Redued-Order Mrogrd Models Frst, we desrbe how the orgnal nnth-order model desrbed n Seton II (wth dynam states x = [δ, P, Q, φ d, φ q, γ d, γ q, od, oq ] T ) an be redued to a ffth-order model (wth dynam states x = [δ, P, Q, od, oq ] T ) by elmnatng the dynamal equatons for the voltage and urrent ontrollers. For the purpose of analyss, we wll fnd t useful to wrte the urrent- and voltage-ontroller dynamal equatons n (7) and () n the followng form K γ p K γ K φ p K φ dγ dt dφ dt ref o = φ + vref = γ + Then, let ε = dag {ε 1, ε } = dag F o, () K φ jω nom L f o. (3) K γ { K φ p K φ, Kγ p K γ } denote the dagonal matrx wth non-zero entres omprsed of the smallperturbaton parameters. Wth ths hoe, the nnth-order model desrbed by (1), (), (), () and (3) an be expressed n the standard sngular perturbaton form n (17)- (1). In ths partular ase, x = [δ, S, o ] T C 3 1 s the

5 Table I: Suessve Temporal Model Reduton wth Sngular Perturbaton Ffth-order model Thrd-order model Sngle-order model Slow varables, x [δ, S, o] T [δ, S] T δ Fast varables, z [φ, γ] T { } o S K φ Small parameter, ε dag {ε 1, ε } = dag p, Kγ p K φ K γ ε 3 = L ε R = 1 ω S+S ω nom m P ω om [ Vetor feld, f( ) ( ω S + ω v o S+S ωnom m ] ) o P ω om S+S R + jω L o + vo v b ω S + ω v ref ω nom m P ω o om o L φ + ref o F o K Vetor feld, g( ) φ (1 + jωε 3 ) o + vref o u S + v γ + vref jω noml f o R ref o o K γ [ ] T QSS soluton of z [φ, γ] T (1 F ) = o v, ref K φ jω noml f o K γ o = vref o u R S+S ω nom m P ω om Redued-order model, f( ) ω S + ω vo ref ( ) o ωnom m P S+S ( ω om ) R + jω L o + vref o u ω S + ω vo ref vo ref u R L ( S = vo ref o = vo ref ) vo ref u R ω nom m P vo ref Re( o) ω om vetor that ollets the slow dynam varables, z = [φ, γ] T C 1 s the vetor that ollets the fast dynam varables, and u = v b,.e., the PCC bus voltage s adopted as the nput. Wth referene to the notaton n (17)-(1), the vetor felds f( ) and g( ) are gven by S+S ω nom m P ω om f = g = ( ω S + ) ω v o o R L + jω φ + γ + vref g o + vo v b L ref o F o K φ jω noml f o K γ, (). () The Jaoban of g( ), z ε=0 = dag{ 1, 1}, whh s nonsngular and has egenvalues wth negatve real parts. Settng ε = 0, the QSS soluton for z s gven by [ ] T z = [φ, γ] T ref o F o v ref jω nom L f o =, K γ. () K φ Addtonally, note that when () s substtuted n (7)-(), we get dφ/dt = 0 and dγ/dt = 0, whh from () mples that v o = vo ref and o = ref o. Ths rgorously justfes that the dynams of the urrent and voltage ontrollers are systematally elmnated n the ffth-order model. Substtutng the QSS soluton for z nto (17), we obtan the redued ffthorder model of a sngle droop-ontrolled nverter: d δ S+S ω nom m P ω om S = ω S + ω vo ref ( ) o, (7) dt R o L + jω o + vref o u L where only the power angle, apparent power, and output urrent dynams are retaned as dynam states, and the dynams of the voltage ontroller and urrent ontroller are systematally elmnated (note they an be reovered usng the algebra relatons n ()). Adoptng the same proedure, we an redue the ffth-order model to a thrd-order model by negletng the dynams of the nverter output-flter urrent. To ths end, x = [δ, S] T, z = o, and the small perturbaton parameter, ε 3 = L R. Fnally, the thrd-order model an be further redued to a sngle order model by elmnatng the dynams of the power ontroller. To ths end, x = δ, z = S, and the small perturbaton parameter, ε = 1 ω. The vetor felds, f( ), g( ), the QSS soluton of z, and the redued order dynamal model, f( ), for the ffth-, thrd-, and frst-order models are lsted n Table I. Conevably, the dfferent redued-order models are useful n dfferent ontexts. For nstane, the orgnal full-order model s useful for the desgn of the urrent and voltage ontrollers. Smlarly, the ffth-order model an be utlzed to analyze the performane of (or desgn) the output ndutve flter. The thrd-order model s relevant n the desgn and verfaton of seondary- and tertary-level ontrollers (see, e.g., [], [3], [3]). Fnally, the frst-order model s applable n long-term performane evaluaton, and dynam relablty assessment [7], []. IV. NUMERICAL CASE STUDIES In Seton III, we outlned the redued-order models of droop-ontrolled nverters n slanded mrogrds. To test the spato-temporal redued order models, a software pakage wth a user-seletable model order has been developed n MATLAB. The smulatons are performed on a personal omputer wth an Intel R Core TM GHz and.0 GB nstalled RAM. The purpose of the ase studes s two fold. Frst, the auray and omputatonal burden of smulatng the redued-order models s evaluated on a modfed IEEE 37-bus system; the results are summarzed n Seton IV-A. Furthermore, based on Kron reduton, a systemat proedure for seletng droop oeffents s presented to mnmze power losses and voltage devatons n the mrogrd eletral network; smulaton results to valdate ths method are presented n Seton IV-B. The standard IEEE 37-bus system was modfed by addng seven nverters at ertan buses as shown n Fg. 3. The test system s studed at the nomnal voltage of v nom = 0 3 V and a nomnal frequeny ω nom = π0 rad s 1. The swthng frequeny of the nverters s set to be f s = khz. Loads are modeled as onstant mpedanes, and seven nverters are added at buses 1, 1,,,, 33 and 3, as shown n Fg. 3. The

6 nverter ontrol parameters are adopted from [] and lsted n Table IV; whle the network and load parameters are olleted n Tables V and VI n Appendx A. A. Orgnal Model Compared to Redued Models To test the dynam behavor of the redued-order models, the load onneted to bus 1 s stepped up (from. Ω to. Ω) at t = 0.1s and then stepped down (from. Ω to. Ω) at t = 1.1s. The tme-doman smulaton results for dfferent redued model orders are ompared. Sne the load s hanged at bus 1, results from four eletrally lose nverters onneted to buses 1, 1,, and 3 are foused on (see Fg. 3). The smulaton results of the termnal voltage angle, atve power, and reatve power of the four nverters are shown n Fg. for the dfferent model orders. From the plots, we an onlude that the dynamal behavors of the ffth-order model math very well wth the orgnal model. The thrd- and sngle-order model have a small steady-state error n the termnal voltage angle and atve power. Ths s prmarly aused by negletng the ndutane L n the thrd- and sngle-order models. Furthermore, the omputatonal effeny of the redued-order models s also evaluated. The total omputaton tme from startup to the load steps (.e., for the total s smulaton tme) for the orgnal nnth-, ffth-, and thrd-order models are.377 s, 1.73 s, and 0.7 s, respetvely. The smulaton tme for the thrd-order model s less than one thrd of that requred for the orgnal model. Ths verfes that the spato-temporal redued-order models an sgnfantly redue omputatonal burden. Two varants of the sngle-order model are tested. In one, the nonlnear algebra equatons (see Table I) are retaned, whle n the other, these are dsarded. If the nonlnear algebra equatons are dsarded, ths amounts to assumng that the termnal voltages are fxed (as s done, e.g., n the dervaton of the lassal model for synhronous generators n bulk power systems). The smulaton tmes for these two varants are 0. s, and 0. s, respetvely. Whle the smulaton tme an be redued to a sxth of the orgnal model by assumng the voltages are fxed, ths varant of the sngle-order model s patently less aurate (a) Termnal voltage angles, δ ( 3 rad), versus tme (se.) (b) Atve power njetons, P (kw), versus tme (se.) () Reatve power njetons, Q (kvar), versus tme (se.) Fgure 3: One-lne dagram of modfed IEEE 37-bus test system (nverters are represented by blue dots) Fgure : Comparng the orgnal and the redued models for the response of four nverters n the modfed IEEE 37-bus system to load steps. Clokwse from the upper left orner are waveforms for nverters onneted to buses 1, 1, 3, and. Full order model s n sold blue, ffth-order model s n dotted blue, thrd-order model s n dotted red, and sngle-order model s n sold red.

7 7 B. Systemat Desgn of Droop Coeffents Typally, droop oeffents for nverters n slanded nverters are seleted to be equal, see, e.g., [], []. From a system-level perspetve, ths mples that loads are meant to be shared equally by the nverters (assumng they have the same ratng). Ths strategy s perhaps justfable n lossless networks. However, n general, mrogrds are resstve-domnant networks, and network power losses beome an mportant desgn onsderaton. A useful strategy then, s to ensure that nverters meet the loads that are eletrally losest to them. Ths would mnmze power flows between nverters effetvely redung losses as well as voltage devatons n the network. To ths end, we propose a systemat strategy to selet the droop oeffents nspred by the nsghts offered by Kron reduton whh learly llustrates the equvalent loads that the nverters have to support, as well as solates the mutual nverter nteratons. Assume the nomnal frequeny (voltage) of all nverters are the same and denoted as ω nom (v nom ). Wth regard to the Kron-redued network, the equvalent load mpedane of a gven nverter s denoted by Z eq (reall from (1) that ths an be reovered from the approprate dagonal entry of Y eq ). Therefore, the equvalent real and reatve power loads for the nverter are approxmately gven by S eq = P eq + jq eq = v nom (Z eq ). () If the droop oeffents m P and n Q are pked to be: m P = ω nom ω om, n Q = v nom vo ref, () P eq Q eq Then t s straghtforward to show from (1)-() that the steady state atve- and reatve-power delvered by the nverter s gven by that demanded by the equvalent loads P eq and Q eq. However, sne we do not a pror know the values of ω om and vo ref, a heurst desgn hoe s to set for eah nverter: m P = ω ω nom P eq, n Q = v nom v nom Q eq, (30) where ω and v nom are the tolerable frequeny and voltage devatons. For the modfed IEEE 37-bus system n Fg. 3, the equvalent loads of the nverters omputed usng (), and the droop oeffents desgned usng (30) are lsted n Table II. To valdate the strategy hghlghted above, we onsder two senaros. In the frst, we use unform droop oeffents for all nverters (m P =. and n Q = 3.3 3, desgned based on equally rated nverters P max =. kw, Q max =.77 kvar). In the seond senaro, we pk the droop oeffents n Table II. A load step at bus 1 s effeted by hangng the load from.3 Ω to.3 Ω. The evoluton of the termnal voltage angle, atve power, and reatve power of the nverters for the two senaros are shown n Fg.. We see that when the droop oeffents are seleted unformly, all the nverters respond unformly to the load step (.e., they ontrbute the same atve power to support the load hange) (see Fg. (a)). Wth the droop oeffents seleted based on (), the nverters ontrbute dfferent atve powers (reatve powers) to support the frequeny (voltage), sne they tend to fous on balanng the equvalent loal loads. The power losses and maxmum voltage devatons n steady state (both before and after the load step) are shown n Table III. Compared to the ase where all nverters have unform droop oeffents, the network power losses and maxmum voltage devaton are sgnfantly lowered wth the suggested desgn strategy. The suggested approah requres knowledge of the Kronredued network. To ths end, the effetve mpedanes (and therefore, the Kron redued rut) an be reovered utlzng off-lne measurements made at the nverter termnals [], [], [7], or by on-lne topology dentfaton approahes when tme-synhronzed measurements are avalable [], []. Table II: Equvalent loal loads and desgned droop oeffents Bus P eq (kw) Q eq (kvar) m P ( ) n Q ( 3 ) Fgure : Step responses of the seven nverters (Clokwse from upleft orner: plots of P (kw, Senaro 1); Q (kvar, Senaro 1); Q (kvar, Senaro ), P (kw, Senaro ). All waveforms are plotted aganst tme (se). Note that the atve power output response of the nverters wth the suggested desgn strategy s dfferent for dfferent nverters based on ther loal equvalent load.

8 Table III: Comparson of power losses and voltage devatons. Senaro 1 Senaro Before After Before After Power losses (W) Max voltage devaton (V) V. CONCLUDING REMARKS AND DIRECTIONS FOR FUTURE WORK In ths paper, model reduton methods are proposed for systematally redung large-sgnal dynam models of droopontrolled nverters n slanded mrogrds. The auray of the redued-order models s verfed wth numeral smulatons to aurately desrbe the orgnal dynams of the system. Also, we note that the redued models offer sgnfantly lower omputatonal burden as ompared to the orgnal nonlnear models. Indeed, wth the nevtable large-sale prolferaton of power eletrons ruts n mrogrds, the omputatonal benefts of model-reduton methods wll be appreated n all aspets of mrogrds modelng, analyss, and ontrol. In addton, spatal model reduton based on Kron reduton s employed to solate the mutual nverter nteratons and learly llustrate the equvalent loads that the nverters have to support n the mrogrd. Based on the Kron-redued network model, a systemat approah to selet the droop oeffents s proposed to mnmze the power losses and voltage devatons. The models proposed n ths paper are expeted to ad future efforts n modelng, analyss, and ontrol of mrogrds. In partular, we wll leverage the redued-order models for the desgn of sparse ontrol arhtetures for seondary-level ontrol. As part of future work, we wll also ntegrate onstantpower loads nto the analytal framework and onsder grdonneted operaton (as well as transtons between grdonneted and slanded modes). Furthermore, note that n the present setup, to leverage Kron reduton we assume quas-statonary operaton of the eletral network. Future efforts ould nvolve leveragng the nsghts n, e.g., [0] to jontly onsder transent behavors whle aomplshng the spatal model reduton objetves afforded by Kron reduton. Fnally, an mportant dreton s to establsh robustness of the strategy suggested to desgn droop oeffents by leveragng nsghts from Kron reduton and algebra power flow relatonshps. Valdaton wth smulatons of the nonlnear models and expermental prototypes wll also be undertaken. REFERENCES [1] M. Chandorkar, D. Dvan, and R. Adapa, Control of parallel onneted nverters n standalone a supply systems, IEEE Trans. Ind. Appl., vol., pp , Jan. 13. [] N. Pogaku, M. Prodanov, and T. Green, Modelng, analyss and testng of autonomous operaton of an nverter-based mrogrd, Power Eletrons, IEEE Transatons on, vol., pp. 13, marh 007. [3] G. S. Ladde and D. D. Sljak, Multparameter sngular perturbatons of lnear systems wth multple tme sales, Automata, vol. 1, no., pp. 3 3, 13. [] S. Bae and A. Kwasnsk, Dynam modelng and operaton strategy for a mrogrd wth wnd and photovolta resoures, Smart Grd, IEEE Transatons on, vol. 3, pp , de. 01. [] A. Bdram and A. Davoud, Herarhal struture of mrogrds ontrol system, Smart Grd, IEEE Transatons on, vol. 3, no., pp , 01. [] A. Bdram, A. Davoud, F. Lews, and J. Guerrero, Dstrbuted ooperatve seondary ontrol of mrogrds usng feedbak lnearzaton, Power Systems, IEEE Transatons on, vol., no. 3, pp , 013. [7] D. Chanots and M. A. Pa, Model reduton n power systems usng krylov subspae methods, Power Systems, IEEE Transatons on, vol. 0, no., pp., 00. [] F. Fretas, J. Rommes, and N. Martns, Graman-based reduton method appled to large sparse power system desrptor models, Power Systems, IEEE Transatons on, vol. 3, no. 3, pp , 00. [] J. Chow, R. Galarza, P. Aar, and W. Pre, Inertal and slow ohereny aggregaton algorthms for power system dynam model reduton, Power Systems, IEEE Transatons on, vol., no., pp. 0, 1. [] E. Abed and R. Slva-Madrz, Stablty of systems wth multple very small and very large parasts, Cruts and Systems, IEEE Transatons on, vol. 3, no., pp , 17. [11] H. Khall, Nonlnear Systems. Upper Saddle Rver, NJ: Prente Hall, 00. [1] H. Khall, Stablty analyss of nonlnear multparameter sngularly perturbed systems, Automat Control, IEEE Transatons on, vol. 3, no. 3, pp. 0 3, 17. [13] G. Xu and V. Vttal, Slow ohereny based utset determnaton algorthm for large power systems, Power Systems, IEEE Transatons on, vol., no., pp. 77, 0. [1] G. Xu, V. Vttal, A. Mekln, and J. Thalman, Controlled slandng demonstratons on the we system, Power Systems, IEEE Transatons on, vol., no. 1, pp , 011. [1] P. V. Kokotov, R. Malley, and P. Sannut, Sngular perturbatons and order reduton n ontrol theory an overvew, Automata, vol. 1, pp , 17. [1] H. K. Khall, Asymptot stablty of nonlnear multparameter sngularly perturbed systems, Automata, vol. 17, no. 1, pp. 77 0, 17. [17] X. Xu, R. M. Mathur, J. Jang, G. Rogers, and P. Kundur, Modelng of generators and ther ontrols n power system smulatons usng sngular perturbatons, Power Systems, IEEE Transatons on, vol. 13, no. 1, pp. 11, 1. [1] J. H. Chow, Asymptot stablty of a lass of nonlnear sngularty perturbed systems, J. Frankln Inst., vol. 30, pp , 17. [1] V. Maran, F. Vasa, and J. M. Guerrero, Analyss of droop ontrolled parallel nverters n slanded mrogrds, n IEEE Internatonal Energy Conferene (ENERGYCON), pp , May 01. [0] J. Kmball and P. Kren, Sngular Perturbaton Theory for DC-DC Converters and Applaton to PFC Converters, IEEE Transatons on Power Eletrons,, vol. 3, pp. 70 1, Nov. 00. [1] A. Davoud, J. Jatskevh, P. L. Chapman, and A. Bdram, Multresoluton modelng of power eletrons ruts usng model-order reduton tehnques, IEEE Transatons on Cruts and Systems I: Regular Papers, vol. 0, pp. 3, Marh 013. [] J. D. Owens and P. L. Chapman, Automat generaton of aurate low-order models for magnet deves, IEEE Transatons on Power Eletrons, vol. 0, pp. 73 7, July 00. [3] A. Davoud, P. L. Chapman, J. Jatskevh, and H. Behjat, Reduedorder dynam modelng of multple-wndng power eletron magnet omponents, IEEE Transatons on Power Eletrons, vol. 7, pp. 0, May 01. [] G. Kron, A method to solve very large physal systems n easy stages, Crut Theory, Transatons of the IRE Professonal Group on, vol. PGCT-, pp. 71 0, 13. [] R. Cr, A. Padlha Feltrn, and L. Ohoa, Power flow n fourwre dstrbuton networks-general approah, Power Systems, IEEE Transatons on, vol. 1, no., pp. 13, 003. [] F. Dörfler and F. Bullo, Kron reduton of graphs wth applatons to eletral networks, Cruts and Systems I: Regular Papers, IEEE Transatons on, vol. 0, no. 1, pp , 013. [7] I.-Y. Chung, W. Lu, D. Cartes, J. Collns, E.G., and S.-I. Moon, Control methods of nverter-nterfaed dstrbuted generators n a mrogrd system, Industry Applatons, IEEE Transatons on, vol., no. 3, pp. 7, 0. [] M. Hassan and M. Abdo, Optmal desgn of mrogrds n autonomous and grd-onneted modes usng partle swarm optmzaton, Power Eletrons, IEEE Transatons on, vol., no. 3, pp. 7 7, 011. [] J. Km, J. Guerrero, P. Rodrguez, R. Teodoresu, and K. Nam, Mode adaptve droop ontrol wth vrtual output mpedanes for an nverterbased flexble a mrogrd, Power Eletrons, IEEE Transatons on, vol., no. 3, pp. 701, 011.

9 [30] S. Dasgupta, S. Sahoo, and S. Panda, Sngle-phase nverter ontrol tehnques for nterfang renewable energy soures wth mrogrd; part : Parallel-onneted nverter topology wth atve and reatve power flow ontrol along wth grd urrent shapng, Power Eletrons, IEEE Transatons on, vol., no. 3, pp , 011. [31] M. Zhu, H. L, and X. L, Improved state-spae model and analyss of slandng nverter-based mrogrd, n Industral Eletrons (ISIE), 013 IEEE Internatonal Symposum on, pp. 1, 013. [3] A. Radwan and Y.-R. Mohamed, Analyss and atve-mpedane-based stablzaton of voltage-soure-retfer loads n grd-onneted and solated mrogrd applatons, Sustanable Energy, IEEE Transatons on, vol., no. 3, pp. 3 7, 013. [33] J. Vasquez, J. Guerrero, M. Savagheb, J. Eloy-Gara, and R. Teodoresu, Modelng, analyss, and desgn of statonary-referene-frame droop-ontrolled parallel three-phase voltage soure nverters, Industral Eletrons, IEEE Transatons on, vol. 0, no., pp. 171, 013. [3] J. W. Smpson-Poro, F. Dörfler, and F. Bullo, Synhronzaton and power sharng for droop-ontrolled nverters n slanded mrogrds, Automata, vol., no., pp , 013. [3] X. Lu, K. Sun, J. Guerrero, J. Vasquez, and L. Huang, State-ofCharge Balane Usng Adaptve Droop Control for Dstrbuted Energy Storage Systems n DC Mrogrd Applatons, IEEE Transatons on Industral Eletrons, vol. 1, pp. 0 1, June 01. [3] F. Dörfler, J. W. Smpson-Poro, and F. Bullo, Breakng the herarhy: Dstrbuted ontrol & eonom optmalty n mrogrds, IEEE Transatons on Control of Network Systems, January 01. Submtted. [37] Q.-C. Zhong, Robust droop ontroller for aurate proportonal load sharng among nverters operated n parallel, Industral Eletrons, IEEE Transatons on, vol. 0, no., pp. 11, 013. [3] J. Guerrero, J. Vasquez, J. Matas, L. de Vuna, and M. Castlla, Herarhal ontrol of droop-ontrolled a and d mrogrds - a general approah toward standardzaton, Industral Eletrons, IEEE Transatons on, vol., no. 1, pp. 1 17, 011. [3] Y. Mohamed and A. Radwan, Herarhal ontrol system for robust mrogrd operaton and seamless mode transfer n atve dstrbuton systems, Smart Grd, IEEE Transatons on, vol., pp. 3 3, june 011. [0] R. A. Horn and C. R. Johnson, Matrx analyss. Cambrdge unversty press, 01. [1] S. V. Dhople, B. B. Johnson, F. Dörfler, and A. O. Hamadeh, Synhronzaton of Nonlnear Cruts n Dynam Eletral Networks wth General Topologes, IEEE Transatons on Cruts and Systems I: Regular Papers, Otober 013. To appear. Avalable at [] M. A. Pa, Power System Stablty: Analyss by the Dret Method of Lyapunov. North-Holland systems and ontrol seres, North-Holland Publshng Company, 11. [3] H. Khall, Stablty analyss of nonlnear multparameter sngularly perturbed systems, n Ameran Control Conferene, 17, pp , 17. [] D. Pepondes, P. V. Kokotov, and J. H. Chow, Sngular perturbatons and tme sales n nonlnear models of power system, Cruts and Systems, IEEE Transatons on, vol., no. 11, pp. 7 77, 1. [] S. V. Dhople, Y. C. Chen, L. DeVlle, and A. D. Domínguez-Garía, Analyss of power system dynams subjet to stohast power njetons, IEEE Transatons on Cruts and Systems I: Regular Papers, vol. 0, pp , Deember 013. [] A. M. Ehrlh, Parallel and seres addton of networks, 1. Teh. report, REU Mathemats Projet, Unversty of Washngton, [Onlne] Avalable at: reu. [7] N. Saksena, The Drhlet problem for nfnte networks, 00. Teh. report, REU Mathemats Projet, Unversty of Washngton, [Onlne] Avalable at: reu. [] S. Nabav and A. Chakrabortty, Topology Identfaton n Dynam Equvalent Models of Power Systems, IEEE Transatons on Power Systems, 013, n revew. [] S. Nabav and A. Chakrabortty, Topology dentfaton for dynam equvalent models of large power system networks, n Ameran Control Conferene, pp , June 013. [0] S. Calskan and P. Tabuada, Kron reduton of power networks wth lossy and dynam transmsson lnes, n IEEE 1st Annual Conferene on Deson and Control, pp., De. 01. A PPENDIX A S IMULATION PARAMETERS F OR I NVERTERS AND N ETWORK Table IV: Inverter ontroller parameters [] Parameter ω Kpφ Kpγ Lf L Value mh 0.3 mh Parameter F Kφ Kγ Rf R Value Ω 0.03 Ω Table V: Branh parameters From To R L From To R L Table VI: Load parameters Bus R L Bus R L Lng Luo reeved the B.S. and M.S. degrees n Eletral Engneerng, n 00 and 007, respetvely, from Huazhong Unversty of Sene and Tehnology, Wuhan, Chna. She reeved an M.S. degree n Eletral Engneerng n 01 from the Unversty of Mnnesota, Mnneapols, MN. Her researh nterests nlude power system stablty and ontrol, dstrbuted generaton, and mrogrd modelng.

10 Saraj V. Dhople (S 0, M 13) reeved the B.S., M.S., and Ph.D. degrees n eletral engneerng, n 007, 00, and 01, respetvely, from the Unversty of Illnos, Urbana-Champagn. He s urrently an Assstant Professor n the Department of Eletral and Computer Engneerng at the Unversty of Mnnesota (Mnneapols), where he s afflated wth the Power and Energy Systems researh group. Hs researh nterests nlude modelng, analyss, and ontrol of power eletrons and power systems wth a fous on renewable ntegraton.