Advanced Modeling, Design, and Control of ac-dc Microgrids

Transcription

1 Lousana State Unversty LSU Dgtal Commons LSU Doctoral Dssertatons Graduate School Advanced Modelng, Desgn, and Control of ac-dc Mcrogrds Hossen Saber horzough Follow ths and addtonal wors at: Part of the Power and Energy Commons Recommended Ctaton Saber horzough, Hossen, "Advanced Modelng, Desgn, and Control of ac-dc Mcrogrds" (2018). LSU Doctoral Dssertatons Ths Dssertaton s brought to you for free and open access by the Graduate School at LSU Dgtal Commons. It has been accepted for ncluson n LSU Doctoral Dssertatons by an authorzed graduate school edtor of LSU Dgtal Commons. For more nformaton, please contactgradetd@lsu.edu.

2 ADVANCED MODELING, DESIGN, AND CONTROL OF AC-DC MICROGRIDS A Dssertaton Submtted to the Graduate Faculty of the Lousana State Unversty and Agrcultural and Mechancal College n partal fulfllment of the requrements for the degree of Doctor of Phlosophy n The Department of Electrcal and Computer Engneerng by Hossen Saber horzough B.Sc., Isfahan Unversty of Technology, 2009 M.Sc., Unversty of Tabrz, 2013 May 2018

3 ACKNOWLEDGEMENTS I would le to express my grattude to my advsor Dr. Shahab Mehraeen and all the commttee members, Dr. Lesze S. Czarnec, Dr. Mehd Farasat, and Dr. Amn Kargaran, and also Dr. Jerry Trahan as the char of Dvson of Electrcal and Computer Engneerng. I also express grattude to all ECE staff specally Ms. Beth Cochran for beng always nd and helpful and would le to than all my lovely frends at Lousana State Unversty who have been supportve throughout all my PhD study, especally Sharareh Hedaran and Hamed Shamhalchenar. Fnally, I am really grateful to my famly for ther nvaluable support durng all my study.

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS... ABSTRACT...v CHAPTER 1. RELIABLE OPERATION OF SMALL-SCALE INTERCONNECTED DC GRIDS VIA MEASUREMENT REDUNDANCY Introducton Interconnected Dc Grd DER and CPL Model Development Decentralzed Output-Feedbac Controller Desgn Smulaton and Hardware Test Results Concluson References...31 CHAPTER 2. A SIMULTANEOUS VOLTAGE AND PHASE CONTROL SCHEME FOR PHOTOVOLTAIC DISTRIBUTED GENERATION UNITS IN SMALL-SCALE POWER SYSTEMS Introducton System Topology and Proposed Control Scheme Smulaton Results Expermental Results Concluson References...60 CHAPTER 3. STABILITY IMPROVEMENT OF MICROGRIDS USING A NOVEL REDUCED UPFC STRUCTURE VIA NONLINEAR OPTIMAL CONTROL Introducton Modelng and Control of the System Nonlnear Optmal Controller Desgn Test Results Concluson References...79 VITA...82

5 ABSTRACT An nterconnected dc grd that comprses resstve and constant-power loads (CPLs) that s fed by Photovoltac (PV) unts s studed frst. All the sources and CPLs are connected to the grd va dc-dc buc converters. Nonlnear behavor of PV unts n addton to the effect of the negatve-resstance CPLs can destablze the dc grd. A decentralzed nonlnear model and control are proposed where an adaptve output-feedbac controller s employed to stablze the dc grd wth assured stablty through Lyapunov stablty method whle each converter employs only local measurements. Adaptve Neural Networs (NNs) are utlzed to overcome the unnown dynamcs of the dc-dc converters at Dstrbuted Energy Resources (DERs) and CPLs and those of the nterconnected networ mposed on the converters. Addtonally, the use of the output feedbac control maes possble the utlzaton of other measured sgnals, n case of loss of man sgnal, at the converter locaton and creates measurement redundancy that mproves relablty of the dc networ. The swtchng between measurement sgnals of dfferent types are performed through usng the NNs wthout the need to further tunng. Then, n a small-scale ac grd, PV-based Dstrbuted Generaton (DG) unts, ncludng dc/dc converters and nverters, are controlled such that mmc a synchronous generator behavor. Whle other control schemes such as Synchronverters are used to control the nverter frequency and power at a fxed dc ln voltage, the proposed approach consders both the dc-ln voltage and the nverter ac voltage and frequency regulaton. The dc-ln capactor stores netc energy smlar to the rotor of a synchronous generator, provdng nerta and contrbutes to the system stablty. v

6 Addtonally, a reduced Unfed Power Flow Controller (UPFC) structure s proposed to enhance transent stablty of small-scale mcro grds. The reduced UPFC model explots dc ln of the DG unt to generate approprate seres voltage and nject t to the power lne to enhance transent stablty. It employs optmal control to ensure that the stablty of the system s realzed through mnmum cost for the system. A neural networ s used to approxmate the cost functon based on the weghted resdual method. v

7 CHAPTER 1 RELIABLE OPERATION OF SMALL-SCALE INTERCONNECTED DC GRIDS VIA MEASUREMENT REDUNDANCY 1.1 Introducton Renewable energy sources have attracted attenton n modern power systems, as clean and abundant sources of energy. Dstrbuted energy resources (DERs) such as Photovoltacs (PVs) and fuel cells generate dc power. Loads that consume dc power such as energy storage unts, electronc loads, chargers, and LED lghtngs are ncreasngly used n the modern power system. Hence, employng a dc grd for nterconnectng the DERs and loads can be more effcent snce they are not challenged by frequency and phase stablty and reactve power flow [1]-[5]. DERs are ntegrated nto the grd through approprate power electroncs nterfaces. Ac sources use an ac-dc nterface to ln to the dc grd whereas dc sources use dc-dc converters. Unle ac grds, dc networs studes are n ther early stages. Several ssues of dc systems need more attenton and nvestgaton. Stablty of the dc grds s one mportant ssue, specfcally n small-scale nterconnected dc grds wth low nerta [6]-[7] that are more vulnerable to faults and dsturbances. Constant-power loads (CPLs) can potentally cause nstablty and voltage collapse n dc grds. The negatve resstance of the CPLs can destablze the grd. Large oscllatons or voltage collapse are observed n the presences of CPLs n dc grds wthout proper controls [8]. CPLs cause more deteroraton specfcally n low-nerta dc grds where only a small error margn between generaton and consumpton s allowed, and total load of the system s close to system generaton lmt. Stored energy n dynamc elements such as capactors and nductors acts as system nerta [9]. However, t s not approprate to ncrease the nerta by selectng overly larger dynamcal elements due to practcal lmtatons on the sze of the components and an ncreased grd settlng tme. The ntermttent nature of DERs also 1

8 threaten the stablty of the small-scale grds. For nstance, a passng cloud can decrease PV s output power abruptly and drastcally. Ths along wth the PV s nonlnear voltage-power characterstcs adversely affects the system behavor. In order to smultaneously regulate voltage of the grd and also optmze the power sharng between all dspatchable DERs a dstrbuted control technque s presented n [10]. However, t does not address the small sgnal stablty problems related to the nonlnear behavor of PVs and CPLs. In [11], authors proposed an optmzed load sharng for dfferent DGs through adjustng the droop gans. Droop-based schemes are nown as decentralzed control approaches that provde proper power sharng. In ths method, the reference voltage of each converter s adjusted usng vrtual resstance Rdroop. The proposed scheme offers desred current sharng and voltage regulaton smultaneously usng a Proportonal Droop Index (PDI). Fnally, the resultant reference voltage s appled usng a PI controller. In [12]-[13] another mproved droop mechansm s proposed that consders lne resstances to acheve more precse power sharng. In addton, a varable droop gan s exploted for an accurate power sharng whle preservng voltage regulaton and power effcency. In [14-18], authors proposed modfed droop based methods to enhance load sharng between dfferent sources. However, all these approaches ncorporate conventonal PI controllers to apply the desred voltage set ponts to the system. The PI controllers do not show promsng performance n dfferent condtons, specfcally n the presence of CPLs n the grd [8]. In [19]-[22] load and source mpedance reshapng have been proposed to enhance the stablty margns. The small sgnal analyss s provded, and the stablty of the system s evaluated for small varatons around the operatng pont. The dc grd s stablty usng nonlnear 2

9 decentralzed stablzaton technques are evaluated n [23]-[27]. Although large sgnal analyss are consdered, a smple grd topology wth sngle voltage level s studed, and nterconnected dc grds wth varous bus voltages across the grd are not consdered n these studes. In [28], the Lnear Quadratc Gaussan (LQG) method s employed where an extended Kalman flter s ncorporated along wth a Lnear Quadratc Regulator (LQR). The CPL current s ntroduced as vrtual dsturbance, and the system response s composed of two parts: demand and dsturbance responses. In [29], the only measured sgnal s the bus voltage where a Kalman flter s used to estmate the other states. However, t s shown that convergence rate of the Kalman flter adversely affects the performance of the controller. In order to stablze a Medum Voltage dc (MVDC) networ that feeds CPLs and resstve loads, lnearzaton through state feedbac s used n [30]-[31]. Whle n the majorty of these wors only a measured sgnal (output voltage) s utlzed, and CPL stablty problem s addressed, these schemes are proposed for a grd topology wth a common dc bus that nterlns all the sources and loads and not an nterconnected grd. In [32], a nonlnear control method s proposed based on sldng mode control. The proposed scheme mtgates destablzng effects of CPLs and provdes an approxmately fxed voltage for a dc bus. In [33], t s amed to enhance stablty of a dc grd conssts of multple dcdc converters and CPLs. The proposed approach based on dentfyng the egenvalues of the Jacoban matrx to stablze the dc system. However, these methods are also applcable to a specfc grd topology wth sngle dc bus. An output feedbac control approach s presented n [34] for dc-dc buc converters. A reduced order observer s employed to estmate unmeasured states of the converter and stablty 3

10 of the entre system s proved. Nonetheless, the proposed method s appled to a sngle converter n a centralzed control manner and does not consder an nterconnected networ comprsng dfferent converters that affect each other. The decentralzed control s preferred n nterconnected systems, where several loads and DERs nteract [35]-[36], because t decreases data exchange, computatonal burden, and tme delays whle enablng easy addton of new DERs. In partcular, nonlnear decentralzed control schemes provde transent stablty and steady-state performance usng local measurements and offer larger stablty margns n comparson wth ther lnear counterparts. Dstrbuted control mechansms have been recently appled to the small-scale power networs; however, a communcaton nfrastructure s requred to collect global data from across the networ [37]. In order to mplement the control, the dscrete-tme models are desred [38]-[40] due to easer dgtal mplementaton and preventon of nstablty due to dscretzaton of contnuoustme systems. Conversely, most of the wors n ths area s based on contnuous-tme modelng [41]. Recent developments n ths feld by the author ncludes [42]-[44] that use state- and output-feedbac methods. In these wors, the nonlnear characterstcs of both the DERs (PVs) and CPLs are taen nto account. The CPLs are bult as resstve loads that are connected to the grd va dc-dc converters and controlled such that absorb constant power. Both DER and CPL converters are controlled by the proposed decentralzed nonlnear controllers. Then, through the Lyapunov stablty method the stablty of all the converters states n the dc grd s proven. When usng output feedbac controller, the other avalable converter states can be potentally used to provde measurement redundancy. However, ths was not studed n the prevous wor [44]. 4

11 In ths paper, two major mprovements are proposed over the authors prevous wor. Frst, usng output-feedbac decentralzed controller, the addtonal avalable states are used for control when the man measurement s lost. It s proven that the states can be utlzed wth no change n the nonlnear controller structure or parameters wth assured overall grd stablty. Second, the proposed output-feedbac fault tolerant controller s mplemented n the hardware usng a 1400W dc crcut that ncludes three photovoltac sources, one CPL, and seven resstve loads. The IEEE 14-bus networ topology s chosen to establsh an nterconnected networ. Adaptve neural networs (NNs) [45]-[47] are employed to overcome the unnown dynamcs of the dc-dc converters at DERs and CPLs and to stablze the entre grd. The approxmaton property of the neural networs can effectvely replace the unnown dynamcs of converters and those of the nterconnected networ mposed on the converters. The dc grd nterconnects dstrbuted energy resources, CPLs, and resstve loads, and the control approach s not restrcted to any specfc topology. Addtonally, t should be noted that the goal of ths wor s short-term stablty enhancements that le n the mcrogrd prmary control [48]. The rest of ths chapter s organzed as follows. The dc grd topology s presented n secton 2. In Secton 3, nonlnear dscrete-tme decentralzed model of photovoltac DER s explaned and the CPL model s developed n Secton 4. The smulaton and expermental results on the low-voltage small-scale dc grd are gven n Secton 5. Secton 6 ncludes the concludng remars of the wor. 1.2 Interconnected DC Grd Dfferent types of DERs can be used as low nerta power sources n a small scale dc grd. Wthout loss of generalty, PV unts are selected as the DERs that are connected to the grd va 5

12 dc-dc buc converters. Resstve loads are connected to the grd drectly whle CPLs are ted to the grd va dc-dc buc converters as shown n Fg The CPL converter eeps the voltage constant across a resstve load such that the power consumpton at the CPL bus remans constant regardless of dc grd voltage. Nonetheless, any type of load that nteract wth the grd as a CPL, such as nducton motors that are connected to the dc grd va dc-ac converters, can be stablzed usng the proposed control scheme. Fg. 1.1 Interconnected dc Networ Although the boost converter poses hgher output voltage, due to the stablty ssues, the buc converter s more relable for the PV system. For ncreasng the voltage level, transformers at the ac sde or topologes le forward converters that employ transformers can be used, wthout affectng stablty of the system [49]. Overall, any converter can be modeled more or less the same way as the buc converter chosen here. Ths wor ams at desgn of stablzng controllers that control the output voltages wth any avalable converter sgnal as opposed to [44] to acheve tolerance aganst loss of measurement. The nonlnear buc converter models are used 6

13 and duty cycle control s performed va nonlnear decentralzed neural networ-based adaptve controller. The proposed controller employs an observer to estmate the unmeasured local states. The dc-dc converters of DERs and CPLs are explaned n [43] and [44] and s brefly revewed n the next secton. 1.3 DER and CPL Model Development The DERs and CPLs are connected to the grd through dc-dc buc converters. In ths secton, frst a DER converter model s developed and then, a model s presented for CPL converter n1. A. DER Converter Model The dc-dc converter that tes the DER to the grd s shown n Fg Index represents the converter number. If 1 n, t refers to a DER converter, and f 1 n1 n, a CPL converter s denoted. The nput-output canoncal equatons n dscrete doman for the converter model s derved. The state varables of the DER converter are nput voltage v n,, nductor current L,, and output voltage v out, as depcted n Fg Only one state (such as v out, ) s utlzed n the control whle the other measured states provde measurement redundancy. The dynamcal equaton of the DER converter n dscrete doman at tme step T s wrtten as [43] ( 1) T ( D ) T 1 vn, (( 1) T ) Cn, [ 1, ( t) dt L, ( t) dt] vn, ( T) T ( D ) T T ( 1) T 1 L, (( 1) T ) L [ vn, ( t ) dt vout, ( t) dt] L, ( T) (1) T T v out ( 1) T T 1, (( 1) T ) Cout, ( L, ( t) 2, ( t)) dt vout, ( T) 7

14 where T s the converter swtchng perod, D s duty cycle, L s the converter s nductor, C n, and C out,, are the nput and output capactances,, 1 and 2, are nput and output currents and s the dscrete step ( functon of PV voltage. ). The njected nput current 1, s a The state error vector s defned as X ( )] ( ) X( ) X o, [ vn, ( ), L, ( ), vout, T wth X )] T ( ) [ vn, ( ), L, ( ), vout, ( and Xo, s the steady-state vector. Input and output current errors are 1, ( ) 1, ) I1, ( and 2, ( ) 2, ( ) I where 2, I1, and I2, are steadystate nput and output currents. The control objectve s to mae the converter measured state error zero by generatng an addtonal duty cycle D () to the steady-state duty cycle Do, where D () D, D ( ). o Hence, the converter state s stablzed around ts reference value after grd dsturbances and faults have cleared. Remar 1. The converter duty cycle reference can be set by the upper level control n the grd (power sharng controller). Short term stablty (less than second) s the goal here and thus slower system dynamcs such as slow cloud effects or load changes are not consdered. Due to faster dynamcs consdered n the proposed control method set pont Do, can be consdered constant. It s notable that system (1) s n a hghly nonlnear form and thus must be converted to Brunovs canoncal form [50], before any nown control mechansm can be appled. x ( 1) x ( ) 1 2 x ( 1) f ( x) g( x) u 2 (2) 8

15 However, the Brunovs canoncal model depends on the selected measured output state;.e., the order and the value of coeffcents n the model vares when the target output state vares ( v n,, L,, or v out, ). Ths s mportant to note when swtchng from one measured state to another durng measurement falures n a redundancy-supported control mechansm. For smplcty, one can conclude from the lnearzed form of system (1), shown n (2), to see ths varaton but the actual controller desgn wll be conducted usng the orgnal nonlnear forms (1): vn, (( 1) T ) T Cn, [ 1, ( T) D L, ( T)] v1, ( T) L, (( 1) T ) T L [ D vn, ( T) vout, ( T)] L, ( T) (3) v out, (( 1) T ) T Cout, [ L, ( T) 2, ( T)] vout, ( T). From (3), one can observe that selectng output voltage v out, leads to a second-order system whereas nput voltage concluson apples to (1). v, or nductor current n L, render frst-order systems. Smlar In addton, the output current 2, n (1) depends on the unavalable states of other converters (due to decentralzed nature of the control). Ths current appears only when v out, s selected as the measured output. Output current 2, reflects the effect of the entre networ on the ndvdual DER converter and s sometmes called an nterconnecton term and wll be dscussed later. For convenence, t can be assumed that the converter model s a second-order dynamcal system so that one can nclude dfferent systems (frst- and second-order systems) when dfferent measured outputs are selected. Note that a frst-order dynamcal system can be converted to 9

16 second-order one by addng an ntegrator;.e., addng the delayed state as a new state. Once the system s avalable n second-order dynamcal form the nonlnear decentralzed controller can be developed and s explaned next. For smplcty, the controller development s descrbed for DER measured output voltage, whch s a second-order system but the approach can be appled to the other measured states, smlarly. State-space equatons (1) can be converted nto general decentralzed form when choosng output voltage error as the output of the converter y ( ) e1, ( ) vout, and defnng the dynamc e 1) e ( ). Next, dynamc 1, ( 2, e2, ( 1) ;.e., output voltage error v ( 2), s obtaned usng the approach presented n [43] as out, e ( 1) e 1, e 2, 2, ( ) ( 1) f ( X ( )) g ( X ( )) u ( ) ( e( )) (4) for 1 n1 where u ( ) D ( ) s the converter duty cycle error (nput to the converter), y ( ) e 1, ( ) s the only converter state that s used for control, f ( X ( )) and g( X( )) are unnown nonlnear functons that orgnate from system (1) nonlneartes, and (e) characterzes unnown nonlnear dependency of output current ( ) to other 2, converters states. Functon ( e( )) depends on the entre system s state vector [ e ( ),, e ( T T wth e n [ e,, e2, ] e ( ) )] 1 1 for 1 n and s the nterconnecton term that s a functon of all the converters states ncludng the DERs and CPLs. DER state errors e (for 1 n 1 ) are presented n (4) whle state errors of the CPL e (for n1 n) wll be gven n the 10

17 followng secton. Note that when nput voltage DER model resembles e e 0, 1, ( 1) e ( 1) 1, ( ) f ( X ( )) g ( X ( )) u ( )) v, or nductor current n L, are selected, the (5) where e ) e ( 1) s the ntegrator (delay) n dscrete-tme. Model (5) can be easly 0, ( 1, represented as model (4) wth zero nterconnecton effect. Also, a new set pont for the created delayed state (measured state error) must be defned. One can convenently choose zero as the set pont, notng that vn, or L, errors are zero at steady state. Remar 2. Output feedbac control system s amed here and requres only one measured output n the proposed scheme. Note that there are three states, whch can be measured and used for DER control. However, the developed dynamcal model s of order one (vn, or L,) or two (for vout,) leavng two or one of the states unobservable, respectvely. The unobservable states are often called zero states and must be assured to be stable once the observable state s proven stable. Once the proposed adaptve controller assures stablty of the observed states, due to propertes of currents and voltages n the dc networ the unobservable states cannot be unbounded [See 42 and 43]. In addton, the nterconnecton effect appears for state dynamc v, whle t s zero for the others ( out v n,, L, ) for whch the nterconnecton effect appears n the zero dynamcs. However, snce the control mechansm s appled to all the DER and CPL converters, all observed states n the networ are stable and thus the unobservable states reman stable due to nherent lmtatons that exst for states of the dc crcut as explaned earler. Thus, one must prove that the observable states of all converters n the networ are stable smultaneously. 11

18 Remar 3. The presented model can utlze one of the three avalable converter states. In practce measurng all three converter states are not very expensve, and thus, the three measurements are avalable. Snce the presented model reles on only one output measurement, the other two measurements provde redundancy and can be used when the orgnal measured sgnal s lost, eepng the converter system up and runnng nsde the control loop wth no change n the controller parameters due to usng and adaptve neural networ controller as wll be explaned. Fg. 1.2 dc-dc buc converter B. Constant-power Load s Converter Model The dc grd feeds the constant-power and resstve loads. Resstve loads are connected to the networ buses drectly whle CPLs are connected va dc-dc buc converters that provde constant power for the load. The dc-dc buc converter s the nterface between the dc bus and the load and eeps the voltage across the load resstance constant. Thus, the absorbed power at the dc bus remans constant regardless of grd voltage varatons. The nput voltage s the bus voltage and the output voltage of ths converter s voltage across the load, as shown n Fg Remar 4. The CPL mechansm s dfferent from that of the DER n two ways. Frst, the nput current ( ) s a functon of the grd voltage and thus a functon of the entre grd state vector. It 1, reflects the nterconnecton effect. Second, the output current ( ) s only dependent on 12 2, vout ( ) and s not an nterconnecton term n CPL. These are just the opposte n DER.,

19 Smlar to DER converter, the state error vector s defned as X ( )] ( ) X( ) X o, [ vn, ( ), L, ( ), vout, T wth X )] T ( ) [ vn, ( ), L, ( ), vout, ( and X o, as the steady-state vector. Also, nput current error s and the output current s defned as 2, vout, RL,. 1, ( ) 1, ( ) I1, A thrd order system can be obtaned for CPL converter smlar to that of the DER converter descrbed n (3) [43]. In addton, smlar arguments to Remar 4 can be brought here to show that the choce of measured output can change resultng system order leadng to a maxmum second-order system and that the nterconnecton term can appear n ether observable state dynamcs or zero states (unobservable dynamcs). Ths dscusson s spped here due to smlarty to the DER case. Consequently, one can propose a converter second-order descrpton as e e 1, ( 2, 1) e ( ) for n 1 n 1 ( 1) f ( X ( )) g ( X ( )) u ( ) ( e( )) (6) 2, where u ( ) D ( ) s the converter duty cycle error and ( e( )) s the nterconnecton effect that can be zero for some choce of measurement. So far, decentralzed canoncal forms for both DER and CPL converters are explaned. The proposed decentralzed output-feedbac controller s explaned n the next secton. 13

20 Fg. 1.3 Constant-power load dc-dc buc converter. 1.4 Decentralzed Output-Feedbac Controller Desgn In the rest of the paper, system model (3) s employed to denote both DER and CPL converter models. The decentralzed output-feedbac controller ntroduced n [43] s appled to system model (3), usng nowledge of only one of the states. The control objectve s to stablze error e() at the orgn (e=0). Snce only one measurement s utlzed ( e 1, for all 1 n ), an observer s utlzed to estmate other states of the system. Two assumptons and one defnton are needed pror to desgn of the output-feedbac controller. Assumpton 1- In the small-scale system, functons g ( X ( )) satsfes 0 g, mn g ( X ( )) g,max (7) where g,mn and g, max are postve constants. It should be noted that vn, (DER nterface voltage) have maxmum value. Also, t s reasonable to assume that solar voltage vn, stays away from zero. Snce g s a functon of vn, and some other lmted parameters of the system, (7) s a realstc assumpton [43]. 14

21 Assumpton 2 [51]- The nterconnecton terms are bounded by a functon of the entre grd states such that N j 1 j ( e) e where 0 j 0 and j are postve constants for 1 n. Ths assumpton s vald for dc small-scale grds [43]. A. Observer Desgn Neural networs (NNs) approxmate general nonlnear functons T T f ( X ) W ( V X ) [52] where s the actvaton functon, W s the deal (target) weght matrx, and s the functonal reconstructon error. The unnown states of the DER and CPL converters n (6) are approxmated through neural networ (7) as e T T ( ) f( X ( 1)) g( X ( 1)) u ( 1) W1, ( V1, M( 1)) 1, ( M ( 1)) (8) 2, where M )] T ( 1) [ e1, ( 1), e2, ( 1), u ( 1 for all n 1. Whle the deal weghts, reconstructon error, and converter states are not avalable, estmatons of the NN weghts and converter states are used n the observer: eˆ 1, eˆ 2, ( ) eˆ 2, ( ) Wˆ ( 1) 1, ( 1) ( V T 1, Mˆ ( 1)) (9) where T e ˆ [ˆ e, eˆ 1, 2, ] s the estmaton of e Mˆ ( 1) [ eˆ ( 1), eˆ ( 1), u ( 1)] T and 1, 2, s estmated for M ( 1) for all 1 n. Also, ˆ 1, W R s the approxmaton of target neural networ weght 1, L 1 matrx W, 1 wth L 1, the number of the hdden layer neurons, and (.) s the neural networ actvaton functon vector. The hdden layer weght matrx V 1, s randomly chosen at the frst T T and ept constant [53]. For smplcty V M ( 1)) and V Mˆ ( 1)) are shown by ( 1) ( 1, ( 1, 15

22 and ˆ ( 1), respectvely. As the converters neural networ target weght matrces are unnown, an update law s assgned to fnd observer NN weghts (begnnng wth an ntal estmaton) as Wˆ ( ) Wˆ ( 1) ˆ( 1) [ Wˆ ( 1) ˆ( 1) l e ( 1)] (10)(10 )(10) T T 1, 1, 1, 1, 1, 1, where 0 1, 1 and l 1, 1 are user defned postve constants and e~ eˆ e s the state estmaton error. Then, the decentralzed output-feedbac controller can be developed. B. Controller Desgn Here, a controller wll be developed to stablze converter model (4) usng the estmated states of observer (9). Snce the models nclude unnown dynamcs (f and g), another neural networ s ncorporated for approxmatng the unnown dynamcs. The error dynamc of the system s wrtten as e, ( 1) f ( X ( )) g ( X ( )) u ( ) ( e( )). m One may am a behavor such as e ( 1) K g X ( )) e 2 ( ) wth K g( X( )) 1 to attan 2, (, asymptotcally stable dynamc for the error when there s no grd effect ; hence, deal 1 stablzng control nput for model (4) can be defned as u g ( X ( )) ( f ( X ( )) K e 2 ( )) where d,, K s a postve desgn constant. Nonetheless, practcally, the nternal dynamcs f ( X ( )) and control gan g ( X ( )) are unnown and u d, s not avalable. Thus, the neural networ functon approxmaton property s utlzed to approxmate u d, as u T T T d, W2, ( 2, 2, 2, 2, where V Y ( )) ( V Y ( )) K e ( ) (11) T 2 s an unnown target NN weght matrx, Y ) [ e ( ), e ( )], and (.) s the W, ( 1, 2, 2, NN functon approxmaton error for all 1 n. Snce W, 2, 2,, and the full converter state 16

23 vector e () are not avalable, u d, s developed by means of approxmate NN weghts along wth the estmated converter states through nonlnear observer (9); that s, ˆ T T u u W ( V Yˆ ( )) K zˆ ( ) (12) where ˆ d, 2, 2, m, ˆ s the NN weght estmaton matrx and T. Smlar to the W 2, Y ˆ ( ) [ˆ e ( ), eˆ ( )] observer case, the hdden layer weght matrx V2, s chosen ntally at random and ept constant. Snce NN weghts are to be estmated, defne the controller NN weght update law as Wˆ ( 1) Wˆ ˆ ( )[ Wˆ ( ) ˆ ( ) l T 2 2( ) 2 2e1 (13) ] 1, 2, where 0 2, 1 and l, 2 1 T are user defned postve constants and V Y ˆ ( )) s shown by ( 2, ˆ ( ) for smplcty. The stablty of the nonlnear dscrete-tme nterconnected system (3) s proven and gven n [44]. The errors e (), the state estmatons errors e ( ), and NN weght estmaton ~ 1 ~ W 2 errors W ( ) and ( ) of the ndvdual converters are stable and bounded n the presence of unnown dynamcs f ( X( )), control gan matrx g ( X ( )), and nterconnecton terms (e) for 1 n (See proof n [44]). Fg. 1.4 depcts the entre control system bloc dagram that s composed of the adaptve observer and controller. ~ 17

24 Fg Observer neural networ, controller neural networ, and adaptve output feedbac controller bloc dagram 18

25 Fg Test system based on IEEE 14-bus topology 1.5 Smulaton and Hardware Test Results In order to verfy effcency of the proposed decentralzed output-feedbac controller, smulatons and hardware experments are carred out n Matlab/Smuln envronment as well as on a lab setup. The test system s a dc networ developed based on IEEE 14-bus topology shown n Fgs Three PV unts are connected as the DERs n the networ whle loads nclude one CPL and a number of resstve loads that are drectly connected to the networ buses. Lnes are seres RL components wth a nomnal grd voltage of 48V. The DER and CPL controllers stablze the output voltages at ther nomnal values n the presence of dsturbances n the networ. The networ specfcatons are gven n Table 1.1. An expermental prototype s bult to examne the performance of the proposed scheme (Fg. 1.6). Sx 280W, 39.5V solar panels are used to buld three solar DERs (two panels n seres per DER). dspace Mcrolabbox control platform s used to apply the control method usng MATLAB. The control sgnals are generated by the Mcrolabbox PWM unt and routed to the optocoupler nterface board. The nterface board provdes proper sgnals to run the IGBTs drvers that n turn run the IGBT swtches. The swtch modules nclude IGBTs and dodes, confgured such that form dc-dc buc converters. Input 19

26 capactors (Cn,) are connected to the swtches nput termnals and the nductors (L) te the converters to the grd buses that are connected together through R-L lnes. Output capactors and local loads at the converters output termnals are n the lower part where the measurement board s also mounted to obtan the locally measured sgnal of each converter. The measured sgnals are sent to the control platform va McrolabBox ADC channels. Table 1.1. System parameters Parameter Value Parameter Value - PV short crcut current I sc =9.71 A P L,1 100W-Resstve - PV open crcut voltage V oc =39.5 V P L,2 100W-Resstve - PV maxmum power V oc =280 W P L,3 100W-CPL - Conv. capactors C n=300, C out=300 μf P L,4 100W-Resstve - Conv. nductance L=20 mh P L,5 100W-Resstve - Swtchng frequency f s=10 Hz P L,6 100W-Resstve - Lne resstances R Lne= 0.3 Ohm P L,7 100W-Resstve - Lne nductances L Lne= 0.3 mh P L,8 100W-Resstve - CPL conv. capactors C n=0.1, C out=0.1 mf P L,9 100W-Resstve - CPL conv. nductance L=10 mh 100W-Resstve P L,10 20

27 Fg. 1.6 Implemented dc grd Varous scenaros are examned to ensure effectveness and robustness of the proposed control scheme. It should be noted that for better stablty all PV unts are operatng n the stable 21

28 operaton zone where vn>vmpp; thus, n order to ncrease the PV delvered power, voltage at the PV termnal (nput voltage) must decrease. Neural networs are utlzed to approxmate unnown dynamcs T T f ( X ) W ( V X ) adaptvely, as explaned n Secton 4. The number of hdden layer neurons n matrx V can be ncreased to mprove the approxmaton precson; however, t s lmted to 10 neurons n all DER and CPL controllers to mnmze computatonal burden. The controllers are tuned for the three measurements (DDC output voltage, nput voltage, and nductor current) ntally and ept unchanged throughout all the followng experments as descrbed n Table 1.2. The controllers need not be altered when dsturbance scenaros change due to adaptve nature of the proposed decentralzed controller. Table 1.2. Controllers parameters NN Controller Parameters K 0.9 l l α 0.1 PI Controller Parameters K p 20 K I 50 Case 1- CPL sudden load change: In ths hardware-smulaton experment, the CPL at bus 3 s stepped up to 120W from nomnal value of 80W. The DER controllers measure output voltage and adjust the converters duty ratos to mantan grd voltage at nomnal values by extractng more power out of the PV unts, whle the CPL controller provdes a constant voltage over the resstve load. Bus 1 voltage (vout,1) and CPL voltage (vout,3) are depcted n Fg In addton, generated power of the DER #1 (Bus 1) and power consumpton at CPL bus (Bus 3) are demonstrated. From Fg. 1.7 voltages are controlled satsfactorly around the nomnal values. Moreover, the ncreased demand of the CPL resulted n addtonal DER power generatons; 22

29 however, only one DER power s llustrated here. Smulaton results and expermental ones reasonably match accordng to the fgures. P (W) P (W) P (W) 350 P1 exp Pcpl P1 exp exp Pcpl P1 sm exp Pcpl P1 sm exp Pcpl P1 sm P1 sm Tme (s) Tme (s) Tme (s) Voltage (V) Voltage (V) Voltage (V) Voltage (V) Voltage (V) Vcpl Vcpl exp V1 Vcpl sm V1 exp exp Vcpl sm V1 exp V1 exp Vcpl sm V1 sm sm Tme (s) Tme (s) Tme Tme (s) (s) Tme (s) Fg. 1.7 Case 1: DER delvered power at bus #1 and CPL absorbed power (located at bus #3) as well as voltage at bus #1 and CPL output voltage (vout1 and vout3) durng CPL power step change (ndex sm denotes smulaton results, whle ndex exp denotes expermental results). Case 2- Loss of generaton: In ths smulaton scenaro, the PV unt at bus 2 s dsconnected from the grd for 0.1 second from t=0.9s to t=1s. Whle the DER generaton at bus 2 s nterrupted, the other DERs compensate the power loss to eep the voltage of the grd at the nomnal value. The nput and output voltages as well as the njected power of the DER at bus 2 s shown n Fg. 1.8 n comparson wth the PI controllers performance that shows the proposed method effcency n regulatng the voltage n the sudden absence of one of the generaton unts usng the measured output voltage. It should be noted that the other sources compensate loss of generaton at one of the buses and mantan the grd voltage. In contrast, the PI controllers cannot retreve the pre-fault condtons usng the measured output voltage. 23

30 80 Vn2 (W) Pn2 (W) Vout2 (V) NN PI Tme (s) Fg. 1.8 Case 2: Proposed controller s performance s compared to that of the PI controller s when DER #2 s nterrupted for 0.1s. The proposed controller s able to retreve the generaton. Case 3- Loss of generaton: In ths next hardware expermental study, the DER power generaton at bus 1 s nterrupted for 0.5 second whle the output voltage s the utlzed measured sgnal of all converters. In absence of one generaton unt, the other sources compensate the loss of power at bus 1 and eep the voltage of the grd and CPL output voltage at ther nomnal values as demonstrated n Fg Due to sudden reducton of generaton at bus 1, the voltage varatons are more severe n ths case n comparson wth the prevous case (passng cloud) that exhbted slower rate of change. 24

31 Fg. 1.9 Case 3: Proposed controller performance durng nterrupton of a DER generaton Case 4- Cloud effect: In ths smulaton scenaro, the PV unt connected to bus 2, experences a generaton reducton due to passng a movng object over the PV panels. Ths reducton wll be for 0.2 second from t=0.9s to t=1.1s when 45% of the seres modules are covered. The DER s controllers are able to eep the voltage stable on the nomnal voltage. For nstance, the voltage at bus 3 remans around 48V durng the generaton reducton of DER at bus 3, whle the other resources compensate the power loss as shown Fg Although the PI controllers are also able to mantan the grd voltage, the output waveforms contan more nose and rpple that can be harmful for senstve loads n the grd. 25

32 Vn2 (V) Pn (W) Vout2 (V) Vout2 (V) Tme (s) NN PI Tme (s) Fg Case 4: Proposed controller s performance n the presence of ntermttent solar power Case 5- Cloud effect: In ths case hardware expermental study s conducted to observe cloud effects where the DER power generaton at bus 2 s reduced due to passng cloud over the panels. The cloud effect s mtated by blocng the sunrays from reachng the panels usng a large object. The employed measured sgnal n the control method s the output voltage. The other sources compensate ths power loss and the grd voltage s stablzed around the nomnal value as depcted n Fg

33 Fg Case 5: Proposed controller performance wth generaton changes due to cloud Case 6- Input voltage control (measurement redundancy): In ths hardware expermental study, performance of the controller s evaluated when the measured/controlled sgnal s the nput voltage at D ER converter of bus 2. It s amed to eep the output voltage around the nomnal value by controllng the nput voltage around the ntal set value whle dsturbances tae place n the system. Stablzng the nput voltage s expected to result n the stablty of the other states. In ths case, a large load (2W at bus 10) s added to the grd and removed after 0.2 seconds. The controller s left unchanged from the prevous case and only the measurement s swtched to nput voltage along wth ts correspondng set pont. Whle the output voltage returns to the nomnal value after the dsturbance has been removed, the nput voltage s stablzed at another operatng pont as shown n Fg Accordng to the presented decentralzed control scheme n Secton 4, the states are only bounded wth fnte steady-state error and asymptotc stablty cannot be guaranteed. Thus, nput voltage may stay stable wth an error. However, ths 27

34 does not affect the output voltage due to the power balance that s enforced by the controller. The slght reducton n the nput voltage leads to an ncreased share of power for the connected DER. Thus, other DER wll reduce power to ensure power balance n the dc crcut. 500 P1 (W) Vn2 Vout2 Voltage (V) Tme (s) Fg Case 6: Proposed controller s performance wth control on vn Case 7- Faulted crcut va voltage/current measurement (measurement redundancy): In ths hardware experment, robustness of the control scheme s evaluated n the fault condtons usng hardware experment. A low mpedance fault (Rfault=1Ω at bus 11) s appled to the grd for 0.35 seconds. After voltage and power dstortons due to the fault occurrence, the adaptve output-feedbac controllers can attan the system stablty usng only the output voltage as depcted n Fg The output voltage and power of the DER at bus 1 as well as the CPL output voltage are shown, whch confrm satsfyng performance of the proposed method n the fault condton. Conversely, the PI controllers are not able to recover the voltage after the fault as depcted n Fg

35 P1 (W) Voltage (V) Vout1 Vcpl Tme (s) Fg Case 7: Proposed controller performance durng a fault at bus 11 and control on vout Voltage (V) Vout1 Vcpl Tme (s) Fg Case 7: PI controller performance wth fault at bus 11 Next, the same ground fault has occurred (at bus 11 for 0.35s) where the nductor current L,1 s employed as the measured/controlled varable at DER converter of bus 1 and the other converters utlze vout as the controlled varable. The controller ams at eepng the current at the ntal set value. The controller s left unchanged n case 7 and only the measurement s swtched to nductor current along wth ts correspondng set pont. After fault removal the proposed decentralzed control scheme adjust the current at ts ntal value leadng to the other states return to ther ntal ponts and the entre system stablty as depcted n Fg Thus, the 29

36 approach s able to assure stablty for the system usng varous measured states whle current control approach can provde better performance n the fault condtons over the voltage snce t prevents hgh amount of currents n the fault condton. The grd and CPL voltage acheve ther nomnal values after the fault removal and system stablty s attaned va current control as depcted. P1 (W) Voltage (V) Tme (s) Fg Case 7: Proposed controller s performance n fault condton wth control on L. 1.6 Concluson In ths wor, a decentralzed dscrete-tme model for the nterconnected dc grd s developed. The dc system conssts of resstve loads as well as constant power loads and the photovoltac sources that are coupled to the dc networ va dc-dc buc converters. Decentralzed adaptve output-feedbac neural networ controllers are proposed to guarantee the entre system stablty and address the destablzng effect of the CPLs, as a crucal ssue n the dc systems. The proposed controller utlze only one measured state of the local converter and the other unnown states and nterconnecton dynamcs are approxmated through actve learnng property of the neural networ to stablze the entre system usng the decentralzed approach. In addton, the 30 Vcpl Vout1

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43 CHAPTER 2 A SIMULTANEOUS VOLTAGE AND PHASE CONTROL SCHEME FOR PHOTOVOLTAIC DISTRIBUTED GENERATION UNITS IN SMALL- SCALE POWER SYSTEMS 2.1 Introducton Renewable energy resources have recently attracted more attenton as clean and abundant sources of energy [1-5]. Harvestng energy from dstrbuted energy resources (DERs) such as solar, wnd, wave, and the tdal power are becomng more effcent and practcal. Mature technologes for nstallng dstrbuted generaton (DG) unts become more avalable and cost effectve. Ths results n an ncreased demand for dstrbuted generaton nstallatons that explot renewable power; especally, solar and wnd power whose consderable share of the total consumed energy s mmnent. The solar power s probably the most avalable renewable energy n resdental and commercal scales. Power electronc converters are used to nterface the photovoltac (PV) unts to the ac grd through dc-dc and dc-ac power electronc converters. It has been a common strategy for PV systems to be controlled such that the maxmum avalable energy s extracted from the solar panels. For ths purpose, the nverter s controlled as a current-source nverter (CSI) to nject the maxmum current to the grd [6]. However, n a grd that the DGs generate a consderable share of energy, ths strategy wll not be effcent. In such a grd, DGs should contrbute n voltage and frequency regulaton, system stablty, and all other ssues that arse n a conventonal power system smlar to the synchronous generators. In the conventonal power systems, synchronous machnes are the man sources of electrc power. Durng the past decades, ths basc part of the system s well studed and analysed by researchers and engneers. Several control schemes are desgned for the synchronous machnes ncludng voltage, frequency, actve, 37

44 and reactve power controls. Due to the exstence of such mature technologes, t s ratonal to utlze them n the nverter-base solar power generaton. Voltage source nverters (VSIs) are then requred to mae possble the mplementaton of the mentoned control mechansms [7, 8]. The modellng, control, and stablty of the parallel VSIs n a mcrogrd s nvestgated n [9]. The control mechansm conssts of two levels: a) a prmary control wth current and voltage loops that s a droop-based control, and b) a secondary control that s a local centralzed control for power sharng and restorng the voltage magntude and mtgate frequency msmatch resulted from the prmary control. Also, a droop-based control s presented n [10], where the system s parameters are estmated frst and then, actve and reactve powers are delvered to the grd ndependently. However, these wors consder a constant-voltage dc ln, an assumpton that may not be feasble n larger PV unts. Note that, capactors are more sutable n the dc ln when power fluctuatons (a factor that reduces the battery s lfetme) are sgnfcant. Droop-based strateges are proposed for ac and dc sub grds and the nterlnng converter n a hybrd system as well as other nverter-based DGs to control the actve and reactve powers and mtgate the frequency and voltage fluctuatons wth tme constants of several seconds n the secondary control loops [11, 12]. Recently, several algorthms have been proposed to control the nverters smlar to synchronous generators for more rapd response to stablty concerns. The concept of Synchronverter frst proposed n [13], s an nverter that s controlled to operate le a synchronous generator for frequency and voltage control. Whle the actve and reactve power control are doable smlar to that n the synchronous machne, n most of these wors the dc ln s connected to a battery that provdes a constant dc-ln voltage, and thus, no voltage 38

45 fluctuatons are present n the dc ln under actve and reactve power dsturbances. By contrast, the photovoltac generaton unts connected to the dc-ln capactors are prone to sgnfcant dcln voltage varatons n the absence of an approprate voltage control scheme. So far, all the above mentoned approaches assume a constant voltage at the dc ln. Installng a battery ban at the dc ln can result n a constant voltage; however, mposng fast transents to the batteres degrades them and reduces ther lfetme sgnfcantly [14]. A capactor should be nstalled at the dc ln to respond to the fast transents such that the battery can be used for longer term applcatons such as load shapng, pea shavng, etc. Smart nverters are recently proposed to add reactve power control to the conventonal nverter-based photovoltac unts n order to mantan a satsfactory voltage level. In [15] a volt/var control of PV nverters s presented for varous rradance and voltages at the Pont of Common Connecton (PCC). The controller modfes the solar unt reactve power accordng to the voltage senstvty ndces to mantan the voltage. In [16], a dstrbuted mechansm based on the conventonal Volt/Var droop control functons s appled to the PV nverters and the stablty of the controller s nvestgated. These strateges can support the reactve power to mantan the voltage wthn the permssble margns and mtgate the voltage fluctuatons. However, the photovoltac unts are not characterzed and modelled n the controller development, and thus, the dc-ln voltage varatons are not dscussed. In contrast, a detaled model of the photovoltac generaton unts s descrbed n ths paper where the dc-ln voltage stablty s explctly consdered and smultaneous output voltage and frequency control are targeted. A dc-dc buc converter s used to connect the photovoltac unt to the nverter. The dc-ln voltage s mantaned va a capactor that provdes stored energy 39

46 and contrbutes to the system stablty n transents, smlar to the synchronous generator rotatng mass. An exctaton-le mechansm s employed to control the nverter ac voltage through the nverter s ampltude modulaton factor. Then, a fast droop controller vares the dc-dc converter s duty cycle to control the power balance at the capactor termnals va photovoltac output power. Once the capactor voltage s controlled, the voltage angle at the nverter termnal, whch s coupled wth the capactor voltage varatons, s stablzed, and frequency stablty s acheved as well [17, 18]. The rest of the paper s organzed a follows: the proposed modellng and control scheme are presented n Secton 2. Secton 3 provdes stablty analyss of the nterconnected grd. Smulaton scenaros and results are gven n Secton 4. Fnally, the concluson remars wll be presented n Secton System Topology and Proposed Control Scheme A. System topology The buc topology s used as the dc-dc converter of the proposed system. Although the boost converter poses hgher output voltage, due to the stablty problems, the buc converter s more relable for the PV system [19, 20]. For ncreasng the voltage level, transformers at the ac sde or topologes le forward converters that employ transformers for steppng up the voltage can be used, wthout affectng stablty of the system [21]. There are two capactors at the nput and output of the dc-dc converter. The nput capactor provdes a smoother voltage at the nput, and the output capactor, as wll be dscussed later, plays an mportant role n system stablty. 40

47 Fg 2.1. The entre system bloc dagram. The prmary source of energy s a PV unt. In order to realze a stable operaton for the system, voltage of the PV should vary between the open crcut voltage (Voc) and voltage at the maxmum power pont (Vmpp). The dc-dc converter duty rato vares accordng to the ac sde voltage and the modulaton factor, to eep the PV s voltage n the permssble range. The dc-ac converson s carred out usng a three-phase two-level PWM nverter. The nverter s connected to the Pont of Common Connecton (PCC) through an LC flter and step up transformers. The entre system bloc dagram s depcted n Fg B. Control Scheme The dc-dc converter s connected to the nverter through the dc ln capactor. The energy stored n ths capactor acts le the netc energy stored n rotatng mass of the synchronous generator. The sze of the capactor determnes the amount of stored energy that s avalable at the dc ln. However, hgher sze of the capactor results n hgher cost for the system. The amount of stored energy s also dependent on dc ln voltage. Dynamc equatons for the power at the dc ln can be wrtten as CV V P P (1) C C n o 41

48 where Pn s the power that s njected to the capactor from the dc-dc buc converter and comes from the prmary source of energy that s a PV here, Po s the delvered power to the nverter, and VC s the capactor voltage. On the other hand, the nverter s delvered power to the grd can be wrtten as Pe BV V sn( ) (2) nv where B s the admttance that connects the nverter to the grd, Vnv and γ are the voltage magntude and phase angle at the nverter s output termnals, and V and θ are the grd bus voltage magntude and phase angle. Neglectng the nverter s losses results n Pe P o. In (2), B s constant, V and θ are also not under control and dctated by the grd, Vnv should also be wthn the allowable range (normally between 0.95 and 1.05pu). Then, the only varable that can be adjusted to control the nverter s output power s γ. A new varable λ s ntroduced to control γ n the nverter as (4) Parameter λ resembles rotor speed (ω) n the synchronous generator. The rotor speed defne the netc energy stored n the rotor as / M( P m P ) (5) 1 elec where M s the nerta coeffcent (2H n seconds), Pm s the nput mechancal power and Pelec s the output electrcal power. In order to relate the new varable λ to the stored energy n the dc ln capactor, t can be defned as 1 C)( P n P ) (6) ( o By usng equaton (1) 42

49 VCV C (7) that yelds 2 2 C V Co ( V ) 2 (8) Hence, λ s the scaled changes n capactor stored energy. Equaton (6) relates the output power at the ac sde to the avalable power that comes from the prmary source of energy at the dc sde. In the synchronous generators, Pm s the nput power that comes from the turbne and Pelec s the generator s output power that s delvered to the grd. Dfference between nput and output powers s the netc energy that s stored n the generator rotatng mass and s a functon of rotor speed and nerta coeffcent. In the DG system, Pn s the nput power that comes from the source of energy at dc sde, Po s the nverter s output power that s delvered to the power system. Dfference between nput and output powers s the stored energy at the dc-ln capactor that s a functon of dc-ln voltage and sze of the capactor. In the synchronous generators, droop mechansm provdes addtonal torque durng dsturbances n the power system. Any devaton from the synchronous speed causes an addtonal dampng torque that opposes any change n the speed of the generator. Hence, the resultant torque helps mantan synchronsm of the machne. Ths technology can be utlzed to mantan the dc-ln capactor voltage. Thus, as an actve power control strategy, any devaton from the steady state value of λ wll result n some change n the dc-dc converter s duty cycle, D. Ths adjustment subsequently changes the operatng pont of the PV unt whch alters ts output power. Thus, any varaton n λ follows by some change n the njected power whch comes from 43

50 the dc sde and helps mantan λ at ts steady state value smlar to operaton of machne governor. So far, the DG system s modeled and controlled to behave smlar to classcal model of a synchronous generator wth a constant voltage source s behnd the transent reactance. In order to provde voltage magntude and consequently reactve power control, an exctaton-le mechansm s employed. Smlar to the rotor flux and feld voltage n a synchronous generator, two new varables, E q and Efd, are defned for the DG system. Thus, a new dynamc s added to the system as d0 E 1 q ( xd xd ) Eq (( xd xd ) xd ) V cos( ) E T fd (9) where x dr, x dr are drect-axs synchronous and transent reactances, respectvely, and T d 0 s drect-axs transent open-crcut tme constant that can be selcted for the proposed model. Here, these parameters can be set as desred n contrast wth the synchronous machnes, where the parameters are constant for each machne. Thus, the exctaton-le system s developed usng preferred parameters. For nstance, T d 0 that defnes the exctaton crcut tme constant, can be set as preferred based on system requrements. Then, Efd s adjusted for voltage and reactve power regulaton of the DG system. Varyng Efd alters E q as the rotor flux that n turn contrbutes to adjustng voltage magntude of the synchronous machne. Tang nto account (9) along wth (4) and (6), the model resembles fluxdecay model of the synchronous generator. The beneft of ths model s to mae possble applcaton of all avalable technologes for a synchronous generator such as automatc voltage 44

51 regulator (AVR), power system stablzer (PSS), etc. The output power of a synchronous generator that s descrbed va flux-decay model s [22] P (1 x ) V [ E sn( ) (( x x ) (2 x )) V sn(2( ))] (10) e d q q d q However, the nverter output power s as descrbed n (1). In order to have an dentcal behavour to the flux-decay model, nverter s output power should be equal to Pe n (10). By equatng (10) and (2), Vnv, the voltage magntude at the nverter s output termnals, s derved as V xq xd Eq sn( ) V sn(2( )) 2 xq x Bsn( ) nv (11) d In order to attan ths voltage, the nverter modulaton factor, nv, s adjusted such that the fundamental harmonc of the voltage at nverter output termnal s Vnv. Therefore, smlar to the flux-decay model of a synchronous generator, accordng to (4), (6), and (9), the DG system s descrbed va a thrd order dynamc equaton where E q, γ, and λ are the state varables, and nput power Pn, and feld voltage, Efd, are control nputs of the system. Fgure 2.2 shows the relaton between dc and ac dynamcs and the actve power control through the droop-le mechansm. Also, any change n the Efd s reflected n Vnv and the nverter s modulaton factor nv, as shown n (11). Thus, the exctaton system controls the voltage ampltude usng modulaton factor, nv. Whle the nput power s controlled through the droop mechansm, the other nput, Efd, can be controlled va an AVR. The AVR receves the error between voltage set value and measured voltage magntudes and generate the approprate feld voltage Efd to attan the voltage at the set value. The voltage magntude and reactve power regulaton can be done through the AVR smlar to the synchronous generator. However, usng an AVR adds another dynamc to the 45

52 system wheren the feld voltage Efd wll be the forth state varable and control nputs are voltage magntude set value and nput power. In addton, the PSS adds a modulaton sgnal to the voltage controller nput. Although more detals can be taen nto account; here, the focus s to propose the central framewor of synchronous generator for DG unts. Fg 2.2. Voltage phase and magntude control dagram Remar 1: The modelng and control presented n ths paper s a short-term analyss and must not be confused wth the longer-term dynamcs ntroduced by the maxmum power pont tracers (MPPTs) and daly load changes. Rather, the proposed approach studes dsturbances that can cause rapd voltage fluctuatons, whch can adversely affect the PV power generaton, and could potentally lead to the entre networ nstablty. Thus, the slow dynamcs of MPPT are gnored. 2.3 Smulaton Results Effcency of the proposed scheme s nvestgated through smulatons. The smulatons are carred out usng Matlab\Smuln. Frst, stable operaton of the proposed method n standalone condton s evaluated n comparson to the Synchronverter model [13]. The parameters of the systems ncludng the Synchronverter and the proposed model are lsted n Table 2.1. In [23], t s shown that the stablty of the Synchronverter s dependent to an external seres resstance at the nverters termnals. Presence of ths seres resstance decrease the effcency of 46

53 the Synchronverter, ths defcency s addressed by modfyng the orgnal Synchronverter model. Case 1- Load change: As the frst smulaton scenaro, the synchronverter and proposed scheme are appled to a stand-alone DG unt that s feedng a local load. At t=2s the load s stepped up from 200W to 250W for half a second and then returns to ts ntal value. Satsfyng performance of the proposed control method s shown n Fg The voltage at the PV termnals decrease to provde more power at the output based on PV characterstcs. Subsequently, the buc converter duty cycle s ncreased to push bac the dc ln voltage to the ntal value. Stable dc ln voltage results n steady operaton of the entre DG unt. In addton, the feld voltage (Efd) s set on the nomnal value to acheve nomnal voltage value at the nverter output termnals. On the other hand, the Synchronverter cannot restore dc ln voltage and faled to feed the load approprately as demonstrated n Fg A PI controller s appled to adjust the dc ln voltage at the ntal value n the Synchronverter model. Table 2.1. System parameters Parameter Value Parameter Value - PV short crcut current I sc =9.71 A 0.2 pu - PV open crcut voltage V oc =39.5 V 0.06 pu - PV maxmum power V oc =280 W x - Conv. capactors C n=1000 q 0.19 pu - Conv. Capactor C out=1000 μf T do 7s - Conv. nductance L=2.5 mh B 5 - Swtchng frequency f s=10 Hz D p Lne resstances R Lne= 0.5 Ohm D q Lne nductances L Lne= 20 mh V grd V (L-L)=200 x d x d f v

54 45 Voltage (V) (a) Voltage (V) (b) Power (W) Tme (s) (c) Fg 2.3. Case 1: The proposed model operaton n stand-alone operaton: a) Voltage at the PV termnals, b) Voltage at the DC ln c) Output power 48

55 45 Voltage (V) (a) Voltage (V) (b) Power (W) Tme (s) (c) Fg Case 1: Synchronverter performance n stand-alone operaton: a) Voltage at the PV termnals, b) Voltage at the DC ln c) Output power Case 2- Load change: In the second smulaton scenaro, effcency of the control scheme s evaluated n the benchmar low voltage mcrogrd that s shown n Fg Two PV-based 49

56 DG unts equpped wth the proposed control system feed fve resstve loads n the grd. DG 1 s composed of three PV panels n seres and DG 2 nclude two panels n seres. As the second smulaton scenaro, Load 4 s connected to the grd at t=1s and removed after half a second whle DGs are feedng the grd under normal condtons. The control method s capable of sharng the load between the DGs and system sustan ts stablty after frst swngs. The voltage at the PV termnal decreases to extract more power out of the PV, then buc converter s duty cycle ncreases to mantan the dc ln voltage at the ntal value as shown n Fg. 2.6 and Fg In addton, the voltage magntude s regulated through the nverter s modulaton factor accordng to exctaton-le mechansm n (11). The modulaton factor (Knv) vares to adjust the voltage magntude at the nomnal value as shown n Fg It should be mentoned that the Efd remans constant; however, t s nternal dynamcs of the synchronous generator-le model that tends to eep the output voltage constant. Fg Benchmar low voltage mcrogrd 50

57 120 PV Voltage (V) dc Ln Voltage (V) Power (W) (a) (b) Tme (s) (c) Fg Case 2: DG#1 performance durng load change: a) PV output voltage, b) dc ln voltage, and c) DG output power 80 PV Voltage (V) (a) dc Ln Voltage (V) (b) Fg Case 2: DG#2 performance durng load change: a) PV output voltage, b) dc ln voltage, and c) DG power (Fg. 2.7 cont'd) 51

58 Output Power (W) Tme (s) (c) 1.1 Voltage (pu) (a) 1.02 Knv Tme (s) 200 (b) Voltage (V) Tme (s) (c) Fg Case 2: Voltage at DG#1: a) Output voltage RMS value, b) Inverter modulaton factor, and c) Zoomed voltage waveform Case 3- Cloud effect: As the thrd smulaton scenaro, cloud effect on PV-based DG unts s nvestgated. In the thrd smulaton scenaro, DG unts are feedng the grd n normal condton whle a movng cloud passes over one of the DG s PV panels and decreases the generaton capacty of the DG by 50% for half a second. Then, voltage over that PV panels decreases to extract more power out of the panels, dc-dc converter duty cycle s also ncreased to mantan the dc ln voltage around ntal set value. On the other hand, voltage over the other 52

59 DG s PV panels dropped to nject more power to the grd and compensate power loss of the other DG unt as shown n Fg. 2.9 and Fg Voltage (V) (a) 62 Voltage (V) (b) Power (W) Tme (s) (c) Fg Case 3: DG#1 performance n the cloud scenaro: a) PV output voltage, b) dc ln voltage, and c) DG output power Voltage (V) (a) Fg Case 3: DG#2 performance n the cloud scenaro: a) PV output voltage, b) dc ln voltage, and c) DG output power (Fg cont'd) 53

60 Voltage (V) Tme (s) (b) Power (W) Expermental Results Tme (s) (c) An expermental setup s bult to valdate stable operaton of the system and dynamc voltage control capablty. The proposed control scheme s appled to the DG unt that s composed of a dc-dc buc converter and a dc-ac converter as depcted n Fg dspace Mcrolabbox control platform s used to apply the control method usng MATLAB. The PV panels outputs are connected to dc-dc buc converters. The buc converter provdes a stable dc voltage at the output. Subsequently, the nverter generate a three phase ac voltage usng the stable dc voltage. In order to elmnate the harmoncs and create a snusodal waveform out of the nverter output LC flters are employed. The resultant voltage feed a local load and also the grd through step up transformers that are arranged n Delta-Star confguraton. The developed mcrogrd s based on benchmar low voltage mcrogrd n Fg

61 (a) (b) Fg Expermental setup a) Detaled graph of a DG unt b) Implemented low voltage mcrogrd 55

62 Case 4- Load Change: In the frst experment scenaro, two PV-based DG unts are feedng the loads n the low voltage mcrogrd. At t=6.75s load 4 s added to the grd and removed after four seconds. Dc ln voltage, dc-dc converter s nductor current, and output power of each DG s shown n Fg and Fg The presented results demonstrate desred performance of the entre system n sharng the added load properly between the sources whle the dc lns voltages reman stable around the ntal values. Voltage of the grd s also shown n Fg that shows satsfactory condton of the grd. 60 Voltage (V) (a) Current (A) (b) Power (W) Tme (s) (c) Fg Case 4: DG#1 performance durng load change: a) Voltage at dc ln, b) dc-dc converter nductor current, and c) DG output power 56

63 33 Voltage (V) (a) Current (A) Power (W) (b) Tme (s) (c) Fg Case 4: DG#2 performance: a) dc ln voltage, b) nductor dc current, c) output power 175 Voltage (V) (a) Voltage (V) Tme (s) (b) Fg Case 4: Voltage at Bus 1: a) Zoomed voltage, and b) fltered voltage waveform Case 5- Cloud effect: In the second experment, effect of a movng cloud on PV-based DG unts s evaluated. The DG unts are feedng the grd n normal condton whle generaton 57

64 capacty of one of the DG unts decreases by 30% due to a passng cloud for 3s. Thus, voltage over the shaded PV panels drops to extract the maxmum avalable power out of the panels, dcdc converter duty cycle s also ncreased to mantan the dc ln voltage around ntal set value. On the other hand, voltage over the other DG s PV panels decreases to nject more power to the grd and compensate power loss of the other DG unt. However, dc-ln voltage s ept at the ntal set value due to the appled control mechansm as shown n Fg and Fg Voltage (V) (a) 6 Current (A) (b) Power (w) Tme (s) (c) Fg Case 5: DG#1 performance n presence of movng cloud: a) Voltage at dc ln, b) dcdc converter nductor current, and c) DG output power 58

65 30 Voltage (V) (a) Voltage (V) (b) Power (W) Tme (s) (c) Fg Case 5: DG#2 performance n presence of movng cloud: a) Voltage at dc ln, b) dcdc converter nductor current, and c) DG output power 2.5 Conclusons A novel control scheme for ntegratng solar power nto the grd s proposed. The photovoltac unt s connected to the grd through a dc-dc buc converter and voltage source nverter. The entre system s controlled such that the photovoltac DG unt mmcs a synchronous generator behavor n whch the capactor nstalled at the nverter dc ln, acts smlar to the synchronous generator rotatng mass and mantans stablty of the system n case of sudden change n the loads or the other system dsturbances. An AVR-le mechansm s also ntroduced to the system that facltates DG s voltage control le a synchronous machne. The proposed control scheme exhbt desred performance n stand-alone condton and n a mcrogrd 59

66 envronment. In addton, a prototype system s mplemented to valdate precse operaton of the control approach. 2.6 References [1] Y. A. R. I.Mohamed and E. F. El-Saadany, Adaptve decentralzed droop controller to preserve power sharng stablty of paralleled nverters n dstrbuted generaton mcrogrds, IEEE Trans. Power Electron., vol. 23, no. 6, pp , Nov [2] J. C. Vasquez, J. M. Guerrero, A. Luna, P. Rodrguez, and R. Teodorescu, Adaptve droop control appled to voltage-source nverters operatng n grd-connected and slanded modes, IEEE Trans. Ind. Electron., vol. 56, no. 10, pp , Oct [3] A. Bdram, A. Davoud, and R. S. Balog, Control and crcut technques to mtgate partal shadng effects n photovoltacs arrays, IEEE J. Photovolt., vol. 2, no. 4, pp , Oct [4] Y. C. Chen and A. D. Garca, A method to study the effect of renewable resource varablty on power system dynamcs, IEEE Trans. Power Syst., vol. 27, no. 4, pp , Nov [5] D. Gautam, V. Vttal, and T. Harbour, Impact of ncreased penetraton of DFIGbasedwnd turbne generators on transent and small sgnal stablty of power systems, IEEE Trans. Power Syst., vol. 24, no. 3, pp , Aug [6] Anand, S., Gundlapall, S. K., Fernandes, B. G.: 'Transformer-less grd feedng current source nverter for solar photovoltac system', IEEE Trans. Industral Electroncs, vol. 61, no. 10, pp [7] Olvares, D. E., Mehrz-San, A., Etemad, A. H., et al.: 'Trends n mcrogrd control', IEEE Trans. Smart Grd, vol. 5, no. 4, pp [8] EPRI., ' Laboratory evaluaton of grd-ted photovoltac and energy storage systems ' (EPRI, Palo Alto, CA), pp [9] Vasquez, J. C., Guerrero, J. M., Savagheb, M., et al.: 'Modellng, analyss, and desgn of statonary-reference-frame droop-controlled parallel three-phase voltage source nverters', IEEE Trans. Industral Electroncs, vol. 60, no. 4, pp

67 [10] Vasquez, J. C., Guerrero, J. M., Luna, A., et al.: 'Adaptve droop control appled to voltage-source nverters operatng n grd-connected and slanded modes', IEEE Trans. Industral Electroncs, vol. 56, no. 10, pp [11] Amelan, M., Hooshmand, R., Khodabahshan, et al.: 'small sgnal stablty mprovement of a wnd turbne-based doubly fed nducton generator n a mcrogrd envronment', Proc. 3rd Int. Conf. Computer and Knowledge Engneerng, Mashhad, Iran, 2013 [12] Loh, P., L, D., Cha, Y. K., et al.: 'Autonomous control of nterlnng converter wth energy storage n hybrd ac dc mcrogrd', IEEE Trans. Industryl Applcatons, 49, (3), pp [13] Zhong, Q., Wess, G.: 'Synchronverters: nverters that mmc synchronous generators', IEEE Trans. Industral Electroncs, 58, (4), pp [14] EPRI., 'The solar-to-battery and communty energy storage project demonstratons at the solar energy acceleraton center' (EPRI, Palo Alto, CA), pp [15] Zhong, Q., Nguyen, P., Ma, Z., Sheng, W.: 'Dstrbuted volt/var control by pv nverters', IEEE Trans. Power Systems, 28, (3), pp [16] Malepour, A. R., Pahawa, A.: 'A dynamc operatonal scheme for resdental pv smart nverters', IEEE Trans. Smart Grd, 2, (99), pp [17] Kazemlou, S., Mehraeen, S.: 'Novel decentralzed control of power systems wth penetraton of renewable energy sources n small-scale power systems', IEEE Trans. Energy Converson, vol. 29, no. 4, pp [18] H. Saber, and S. Mehraeen, A Smultaneous Voltage and Frequency Control Scheme for Photovoltac Dstrbuted Generaton Unts n Small-scale Power Systems, Proc. Energy converson Congress & Expo (ECCE), [19] R. C. N. Plawa-Podgurs and D. J. Perreault, Submodule ntegrated dstrbuted maxmum power pont tracng for solar photovoltac applcatons, IEEE Trans. Power Electron., vol. 28, no. 6, pp , Jun [20] G. R. Waler and P. C. Serna, Cascaded dc dc converter connecton of photovoltac modules, IEEE Trans. Power Electron., vol. 19, no. 4, pp , Jul

68 [21] N. Mohan, T. M. Undeland, and W. P. Robbns, Power Electroncs: Converters, Applcatons, and Desgn. Hoboen, NJ, USA: Wley, [22] P.W. Sauer and M. A. Pa, Power system dynamcs and stablty, Stpess, Jan [23] Pya, P., Karm-Gharteman, M.: 'A stablty analyss and effcency mprovement of synchronverter', Proc. Appled Power Electroncs Conf. and Exposton,

69 CHAPTER 3 STABILITY IMPROVEMENT OF MICROGRIDS USING A NOVEL REDUCED UPFC STRUCTURE VIA NONLINEAR OPTIMAL CONTROL 3.1 Introducton Applcaton of Flexble AC Transmsson System (FACTS) devces are ncreasng for power qualty and stablty mprovement n the power systems. One of the most attractve devces s Unfed Power Flow Controller (UPFC) that s used n transmsson systems for power flow control and transent stablty mprovement. Also, t can be utlzed n dstrbuton systems and mcro grds to mprove transent stablty n addton to power flow control. In small-scale power systems and mcro grds wth low stored energy level, transent stablty of the system s of utmost mportance whle varous dsturbances threaten stablty of the system. The UPFC conssts of two man parts: shunt and seres branches that are coupled together through a dc ln. The shunt branch control voltage at the shunt bus through absorbng or njectng requred reactve power. It also absorb the necessary actve power from the grd to provde a constant voltage at the dc ln. Although some dynamc varaton are observable at the dc ln voltage, t should be ept constant n steady state to acheve desred performance of the UPFC [1]. The seres branch nject a seres voltage to the power lnes that s controlled for dfferent purposes such as drect voltage njecton, phase angle shfter and lne mpedance emulator, and automatc power flow control. The automatc power flow control can also be employed for oscllaton dampng and stablty mprovement. Thus, the njected seres voltage can be controlled such that system stablty s enhanced. Dfferent controllers can be used to determne the requred seres voltage based on states of the systems. Prevously, lnear approaches has been utlzed to control the power systems [2-63

70 7]. In those methods, the system s lnearzed around an operatng pont so that lnear methods are applcable provdng the system remans n a small neghborhood around the operatng pont that s not always a correct assumpton. However, snce the power systems are nherently nonlnear, nonlnear approaches are more approprate. In addton, employng an optmal nonlnear control method mnmzes the cost that s defned as a functon of states and control varables [8-11]. The optmal nonlnear control method can be derved by solvng the Hamlton Jacob Bellman (HJB) equaton. Recently, renewable energy resources, especally Photovoltac (PV) unts, have attracted more attenton as envronment-frendly and abundant resource of energy. In order to ntegrate solar energy nto the grd several methods and technologes are presented such as mcronverters [12-13]. The man goal has been extractng maxmum power out of the PV sources and nject t to the grd properly. Those methods are approprate when the share of energy that comes from renewable energy resources s neglgble. However, n a grd that the renewable energy resources generate a consderable share of energy Dstrbuted Generaton (DG) unts should also contrbute n power system control and stablty [14-17]. In ths paper, a small-scale mcro grd that conssts of DG unts as the sources of energy s studed. The DG unts are Photovoltac (PV) unts that are connected to the grd va dc-dc buc converters and nverters. The DG unts are modeled and controlled to operate smlar to a synchronous generator classcal model n the power system. In order to mprove the transent stablty of the system, a novel reduced UPFC structure s proposed that decreases the mplementaton cost of the UPFC consderably. Also, an optmzed mechansm s utlzed for the 64

71 UPFC control to stablze the power system by mposng the mnmum cost. The mnmzed cost functon results n lower stress on power electroncs devces that ncrease ther lfetme. 3.2 Modelng and Control of the System The DG unt utlzes a PV source as prmary source of energy. A dc-dc buc converter connects the PV unt to the nverter through a dc ln capactor. The dc-dc converter provdes a constant voltage at the dc ln regardless of voltage varaton at the PV s output termnals. The regulated dc voltage s also used by the reduced UPFC to generate the requred seres voltage and nject t to the grd s power lnes to enhance system stablty. In ths secton frst the DG unt modelng and control s presented and then, modelng and control of the UPFC s developed. A. DG Unt Modelng and Control The DG unt s modeled and controlled such that mmc behavor of a synchronous generator. Dynamc equatons for the power at the dc ln can be wrtten as CV V P P (1) C C n o where Pn s the power that s njected to the capactor from the dc-dc buc converter, Po s the delvered power to the nverter, and Vc s the capactor voltage. On the other hand, the nverter s delvered power to the grd wll be Pe BV V sn( ) (2) s 1 where B s the admttance that connect the nverter to the grd bus, Vs and φ are the voltage magntude and phase angle at the nverter s output termnals, and V1 and θ are the grd bus s voltage magntude and phase angle. Neglectng the nverter s losses results n P P e o 65

72 In (2), the angle φ behaves smlar to the stator angle δ n the synchronous generator and the nverters output power can be adjusted through changng ths varable. A new varable λ s ntroduced to control φ n the nverter as (3) λ s smlar to the rotor speed (ω) n the synchronous generator. The rotor speed defne ts stored netc energy as 1 M )( P m P ) (4) ( elec where M s the moment of nerta, Pm s the nput mechancal power and Pelec s the output electrcal power. In order to relate the new varable λ to the stored energy n the dc ln capactor, t can be defned as 1 C)( P n P ) (5) ( o Equatons (3) and (5) resemble the equatons of a synchronous generator classcal model. In order to mmc the droop mechansm n synchronous generators, any devaton from the steady state value of λ wll result n some change n the dc-dc converter s duty factor, D. Ths adjustment subsequently changes the operatng pont of the PV unt whch alters ts output power. Thus, any varaton n λ follows by some change n the njected power whch comes from the dc sde that pull bac the λ to ts steady state value. Fg System composed of DG and UPFC 66

73 B. Modelng and Control of the System n Presence of the UPFC The vector dagram of the system shown n Fg. 3.1 s depcted n Fg The seres voltage of the UPFC s composed of two components Vup and Vuq. The voltage components are consdered to be proportonal to the voltage at the pont of connecton of the UPFC. Consequently, V uq V (t), V V (t) r up (6) r where β(t) and γ(t) are two control varables. When the UPFC s taen nto account, the electrcal power n (2) can be wrtten as Pe VsV1 Bsn( ) VsV1 B(sn cos cos sn) (7) Based on the dagram n Fg.2, Vuq V cos V 1 r (1 ) V V 1 r sn V ( ) V V up r (8) V1 1 Substtutng (8) and (9) nto (7), dynamcs of the varable λ can be expressed as C P V V Bsn( ) D V V Bcos( ) ( t) V V Bsn( ) ( t) (9) n s r s r s r Accordng to the (3) and (10) the space state equatons of the system wll be wrtten n the form of x f ( x) g( x) u( t) as follows ( Pn VsVr Bsn( ) D) VsVr Bcos( ) VsVr Bsn( ) C C C (10) By assumng the tme step T, the system dynamc equatons can be approxmated n dscrete-tme by 67

74 T ( ) 0 0 ( 1) T T T ( 1) ( Pn VsVr Bsn( ( )) D( )) ( ) VsVr B cos( ) VsVr Bsn( ) C C C (11) Next, an optmal nonlnear control approach s appled to create the nputs β and γ accordng to the system states γ, λ. Fg Vector dagram of DG and UPFC 3.3 Nonlnear Optmal Controller Desgn A. Control Objectve Consder the affne nonlnear state feedbac dscrete-tme system as x 1 f g u (12) n n where x x( ) s the state vector at step, f f x ), f (.), g g x ), are nput gan ( ( functons whch are smooth and defned n a neghborhood of the orgn, u m u( x ), s the control nput. Assume that f g u s Lpschtz contnuous and there exsts a control polcy that can asymptotcally stablzed the system. Then, the tas s converted to fnd a control nput u that mnmzes the generalzed quadratc dscrete tme cost functon T T ( Q( x j ) u j Ru j ) Q( x ) u Ru V 1 j V (13) 68

75 where Q and R are postve defnte matrces. Meanwhle, the control polcy u needs not only stablzes the system, but guarantees the cost functon V s fnte, whch means u s admssble. Defnton 1. (Admssble Control): a pecewse contnuous state feedbac control u( x( )) s sad to be an admssble control f t can not only stablze the system but also guarantee the cost functon to be fnte, whch means J J x(0), u ). ( Note that, admssble nput can mae sure the system convergence, however, not all converged control nput are admssble [8]. Thus, defne the Hamltonan functon as T H( x, u ) V( f g u ) V( x ) Q( x ) u Ru (14) Accordng to the Bellman optmalty prncple [9] the goal s to mnmze the Hamltonan functon, and when t goes to zero, we wll have dscrete-tme HJB equaton [8] T V ( f g u ) V( x ) Q( x ) u Ru 0 (15) where the optmal control polcy u s the soluton of (15). By utlzng the statonary condtons, the optmal control polcy u can be realzed from H( x, u, w ) / u 0 as 1 T 1 2R g V 1 x 1 u (16) Next, by substtutng (16) nto (15), the HJB equaton becomes T T * V V Q( x ) 1 4 V x g R g( x ) V x 0 (17) Due to the nature of partal dervatves V 1 x and nonlnearly of the equaton, t s stll 1 a challengng tas to solve (17). Therefore, applyng Tylor seres expanson s one way to 69

76 overcome the obstacle. The dfference of two adjacent cost functon V can be expanded by remanng the frst and second order terms n the Taylor Seres and gnorng hgher order terms. In other words, hgher order terms can be treated as small dsturbances around the operatng ponts. Thus, we obtan V T 1 T 2 V 1 V V ( x1 x ) ( x1 x ) V ( x1 x ) (18) 2 where V and 2 V are the gradent vector and Hessan matrx, respectvely. Then, we obtan generalzed HJB equaton (19), by substtutng (18) nto (14). T 1 T 2 Q( x ) u Ru V ( f gu x ) ( f gu x ) V ( f gu x ) 0. (19) 2 wth the new GHJB equaton, another Hamltonan functon needs to be defned as Defnton 2. Defnton 2. (Pre-Hamltonan Functon): A sutable pre-hamltonan functon for the affne nonlnear dscrete-tme system (12) s defned as 1 T 2 T H( x, V, u ) V ( f gu x ) ( f gu x ) V ( f gu x ) Q( x ) u Ru. (20) 2 The new optmal control polcy u can be derved as T 2 1 T T 2 g V g 2R g V V ( f h w x ) u (21) Now, the optmal control polcy contans only the current step states 70 x and get rd of x1 compare to polcy (16). However, the soluton of GHJB s stll challengng to solve and the value functon remans unnown. A successve approxmaton methodology s utlzed by updatng the approxmaton of GHJB and controller every teraton untl t reaches to the optmal polcy. In each teraton, a crtc NN s appled to approxmate the value functon to adjust the weghts vector.

77 B. Value Functon Approxmaton usng Neural Networ As prevously dscussed, the successve approxmaton method can be used to fnd the optmal control polcy wth modfed lnear GHJB (20). However, recursvely solvng GHJB and updatng controller s mpossble wthout nowng V. Before the teratve based method s mplemented, V s approxmated by NN. The NN s well nown for smooth functon approxmaton and been wdely used n prevous research. Thus, t s an approprate approach for our dscrete-tme problem. A nonlnear functon can be approxmated by a NN as L V ( x) ( x) W L l 1 l l T L ( x) L (22) where L s the number of neurons n the hdden layer, ωl, are NN weghts and the actvaton functon vector (x).then the actvaton functon and weghts are vectored as T and W T L 1 l L L 1 L. The resdual error wll be mnmzed by adjustng the weghts n each teraton and be evaluated by least squares method. The soluton wll guarantee the lowest weghts resdual error. A resdual error s set as GHJBV ( ) L L ( ) l l ( x), u el ( x) (23) l1 where VL denotes the approxmaton of the cost functon. The weghts ωl are obtaned by el ( x) 0 W.e. L x ϵ Ω,whch means the error remans unchanged wth respect to current weghts. 71

78 el ( x), el( x) 0 (24) W L Substtutng (18) and (22) nto (24) yelds 1 T 2 1 T 2 T 1 T 2 Lx x Lx, Lx x Lx. WL Q( x) u Ru, Lx x Lx 0 (25) where the terms L and 2 L are gradent vector and Hessan matrx of L (x) wth respect to x, respectvely, and x f ( x) g( x) u( x) x. By defnng L T 1 T 2 j x x jx 2 as 1, the current weght vector can be wrtten as W L 1 T, Q( x) u Ru, (26) In [8], the term, s shown to be full and therefore, t can be nverted, whch means a unque soluton WL exsts. In Hlbert space, nner product s defned as N a( x), b( x) a( x) b( x) dx a( x) b( x) x (27) 1 Calculatng the ntegraton n (27) s computatonally costly. However, by utlzng the Remann defnton of ntegraton, t s reasonable to approxmate the ntegraton n a acceptable degree. Ths leads to a nearly optmal soluton algorthm. A mesh s generated on Ω, where the mesh sze s x, that s around the operaton ponts,(26) can be smplfed as W L T 1 X X XY (28) where X and Y are n vector form as T 2 T 2 x x x / 2 x x x / X 2 (29) L L xx 1 L L xx p T 72

79 ( ) T ( ) Q( x) u Ru xx1 Y ( ) T ( ) Q( x) u Ru x xp (30) where p n xp denotes the number of ponts of the mesh. Ths number ncreases as the mesh sze s reduced. Note that actvaton functon should be contnuous and lnearly ndependent. The mesh must generate more ponts compare to the order of the approxmaton L to guarantee the convergence. These condtons mae sure ( X T X ) s full ran [10]. C. Successve Approxmaton of the Approxmate HJB Equaton polcy The successve approxmaton procedure starts wth a selected ntal admssble control (0) u, teratvely updates the control nput Hamltonan equaton (20) s solved forv () as () u n a loop wth ndex. Then, the pre- ( ) ( ) 1 ( ) T 2 (, j) ( ) ( ) T ( ) V ( f gu x ) ( f gu x ) V ( f gu x ) Q( x ) u Ru 0. (31) 2 Next, () u s updated as ( 1) T 2 ( ) 1 T ( ) T 2 ( ) u [ g ( V ) g 2 R] g [ V V ( f x )] (32) Then, next step value functon s calculated by solvng (31) for () V. The teratons proceed untl t converges.e. ( ) ( 1) ( ) V V V. The convergence s proved usng the followng theorem. () Theorem 1. Wth u be an admssble control for system (10) on the compact set. Then, the ( 1) nonlnear system x f ( x ) g( x u satsfes ( ) ( 1) ( ) ( ) ( ) () V V V and lmv where V solves the 1 ) ( ) ( 1) V ( ). approxmate GHJB equaton (13). Also, f V V, then V V 73

80 An advantage of successve approxmaton s mproved control polcy accordng to the last step value functon approxmaton that can guarantee the next step control polcy to be admssble controller. Also, GHJB avods the nonlnearty of the HJB that contans nonlnear partal dfferental equatons. Now, usng the successve approxmaton method and the approxmaton of GHJB by NN, the algorthm s presented as the followng 5 steps: L 1) Defne a NN as V l l l ( x) to approxmate smooth functon of V(x). Choose an admssble state feedbac control polcy u (0). 1 2) For th step, fnd V () assocated wth u () to assure the approxmate HJB by utlzng least square manner to fnd the NN weghts W ; 3) Update the control as 2 1 T 2 ( 1) T ( ) ( ) ( ) u g V. g 2 R g V V ( f x ) (32) 4) Fnd V (+1) assocated wth u (+1) to assure the approxmate HJB by utlzng least square method to fnd the NN weghts W +1 ; 5) If V () (0) - V (+1) (0) ε, where ε s the stoppng crtera as a small enough constant, then V * = V () and stop. Otherwse, =+1, go to step2. u * Once V * s obtaned, the optmal state feedbac control polcy wll be T 2 * 1 T * 2 * g V. g 2R g. V V ( f x ) (33) 3.4 Test Results The smulatons are carred out n Matlab/Smuln envronment. The DG unt that s equpped wth the UPFC s connected to a mcro grd as shown n Fg

81 Fg. 3.3 Low voltage mcrogrd wth DG and UPFC Table 3.1. System parameters Parameter Value - PV short crcut current I sc =9.71 A - PV open crcut voltage V oc =39.5 V - PV maxmum power V oc =280 W - Conv. capactors C n=8200 μf - Conv. Capactor C out=8200 μf - Conv. nductance L=2.5 mh - Swtchng frequency f s=10 Hz - Flter Capactance C f= 47μF - Flter nductor L f= 3mH - Lne resstances R Lne= 0.5 Ohm - Lne nductances L Lne= 20 mh - V grd V (L-L)=200 - S base S b=2kva In addton, the mcrogrd ncludes another PV-based DG unt, fve loads at fve buses of the system, and transmsson lnes that are modeled as seres RL branches. All the system defntons are gven n Table 3.1. As a case study, a fault happens on bus 1 where the DG wth UPFC unt s connected. The fault occurred at t=3s and s removed after 0.2s. The presented optmal nonlnear controller regulate the njected seres voltage of the UPFC n order to damp the after fault oscllatons of the system. The offlne tranng s performed n order to approxmate NN weghts that are used n cost functon calculatons. The resultant NN weghts are: 75

82 W [ ] T. L The proposed method can stablze the after fault oscllatons of the system s states effectvely as shown n Fg It s also shown that when the weghts are not approxmated properly n the cost functon, dampng effect of the UPFC s deterorated that results n more overshoot and settlng tme for the system states. Fg Angle and angular speed of DG+UPFC wth properly approxmated and untuned weghts Fg Actve power of transmson lne and njected power of the UPFC 76

83 Fg Injected seres voltage of the UPFC The actve power of the transmsson lne and also actve power of the UPFC unt s depcted n Fg It s shown that the UPFC s actve power s less than 2% of the nomnal power. In addton, njected seres voltage of the UPFC s less than 10% of the nomnal voltage of the grd as demonstrated n Fg Furthermore, an expermental setup s developed to test the theoretcal analyss. The proposed modelng and control method s mplemented usng a dspace Mcrolabox control platform as shown n Fg The PV unt output voltage s delvered to a dc-dc buc converter. The buc converter provde a stable dc ln voltage at the output. Ths dc voltage provde power for the three-phase nverter that feed the grd and also the nverter that generate seres voltage of the UPFC unt. The LC flters are utlzed to reduce voltage harmoncs at the nverters output termnals. The output voltage of the man nverter feed a local load and the grd through a step up transformers. The output voltage of the UPFC nverter s also fltered, and then njected to the transmsson lne through the transformers that are postoned n seres wth the transmsson lnes. The dc ln voltage that s depcted n Fg. 3.8, shows voltage stablty at the dc sde. In addton, voltage of the grd at Bus 1 and also njected seres voltage of the UPFC unt n normal condton of the mcrogrd s presented n Fg. 3.9 that dsplays stable operaton of the entre system. 77

84 Fg The DG+UPFC unt that s connected to the grd at Bus1 to enhance system stablty 75 Voltage (V) Tme (s) Fg Dc ln voltage that feed both nverters of the DG+UPFC unt Voltage (V) Voltage (V) Tme (s) Fg Injected seres voltage of the UPFC and grd voltage at Bus 1 78