Dynamic Phasor-Based Modeling of Unbalanced Radial Distribution Systems

Transcription

1 1 Dynamic Phasor-Base Moeling of Unbalance Raial Distribution Systems Zhixin Miao, Senior Member, IEEE, Lakshan Piyasinghe, Stuent Member, IEEE, Java Khazaei, Stuent Member, IEEE, Lingling Fan, Senior Member, IEEE Abstract This paper evelops an analytical moel of an unbalance raial istribution system consisting of a single-phase PV, a three-phase inuction machine loa, a three-phase power factor correction capacitor (PFC) an a loa. The analytical moel is base on ynamic phasors (DP) for abc phases. The single-phase PV moel inclues inverter current control (Proportional Resonance (PR) controller), an L or an LCL filter. The inuction machine moel is base on positive-, negative-, an zero-sequence components ynamic phasors. The sequencebase inuction machine moel was converte to the DP-abc reference frame an interconnecte with other gri components. The evelope analytical moel is capable of small-signal analysis an can be use to ientify variety of stability an/or harmonic issues in istribution networks, e.g., instability ue to weak gri. Impact of unbalance on system ynamic performance can also be investigate using this moel. The analytical moel is benchmarke with a high-fielity moel built in Matlab/SimPowerSystems where power electronic switching etails are inclue. The small-signal analysis results are valiate via Matlab/SimPowerSystems time-omain simulations. Inex Terms Dynamic Phasor (DP), Single-Phase Photovoltaic, Inuction Machine, Small-Signal Analysis I. INTRODUCTION Increasing efficiency an ecreasing cost of solar technology promotes substantial growth of Photovoltaic (PV) power integration in moern power systems. The total capacity of gri connecte PV systems has increase from 3 MW in 2 to 21 GW in 21 [1]. PV has share a fair amount of renewable energy penetration in microgris where the PV power supplies electrical loas for local communities [2] [4]. New government policies an incentives encourage more an more single-phase PV systems to be connecte. In aition, inuction machine base loas are ominant in istribution systems. Analytical moels of such unbalance istribution systems woul provie insights of the system an further be use for small-signal an large-signal stability analysis. The analysis will help ientify stability issues an mitigate relate problems. For unbalance systems, abc frame-base ynamic moels can be use for ynamic performance examination. Simulation packages such as PSCAD [5] an Matlab/SimPowerSystems [6] are base on simulation moels with instantaneous variables, e.g., instantaneous voltages an currents in three phases. Conventional linearization at an operating conition cannot be applie to these moels ue to the perioic varying state Z. Miao, L. Piyasinghe, J. Khazaei, an L. Fan are with Department of Electrical Engineering at University of South Floria, Tampa, FL 3362 ( zmiao@usf.eu). variables. The necessary conition for small-signal analysis is to have constant values for state variables at steay state [7]. Transforming the moels to a synchronous rotating reference (q) frame is the most common technique utilize to overcome the above problem [7]. However the negative-sequence components presente in an unbalance system will be converte to 12 Hz ac variables in a q-reference frame. Hence, qreference frame-base moels o not offer the capability of small-signal analysis uner unbalance topology an operating conitions. On the other han, ynamic phasor (DP)-base moeling, an averaging technique, has been emonstrate to be capable of converting perioic varying state variables into c state variables [8] [16]. It has been use in electrical machines analysis [8], [9], c/c converter analysis [15], istribution system analysis [13], an HVDC an FACTS system moeling an analysis [11], [12], [14]. For example, [14] presents an average moel of LCC base HVDC system which is capable of representing low frequency ynamics of the converters at both AC an DC sies. DP moels make small-signal analysis feasible. Small-signal analysis base on DP moels have been carrie out in [13], [16]. DP moeling technique also provies very accurate simulations for larger time steps [9], [14], [15]. Moreover, DP-base moeling can take into consieration of unbalance. For example, in [9], an inuction machine (IM) moel as well as a permanent magnet synchronous generator (PMSG) moel in unbalance conitions have been evelope base on positive, negative, an zero (PNZ) variables. In [11], the ynamic phasor-base moeling technique is use to moel the Unifie Power Flow Controller (UPFC). The moel is expresse in PNZ variables an can be use to stuy the effect of unbalance operation. [16] presents a DP-base moel taking into consieration of positive-, negative, an zero-sequence elements of a synchronous generator an its voltage controls. Tests on a single-machine infinite-bus system is conucte. The above mentione references [9], [11], [13] consier only unbalance operation instea of unbalance topology.the objective of this paper is to moel, analyze an simulate an unbalance istribution system with both complex loas such as IM an renewables such as inverter interface PVs. To the authors best knowlege, the literature lacks a comprehensive moeling approach for unbalance istribution systems that is suitable for both small-signal analysis an nonlinear timeomain simulation. An effort has been taken in [17] to moel a istribution system with unbalance topology. The system moel is ex-

2 2 presse by DPs of funamental frequency in in abc frame an the system contains a synchronous generator an a singlephase inverter. Our propose work will improve the moeling strategy in two aspects an therefore tackle comprehensive ynamic phenomena of unbalance istribution systems. A more comprehensive inverter control will be moele in this paper. The inverter is moele in [17] as a PQ controlle voltage source. Current controls are ignore. Interactions between the current controls of inverters an the gri can lea to resonances [18]. Therefore, ignoring current controls of a gri-connecte inverter will lea to the omission of certain ynamics. The current controls of a PV inverter will be moele in our paper. A more comprehensive system moel will be pursue in this paper. The network moel is treate as algebraic equations in [16], [17] with ynamics of inuctors an capacitors ignore. Each source in the system is treate as a voltage source. The sources are interface through voltage/current algebraic relationship erive base on the network moel. In our paper, a ifferent moeling strategy is aopte. Each source is moele as a current source. The sources are then interface through network ynamics. Using this strategy, the network ynamics ue to inuctors an capacitors will not be omitte. In summary, this paper investigates comprehensive moeling of unbalance istribution systems using ynamic phasors. The system consists of a three-phase inuction machine loa, three-phase resistive loas, a PFC an a single-phase PV system. The three-phase inuction machine, will be moele base on pnz ynamic phasors an then be converte to abc phasors. A single-phase PV system at phase a will be moele in phase a phasors. The entire system moel will be obtaine through a current source base integration technique. The major contributions of this paper inclue a comprehensive ynamic moeling approach for unbalance raial istribution systems. Applications of the moel will be emonstrate in small-signal analysis an fast an accurate simulation. Though the stuy system has only one single-phase source, the propose moeling metho can be applie to raial systems with multiple unbalance components. The easiness of integrating multiple unbalance elements will be shown in Section III. The moel is base on first harmonic DPs an therefore is suitable to stuy unbalance in the ac network an the resulting 12 Hz harmonics in the torque an spee in a three-phase inuction machine. The rest of the paper is organize as follows. In Section II, the ynamic phasor concept will be escribe an the principle of unbalance poly-phase DP moeling will be summarize. The istribution system configuration is presente in Section III. Dynamic Phasor moeling of components, incluing a PV, an inuction machine, an the gri with a PFC, is investigate in Sections III as well. Section IV presents case stuies to emonstrate the capability of the evelope analytical moel in small-signal analysis an time-omain simulations. Section V conclues the paper. II. DYNAMIC PHASOR CONCEPT Dynamic Phasor (DP) moels provies abunant merits, incluing: (i) the capability of small-signal analysis an (ii) availability of large step size simulations. The main iea of DP comes from escribing the waveform x(τ) on interval [t T, t] by Fourier Series [8]: x(τ) = k= X k (t)e jkωst (1) where ω s = 2π/T an X k is the k-th complex Fourier coefficient or Dynamic Phasor (DP). Due to the fact that these coefficients are constant at steay state, the DP moel can be linearize for small-signal analysis. The k-th DP of the time varying signal x(τ) can be obtaine by (2) [8]: X k (t) = 1 T t t T x(τ)e jkωsτ τ = x k (t). (2) where. k enotes the kth harmonic DP. The main characteristics of the DP moeling are escribe as follows [8]. x t = X k k t + jkωx k x.y k = (3) (X k l.y l ) l= Equation (3) escribes the relationship between the DP of a erivative versus the DP of the original signal while (3) escribes the relationship between the DP of a prouct versus the DPs of the iniviual variables. In this stuy, the main aim is to erive the DP moel of a istribution system compose of a single-phase PV, a three-phase inuction machine, a PFC an istribution lines represente by RL circuits. The DP moels in the abc frame can be erive by converting the DP moel from the positive-, negative-, an zero-sequence (pnz) reference frame [9]. The original signals in the abc frame can be expresse by pnz DPs as follows. x a x b x c (τ) = l= e jkωsτ α α 1 α α 1 } {{ } M X p,l X n,l X z,l (4) where l stans for the harmonic component inex, M is the transformation matrix (M H = M 1 ) from pnz to abc an p, n, an z stan for positive-, negative-, an zero-sequences. It is easy to see that the DPs of abc variables have the following relationship with the DPs of pnz-sequences. X a,l X b,l X c,l = M X p,l X n,l X z,l (5) In the next section, after introucing the entire system topology, the DP moels of each element will be presente one by one, which will be integrate into the overall system moel.

3 3 III. SYSTEM CONFIGURATION AND MODELING The parameters presente in [17], [19] are utilize for the propose stuy system shown in Fig. 1. The istribute system consists of a single-phase PV station installe in phase a of the system, a 3-phase inuction machine, a 3-phase PFC an a 3-phase loa. VC L2 L1 L 3 I I 1 2 I3 V c1 VG VC (a) (b) VG 2 kva E 2 kv/ 4V N I V G I m V m 2 kw IM PFC 2 kw 3 kw AC/DC I PV Single Phase PV 2kW Fig. 3. Simplifie PV moel with ifferent filters. (a): LCL filter, (b): L filter harmonic is consiere for the erivation of ynamic phasor coefficients. I 1 = 1 L 1 (V c1 V G ) jω s I 1 I 2 = 1 L 2 (V con V c1 ) jω s I 2 (7) V c1 = 1 C 1 (I 2 I 1 ) jω s V c1 Fig. 1. The stuy system. A. DP Moel of a Single-Phase PV Traitionally, two-stage converters (a DC-AC converter after a DC-DC converter) have been use for PV systems. Twostage converters nee aitional evices compare with singlestage converters. Therefore, single-stage converters have been implemente in PV gri integration [2] [23]. The basic configuration of a single-phase PV is illustrate in Fig. 2. The main elements of the single-stage PV are the Proportional Resonant (PR) current controller an the output filters. Lb La It shoul be note that V con is the DP of the output voltage of PV inverter. For an L filter enable PV system, the basic DP equation of the PV system connecte to gri is: I 3 = 1 L 3 (V con V G ) jω s I 3. (8) A etaile control block iagram of the single-stage single phase PV is illustrate in Fig 4. It is compose of a Maximum Power Point Tracking (MPPT) unit, a Proportional Resonant (PR) controller, a Phase-Locke-Loop (PLL), an a Pulse With Moulation (PMW) pulse generation unit. In this paper, the effect of the MPPT ynamics an the PLL has been neglecte for simplicity an special attention has been eicate to the PR controller an the LCL filter. Fig. 2. Basic configuration of PV system. Cap V g I c V c Ic.Vc. 2 V (rms) G MPPT i 1 i 3 PI LCL L vg PLL sin * i PR PWM Controller Fig. 2 shows the basic configuration of a LCL filter in a single-phase PV. It is compose of two inuctances an one capacitor connecte to the gri through a single-phase transformer. The simplifie moel of PV connecte to the gri with an L or an LCL filter has been illustrate in Fig. 3. The output voltage of the DC-AC converter is v con, the filter inuctances are L a an L b, an the gri sie voltage is v G. Note that the transformer can be represente by an inuctor L T. Therefore, L 2 = L b an L 1 = L a + L T. Furthermore, for a PV connecte to an L filter, if L f is use for the L filter inuctance, L 3 = L f + L T. The time-omain equations of the system for the LCL filter are liste as follows. i L 1 1 t = v c1 v G i L 2 2 t = v con v c1 (6) v C c1 1 t = i 2 i 1 The ynamics of the PV system with LCL filter in DP is expresse as follows. It shoul be note that only the first Fig. 4. Basic control of a single-phase PV. 1) DP moel of a PR controller: PR control is use to track ac signals. The PR controller in Fig. 4 tries to provie unity power factor power from the PV. Therefore, the current reference is synchronize with the gri voltage through a PLL. The ynamics of a PR controller consiering only the funamental harmonics can be expresse as: K r s v con = (K p + s 2 + (ω s ) 2 )(i i 1 ) [.5 = K p + K r ( +.5 ] ) (i i 1 ) (9) s + jω s s jω s where i is the reference current comes from PV array. In the analytical moel, the ynamics of PLL an MPPT are neglecte for simplicity of analysis. i 1 is the gri current when the PV enhance with LCL filter. In a case where the PV is interconnecte with an L filter, i 1 will be replace by i 3, which is the gri current. The rest of the moeling part has consiere

4 4 the LCL filter only. Define intermeiate state variables x 1 an x 2, where { (s + jω s )x 1 =.5(i i 1 ) (s jω s )x 2 =.5(i i 1 ). Rewriting (1) in time omain gives (11). { x1 t + jω s x 1 =.5(i i 1 ) x 2 t jω s x 2 =.5(i i 1 ) (1) (11) Applying the characteristics of DP, the DP relationship can be erive { X1 t =.5(I I 1 ) 2jω s X 1 X 2 t =.5(I I 1 ) (12) The DP of the converter output voltage s funamental frequency component can be expresse as: V con = K p1 (I I 1 ) + K r1 (X 1 + X 2 ) (13) The DP moel of a single-phase PV consists of (7), (12) an (13). The current reference I comes from this equation: I = 2V c I c V G (rms) = 2Pref V G (rms). B. DP moel of an inuction machine Since the single-phase PV will introuce unbalance in the istribution system, the inuction machine will be moele to inclue unbalance effect. Negative-sequence components in the stator voltage can cause a clockwise rotating stator flux. When this flux is interacting with the counter-clockwise rotating rotor flux, a 12 Hz torque ripple will appear. In turn, the rotating spee will have ripples with 12 Hz frequency. To count in the negative effect, the ynamic moel of a three-phase inuction machine in [9] base on pnz-sequence components is aopte in this paper. The space-vector moel of a squirrel-cage inuction machine with magnetic saturation an slot harmonics neglecte is presente as follows. v s = (r s + L s t ) i s + L m ti r = L m ti s + (r r + L r t ) P i r jω r 2 (L m i s + L r i r ) J t ω r = 3P 4 L mi( i s i s) Bω r T L (14) where v s, i s, i r enote the stator voltage, stator current an rotor current respectively. T L is the mechanical torque an ω r is the rotor spee. s an r enote the stator an rotor quantities, respectively. I enotes the imaginary part. The DP moel of an inuction machine can be erive by consiering the positive- an negative-sequence components in stator/rotor voltages an currents, as well as the c an the 12 Hz components in the rotating spee [9]. V ps = (r s + jω s L s + L s t )I ps + (jω s L m + L m t )I pr = (jω s L m + L m t )I ps + (r r + jω s L r + L r t )I pr jω r P 2 (L mi ps + L r I pr ) jω r2 P 2 (L mi ns + L r I nr) V ns = (r s jω s L s + L s t )I ns (jω s L m L m t )I nr = (L m t jω sl m )Ins + (r r jω s L r + L r t )I nr jω r P 2 (L mi ns + L r I nr) jω r2 P 2 (L mi ps + L r I pr ) J t ω r = 2P 4 L mi(i ps I pr + I nsi nr ) Bω r T L J t ω r2 = 2P j8 L mi(i ps I nr I ns I pr ) (B + j2jω s )ω r2 (15) where subscripts p an n stan for positive an negative sequence components, respectively. Since the DP moel for the PV system is base on phase a, to integrate the inuction machine moel with the PV system moel, the above pnz moel will be converte from an to the abc frame using the relationship presente in (5). The block iagram of the conversion has been illustrate in Fig. 5. V a V b V c ABC to PNZ V p V n V z IM Moel I p I n I z PNZ to ABC Fig. 5. Conversion from abc to pnz an back to abc for an inuction machine. C. PFC an the integrate system Consiering that there is a three-phase PFC in parallel with the PV, the circuit moel of the istribution system can be illustrate as in Fig. 6, where C enotes the capacitance of the PFC, I M is the inuction machine s stator current, I is the line current, R an L are the istribution line s parameters, R L is the loa moel, an E is the system voltage. Ipva Ima VGa C (a) R R L I a L E a I mp VGp C R R L (b) L I p E p Fig. 6. (a): Circuit moel of the istribution system with PV in phase a, (b): Circuit moel of the istribution system in phase b an c I a I b I c

5 5 For phase a, the DP moel of the integrate system can be expresse as { t I a = 1 L ( (jω sl + R)I a + V Ga E a ) t V Ga = 1 C ( ( ) ) jω s C + 1 R L V Ga I ma + I P V I a (16) where two state variables (gri voltage an gri current) have been introuce. For phase b an c, the DP moel of the integrate system can be expresse as: { t I p = 1 L ( (jω sl + R)I p + V Gp E p ) t V Gp = 1 C ( ( ) ) (17) jω s C + 1 R L V Gp I mp I p where p represents either phase b or phase c. In the propose system integration approach, iniviual elements such as PV systems, inuction machine loas an resistive loas are moele as current sources or passive elements. Through the PFC ynamics an the gri inuctor ynamics, the iniviual current sources are then integrate with the gri voltage. As long as the istribution system is raial, aitional unbalance elements can be moele as shunt current sources or passive elements an easily integrate into the overall system moel. The entire system is compose of a PV system, an inuction machine, a PFC an the RL line. As a total, 17 complex state variables are presente, incluing the line currents (I abc ), PFC voltage (V G,abc ), inuction machine stator currents(i s,pn ), inuction machine rotor currents (I r,pn ), inuction machine rotor spee (ω r, ω r2 ), PV system state variables (output current (I 1 ), current before filter (I 2 ), voltage across the LCL filter capacitor (V c1 ), the stator variables in the PR controller of the PV (X 1 an X 2 ). The complex state variables will be separate into real an imaginary components. Therefore, as a total, 34 real state variables are introuce for this DP moel an small-signal analysis will show 34 eigenvalues. IV. CASE STUDIES The analytical moel for the entire istribution system has been erive in Section III. The moel was been built in Matlab/Simulink. The nonlinear analytical moel can be linearize base on a certain operating point using Matlab function linmo. Small-signal analysis can then be carrie out for the linearize moel. The same stuy system was also built in Matlab/SimPowerSystems base on the physical circuit connection. The Matlab/SimPowerSystems moel captures power electronic switching etails an therefore is consiere high-fielity simulation moel. Three case stuies have been carrie out. In the first case, the analytical moel in Simulink is benchmarke with the high-fielity moel in SimPowerSystems. Dynamic simulation results are compare for the same ynamic event: a step change in loa torque of the inuction machine. In the secon case, the effect of unbalance on the ynamic performance is investigate by applying a ramp change in irraiance of the PV. This ynamic event emulates the clou effect on a PV an a istribution system. In the thir case, the effect of the gri-line length on stability is investigate. Small-signal analysis analysis an time-omain simulation are carrie out. A. Case Stuy 1 In this part, the analytical moel-base simulation results are compare with the Matlab/SimPowerSystems moel-base simulation results (in short, Simpower). A single-phase PV is connecte to the phase a of the system at the Point of Common Coupling (PCC) shown as in Fig. 1. At t = 4s, the inuction machine s mechanical torque was applie a step change from 28 N.M to 23 N.M. T e (N.M) ω r (RPM) Simpower Simulink Fig. 7. Simulation results of torque an rotor spee ue to a step change in mechanical torque (from 28 N.M to 23 N.M). The simulation results of the electromagnetic torque an the rotor spee of the inuction machine have been presente in Fig. 7. The ynamic responses from the two moels match each other well. The simulation results for the line current, the line voltage an the PV current are presente in Fig. 8. The results of the line current an the line voltage from both moels match well, which emonstrates the accuracy of the analytical moel erive in this paper. The PV current from SimPowerSystems simulation has ynamics relate to MPPT control an the csie capacitor. In the analytical moel, MPPT effect an csie ynamics are neglecte. While the PV power is constant an V G has negligible variation, therefore the PV current of the analytical moel is almost constant. B. Case Stuy 2 Eigenvalue analysis for the system with an without PV has been conucte an the results are presente in Tables I an II. It can be observe from the tables that, ue to the PV, five pairs of eigenvalues are introuce an these eigenvalues are relate to PV state variables such the LCL capacitor voltage V c1, PR controller state variables X 1, X 2 an the PV current I P V.

6 6 I a RMS (A) V Ga RMS (V) I PV RMS (A) Simpower Simulink Fig. 8. Simulation results of the IM stator current, stator voltage an PV current ue to a step change in mechanical torque (from 28 N.M to 23 N.M). TABLE I EIGENVALUES OF THE SYSTEM WITHOUT PV Eigenvalue amping ratio % frequency(hz) ominant state -959 ± 5768i ± 5935i ± 5935i V -959 ± 514i G, I -966 ± 5181i ± 5181i ± 38i I ps ± 117i I pr ± 445i I ns ± 637i I nr -.76 ± ω r ± i 1 ω r TABLE II EIGENVALUES OF THE SYSTEM WITH PV Eigenvalue amping ratio % frequency(hz) ominant state -966 ± 5935i ± 5848i ± 567i V -993 ± 4853i G, I -966 ± 5181i { -967 ± 594i} { 18.65} { 81 } -855 ± 8475i V -855 ± 7721i c1 { } { } { } ± 754i X ±.1i , X ± 377i I pv ± 38i I ps ± 117i I pr ± 445i I ns ± 637i I nr -.75 ± ω r ± i 1 ω r a) PV irraiance change: In this part, the effect of PV irraiance change will be simulate in both Matlab/Simulink an Matlab/SimPowersystems. The PV irraiance was set to 1 W/M 2 previously. A ramp change will be applie at t = 4s to ecrease the irraiance to 2 W/M 2 in.2 sec. Then after 1.4 sec, the irraiance will be set back to 1 W/M 2. The change of irraiance has been illustrate in etail in the first figure of Fig. 9. The secon figure of Fig. 9 shows Irr (W/M 2 ) P PV (kw) T e (N.M) I PV (A) 1 5 Simpower Simulink Fig. 9. Simulation results for the effect of irraiance change. the PV power which follows the irraiance comman. The PV power of the analytical moel is set to follow the change of the irraiance. It is notice that the maximum power level (2 kw) is obtaine when the irraiance is set to 1 W/M 2. The thir figure shows the electrical torque of the inuction machine. When the irraiance is ecrease ue to clous, the PV power is ecrease, which leas to the ecrease in the unbalance injection level to the system. The magnitue of the 12 Hz ripple has been ecrease uring the interval of 4 to 6 secons. The last figure shows the PV current which has been ecrease ue to the irraiance change. C. Case Stuy 3 In Case 3, impact of line length on system stability was investigate by both eigenvalue analysis an time-omain simulation in Matlab/SimPowerSystems. The eigenvalues of the system are presente in Table III. The movement of the ominant moes is presente in Fig. 1. One of the ominant moes is relate to a real-axis eigenvalue. The gri line length has been change from 3 km to 15 km in orer to observe its effect on ynamics. It is worth mentioning that increasing the line length more than 15 km causes non-convergence of the sweeping metho for initialization. Therefore the results are only shown for the initial conitions where the system is able to converge. It can be foun from Fig. 1 that as the line length increases, the real-axis eigenvalue will move to the right half plane. This can cause voltage instability of the system.

7 7 TABLE III EIGENVALUE COMPARISON FOR DIFFERENT GRID LENGTH l = 3km l = 1km l = 15km ominant State Eigenvalue amping ratio % frequency (Hz) Eigenvalue amping ratio % frequency (Hz) Eigenvalue amping ratio % frequency (Hz) { } { } { } { } { } { } { } { } { } -967 ± 5848i ± 3942i ± 3411i V -993 ± 567i ± 37i ± 312i G, I ± 38i ± 321i ± 328i I ps ± 117i ± 17i ± 13i I pr ± 445i ± 432i ± 426i I ns ± 637i ± 647i ± 651i I nr -.75 ± ± ± ω r ± i ± i ± i 1 ω r Imaginary Increase in the line length 3 Km 1 Km 15 Km magnitue, for the inuction machine, its electromagnetic torque will ecrease an its rotor spee will ecrease as shown in Fig. 13 (a) an (b). For the PV, since the reference power is kept intact, the reference current increases ue to the ecrease of the voltage. In turn, the PV current s magnitue increases as shown in Fig. 13. The entire system becomes unstable Real Fig. 1. The ominant moes (12 Hz, an the voltage stability moe) by increasing the line length. In time-omain simulations, a ynamic event to increase the gri line from 3 km to 3 km was triggere. Initially the gri connection consists of two parallel lines. At t = 4s, a breaker of one line is opene so the effective line impeance increases suenly. Such an event causes the voltage stability moe to move to the right half plane (RHP). The stator voltage of the inuction machine ecreases significantly as shown in Fig. 11. T e (N.M) ω r (RPM) Km Line I PV (A) 2 V IM (V) Fig. 13. Simpowersystems simulation results for the effect of gri line length increase. (a) torque; (b) rotating spee; (c) instantaneous current from PV. Fig. 11. Stator voltage when the gri line length increases from 3 km to 3 km. Simulation results were prouce by Matlab/SimPowerSystems. Fig. 12 presents the ynamic response of the RMS value from 3.5 secons to 6 secons, which clearly shows the ecline of the voltage magnitue. V IM RMS (V) Fig. 12. RMS stator voltage when the gri line length increases from 3 km to 3 km. Simulation results were prouce by Matlab/SimPowerSystems. Due to the ecrease of the stator voltage an system voltage D. Remarks on simulation time In the previous sections, it has been emonstrate that timeomain simulation can be carrie out by the Simulink moel an the etaile moel in SimPowerSystems. Due to its moeling etails, the simulation time for the SimPowerSystems is long. A comparison (Table IV) has been carrie out to show the simulation time ifference between the two moels. Table TABLE IV COMPARISON OF SIMULATION TIME BETWEEN TWO MODELS Time to be simulate SimPowerSystems Simulink 2 sec 4 min an 12 sec 2 sec 4 sec 9 min an 55 sec 4 sec 8 sec 18 min an 33 sec 6 sec 1 sec Memory error 58 sec IV shows that the analytical moel can be simulate in a much

8 8 shorter perio of time compare with the SimPowerSystems moel. V. CONCLUSION In this paper, a ynamic phasor-base ynamic moel was erive for an unbalance istribution system consisting of a single-phase PV, a three-phase inuction machine an a threephase power factor correction capacitor. The moel is capable of fast time-omain simulation an small-signal analysis. The moel s accuracy in capturing time-omain ynamics has been valiate by Matlab/SimPowerSystems base simulation. The moel s capability of small-signal analysis was also emonstrate. The eigenvalue analysis results corroborate with timeomain simulation results in Matlab/SimPowerSystems. APPENDIX TABLE V PARAMETERS OF THE INDUCTION MACHINE Total capacity 5.5 kva Nominal voltage 4 V Frequency 6 Hz R s 2.52 Ω R r 2.67 Ω X ls 3.39 Ω X lr 3.39 Ω X M 197 Ω J.486 kg.m 2 P (poles) 4 TABLE VI PARAMETERS OF PV Total capacity 2 W Frequency 6Hz L a.1 H L b.2 H Cap 3 µh K p(p LL) 18 K i (P LL) 32 K p(p R) 2 K i (P R) 15 TABLE VII LINE DATA OF THE NETWORK Line No Line Type Z (Ω/km) Length(m) 1 Z Gri.579+j Z IM.497+j Z P V.462+j [4] K. Tan, P. So, Y. Chu, an M. Chen, Coorinate control an energy management of istribute generation inverters in a microgri, IEEE Trans. Power Del., vol. 28, no. 2, pp , Apr [5] PSCAD Manual, Manitoba HVDC Research Center. [Online]. Available: [6] SimPowerSystems For Use with Simulink R, TransÉnergie Technologies Hyro-Québec. [Online]. Available: [7] J. Sun, Small-signal methos for ac istribute power systems a review, IEEE Trans. Power Electron., vol. 24, no. 11, pp , 29. [8] A. M. Stankovic, B. C. Lesieutre, an T. 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Abollahi, Online overvoltage prevention control of photovoltaic generators in microgris, IEEE Trans. Smart Gri, vol. 3, no. 4, pp , Dec REFERENCES [1] M. Braun, G. Arnol, an H. Laukamp, Plugging into the zeitgeist, IEEE Power Energy Mag., vol. 7, no. 3, pp , 29. [2] A. K. Abelsalam, A. M. Massou, S. Ahme, an P. N. Enjeti, High-performance aaptive perturb an observe MPPT technique for photovoltaic-base microgris, IEEE Trans. Power Electron., vol. 26, no. 4, pp , 211. [3] H. Kanchev, D. Lu, F. Colas, V. Lazarov, an B. Francois, Energy management an operational planning of a microgri with a PV-base active generator for smart gri applications, IEEE Trans. In. Electron., vol. 58, no. 1, pp , 211.

9 9 Zhixin Miao (S M 3 SM 9) receive the B.S.E.E. egree from the Huazhong University of Science an Technology, Wuhan, China, in 1992, the M.S.E.E. egree from the Grauate School, Nanjing Automation Research Institute, Nanjing, China, in 1997, an the Ph.D. egree in electrical engineering from West Virginia University, Morgantown, in 22. Currently, he is with the University of South Floria (USF), Tampa. Prior to joining USF in 29, he was with the Transmission Asset Management Department with Miwest ISO, St. Paul, MN, from 22 to 29. His research interests inclue power system stability, microgri, an renewable energy. Lakshan Piyasinghe is a Ph.D. stuent at University of South Floria (USF). He receive his Bachelor egree in Electrical Engineering in 26 from University of Moratuwa, Sri Lanka. He starte his Ph.D. stuy at USF in Fall 21 an his research interests inclue ynamic moeling an analysis of power electronic systems an power systems. Java Khazaei is a Ph.D stuent at University of South Floria (USF). He receive his Bachelor egree in Electrical Engineering from Mazanaran University (29) an Master egree from Urmia University (211) in Iran. He starte his Ph.D stuy at USF in Summer 213 an his research interests inclue Smart Gri moeling, Renewable Energy Integration, an Power Electronics. Lingling Fan (SM 8) receive the B.S. an M.S. egrees in electrical engineering from Southeast University, Nanjing, China, in 1994 an 1997, respectively, an the Ph.D. egree in electrical engineering from West Virginia University, Morgantown, in 21. Currently, she is an Associate Professor with the University of South Floria, Tampa, where she has been since 29. She was a Senior Engineer in the Transmission Asset Management Department, Miwest ISO, St. Paul, MN, form 21 to 27, an an Assistant Professor with North Dakota State University, Fargo, from 27 to 29. Her research interests inclue power systems an power electronics. Dr. Fan serves as a technical program committee chair for IEEE Power System Dynamic Performance Committee an an eitor for IEEE Trans. Sustainable Energy.