A study of the use of synchronverters for grid stabilization using simulations in SimPower

Transcription

1 See dicuion, tat, and author proile or thi publication at: A tudy o the ue o ynchronverter or grid tabilization uing imulation in SimPower THESIS AUGUST 2015 DOWNLOADS 3 2 AUTHORS, INCLUDING: George Wei Tel Aviv Univerity 140 PUBLICATIONS 3,235 CITATIONS SEE PROFILE Available rom: George Wei Retrieved on: 17 Augut 2015

2 TEL AVIV UNIVERSITY The Iby and Aladar Fleichman Faculty o Engineering The Zandman-Slaner Graduate School o Engineering A tudy o the ue o ynchronverter or grid tabilization uing imulation in SimPower A thei ubmitted toward the degree o Mater o Science in Electrical and Electronic Engineering by Eitan Brown July

3 TEL AVIV UNIVERSITY The Iby and Aladar Fleichman Faculty o Engineering The Zandman-Slaner Graduate School o Engineering A tudy o the ue o ynchronverter or grid tabilization uing imulation in SimPower A thei ubmitted toward the degree o Mater o Science in Electrical and Electronic Engineering by Eitan Brown Thi reearch wa carried out in the Department o Electrical Engineering Sytem Under the uperviion o Pro. George Wei July

4 Acknowledgement I would like to thank my advior, Proeor George Wei, who introduced me the ield o control and power electronic. Proeor Wei ha guided me throughout my tudie both in reearch and during my coure election. He had alo helped me inancially by increaing my univerity cholarhip rom hi grant. The company Synvertec ha ponored my reearch by paying my burary or about one year, and thi i grateully acknowledged. Thi help ha given me the opportunity to devote my time to reearch. I alo wih to thank Vivek Natarajan, a pot-doc who gave me guidance and advice throughout my work and alo in prooreading the drat o thi thei. Finally I mut mention Elad Venaezian, an MSc tudent with whom we collaborated on many apect o thi work. 3

5 Abtract Synchronou generator (SG) have the ollowing ueul property: once ynchronized, they tay ynchronized even without any control, unle trong diturbance detroy the ynchronim. Thi i one o the eature that have enabled the development o the AC electricity grid at the end o the XIXth century. Today, the tability o network o ynchronou generator that are coupled with variou type o load and other type o power ource (uch a renewable) and are operated with the help o multiple control loop, i an area o high interet and intene reearch, ee or intance [1], [3], [12], [16], [26]. Thi i partly due to the prolieration o power ource that are not ynchronou generator, which threaten the tability o the power grid. Thee power ource ue inverter to deliver power to the grid. An inverter which behave like a ynchronou generator can impliy the modelling o the overall ytem and the overall ytem behavior during udden diturbance or ault will be a table a it would be without the inverter or even better. Such inverter, ometime called ynchronverter, have been propoed in [2], [6], [27], [32]. (Actually, the control algorithm propoed in thee paper are dierent and the term ynchronverter reer to inverter controlled a in [33].) The grid operator can relate to ynchronverter in the ame way a to claical generator, which make the tranition to the maive penetration o renewable and other ditributed energy ource eaier and moother. In thi work I will briely introduce the ynchronou machine and derive the equation o a ynchronou generator with a contant mechanical torque, i.e. no prime mover or mechanical control o any ort i modelled. The prime mover ytem will not be dicued here but one can ind additional inormation on [16]. Ater deriving the mathematical model o a ynchronou generator, I will explore the local tability o a ynchronou generator connected to an ininite bu and introduce a reduced model with model reduction technique (additional ino on uch technique can be ound in [13]). The concept o ynchronverter will be introduced. I will preent ome known problem and olution regarding power grid and demontrate how ynchronverter help the grid to recover rom ault and reduce unwanted ocillation better than popular known olution. The end o my thei will dicu an additional practical ue o the ynchronverter or energy torage unit. Relevant imulation with a uggeted explanation a to how a ynchronverter can ue the energy torage or increaing grid robutne are included. 4

6 Content Acknowledgement... 3 Abtract... 4 Content... 5 Lit o Figure... 6 Lit o Table Introduction Model o a ynchronou generator Electrical part Mechanical part Model o a ynchronou generator connected to ininite bu Linearization o the SG model Reduced order SG model Linearization o the reduced order SG model Introduction to ynchronverter Addtional improvement to the ynchronverter algorithm Some imulation reult or ynchronverter Inter-area ocillation Power ytem tabilizer Synchronverter with virtual riction The Pogaku-Prodanovic-Green (PPG) inverter and inter-area ocillation Eilat Ktura grid ection: a cae tudy A ynchronverter baed energy torage device or grid tabilization Future work and concluion Bibliography Appendix - ynchronverter parameter Baic ytem or detecting grid ault

7 Lit o Figure Figure 1. Structure o an idealized three-phae round-rotor ynchronou generator Figure 2. Linearization o ynchronou generator connected to ininite bu devided.. 19 Figure 3. Bode diagram o the linearized ubytem Figure 4. Nyquit diagram o the Linearized ubytem Figure 5. Bode diagram o the linearized ubytem decribed in (3.18) Figure 6. Nyquit diagram o the linearized ubytem decribed in (3.18) Figure 7. Electronic part o a ynchronverter (without control) Figure 8. Propoed controller or a el-ynchronized ynchronverter Figure 9. Scheme o a 2-level 3 phae inverter with neutral line Figure 10. Neutral clamp inverter topology or one phae o a 3 phae inverter Figure 11. Schematic o a 2-level 3 phae inverter connected to the grid via an LCL.. 32 Figure 12. Reactive power proile o a ynchronverter connected to an ininite bu Figure 13. Active power proile o a ynchronverter connected to an ininite bu Figure 14. Frequency o a ynchronverter connected to an ininite bu with PWM Figure 15. Updated cheme o the ynchronverter algorithm Figure 16. Synchronverter grid requency tracking, requency decreae Figure 17. Synchronverter active power proile when traking grid requency Figure 18. Synchronverter grid requency tracking, requency increae Figure 19. Synchronverter active power proile when traking grid requency Figure 20. Voltage proile o the ininite bu and the ynchronverter terminal Figure 21. Synchronverter requency remain contant when the voltage change Figure 22. The reactive power o the ynchronverter tabilize Figure 23. Two ynchronverter micro grid cheme Figure 24. Two ynchronverter grid active power proile Figure 25. Two ynchronverter grid requency proile Figure 26. Two ynchronverter grid voltage proile Figure 27. The imple power network (without ynchronverter) Figure 28. Power ocillation in the ytem rom Figure Figure 29. General model o a PSS Figure 30. Comparion o power ocillation uing three type o PSS Figure 31. The network rom Figure 27 ater adding ynchronverter Figure 32. Comparion o power ocillation with and without virtual riction Figure 33. Comparion o ytem power low with ynchronverter and PSS Figure 34. Comparion o ytem voltage proile with ynchronverter or PSS Figure 35. Comparion o ytem power low with ynchronverter and PSS Figure 36. Comparion o ytem voltage proile with ynchronverter or PSS Figure 37. Synchronverter with PSS controller Figure 38. Synchronverter with PSS controller, proo o concept Figure 39. Synchronverter with PSS controller and virtual riction, comparion Figure 40. PPG inverter general cheme Figure 41. Power controller Figure 42. Voltage controller Figure 43. Curent controller Figure 44. Comparion o PPG inverter and ynchronverter Figure 45. Schematic o Eilat-Ktura grid ection Figure 46. Updated power controller o the inverter deigned in [23] Figure 47. Synchronization o the inverter at Ktura to the grid Figure 48. Frequency in Eilat 10Ω hort circuit at Eilat Figure 49. Frequency in Ktura 10Ω hort circuit

8 Figure 50. Voltage in Eilat 10Ω hort circuit at Eilat Figure 51. Voltage in Ktura 10Ω hort circuit at Eilat Figure 52. Power upply in Ktura 10Ω hort circuit at Eilat Figure 53. Reactive power upply in Ktura 10Ω hort circuit at Eilat Figure 54. Frequency in Eilat 1Ω hort circuit at Eilat Figure 55. Frequency in Ktura 1Ω hort circuit at Eilat Figure 56. Voltage in Eilat 1Ω hort circuit at Eilat Figure 57. Voltage in Ktura 1Ω hort circuit at Eilat Figure 58. Power upply in Ktura 1Ω hort circuit at Eilat Figure 59. Reactive power upply in Ktura 1Ω hort circuit at Eilat Figure 60. Frequency in Eilat 0.1Ω hort circuit at Eilat Figure 61. Frequency in Ktura 0.1Ω hort circuit at Eilat Figure 62. Voltage in Eilat 0.1Ω hort circuit at Eilat Figure 63. Voltage in Ktura 0.1Ω hort circuit at Eilat Figure 64. Power upply in Ktura 0.1Ω hort circuit at Eilat Figure 65. Reactive power upply in Ktura 0.1Ω hort circuit at Eilat Figure 66. Ktura-Eilat imulation with a ynchronverter in Ktura and power delay Figure 67. Scheme o power grid with low, medium and high voltage part Figure 68. Output power o ynchronou generator and ynchronverter attached Figure 69. Input power o 1MW ynchronou motor and 5 MW load Figure 70. Synchronverter with any virtual riction how a mall improvement Figure 71. Synchronverter without one ided virtual riction generator requency Figure 72. Synchronverter with one ided virtual riction generator requency Figure 73. Synchronverter without one ided virtual riction motor requency Figure 74. Voltage amplitude at motor terminal no one ided virtual riction Figure 75. Voltage amplitude at motor terminal with one ided virtual riction Figure 76. Comparion o one ided virtual riction to increae o the droop actor

9 Lit o Table Table 1. Typical value or a 5kW ynchronou generator...19 Table 2. Parameter or a 40 MW PPG inverter...54 Table 3. Summary o the 6 teted cenario or ynchronverter...69 Table 4. Parameter or 5 kw ynchronverter...77 Table 5. Parameter or PSS and virtual riction in the ynchronverter...78 Table 6. Parameter or a 1 MW ynchronverter...79 Table 7. Parameter or additional ilter or the ynchronverter

10 1. Introduction Synchronou generator are a undamental component o the AC power grid. From a control ytem perpective, ynchronou generator have the ollowing ueul eature: once the generator are ynchronized to the grid, they remain ynchronized without need or any external control. The AC power grid deliver high power at high voltage uing AC/AC tranormer which, a o today, i very diicult to accomplih uing a DC grid with DC/DC tranormer due to the limitation o the electronic witche they ue. The power grid mut maintain contant requency and voltage. In a phyical grid, where the load vary requently, thi i achieved uing control algorithm that regulate grid operation. One control loop in the ynchronou generator modiie the power et point o the generator in repone to requency change in the grid uing a contant droop coeicient. Another control loop regulate the voltage through the reactive power (although it i more complex). In the European grid the nominal requency i 50 Hz and the droop coeicient i 3%, thi mean that when the grid requency drop by 3% rom the nominal value, the power output o the generator i required to go up by 100%. Similarly a 3% requency rie mut lead to a 100% power decreae. The power i changed o a to bring the requency back to the nominal value. There are phyical limitation to the perormance o thee control loop. For intance, there i a delay in the loop due to the low repone o the mechanical component and the exce energy initially come rom inertia, meaning the tored mechanical energy in the rotor. In order to tabilize the voltage the generator upplie or aborb reactive power rom the grid. The regulation here i le critical, ince there are other device in the power grid or tabilizing the voltage (by upplying or aborbing reactive power). There i alo a relation between voltage and requency but uually the control ytem decouple them. Inverter are ued or connecting renewable energy ource to the grid. They try to participate in the requency and voltage regulation in the grid, but due to limitation related to the ytem architecture and inancial contraint (renewable energy a o today i more expenive) oten the regulation objective are acriiced in avor o other objective uch a harveting more energy. Inverter are not only ued by renewable ource. Today in ome countrie there are high power DC line. Thee line are good or connecting electrical grid rom dierent countrie with dierent regulation or or imply delivering electricity without radiation in the cable. In the near uture, energy torage unit will alo play a role in the power grid to enure that power i available during high demand and generator ault. Thee unit will alo require inverter to connect to the grid. Another igniicant change today i in the nature o the load. While in the pat load were uually linear reitive (lighting bulb, etc.) and linear inductive (wahing machine or example), today we have computer, and large batterie (electrical car or torage unit) which are nonlinear load. Thee new load require converter (AC/DC) which have their own control ytem and limitation. Thi mean that unlike in the pat, the grid today i le predictable and experience ater load change, i ubject to harmonic and interact with many dierent controller. For now when the renewable energy market i mall, around 10% in mot countrie, the inluence o introducing inverter to the grid i negligible and can be handled by trict regulation or any power ource in the grid. In the uture it i likely that the percentage contribution o renewable energy will be much larger and the grid will contain a large number o nonlinear load. There are two approache to olve thi iue, changing the concept o power grid operation and power grid control or introducing a controller or inverter and converter that orce them to behave like a ynchronou generator. 9

11 In thi work I will briely introduce the ynchronou machine and derive the equation o a ynchronou generator with a contant mechanical torque, i.e. no prime mover or mechanical control o any ort i modelled. The prime mover ytem will not be dicued here but one can ind additional inormation on [16]. Ater deriving the mathematical model o a ynchronou generator, I will explore the local tability o a ynchronou generator connected to an ininite bu and introduce a reduced model with model reduction technique (additional ino on uch technique can be ound in [13]). The concept o ynchronverter will be introduced. I will preent ome known problem and olution regarding power grid and demontrate how ynchronverter help the grid to recover rom ault and reduce unwanted ocillation better than popular known olution. The end o my thei will dicu an additional practical ue o the ynchronverter or energy torage unit. Relevant imulation with a uggeted explanation a to how a ynchronverter can ue the energy torage or increaing grid robutne are included. All the variable in thi thei are in SI unit, unle peciied otherwie. 10

12 2. Model o a ynchronou generator In thi ection, we develop a model or a round rotor ynchronou generator tarting rom irt principle. We aume that the generator ha one pair o pole per phae and one pair o pole on the rotor, i.e., p 1. The machine i aumed to be perectly built and thereore we ignore magnetic-aturation eect, iron core loe and Eddy current in our model. Section 2.1 conider the electrical part o the generator while Section 2.2 ocue on the mechanical part. Our derivation ollow Grainger and Stevenon [11], Zhong and Wei [33]. A more detailed model taking into account cogging and other nonlinear eect i in Mandel and Wei [18], ee alo Sauer and Pai [24] Electrical part The tructure o an idealized three-phae generator i hown in Figure 1. The current in the three tator winding are denoted by i a, i b and i c and the voltage acro them by v, a v b and v c. The current and voltage acro the rotor winding are i and v, repectively. The tator winding can be regarded a concentrated coil with elinductance L, mutual inductance - M and reitance R. Deine L M L. The rotor winding are alo regarded a a concentrated coil with el-inductance L and reitance R. M i the amplitude o the mutual inductance between the rotor and each o the tator winding. The rotor angle i denoted by, a hown in Figure Figure 1. Structure o an idealized three-phae round-rotor ynchronou generator with p 1, modiied rom [11], Figure 3.4. We aume that the neutral line i abent and thereore

13 Denote the tator luxe by a, b and ia ib ic 0. c. Deine the vector ia a va i i b, b, v v b, i c c v c co in co co 2 3, in in 2 3. co 4 3 in 4 3 The tator lux linkage are given by and the rotor lux linkage L i M i co i given by (2.1) L i M i,c o. The phae terminal voltage vector v atiie (2.2) where T (2.3) a b c d i v e Ri L, d t e e e e i the back EMF due to the rotor movement and it i given by di e M i in M co, d t where i the angular velocity o the rotor. e i alo called the ynchronou internal voltage. Similarly, the ield terminal voltage i d (2.4) v R i. dt By applying the Park tranormation (2.5) to (2.2), we get the ollowing expreion: (2.6) 2π 2π co co co π 2π U ( ) in in in di LU ( ) RU ( ) i U ( ) e U( ) v. dt We denote e U( ) e, whoe component are, dq e d q v dq and i dq are deined imilarly. It i eay to veriy that e and e 0 (in thi order). The vector 12

14 (2.7) which can be rewritten a id iq d di i q U ( ) i d d d i 0 0 (2.8) Uing thi, we can rewrite (2.6) a ollow: (2.9) id iq d di i q U ( ) i d. dt dt i 0 0 id iq id ed vd d L i q L i d R i q eq v q dt i 0 i e v Since there i no neutral line, i 0 0 and hence e. 0 v 0 From (2.9) d id iq R i d 1 ed vd (2.10) d i t q i i d L q L e q v q Applying the Park tranorm to (2.3) we get di di ed 3 M 3 (2.11) dt M dt e. q 2 2 M i i.. In (2.4), ubtituting or rom (2.1) and uing the identity,co 3/ 2 d i i (recall that i U( ) i where U( ) i deined in (2.5)) it ollow that dq di 3 did (2.12) L M R i v. dt 2 dt Uing the equation or d id dt (2.13) 2 where M L L rom (2.9) in (2.12) we get 1 2 d 2 3M d i 3 M R 1 R 1 iq id vd i v L L t 2 L L L L L 1 0 and L L 3. By deining 2, (2.14) (2.13) can be written a (2.15) 2 3M 1 2 LL 1, 3 m M, 2 di m R 1 R i i v i v dt L L L L L q d d. 13

15 Now we obtain the ollowing equation or i d and i q rom (2.10) by ubtituting or e d and e rom (2.11) and or di dt rom (2.15): q (2.16) m R 1 R d i d i q R i d m i q i d v d i v L L L L L. dt i q i i d L q L i 2.2. Mechanical part 14 The energy tored in the machine magnetic ield i E i, i i, L i M i i,co L i. mag The electromagnetic torque can be calculated a hown below (ee [7], [10] and [11] or detail), where we ue the relation pm, m i the mechanical rotor angle and p 1(a explained at the beginning o thi chapter): (2.17) Emag Emag Emag Te p m, contant m ii, contant ii, contant co 3 pm i i, pm i i,in M i iq mi iq. 2 Uually the mechanical torque i produced/regulated by a prime mover which oten conit o a turbine and a controller, ee Chapter 9 in [16] or dierent type o prime mover. In our model, we ignore the dynamic o the prime mover and intead aume that the peciied mechanical torque T m (plu the droop correction, explained below) act on the rotor. I we aume no cogging torque, which i the torque due to the interaction between the permanent magnet or electromagnet o the rotor and the tator lot, then the mechanical part o the machine i governed by (2.18) JT T D, m e p where J i the moment o inertia o all part rotating, rotation requency i the irt derivative o the rotor angle. The term D p i the damping actor and the Dp i due in part to vicou riction, but motly it i due to the droop controller o the prime mover, which adjut the active torque depending on. The kinetic energy o the rotor i E 2 kin J 2. It ollow that. E J T T D T mi i D kin m e p m q p In the abence o any external torque ( Tm 0 ) and hort circuit in all terminal, the change in the total energy E Emag Ekin o the ytem, mut be equal to the lo o energy due to mechanical damping (including the droop correction) and electrical reitance. It i eay to check that

16 (2.19) E E E R i i R i D mag kin d q - p 0. Thi validate our energy calculation. Note that, i we neglect the vicou riction, then the active torque acting on the generator i Tm Dp. Uing (2.15), (2.16), (2.17) and (2.18) we obtain the ollowing ytem which model the dynamic o a perectly built ynchronou generator with one pair o pole: L 0 id R L mr id d L 0 i L R mi q i q dt L mr L m R 0 i i J 0 mi 0 Dp (2.20) 0 ml 0vd v q. ml 0 0v Tm We mention that thi ytem can be repreented a a port-hamiltonian ytem, ee Fiaz, Zonetti, Ortega, Scherpen and van der Schat [9]. 3. Model o a ynchronou generator connected to an ininite bu In thi ection we develop a model or a ingle ynchronou generator (SG) connected to the power grid. The grid i aumed to be an ininite bu, i.e., a contant three-phae AC voltage ource. Thi i a reaonable aumption ince the inluence o a ingle generator on a grid coniting o many equivalent generator i typically mall. To impliy the preentation, the contant L and R are redeined to include the reitance and inductance o the line leading to the bu. Thu the tator terminal voltage v i the voltage o the bu. The bu voltage T v v v v depend on the bu angle bu a ollow: a b c v 2 3 V co, v 2 3V co( 2 3), v 2 3V co ( 2 3). a bu b bu Here V 0 i the line to line rm voltage. Deine 2, where i the rotor angle o the ynchronou generator. Thi i the well-known power angle, the angle by which the ynchronou internal voltage e i ahead o the bu voltage v. We apply the Park tranorm (2.5) to v. Uing the deinition v U( ) v, we get that (3.1) v V in, v V co. It i eay to ee that (3.2) g, d bu q c dq bu 15

17 where i the contant bu requency. Thu. From (2.17) and (2.18) we get 16 g (3.3) J D mi i T. p g q m I we can expre i q a a (poibly approximate) unction o, then rom the above equation an ODE in that reemble the claical wing equation can be obtained. We will derive uch an ODE uing model reduction in Section 3.2. We remember (2.20) or the dynamic o the ull ytem. Now we add the tate variable with the tate equation (3.2) and aign the value o v d and v q rom (3.1) to get the ollowing ODE ytem with 5 tate variable: (3.4) Lid R L mr L 0 0 id Li q R 0 mi 0 i q d L i mr L m R 0 0 i dt J 0 mi 0 Dp ml 0 0 in V V co ml v Tm g 3.1. Linearization o the SG model on an ininite bu We analyze the local tability o (3.4) by linearizing it near it equilibrium point in thi ection. We aume that i i contant (thi aumption i made to impliy the calculation). Then the nonlinear equation (3.4) reduce to (3.5) We compute the equilibrium point L in id R L id V d L co i q L V R mi i q. dt J 0 mi -Dp 0 Tm g id0 iq0 0 0 T or (3.5) by etting the derivative o all the tate variable to be 0. Letting 0, it ollow rom the ourth equation in (3.5) that 0 g. The third equation in (3.5) (etting 0 ) give mi - 0 iq0 Dpg Tm 1 i T D q0 m p g mi It ollow rom the econd equation in (3.5) that R i L i V in 0 d 0 g q0 0.

18 V L i in T D. g d 0 0 m pg R Rmi Finally we get rom the irt equation in (3.5) that (3.6) L i 0 R i 0 mi V co 0 0. g d q g We deine Z and a ollow: Z R j L Z e j, o that g (3.7) Z R gl gl R 2 2 2, arctan. Ater ubtituting or i d 0 and i q0 in (3.6), uing the deinition in (3.7), the power angle at the equilibrium i given by the expreion (3.8) mi gr Z T m Dp g 0 arcco. Z V Vmi Thi equation may have 0, 1 or 2 olution modulo 2. I there are two olution, then there are two equilibrium point and they are either both untable or one table and one untable. We want to linearize the ytem near a table equilibrium point. For thi we compute the equilibrium point and check i any o them i table. The linearized ytem will be o the orm (3.9) x Ax.. Deine i i, i i,, d, d id0 q iq0 T q 0 0 T where xeq id 0 iq0 0 0 i an equilibrium point o (3.5), and denote x [ idiq]. Deine d q T z i i and rewrite (3.5) in the orm z ( z), then A in (3.9) i the Jacobian deined by A ij ( z) z j i zxeq. We calculate the Jacobian rom (3.5) and get R L g iq0 V co0 L g R L id 0 mi L V in0 L A. 0 mi J Dp J We need to aign parameter value to compute the equilibrium point o (3.5) and check i they are table, by computing the eigenvalue o the reulting Jacobian matrix. For the nominal value o the parameter hown in Table 1, we obtain a pair o equilibrium point xeq id 0 iq0 0 0 hown below: 1 2 x x For eq, 1 x eq the matrix A i eq A

19 and the eigenvalue o A are: i, j, = j, = j For 2 x eq the matrix A i A and the eigenvalue o are: j, j, =-47.23, = Notice that in both cae 1 and 2 are almot equal to R L jg. Hence (3.5) ha two equilibrium point, where 1 x eq i table and 2 x eq i untable. Thi i true or typical value o the parameter. For ome et o parameter vector, it i poible to how that the ynchronou generator model (3.5) i almot globally aymptotically table, meaning that almot all the trajectorie converge to the table equilibrium point, ee [20]. We remark that (3.4) can be linearized without the aumption o contant rotor current, but the expreion become extremely complex and the model i very diicult to analyze. In order to compare (3.5) to a reduced model that will be introduced in Section 3.2 we will manipulate the linearized ytem x Ax, where calculated above. We can write x input and output ) with the third order ytem (3.10) z A3 z B3 z, y C3z, x T [ id iq ] and A i a Ax a a eedback interconnection o an integrator (with where z i i T [ d q ] and R L g i q0 V co0 L A3 g R L ( id 0 mi L ), B 3 V in 0 L, C3 0 0 mi J Dp J 0 0 1, a hown in Figure 2. δ ω g Figure 2. The linearization o a ynchronou generator connected to an ininite bu, divided into two ubytem. 18

20 The traner unction o the third order ytem i 1 G C I A B The Bode and Nyquit plot or thi traner unction, with A, 3 B 3 and C3 evaluated at x 1 eq, are in Figure 3 and 4. The eigenvalue o A 3 or The eigenvalue o A 3 or 1 x eq are: j, j, = x eq are: j, j, 3 = In Section 3.3 we will derive a eedback interconnection, imilar to that hown in Figure 2, but uing a reduced model or the SG. We plot it Bode and Nyquit plot and compare it with the plot in Figure 3 and 4. We do thi comparion to acertain the idelity o the reduced model. Parameter Value V - line to line rm value V Pn g R L R L M 5kW, Tm Pn g Dp Nm g J (or 100π Ω 4.4 mh 0.6 Ω 0.3 H H J 2 / 2P 2econd ) 0.2 Kgm 2 D p (100% increae o P or 3% decreae o ) i 1.7 J A Table 1. Nominal value or a 5kW ynchronou generator connected to a 50 Hz low voltage line in a European or imilar power grid. The reulting parameter,, m, and mi at equilibrium are: 3.451, m H, mi 1.304V, e M i V V. rm Some o the value in Table 1 are obtained rom regulation requirement in Europe (or intance the value o D p, J, g and V ). The value o R and L are elected to be the ame a thoe o the 5 kw ynchronverter on which experiment are being conducted at Tel Aviv Univerity. The value o R, L and M were choen intuitively to match the other parameter. The value o mi wa ound via imulation o a imple ytem coniting o a ynchronou generator and an ideal voltage ource. The Bode and Nyquit plot below were obtained uing tandard MATLAB command. 19

21 Imaginary Axi Phae (deg) Magnitude (db) 50 Bode Diagram Frequency (rad/) Figure 3. Bode diagram o the linearized ubytem decribed in (3.10) around the table equilibrium point x o the nonlinear model (3.5). 1 eq 150 Nyquit Diagram Figure 4. Nyquit diagram o the linearized ubytem decribed in (3.10) around the table equilibrium point Real Axi 1 xeq o the nonlinear model (3.5) The reduced order SG model 20 We develop a reduced order model or the ith order ytem in (3.4). A mentioned below (3.3), the idea i to expre i q a a unction o. Since ollowing the ingular perturbation theory (ee Chapter 11 in [13]), we let L i typically mall, Ld i / d 0 d t

22 and Ld iq /dt 0 in (3.4). In other word, i d and i q are aumed to be at variable. Thi give: (3.11) L mr Vin m i i i v R R L R L R d q, (3.12) L m Vco iq id i. R R R We deine the tator impedance Z R jl, o that Solving or i q rom (3.11) and (3.12) give Z R L m m L iq 2 R L R L i V Rco Lin v. Z L L Subtituting or i q in (3.3) give J D p g mi m m L Tm 2 R L R L i V Rco Lin v, Z L L which can be equivalently expreed a (3.13) J D T R L R L p g m VR VL L Z Z Z L Z L By deinition L aume that co in mi. 2 2 m i 2 m i 2 v g Z in and R Z co ( Z and are deined beore (3.7)). We g and thereore Z Z. By replacing Z with Z in (3.13) and uing the identity co( ) co co in in and, we get: (3.14) 2 m R 2 L RL 2 ml mi V J Dp i co 2 i 2 v Z L Z Z L m 2 R L RL 2 2 ml Tm Dp i. 2 i 2 v g Z L Z L Thi ODE i coupled with another ODE that expree i, ee (3.15) below. From (3.4), the dynamic o i i decribed by the equation di mr m R v m id iq i Vin. dt L L L L L L L Ater ubtituting i q and i d with their reduced model expreion (3.11) and (3.12), we get: g 21

23 which can be impliied to (3.15) 1 i Vin v, 2 di R m m dt L L L L L L L di R 1 i v. dt L L Irrepective o initial condition, i tabilize to a inal value that depend only on v and the value o the rotor parameter. The inal value i lim i () t v R. t In other word the equilibrium value i i 0 v R. Since we are intereted in the long time behavior o (3.14), we replace i with i 0 and rewrite it a (3.16) 2 2 m R mv V 2 m R 2 J Dp v co. 2 2 Tm Dp v 2 2 g Z R ZR Z R For the almot global tability o (3.16) it i neceary that D p 2 mr 2 v 0, 2 2 ZR which hold trivially ince both the term on the let ide are poitive. For more inormation on the tability o pendulum like equation, ee [17]. To calculate the equilibrium point, we let 0 and 0 in (3.16) and get the ollowing expreion or the equilibrium value o : Tm Dpg Z R mv gr 0 arcco, mvv V Z R which i equivalent to (3.8) i we ue i v R tability condition: Tm Dpn Z R mv gr 1 1. mvv V Z R. From here we get an additional Thi condition appear to hold or mot typical generator parameter value and in mot cae the ytem (3.16) ha one table equilibrium point. being In the above analyi we have made everal aumption, the mot igniicant o them i v R. In the abence o thi aumption, but till uing the teady tate value g, (3.16) had to be replaced with 2 2 m R mv V 2 m R 2 J Dp v c, 2 2 Tm Dp v 2 2 g Z R R Z Z R (3.17) o where i the tator impedance angle, L R arctan. 22

24 The global tability o thi ytem i an open quetion. But it i quite traightorward to analyze the local tability o thi ytem around an equilibrium point (the procedure or local tability analyi wa demontrated in Section 3.1 or a more complex model). Future work will ocu on etimating the domain o attraction o the table equilibrium point, whenever it exit. Note that in thi ection the rotor current wa not aumed to be contant like it wa in Section 3.1. However a een rom (3.15) the rotor current doe not depend on other tate variable in the reduced model. Thereore in the tability analyi, the rotor current can be regarded a a contant with value equal to it equilibrium value i Linearization o the reduced order SG model While analyzing the dynamic o a network o ynchronou generator it i convenient to ue reduced order model. But it i not clear i the reduced model capture the primary dynamic o the network. In Section 3.2 a reduced model (3.13), (3.15) wa obtained or a ynchronou generator connected to an ininite bu. In thi ection we try to evaluate the quality o thi reduction. To do thi we linearize the reduced model around a table equilibrium point and repreent it a a eedback interconnection o a econd order ytem and an integrator (imilar to that in Figure 2). We plot the Bode and Nyquit plot o the econd order ytem and compare it with the correponding plot o the third order ytem in Figure 2 o a to compare the reduced model to the ull model with contant i, (3.5). In the reduced model i i not aumed to be contant, but in the ull model in Section 3.1 it i aumed to be contant. T Deine xeq i to be an equilibrium point o the reduced model (3.13), (3.15) and deine T z i. We calculate the Jacobian Ai j ( z) i, where z i R L v L 2 2 i mv R co L in m i RL R L i v L L ( y) T 2 m Dp Z g i deined uing (3.15), (3.2) and (3.13). We get that where A j zxeq R L 0 0 A A21 A22 mv i 0 in( 0 ) J Z, co in L mv R L v L m m i R L R L g g g 21 2 J Z L 23

25 A 2 2Lg mvl Ri 0co J Z L ( R L ) m i R L R L m i v L L i mvl in D 4 J Z L J g p. The equilibrium value o i i and ubtituting v with i 0R to get : v R and or we ue (3.17) by letting 0, 0 i 0Rm Z T g m Dp g 0 arcco, V Z mvi 0 which i equivalent to (3.8). Again we have ued the value rom Table 1 and have checked i there are equilibrium point. We have ound two equilibrium point, where the 1 irt point i x T 2 and the econd i x eq eq T. We can ue (3.11) and (3.12) to compute the value o i d and i q according to d 1 x eq and get i and iq We ee that the value o i d and i q along with and have identical value to the value id0, iq0, 0 and 0 in the table equilibrium point o the ourth order ytem (3.5). For the irt equilibrium and the eigenvalue are: For the econd equilibrium and the eigenvalue are: A , xeq the matrix A i j, j, =-2. 2 xeq the matrix A i 19.26, , =-2. We ee that correpond to the imilarity in value. Deine x A T [ i ], where i, i i 0 0 the linearized ytem x Ax, where x 1 x eq i the table equilibrium, which 1 x eq and 0. We examine T [ i ] and A i the 3x3 Jacobian matrix calculated previouly. We can write thi ytem a a eedback interconnection o an integrator (with input and out output ) with the econd order ytem (3.18) z A2 z B2 z, y C2z, where z T [ i ] and 24

26 Imaginary Axi Phae (deg) Magnitude (db) R 0 L 0 A2, B2, C2 0 1, A mvi 21 A 22 0in 0 JZ identical to what wa done in Figure 2 and receive the ollowing traner unction or the econd order plant: G C I A B We plot Bode and Nyquit diagram uing the value rom Table 1 and get: Bode Diagram -45 Figure 5. Bode diagram o the linearized ubytem decribed in (3.18) around the table equilibrium point Frequency (rad/) 1 x eq o the nonlinear reduced model (3.13), (3.15). Thi hould be compared to Figure Nyquit Diagram Real Axi 25 Figure 6. Nyquit diagram o the linearized ubytem decribed in (3.18) around the table equilibrium point 1 x eq o the nonlinear reduced model (3.13), (3.15). Thi hould be compared with Figure 4. The reduced model i almot identical to the ull model in the low requency region. Thi mean that or the generator connected to the ininite bu, near an equilibrium point we can ue the model reduction to analyze the ynchronou generator. In the uture, we hope to how that taking the rotor current to be contant (or lowly varying) i a good approximation o the ull ith order ytem in (3.4).

27 4. Introduction to ynchronverter In thi chapter we will ue the model o the ynchronou generator rom Chapter 1 to build the model o a ynchronverter. The ynchronverter i an inverter with a pecial control algorithm that caue it to mimic the operation o a ynchronou generator. We ue the ame model o a ynchronou machine a in Figure 1 with no neutral line connected. (However we mention that ynchronverter can be built alo with a neutral line connected to the grid.) Now we can take equation (2.2), (2.3), (2.4), (2.17) and (2.18) rom Chapter 2 or our ynchronverter model. We add an additional aumption o a contant rotor current to impliy calculation. A mentioned in Section 3.3 thi i not necearily a good approximation but it i required to impliy the deign o the controller. I we conider the ynchronverter to be a mall inverter (power wie) in a very large grid then the grid may be regarded a an ininite bu, o that the equation o the ytem hould be imilar to (3.5). Future work hould include a comparion o thee two model. We get the ollowing equation or our ynchronverter: di (4.1) v Ri L e, dt (4.2) e M i in, (4.3) T M i,i n, e i (4.4) J T T D, m e p. The active and reactive power are deined by P i, e and Q i, e quad, where e M i co. We can calculate the active power and the reactive power uing the quad imple equation: P M i i,in, (4.5) Q M i i,co. Thee equation give u the baic algorithm o the ynchronverter hown in Figure 7: 26

28 Eqn (4.2) Eqn (4.3) Eqn (4.5) 27 Figure 7. Electronic part o a ynchronverter (without control), typically running on a DSP. Note that, i we aume no vicou riction, then the virtual active torque acting on the rotor i T D. The act that thi expreion depend on mean that it i actually a m p eedback loop, called requency droop. A wa mentioned in Section 2.2 (about the model o the ynchronou generator), we conider the torque T m to be contant or an input which change by uer demand. To avoid harp tranient, we add an LPF or the ignal T m. I we denote Tn Tm Dpn, which i the active torque at the nominal requency n, then. T T D m n p n Since the requency o the ynchronverter ollow the requency o the grid, the ynchronverter will change it output power o that or requencie above nominal the power will be le then the nominal power, while or requencie below nominal, the power will be more than the nominal power. We would like to chooe the droop coeicient D p o that it matche the real droop coeicient ued in ynchronou generator. For example or mot generator in the European grid, or a 3% change in requency the power will change 100% rom nominal. Obviouly or a real ytem thi cannot alway be obtained. We don't alway have enough energy tored or the ability to aborb energy in our ytem. Later on we will how how to handle thi problem. The importance o J hould alo be noted. J hould receive a value that allow the machine to have an inertia time contant that it an actual ynchronou generator. For the ynchronverter we preer the lowet o value o J, ince it low down ytem repone time and may caue tability problem. Let H be the inertia time contant o a ynchronou generator, deined a Thi lead to J 1 2 H Jn Pn. It i known that: 2ec H 12ec P n n and Dp Pn n droop rate, where the droop rate i the proportion o n that will caue a 100% change in T m (typically around 0.03). For voltage and reactive power control an additional control loop i required. The rotor current i conidered contant or at leat lowly changing, and a we have een the rotor lux ha a trong inluence on the voltage amplitude in the ynchronou generator, ee

29 (4.2). Indeed, the ynchronou internal voltage e i the mot igniicant component in the voltage v rom (4.1) ince we can aume mall reitance and inductance o the tator. We ue the approximate relation VE X (4.6) Q 3 co 2 V X derived in [16], where V i the rm voltage value on one phae o the ynchronverter terminal, E i the rm value o e, i the power angle a deined in (3.1) and i the abolute value o the tator impedance when X L R i neglected. We ee rom (4.2) that we can ue the rotor lux to control E and o alo the reactive power uing a imple integral controller. We can ee thi alo directly rom (4.5). We want to enable a change in Q uing a change o rotor lux with minimum change to P, which we have already controlled uing the requency droop part. Mot ynchronou generator operate in a PV (power-voltage) control cheme. Thi mean that the generator will help tabilize voltage by upplying or aborbing reactive power. For thi we add another droop loop or voltage dependent reactive power Here Q i the output reactive power, D q QQet Q 0. V V V n Q et i the deired reactive power or nominal voltage, V i the meaured rm voltage on the ynchronverter terminal and V n i the nominal rm voltage. The integrator on the rotor lux control loop (hown in Figure 8) ha a gain1 K 0. Chooing K and a lower ytem but uually more table. Larger D q depend on the application, by tuning. Larger K mean D q mean that the ynchronverter will try harder correcting the voltage at the expene o accurately tracking the deired Q et. In order to work in a PQ (power-reactive power) control cheme, only reactive power control, the uer can take Dq 0. Meauring the amplitude o the voltage on the ynchronverter terminal require extra work in the algorithm. Here we make the aumption that the ytem i ymmetric, balanced and ha no higher harmonic. Under thee aumption the voltage amplitude detection i very imple (ee [33] or more inormation): (4.7) V v v v v v v where va, v b and 2 3, g a b b c c a v c are the meaured voltage on the ynchronverter terminal. In otware implementation one hould check that the um inide the quare root i indeed poitive to avoid Nan (not a number) error. Since thi expreion doen t alway hold, i the grid i unbalanced, it i recommended to pa the meaured voltage through an LPF. The ame i true or Q and Te, ee Figure 8. Thi LPF i a tunable ilter, my choen value or imulation will be dicued in Section 12. The initial condition or the rotor lux hould be the approximate teady tate value which can be calculated rom (4.8) M i e n, 28

30 where e i the expected amplitude o e rom (2.3) (about 10% higher than the amplitude o the nominal grid voltage). Synchronization o the ynchronverter to the grid can be done by uing a PLL, but there i a better way. The paper [32] preent the deign o a el-ynchronizing ynchronverter. Thi i implemented by adding 3 otware implemented witche. The irt witch, SQ turn o the voltage control part, D q. The econd witch inductor SC connect a virtual reitor and R vir and L vir, where a virtual current i calculated rom meaured grid voltage, intead o a meaured current. Thi i done ince without ynchronization we can't connect the ynchronverter to the grid, to avoid a large current and a large tranient at the moment o grid connection. In our equation, ee (4.3) and (4.4) we ee that the meaured current play an important role. Thi mean we mut ind a uitable replacement. Uually we elect the virtual reitor to be twice the tator reitor and the virtual inductor i identical in ize to the actual inductor. The calculation o the current i a ollow: where a iˆ vir eˆ( ) vˆg ( ) (), L R vir denote the Laplace tranorm, i the complex variable, e i the internal ynchronou voltage vector, replacing i. The third and lat witch vir v g i the meaured grid voltage vector and i vir i a vector S P connect a PI controller to the requency droop loop in order to replace the nominal requency o the grid, which i 50Hz or Europe and 60Hz in North America, with the actual grid requency g etimated via r. Thi calculation i required becaue the grid requency may deviate rom nominal value by a mall portion. During the entire ynchronization proce P et and Q et are et to be 0. The purpoe i to make the ynchronverter create a mirror image o the grid where no power i delivered to or rom the grid. The reult i a mooth connection to the grid and aterward we can lowly increae Pet and Q et. It i recommended to give dierent value or the ynchronverter contant when ynchronizing. Smaller inertia contant will igniicantly reduce the ynchronization time. Increaing the integrator contant K in the reactive power control loop will low ynchronization but will help tabilize the proce. Chooing thee new value mut be done careully ince the ytem can become untable! For intance, i glvir i mall compared to R vir and i K i relatively mall, then the ytem will ocillate. 29

31 Figure 8. Propoed controller (electronic part) or a el-ynchronized ynchronverter, ollowing [32]. Finally we mention briely the PWM (Pule Width Modulation, ee [8]) part and the ilter that connect the inverter to the grid. The deired ignal e computed by (4.2) i ent to a PWM ignal generator that open and cloe electronic witche (IGBT or MOSFET, ee [14]) in order to generate an approximation o a ine wave in the low requency range. The witching requency mut atiy two condition. We want the witching requency to be high enough o that the Total Harmonic Ditortion (THD) will be uiciently low, the exact value depend on regulation and application. We want our ilter reonant requency to be at leat ive time lower than the witching requency, o that witching noie won't be increaed due to reonance. A an upper bound we want to minimize our loe on the witche which we know increae with requency. A witching requency o 10 khz i good enough or a 5 kw inverter that wa teted in our imulation and later on in the lab. Beore giving a detailed decription o the PWM ytem ued, we will give a hort background on three phae inverter topology. The topology hown below in Figure 9 i baed on the model o the canonical witching cell, ee [22] and it i ued or mot power converter. Thi allow a two way power low. 30

32 Figure 9. Scheme o a 2-level 3 phae inverter with neutral line baed on the model o the baic witching cell. The igure alo how a neutral line. In our imulation that line wa not included. In order to reduce loe and noie we have improved the witche or each phae and implemented a neutral point clamped inverter topology with three DC voltage level (the neutral i repreented a ground): Figure 10. Neutral clamped inverter topology or one phae o a 3-level, 3 phae inverter. On each phae there i a at witch between neutral and V when the deired ine wave e i poitive, and between neutral and V when e i negative. Uually the voltage are ymmetric, meaning that V V in Figure 10 i operated a ollow:, to generate a ymmetric ine wave around 0. The circuit V V S P 1 and 2 VP 0 S2 and 3 V V S P 3 and 4 S are cloed, S are cloed, S are cloed. The ilter ued to connect the inverter to the grid i an LCL ilter. Thi i a very wellknown ilter ued in many inverter. Thi ilter alo ha the role o being imilar to the tator R, L i we neglect the capacitor (which i needed to reduce the ripple). 31

33 Figure 11. Schematic o a 2-level 3 phae inverter connected to the grid via an LCL ilter and a circuit breaker. The LCL ilter i mainly meant to reduce harmonic and make the output current moother. There are a ew baic rule or electing the ilter capacitor and inductor. We want le than 5% current ripple, le than 10% voltage drop on the inductor and a reonance requency o about 20 time more than the grid requency 50 Hz. Thi will allow econd and third harmonic to pa but higher harmonic, which include harmonic caued by the witching, will be blocked. The ormula or Lg and C below were calculated by a baic calculation o the ripple o the current and voltage drop on both inductor, the ormula or L i taken rom the model o the baic witching cell: Here 2Δ, L V i L and C L L L V T i i i the current ripple, period and g L n n g n V L i the voltage on the two inductor, T i the witching n i the nominal requency. The lat ormula inure that the reonant requency o the LCL ilter i around 20 n. Typically V L i 5% - 10% o V n (the nominal rm phae voltage). Once the ilter value L, Lg, C are calculated or an inverter that traner power P 1, the ilter value L2, Lg2, C 2 or another inverter that traner power P 2, with identical ripple characteritic and witching requency, are given by: L, 2 L P1 P2 Lg 2 Lg P1 P2 and C2 CP2 P1. Since imulation with the exact witching model take a very long time to run, it i deirable to replace the witching model with an average model average model that contain a controlled voltage ource intead o witche. To validate thi replacement, we have run imulation o a ynchronverter with the witche and with the average model. In the imulation we have ued a el-ynchronized ynchronverter with the nominal power 5kW connected to an ininite bu. The olver ued wa ode23tb, which wa recommended when working with SimPower element. We let P 3.5kW and Q 500VAr. The voltage wa V line to line rm. Figure 12 and 13 how the active power and reactive power proile rom the two imulation: et et 2 32

34 Figure 12. Active power proile o a ynchronverter connected to an ininite bu imulated with PWM and with an average model uing a controlled voltage ource. Figure 13. Reactive power proile o a ynchronverter connected to an ininite bu imulated with PWM and with an average model uing a controlled voltage ource. Figure 14 how the requency rom the two imulation: Figure 14. Frequency o a ynchronverter connected to an ininite bu imulated with PWM and with an average model uing a controlled voltage ource. 33

35 5. Additional improvement to the ynchronverter algorithm Chapter 4 preented the background on ynchronverter, mainly baed on [32] and [33]. In thi chapter we propoe everal improvement or the ynchronverter model. A mentioned earlier the power upplied by any real ource cannot change intantaneouly. In order to prevent the inverter rom requiring the battery or PV array to change their output intantaneouly, we ilter P et and Qet bandwidth ha been choen to be 4 rad/. with LPF beore urther proceing. The ilter Another practical problem arie when the requency deviation rom the nominal value i large and the power that the ynchronverter i required to upply or aborb i more than phyically poible. To olve thi problem we divide the requency droop loop into two branche: a high and a low pa branch. All the torque except the torque rom the high pa branch and T e are aturated (ee Figure 15) thereby limiting the power requirement. We choe 20 rad/ec to be the cuto requency or the ilter in the low branch o the requency droop loop. The inal change made to the algorithm i the addition o an over current protection beore the PWM. We etimate the current by irt computing the vector e v (the notation i a in (2.2) o that v i now the voltage vector on the ilter capacitor). In order to calculate the current i, we alo need to etimate the impedance o the inductor nearet to the witche. We aume that the requency i almot the ame a the nominal grid requency and compute the inductor impedance Z uing (3.7). It ollow that e v Z i. From here we get that the maximal voltage dierence e v allowed, baed on the maximal current allowed to pa through the inverter uing. max max e v Z i Uing a aturation block, we enure that e v Z i componentwie. Thi doe not replace the uual overcurrent protection or the inverter, but it can give an additional layer o protection provided that the current can be aturated without ditorting the ignal igniicantly and that our aumption o the requency, i.e., it i near nominal grid requency, i correct. All thee new change give u the updated ynchronverter cheme hown in Figure

36 Figure 15. Updated cheme o the ynchronverter algorithm including ilter on aturation o output power and extra protection rom high current. P et and Q et, Another poible change to the ynchronverter tructure i an additional LPF beore the aturation block o the torque loop. Thi LPF can imulate the delay caued by additional mechanim uch a an MPPT (controller that track the mot eicient voltage to be acting on a PV cell or maximum power generation) or a PV array (due to the act that the DC/DC tranormer that deliver power to the inverter i delayed by the MPPT control in cae additional power rom a torage unit i alo required) or a DC/DC tranormer that i uppoed to dicharge an energy torage unit, uch a a battery or uper capacitor. Thi LPF hould be conditioned to work only i the required output power i greater than the power available nominally to the ynchronverter. Thi i becaue we will not need to dicharge energy rom a torage unit i the required power i maller to or equal to the power generated nominally by the energy ource, uually a PV array. 35

37 6. Some imulation reult or ynchronverter In thi chapter we will how that the ynchronverter work like a ynchronou generator in a imple grid containing either another ynchronverter or an ininite power ource (ininite bu) with a tranormer o ratio 1:1 and load modeled a power ink equal to more or le the nominal power o the ynchronverter. Thi will be done by imulation in Matlab uing the average model o a ynchronverter deigned or 5kW nominal output power. We will how 4 cenario. The irt 3 conider a ingle ynchronverter and a grid with load and a tranormer o ratio 1:1 a decribed above with an additional line impedance o 1 mω and 2 mh between the ininite bu and the load. The lat one conider a grid compoed o 2 ynchronverter, where the irt generate the grid by working in iland mode with contant voltage and requency reerence and the econd ynchronverter ynchronize and join. In the irt cenario we connect the ynchronverter to an ininite bu and lowly decreae the bu requency at the rate o 1Hz per econd until it reache 48 Hz. We would like to ee that the ynchronverter track the grid requency and maintain a power-requency droop o 100% power increae per 3% requency decreae. We would alo like to ee that the power doe not exceed a maximum o 8kW. The active power o the ynchronverter will be et to 3kW, while it nominal power i 5kW. So when the grid requency decreae to 48.5Hz ynchronverter hould reach 8kW and aturate. Figure 16. The ynchronverter connect to the ininite bu at t=0.14. The grid requency tart decreaing at t=2 at a rate o 1 Hz per econd until it reache 48 Hz at t=6. At t= 7 the grid requency tart climbing back to it nominal value at a rate o 1 Hz per econd. We ee that the ynchronverter track the grid requency change. 36

38 Figure 17. The ynchronverter connect to the ininite bu at t=0.14. The active power tabilize to P et = 3kW. When the grid requency tart decreaing at t = 2, the ynchronverter tart increaing it power until reaching aturation at t=5.5, while the grid requency goe down to 48.5 Hz. The grid requency tart increaing to it nominal value at t = 8 and hortly aterward the ynchronverter tart decreaing power back to it nominal value. In the econd cenario a ynchronverter i connected to an ininite bu and we lowly increae the bu requency at the rate o 1Hz per econd until it reache 52 Hz. We would like to ee that the ynchronverter track the grid requency and maintain a powerrequency droop o 100% power decreae per 3% requency increae. We would alo like to ee that the power doe not go below the minimum o 0kW. The active power o the ynchronverter will be et to 3kW and the nominal power i 5kW. So when the grid requency increae to 51.5Hz ynchronverter hould reach 0kW and aturate. Figure 18. The ynchronverter connect to the ininite bu at t=0.14. The grid requency tart increaing at t=2 at a rate o 1 Hz per econd until it reache 52 Hz at t=6. At t=7 the grid requency tart alling back to it nominal value again at a rate o 1 Hz per econd. We ee that the ynchronverter track the grid requency change. 37

39 Figure 19. The ynchronverter connect to the ininite bu at t=0.14. The ynchronverter active power tabilize to P et = 3kW. When the grid requency tart increaing, the ynchronverter decreae it output power until reaching aturation at t= 4 at which time the grid requency rie above 51 Hz. When the grid requency tart decreaing to it nominal value, the ynchronverter output power tart increaing back to it nominal value. From thee two cenario we ee that the ynchronverter i indeed capable o tracking grid requency under a requency-power droop loop whoe rate can be altered according to practical need. We have alo een that we can aturate the ynchronverter power according to application requirement which in our cae wa 8kW maximum and 0 kw minimum. Another important attribute o a ynchronverter i that it can tabilize voltage uing reactive power, like a ynchronou generator. To veriy thi in imulation we connected the ynchronverter to an ininite bu via a line impedance o 1mΩ and 2mH and a tranormer with ratio 1:1, and lowly changed the voltage rom nominal value to 90% o thi value. The voltage wa decreaed at a rate o 16.5 volt per econd and wa later on increaed at the ame rate until it reached the nominal value. Since thi i an actual electrical ytem with load and a tranormer (1:1 tranormation rate) we ee that the voltage in the ynchronverter terminal i not identical to the voltage o the ininite bu. The reult o thi imulation are hown in Figure 20, 21 and 22. Figure 20. Voltage proile o the grid meaured on the terminal o the ininite bu and on the ynchronverter terminal. The grid voltage decreae rom t=2 until t =4 and then climb back to it nominal value. 38

40 Figure 21. In the ame experiment a in Figure 20, the ynchronverter connect to the grid at t=0.14 and it requency remain contant when the voltage change. We ee in Figure 20 that the ynchronverter i able to lightly improve the voltage drop (the voltage on the ynchronverter terminal drop to 305 Volt and not 300 Volt). Thi i achieved without changing the requency (a een in Figure 21) by producing reactive power. Figure 22. The reactive power o the ynchronverter tabilize at it nominal value at t = 2. It then increae becaue grid voltage drop. Ater t = 4 the reactive power decreae becaue the grid voltage climb back to it nominal value. The nominal reactive power o the ynchronverter wa et to 500 VAr, but the ynchronverter doe not tabilize at 500 VAr due to the voltage drop on the line impedance. It i important to note that the behavior hown in Figure 20, 21 and 22 i not a Fault Ride Through. A Fault Ride Through mechanim mean changing the algorithm to make ure certain regulation are met. Since thi algorithm wa not deigned or a peciic regulation uch a mechanim wa not deigned. A imple check i done to detect voltage drop (will be dicued in Section 12.1) and hort circuit. It i let or uture deigner o the ynchronverter to decide what to do when uch a ault occur. For now only the mot baic regulation i kept, i.e. the ynchronverter diconnect ater a ault o more than 30 milliecond with the terminal voltage amplitude under 30% o the nominal value, or i the 39

41 requency change rate exeed 8 Hz per econd. An additional check or correct requency range i alo done routinely in the algorithm to make ure ynchronverter requencie are not above or below allowed grid requencie. A we ee it i very eay to it the ynchronverter algorithm to any given regulation. Thi implicity i very important in the at changing world o renewable and mart grid application. Next we will invetigate a micro grid (in imulation) compoed o two ynchronverter and two load. We expect that thi grid behave like an interconnection o 2 ynchronou generator. To tet thi we perorm ome load change and track the requencie at which the micro grid tabilize. We have deigned thi experiment o that the requency will be around 50 Hz. We chooe a non-ymmetric ytem in which the ynchronverter that tart the micro grid, called the grid ynchronverter, i working lightly above ull capacity, i.e. it i deigned to upply 5kW at the nominal requency 50 Hz, but it upplie 5.5kW to the two load and o cannot work at the nominal requency. The irt load cloet to the grid ynchronverter i 4kW and the econd i 1.5kW. Between the two load there i a mall impedance with reitance 1 mω and inductance 2mH. Figure 23 below decribe thi ytem. Figure 23. The two ynchronverter micro grid. The irt ynchronverter with P et = 5kW tart the micro grid. The econd ynchronvertert connect ater 4 econd. The 1 kw load i connected at t = 9, 5 econd ater the econd ynchronverter connect to the micro grid. Ater 4 econd we allow the econd ynchronverter to connect to the micro grid and ee that indeed the micro grid requency rie lightly above 50 Hz (ee Figure 25) and the load are hared between both ynchronverter a expected. The irt ynchronverter ha an active power o 4.5kW while the output o the econd i 1kW, ee Figure 24. Another 5 econd later we add an additional load o 1kW and thi caue the grid requency to drop and the extra load i hared between the two ynchronverter. Thi additional load caue the demand or power in the micro grid to be equal to the um o the nominal power o the grid ynchronverter (5kW) and the econd ynchronverter (1.5kW). Thi caue the ytem requency to tabilize to it nominal value 50 Hz. Both ynchronverter have identical parameter except their active power et point. Both ynchronverter were et to output 500 VAr. 40

42 Figure 24. Synchronverter 1 tart building the micro grid at t=0 and tabilize to provide 5.5kW. At t =4.2 ynchronverter 2 i added and the power rom ynchronverter 1 drop to 4.5kW while ynchronverter 2 upplie 1kW. Near t=4 we notice inter-area ocilation which quickly decay. At t=9 another load o 1kW i connected and the power output o both ynchronverter rie by 500 watt until they both output their nominal power which i 4.5 kw or the grid ynchronverter and 1.5 kw or the econd ynchronverter. Figure 25. Synchronverter 1 tart building the micro grid at t=0 and it requency tabilize lightly below 50Hz ince the load i above the ynchronverter et active power. At t =4.2 ynchronverter 2 i added to the micro grid and the requency increae lightly above 50 Hz. Near t=4.5 we notice inter-area ocillation which decay quickly. At t=9 another load o 1kW i connected and the requency decreae to 50 Hz. At thi point the load are balanced by the et active power o the ynchronverter o that the ytem tabilize to the nominal requency o 50 Hz. 41

43 Figure 26. Synchronverter 1 tart building the micro grid at t=0. It terminal voltage tabilize lightly below the nominal value o 330 V (amplitude). At t=4.2 ynchronverter 2 i connected and the voltage tart luctuating, it increae until t=9 when another load o 1kW i connected. 7. Inter-area ocillation We hope that the ynchronverter can be ued to olve practical problem in an actual power grid. To veriy thi we need to conider peciic problem, the current approach to olve them and then propoe new olution uing the ynchronverter. We have choen to ocu on inter-area ocillation. We will explain what they are and how they are dealt with today. Finally we will preent our olution to inter-area ocillation uing ynchronverter. Local ocillation are aociated with a ingle generator, due to it dynamic vaguely reembling the dynamic o a damped pendulum (the wing equation). The requency range o thee damped ocillation i approximately 0.7 to 2 Hz. In contrat inter-area ocillation involve group o generator which ocillate againt one another when the phyical link between them i weak (i.e., it ha relatively high impedance), ee [15] or detail. Thee ocillation are much more complex ince they involve everal non-linear ytem. The ocillation requencie range rom 0.1 Hz to 0.8 Hz. A key eature o thee ocillation i that the power variation in one (group o) generator will be oppoite to the power variation in the other (group o) generator. Inter-area ocillation are caued by the electro-mechanical nature o the generator. They contitute a well-known problem, ee or intance [16], [15], [21] and [29], and the uual olution i to regulate the ield voltage with a controller called Power Sytem Stabilizer (PSS). Thi require additional hardware and each type o PSS ha it own pro and con. Deigning a PSS controller i a hard tak which require comprehenive knowledge o the ytem, including all mode o ocillation involved, and careul thinking about which o the everal generator hould have a PSS intalled. We explore inter-area ocillation by imulation experiment: we look at a imple grid compoed o two ynchronou machine o 1 GW nominal power, each uing a team 42

44 turbine and a governor with an IEEE type 1 ynchronou machine voltage regulator and exciter, ee Figure 27. Thee generator operate at 22 kv and are connected to a 22/130 kv tranormer. Each generator ha a local load o 300 MW and both generator are connected via a 220 Km line with 0.05W/Km reitance and 1.4 mh/km inductance. The capacitance o the line wa neglected to improve the imulation run time (including the capacitance doe not change the reult igniicantly according to our tet). In the middle o the line there i a ubtation compoed o a 130/22 kv tranormer and a mall load o 55 MW. Another mall load o 60 MW located near one o the generator i witched on ater 10 econd. Thi udden change caue power ocillation between the generator until a new equilibrium i reached. Our ytem vaguely reemble the one ued in the example in the imulation package SimPower (the example i called power_pss). Figure 27. The imple power network (without ynchronverter) exhibiting inter-area ocillation. The model include team turbine a prime mover or the two ynchronou generator. The ocillation are triggered by cloing the witch connecting an additional load o 60 MW. Figure 28. Power ocillation in the ytem rom Figure 27. Figure 28 how that in our model, the active power output o the two generator in Figure 27 ocillate (with damping) againt each other when the witch i cloed connecting the additional 60 MW load at t=10.5. The requency o thi ocillation i Hz, which i well within the range o inter-area ocillation and below the range o local ocillation. The 43

45 ytem wa table beore the load jump and no power wa tranerred rom one generator to the other. The ocillation lat or more than 20 econd and even ater 10 econd they have an amplitude o 10 MW which can caue heating in the line and tranormer Power ytem tabilizer There are olution to the problem o inter-area ocillation, a we tated earlier, but they are ar rom perect and require change to the generator themelve. The mot common olution i the PSS which we will now review. The excitation control ytem in a conventional SG control the rotor current i and it main purpoe i to make ure that the generator maintain a terminal voltage cloe to the nominal voltage. The reactive power o a generator i cloely related to the terminal voltage: in inuoidal teady tate, VE Q 3 co X 2 V X where V and E are rm value a in equation (4.6). Thi expreion i undamental in the ield o AC power ytem, or more inormation ee [16]. The excitation control ytem provide the direct current to the ield winding which in turn determine the voltage o the generator and the reactive power. I we look at the ormula 3VE or the active power, which in teady tate i P in, (E, X, V and have the X ame value a in equation (4.6)) we ee that P depend on the voltage V and E. However, ince the variation in V and E are mall, P i mainly changed via (which change a a reult o change in the active torque o the prime mover). Power Sytem Stabilizer (PSS) ue the dependence o P on E to dampen inter-area ocillation a well a local ocillation, ee or intance [16], [30]. The general tructure o a PSS hown below i taken rom [30]: Figure 29. General model o a PSS. We have demontrated the eiciency o variou PSS by adding them to our imulation model rom Figure 27. We have ued 3 type o PSS available in the Matlab SimPower library. The irt i called Multi Band PSS (MB PSS) and the input to it i the rotor angle velocity deviation rom nominal requency. It handle 3 band o requencie (low, intermediate and high) adjuted in advance. The econd i a generic PSS the input to which i the rotor peed deviation (rom nominal requency) multiplied with a contant gain and iltered, a hown in Figure 29. The lat PSS i almot like the econd, but work on the dierence between electrical and mechanical power intead o rotor angle velocity deviation. 44

46 Figure 30. Comparion o power ocillation uing three type o PSS, a well a no PSS, in the ytem rom Figure 27. We can ee that the variou PSS perorm well, but have certain diadvantage. MB PSS i the bet o thoe teted here but it require very precie deign baed on an accurate model o the ytem. The econd PSS (alo called dω PSS) take a long time to reach the new equilibrium point. Even ater 20 econd it i ar rom ettled. The third type o PSS (alo called Pm Pe PSS) caue large and very low requency power ocillation. All three PSS have the diadvantage o having an advere eect on voltage regulation ince they work by changing the ield current and hence E. Maintaining a contant voltage i important or tranormer and certain load. Ater preenting our olution to the inter-area ocillation problem, we will how the voltage proile o the variou PSS Synchronverter with virtual riction Frequency droop baed olution 45 We would like to olve the problem o inter-area ocillation by uing the ynchronverter which i an available reource rather than modiying the generator by adding a PSS unit. Adding a PSS unit to a ynchronou generator i not a imple proce. It require identiying the generator on which the extra controller mut be added. Thi mean deciding which i the mot important generator rom a power tand point and rom a voltage regulation tand point ince PSS damage voltage control. From (4.4) we ee that the droop actor D p play a main role in the damping o requency ocillation in the ynchronverter. The requency droop loop in the ynchronverter i at ince it ha no mechanical part, only electronic component. Thi i unlike a real generator in which the droop loop act on the prime mover and thereore ha delay due to phyical limitation (o the order o one minute). Thi mean that generator are more enitive to ocillation and adding our ynchronverter to the grid, even without any change, can potentially improve ytem behavior. To tudy the above idea, two ynchronverter were added to the mall grid rom Figure 27, one near each generator, uing tranormer (inverter do not work at high voltage due

47 to limitation o the electronic witche). Each ynchronverter wa deigned to give 45 MW maximum power and we operate them at 35 MW. Thi mean that the ynchronverter upply roughly 10% o the power required by the load. 46 Figure 31. The network rom Figure 27 ater adding ynchronverter next to each ynchronou generator. From the imulation o thi ytem, we ee that adding the ynchronverter improve the overall repone o the ytem (ee Figure 32). Obviouly thi i not a igniicant improvement. In the next ection we will make an additional change to the ynchronverter algorithm which will greatly improve the perormance o the ytem when combating interarea ocillation Virtual riction baed olution I there are everal generation area and each contain a ynchronverter, an additional torque can be introduced that act a i there wa vicou riction between the imaginary rotor o the ynchronverter. Thi will require at communication between ynchronverter which may incur cot, however, communication ytem have improved and have become very cot eicient. The additional virtual riction torque will be added to the total torque acting on the imaginary rotor o the ynchronverter. To make thi more precie, conider or implicity a micro grid with only two ynchronverter with rotor angle 1 and 2. The mechanical equation including the virtual riction torque are: J1 Tm 1 Te 1 D ( ), p1 1 r Dpv 1 2 J2 Tm 2 Te 2 D ( ). p2 2 r Dpv 2 1 Here Dp 1, Dp2 0 are the droop contant o the ynchronverter, while D 0 i the virtual riction coeicient that help damping unwanted inter-area ocillation. In teadytate operation the virtual riction torque i zero. When there i a udden load change and the ytem tart to go o balance then the virtual rotor o the ynchronverter rotate againt each other and the virtual riction torque tart working to counteract thi. Thi idea wa irt reported in our conerence paper [5]. pv

48 Notice that the rotor o a real ynchronou generator tore the kinetic energy J 2 2, proportional to it moment o inertia J. The importance o inertia in power ytem i dicued or intance in [26]. Inverter have no moving mechanical part, o they cannot tore kinetic energy. To imitate the eect o the kinetic energy, a ynchronverter need large capacitor on it DC bu or torage batterie (a better but more expenive olution). Conider the tet ytem in Figure 31 with virtual riction added to the ynchronverter. Figure 32. Sytem with two ynchronverter with and without virtual riction. In Figure 32 we ee the eect o the requency droop o the ynchronverter (in red) on the active power ocillation in one o the generator. Thi eect i mall but conidering that mot o the power in the tet grid come rom the generator and not rom the ynchronverter it i not negligible. The eect o virtual riction on the ocillation in the active power o the generator i trong enough to dampen them completely within 10 econd, at t= 20. A mentioned earlier the PSS damage the voltage regulation. Figure 33 how the active power while uing variou type o PSS and alo while uing ynchronverter with virtual riction or damping inter-area ocillation. Figure 34 how the correponding voltage proile near generator 1. 47

49 Figure 33. Comparion o ytem power low with ynchronverter and PSS. Figure 34. Comparion o ytem voltage proile with ynchronverter and PSS. We wanted to take the ytem to an extreme cenario, to generate tronger inter-area ocillation and demontrate how the voltage proile can be ditorted by the PSS while the ynchronverter olve the problem without voltage ditortion. The tet ytem in Figure 31 i changed to caue tronger inter-area ocillation. Thi i done by increaing the ubtation load to 150 MW rom 55MW and alo by increaing the witched load to 150 MW rom 60 MW. Figure 35 how the ocillation in active power o generator 1 under the change mentioned above. A een in Figure 35 the ocillation in power are better damped by the ynchronverter with virtual riction than the PSS, but not igniicantly. 48

50 Figure 35. Comparion o ytem power low with ynchronverter and PSS, with the witched load increaed to 150 MW (rom 60 MW) and the ubtation load increaed to 150 MW (rom 55 MW). Figure 36. Comparion o ytem voltage proile with ynchronverter or PSS, under the ame condition a in Figure 35. A een in Figure 36 the voltage proile on the generator terminal i ditorted igniicantly when the PSS olution i applied during trong inter-area ocillation. Figure 36 how that even or trong inter-area ocillation the voltage proile o the generator terminal remain unchanged when the ocillation are damped uing ynchronverter and virtual riction. One mut remember that in thi example the PSS work on generator which upply 90% o the power while the virtual riction work on inverter upplying only 10% o the power. In cae where the ynchronverter upply much more power the reult will be much better in avor o the ynchronverter. Since we have compared ynchronverter with virtual riction to ynchronou generator with a PSS controller, it i hard to ay which olution would work better i both o them are implemented on the ame power ource, either a ynchronverter or a ynchronou generator. We would like to tet 49

51 both olution on the ame power ource. We cannot create virtual riction between two ynchronou generator, however we can add an additional PSS controller to our ynchronverter algorithm. We aume that the ynchronverter behave imilar enough to a ynchronou generator, o that we could actually apply to it a PSS control. The PSS controller work on the induced voltage in the rotor. In our ynchronverter algorithm we aume that the derivative o the rotor current i 0 which i equivalent to auming that the voltage acro the rotor i contant. But in reality mall luctuation in the rotor current inluence the lux in the rotor and thereby change the rotor voltage. Thi i good or u ince it mean we can ue the current to change the voltage on our imaginary rotor to tabilize requency luctuation caued by inter-area ocillation uing PSS control. In Figure 37 we can ee where we place our PSS controller in our algorithm. The proportional gain D q tabilize the voltage amplitude in the ynchronverter terminal. By adding another reerence voltage rom PSS control which depend on requency luctuation we couple the voltage amplitude E with the requency diturbance, Δ. The wahout ilter contant T w in Figure 37 i choen in the range o 1 to 10 econd. The ynchronverter algorithm i run on a proceor o it i dicrete and not continuou. The contant 50 Figure 37. Synchronverter with PSS controller K p hould be choen uch that it ampliie the diturbance in requency

52 uiciently, but not higher than what the ynchronverter can handle. We have ound K 10 to be uitable. It i important to aturate the additional voltage rom the PSS p control to be not more than 5% o the nominal voltage and not le than -10% o the nominal voltage. The implementation in Figure 37 i a baic tandard control tructure o a PSS. There are better implementation o PSS controller but thi i good enough or our tet. In order to check that thi PSS controller actually doe what it i uppoed to do we deigned a tet ytem o two ynchronverter with a witched load between them connected via a high impedance line. Thi hould generate inter-area ocillation when the load are witched. I will not give an exact decription o thi imple ytem ince the exact parameter are not important, only the act that inter-area ocillation are generated between 2 ynchronverter. The igure below preent the phae dierence between the ynchronverter virtual rotor angle,. Figure 38. Proo o concept or PSS implementation in a weakly connected two ynchronverter ytem. We ee in Figure 38 that until t=8 there are no ocillation. At t=8 the load change and the load haring between both ynchronverter change a well. We ee that the blue line repreenting the cae where the ynchronverter have neither PSS nor virtual riction, ha ocillation that do not decay until the load change back to it previou value at t=16. The red line repreent the cae where both the ynchronverter have a PSS controller. We ee that the ocillation decay but the haring o the load, located between the two ynchronverter in the middle o the line connecting them, i dierent. It i hard to ay whether thi i good or bad ince it depend on the application. In uch a imple ytem there i no ideal load haring. The teal line repreent the cae where the ynchronverter have virtual riction between them and we obtain better decay o the ocillation. The black line repreent the cenario where the ynchronverter have virtual riction between them and alo a PSS controller. There i a light improvement in the decay o ocillation but it i hard to tell whether thi i a real improvement (compared to only having virtual riction) or ome ide eect due to imulation approximation. To ummarize we ee that the propoed 51

53 PSS controller doe indeed work a it hould in a ynchronverter. We would like to tet i the ynchronverter with PSS controller perorm a well a the ynchronverter with virtual riction in the ytem decribed in Figure 31. Then we can truly undertand which olution i better or the ynchronverter operating in a ytem with ynchronou generator. Here we wanted to ee the perormance o each olution clearly o we run the imulation under the extreme condition decribed in Figure 35. Figure 39. Comparion o the ynchronverter rom the tet ytem in Figure 31 in 4 dierent operation mode, with and without PSS and virtual riction. We ee in Figure 39 that the dierence between the blue line which decribe the reult under no PSS or virtual riction between ynchronverter, and the green line with PSS only i very mall. Since the PSS i placed in the ynchronverter which only upply 10% o the power and are not the caue o the ocillation it doe not do much. Thi i unlike the imulation in Figure 38 where only ynchronverter (and no ynchronou generator) are involved with PSS added to them. The red line in Figure 39 how that uing virtual riction dampen inter-area ocillation like in Figure 35. The black line which combine both PSS and virtual riction how very little improvement. The reult in Figure 39 are dierent rom the reult or the two ynchronverter ytem in Figure 38, ince here the ynchronverter are not the caue o ocillation, the ynchronou generator are. We ee that not only virtual riction i a better olution than PSS (when they both operate on the ynchronverter) but alo that it can help in cae where the PSS doe not help. It i important to note that communication ytem have delay. Thee delay may interere with our virtual riction implementation. It ha been checked that or mall delay o up to 2 milliecond the reult do not dier by much. I have choen not to attach thee reult ince thi iue require more reearch then I have conducted o ar. 52

54 7.3. The Pogaku-Prodanovic-Green (PPG) inverter and inter-area ocillation We would like to invetigate the perormance o the ytem decribed in Figure 31 i intead o ynchronverter we ue a conventional inverter. For thi purpoe we have elected a very advanced inverter controller developed by N. Pogaku, M. Prodanovic and T.C. Green in [23]. Thi controller ha requency-power droop control a well a advanced load haring capabilitie. Below are the control block diagram o thi inverter a they appear in [23]: Figure 40. General inverter block diagram. P i the Park tranorm and an invere Park tranorm i done ater the calculation in the current controller, beore the PWM generator and witche. ω PPG 53 Figure 41. Power controller o th PPG inverter. ω c i an LPF cut-o requency or high harmonic illtering o active and reactive power. ω n i a reerance requency rame within the range o allowed operational requencie, that will et the power P n tranerred to the grid at nominal grid requency correponding to the equation ω n m p P n = ω grid, where ω grid i the grid nominal requency and P n i the active power ater the LPF. The actual power that will low to the grid i bigger. In order to calculate the actual power upplied to the grid we mut alo add the power that i cut by the ilter. Thi i not eay to compute, ince we do not know the eect o the witche and other non linear element in the grid. We imply tuned the value o ω n until we got the wanted nominal power or the imulation. m p i a droop gain calculated by m p = (ω max ω min ) P max where i we conider negative requency drooping (extra power or requency below nominal) we need to put 2P max. V n i the nominal voltage amplitude (phae to ground) o the inverter and n q i calculated by (V max V min )/Q max. Thi value repreent the voltage drooping, the higher the value o n q, the le companation in VAr i given or maintaning contant voltage. Thi parameter require tuning. θ PPG i the etimeted grid angle, ued in the Park and invere Park tranormation.

55 Figure 42. Voltage controller o the PPG inverter. K pv and K iv are PI controller contant or the voltage controller and C i The value o the ilter capacitor, C ued to compute the eect o ilttering on the output voltage and F i the current eed-orward gain. F i ued to improve the diturbance rejection o the inverter ytem. Figure 43. Current controller o the PPG inverter. K pc and K ic are PI controller contant or the current controller and L i the value o the output inductor, L c. The value o thi inductor i ued ince we want to control the output ilter inductor current and not the current beore the ilter. Value For 40 MW PPG inverter Parameter L C Value nh 5.52 mf L c nh K pv 500 K iv 1000 K pc 10.5 K ic 5000 F 0.75 ω n V n rad/econd 563 V 54

56 ω c m p n q 31.4 rad/econd 0.35 ( rad econd ) /MW 60 V/GVAr Table 2. Value o parameter or a 40 MW PPG inverter working in a 50 Hz grid. One can eaily ee that thi inverter ue an LCL ilter like the ynchronverter and it alo ha requency-power drooping and voltage-reactive power drooping like in the ynchronverter. Thi make it eay or u to build an inverter model with parameter equal to thoe o the ynchronverter. An additional ynchronization block hould alo be added to thi inverter. We have deigned a ynchronization proce which i almot identical to the one in our elynchronized ynchronverter. We ound that the reulting ytem behave imilarly to the one containing ynchronverter without virtual riction. Thi mean that ynchronverter have the advantage o other advanced inverter, and thi i probably due to their at requency-power droop, which doe not uer rom mechanical limitation and delay like the requency-droop loop o a real generator. The ollowing igure i the comparion o the ynchronverter and thi inverter, which we will reer to a PPG inverter, when we put them in the ytem hown in Figure 31. Figure 44. For the ytem in Figure 31, comparion o behaviour with inverter and ynchronverter (without virtual riction). 8. Eilat Ktura grid ection: a cae tudy So ar we have teted the ynchronverter in imulated ytem, ome o which we deigned ourelve or ome peciic tet and other were baed on ytem in SimPower example. To veriy that the ynchronverter actually help in a real ytem we hould tet it on a real ytem. For thi purpoe we have choen the grid ection between Eilat and Ktura. Eilat i the outhernmot city in Irael. Thi mean that electricity produced motly in Hadera power tation ha to travel long ditance to reach Eilat. Thi i very cotly which make Eilat a perect candidate or renewable ource, along with the act Eilat i unny 55

57 mot o the year. We obtained data rom the IEC concerning line parameter, ynchronou generator ound in Eilat, renewable ource currently located in Ktura and other ource that will be available in the uture. The north line rom the center o Irael to Ktura wa given imilar value a the line rom Ktura to Eilat, only that the ormer i ix time longer. The grid rom the north i modeled a an ininite bu. In Figure 45 we ee the ull cheme o the ytem. Figure 45. Schematic o Eilat-Ktura grid ection. The imulation cenario i a ollow: irt we connect the generator in Eilat to the ininite bu, wait or 8 econd until the ytem tabilize. Ater the ytem tabilize, at t=8, the inverter in Ktura i connected. It take another 8 econd or the ytem to tabilize. Aume that when the inverter in Ktura i connected along with the generator in Eilat, we get a cenario where the local load i balanced by the local power production. Thi i a reaonable cenario ince it i expenive to bring power rom the north to Eilat. Two econd later at t=18 the line rom the north to Ktura i diconnected thereby creating an electrical iland. Thi could happen or maintenance reaon or ault omewhere along the line rom Ktura to the north. It i important to note that currently the IEC doe not allow ilanding but thi ituation would probably change in the uture. The newly created electrical iland in Eilat-Ktura grid ection tart ocillating in requency and power. It take almot 15 econd until the ocillation decay. Ater another 17 econd at t=35 a three phae ault occur in Eilat. Thi i an extreme but poible cenario. Short circuit occur requently in a power grid. Mot o them are one phae ault that lat or a hort duration and conume little power. We conider thi three phae hort circuit in three cenario: with the reitance between the phae (in Eilat) being 10Ω, 1Ω and 0.1Ω. In the irt cenario the three phae ault caue a 2% requency decreae at the Ktura and Eilat generator, due to the extra power diipated in the high voltage line and alo in the reitor o the hort circuit. The requency in Eilat alo decreae in the cae o 1Ω hort circuit, but it increae or 0.1Ω, a i normal or generator near a total hort circuit (where terminal voltage drop near zero). In each o the three cae, a phae dierence develop between Eilat and Ktura during the hort circuit and the ilanded ytem ocillate ater the hort circuit i removed. The recovery o the ytem rom the ault depend on the type o inverter ued in Ktura. We want to compare the ynchronverter with our PPG inverter 56

58 decribed in the previou chapter. Today there are no inverter that allow poitive requency-power drooping, meaning that i the requency drop the inverter cannot upply additional power ince it ha no energy torage. Thi i why we need to update the PPG inverter to only upport negative power-requency drooping. Thi i done by aturating the power output i the ignal P rom the requency droop i larger than P n a hown in Figure 45 below. Notice that P n i the nominal power ater the LPF, thi i lower than the actual power lowing to the grid at nominal grid requency, ee Figure 41 or more detail. Figure 46. Updated power controller o the inverter deigned in [23] (compare with Figure 41). In Figure 46 we ee that i the power the inverter ha to generate (due to requency drop) i larger than the nominal power then, the requency-power droop characteritic change o that the requency i till tracked by the inverter but the power doe not exceed nominal power. We have run three imulation: one with the ynchronverter a the inverter at Ktura, the econd with the PPG inverter a the inverter at Ktura and a third with a ynchronou generator intead o the inverter at Ktura. We have run thee imulation to compare the perormance o the ynchronverter and the PPG inverter. The ynchronization proce (ee below in Figure ) i equally good or the ynchronverter and the updated PPG inverter and they both ynchronize better than a ynchronou generator. Figure 47. Synchronization o the inverter at Ktura to the grid (requency in Ktura). Auming that 89 MW are generated by ynchronou generator in Eilat, and 40 MW are generated by an extra unit 52 km to the north in Ktura, generation and conumption are balanced. We have imulated 3 option or thi extra unit: ynchronverter, updated PPG inverter and ynchronou generator. The Eilat-Ktura grid ection i ilanded at t = 18 and at t = 35 a 150 m hort circuit occur. In each o the ollowing cenario when the extra unit in Ktura i a ynchronou generator the local grid o Eilat-Ktura loe tability, leading 57

59 to a blackout. We will irt how the reult o the irt cenario, a 10Ω hort circuit reitance. Figure 48. Frequeny in Eilat ater ilanding and hort circuit at Eilat when the inverter in Ktura i a ynchronverter, PPG inverter or a ynchronou generator. Both ynchronverter and PPG inverter tabilize the ytem, the ynchronverter tabilize ater. The ynchronou generator trip the ytem (intability ater the hort circuit). Figure 49. Frequeny in Ktura ater Ilanding and hort circuit at Eilat when the inverter in Ktura i a ynchronverter, PPG inverter or a ynchronou generator. Similar to the behavior een in Eilat in Figure 48 (thi i the ame imulation). Figure 50. Voltage rm value per unit on the terminal o the ynchronou generator in Eilat under the ame cenario a in Figuree 48 and 49. We ee very mall inluence o the type o power ource in Ktura, when a ynchronou generator i the extra power unit in Ktura the voltage ignal o the generator in Eilat ha ripple. 58