Short-Circuit Calculation in Distribution Networks with ... - MDPI

energies Article

Short-Circuit Calculation in Distribution Networks with Distributed Induction Generators Niancheng Zhou 1 , Fan Ye 1, *, Qianggang Wang 1,2 , Xiaoxuan Lou 1,3 and Yuxiang Zhang 2 1

2 3

*

State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, Chongqing 400044, China; [email protected] (N.Z.); [email protected] (Q.W.); [email protected] (X.L.) School of Electricaland Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore; [email protected] Department of Electrical Engineering, University of Wisconsin-Milwaukee, Milwaukee, WI 53211, USA Correspondence: [email protected]; Tel.: +86-136-1768-5854; Fax: +86-23-6511-2739

Academic Editor: João P. S. Catalão Received: 19 January 2016; Accepted: 31 March 2016; Published: 9 April 2016

Abstract: This paper presents an improved current source equivalent model method to determine the short-circuit current of a distribution system with multiple fixed-speed and variable-speed induction generators (IGs). The correlation coefficients of flux components between stator and rotor under the unsymmetrical fault are analyzed using the positive and negative sequence steady-state equivalent circuits of an IG. The terminal voltage and current responses of fixed-speed and variable-speed IGs with and without the rotor slip changes under different penetration levels are compared to investigate the coupling relation between the short-circuit currents of IGs and the nodal voltages in the distribution network. Then the transient equivalent potential of an IG at the grid fault instant is derived. Sequence components of the short-circuit current in the network can be determined using the proposed technique. The correctness of the proposed method is verified using dynamic simulation. Keywords: short circuit calculation; induction generator; symmetrical components; rotor slip; steady-state equivalent circuit; distribution network

1. Introduction The increased penetration of renewable power has resulted in more distributed generators (DGs) embedded in distribution networks. The integration of large-scale DGs into the distribution network changes the distribution of power flow and short-circuit currents, and leads the short-circuit current contribution of the DGs to affect the power system protection and reclosing [1]. DGs include synchronous generators, induction generators (IGs), and other power sources with electronic interfaces; of these types, the squirrel-cage IG has received increased attention in distribution networks because of its low cost, small size, and low maintenance requirements [2]. When short-circuit faults occur, IGs are unable to maintain their terminal voltage without an external excitation current, because they have different short-circuit current characteristics compared with synchronous generators [3]. IG technology is based on the relatively mature induction motor (IM), and the difference is that the IG tends to excessively accelerate instead of the IM stalling that occurs during faults. Short-circuit calculation of the IM usually uses the voltage source equivalent method [4]. The similar method used for IG cannot calculate the short-circuit currents accurately during the fault process, because it does not consider the electromagnetic transient characteristics of IGs [5,6]. Therefore, understanding the dynamic characteristics and short-circuit current calculation of IGs under different fault conditions is essential. This work aims to study the short-circuit calculation of a multi-IG distribution network. Surge current in the stator windings of IGs is driven mainly by the stator transient DC flux linkage. At the same time, the extra transient AC component of the stator current is produced by Energies 2016, 9, 277; doi:10.3390/en9040277

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the rotor flux linkage when a short-circuit fault occurs in the distribution network. The short-circuit behavior of IGs has been extensively investigated using dynamic simulation software [7–9], physical experiments [10] and real-time simulation tools [11]. With the assumption of a fixed rotor speed during fault, an approximate formula based on the generator state equations at dq0 coordination has been derived for three-phase short-circuit current calculation in [12]. Owing to the short-current contribution of the healthy phases during an unsymmetrical faults, the short-circuit characteristics of IGs under symmetrical and unsymmetrical faults are different from one other. For unsymmetrical voltage sags, the analytic expressions of transient stator voltage and current and transient rotor voltage and current have been derived in [13–15]. The above studies are limited to the mathematical and time-domain transient studies. Although the short-circuit current of a distribution network with IGs can be determined by using simulation software, but a very long computation time is required. Additionally, the physical transient process of IGs during a fault cannot be explained theoretically. The implicit simulation is usually adopted for the numerical stability and errors and it is difficult to obtain the analytical relationship between the short-circuit current of IGs and their parameters. A more accurate sequence network circuit representation of IG is derived for the short-circuit calculations on the basis of the sequence network in [16]. The transient reactance and short-circuit current of a single fixed-speed IG is studied, however the work cannot calculate the trend of short-circuit current changing with time and the short-circuit current of multiple fixed-speed and variable-speed IGs. Those studies are not suitable for fault analysis in a distribution network with multiple IGs because of without considering the coupling between the short-circuit current of IGs and the distribution network. The fundamental frequency components of the current and voltage in a power system for the first few cycles after a fault are the primary focus of protection engineers. The sequence component method is usually adopted to calculate the short-circuit current because of the smaller computational burden and problem size compared with those of other methods [17]. The present work analyzes the transient behavior of fixed-speed and variable-speed IGs when an unsymmetrical fault occurs in the distribution network and the sequence components current source models of fixed-speed and variable-speed IGs considering the rotor slip change during a fault is established. The terminal voltage and short-circuit current responses of IGs under different integrated capacity, with and without considering slip change are investigated. Then an analytical method to calculate the shortcircuit currents of a multi-IG system based on the current source model of IG and the parameters of the distribution network is then proposed. The technique is a significant improvement of the one in [16] and can significantly reduce computational burden. The method is verified using the simulation software PSCAD/EMTDC where IGs are presented by the fifth-order differential model. The errors between analytical and simulation results are analyzed. 2. Sequence Component Current Model of Induction Generators (IGs) 2.1. Stator and Rotor Flux of IGs during Grid Faults All parameters in this paper are per unit. The stator short-circuit current of the IG is determined by the stator and rotor fluxes when an unsymmetrical fault occurs. The stator and rotor fluxes of IG are not affected by the zero-sequence component of the voltage [13], whereas the forced sinusoidal component is driven only by the positive and negative sequence components of the IG terminal voltage. With the motor convention, the stator voltage equation of the IG under an unsymmetrical fault using a space vector can be written as: .

.

us ptq “ u1s ptq ` u2s ptq “ U 1s e jωs t ` U 2s e´ jωs t “ Rs ri1s ptq ` i2s ptqs `

d rψ ptq ` ψ2s ptqs dt 1s

(1)

where u1s (t) and u2s (t) are the positive and negative voltage space vectors, i1s (t) and i2s (t) are the positive and negative current space vectors, Rs is the stator resistance, ψ1s (t) and ψ2s (t) are the positive . . and negative fluxes after faults, U 1s and U 2s are the sequence component phasors of the stator voltage,

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positive and negative fluxes after faults, U1s and U 2s are the sequence component phasors of the and ωs isvoltage, the synchronous electric angular velocity. When the stator resistance is disregarded, the forced is stator and ωs is the synchronous electric angular velocity. When the stator resistance . . jω t ´ t {jω s j  components of the can be expressed as ψ1s (t) =can U 1sbe e expressed {jωs andasψψ (t) U 2see s tjωjs s 2s1s disregarded, the stator forcedflux components of the stator flux (t)==´U 1s s and by solving Equation  js t (1). According to the flux linkage conservation, the post-fault stator flux also . ψ2s(t) = U 2se conservation, the js by solving Equation (1). According to the flux linkage contains a DC component. Assuming the normal operation voltage us (t) = U s e jωs t , then the stator post-fault stator flux also contains a DC component. Assuming the normal operation voltage us(t) = post-fault flux linkage ψs (t) of IG can be given as: U s e js t , then the stator post-fault flux linkage ψs(t) of IG can be given as: .

.

.

.

.

U 1s jω UU UU1s U s 2s U ´ jωs t Us t jU U 2s qse´t /tT{s Ts ψs ptq “ s t r st ψsjω (t )se 1s ´ e jω s e 2s e j` jω [ s s´p (jω1ss ´ jω2ss )] e js js js js js

(2) (2)

where Ts T=s (L Lmm22/L )/Rs sisisthe thestator statortime timeconstant, constant,LLmmisisthe theexcitation excitationinductance, inductance,LLs s= =LlsLls+ + s´ where = (L s − L /Lrr)/R LmL, mL,r = Lr L=lr L+lrL+m,Land , and L and L are the leakage inductances of stator and rotor. The post-fault forced m ls Llr are lr the leakage inductances of stator and rotor. The post-fault forced flux Lls and flux components thestator statorare are the the steady-state steady-state response, components in in the response, and and the thepositive positiveand andnegative negativeforced forced components in the rotor can be derived through the steady-state equivalent circuits in Figure 1. The components in the rotor can be derived through the steady-state . . . .equivalent circuits in Figure 1. relationship between between the rotor current I 1r , current I 2r and the I 1s , stator I 2s cancurrent be expressed The relationship the rotor , I 2r current and the can be I 1r stator I 1s , as I 2sfollows:

expressed as follows:

.

I 1r

I1s U 1s

Rs

jωsL ls

jωsL m

(a)

.

.

. jωs Lm I 2s ´jωs Lm I 1s “ , I 2r “  j  L I jsq R {s ` jω L R {p2 ´ s s mr 1s s L´ m I jω 2s s Lr I1r r , I 2r  r Rr s  js Lr Rr (2  s)  js Lr

jωsL lr

I1r

I 2s Rr s

U 2s

Rs

-jωsL ls

-jωsL m

(3)

(3) -jωsL lr

I 2r Rr 2s

(b)

Figure 1. Positive and negative sequence steady-state equivalent circuits of an IG: (a) Positive Figure 1. Positive and negative sequence steady-state equivalent circuits of an IG: (a) Positive sequence sequence circuit, (b) Negative sequence circuit. circuit, (b) Negative sequence circuit.

where Rr is the rotor resistance, the rotor slip s = (ωs − ωr)/ωs, and ωr is the rotor angular velocity. where Rr is theψrotor resistance, the rotor slip s = (ωs ´ ωrirr)/ω ωr is theforced rotor components angular velocity. s , and Combining s(t) = Lsis(t) + Lmir(t) and ψr(t) = Lmis(t) +L (t), the post-fault in the Combining ψ (t) = L i (t) + L i (t) and ψ (t) = L i (t) + L i (t), the post-fault forced components in the s s s m r r m s r r rotor can be calculated as follows: rotor can be calculated as follows: Lm Rr s  ψ1s (t ) ψ1r (t )   1rs ( s)ψ1s (t ) 2 Lm Rr {s¨ψ1s ptq L R ) η1rs psqψ1s ptq ψ1r ptq “ sL R r s  j s ( Ls Lr 2 Lm“ s r {s` jωs p Ls Lr ´ Lm q (4) (4) Lm LRm rR{pr 2´ (2sq¨ψs2s) ptψq 2s (t ) “ η psqψ ptq ψψ ptq “ 2r2r (t )  L R {p2´ 2s 2 2rs   ( s ) ψ ( t ) sq´ jω p L L ´ L q 2rs 2s Ls s Rr r (2  s) s js s (r Ls Lmr  L2m ) With a 3MW IG (the parameters are given in [18]) as an example, Figure 2 illustrates the amplitude With a 3MW IG (the are given sequence in [18]) as an example, Figure η2 illustrates the and phase characteristics of theparameters positive and negative correlation coefficients 1rs and η2rs in amplitude and phase of flux. the positive and negative coefficients Equation (4) between thecharacteristics stator and rotor The amplitude of η1rssequence is near 1.0correlation when the rotor slip η 1rs and η2rs in Equation (4) between the stator and rotor flux. The amplitude of η1rs is near 1.0 when is small in the normal operation [2]. When the slip drops to ´0.1 during the fault, the amplitude of the rotor slip isby small normal operation [2]. decrease When the slip drops the fault, η1rs will decrease 90%,inasthe shown in Figure 2. This indicates thattoa −0.1 smallduring slip change has athe amplitude of η1rs will by 90%, as shown Figure 2. in This indicatesthe that a small slip significant influence on decrease the positive sequence forcedinrotor flux thedecrease rotor. Therefore slip change change has a significant influence on the positive sequence forced rotor flux in the rotor. Therefore should be taken into consideration in the rotor flux calculation. theThe slipnegative change should beflux taken into consideration in the rotor flux calculation. sequence component in rotor windings induced by the stator negative sequence The negative sequence flux component in rotor windings induced byfor the stator negative forced flux can be neglected, because η2rs is near zero under the different rotor slips a fixed-speed IG sequence forced flux can be neglected, because η 2rs is near zero under the different rotor slips for a in Figure 2a. However, there are still negative sequence short-circuit currents of IGs caused by the grid fixed-speed IG involtages, Figure 2a. However, are still negative sequence short-circuit currents of IGs negative sequence especially, thethere negative sequence current of a variable-speed IG is mainly caused by the grid negative sequence voltages, especially, the negative sequence current of a influenced by the control structures and objectives in its machine side and line side converters [19]. variable-speed IG is mainly influenced by the control structures and objectives in its machine side The rotor resistance of a variable-speed IG will increase significantly after its crowbar protection and line side converters [19]. The rotor resistance of a variable-speed IG will increase significantly

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activation duringprotection grid faultsactivation and the magnitude η2rs will in the range 0.2–0.3 pu under after its crowbar during gridoffaults and stay the magnitude ofof η2rs will stay in the different slips, shown in Figure 2b. In this as figure, the in rotor resistance 25 times of range of rotor 0.2–0.3 pu as under different rotor slips, shown Figure 2b. Inrises thistofigure, thethat rotor the fixed-speed IG,25and so athat non-ignorable flux component inathe rotor windings a variable-speed resistance rises to times of the fixed-speed IG, and so non-ignorable fluxofcomponent in the IG can be generated by the negative component of the statorsequence flux. Besides the coupling rotor windings of a variable-speed IGsequence can be generated by the negative component of the differences flux forced components between variable-speed IGs,fixed-speed the decay time stator flux. of Besides the coupling differences offixed-speed flux forced and components between and 2 constant of the IGs, rotorthe flux DC component, Tr of = (L Lm /L will also be changed by increase in variable-speed decay time constant the rotor flux DCr , component, Tr = (Lr − L m2/L s)/Rr, will r´ s )/R rotorbe resistance IGsresistance during a grid fault. also changedof byvariable-speed increase in rotor of variable-speed IGs during a grid fault. 90 0. 9 80 70

 1rs  2rs

0. 7

Phase / deg

Magnitude/pu

0. 8

0. 6 0. 5 0. 4

50 40

0. 3

30

0. 2

20

0. 1

10 0

0 - 0. 1

- 0. 08

- 0. 06

s

- 0. 04

- 0. 02

- 0. 1

0

 1rs  2rs

60

- 0. 08

- 0. 06

s

- 0. 04

- 0. 02

0

(a) 80

0.9 60

 1rs  2rs

0.7

 1rs  2rs

40

Phase /deg

Magnitude/pu

0.8

0.6 0.5 0.4

20 0 -20

0.3

-40

0.2

-60

0.1 -80

0

-0.2

-0.1

0

0.1

0.2

0.3

s

-0.2

-0.1

0

0.1

0.2

0.3

s

(b) Figure and negative negativesequencecorrelation sequencecorrelationcoefficients coefficients between stator Figure2.2. Positive Positive and between stator andand rotorrotor flux: flux: (a) a (a)fixed-speed a fixed-speed IG, (b) a variable-speed IG. IG, (b) a variable-speed IG.

The stator flux ψs(t) has both the positive and negative sequence forced components ψ1s(t) and The stator flux ψs (t) has both the positive and negative sequence forced components ψ1s (t) and ψ2s(t) and the DC component ψdcs(t) in Equation (2). The reverse rotation flux with the rotor speed ψ2s (t) and the DC component ψdcs (t) in Equation (2). The reverse rotation flux with the rotor speed will be generated in the rotor windings because of ψdcs(t) cutting the rotor windings. The stator flux will be generated in the rotor windings because of ψdcs (t) cutting the rotor windings. The stator flux DC component is difficult to analyze in the stator reference frame, but it can be discussed in the DC component is difficult to analyze in the stator reference frame, but it can be discussed in the rotor rotor side by the frequency transformation [2]. The equivalent circuit in the rotor reference frame side by the frequency transformation [2]. The equivalent circuit in the rotor reference frame for the for the stator flux DC component is shown in Figure 3, . where. I dcs and I dcr are the DC components stator flux DC component is shown in Figure 3, where I dcs and I dcr are the DC components of the stator of the stator and rotor .currents, and U dcs is a virtual voltage caused by the stator flux DC and rotor currents, and U dcs is a virtual voltage caused by the stator flux DC component. With the component. With the relationship of the stator current I dcs and the rotor current I dcr in Figure 3, the rotor flux component ψdcr(t) caused by the stator flux DC component ψdcs(t) can be obtained as:

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.

relationship of the stator current I dcs and the rotor current I dcr in Figure 3, the rotor flux component ψdcr (t) caused by the stator flux DC component ψdcs (t) can be obtained as:

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Lm Rr ¨ ψdcs ptq “ ηdcrs psqψdcs ptq ψdcr ptq “ Ls Rr ´ jωLs p1 ´ sqpLs Lr ´ L2m q m Rr  ψ dcs (t ) ψdcr (t )   dcrs ( s)ψdcs (t ) L R  ψdcs (t ) Ls Lr  L2m2 )  dcrs ( s)ψdcs (t ) ψdcr (t )  Ls Rr  jms (1r  s)( Ls Rr  js (1  s)( Ls Lr  Lm )

I dcs I dcs

Rs Rs

-jωrL ls -jωrL ls

U dcs U dcs

-jωrL lr -jωrL lr

(5)

(5) (5)

I dcr I dcr

-jωrL m -jωrL m

Rr Rr

Figure Figure 3. 3. An Anequivalent equivalentcircuit circuit for for stator stator flux flux DC DC component component in in the the rotor rotor reference reference frame. frame. Figure 3. An equivalent circuit for stator flux DC component in the rotor reference frame.

0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0

Fixed-speed IG Fixed-speed IGIG Variable-speed Variable-speed IG

-0.2 -0.2

-0.1 -0.1

0

s0 s

0.1 0.1

0.2 0.2

Phase Phase /deg /deg

Magnitude/pu Magnitude/pu

The amplitude and phase characteristics of the DC component correlation coefficient ηdcrs in The amplitude and phase characteristics of correlation coefficient ηdcrs in The amplitude of the the DC DC component component coefficient in Equation (5) betweenand thephase stator characteristics and rotor fluxes are shown in Figurecorrelation 4. The reverse rotationηdcrs flux Equation (5) between the stator and rotor fluxes are shown in Figure 4. The reverse rotation flux Equation (5) between the stator and rotor fluxes are shown in Figure 4. The reverse rotation flux component ψdcr(t) in the rotor windings of a fixed-speed IG is approximately zero because of its component ψ (t) in a fixed-speed IG is zero zero because of its small component ψdcr dcr(t) in the therotor rotorwindings windingsofof a fixed-speed IGapproximately is approximately because of its small rotor resistance. Meanwhile, the amplitude changing range of ηdcrs for a variable-speed IG is rotor resistance. Meanwhile, the amplitude changing range of η for a variable-speed IG is dcrs small rotor resistance. Meanwhile, the amplitude changing range of ηdcrs for a variable-speed from IG is from 0.4 pu to 0.6 pu as the rotor slips increase. The rotor flux of variable-speed IG will also appear 0.4 pu to 0.6 pu as the rotor slips increase. The rotor flux of variable-speed IG will appear as a from 0.4 pu to 0.6 pu as the rotor slips increase. The rotor flux of variable-speed IG also will also appear as a significant reverse rotation component in the rotor flux. significant reverse rotation component in theinrotor flux. flux. as a significant reverse rotation component the rotor

0.3 0.3

90 90 80 80 70 70 60 60 50 50 40 40 30 30 20 20 10 10 0 0

Fixed-speed IG Fixed-speed IG Variable-speed IG Variable-speed IG

-0.2 -0.2

-0.1 -0.1

0

s0 s

0.1 0.1

0.2 0.2

0.3 0.3

Figure 4. DC component correlation coefficient between stator and rotor flux. Figure 4. DC DC component component correlation correlation coefficient between stator and rotor flux. Figure

2.2. Sequence Component Current Model of Fixed-Speed IGs 2.2. Sequence Component Current Model of Fixed-Speed IGs Assuming the initial slip of a fixed-speed IG is s0 before the short-circuit fault occurrs, the rotor Assuming the initial slip of a fixed-speed IG is s00 before the short-circuit fault occurrs, the rotor flux linkage ψr(t) of the fixed-speed IG in the stator reference frame is derived as in Equation (6) flux ψ rr(t) of the fixed-speed IG linkage ψ (t) of the fixed-speed IG in in the stator reference frame frame is derived as in Equation (6) (6) according to the flux linkage conservation: according to the flux linkage conservation: U U s  U1s t /Tr jr t .  U. e ψr (t )  1rs ( s.) U1s1s e jjs tt  1rs ( s0 ) U e (6) s j   U1s et /Tr e jr t ψr (t )  1rs (U s)1s sjωes t  1rs ( s0 )U ssj´ (6) ψr ptq “ η1rs psq j es ` η1rs ps0 q jss 1s e´t{Tr e jωr t (6) jωs jωs 2 where Tr = (Lr − Lm /Ls)/Rr is the rotor time constant. With Equations (2) and (6) substituted into is(t) = where Tr = (Lr − Lm2/Ls)/Rr is the rotor time constant. With Equations (2) and (6) substituted into is(t) = (Lrψs − Lmψr)/(LsLr − Lm2), the general expression for the post-fault stator short-circuit current of a (Lrψs − Lmψr)/(LsLr − Lm2), the general expression for the post-fault stator short-circuit current of a fixed-speed IG can be given as: fixed-speed IG can be given as:  U 1s js t U s  U 1s t /Tr jr t  1 is (t )  1 2 [ Lr  Lm 1rs ( s)] U 1s e js t  Lm 1rs ( s0 )  U s  U 1s e t /Tr e jr t  j  L L  L is (t )  s r [ Lr  Lm 1rs ( s)] s e  Lm 1rs ( s0 )  js e e  m  js js Ls Lr  L2m  (7)  U 2s  js t Us U 1s U 2s  t /Ts Lr Lr (7)  e  [  (  )] e U 1s U 2s  t /Ts L 2 U 2s  js t Lr 2 U s j  j  j  L L  L  Ls Lr r Lm2  j e  [  (  )] e s s s s s r m js js Ls Lr  Lm  js Ls Lr  L2m js

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where Tr = (Lr ´ Lm 2 /Ls )/Rr is the rotor time constant. With Equations (2) and (6) substituted into is (t) = (Lr ψs ´ Lm ψr )/(Ls Lr ´ Lm 2 ), the general expression for the post-fault stator short-circuit current of a fixed-speed IG can be given as: is ptq “

1

Ls Lr ´ L2m

" . U 1s jωs t rLr ´ Lm η1rs psqs jω e ´ Lm η1rs ps0 q¨ s

.

`L

U 2s ´ jωs t ´ jωs e

Lr

2 s Lr ´ L m

`

.

.

Lr

Ls Lr ´ L2m

.

.

*

U s ´U 1s ´t{Tr jωr t e jωs e

(7)

.

U 1s Us r jω ´ p jω ´ s s

U 2s ´t{Ts jωs qse

The short-circuit current of the fixed-speed IG is composed of three components. The first component is the steady periodic component, which includes both positive and negative components. The second component decays exponentially with the time constant Tr and rotates with ωr in the stator reference frame because of the natural rotor flux. The third component is the DC component decaying exponentially with the time constant Ts because of the natural stator flux. Given that the neutral point of a distribution network is commonly not grounded or grounded by arc-suppression coil, the zero sequence is not considered here. When a short-circuit fault occurs in distribution networks with multiple IGs, the short-circuit currents of IGs inject into the distribution network in the form of a variable current source based on symmetrical components, and the DC component is not within the symmetrical network. The change law of the short-circuit current with time and the relation between the current and terminal voltage sequence components of IG can be determined by Equation (7). With a terminal three-phase short circuit, the rotor speed after 10 cycles is less than 1.10 when a rated torque is given [20]. Thus the AC component of the short-circuit current rotating with ωr can be considered a synchronous frequency component. Assuming the slips of m sets of fixed-speed IGs are si (i = 1,2, . . . ,m), the positive and negative sequence phasors of the short-circuit current for the ith fixed-speed IG can be derived as in Equations (8) and (9): .

I 1si

1 “ Lsi Lri ´ L2mi

#

.

.

.

U U ´ U 1si ´t{Tri rLri ´ Lmi η1rsi psi qs 1si ´Lmi η1rsi psi0 q si e jωs jωs

+ (8)

.

.

I 2si

U 2si Lri “ 2 Lsi Lri ´ Lmi ´jωs

(9)

2.3. Sequence Component Current Model of Variable-Speed IGs Besides considering the effect of positive sequence forced component ψ1s (t) in the stator flux, the post-fault rotor flux of the variable-speed IG has to include the flux components produced by the negative sequence forced component ψ2s (t) and the DC component ψdcs (t). According to the flux linkage conservation before and after a short-circuit fault occurs, the post-fault rotor flux ψr (t) of the variable-speed IG in the stator reference frame can be given as: .

# ψr ptq

“

ψr0

“

.

.

.

U 1s jωs t U 1s U 2s ´jωs t Us η1rs psq jω e ´ η2rs psq jω e ` ηdcrs psqr jω ´ p jω ´ s s s s .

.

.

.

.

´U 1s U 1s U 2s Us η1rs ps0 q U sjω ` η2rs ps0 q jω ´ ηdcrs ps0 qr jω ´ p jω ´ s s s s

.

.

U 2s ´t{Ts jωs qse

` ψr0 e´t{Tr e jωr t

U 2s jωs qs

(10)

The stator current can be obtained as is (t) = (Lr ψs ´ Lm ψr )/(Ls Lr ´ Lm 2 ) through the relationships among the flux and current in the stator and rotor windings. Combined with the formulas of ψs (t) and ψr (t), the short-circuit current of a variable-speed IG can be written as: is ptq “

1

Ls Lr ´ L2m

* " . U 1s jωs t jω t ´ t { T r r rLr ´ Lm η1rs psqs jωs e ´ Lm ψr0 e e

.

`

Lr ´ Lm η2rs psq U 2s ´ jωs t e Ls Lr ´ L2m ´ jωs

.

`

Lr ´ Lm ηdcrs psq U s r jωs Ls Lr ´ L2m

.

U 1s ´ p jω ´ s

.

U 2s ´t{Ts jωs qse

(11)

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The positive sequence steady component of the short-circuit current of variable-speed IG is the same as that of fixed-speed IG, but the positive sequence steady component, transient decay AC component rotoring with ωr , and DC component are different from those in Equation (7) because of the nonignorable flux correlation coefficients η2rs and ηdcrs In consideration of the AC component rotoring with ωr as a synchronous frequency component, the positive and negative sequence phasors of the short-circuit current for the ith variable-speed IG can be derived as follows: Energies 2016, 9, 277 .

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„ ˆ . . . . . U 1si U ´U U 2si U rLri ´ Lmi η1rsi psi qs jω ´ Lmi η1rsi psi0 q sijωs 1si ` η2rsi psi0 q jω ´ ηdcrsi psi0 q jωsis ´ s s

. U 1si jωs

. U 2si jωs

˙

*

(12) Lri  Lm i2rsi ( si ) U 2si I 2si  (13) . js LsiLLri η L2mi ps q U . Lri ´ mi 2rsi i 2si I 2si “ (13) L2mi ´jωscurrent, network equation and rotor si Lri ´ According to the interaction among the L stator short-circuit movement equation, terminalamong voltagethe and slip short-circuit can be updated iteratively in each dataand window According to thethe interaction stator current, network equation rotor during the fault. The calculation process similar that of the simulation Thewindow IGs are movement equation, the terminal voltageisand slipwith can be updated iteratively software. in each data represented by the analytical expressions Equations (12) andsoftware. (13) in the proposed during the fault. The calculation process isincluding similar with that of (8), the (9), simulation The IGs are method, while software adopts the fifth-order as the represented by the the simulation analytical expressions including Equationsdifferential-algebraic (8), (9), (12) and (13) equations in the proposed model IGs. Then the time series for adopts the positive and negative sequence components of the method,ofwhile the simulation software the fifth-order differential-algebraic equations as short-circuit current can be determined based on the dynamic phasor and analytical expression the model of IGs. Then the time series for the positive and negative sequence components of the during the fault process from sub-transient steady-state. short-circuit current can be determined basedtoon the dynamic phasor and analytical expression during the fault process from sub-transient to steady-state. 3. Coupling Relationship between Short-Circuit Current of IGs and the Distribution Network 3. Coupling Relationship between Short-Circuit Current of IGs and the Distribution Network When the integrated capacity of IGs is small, the terminal voltages of IGs will drop to a When the integrated capacity of fault IGs is occurs small, the voltagesnetwork of IGs will drop toHowever, a steady-state steady-state value as soon as the in terminal a distribution [13–15]. the value as soon as the fault occurs in a distribution network [13–15]. However, the terminal voltages terminal voltages will be significantly changed by their own short-circuit current injections in a will be significantly bypermeability their own short-circuit current in a between distribution network distribution networkchanged with high of IGs. In this case,injections the coupling short-circuit with highofpermeability of IGs. In network this case,have the coupling between short-circuit currents of IGs and currents IGs and distribution to be considered. A distribution network with a distribution network have be considered. A distribution network a short-circuit short-circuit capacity of 240toMW is analyzed, and the short-circuit ratiowith (SCR) [21] betweencapacity IG and of 240 MW analyzed, and thesimulation short-circuit ratio (SCR) [21]results between network IG is 80. The network is is80. The dynamic and calculation of IG a and fixed-speed with a dynamic simulation and calculation results of a fixed-speed IG with a three-phase short-circuit fault at three-phase short-circuit fault at the end of the feeder downstream are shown in Figure 5, where ε is of the feeder downstream shownByinsetting Figurethe 5, where ε is theofrelative error of the calculation the end relative error of the calculationare results. impedance the downstream feeder equal results. By setting the impedance of the downstream feeder to the short-circuit impedance of to the short-circuit impedance of the network, the voltage dipequal magnitude at the generator terminals the network, the voltage dip magnitude at the generator terminals during a fault can be fixed to a during a fault can be fixed to a proportion p = 0.5 (the .voltage changes from U s to pU s , where p is . proportion p = 0.5 (thepost-fault voltage changes U sthe to pre-fault pU s , where p is thethe proportion of the post-fault the proportion of the voltage from versus voltage), rotor speed increases in a 2 voltage versuswith the pre-fault rotor speed in mechanical a linear manner linear manner a slope ofvoltage), (p − 1)Tmthe /(2H), where Tm increases = −1.0 is the torquewith and aHslope = 5.04ofs 2 ´ 1)T /(2H), where T = ´1.0 is the mechanical torque and H = 5.04 s is the inertia constant. (p m m is the inertia constant. I 1si “

1 Lsi Lri ´L2 mi

`

e´t{Tri

20 Variable slip Initial slip Zero slip Simulation

1 .6

10 0

1 .4

/%

I1s /pu

- 10 1 .2

- 20

1 .0

- 30

0 .8

- 40 - 50

0 .6 0 . 95

1 .0

1 . 05

1 .1 t/s

1 . 15

1 .2

- 60 0 . 95

1 .0

1 . 05

1 .1

1 . 15

1 .2

t/s

Figure a Figure5.5.Positive Positiveand andnegative negativesequence sequencecorrelation correlationcoefficients coefficientsbetween betweenstator statorand and rotor rotor flux flux of of a fixed-speed IG. fixed-speed IG.

The short-circuit current can be calculated by substituting the pre-fault and post-fault terminal voltages and slip variation of IGs into Equation (8). The calculation results under the variable slip and fixed slip (including both the initial slip s0 = −0.0051 pu and zero slip) are shown in Figure 5. The calculation error of fixed slip increases with time. The calculation accuracy with initial slip is above that of zero slip, although its relative error can be over 10%. The comparison results show that the

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The short-circuit current can be calculated by substituting the pre-fault and post-fault terminal voltages and slip variation of IGs into Equation (8). The calculation results under the variable slip and fixed slip (including both the initial slip s0 = ´0.0051 pu and zero slip) are shown in Figure 5. The calculation error of fixed slip increases with time. The calculation accuracy with initial slip is above that of zero slip, although its relative error can be over 10%. The comparison results show that the calculation results with variable slip is much closer to the simulation results. When a 277 phase B to phase C fault occurs at the end of the feeder downstream, the sequence Energies 2016, 9, 8 of 20 . . . . components of the terminal voltage are U 1s “ 3p{2U s and U 2s “ p{2U s , and the rotor speed increases 2 /2 ´ 1)T /(2H), where T = ´0.5. Figure 6 shows the sequence components of with the slopeofofthe (5pshort-circuit components and variable-speed IGs in the case of two-phase fault. m of fixed-speed m the of negative fixed-speed and variable-speed the case of current two-phase positiveby anda Theshort-circuit positive and sequence componentsIGs of in short-circuit of fault. IG areThe extracted negative components of short-circuit current of IG are extracted a sequence filter based on sequencesequence filter based on the second-order generalized integrators [22].byThe output of filter is the the second-order generalized integrators The output of filter is theand instantaneous waveform instantaneous waveform of positive and[22]. negative sequence current, then the RMS value of positive and negative sequence andthrough then thea RMS of instantaneous waveform can be instantaneous waveform can becurrent, computed RMS value block.The negative sequence component computed through RMS block.The negative sequence component I2schange of fixed-speed its I2s of fixed-speed IGareaches its steady-state value directly and doesn’t with the IG slipreaches in Figure steady-state directly and doesn’t changewith withthe theincrease slip in Figure 6a,duration while I2sbecause of variable-speed IG 6a, while I2s value of variable-speed IG decreases of fault of the rising decreases with the increase rotor speed in Figure 6b. of fault duration because of the rising rotor speed in Figure 6b. 0.9

0.8 Variable slip Initial slip Zero slip Simulation

0.75 0.7

0.8 0.7 0.6

I2s/pu

I1s/pu

0.65 0.6

0.5 0.4

0.55 0.3 0.5

0.2

0.45 0.4 0.95

0.1 1

1.1

1.05

1.15

0 0.95

1.2

1

1.1

1.05

t/s

1.15

1.2

t/s

(a) 1

1 Simulation Variable slip Initial slip Zero slip

0.9

Simulation Variable slip Initial slip Zero slip

0.9 0.8

0.8 0.7 0.6

I2s/pu

I1s/pu

0.7 0.6

0.5 0.4

0.5

0.3 0.4 0.2 0.3 0.2 0.95

0.1 1

1.05

1.1

1.15

1.2

t/s

0 0.95

1

1.05

1.1

1.15

1.2

t/s

(b) Figure 6. Two-phase short-circuit current with and without rotor slip change: (a) a fixed-speed IG, Figure 6. Two-phase short-circuit current with and without rotor slip change: (a) a fixed-speed IG, (b) (b)a avariable-speed variable-speedIG. IG.

The steady-state short-circuit currents of variable-speed IGs in Figure 6b are less than that of The steady-state short-circuit currents of variable-speed IGs in Figure 6b are less than that of fixed-speed IGs because of the offsetting effect from the short-circuit currents caused by the positive fixed-speed IGs because of the offsetting effect from the short-circuit currents caused by the positive and negative sequence forced fluxes in rotor windings, which are markedly influenced by the rotor resistance and the rotor speed. For the steady-state short-circuit currents, the signs of η1rs and η2rs are minus in Equations (12) and (13) where the change of η1rs with rotor slip becomes slow and the magnitude of η2rs stays in the range of 0.2–0.3 pu due to the rising of rotor resistance of variable-speed IGs. The calculation results of fixed initialand zero slips can not match the real trend of I2s of variable-speed IG during a grid fault. Although the calculation error with a fixed initial slip

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and negative sequence forced fluxes in rotor windings, which are markedly influenced by the rotor resistance and the rotor speed. For the steady-state short-circuit currents, the signs of η1rs and η2rs are minus in Equations (12) and (13) where the change of η1rs with rotor slip becomes slow and the magnitude of η2rs stays in the range of 0.2–0.3 pu due to the rising of rotor resistance of variable-speed IGs. The calculation results of fixed initialand zero slips can not match the real trend of I2s of variable-speed IG during a grid fault. Although the calculation error with a fixed initial slip (s0 = ´0.0023 pu) is slightly reduced because of the smaller rotor slip differences compared to that of zero2016, slip,9,the and Energies 277 errors are still larger than that with a variable slip in both cases of fixed-speed 9 of 20 variable-speed IGs. Figure 7 shows the the terminal terminal voltage voltage and and short-circuit short-circuit current current response of fixed-speed IGs with a feeder downstream downstream when an IG, four IGs, IGs, and and eight eight IGs IGs are are integrated integrated three-phase fault at the end of feeder capacity of 240 MWMW separately. Furthermore, the peak into a distribution distributionnetwork networkwith witha ashort-circuit short-circuit capacity of 240 separately. Furthermore, the values I of the short-circuit current of fixed-speedand variable-speed IGs under different SCRs are peak values I1sp of the short-circuit current of fixed-speedand variable-speed IGs under different 1sp shownare in the figure. large difference is observed the calculation results of fixed-speed SCRs shown in A the figure. A large differencebetween is observed between the calculation resultsIGs of using Equation (8) without considering the transient variation of terminal voltage and the simulation fixed-speed IGs using Equation (8) without considering the transient variation of terminal voltage results a small SCR. to theSCR. variation of I1sp to with it is unnecessary consider and thewith simulation resultsAccording with a small According theSCR, variation of I1sp withtoSCR, it is the effect of the short-circuit current on their terminal when SCR ě 20 for fixed-speed unnecessary to consider the effect of of theIGs short-circuit currentvoltage of IGs on their terminal voltage when IGs and ě 40 for variable-speed the current of However, IGs is determined by of both SCR ≥ 20SCR for fixed-speed IGs and SCRIGs. ≥ 40However, for variable-speed IGs. the current IGsthe is transient characteristics the IG generator and the power system network equations when SCR < 20 determined by both the of transient characteristics of the IG generator and the power system network for fixed-speed SCR 40 for variable-speed IGs. equations whenIGs SCRand < 20 for