innovative predictive current control for synchronous...
TRANSCRIPT
❯❱ ❯ P❱
♣rt♠♥t♦ ♥♥r ♥str
♦ ♦tt♦rt♦ r ♥ ♥♥r ♥str
rr♠ ♥ ♥♥r ♥r ttr
♦ ❳❳❳
♥♥♦t Prt rr♥t ♦♥tr♦ ♦r
②♥r♦♥♦s t♥ ♥s
rtt♦r ♦
♠♦ Pr♦ P♦♦ ♦♦♠♦
♦♦r♥t♦r ♥r③③♦
Pr♦ ♦rt♦ rr
♣rs♦r
♠♦ Pr♦ r♦ ♦♦♥♥
♦tt♦r♥♦ ♥ ù
tt♦r
♦♥t♥ts
Pr
②♥r♦♥♦s t♥ ♥
♥tr♦t♦♥
♥r ♠♦ ♦ t ♦② tr ♣s ♠♥
♥r② qt♦♥s
qt♦♥ t♦t strt♦♥
♥r ♠♦ ♦ t ♦② t♦ ♣s ♠♥
♥r② qt♦♥s
♦ t♦t strt♦♥
②♥r♦♥♦s t♥ ♥
♥r② qt♦♥s
♦ t♦t strt♦♥
♥ t s♥s♦② strt ♥♥s
♦♥s♦♥s
Prt rr♥t ♦♥tr♦
♥tr♦t♦♥
Prt rr♥t ♦♥tr♦
♦st ♥t♦♥
♦s P
t ♦ ♣r♠tr ♠s♠t ♥ rt♦♥
♦r P
♦♥s♦♥
♦♥t♥ts
♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦
♥tr♦t♦♥
♥②ss ♦ st♥t♦♥ ts
Pr♦♣♦s ♥tst♥t♦♥ ♠t♦
q♥ ♥tt♦♥
♦ t♦♥
♠tt♦♥s ♥ ♥r♥t st♥t♦♥
trt♣
♦♥s♦♥s
♦ Prt ②strss rr♥t ♦♥tr♦
♥tr♦t♦♥
Pr♦♣♦s ♦♥tr♦ s♠
♣ ♦♦♣
♣rt♥ ♥
P ♦
❱♦t ♦♦♣
♠♣♠♥tt♦♥ ♥ sts
♦♥s♦♥s
♦♥s♦♥s
tr ♦rs
♦r♣②
♥♦♠♥ts
Pr
s Pr srs t ♠♦tt♦♥ ♥ t ♠♥ ♦♥trt♦♥s ♦ t tss ♦♥t♥ts ♦ ♣tr ♦ t tss r r② s♠♠r③ ♥② st ♦ ♣t♦♥s♦ t t♦r s r♣♦rt
r♦♥
♥ r♥t s t s ♦ ♣♦r ♦♥rtrs s ♦♠ r② ♣♦♣r ♥ t ♦tr rs r ♦♥tr♦ t♥qs ♥ ♣r♦♣♦s ♦r ♣♦r ♦♥rtrs ♥r② ②r t ♦♥♦♥ rsr ♥ t ②s ♠♦r ♣♦r ♠r♦♣r♦ss♦rs t♦ ♥ ♣r♦r♠♥ s♦t♦♥s s♣t ts s♥ t ♦t♣t ♦ t ♦rrsr ♦t♥ rsts ♥ ♦♠♣① ♥ r② ♣♣ s♦t♦♥s ♦tr stst♥qs s s ♥r ♥ ②strs②s ♦♥tr♦ t ♣st ♠♦t♦♥ r stt ♠♥ ♦ ♥ rt ♥♠r ♦ ♥str ♣♣t♦♥s rs♦♥s ♦ trrs♣ ♥ ♦♥ ♦♥sr♥ t rtrsts ♦ ts ♠t♦s ♦♥ ♦♥ ss♠♣t② ♦ ♦♠♣r♥s♦♥ ♥ ♠♣♠♥tt♦♥ ♥ ♦♥ t ♦tr s♥t② ♦♦♣r♦r♠♥ ♥ r♦st♥ss t♦ ts r♥t trs s♣t tr s st ①t♥s r♦♦♠ ♦r ♠♣r♦♠♥ts t s ♥♦t ♣♥ss t♦ ♣r♦♣♦s s♦t♦♥s tt ♥ ttrt♦r ♣♦♣ ♦r♥ ♥ ♥str② ♥ t♦ ♦♠♣t t ♥ ♣♦ss② t♦ r♣ trt♦♥♠t♦s
sr② ♦♥tr♦ ♦rt♠ ♦r tr rs s t♦ s♠♣ ♥ s② ♥rst♥ ss t s t♦ st ♦r rt♠ ♣♣t♦♥s ♦st♥ss ♥rt② ②♦♥ tt ♣r♦r♠♥ t♦ r♥t s♥ t ♥tr ♦ tr♥t ♣♣t♦♥s ♦♠ ♣♣♥s ♥ t♦♠♦t
♥ ts ♣rs♣t Prt ♦♥tr♦ ♦ r♣rs♥t ♥t t♦ ♥tr♦ ♠♣r♦♠♥ts ♥ ♥s ♥ t ♦r♠♥t♦♥ ♥str ♣♣t♦♥s Prt ♦♥tr♦s ss ♦ ♦♥tr♦rs tt ss t ♠♦ ♦ t s②st♠ ♦r t ♣rt♦♥ ♦ ttr ♦r ♦ t ♦♥tr♦ rs s ♥♦r♠t♦♥ s s ② t rt♦r ♥♦rr t♦ ♦t♥ t ♦♣t♠ tt♦♥ ♦r♥ t♦ ♣r♥ ♦♣t♠③t♦♥ rtr♦♥r♣rs♥t ② ♦st ♥t♦♥ s ♦♥tr♦ tq s s ♦♥ ♦♥♣ts tt r①tr♠② s♠♣ ♥ ♥tt ♥ ss ♣♥♥ ♦♥ t t②♣ ♦ ♣rt ♦♥tr♦t ♠♣♠♥tt♦♥ ♥ s♦ s♠♣ ♥ ♣rtr ♥t ♦♥tr♦ t ♦s ♦♥sr♥ t srt ♥tr ♦ t ♣♦r ♦♥rtr ♥ rsts ♥ ♥ ①tr♠② s♠♣♠♣♠♥tt♦♥ ②♦♥ s♠♣t② ♦tr ♥ts ♥ r♦♥③ rst t
Pr
♣rt ♦♥tr♦ t s ♣♦ss t♦ ♦ t s strtr ♦t♥♥ r② sttr♥s♥t rs♣♦♥s ss ♥♦♥♥rts ♥ ♥ ♥ t ♠♦ ♦♥ t♥ ♦ ♥r③♥ t ♠♦ ♦r ♥ ♦♣rt♥ ♣♦♥t ♥ ♠♣r♦♥ t ♦♣rt♦♥ ♦t s②st♠ ♦r ♦♥t♦♥s ♥② t s ♣♦ss t♦ ♥ ♠tt♦♥s ♦ t rs♥ s♥♥ t rt♦r
♠ ♦ ts tss s t♦ st② Prt ♦♥tr♦ ♣♣ t♦ t rr♥t ♦♥tr♦♦ s②♥r♦♥♦s ♠♥s ♥②s♥ ♥ rss♥ s♦♠ ♦♣♥ rsr t♦♣s rr♥ts ♥ ♦ ♦♥tr♦ ♥ ♣rtr t♦ ♠♥ s♣ts r st ♥♠② t ♥ ♦ ♣rs ♥♦ ♦ t ♠♥ ♠♦ ♥ t ♣♦sst② t♦ r s②♥r♦♥♦s ♠♥ ♦♥ t ①♠♠ ♦rq ♣r ♠♣r t ① ❲♥♥ ♥ t ①♠♠♦rq ♣r ❱♦t ♦♣rt♦♥s
♣r♦r♠♥ r strt② rt t♦ t r② ♦ t ♠♦ s ♦r t♣rt♦♥ ♥ s ♦ ♣r♠trs ♠s♠t ♦r rt♦♥ rtr t♥ ♦tr ♠♦ ♥qs t ♣rt♦♥ ♦ t s♥ ♦rs♥♥ ♦ t ♦r ♦r♦ t r rst ♣rt ♦ ts ♦r s ♦♠♠t t♦ st② ts s♣t ♥②s♥ tts ♦ ♠s♠ts ♥ rt♦♥s ♦s♥ ♥ ♣rtr ♦♥ t tr♠♥t ts ♦r♦♥ strt♦♥ ♥♦ ♠♦r s♦t♦♥ s ♣rs♥t t♦ ♦r♦♠ t ♠tt♦♥s♥ ② ♥ ♥qt ♠♦
♥ ♦rr t♦ ② ①♣♦t t rtrsts ♦ t r ssr♥ t ♦st♣♦r ♦sss ♥ r② ♦r♥ ♦♥t♦♥ ♣r♦♣r ♦♥tr♦ ♦rt♠ s t♦ s♥ t s♦♥ ♣rt ♦ ts ♦r ♣rt rt♦r t♦ tr t ♠♦st sttrt♦r② ♣♥♥ ♦♥ t ♠♥ ♦♣rt♦♥ s ♣rs♥t ♣r♦♣♦s s ♦♠♥t♦♥ ♦ ♣rt ♦♥tr♦ ♥ ②strss ♦♥tr♦ s♥ ts ♠ s t♦ ♣ t rr♥trr♦r t♥ rt♥ ②strss ♥
s st② s rr ♦t ♦♥sr♥ Prt rr♥t ♦♥tr♦ ♦r ②♥r♦♥♦st♥ ♠♥s s ♥ ♦ ♠♥ s ♦ rt ♥trst t♦ t t tt ttrs ♣♦r ♥st② s♣r♦r rt② ♥② ♥ t s ♦st t t♦ t s♥ ♦ ♣r♠♥♥t ♠♥ts ♥ rts ♥ t r♦t♦r ss s♥ ts♥♥t r♦♥ strt♦♥ ts ♦♥tr♦ r♣rs♥ts ♥ ♥ ♣rtr ♦r ♣rt♦♥tr♦ s♠s ♥ ♦r ts rs♦♥ t s ♣rt s st②
t♥ ♦ t tss
rtr t ♦♥t♥ts ♦ t ♣tr ♦ t tss r r② sr
♣tr ♠♦ ♦ t s②♥r♦♥♦s rt♥ ♠♥ s r ♥ sr tr qt♦♥s r rst ♦t♥ ♦♥sr♥ ♦② tr ♣s♠♥ tt s t ♠♦st ♥r r♣rs♥tt♦♥ ♦r ♥ tr ♠♥ ♠♦ s ♦♣ t t ♦♥② ss♠♣t♦♥ ♦ ♥♦ ②strss ♥ ♥♦ ② rr♥ts tr tt t s ♦ ♥rt② t ♥♦ r♦♥ strt♦♥ ts sst tr ♣s s②st♠ s tr tr♥s♦r♠ ♥ t q♥t t♦ ♣s♠♥ ♥② s♣ r♣rs♥tt♦♥ ♦r t rt♥ ♠♥ s ♦t♥② ♠♣♦s♥ ♣♣r♦♣rt ♦♥t♦♥s ♦♥ t r♦t♦r rr♥ts
♣tr Prt rr♥t ♦♥tr♦ s ♣rs♥t strt♥ t ♥ ♦r ♦t ♠♥ trs tr tt t s②st♠ s sr ♦♥sr♥ t s♣
Pr
s ♦ s②♥r♦♥♦s rt♥ ♠♥ ♣♣t♦♥s ♦♠ ♦♥srt♦♥s ♦♥t ♦ ♦ t ♦st ♥t♦♥ r ♣r♦ t♦ ♦ ♠♣r♦♣r ♦♣rt♦♥s ♥♦♥tr♦♥ ♥ ♥s♦tr♦♣ ♠♥ r♠♥r ♦ t ♣tr srs ♦s ♥ ♦r ♣rt♦♥ ♦r♠r s ♣rs♥t ♥ ♥②s ♥♥t t ♦ ♣r♠trs ♠s♠t ♥ ♣rtr t ts ♦ r♦♥ strt♦♥♦♥ t ♣rt rt♦r r ♣rs♥t ♦r ♣rt♦♥ s ♥tr♦t♥ t ♠rts ♦ ts t♥q ss t s♦ ♦t t♦rst♥t♦♥ s ♥②③
♣tr ♦r ♣rt s♠s sr r♦♠ ♦t t♦r st♥t♦♥ ttr ♦rs ♥ ♦ rr♥t rt♦♥ s ♥♦t ♣t ♦r sr st♣ss ♣♥♦♠♥♦♥ s t♦ ♥ ♠♣rs rr♥t ♣rt♦♥ ♥ ts ts ♣rsst♥stts ♥ ♦ss ♦ ♦♥tr♦ ♦ ♦r ♥ ts ♣tr ♥♦ ♥tst♥t♦♥s♦t♦♥ s ♣rs♥t ♣r♦♣♦s ♠t♦ ♦s ♦♥t♥♦s ♣t t♦t♦r♥ ♥♦♥ ♦♣t♠ ♦t t♦rs
♣tr ♣rt ♦♥tr♦ s♠ ♦r t s♣ ♦♣rt♦♥ ♦ t s②♥r♦♥♦srt♥ ♠♥ s ♣rs♥t ♦rt♠ ♦♥s t♦ ♦t ♣rt ♦♥tr♦♥ ②strss ♦♥tr♦ s♥ ts ♠ s ♣♥ t rr♥t rr♦r t♥ t♦r♥♥ ♣r♦♣♦s s♠ s t♦ ♦♥tr♦ t ♠♥ ♦♥ t ①♠♠♦rq ♣r ♠♣r t ①❲♥♥ ♥ t ①♠♠ ♦rq ♣r ❱♦ttrt♦rs r♥t ♦♣rt♦♥s r ♦t♥ ♥tr♦♥ t ♦♣rt♥ ♥ ♦s ♦ ♦♣rt♥ ♣♦♥ts ♦t t♥ t P ♥ t P❱tr♥st♦♥ t♥ t r♥t ♦ s ♦t♥ ② ♠♥ ♦ ♥ ♥♥♦t ♦t♦♦♣ tt ♦s ♥♦t rqr t ♥♦ ♦ t s s♣
st ♦ ♣t♦♥s
r ♣rts ♦ ts P tss ♥ ♣rs♥t ② t t♦r r♥ s P♦rs ♥ ♥tr♥t♦♥ ♦♥r♥s ♥ ♦r♥s st ♦ ♣t♦♥s s r r♣♦rt
❼ ♦rs ù ♥ ♦♦♥♥ ♦st rr♥t ♦♥tr♦ s♦♥ Pr♦♣♦rt♦♥♥tr srrs ♦r Pr♠♥♥t ♥t ②♥r♦♥♦s ♥s r♥st♦♥s ♦♥ ♥str② ♣♣t♦♥s ♥ Prss
❼ ♦rs ù ♥ ♦♦♥♥ t♠ str♥ ♦♠♣♥st♦♥ ♦rt♠ ♦r t rr♥t ♦♥tr♦ ♦ P rs ♥r②♦♥rs♦♥ ♦♥rss ♥ ①♣♦st♦♥ ♥♥♥t t♦r
❼ ❱ ♥③♦♥ ù ♥ ♦♦♥♥ ♥ t ❱♦t ♦♥tr♦ ♦♦♣ ♦r ♣ ①❲♥♥ ♥ P ②♥r♦♥♦s ♦t♦r rs ♥r②♦♥rs♦♥ ♦♥rss ♥ ①♣♦st♦♥ ♥♥♥t t♦r
❼ ♦s♦ ù ♥ ♦♦♥♥ ♦♥ ♦r③♦♥ st♠t♦r ♦r t♣ ♥ ♦t♦r P♦st♦♥ ♦ ♥s♦rss P r ②♠♣♦s♠ ♦♥♥s♦rss ♦♥tr♦ ♦r tr rs t♥ t② ♣t♠r
Pr
❼ ù P♦t♦ ♥ ♦♦♥♥♦r Prt rr♥t ♦♥tr♦♦r ②♥ r s ♦♥ ♥ t ♣t ♦ ♠sr rr♥t rs♣♦♥ss ②♠♣♦s♠ ♦♥ Prt ♦♥tr♦ ♦ tr rs ♥ P♦r tr♦♥s P Ps♥ ③ ♣ ♣t♠r
❼ ❱ ♥③♦♥ ù ♥ ♦♦♥♥ ♥ ♦♥tr♦ strt② ♦r ♥② s♣ r♥ s②♥r♦♥♦s rt♥ ♠♦t♦r rs ♥tr♥t♦♥ tr ♥s ♥ rs ♦♥r♥ ♠ ♣♣ ♦
❼ ù ♦r♥♥ ♦♦♥♥ ♥ st♦ st ♥ ♦r ♠t♥ rt② ♦ s♥t r♦t♦r tr ♣r♦♣s♦♥ ♠♥s t s♥ ♣r♠♥♥t♠♥t s②♥r♦♥♦s ♠♥ r ♥ tr ②st♠s ♥ r♥s♣♦rtt♦♥♦ ♥♦ ♣♣ ♦ tst
❼ ♦rs ù ♥ ♦♦♥♥ r♦st rr♥t ♦♥tr♦ s♦♥ ♣r♦♣♦rt♦♥♥tr ♦srrs ♦r ♣r♠♥♥t ♠♥t s②♥r♦♥♦s ♠♥s ♥r② ♦♥rs♦♥ ♦♥rss ♥ ①♣♦st♦♥ ❲ ♣♣ ♦
❼ ù ♦r♥♥ ♦♦♥♥ ♥ st♦ ♦ Prt ②strss rr♥t ♦♥tr♦ ♦r s♣ ♦♣rt♦♥ ♦ ②♥r♦♥♦s t♥ ♠♥ r ♥ ♥♥ ♦♥r♥ ♦ t ♥str tr♦♥s ♦t② ♦r♥ ♣♣ ♦
❼ Pr♥t ù ♥ r ♥ ♦♥r②s♥s ♦♣♥ t♥q ♦r ss♥s♥ ♥trt ♠trs ②♠♣♦s♠ ♦♥ ♥s♦rss ♦♥tr♦ ♦r tr rs ♣♣ ♦
❼ ♦r♥♥ ù ♦♦♥♥ ♥ ♥ ♥ ♥trt trtrtr♥t♦r s ♦♥ ♥s♦rss ②♥r♦♥♦s t♥ ♥ r ❱ P♦r ♥ Pr♦♣s♦♥ ♦♥r♥ ❱PP ♦♥tr ♣♣ ♦ ❱PP
❼ ù ♦r♥♥ ♦♦♥♥ ♥ st♦ ♥ P ♦t♦r rrss s P ♦t♦r r ♦r ① st ♥ ♦ ♥t ♦t♦r Pr♦♣s♦♥ ♥s ❱ P♦r ♥ Pr♦♣s♦♥ ♦♥r♥ ❱PP♦♥tr ♣♣ ♦ ❱PP
❼ ♦r♥♥ ù ♦♦♥♥ ♥ st♦ tst ♥ ♦r ②r♣r♦♣s♦♥ tr♥ rsr ♥ ♦♣♠♥t ♥tr♥t♦♥ tr ❱ ♦♥r♥ ❱ ♦r♥ ♣♣ ♦ ❱
Chapter 1
②♥r♦♥♦s t♥ ♥
♥②s♥ ♥ ♦♣♥ ♦♥tr♦ s♠s rqrs ♣ ♥rst♥♥ ♦ t ♣♥t♣②ss s ♣tr ♥s t ♥②ss ♦ t tr ♠♦ ♦ s②♥r♦♥♦srt♥ ♠♥ ♥ t s t ♥♠♥t ss ♦ ts ♦r
♥tr♦t♦♥
♥ ts ♣tr t ♠♦ ♦ ②♥r♦♥♦s t♥ ♥ ②♥ s r♥ sr ♦r t s ♦ ♥rt② ♥ ♥rs ♠♦ ♦ s②♥r♦♥♦s ♠♥s r t rst t s ♦t♥ ② t qt♦♥s ♦ ♦② tr ♣s ♠♥♣rt♦r③ ♦r♥② ttr s t ♠♦st ♥r r♣rs♥tt♦♥ ♦r ♥ tr♠♥ s♥ t s tr ♣s ♥♥ ♦♥ ♦t t stt♦r ♥ t r♦t♦r ♦r♥t♦ ts ♠♦ t ♠♥ s sr ② ♠♥ ♦ qt♦♥s rt♥ t stt♦r ♥r♦t♦r ♦ts t t rr♥ts ♥ ①s ♠♦ s ♦♣ t t ♦♥②ss♠♣t♦♥ ♦ ♥♦ ②strss ♥ ♥♦ ② rr♥ts ss t tr qt♦♥s rrst r ♦♥sr♥ ♥r rt♦♥s♣ t♥ ① ♥ rr♥ts ♥ t♥ ②♥tr♦♥ t ②♣♦tss ♦ ♥rt② r♦♥ strt♦♥ s ♥t
tr ♣s s②st♠ s tr tr♥s♦r♠ ♥ t t♦ ♣s s②st♠ ♦t♥♥ tqt♦♥s ♦ t ♦② t♦ ♣s ♠♥ s♠ ♦♥srt♦♥s ♠ ♦r ttr ♣s ♠♥ r ①t♥ ♦r t t♦ ♣s s②st♠ strt♥ t ♥r♠♦ ♦t♥ t ♥♦ ss♠♣t♦♥s ♦♥ t ♠♥t ♦r ♦ t ♠♥ ♥t♥ ♥tr♦♥ t ②♣♦tss ♦ ♥rt②
♥② s♣ r♣rs♥tt♦♥ ♦r rt♥ ♠♥ s ♦t♥ ② ♠♣♦s♥♣♣r♦♣rt ♦♥t♦♥s ♦♥ t r♦t♦r rr♥ts ♥ t ♠♦ s ♦♣ strt♥r♦♠ t ♠♦st ♥r r♣rs♥tt♦♥ tr tt t ②♣♦tss ♦ ♥♦ r♦♥ strt♦♥ s♥tr♦ ♥ t♥ t qt♦♥s r r ♦♥sr♥ s♥s♦ strt♦♥ ♦t ♥♥ t t r♣
②♥r♦♥♦s t♥ ♥
♥r ♠♦ ♦ t ♦② tr ♣s ♠♥
isa
sa
sb
sc
ra
rb
rc
isc
isb
ira
irb
irc
m
r ♦② tr ♣s ♠♥ r♣rs♥tt♦♥
♥r ♠♥ ♦♥sr ♥ ts st♦♥ ♣rs♥ts tr ♣s ♥♥ ♦♥♦t t stt♦r ♥ t r♦t♦r s s♦♥ ♥ ♠♥ ♣♦st♦♥ ♦ t r♦t♦rϑm s ♥♥t t t ♥ t♥ ♥♥ sa ♥ ra ♥ s ♦ ♠♥s t♠t♣ ♣♦s t ♣♦ ♣r ♥♠r p > 1 t rt♦♥s♣ t♥ t tr♥ t ♠♥ ♣♦st♦♥ ♥ ①♣rss s ϑ = pϑm ss t ♠♥s♣ ♦ t r♦t♦r s stt t ωm ♥ t tr s♣ s ω = pωm trqt♦♥s rt♥ t ♦ts u t rr♥ts i ♥ t ① ♥s λ r
us = Rsis +ddtλs
ur = Rrir +ddtλr
r us = [usa usb usc]t is = [isa isb isc]
t ♥ λs = [λsa λsb λsc]t r t stt♦r
♦t rr♥t ♥ ① t♦rs rs♣t② stt♦r rsst♥ ♠tr① s ♥ sRs = Rs · I3×3 r I3×3 s t ♥tt② ♠tr① r♦t♦r q♥ts r s② ♥s♥ ssr♣t r
st ♦ s① qt♦♥s r♣rs♥ts t tr ♥ ♦ ♣s ♦t t t tr♠♥s s t s♠ ♦ t rsst ♦t r♦♣ ♥ t t♠ rt♦♥♦ t ① ♦♣ t t ♦♥sr ♥♥ tt♦r ♥ r♦t♦r qt♦♥s ♥ ♥ ♥ ♦♥ ①♣rss♦♥ s
u = Ri+d
dtλ
r u = [us ur]t i = [is ir]
t λ = [λs λr]t r t♦rs t s③ 6 × 1 ss t
rsst♥ ♠tr① R ♥ rtt♥ s ♦ ♠tr①
R =
[
Rs 03×3
03×3 Rr
]
r 03×3 s t ♥ ♠tr①
♥ t ♠♦ s ♦♣ ♥r t ss♠♣t♦♥ ♦ ♥♦ ②strss ♥ ② rr♥ts t rt♦♥s♣ tt ♥s t rr♥t ♥ t ① ♥ s ♥♦ ♥
♥r ♠♦ ♦ t ♦② tr ♣s ♠♥
♣rtr
λ = λ(is, ir, ϑ) =
[
λs(is, ir, ϑ)
λr(is, ir, ϑ)
]
♥
i = i(λs,λr, ϑ) =
[
is(λs,λr, ϑ)
ir(λs,λr, ϑ)
]
♦r rt♥ ♣♦st♦♥ ϑ t rt♦♥s♣ t♥ t ① ♥ t rr♥t ♥ r♣rs♥t ② t ♠♥t③t♦♥ r qtt② r♣♦rt ♥
0.5
Current
Flu
x
wm
wm'
r ♥t③t♦♥ r
① t ♣♦st♦♥ ϑ t ♠♥t ♥r② wm ♥ ♦♥r② w′
m r ♥ s
wm(λ, ϑ) =
∫ λ
0i(λ, ϑ)dλ
w′
m(i, ϑ) =
∫ i
0λ(i, ϑ)di
♦♠tr ♠♥♥ ♦ ts t♦ q♥tts s r♣rs♥t ♥ ♥ trt♦♥s♣ t♥ t♠ s
itλ = wm + w
′
m
♥ t ① ♥ λ s ♥t♦♥ ♦ t rr♥ts i ♥ t ♣♦st♦♥ ϑ λ = λ(i, ϑ) t tr qt♦♥ ♥ rrtt♥ s♣tt♥ t ① rt♦♥sr♥ t ♣rt rts t rs♣t t♦ t rr♥t ♥ t ♣♦st♦♥ t rsts
u = Ri+ ℓd
dti+
∂
∂ϑλd
dtϑ
r ℓ s t 6× 6 r♥t ♥t♥ ♠tr① ttr r♣rs♥ts t s♦♣♦ t ♠♥t③t♦♥ r t t ① rr♥t ♥ t ♥ ①♣t② rtt♥ s
ℓ = ℓ(i, ϑ) =
∂λsa
∂isa. . . ∂λsa
∂irc
∂λrc
∂isa. . . ∂λrc
∂irc
②♥r♦♥♦s t♥ ♥
♥r② qt♦♥s
♥r② ♥ ♥ ♦t♥ ② ♠t♣②♥ ② idt t ♥ ①♣rss s
itudt︸ ︷︷ ︸
dwin
= itRidt︸ ︷︷ ︸
dwJ
+ itdλ︸︷︷︸
dwm+dwem
r dwin s t ♥♣t tr ♥r② dwJ s t ♦ ♦ss ♦♠♣♦♥♥t dwm s trt♦♥ ♦ t ♠♥t ♥r② st♦r ♥ t s②st♠ ♥ dwem s ♥r② ♦♥rtr♦♠ tr t♦ ♠♥ ❯♥r t ss♠♣t♦♥ ♦ ♥♦ ♠♥ ♦sss t ttr♦rrs♣♦♥s t♦ t ♦t♣t ♠♥ ♦r t♦ ttr tr♠s ♦ ♥ s② ♥t strt♥ r♦♠ ♥ ♠t♣②♥ t ② idt s s♦♥ ♥
itudt︸ ︷︷ ︸
dwin
= itRidt︸ ︷︷ ︸
dwJ
+ itℓdi︸︷︷︸
dwm
+ it∂λ
∂ϑdϑ
︸ ︷︷ ︸
dwem
♦♠♣r♥ ♥ ♥ ♠♣♦s♥ dwem = τdϑm t s ♣♦ss t♦ rt
itdλ = dwm + τdϑm
r τ s t ♥st♥t♥♦s t♦rq
♥ t ♠♥t ♥r② s ♥t♦♥ ♦ t stt♦r ♥ r♦t♦r rr♥t ②♦♥ ttt ♣♦st♦♥ t ♥♥ts♠ rt♦♥ dwm ♥ ①♣rss s
dwm =∂wm
∂λdλ+
∂wm
∂ϑdϑ =
∂wm
∂λsadλsa + · · ·+
∂wm
∂λradλra + · · ·+
∂wm
∂ϑdϑ
♦♠♥♥ ♥ t s ♣♦ss t♦ rt
itdλ− τdϑm =
∂wm
∂λdλ+
∂wm
∂ϑdϑ
♥ t stt rs r ♥♣♥♥t q♥tts ♥ s♣rt ♥ ts t♦♦♠♣♦♥♥t ♥ ♥t s
i(λ, ϑ) =∂wm
∂λ
τ(λ, ϑ) = −p∂wm
∂ϑ
r t ♦r♠r r♣rs♥ts t ♥♥ rr♥t ♥ t ttr t t♦rq ①♣rss♦♥s ♥t♦♥ ♦ t ♠♥t ♥r②
t♦rq ♥ s♦ ①♣rss s ♥t♦♥ ♦ t rr♥t i ♥ t ♣♦st♦♥ ϑs ♥ ♦♥ ② ♦♥sr♥ t ♠♥t ♦♥r② ♥ ♥ ♣rtr
w′
m(i, ϑ) = itλ− wm(λ, ϑ)
♥ ①♣rss♦♥ ♦r t t♦rq ♥ ♦t♥ ② r♥tt♥ ♥♦♠♥♥ t t rst s r♣♦rt ♥
τ(i, ϑ) = p∂w
′
m
∂ϑ
♥r ♠♦ ♦ t ♦② tr ♣s ♠♥
qt♦♥ t♦t strt♦♥
0 1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1
Current [A]F
lux
[Vs]
w/i iron saturationw/o iron saturation
r ♥t③t♦♥ r ♥ s ♦ ♥rt②
♠♦ ♦♣ ♥ t♦♥ ♦♥srs ♥r ♦② tr ♣s♠♥ qt♦♥s r ♥♦ s♣ ♦r ♥r ♠♥ t r♦♥ strt♦♥s ♥t ♠♥t③t♦♥ r s qtt② r♣♦rt ♥ ♥ ts st ① ♥ ①♣rss s
λ(i, ϑ) = L(ϑ)i
r L s t 6 × 6 ♥t♥ ♠tr① tt ♣♥s ♦♥ t ♣♦st♦♥ ♥ ♥♦t ♦♥ trr♥ts t ♥ ①♣rss s
L =
[
Ls(ϑ) Msr(ϑ)
Mrs(ϑ) Lr(ϑ)
]
♥ ts s♠trs r
Ls(ϑ) =
Ls,a(ϑ) Ms,ab(ϑ) Ms,ac(ϑ)
Ms,ba(ϑ) Ls,b(ϑ) Ms,bc(ϑ)
Ms,ca(ϑ) Ms,cb(ϑ) Ls,c(ϑ)
Msr(ϑ) = Mtrs(ϑ) =
Msr,aa(ϑ) Msr,ab(ϑ) Msr,ac(ϑ)
Msr,ba(ϑ) Msr,bb(ϑ) Msr,bc(ϑ)
Msr,ca(ϑ) Msr,cb(ϑ) Msr,cc(ϑ)
Lr(ϑ) =
Lr,a(ϑ) Mr,ab(ϑ) Mr,ac(ϑ)
Mr,ba(ϑ) Lr,b(ϑ) Mr,bc(ϑ)
Mr,ca(ϑ) Mr,cb(ϑ) Lr,c(ϑ)
r s ♥t♥s r stt t L(ϑ) M(ϑ) s s ♦r t ♠t ♥t♥ t♥ t♦ ♥♥s
t s ♦rt t♥ tt Ls(ϑ) Lr(ϑ) ♥ Msr(ϑ) r r♥t ♠tr① ♥ r♦ t♦r s st ♦♥ ♠♥t t♦ t rt t rs♣t t t ♣r♥t r♦t♦r s ♥ ①♠♣
❼ Ms,ab(ϑ) Ms,ba(ϑ)
❼ Ls,b(ϑ) = Ls,a(ϑ− 2/3π) ♥ Ls,c(ϑ) = Ls,a(ϑ− 4/3π)
②♥r♦♥♦s t♥ ♥
stt♦r ♥ r♦t♦r ① ♥ ♥ ①♣rss s
λs = Lsis +Msrir
λr = Mrsis + Lrir
♥ s ♦ ♥rt② t ♠♥t ♥r② ♥ ♦♥r② r q ♥ t♦♥ ♦♥sr♥ wm = 1/2itλ t ♣♦ss t♦ rt
τ(i, ϑ) = p∂
∂ϑw
′
m(i, ϑ) = p∂
∂ϑwm(λ, ϑ)
=1
2pit
∂
∂ϑλ =
1
2pit
d
dϑLi
t♦rq ♥ s♣tt ♥
τ(i, ϑ) =1
2pits
[d
dϑLs(ϑ)
]
is
︸ ︷︷ ︸
τr
+ pits
[d
dϑMsr(ϑ)
]
ir
︸ ︷︷ ︸
τed
+1
2pitr
[d
dϑLrϑ
]
ir
︸ ︷︷ ︸
τc
r τr s t rt♥ t♦rq τed t tr♦②♥♠ t♦rq ♥ τc s t ♦♥t♦rq
rt♥ t♦rq τr s rt t♦ t rt♦♥s ♦ t stt♦r ♥t♥ ♠tr①t t r♦t♦r ♣♦st♦♥ tt ♦rs r② t♠ t r♦t♦r s ♥ ♥s♦tr♦♣ strtr t♦ ♣♦ s♥s ♦r ① rrrs t ♥ s♥ s
τr =1
2p∂λt
s(is, 0, ϑ)
∂ϑis
=1
2pits
∂
∂ϑλs(is, 0, ϑ)
= p∂
∂ϑwm(is, 0, ϑ)
r λs(is, 0, ϑ) s t stt♦r ① ♥ t♦ t stt♦r rr♥t ♦♥② ir = 0
tr♦②♥♠ t♦rq s rt t♦ t rt♦♥ ♦ t ♠t ♥t♥st♥ t stt♦r ♥ t r♦t♦r ♥♥s t t r♦t♦r ♣♦st♦♥ ♠♥ rs♦♥♦r ts rt♦♥ s t ♠♦♠♥t ♦ t r♦t♦r t rs♣t t♦ t stt♦r t ♥ ①♣rss s
τed(is, ir, ϑ) = pits∂Msr(ϑ)
∂ϑir
= p∂λt
sr(0, ir, ϑ)
∂ϑis = pits
∂
∂ϑλtsr(0, ir, ϑ)
= p∂λt
rs(is, 0, ϑ)
∂ϑir = pitr
∂
∂ϑλrs(is, 0, ϑ)
r λsr(0, ir, ϑ) s t stt♦r ① ♥ t♦ t r♦t♦r rr♥t ♦♥② is = 0♥ λrs(is, 0, ϑ) s t r♦t♦r ① ♥ t♦ t stt♦r rr♥t ♦♥② ir = 0
♥r ♠♦ ♦ t ♦② t♦ ♣s ♠♥
♦♥ t♦rq s t♦ stt♦r ♥s♦tr♦♣s ♠♥② s ② t s♦ts ♦♣♥♥t ♥ ①♣rsss s
τcog(ir, ϑ) =1
2pitr
∂Lr(ϑ)
∂ϑir
=1
2p∂λt
r(0, ir, ϑ)
∂ϑir
=1
2pitr
∂λtr(0, ir, ϑ)
∂ϑ
= p∂wm(0, ir, ϑ)
∂ϑ
r λr(0, ir, ϑ) s t stt♦r ① ♥ t♦ t r♦t♦r rr♥t ♦♥② is = 0
♥r ♠♦ ♦ t ♦② t♦ ♣s ♠♥
iryirz
ry
isy
isz
sy
rz
sz
r ♦② t♦ ♣s ♠♥ r♣rs♥tt♦♥
tr♣s ♥♥ ♦♥ t stt♦r ♥ ♦♥ t r♦t♦r ♥ st ♦♥sr♥♥ q♥t t♦ ♣s s②st♠ yz r♣rs♥t ♥ r ♦t ♥♥s rtr♥s♦r♠ ♥ t q♥t t♦ ♣s s②st♠
❼ tr♥s♦r♠t♦♥ s ①♣rss ♦r ♥r tr♣s ♥♥ abc r♦tt♥t ♥ ♥r s♣ ωabc ♣♦st♦♥ ♦ t ♣s a t rs♣t t♦ tsss ①s s ϑabc
❼ t♦♣s rr♥ r♠ yz s r♦tt♥ t ♥ ♥r s♣ ♦ ωyz ♣♦st♦♥ ♦ t ♣s y t rs♣t t♦ t sss ①s s ϑyz
❼ t ♦♠♦♣♦r ♦♠♣♦♥♥t s ss♠ t♦ ③r♦
rt♦♥s♣ t♥ t tr♣s abc ♥ t t♦♣s yz s②st♠s ♥ ①♣rss s
fyz = Tfabc =⇒ fabc = T−1
fyz
r f s s t♦ r♣rs♥t ♥r q♥tt② t ♦t t rr♥t ♥ t① ssr♣ts abc ♥ yz rr t♦ t tr♣s ♥ t t♦♣s s②st♠
②♥r♦♥♦s t♥ ♥
rs♣t② T s t tr♥s♦r♠t♦♥ ♠tr① t s③ 2 × 3 ♥ T−1 s t ♥rs
♠tr① ② ♥ ①♣rss s
T(∆ϑ) =
[
cos[∆ϑ] cos[∆ϑ− 2/3π] cos[∆ϑ+ 2/3π]
−sin[∆ϑ] −sin[∆ϑ− 2/3π] −sin[∆ϑ+ 2/3π]
]
T−1(∆ϑ) =
cos[∆ϑ] −sin[∆ϑ]
cos[∆ϑ− 2/3π] −sin[∆ϑ− 2/3π]
cos[∆ϑ+ 2/3π] −sin[∆ϑ+ 2/3π]
r ∆ϑ = ϑyz −ϑabc ♥ t ♦♦♥ t ♣♥♥② ♦♥ ∆ϑ s ♦♠tt t s ♦rtt♥ tt t ♥rs ♠tr① ♥ ♦♥ ② r♥tr♦♥ t tr r♦ ♥ t♠tr① T ♦♥sr♥ t ♦♠♦♣♦r ♦♠♣♦♥♥t ♥ ts ② t trs♦r♠t♦♥♠tr① s sqr 3 × 3 ♠tr① ♥ ts ♥rs ♥ rtt♥ s ♥ r t♦♠♦♣♦r ♦♠♣♦♥♥t s ♥ ♦♠tt tr t ♥rs♦♥
trt♥ r♦♠ t qt♦♥ ♦ t tr♣s ♥♥ ♥ s♥ t ♣♦sst♦ rt
uabc = Riabc +d
dtλabc
T−1
uyz = RT−1
iyz +d
dt
(T
−1λyz
)
T−1
uyz = RT−1
iyz +T−1 d
dtλyz +
d
dt
(T
−1)λyz
rt ♦ t tr♥s♦r♠t♦♥ ♠tr① s ♦t♥ ② ♥ ♠♥t②♠♥trt♦♥
d
dt
(T
−1)= (ωyz − ωabc)
−sin[∆ϑ] −cos[∆ϑ]
−sin[∆ϑ− 2/3π] −cos[∆ϑ− 2/3π]
−sin[∆ϑ+ 2/3π] −cos[∆ϑ+ 2/3π]
= (ωyz − ωabc)T−1
J
r
J =
[
0 −1−1 0
]
♦♠♥♥ ♥ t s ♣♦ss t♦ rt
T−1
uyz = RT−1
iyz +T−1 d
dtλyz +T
−1Jλyz(ωyz − ωabc)
uyz = Riyz +d
dtλyz + Jλyz(ωyz − ωabc)
t s ♦rt ♠♥t♦♥♥ tt s ♦r ♦t t stt♦r ♥ t r♦t♦r ②♥ r♥tt ② ♠♣♦s♥ t s♣ ♦ t t♦ s②st♠s ♥♠② ωabc = 0 ♥ωabc = ω ♦r t stt♦r ♥ t r♦t♦r rs♣t②
♥ t ♥②ss ♦ t s②♥r♦♥♦s ♠♥ t♦ ♣rtr tr♥s♦r♠t♦♥s r ①t♥s② s ♥♠② t r ♥ t Pr tr♥s♦r♠t♦♥s rst s ♦t♥
♥r ♠♦ ♦ t ♦② t♦ ♣s ♠♥
♦♥sr♥ t♦♣s stt♦♥r② ♥♥ ♥♠ αβ t ωyz = 0 ♥ ts s t①s r♣rs♥t♥ t ♦ α s ♥ t t ♣s a tr♥s♦r♠t♦♥ ♠tr① ♥ rrtt♥ ♦♥sr♥ ϑyz = ϑabc ♥ ωyz = ωabc = 0 ♥ t qt♦♥s r
us,αβ = Ris,αβ + ddtλs,αβ
ur,αβ = Rir,αβ + ddtλr,αβ − ωJλr,αβ
s♦♥ s tr♥s♦r♠t♦♥ s ♦t♥ ♦♥sr♥ t♦♣s ♥♥ ♥♠dq s②♥r♦♥♦s② r♦tt♥ t t r♦t♦r ♥ ts s t tr♥s♦r♠t♦♥ ♠tr① ♥ ♦t♥ t ωyz = ω ♥ t qt♦♥s r
us,dq = Ris,dq +ddtλs,dq + ωJλs,dq
ur,dq = Rir,dq +ddtλr,dq
♦ r ♠♦r ♥r ①♣rss♦♥ tt ♥s ♦t t stt♦r ♥ t r♦t♦rqt♦♥s t tr♥s♦r♠t♦♥ ♠tr① P t s③ 4× 6 s ♥ s
P =
[
T(ϑyz) 02×3
02×3 T(ϑyz − ϑ)
]
♥ ts ♥rs
P−1 =
[
T−1(ϑyz) 03×2
03×2 T−1(ϑyz − ϑ)
]
tt r ♦t♥ ♦♥sr♥ t ♣♦st♦♥ ♦ t stt♦r ♥♥ ϑabc = 0 s r♣rs♥t♥ ♥rs♦♥ ♦ P ♥ ♦♥ ② r♥tr♦♥ t ♦♠♦♣♦r ♦♠♣♦♥♥t ♥ ts ② t ♠tr① s sqr t s③ 6 × 6 tr t ♥rs♦♥ t♦♠♦♣♦r ♦♠♣♦♥♥t s ♦♠tt ♥ ♥ ts ② ♥ rrtt♥ s
uyz = Riyz +d
dtλyz +Gλyz
r
G =
0 −ωyz 0 0
ωyz 0 0 0
0 0 0 −(ωyz − ω)
0 0 ωyz − ω 0
♥r② qt♦♥s
♥r② ♥ ♦r t♦ ♣s ♠♥ ♥ ♦t♥ ♠t♣②♥ ②iyzdt
ityzuyzdt︸ ︷︷ ︸
dwin
= ityzRiyzdt︸ ︷︷ ︸
dwJ
+ ityzdλyz + i
tyzGλyzdt
︸ ︷︷ ︸
dwm+dwem
rt♦♥ ♦ t st♦r ♥r② ♥ t ♠♥ ♥r② ♦t♣t ♥ rtt♥ s
3
2ityzdλyz + i
tyzGλyzdt = dwm + τdϑm
②♥r♦♥♦s t♥ ♥
tr♦♠♥t t♦rq τ ♥ ①♣rss strt♥ r♦♠ t ♥r② ♥ s
τ =1
ωm[ωyz(λsyisz − λszisy) + (ωyz − ω)(λryirz − λrziry)]+pityz
∂λyz
∂ϑ−p d
dϑwm
r ωm s t ♠♥ s♣ ♦ t r♦t♦r ♥ s ωm = ω/p
ss ♦♥sr♥ t rt♦♥s♣ t♥ ♠♥t ♥r② ♥ ♦♥r② wm +w
′
m = itλ t qt♦♥ ♦♠s
τ =1
ωm[ωyz(λsyisz − λszisy) + (ωyz − ω)(λryirz − λrziry)] + p
∂
∂ϑw
′
m
♦ t♦t strt♦♥
s♠ ♦♥srt♦♥s ♣rs♥t ♥ t♦♥ ♦r t ♥strt tr ♣s ♠♥ ♥ ♦♥ ♦r t t♦ ♣s q♥t ♠♥ ♥ ♣rtr t rt♦♥s♣ ♥♥ t ① ♥ t rr♥t s st ♥ t t♦ ♣s s②st♠ ♥ ♦t♥ ② ♠♥ ♦ t tr♥s♦r♠t♦♥ ♠tr① P s ♦♦s
λ(i, ϑ) = L(ϑ)i
P−1
λyz(i, ϑ) = L(ϑ)P−1iyz
λyz(i, ϑ) = PL(ϑ)P−1
︸ ︷︷ ︸
Lyz(ϑ)
iyz
♥ t ♥t♥ ♠tr① ♥ ①♣rss s
Lyz = PL(ϑ)P−1
=
[
T(ϑyz) 02×3
02×3 T(ϑyz − ϑ)
][
Ls(ϑ) Msr(ϑ)
Mrs(ϑ) Lr(ϑ)
][
T−1(ϑyz) 03×2
03×2 T−1(ϑyz − ϑ)
]
=
[
T(ϑyz)Ls(ϑ)T−1(ϑyz) T(ϑyz)Msr(ϑ)T
−1(ϑyz − ϑ)
T(ϑyz − ϑ)Mrs(ϑ)T−1(ϑyz) T(ϑyz − ϑ)Lr(ϑ)T
−1(ϑyz − ϑ)
]
=
[
Ls,yz Msr,yz
Mrs,yz Lr,yz
]
rLs,yz = T(ϑyz)Ls(ϑ)T
−1(ϑyz)
Msr,yz = Mtrs,yz = T(ϑyz)Msr(ϑ)T
−1(ϑyz − ϑ)
= T(ϑyz − ϑ)Mrs(ϑ)T−1(ϑyz)
Lr,yz = T(ϑyz − ϑ)Lr(ϑ)T−1(ϑyz − ϑ)
t s ♦rt ♠♥t♦♥♥ tt ♥ r t♦t② ♥r ♥ r ♦rr② ♦ ♦ t rr♥ r♠ ♠tr①s ♥ r ♥♦t ①♣t② ①♣rsss♥ tr ♦♠♣t ♦r♠ ♦s ♥♦t ♣r♦ s ♥♦r♠t♦♥s ♣rs ①♣rss♦♥ ♥ ♦♥sr♥ t r ♥ Pr tr♥s♦r♠t♦♥s
♥r ♠♦ ♦ t ♦② t♦ ♣s ♠♥
t r tr♥s♦r♠t♦♥ s ♣r♦r♠ ωyz = 0 t q♥t ♥t♥♠tr① s
Lαβ =
[
Ls,αβ Msr,αβ
Mrs,αβ Lr,αβ
]
r
Ls,αβ =
[
Ls,α Ls,αβ
Ls,βα Ls,β
]
Msr,αβ = Mrs,αβ =
[
Msr,αα Msr,αβ
Msr,βα Msr,ββ
]
Lr,αβ =
[
Lr,α Lr,αβ
Lr,βα Lr,β
]
♥
Ls,α =1
6(4Ls,a + Ls,b + Ls,c − 4Ms,ab − 4Ms,ac + 2Ms,bc)
Ls,αβ = Ls,βα =−Ls,b + Ls,c + 2Ms,ab − 2Ms,ac
2√3
Ls,β =1
2(Ls,b + Ls,c − 2Ms,bc)
Msr,αα =1
6(4Msr,aa +Msr,bb +Msr,cc − 4Msr,ab − 4Msr,ac + 2Msr,bc)
Msr,αβ = Msr,βα =−Msr,bb +Msr,cc + 2Msr,ab − 2Msr,ac
2√3
Msr,ββ =1
2(Msr,bb +Msr,cc − 2Msr,bc)
Lr,α =1
6(4Lr,a + Lr,b + Lr,c − 4Mr,ab − 4Mr,ac + 2Mr,bc)
Lr,αβ = Lr,βα =−Lr,b + Lr,c + 2Mr,ab − 2Mr,ac
2√3
Lr,β =1
2(Lr,b + Lr,c − 2Mr,bc)
♥ ts rr♥ r♠ t t♦rq ♥ rrtt♥ s
τ(iαβ , ϑ) =1
2pits,αβ
[d
dϑLs,αβ(ϑ)
]
is,αβ
︸ ︷︷ ︸
τr
+ pits,αβ
[d
dϑMsr,αβ(ϑ)
]
ir,αβ
︸ ︷︷ ︸
τed
+1
2pitr,αβ
[d
d(ϑ)Lr,αβϑ
]
ir,αβ
︸ ︷︷ ︸
τc
r♥t② t Pr tr♥s♦r♠t♦♥ s ♣r♦r♠ t ♥t♥ ♠tr① ♥ rtt♥ s
Ldq =
[
Ls,dq Msr,dq
Mrs,dq Lr,dq
]
②♥r♦♥♦s t♥ ♥
trt♥ r♦♠ ♥ ♦♥sr♥ ωyz = ω t t♦rq ♥ rrtt♥ s
τ =1
ωm[ω(λsdisq − λsqisd)] + p
∂
∂ϑw
′
m
τ = p(λsdisq − λsqisd) + p∂
∂ϑw
′
m
❯♥r t ss♠♣t♦♥ ♦ ♥rt② t ♥r② ♥ t ♦♥r② r q ♥
∂
∂ϑw
′
m =∂
∂ϑwm =
1
2it ∂
∂ϑλ =
1
2it ∂
∂ϑLi
♦♥r② ♥ rtt♥ s
∂
∂ϑw
′
m =1
2its,dq
d
dϑLs,dqis,dq + i
ts,dq
d
dϑMsr,dqir,dq +
1
2ir,dq
d
dtLr,dqir,dq
Ptt♥ t♦tr ♥ t s ♣♦ss t♦ ♦t♥
τ = p(λsdisq − λsqisd)
+1
2pits,dq
d
dϑLs,dqis,dq + pits,dq
d
dϑMsr,dqir,dq +
1
2pir,dq
d
dtLr,dqir,dq
②♥r♦♥♦s t♥ ♥
♠♦ t♦ sr rt♥ ♠♥ s ♦t♥ stt♥ t ①tt♦♥ rr♥ts♦ t r♦t♦r q t♦ ③r♦ t ♦♦s tt ♥ t tr ♣s s②st♠ t rr♥ts r
ir =[
0 0 0]t
♠r② ♥ t q♥t t♦ ♣s ♠♥
ir,yz =[
0 0]t
♦s② t tr qt♦♥ ♥ s ♦ ②♥ ♥ ♥ ♥r ♦r r②♠♥ t♦t r♦t♦r ♥♥s ♦♥② ♥s t stt♦r q♥tts
us = Rsis +d
dtλs
♥ ♦♥② ♦♥ ♥♥ s ♣rs♥t t ♦♥ ♦♥ t stt♦r t ssr♣t s s ♦♠tt♥ t♦r ♥♦tt♦♥ rtr
s♠ rt♦♥s♣ t♥ ① ♥ rr♥t ♥ ①♣rss s ♥ t s ♦♦② tr ♣s ♠♥
λ = λ(i, ϑ)
♥i = i(λ, ϑ)
②♥r♦♥♦s t♥ ♥
♦♠♥♥ t tr qt♦♥ t t rt♦♥s♣ ♥♥ rr♥t ♥ ① ts♠ ①♣rss♦♥ ♥ ♦t♥ ♦♥② r♥ s tt ♥ s ♦ ②♥t r♥t ♥t♥ ♠tr① s s③ 3× 3 ♥ ♥ ①♣rss s
ℓ = ℓ(i, ϑ) =
∂λsa
∂isa. . . ∂λsa
∂isc
∂λsc
∂isa. . . ∂λsc
∂isc
♥r② qt♦♥s
rsts ♦t♥ t t s ♦ ♦② tr ♣s ♠♥ ♥ s②①t♥ t♦ t s ♦ ②♥ ♥r② ♥ s ♦t♥ ♠t♣②♥ t trqt♦♥ ② idt rst s
♥ t s♠ ② t s ♥ ♦♥ ♥ t s ♣♦ss t♦ rt
itdλ = dwm + τdϑm
r♥t② s t♦ ♠♦ s♥ t ♠♥t ♥r② s ♥t♦♥ ♦ t♣♦st♦♥ ♥ t stt♦r rr♥t ♦♥②
dwm =∂wm
∂λdλ+
∂wm
∂ϑdϑ =
∂wm
∂λsadλsa + · · ·+
∂wm
∂λscdλsc + · · ·+
∂wm
∂ϑdϑ
♦♠♥♥ ♥ t s ♣♦ss t♦ rt
itdλ− τdϑm =
∂wm
∂λdλ+
∂wm
∂ϑdϑ
tt ♥ s♣rt ♥ ts t♦ ♦♠♣♦♥♥t
i(λ, ϑ) =∂wm
∂λ
τ(λ, ϑ) = −p∂wm
∂ϑ
r t ♦r♠r r♣rs♥ts t ♥♥ rr♥t ♥ t ttr t t♦rq ①♣rss♦♥
t♦rq ♥ s♦ ①♣rss s ♥t♦♥ ♦ t rr♥t i ♥ t ♣♦st♦♥ ϑ♦t♥♥
♦ t♦t strt♦♥
♥ ts st♦♥ t ♠♦ ♦ ②♥ s ♦♣ ♦♥sr♥ ♥r ♠♥t rs ♥ rsts r ♦t♥ s ♥ ①t♥s♦♥ ♦ t ♦♥srt♦♥s ♠ ♦rt ♦② tr ♣s ♠♥
♥ s ♦ ♥♦ strt♦♥ t ① ♥ ①♣rss s r L s t 3 × 3♥t♥ ♠tr① tt ♥ rtt♥ s
λ(i, ϑ) = L(ϑ)i
②♥r♦♥♦s t♥ ♥
r L s t 3× 3 ♥t♥ ♠tr① tt ♥ rtt♥ s
L =
Ls,a(ϑ) Ms,ab(ϑ) Ms,ac(ϑ)
Ms,ba(ϑ) Ls,b(ϑ) Ms,bc(ϑ)
Ms,ca(ϑ) Ms,cb(ϑ) Ls,c(ϑ)
♥ s ♦ ♥rt② t ♠♥t ♥r② ♥ ♦♥r② r q t♦rq ♥ ♥ rtt♥ s ♥ t s ♦rt t♥ tt s♥♥t r♥ ♥ ♦♥ t♥ t ♥r ♦② ♠♥ ♥ t ②♥ ♥ s♣tt♥ tt♦rq ♥ ts ♦♠♣♦♥♥ts ♥ t s ♥ s♦♥ tt t t♦rq ♦ ♦② tr ♣s ♠♥ ♥s t rt♥ t tr♦②♥♠ ♥ t ♦♥t♦rq ♥ t ♦♥trr② t t♦rq ♣r♦ ② t ②♥ s ♦♥② ♦♥ tr♠ ♥♠②t rt♥ t♦rq tt ♥ ①♣rss s
τ(i, ϑ) =1
2pit[d
dϑL(ϑ)
]
i
︸ ︷︷ ︸
τr
tr♦②♥♠ t♦rq s ♣r♦ ② t rt♦♥ ♦ t ♠t ♥t♥st♥ t stt♦r ♥ t r♦t♦r ♥♥s s tr♠ s ♣rs♥t ♥♦t ♦♥② ♥ ♠♥st r♦t♦r ♥♥ t ♥ ♣r♠♥♥t ♠♥t ♠♥s s ♥ ts s t t♦rqs ♣r♦ ② t ♥trt♦♥ ♦ t ♠♥t ① ♥ t t stt♦r ♥♥s s♦♠♣♦♥♥t ♦ t♦rq s ♥ ♥ s ♦ ②♥ s♥ t r♦t♦r s ♥♦ ♣r♠♥♥t ♠♥t♥tr ♥♥s
♦♥ t♦rq s s ② t ♥trt♦♥ t♥ t r♦t♦r ① ♥ ♥t stt♦r ♥s♦tr♦♣s ♠♥② s ② s♦ts ♦♣♥♥ s tr♠ s ♣rs♥t ♥ ♦t♦② ♥ ♣r♠♥♥t ♠♥t ♠♥s ♥ t ♦♥trr② ②♥ ♦ ♥♦t tst♦rq ♦♠♣♦♥♥t
♦♥② ♦♠♣♦♥♥t ♥ ♥ t t♦rq ①♣rss♦♥ ♦ t ②♥ s trt♥ t♦rq ttr s rt t♦ t rt♦♥ ♦ t ♥t♥ ♠tr① tt r♦t♦r ♣♦st♦♥ ♥ t s ♣rs♥t ♥ r② ♠♥ ♣rs♥t♥ ♥ ♥s♦tr♦♣ r♦t♦r♥ ② ♣♦ s♥s ♦r ① rrrs t ♥ ①♣rss s
♥ t s♥s♦② strt ♥♥s
♥ ♠♥s t stt♦r ♥♥ s s♥ t♦ ♥♥ t ♥♠♥t ♦♠♣♦♥♥t♦ t r♣ ① strt♦♥ ② t ♥♥ s s♥s♦ strt♦♥r♦♥ t r♣ t rst♥ ① ♥ s s♥s♦ t t r♦t♦r ♣♦st♦♥ ♥ tsst♦♥ t qt♦♥s ♦ t ②♥ r ♥②③ ♥r t ss♠♣t♦♥ ♦ s♥s♦strt♦♥ ♦ t stt♦r ♥♥s
❯♥r t ss♠♣t♦♥ ♦ strt ♥♥s t 3 × 3 ♥t♥ ♠tr① ♦ ttr ♣s ♠♥ ♥ ①♣rss s
L =2
3
LΣ + L∆♦s2ϑ −LΣ
2 + L∆♦s2ϑ2 −LΣ
2 + L∆♦s2ϑ1
−LΣ
2 + L∆♦s2ϑ2 LΣ + L∆♦s2ϑ1 −LΣ
2 + L∆♦s2ϑ
−LΣ
2 + L∆♦s2ϑ1 −LΣ
2 + L∆♦s2ϑ LΣ + L∆♦s2ϑ2
②♥r♦♥♦s t♥ ♥
r ϑ1 = ϑ− 2/3π ♥ ϑ2 = ϑ− 4/3π LΣ = (Ld + Lq)/2 ♥ L∆ = (Ld − Lq)/2
rr♥t ♥ ss♠ ② t ♥r ①♣rss♦♥
ia = IM cos(ϑ+π
2− γ)
ib = IM cos(ϑ1 +π
2− γ)
ic = IM cos(ϑ2 +π
2− γ)
tr ♣s ① ♥s ♥ ♦t♥ strt♥ r♦♠ ♥ sλ = Li ♦t♥♥
λa = IM [−LΣ sin(ϑ− γ) + L∆ sin(ϑ+ γ)]
λb = IM [−LΣ sin(ϑ1 − γ) + L∆ sin(ϑ1 + γ)]
λc = IM [−LΣ sin(ϑ2 − γ) + L∆ sin(ϑ2 + γ)]
rst s ♦♥sst♥t t t ss♠♣t♦♥s tt t ① ♥s ♥ srs s♥s♦ ♥t♦♥
① qt♦♥ ♥ s♠r② r ♦r t q♥t t♦ ♣s ♠♥ ♥ ♣rtr ② ♣♣②♥ t r tr♥s♦r♠t♦♥ t♦ t s ♣♦ss t♦①♣rss t ♥t♥ ♠tr① ♥ t stt♦♥r② rr♥ r♠ ♦t♥♥
L =
[
LΣ + L∆ cos 2θ L∆ sin 2θ
L∆ sin 2θ LΣ − L∆ cos 2θ
]
s♠ tr♥s♦r♠t♦♥ ♥ ♣♣ t♦ ♦t♥♥
iα = −IM sin(θ − γ)
iβ = IM cos(θ − γ)
① ♥ ♥ ♦t♥ strt♥ r♦♠ ♥ s λ = Li ♦t♥♥
λα = IM [−Lq cos(γ) sin(θ) + Ld sin(γ) cos(θ)]
λβ = IM [Lq cos(γ) cos(θ) + Ld sin(γ) sin(θ)]
♠r② t qt♦♥ ♦ t t♦ ♣s ♠♥ ♥ r ♣♣②♥ tPr tr♥♦r♠t♦♥ ①♣rss♥ t q♥tts ♥ t rr♥ r♠ r♦tt♥ s②♥r♦♥♦s② t t r♦t♦r
♥t♥ ♠tr① ♦t♥ t ts tr♥s♦r♠t♦♥ s
Ldq =
[
Ld 0
0 Lq
]
♥ t q rr♥ r♠ t rr♥ts ♥ ①♣rss s
id = IM sin(γ)
iq = IM cos(γ)
②♥r♦♥♦s t♥ ♥
♥② r♣t♥ t s♠ t♦♥ ♦♥ ♦r t tr ♣s s②st♠ ♥ ♦r tt♦ ♣s ♠♥ ♥ t stt♦♥r② rr♥ r♠ t ① ♥ ♥ ①♣rsss t ♣r♦t t♥ ♥
λd = IMLd sin(γ)
λq = IMLq cos(γ)
t s ♦rt ♥♦t♥ tt ♥ ts s t s♥s♦ ♥♥ strt♦♥ tt r♣ t s ♥t♥s r ♦♥st♥t ♥ t ♠t ♥t♥s Ldq = Lqd r♥ ♦r ts rs♦♥ strt♥ r♦♠ t t♦rq ♥ ①♣rss s
τ = p(λdiq − λqid)
♦♥s♦♥s
♥ ts ♣tr t ♠♦ ♦ t s②♥r♦♥♦s rt♥ ♠♥ s ♥ ♦♣rst t qt♦♥s ♥ r ♦♥sr♥ t ♠♦st ♥r s ♦ tr ♣str ♠♥ t ♦② tr ♣s ♠♥ s ♠♥ ♣rs♥ts♥♥s ♦♥ ♦t t stt♦r ♥ t r♦t♦r t s ♦rt t♥ tt ts ♠♦♦ r♣rs♥t t ♦r ♦ r② ♥ ♦ s②♥r♦♥♦s ♠♥ ♥♥ ♣r♠♥♥t ♠♥t ♥ rt♥ ♠♥s ♠♦ s ♥ ♦♣ t t ♦♥②ss♠♣t♦♥ ♦ ♥♦ ②strss ♥ ♥♦ ② rr♥ts tr tt t qt♦♥s ♥♦t♥ ♥tr♦♥ t ②♣♦tss ♦ ♥rt② ♥t♥ r♦♥ strt♦♥
tr ♣s s②st♠ s ♥ tr♥s♦r♠ ♥t♦ ♦② t♦ ♣s ♠♥ s♠ ♦♥srt♦♥s ♠ ♦r t tr ♣s ♠♥ ♥ r♣t s♣②♥ t qt♦♥ ♦r t t♦ ♣s s②st♠ ♥ t ♠♦ s ♥ rst ♦♣t ♥♦ rstrt♦♥ ♦♥ t ♠♥t ♦r ♦ t ♠♥ ♥ tr ♥tr♦♥ t②♣♦tss ♦ ♥♦ r♦♥ strt♦♥
♥② t rsts ♦t♥ ♦r t ♦② ♠♥s ♦t t tr ♣s♥ t t♦ ♣s ♥ s t♦ ♦♣ t ♠♦ ♦ t ②♥ ② ♠♣♦s♥s♣ ♦♥t♦♥ ♦♥ t r♦t♦r rr♥ts ♥ ♣rtr t♥ ♥ r♦t♦r rr♥ts ♥ t♠♦ ♦ t ♦② ♠♥ t s ♣♦ss t♦ r♣rs♥t s②♥r♦♥♦s rt♥♠♥ qt♦♥s ♥ ♦♣ ♦♥sr♥ t ♠♦st ♥r s t♥♥tr♦♥ t ②♣♦tss ♦ ♥rt② ♥ ♥② ♦♥sr♥ s♥s♦ strt♦♥♦ t ♥♥s
Chapter 2
Prt rr♥t ♦♥tr♦
Prt rr♥t ♦♥tr♦ s♠s str♦♥② r② ♦♥ t ♥♦ ♦ t ♣♥t ♠♦ r② ♦ t ♦s rr♥t ♣rt♦♥ ♦ t ② ♣r♠trs rt♦♥♦r ♠s♠t ♥♦♥ ts ♥ ♦tr ♠♦ ♥qs ♥ ②♥r♦♥♦s t♥♥ ts t ♦ ♣rtr② rt s♥ ts ♥r♥t ♥t♥s r♦♥ strt♦♥ss t rt♦♥ ♦ t ♥t♥s ♥ r♥ ♥t② ♦r ♣rt♦♥s ♥ ♣rs♥t s ♥ t s♦t♦♥ t♦ ♦r♦♠ t ♠♦ ♥qs ♦♥tr♦s♠s s ♦♥ ts ♦ ♥♦t rqr ♠♦ ♦r t ♣rt♦♥ s♥ t② r② ♦♥ tst♦r rr♥t rt♦♥s ♦t♥ ② ♠sr♠♥ts s ♥♥♦t ♦♥tr♦ tqr♥ts r♦st♥ss t♦ ♣r♠tr rt♦♥s ♥ ♠s♠ts ♥ t ♦tr ♥ rq♥t ♣t ♦ t st♦r rr♥t rt♦♥s s ♠♥t♦r② t♦ ♦ t s♦ ♦t t♦r st♥t♦♥ tt ts t ♣r♦r♠♥ ♥ t♦ t ♦ss ♦ t ♦♥tr♦♥ ts ♣tr Prt rr♥t ♦♥tr♦ s♠s r ♥②s ♥ ♣rs♥t ♦s♥♦♥ ♦t ♦s ♥ ♦r ♣rt♦♥
♥tr♦t♦♥
♥ t♦ ts s♠♣t② ♦ ♦♠♣r♥s♦♥ ♥ ♠♣♠♥tt♦♥ ts ♠♣r♦ ②♥♠s♥ st rs♣♦♥s Prt ♦♥tr♦ P s ♥ ② ♥stt ♥ ♠♣♠♥t s ♥ ♦ ♦♥tr♦ s t♦ ♣rt t tr ♣♥t ♦r ♥ ss ts♣t② t♦ st t ♦♥tr♦ t♦♥ ♦ ♦ t ttr s s ♦♥ t t♦♥ ♦ ♦st ♥t♦♥ ♥ t s ♦ ♣♦r tr♦♥s ♥ tr rs Pss♠s ♥ ♦t♥ s t♦ ♦♥tr♦ t rr♥t ♦♥ ♥ t ♦ ♥ tr ♠♦t♦r ♦♥♥t t♦ t ♣♦r ♦♥rtr ❬❪ s ♥ ♦ s♠s r rrr sPrt rr♥t ♦♥tr♦ P ♥ ♥ ss ♥ t♦ t♦rs ♣♥♥ ♦♥♦ t st♥ s♥s ♦r tt♥ t ♠♥ ♦ts r ♥rt rst♣♣r♦ ♥♠ ♦♥① ♦♥tr♦ t ♣r♦s ♥② ♦t t♦r ♦t♥ t ♦♠♥t♦♥ tr♦ ♣s t ♠♦t♦♥ st ♦ t♦ ♦r tr s t♦rs ♦t ♥rtr s♦♥ ♥t ♦♥tr♦ t ①♣♦ts t srt ♥tr♦ t ♣♦r ♦♥rtr ♥ ♦s ♥♦t rqr ♠♦t♦r s♥ t ss ♦♥② t st♦rs
♣rt♦♥ ♦♠♠♦♥② rs ♦♥ t ♣♥t ♠♦ ♥ rs t♦ ♦s P
Prt rr♥t ♦♥tr♦
P t♦ ts ♥tr ts ♣r♦r♠♥ r② ♣♥s ♦♥ t r② ♦ t♠♦ t ♦ ♣r♠tr ♠s♠t ♥ t rr♥t ♣rt♦♥ ♦ P ♦r t♦ ♦♥rtr r ♣② ♥stt ♥ ❬❪ ♥ ts ♦r t s s♦♥ tt t♣rt♦♥ rr♦r ♣♥s ♥♦t ♦♥② ♦♥ t ♥rt♥t② ♦ t ♠♦ ♣r♠trs t s♦♦♥ t ♥st♥t♥♦s s ♦ ♦ rr♥t ♥ ♦t♣t ♦t ♦ t ♥rtr st s ♠♦r sr ♥ ♣rs♥ ♦ ♥t♥ ♠s♠ts ♥ ♥ ♣rtr ♥ t♣rt♦♥ ♠♦ ♦rst♠ts t r ♦♥sr♥ ♥ tr ♠♦t♦r s t♣♥t t ♦ ♠♥t♦♥ stt♦♥ ♦ ♦r t♦ r♦♥ strt♦♥ ♥ t ♦ ♣rtr② rt ♥ ②♥ rs♦♥ s tt t♦ t ♦♠tr② ♥ t r♥ ♦♣rt♦♥ ♣t② t ♥t♥s r str♦♥② ♣♥♥t ♦♥ t rr♥ts ♥t② r② ♥ r♥ ♥tr♦♥ s♥♥t rr♦rs ♥ t ♣rt♦♥ ttr♥ tr♥ ♠♣s tr♦rt♦♥ ♦ t r ♣r♦r♠♥
r♦st♥ss ♦ P s ♥ rss ♥ r♥t ②s s s ② ♠♥♦ str♥ stt ♦srr ♦r ♦♥♥ ♣r♠tr st♠t♦♥ ♥ ❬❪ ♥ ❬❪ ♥ ♦srrs s♠ s s t♦ ♠♣r♦ t ♣r♦r♠♥ ♦ P s♠ ♦r ♥ ♥t♦♥♠♥ ♥ t ♦r♠r ♥rr ♦srr s s t♦ ♦♠♣♥st ♦r t ♥♥♦s ②♥♠s t ttr ①♣♦ts s♦♥ ♦rr s♥ ♠♦ ♦srr ♦tts ♦rs ♦♠♣r t ♣r♦r♠♥ t trt♦♥ P s♠ ♥r tt ♦ ♣r♠tr ♠s♠t s♦♥ t t♥ss ♦ t ♦srrs ♥ ❬❪ r♦st ♣rt rr♥t ♦♥tr♦ s♠ ♦r Pr♠♥♥t ♥t ②♥r♦♥♦s ♥P s ♣rs♥t ♥ ♣rtr t ♣r♦r♠♥ ♦ t ♦♥tr♦r r ♠♣r♦ ♥st ♣r♠tr rt♦♥s ② ♠♥ ♦ ♥ ♥tr tr♠ ♥ ❬❪ ♥ ♣tP ♦r P s ♣rs♥t r t ♥t♥s r st♠t ♦♥♥ ② ♠♥ ♦♥ ♣t ♦srr ♥ r s ♥ t ♠♦ ♣rt♦♥ ♥ ♥ t rr♥t rr♥♥rt♦♥ s P♣r ❬❪ ♥②ss t t ♦ ♣r♠tr rt♦♥s ♥ P♦r P ♥ tr tt ♣r♦♣♦ss ♥ ♥tt♦♥ ♣r♦ss s ♦♥ st sqr♠♥♠③t♦♥ t♦ ♥♥ t ♣r♦r♠♥ ♦ t s♠
♥ t s♦t♦♥ rrr s ♦r P P t♦ ♦r♦♠ t ♦♥sq♥s ♦ t ♠♦ ♥qs s ♥ ♣r♦♣♦s ♥ ❬❪ ♥ t♥ ♠♣r♦ ♥ ❬❪ ♥ts ♦rs t rr♥t rt♦♥s rt t♦ rt♥ ♦t t♦r r r♦r ♥st♦r ♥ ♦♦♣ ❯ rr♥t ♣rt♦♥ s tr ♦t♥ ② ss♥t ❯ ♥ t ♣♦st♦♥ rt t♦ t ♦♥sr ♦t t♦r rsts s♦♥♥ ❬❪ t tt r rr♥t ♣rt♦♥ ♥ ♦♥② t ❯ srq♥t② ♣t ♦♥ t ♣♥♦♠♥♦♥ ♦t t♦rs st♥t♦♥ s♠♣ ♥tst♥t♦♥ s♦t♦♥ ♣r♦♣♦s ♥ ❬❪ ♦♥ssts ♥ ♠♦♥t♦r♥ t ♦t ♦t t♦rs ♥ tr ♦♥ ♦ t♠ ♦r♦♠s ① trs♦ ♣♣②♥t t♦t t s ♦ ♠♣♠♥tt♦♥ ts s♦t♦♥ s s♦♠ s♥♥t rsrst② t ♣r♦♣♦s ♥tst♥t♦♥ ♠t♦ ♠♣s t ♣♣t♦♥ ♦ ♥♦♥ ♦♣t♠ t♦rs s♥ t② r ♥♦t t♥ r♦♠ t ♦st ♥t♦♥ ♠♥♠③t♦♥ ss t ♦♦ t trs♦ s ♠ ♦♥ t s ♦ ♠♣r ♦♥srt♦♥s t ♦ ♥♦t st ♦r r② ♦r♥ ♦♥t♦♥ ♥ t r♣rs♥ts tr♦ t♥ t rt②♦ t ♣rt♦♥ ♥ t ♦♠♣♥ t t ♦st ♥t♦♥ ♠♥♠③t♦♥
Prt rr♥t ♦♥tr♦
SynRMCost Function
Minimization ~
UDC
ikPrediction
at k+1
iabck
22
i z,k2|k16
Prediction
at k+216
i*
2 2
uk
ik1|kuz
r Prt rr♥t ♦♥tr♦ s♠
Prt rr♥t ♦♥tr♦
♥r r♣rs♥tt♦♥ ♦ t P s♠ s r♣♦rt ♥ ♥P s t♦ ♣rt t tr ♦r ♦ t s②st♠ ♥ t♦ ♦♦s t ♥①t ♦♥tr♦t♦♥ ♦r♥ t♦ rt♥ ♦♣t♠ rtr♦♥ ♥ ♣rtr ♥ P t ♦♦s♥♦t t♦r s t ♦♥ ♠♦♥ t t ♣♦ss ♥rtr stts u
z = [uzd uzq ]t t
z ∈ Q ♥ Q = 0, . . . , 7 tt ♠♥♠③ t ♦st ♥t♦♥ ttr s t♦ r② s♥ s♥ t ♦r ♦ t ♦ s②st♠ s s ♦♥ t r♥t♦t ♥t♦♥s ♥ s ♥ trtr s ♥②s ♥ ♥ ts♦r t ♦♥sr ♦st ♥t♦♥ s
♠♥z∈Q J =[
i∗d − iz,k2|kd
]2+[
i∗q − iz,k2|kq
]2
t st♣ t rr♥t s♠♣s ik = [ikd ikq ]
t ♥ t st ♦t t♦r ♣♣
uk = [ukd ukq ]
t r s t♦ ♣rt t ♥①t s ♦ rr♥t ik1|k = [i
k1|kd i
k1|kq ]t
♣rt♦♥ t ♥st♥t k1 = k + 1 ♥ ♥r② ①♣rss s
ik1|kd = ikd +∆ikd
ik1|kq = ikq +∆ikq=⇒ i
k1|k = ik +∆i
k
r ∆ik = [∆ikd ∆ikq ]
t s t rr♥t rt♦♥ t♥ st♣s k ♥ k1 ♣rtrr♥ts ik1|k t♦tr t t t ♥rtr stts uz t z ∈ Q r s ♥ tr♥t♦ st♠t t t ♣♦ss rr♥ts iz,k2|k = [i
z,k2|kd i
z,k2|kq ]t t st♣ k2 = k + 2
ttr r ♦♠♣t s
iz,k2|kd = ik1d +∆iz,k1d
iz,k2|kq = ik1q +∆iz,k1q
=⇒ iz,k2|k = i
k1 +∆iz,k1
s ♦♠♣t t r s t♦ t t ♦st ♥t♦♥ ♥ t♦ ♥t st ♦t t♦r tt ♣♣ t t ♥①t st♣ t♦ st♣ ♣rt♦♥s rqr t♦ ♦♠♣♥st ♦r t ♥r♥t ② ♥ t t ♠♣♠♥tt♦♥ ♦ tP s♠ ❬❪ s♥♥t r♥ t♥ P ♥ P st♥s ♥♦ t rr♥t rt♦♥s ∆i
k ♥ ∆iz,k1 ♥ ♥ r t s
sr ♥ t ♥①t st♦♥s
Prt rr♥t ♦♥tr♦
♦st ♥t♦♥
♦ ♦ t ♦st ♥t♦♥ r♣rs♥ts r s♣t ♥ t s♥ ♦ ♦♥tr♦s♠ r♥t ♦st ♥t♦♥s ♥ s ♥ trtr ♥ ❬❪ ♦st ♥t♦♥s ♦♥ t s♦t rr♦r s ♠♣♦② ♥ t abc tr ♣s s②st♠ ♦r ♥ ♥tr♦rPr♠♥♥t ♥t P ♠♥s ♥ ❬❪ t s♠ s ♠♣♦② ♥ t αβ stt♦♥r②rr♥ r♠ ♦r ②♥ s♠ s s ♥ ❬❪ ♦r r♠♦♥t Pr♠♥♥t♥t P ♠♥ ♥ ❬❪ t ♦st ♥t♦♥ s s ♦♥ t rr♥t sqr rr♦r♦r ♥ P ♠♥ t r ♦ t ♣r♦♣r ♦ ♦ t ♦st ♥t♦♥ ♥ ♦♥ ♥ ❬❪ r r♥t s♦t♦♥s r ♥②s ♥ s ♦♥srt♦♥s r ♥♦♥ t t♥ t♦rs ♥ t r ♦♥str♥ts rr♥t ♠ts ♥ ♥ t♦st ♥t♦♥
♥ trt♦♥ r♥t ♦♥tr♦ s♠s t q rr♥ts r ♦♥tr♦② t♦ ♥♣♥♥t rt♦rs s r♥ts tt ♥♦ ♠tt♦♥s r ♥♦♦t t rr♥s r ♣rs② tr ♥ ♥r ②♥♠s ♦r♥ t♦ t ♥t ♦ t t♦ ♦♦♣s P rs ♦♥ t ♠♥♠③t♦♥ ♦ t ♦♥② ♦st ♥t♦♥ tts t♦ ♥ ♦t t rr♥t rr♦rs ♦♥sr♥ t ②♥ ♣♣t♦♥ t r♠r ♥s♦tr♦♣② ♦ rst ♥ t♦♦ r♥t ♦r ♥ t t♦ ①s ♥ ♣rtr♥ ♠♣r♦♣r ♦ ♦ t ♦st ♥t♦♥ ♦ rst ♥ s ♦♥tr♦ t♦♥ ♥ t♣r♦rt② t♦ ♦♥ ♦ t♦ ①s tt ♦ ♣rtr② tr♠♥t r♥ t ♠♥♦♣rt♦♥ ♦♠ ♦♥srt♦♥s ♦♥ ts s♣t r r♣♦rt ♥ ts st♦♥
rst ♦st ♥t♦♥ ♦♥sr s t ♦♥ s ♦♥ t s♦t rr♦r tt ♥ rtt♥ s
♠♥z∈Q J = |i∗d − iz,k2|kd |+ |i∗q − iz,k2|kq |
♦♥sr♥ t s ♣♦♥t s t rr♥ t s♦ rs ♦ ts ♦st ♥t♦♥r ♣♦tt ♥ t s ♥s
♥ ♦rr t♦ r② ts stt② t P s ♦♥ s ♥ tst ♥ s ♦st ②♥♠s ♥♦♥ rr♥t st♣s ♥ ♣rtr t t ♠♥ t st♥st s♣ st♣ s ♥ s rr♥ t♦ t s♣ rt♦r s ♦♥sq♥ ts ♦t♣tstrts t♦ t ♠①♠♠ rr♥t ♠♣t t s ♣♦♥t r♣rs♥ts trr♥t rr♥ rsts ♦ t s♠t♦♥ r r♣♦rt ♥ ♥ ♣rtrt q rr♥t ♦r♠s t t♦rq ♥ t ♦t t♦r ♥①s r s♦♥ ♥ t s ♦rt t♥ tt ♥ t ♥t st t rt♦r ♦♥② ts t♦♥② t ①s rr♦r s ♥ s♥ s♥ t st♣ t ♦t t♦r t ♥①4 s ♦s♥ ♥ ϑ = 0t ①s rr♥t r♣② rs t ♥t rr♥ t q①s s st ③r♦ r r♣rs♥tt♦♥ s ♥ ② t ♣♦♥t ♦ ♦r trt♦r②♥ t q ♣♥ s♦ ♥ ♥ ♥ t rt rr♥t rs trr♥ i∗d = −1/
√2 ❬♣❪ t ♣rt rt♦r strts ♦♦s♥ t♦rs t♦
♥rs t qrtr rr♥t ♥①s 2 ♥ 3 ♦♥tr♦ t♦♥ s r②t♦♦ t rr♥t t t ♦r ♥t♥ s rst② ♣s t♦rs ts rr♥♥ ♦♥② tr t rr♦r ♦ t s♦♥ rr♥t s ♠♥♠③ ♦tr t stt t t♦rq ♣r♦ ② t ♠♥ r♠♥s ③r♦ ♦♥ ♦ t t♦ rr♥t s♥ ♥ ts ♥ tr♥ ♠♣s s♦r s♣ rs♣♦♥s
♦♥srt♦♥s ♠ s♦ r ♥ rs♠ s ♦♦s t♦ t s♥♥tr♦t♦r ♥s♦tr♦♣② t ♦♥tr♦ t♦♥ ♣rs♥ts s ♥ ♣rtr t stst ② t♦r t rr♥t rr♦r s t♦ st t ♣♦♥t ♦ ♦r rst ♦♥ t ①s t t ♦r
Prt rr♥t ♦♥tr♦
0 100 200−1
−0.5
0
d−ax
is c
urre
nt
[p.u.]
ActualRef
0 100 2000
0.5
1
q−ax
is c
urre
nt
0 100 2000
0.5
1
Time step [−]
Tor
que
0 20 400
1
2
3
4
Time step [−]
Vol
tage
vec
tor
inde
x [p
.u.]
♦♥tr♦ q♥tts
−1 −0.8 −0.6 −0.4 −0.2 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.40.6
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
0.81
1
1.2
d−axis current [p.u]
q−ax
is c
urre
nt [p
.u]
q rr♥t ♣♥
r ♦st ♥t♦♥ s ♦♥ s♦t rr♦r
♥t♥ s ♦♥sq♥ ♦♥② ♦♥ ♦ t t♦ rr♥ts r♣② rs t rr♥t s ♦rt ♠♥t♦♥♥ tt ts ♥sr ♦r s ♦t♥ ♥♦t ♦♥② t t strt♣♦ t ♠♥ s ♥ t rsts ♦ t ♥ t♦s ♦♣rt♦♥s ♥♦♥ rr♥t rr♦rs s t s ♣rtr② ♥t ♥ t ①s s ♥ t ♦♥ ♦t s① t ♦t t♦rs ♦ t ♥rtr ♥ ts stt♦♥ t ♣rt rt♦r♦♦ss t ♥ t♦r ♦r ts ♦♣♣♦st ♣♥♥ ♦♥ t s♥ ♦ t rr♥t t♦st② r② t ①s rr♥t ♣♥ ♦♥st♥t qrtr rr♥t
rst t♦ ♦r♦♠ ts ♥sr ♦r s t♦ t t t♦ rr♦r ♦♠♣♦♥♥ts ♥ t ♦st ♥t♦♥ ♥ ♣rtr s♥ ♥ r ♦r t q①s ♦ ♠♦r ♠♣♦rt♥ t♦ ts rr♦r ♦♥ tt t rr♥t r♠♥s t♦♦ st♥t r♦♠ trr♥ ♥ t s s ♥ ② t ♥s♦tr♦♣② rs♦♥ tt♠♣t s t♦ tt rr♦rs ♥ t ♦st ♥t♦♥ t t s♥② rt♦ ♦t♥♥
♠♥z∈Q J = |i∗d − iz,k2|kd |+ Lq
Ld|i∗q − iz,k2|kq |
♠♣ ♦ t ttr s r♣♦rt ♥ t s ♥s rsts ♦ tP s ♦♥ r r♣♦rt ♥ ♦r ♦t♥ t t ♥ ♦st♥t♦♥ s ♦♠♣t② r♥t t ♥ s♥ tt ♥ t ♥t st t ♣rt♦♥tr♦r ♦t♣ts t ♦t t♦r t ♥① 3 ttr ♣ss ♦t t rr♥tst♦rs tr rr♥ ♦♥ t t ♦t♥ s♥ s r② s♦♥ ♥ t♦t t rt rr♥t s ♥ssr② str t ♣♦♥t ♦ ♦r ♦s ♥♦t♠♦ ♦♥ t ①s s ①♣t t t♦rq rs♣♦♥s s str t strts rs♥r♦♠ t rst st♣
♥♦tr ♦st ♥t♦♥ s s ♦♥ t sqr rr♦r ♥ t ♥ rtt♥ s
♠♥z∈Q J =[
i∗d − iz,k2|kd
]2+[
i∗q − iz,k2|kq
]2
r♣♦rts t s♦rs ♦ t ♦♥sr ♦st ♥t♦♥s ♦ s♦♦ tt t ♣♦♥t ♦ ♦r ts t♦♦ r r♦♠ t rr♥ ♥ t rr♦rs r sqr♥ t ♦st ♥t♦♥ s s♦♦♥ s ♦♥ ♦ t t♦ rr♥ts ♠♦ r♦♠ ts rr♥ ts tr♠♦♠s ♠ r t♥ t ♦tr ♦♥ ♦r♥ t ♦st ♥t♦♥ t♦ r t s
Prt rr♥t ♦♥tr♦
0 100 200−1
−0.5
0
d−ax
is c
urre
nt
[p.u.]
ActualRef
0 100 2000
0.5
1q−
axis
cur
rent
0 100 2000
0.5
1
Time step [−]
Tor
que
0 20 400
1
2
3
Time step [−]V
olta
ge v
ecto
r in
dex
[p.u
.]
♦♥tr♦ q♥tts
−1 −0.8 −0.6 −0.4 −0.2 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.50.5
0.5
0.50.5
1
1
1
1
1
1
1
1.5
1.5
1.5
1.52
2
2
2
2.5
2.5
d−axis current [p.u]
q−ax
is c
urre
nt [p
.u]
q rr♥t ♣♥
r ♦st ♥t♦♥ s ♦♥ t s♦t rr♦r
♦st ♥t♦♥ ♥ rrtt♥ s
♠♥z∈Q J =[
i∗d − iz,k2|kd
]
·[
i∗d − iz,k2|kd
]
+[
i∗q − iz,k2|kq
]
·[
i∗q − iz,k2|kq
]
= wd(εd) ·[
i∗d − iz,k2|kd
]
︸ ︷︷ ︸
εd
+wq(εq) ·[
i∗q − iz,k2|kq
]
︸ ︷︷ ︸
εq
♥ t ♥ s♥ s t s♠ ♦ t rr♥t rr♦rs t r ts wd(εd)♥ wq(εq) ♣♥♥ ♦♥ t rr♦rs εdεq t♠ss t ts r ts q t♦ trr♦rs
rsts ♦t♥ t r r♣♦rt ♥ s ♦r t ♦st ♥t♦♥t t s♦t rr♦r t rst st♣s r ♦♠♠t t♦ r t ①s rr♦r ♦ ♦ts t ♣rt rt♦r ♦♦ss t ♦t t♦r t ♥① 4 tt ♦s tstst rs ♦ id ♣♥ t qrtr rr♥t t ③r♦ ♦♠♣rs♦♥ t s♦s tt ts ♦♣rt♦♥ sts ♦r ♦r ♥♠r ♦ st♣s s♥ t sqrrr♦r rs♦♥ s tt s s♦♦♥ s t rr♦r ♦ t ①s ♦♠s ♦r t♥ rt♥trs♦ t q①s rr♦r rsts t ♦♠♥♥t ♦♠♣♦♥♥t ♥ t ♦st ♥t♦♥ s ♦♥sq♥ tr t s♦rt ♥t st t ♦♦s♥ ♦t t♦r s t ♦♥ t♥① 3 ❲t t ♦st ♥t♦♥ s tr♦r ♣♦ss t♦ s♦rt♥ t s ♦♥tr♦t♦♥ t ♥ q①s rr♥t ♥ t♦rq
0 100 200−1
−0.5
0
d−ax
is c
urre
nt
[p.u.]
ActualRef
0 100 2000
0.5
1
q−ax
is c
urre
nt
0 100 2000
0.5
1
Time step [−]
Tor
que
0 20 400
1
2
3
4
Time step [−]
Vol
tage
vec
tor
inde
x [p
.u.]
♦♥tr♦ q♥tts
−1 −0.8 −0.6 −0.4 −0.2 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
10.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.3
0.3
0.30.3
0.4
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.50.5 0.6
0.6
0.7
0.8
0.9
d−axis current [p.u]
q−ax
is c
urre
nt [p
.u]
q rr♥t ♣♥
r ♦st ♥t♦♥ s ♦♥ sqr rr♦r
st ♦st ♥t♦♥ t ♥ ts st♦♥ s ♦t♥ ♦♠♥♥ t ♥
Prt rr♥t ♦♥tr♦
ts ♦ s♥ t s♥② rt♦ s t ♥ t sqr rr♦r rst♥ ♦st♥t♦♥ s
♠♥z∈Q J =[
i∗d − iz,k2|kd
]2+
Lq
Ld
[
i∗q − iz,k2|kq
]2
s♦s t rsts ♦t♥ t t ♥ s♥ tt t ♣r♦r♠♥r ♠♣r♦ t rs♣t t♦ t♦s ♦ ♥ r t s♠ ♦
0 100 200−1
−0.5
0
d−ax
is c
urre
nt
[p.u.]
ActualRef
0 100 2000
0.5
1
q−ax
is c
urre
nt
0 100 2000
0.5
1
Time step [−]
Tor
que
0 20 400
1
2
3
Time step [−]
Vol
tage
vec
tor
inde
x [p
.u.]
♦♥tr♦ q♥tts
−1 −0.8 −0.6 −0.4 −0.2 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
10.2
0.2
0.2
0.20.2
0.4
0.4
0.4
0.40.4
0.6
0.6
0.60.6
0.60.8
0.80.8
1
111
1.21.21.2 1.41.41.4
1.6
1.6
1.61.8
2
d−axis current [p.u]q−
axis
cur
rent
[p.u
]
q rr♥t ♣♥
r ♦st ♥t♦♥ s ♦♥ t sqr rr♦r
rsts ♥ t ♦♥srt♦♥s ♣rs♥t ♥ ts st♦♥ t tt rt♥♦st ♥t♦♥ ♦ ♥♦t st ♦r r② ♦♣rt♦♥ ♦ t ②♥ ♥ ♣rtr tr♦t♦r ♥s♦tr♦♣② ♦ ♥tr♦ s ♥ t ♦♥tr♦ t♦♥ tt ♥ tr♥ ♦ tt ♣r♦r♠♥ r♥ ②♥♠s t s ♦rt ♠♥t♦♥♥ tt t ♦♥sr ♦st♥t♦♥s ①t ♦♠♣r st② stt ♣r♦r♠♥
♦ ♦ ♥tr♦♥ t s♥② rt♦ s t ♥ t ♦st ♥t♦♥ s♦♥ ♣r♦r♠♥ ♥ ♣rtr ♥r strt ②♥♠s ♥ ♦t ①s t ♦tt♦r tt ♣ss ♦t t rr♥t t♦rs tr rr♥ s ♦s♥ ♠♥ r ♦ ts t♥q s tt t ♥♦ ♦ t s♥② rt♦ s rqr t♥t♥s t♦ ♣rs② ♥♦♥ ss t♦ t s♥♥t r♦♥ strt♦♥tr s r ♥♦t ♦♥st♥t t ♣♥ ♦♥ t rr♥ts ♦r ts rs♦♥ ♣rt♦♠♣♥st♦♥ ♦ rqr t ♣rs ♠♥t rtr③t♦♥ ♦ t ♠♥ ♥r② ♦r♥ ♦♥t♦♥ ♥ s ♦♥ ts r ts s♦ s
♥ ts ♣rt ♦ t ♦r t ♦st ♥t♦♥ s ♦♥ t sqr rr♦r s ♥♦♥sr ②♥♠ tst s♦ tt t ♦s s♥t② ♦♦ ♣r♦r♠♥ rtr♠♦r t ♦s ♥♦t rqr t ♥♦ ♦ t ♠♥ ♣r♠trs s r♥tsr♦st♥ss ♥ts ♣r♠tr ♥rt♥t② ♥ rt♦♥s ♥ ♦♥sr♥ t ♣r♣♦s♦ ts ♣rt ♦ t ♦r t r♣rs♥ts t ♠♦st st ♦
Prt rr♥t ♦♥tr♦
♦s P
t♥ t t ♦ r♦ss strt♦♥ ♥ ♦♥sr♥ s♥s♦ strt♦♥ ♦ tstt♦r ♥♥s s ♣tr t q♥t ♠♦ ♦ t ②♥ ♥ ①♣rss♥ t s②♥r♦♥♦s r♦tt♥ rr♥ r♠ s
∆ik = Ai
k +Buk
A =
[
−R/ℓd · Ts ωLq/ℓd · Ts
−ωLd/ℓq · Ts −R/ℓq · Ts
]
B =
[
Ts/ℓd 0
0 Ts/ℓq
]
r R st♥s ♦r t stt♦r ♥♥ rsst♥ Ld(ikd) ♥ Lq(i
kq ) r♣rs♥t t
♣♣r♥t ♥t♥s ℓd(ikd) ♥ ℓq(ikq ) r t r♥t ♥t♥s ♥ ω s t
tr s♣ ♦r ♦♠♣t♥ss t rr♥t ♣♥♥s r ♦♠tt ♥
trt♥ r♦♠ t rr♥t s♠♣s t ♥st♥t k ♠♦ s ♣rt♦♥ ♦t rr♥ts t t ♥①t st♣ ♥ ♦t♥ ♦♥sr♥ ♥ ②♥ ♠♦ ♥ t ♥t♥s r ♦♥st♥t ♥ ℓd = Ld ♥ ℓq = Lq s ♦s ♦t♥♥t ①♣rss♦♥ ♦ t rr♥t rt♦♥s ∆i
k t♥ t♦ ♦♥st s♠♣s s
∆ik = Ami
k +Bmuk
Am =
[
−R/Ld · Ts ωLq/Ld · Ts
−ωLd/Lq · Ts −R/Lq · Ts
]
Bm =
[
Ts/Ld 0
0 Ts/Lq
]
s ♥r③ ♠♦ s ♦t♥ s ♥ trtr t♦ ♣rt t rr♥t ♦ s②♥r♦♥♦s♠♥s t s ♦rt ♠♥t♦♥♥ tt t ②♣♦ts②s ♦ ♥♦ r♦♥ strt♦♥ s ♠♦rst ♦r P ♠♥s r t t ♦ r♦♥ strt♦♥ s ss s♥♥t t♥ ♥P ♦r ②♥
♥ t rr♥ts ik1|k r ♦t♥ t t② r s t♦ t t
♣♦ss ♥①t rr♥t rt♦♥s ∆iz,k1 t z ∈ Q s
∆iz,k1 = Ami
k1|k +Bmuz
♥ tr♥ ♦s ♣rt♥ t rr♥ts iz,k2|k t ttr r s ♥
t t♦♥ ♦ t ♦st ♥t♦♥
r♥ t r♥t ♠♥ ♦♣rt♦♥s ♠s♠t t♥ t r ♥ t①♣t ♣r♠trs s s t♥ r♥t ♥ ♣♣r♥t ♥t♥s ♦♦r ss ②♥ ♣rs♥ts ♠♥t rtrsts tt r str♦♥② ♥♦♥ ♥r♠♥ t ♠♦ s ♦r t ♣rt♦♥s ♥ ♥st
Prt rr♥t ♦♥tr♦
t ♦ ♣r♠tr ♠s♠t ♥ rt♦♥
♥ ts st♦♥ t ts ♥ ② ♣r♠tr ♠s♠t ♥ rt♦♥ r r② ♥stt t♦ t ♥♦♥♥r ♥tr ♦ P t s ♥♦t ♣♦ss t♦ ♥②s ts ♦r ♥ tr♠s ♦ stt② t trt♦♥ ♠t♦s s ♦r ♥r ♦♥tr♦ s♠s t r♦♦t ♦s ♥ t ♦s ♦♦♣ ♥②ss ♥ t ♦♥sq♥s ♦ ♠♦♣r♠tr ♠s♠t s ♥ ♠♣r② t ② ♥②s♥ t ♣r♦r♠♥ ♦t P ♥r r♥t t♥ ♥rt♥ts s sts ♥ rr♦t ♥ ♣♣t♦♥s ♥♦♥ rr♥t ♣rt♦♥ ♥ ♦♥tr♦ t tr♣s t♦♥rtrs ❬ ❪ ♠t♣s rs ❬❪ ♥ t r♦♥t ♥ ♣♣t♦♥s ❬❪ ♦♠♦ t ♦ t ♦rs ♦♥ tt t ♥rt♥t② ♥ t ♦ rsst♥ s ss r♥t ♦r ♥ ♥♥ ♦♥ t P ♦♣rt♦♥ ♥ ♦tr ♦rs t ♥♥ ♦ rsst♥ ♥ ♥t♥ ♠s♠t s ♥ st s♣rt② ♥t②♥ t♦r♥t ♦r rsst♥ rt♦♥s r rs♣♦♥s ♦r st②stt rr♦rs ♥rt♥ts ♦♥ t ♦ t ♥t♥ r rt t♦ rr♦r ♥ ♦t st②stt♥ ②♥♠s ♣rt rr♥ts ♠t♠t ♥②ss ♦♥ t t ♦ ♣r♠tr♠s♠t ♥ t rr♥t ♣rt♦♥ s ♥stt ♥ ❬❪ ♣rt♦♥ rr♦r s ♠♦♥strt t♦ ♠♦r sr ♥ t t ♥t♥ s ♦r t♥ t ①♣tt♦♥s t ♥ ♦rst♠t ♠♦ ♥t♥ ss t r ♣r♦r♠♥ r♦rs♥ ② ♥ ♥rs r♣♣ ♥ t ♦♥tr♦ rr♥ts
♦r ♣rs♥t ♥ ❬❪ ♦♥srs rsst♥ ♥ ♥t♥ ♠s♠t t♥t s s ♦r t ♣rt♦♥ ♥ t t ♦♥ ♥ s ♦ ♥ ♦ ♦♠♦♥srt♦♥s t♦ ♦♥ t♦ ①t♥ t rsts ♦t♥ ♥ ❬❪ t♦ t s ♦ ②♥♥ ♣rtr ts ♥ ♦ ♠♥s ①ts ♥ ♥r♥t s♥♥t r♦♥ strt♦♥tt ♥ ♥t s t ♠♥ rs♣♦♥s ♦r ♠♦ ♥rs ❲tr r♦♥strt♦♥ ♦rs t t s ♥♦t ♦♥② rt♦♥ ♥ t ♥t♥ s t t♠♥ ♠♦ ts s r♥t Prs② t ①t ♠♦ ♦ t strt ♠♥♥t♥ r♦ss strt♦♥ s t ♦♥ r♣♦rt ♥ ♥ s ♦ r♦♥ ♥rt②t ①t ♠♦ s
t s ♦rt t♥ t ♣rt♦♥ rr♦r ♥ s ♥r ♠♦ s s ♦r trr♥t ♣rt♦♥ t♦ ♦♥tr♦ strt ②♥ s ♠t♦ s ♦t♥ s ♦r ts s♦ ♠♣♠♥tt♦♥ ♥ s t ♦s ♥♦t rqr t ♣rs ♠♥t rtr③t♦♥♦ t ♠♥ rr♦r ♥ ♦♠♣t s t r♥ ♦ ♥ ♦t♥♥
ε = ∆A ik +∆Bu
k
∆A = A−Am =
R(ℓd−Ld
ℓdLd
)
ωme
(LdLq−Lqℓd
ℓdLd
)
ωme
(Ldℓq−LdLq
ℓqLq
)
R(ℓq−Lq
ℓqLq
)
Ts
∆B = B−Bm =
(Ld−ℓdℓdLd
)
0
0(Lq−ℓq
ℓqLq
)
Ts
r ℓ ♥ L st♥ ♦r t t r♥t ♥ ♣♣r♥t ♥t♥ L s s♦r t ①♣t ♦ t ♣♣r♥t ♥t♥ s ♥ t ♥r ♠♦ ♦r t rr♥t ♣rt♦♥ rr♦r ①♣rss ♥ ♥s ♦t t ♦♠♣♦♥♥t ♥ ②t r♥ t♥ t ♥r ♥ t strt ♠♦ ℓ 6= L ♥ t ♦♠♣♦♥t♥t
Prt rr♥t ♦♥tr♦
♥ ② t ♣♣r♥t ♥t♥ ♠s♠t L 6= L rsts ♦t♥ ♥ ❬❪ r st ♥ s ♦ ②♥ ♦♥sr♥ s♠ rt♦♥ ♥②ss ♥ ♣rtr t ♦♥srt♦♥ss ♠ ♥ ❬❪ ♥ ①t♥ ♥r③♥ t ♠♦ ♦t ♦r♥ ♣♦♥t ♦♥sr♥ ♥r ♠♥ t ♥t♥ q t♦ t r♥t ♥t♥s♥ t ♦♥sr ♦♣rt♥ ♣♦♥t
ts ♦ r♦♥ strt♦♥ r r ♥ r t ♣r♦r♠♥ ♦ ♥ ♥ strt ♠♥ r s♦♥ ♥ ♦t ss t ♥rs ♠♦ ss t♦ ♣rt t rr♥ts tst s ♣r♦r♠ t ♦ r♦t♦r ♥ ♣r♦♥ t♦rr♥t st♣s ①♣t ♥rs r♣♣ s ♥t ♥
0 1000 2000−0.8
−0.6
−0.4
−0.2
0
d−ax
is c
urre
nt [p
.u.]
Actual Ref
0 1000 20000
0.5
1
q−ax
is c
urre
nt [p
.u.]
−1 −0.5 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
d−axis current [p.u]
q−ax
is c
urre
nt [p
.u]
♠♥
0 1000 2000−1
−0.5
0
d−ax
is c
urre
nt [p
.u.]
Actual Ref
0 1000 20000
0.5
1
q−ax
is c
urre
nt [p
.u.]
−1 −0.5 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
d−axis current [p.u]
q−ax
is c
urre
nt [p
.u]
trt ♠♥
r ♦♠♣rs♦♥ t♥ ♥r ♥ strt ♠♥
t s r r♦♠ tt s♥ ♥rs ♠♦ t♦ ♦♥tr♦ strt ♠♥♠♣s ♥ st♦rt rr♥t ♦r♠ rs♦♥ s tt s♥ t ①♣t♦r s t♦t② r♥t r♦♠ t t ♦♥ t ♣rt rt♦r ♦s ♥♦t ♦r♣r♦♣r② s ♦♥sq♥ t ♦t t♦rs ♦s♥ ② t ♦st ♥t♦♥ ♠♥♠③t♦♥r r♥t r♦♠ t s s♦s t ♦ t rr♥t ♣rt♦♥r② ♥ t ♦t t♦r st♦♥ s ♥ s♥ t ♣rt♦♥ ♥ tsqr ♠rrs s ♣rs ♥ r ♥ s ♦ ♠♥ ♥ t ♦tr ♥t s r tt t ♣rt♦♥ t t strt ♠♥ s ♥♦t ♣rs ♥ ♣rtrt rr♥t rt♦♥s r ②s ♥rst♠t s ♦♥sq♥ t ♦t t♦rst♦♥ s r♥t ♥ t t♦ ss ♥ t ♦♥sr ①♠♣ t t ♠♥t ♣rt rt♦r ♦♦ss ♥①s 3 ♥ 5 t♦ r ♠♦r ♥t s trt rr♥t ♦r♠r ♦s ♥ ♥rs t ttr rs ♦ t q①srr♥t ♥ stt ♥① 0 s s t♦ ♥rs ♠♦r ♣♦st s t ①s s♦② r♥ t qrtr rr♥t ♦♥sr♥ t strt ♠♥ t♦♦♣♣♦st t♦rs 1 ♥ 4 r s t♦ ♥rs ♥ r t rt rr♥t rs♣t② r♥ t qrtr rr♥t ❱t♦rs 2 ♥ 3 ♦s ♥rs♥ t q①s
s s♦♥ ♥ t sq♥ ♦ ♦t t♦rs s ♦♠♣t② r♥t ♥ tt♦ ss s ♠♣s r♥t ♥♠r ♦ st♥ s♦s t ♥♠r ♦st♥ t st♣ ♥ t t♦t ♥♠r t ♥ s♥ tt t st♥ ♣ttr♥t t ♠♥ ♥♦ ♦♥② ♦♥ ♥ t ♦♥trr② ♥ s ♦ strt♠♥ t st♣ ♦r s ♥ ♥♦ s ♦♥sq♥ t t♦t ♥♠r♦ ♦♠♠tt♦♥s s r ♥ t s ♦ strt ♠♥ t s ♦rt ♠♥t♦♥♥ ttt ttr ♦♥srt♦♥ s ♥♦t ♥r t s ♥♦t sr tt s♥ t ♥♦♠♥ ♠♦ ♥
Prt rr♥t ♦♥tr♦
300 305 310 315 320 325−0.72
−0.7
d−ax
is [p
.u.]
300 305 310 315 320 3250.69
0.7
0.71
q−ax
is [p
.u.]
300 305 310 315 320 325012345
Vol
tage
vec
tor
Time step [−]
♠♥
300 305 310 315 320 325
−0.8
−0.7
−0.6
d−ax
is [p
.u.]
300 305 310 315 320 325
0.65
0.7
0.75
q−ax
is [p
.u.]
300 305 310 315 320 3250
1
2
3
4
Vol
tage
vec
tor
Time step [−]
trt ♠♥
r ♦♠♣rs♦♥ t♥ ♥r ♥ strt ♠♥
t ♣rt♦♥ ♦♥tr♦♥ strt ♠♥ ♦s ♥♦t ♠♣② ♥ r ♥♠r ♦♦♠♠tt♦♥s ♥ ♥r t st♥ ♣ttr♥ ♦t♥ t ♥ ①t ♠♦ ♥ t♥ ♠♣rs ♦♥ s r♥t tr tt t t♦t ♥♠r ♦ ♦♠♠tt♦♥s ♣♥s ♦♥t ♥ ♦ ♦♣rt♦♥ ♦♥ t ♦st ♥t♦♥ tt ♦ ♥ tr♠ t♦ ♣♥③ t♥♠r ♦ st♥s ♥ ♦tr t♦rs
0 500 1000 1500 20000
1
2
3Commutations
Eac
h st
ep [−
]
0 500 1000 1500 20000
500
1000
1500
Time step [−]
Tot
al [−
]
♠♥
0 500 1000 1500 20000
1
2
3Commutations
Eac
h st
ep [−
]
0 500 1000 1500 20000
1000
2000
3000
Time step [−]
Tot
al [−
]
trt ♠♥
r ♦♠♣rs♦♥ t♥ ♥r ♥ strt ♠♥
♦r P
♦♥♣t ♦ P s ♥ rst ♣rs♥t ♥ ❬❪ ♥ t♥ ♠♣r♦ ♥ ❬❪ ♠ ♦ ts ♦♥tr♦ s♠ s t♦ r t ♥♦ ♦ ♣r♠trs rqr ② P♥ t♦ ♠♣r♦ t ♣r♦r♠♥ ♦ rr♥t tr♥ ♥ ♥r t t ♦ str♦♥ r♦♥strt♦♥ ♥ ♦ ♦rs ♥r t ts ♦ ♦tr ♣♥♦♠♥ t♥ t ♣rt♦♥
s ♦ P ♥ sr s ♦♦s tr ♦t t♦r s♦t♣t r♦♠ t ♦st ♥t♦♥ ♠♥♠③t♦♥ t rst♥ rr♥t rt♦♥s ♥ t t♦①s r r♦r ♥ st♦r ♥ t♦ r♥t ❯s s r♣rs♥t ♥ ♥ts ② t s ♦ ∆izd ♥ t s ♦ ∆izq r ♥ t♦ t ♦rrs♣♦♥♥♥rtr stt uz t z ∈ Q rr♥t ♣rt♦♥s ♥ r s② ♦t♥② ss♥ t s st♦r ♥ t ❯s ♦rrs♣♦♥♥ t♦ t ♦♥sr ♦tt♦r s ♥ t s ♦ ∆i
z r ♦t♥ r♦♠ ♠sr♠♥ts t②
Prt rr♥t ♦♥tr♦
♥r♥t② ♦♥t♥ t ♥♦r♠t♦♥ ♦t t r ♦r ♦ t ♠♥ ♥ ♦♥t♦♥s♥r t♦ t t ♦♥s
u1 u
4
iΔ1
iΔ4
iΔ1
iΔ4
t♦r♥ rr♥t tt♦♥
iΔ1
iΔ4
iΔ7
iΔ6
iΔ5
iΔ3
iΔ2
iΔ0
♦r ♣rt♦♥
r ♣rs♥tt♦♥ ♦ t ♦r ♣rt♦♥
t♦ t s ♣r♥♣ s t s♠ r♥t ♠♣♠♥tt♦♥ s s ♥ ❬❪ ♥♥ ❬❪ ♦r♠r s r♣rs♥t ♥ r♥ s♠♣ t♠ t♦ rr♥t♠sr♠♥t r ♣r♦r♠ ♥ ♦rr t♦ ♦t♥ t ♦rrt r♥ s♥ ♥ t♦♦ t rr♥t s♣s ♥ ② t sts t rst rr♥t tt♦♥ s ♥s♦r t t st♥ stt s ♣r♦r♠ t s♦♥ tt♦♥ s ② ② ① t♠ ♣r♦ tr t t st♥ s ♣r♦r♠ s ♥ s♥ ♥ t strt♥ ♣♦♥t ♦ t st♥ ♥tr s ♥ ♥ tt ♦ t ♦rrs♣♦♥♥s♠♣♥ ♥tr ② ① ♠♦♥t ♦ t♠ s ♦♥sq♥ t stt♦r rr♥t t t♥ ♦ t t st♥ ♥tr ♠sr ♥ t t s♠♣♥ ♥trs rst t rr♥t r♥ ♦rrs♣♦♥♥ t♦ t t st♥ ♥tr ♥♥♦t t ♥ t t s♠♣♥ ♥tr ♥ ♦tr ♦rs t rr♥t r♥♦rrs♣♦♥♥ t♦ t ♣r♦s st♥ stt ♥ t ♥ t ♣rs♥t s♠♣♥♥tr
Current sampling
Ts
rr♥t r♥ tt♦♥ t♥q
1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6
ia(t) iak-2,1
iak-2,2
iak-1,1
iak-1,2
iak,1
iak,2
uk-3 uk-2 uk-1 uk
1: Read current and rotor position
2: Output trigger signals of new switching state
3: Read current
4: Calculate and update current difference
5: Estimate future current difference
6: Compute current command
7: Predict stator current, calculate cost and choose next switching state
♠ ♦♣rt♦♥s
r P rr♥t rt♦♥ tt♦♥ ♥ ♦♣rt♦♥ s ♥ ❬❪
rr♥t tt♦♥ t♥q r♣rs♥t ♥ rqrs t♦ s♠♣s t ♥tr ♦ ♦ t tt♦♥ ♦ s♣s t s♦♥ rrr♥t ♠sr♠♥t s② ♦ ① ♥tr ♦ t♠ tr t ♥ ♦♥t♥ ♠♦ s ♣♣ ♦rs♥ tr r ♥♦ ♥r ♥s rr♥ t ♦ ♦ t t♠ ② t r②♣♥s ♦♥ ♠♣r ①♣r♥ ♥ t ♥rtr rtrsts ♠♣r♦♣r ♦ ♦t ttr ♦ rst ♥ ♣r♠♥♥t ♠s t♦ t r s②st♠ ♦ ♥② rs♦ tsss s♥ rr♥t tt♦♥ ♠t♦ s ♥ s ♥ ❬❪ ♥ s♠♣♥ ♣r♦t rr♥t tt♦♥ s ♣r♦r♠ ♠♠t② ♦r ♦♥t♥ ♠♦ s ♣♣ s
Prt rr♥t ♦♥tr♦
s♦♥ ♥
Current sampling
Ts
rr♥t r♥ tt♦♥ t♥q
1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
ia(t) iak
iak1|k
iak2|k
uk-1 uk uk+1 uk+2
1: Measure stator current
2: Output new switching state
3: Calculate and update old current variation
4: Estimate current variation
5: Compute current command
6: Compute current command
7: Execute current prediction, calculate cost and choose next switching
state
♠ ♦♣rt♦♥s
r P rr♥t rt♦♥ tt♦♥ ♥ ♦♣rt♦♥ s ♥ ❬❪
❲tr ♥ ♥rtr stt s ♥♦t s ♦r sr s♠♣♥ ♥trs t rtrr♥t rt♦♥s r ♥♦t ♣t ♦r ♦♥ t♠ ♥ ❬❪ t t♦rs rr t♦ ts sss t st♥t♦♥ ♦ rr♥t rt♦♥ ♣t ♥ t② ♣r♦♣♦s s♠♣ s♦t♦♥ t♦rss t ♥ ♣rtr ♦t t♦r s ♥♦t ♥ ♣♣ ♦r ♥♠r Nold ♦♦♥st st♣s ts t♦r ♦r t t ♥①t ♥tr ♥ ts ② t ♦t♣t♦ t ♦st ♥t♦♥ ♠♥♠③t♦♥ s sr ♥ t ♠♥ t ♥♦♥♦♣t♠♦t t♦r ♦♠ ♦♥srt♦♥s ♦♥ ts s♣t ♥ tr ♥ ts st♦♥
t♥ss ♦ t ♠♦r ♣rt♦♥ s r② s♦♥ ♥ rP ♥ t P ♣r♦♣♦s ♥ ❬❪ r ♦♠♣r ♦♥sr♥ t strt ♠♦♦♥ t ②♥ rsts r ♦t♥ t trs♦ Nold = 200 ♣♦ts s♦t rsts ♦t♥ t s♣ st♣ tst t rr♥ ♦ 100 ❬r♣♠❪ strt♥ r♦♠st♥st s r② sr s t P ♣rs♥ts ♥ ♥rsrr♥t r♣♣ t♦ t ♠♦ ♥q② ♥ t ♦♥trr② s♥ t P ①♣♦tst rr♥t ♠sr♠♥ts t ♥♦r♠t♦♥ ♦♥ t ♠♦ r ♦t♥ r♦♠ tt ♠♥ t ①ts ttr rr♥ tr♥ ♥ ♦ r♣♣ s ♥ t s♦ ♥♦♥ strt ♠♥ s♠ ♦♥srt♦♥s ♥ ♠ ♦♥ t st♥ t ♠♦r ♣rt♦♥ ♣rs♥ts r♥t st♥ ♣ttr♥ tt ♥ ts tst ♦s♥ ♦r ♥♠r ♦ ♦♠♠tt♦♥s s s♦♥ ♥
0 200 400 600 800 1000−1
−0.5
0
d−ax
is c
urre
nt [p
.u.]
MB MB−ref MF MF−ref
0 200 400 600 800 10000
0.5
1
q−ax
is c
urre
nt [p
.u.]
Time step [−]
rr♥t ♦r♠s
0 200 400 600 800 10000
200
400
600
800
1000
1200Commutations
Time step [−]
t♥
r ♦♠♣rs♦♥ t♥ P ♥ P t Nold = 200
s ♥ s♥ ♥ t rr♥t ♦r♠s ♦ t P ♣rs♥t s♦♠♥t s♣s s r ♥ ② t s♠♣ ♥tst♥t♦♥ s♦t♦♥ ♣r♦♣♦s ♥ ❬❪
Prt rr♥t ♦♥tr♦
♥ r♣♦rt t rr♥t ♦r♠s ♦t♥ t Nold = 100♥ Nold = 50 rs♣t②
0 200 400 600 800 1000−0.8
−0.6
−0.4
−0.2
0d−
axis
cur
rent
[p.u
.]
0 200 400 600 800 10000
0.5
1
q−ax
is c
urre
nt [p
.u.]
Time step [−]
Nold = 100
0 200 400 600 800 1000−0.8
−0.6
−0.4
−0.2
0
d−ax
is c
urre
nt [p
.u.]
0 200 400 600 800 10000
0.5
1
q−ax
is c
urre
nt [p
.u.]
Time step [−]
Nold = 50
r st t r♥t s ♦ t trs♦ Nold
♥tr♦♥ ♥♦♥♦♣t♠ ♦t t♦rs ♦s ♦♥ t ♦t t♦r st♥t♦♥ t t t s♠ t♠ ss s♣s ♥ t rr♥t ♦r♠ ♠♣t ♦ts s♣s ♣♥s ♦♥ sr s♣ts t r♦t♦r ♣♦st♦♥ ♥ t s ♦ ♥t♥ ♦r t s ♦ rt② t rsts ♦t♥ t Nold = 50 r ♥②s ♥ t♥ ♣rtr t♠ ♥tr ♦ 100 st♣s s ♦♥sr ♥ ♣♦st♦♥ ♦t q①s t t ♥♥♥ ♦ ts ♥tr s s♦♥ ♥ r♥ t ♥♦r♠♦♣rt♦♥ t ♣rt rt♦r ♦t♣ts t rsts ♦ t ♦st ♥t♦♥ ♠♥♠③t♦♥Prs② ♦♥sr♥ t r♦t♦r ♣♦st♦♥ t tr ♦♣t♠ t♦rs
❼ ♥① ♦s s♠ rt♦♥ ♦ t rt rr♥t ♥ ♥rss t q①srr♥t
❼ ♥① rss t ①s rr♥t ♦t♥♥ ♥t rt♦♥ ♦ tqrtr rr♥t
❼ ♥① t s ♦s♥ t♦ ♦t♥ s♠ rt♦♥s t♦rs t ♦r♥ ♦ ♦t trr♥ts s ♥ t ♣♦♥t ♦ ♦r s ♦s t♦ t rr♥
ttr ♥ t rt rr♥t rt♦♥s r ♣♦♥t t r ♦♦r ♥ ♥ t s♦♥ ♣♦t ♦ ♥ ♥ ♥ t♦♥ t♦ t ♦st ♥t♦♥♠♥♠③t♦♥ t ♦rt♠ ♥tr♦s r♥t ♥①s t♦ r♥t rq♥t ❯♣t rst ♣♦t ♦ s♦s t ♦ t t t♦rs t♠ t♦r s ♦♥ t♦ ♦ ♥ t s ♥♦t ♥ ♣♣ ♦r 50 ♦♥st st♣s t♦rt♠ ♣♣s t t ♥ s♥ tt ♥ ts s t st♥t t♦rs r 1 2 ♥ 3r♥ t ♦♦r tr t ♣♣t♦♥ ♦ ♦♥ ♦ ts tr ♥♦♥ ♦♣t♠ t♦rs ♥♦tr ♥♦♥♦♣t♠ t♦r s ♣♣ t♦ r♥ t rr♥ts t♦ t rr♥s t♦r ♥① 5 ♥ ts ①♠♣ t ♦♦r s ♥♦t ♦r ② t ♥tst♥t♦♥♦rt♠ s♥ t ♦s ♥♦t r t trs♦ Nold t t s ♦s♥ ② t ♦st ♥t♦♥t♦ r♥ t ♣♦♥t ♦ ♦r t♦ t rr♥ ♥ ♦tr ♦rs ts ♦t t♦r r♣rs♥ts ♥ ♦♣t♠ ♦ ♥ t ♠♦ stt♦♥ tr t ♣♣t♦♥ ♦ ♦ t♦rst t s ♥♦t t♥ s ♦♣t♠ r♥ t ♥♦r♠ ♦♣rt♦♥ s ①♠♣ s♦s ttr♥ t st② stt ♦♣rt♦♥ t ♥tst♥t♦♥ ♠t♦ ♣r♦♣♦s ♥ ❬❪ ♦rs
Prt rr♥t ♦♥tr♦
760 780 800 820 840 860−0.4
−0.35
−0.3
−0.25
−0.2
d−ax
is c
urre
nt [p
.u.]
760 780 800 820 840 860
0.32
0.34
0.36
0.38
0.4
q−ax
is c
urre
nt [p
.u.]
Time step [−]
rr♥t ♦r♠s
760 780 800 820 840 8600
20
40
60
Age
[−]
0 1 2 3 4 5 6 7
760 780 800 820 840 8600
2
4
6
Vol
tage
vec
tor
inde
x [−
]
Time step [−]
♥ ♦t t♦r ♥①
r t ♦ t P ♦♣rt♦♥
4
5 6
0 1
23
d
q
dq-axes
stagnantnew after stag.optimal
r t ♦ t P ♦♣rt♦♥
♦t t♦rs tt r ♥♦t ♦s♥ ② t ♦st ♥t♦♥ ♠♥♠③t♦♥ ♥ tt ♦ ♥ ♥ ♦♣♣♦st♦♥ t t ♦♣t♠ t♦rs
♦ ♦ Nold s s ♦♥ ♠♣r ♦♥srt♦♥s ♥ t r♣rs♥ts tr♦ t♥ t ♦♠♣♥ t t ♦st ♥t♦♥ ♠♥♠③t♦♥ ♥ t ❯ ♣t♥rq♥② s rt② ♥ t t ♣rt♦♥ r② ♥ ♥ tr♥ t t stt② ♦ t ♦♥tr♦ s♠ ♦r t trs♦ t ♠♦r rq♥t t ♣t ♥t r t ♥♠r ♦ ♥♦♥ ♦♣t♠ t♦rs ♣♣t♦♥s r t trs♦t r t ♦r♥ t t ♦st ♥t♦♥ ♦t♣t t t ♦r t ♣rt♦♥r② s ♦♥srt♦♥s r qtt② r♣♦rt ♥ t ♣♦t ♦
s♠♥ t ♦♥srt♦♥s ♥ t rsts s♦♥ s♦ r t t ♣♦ss t♦ ttt t ♥tst♥t♦♥ s♦t♦♥ ♣r♦♣♦s ♥ ❬❪ s ♠♥ r t ♣♣t♦♥ ♦ ♥♦♥ ♦♣t♠ ♦t t♦rs ♥ r② t♠ tt ♥ ♥rtr stt s♦♥ t♦ ♦ t ♦rt♠ ♣♣s t sr♥ t ♦t♣t ♦ t ♦st ♥t♦♥♠♥♠③t♦♥ ss t ♦ ♦ Nold r♣rs♥ts tr ♦ t♥ t qt② ♦t ♣rt♦♥ ♥ t ♦♠♣♥ t t ♦st ♥t♦♥ ♠♥♠③t♦♥ ♥ ♦tr ♦rs s♠ ♦ Nold ♠♣s rq♥t t ♣t ♥ r② ♦s st♥t♦♥ ♦t♥② ♣♣②♥ ♥♠r ♦ ♥♦♥ ♦♣t♠ ♦ts ♥ t ♦tr ♥ ♦♦s♥ r ♦ Nold ♦s ♣rsr♥ t ♦♣t♠ ♦♥tr♦ t t rr♥t ♣rt♦♥♦ sr② ♦rs♥ ♥② Nold s ♦♥ s ♦♥ ♠♣r ♦srt♦♥s ♥ts ♦ ♥♦t r♥t t stt② t r② ♦r♥ ♦♥t♦♥
Prt rr♥t ♦♥tr♦
Nold1
Optimality
Update frequency,
prediction accuracy,
stability
r tt r♣rs♥tt♦♥ ♦ t ♦ ♦ Nold
♦♥s♦♥
♥ ts st♦♥ t ♥r Prt rr♥t ♦♥tr♦ s sr ♥ ♥②s ♦♠♦♥srt♦♥s ♦t t ♦ ♦ t ♣r♦♣r ♦st ♥t♦♥ ♥ s ♦ s♥t ♠♥s ♥ ♣r♦ ♥ ♣rtr t t ♥ ② t r♦t♦r ♥s♦tr♦♣② ♥♥②s r♥ tr♥s♥t ♦♣rt♦♥s ♥♦♥ rr♥t st♣s t ♠♥ strt♣t s ♥ s♦♥ tt t ♦st ♥t♦♥ s ♦♥ t s♦t rr♦r ♦ rst ♥ s ♦r ♥ t ♣r♦rt② t♦ t ①s t t ♦r ♥t♥ rst stt r♥ t rst st♣s t rr♥t ♦♥ t ①s t t ♦r ♥t♥ st②rs ts rr♥ t ♦tr ♦♥ r♠♥s ③r♦ ♦tr t s tt t t♦rq♥ ♥ r♥ ts ♥tr rst♥ ♥ s♦r s♣ rs♣♦♥s t s ♥ s♦♥tt ♥tr♦♥ t♥ t♦r q t♦ t s♥② rt♦ ♦s ♦r♦♠♥ ts♦r ❲t ts s♦t♦♥ t s s ♦♠♣♥st ♥ t rr♦r ♦♥ t t♦ ①sr r ♥ ♥ ② ♠♥ r ♦ ts s♦t♦♥ s tt t rqrst ♥♦ ♦ t s♥② rt♦ t s ♦ t ♥t♥ ♥ ②♥ tss r str♦♥② ♣♥♥t ♦♥ t rr♥t t♦ r♦♥ strt♦♥ ♦r ts rs♦♥s♥ ♦♥st♥t t♥ t♦r ♦♥sr♥ t ♥♦♠♥ ♣♣r♥t ♥t♥s ♦♥♦t st ♦♠♣♥st♦♥ r♥ t ♠♥ ♦♣rt♦♥s rsts ♦t♥t t ♦st ♥t♦♥ s ♦♥ t sqr rr♦r s♦ s♥t② ♥♦r ♦♥ ♦t ①s ss t ♦s ♥♦t rqr t ♥♦ ♦ t ♠♥♥t♥s ♦r ts rs♦♥s ts ♦st ♥t♦♥ s ♥ ♦s♥ ♦r ts ♣rt ♦ t♦r
♦♣rt♦♥ ♦ t ♣rt rt♦r s ♥ sr ♦♥sr♥ t t♦st♣s ♣rt♦♥ rqr t♦ ♦♠♣♥st ♦r t ②s ♥tr♦ ② t t ♠♣♠♥tt♦♥ tr tt ♦t t ♦s ♥ t ♦r ♣rt♦♥ ♥♣rs♥t ♥ t ♦r♠r s t t ♦ ♣r♠tr ♠s♠t ♥ rt♦♥ ♥♥stt ♥ ♣rtr ♥ ②♥ ♣♣t♦♥s r♦♥ strt♦♥ s ♥ r♦♥ss t ♠♥ rs♣♦♥s ♦r ♠♦ ♥qs
Chapter 3
♦♥ st♥t♦♥ ♥ ♦r
Prt rr♥t ♦♥tr♦
t♦ ts ♥tr ♦r ♣rt♦♥ s ♥r♥t② r♦st t♦ ♣r♠tr rt♦♥s♥ t rr♥t rt♦♥s s ♦r t ♣rt♦♥ r rt② ♦t♥ r♦♠ ♠sr♠♥ts t ♣rt rt♦r ♣rs② ♥♦s t t ♠♦ r s♣t ♦r t♣r♦♣r ♦♣rt♦♥ ♦ t r s t ❯ ♣t ♥ ♣rtr ♣rs ♣rt♦♥ rqrs tt t ttr s ♣r♦r♠ rq♥t② ♦trs t s st♦r ♥ t ❯ r♥♦t r ♥②♠♦r ♥ t② ♦ t♦ sr ♦rs♥♥ ♦ t r ♣r♦r♠♥♥ ♥ t ♦rst s t ♦ss ♦ ♦♥tr♦ ❲tr ♦t t♦r s ♥♦t ♣♣ ♦rsr ♥trs t rt rr♥t rt♦♥ s ♥♦t ♣t ♥ t ♣rt♦♥ s♦♥ ts s ①tr♠② r♥t r♦♠ t t ♦r s ♣♥♦♠♥♦♥ s rrr s st♥t♦♥ ♦ t ♦t t♦r ♥ ts ♣tr ♥ ♥②ss ♦ t ts ♦t ♦t t♦r st♥t♦♥ s ♣r♦ tr tt ♥♦ ♥t st♥t♦♥ ♠t♦ s♣r♦♣♦s ♥ tst
♥tr♦t♦♥
P s ♥ ♦♣ ♥ ❬❪ ♥ ❬❪ ♥ ♠♣♠♥t ♦r t rr♥t ♦♥tr♦♦ ♥ P ♥ ②♥ rs♣t② s ♦rs s♦ t rt ♥ts ♥② t ♦r ♣rt♦♥ ♥ t ♦♥sq♥t ♥♥ ♣r♦r♠♥ ♦ t r rsts ♠♦♥trt t st② ♥ t t♥ss ♦ ts ♥ ♦ ♦♥tr♦s♦♥ tt P s ♣r♦♠s♥ t♥♦♦② t♦ r♥t ♥s t♦ ts ♥r♥t r♦st♥ss t♦ ♣r♠tr rt♦♥s ts s♣r ♦ ♣r♦ rt ♥ts ♥♥str ♣♣t♦♥s rst ♦ t ♦ ♣♣t② ♦ ♣rt ♦♥tr♦ ♦ rt② ①t♥ s♥ t s rr♥t② ♠t ② t ♠♥♥ ♥ ♦ t ♣rs♥♦ ♦ t ♠♦ ❲t♦t t trt♦♥ ♣rt s♠s ♦s tr t♥ss ♥ ♦♠ ♥st ♠♥ ♦tr sts ♠t♦s ♥r ♦♥tr♦t rst ♦ ♥ ♥str② ♥ ♥ ♣rtr ♥ rs ♣♣t♦♥s rtr♠♦r ts♥ ♦ ♥② rqr♠♥ts ♦♥ t ♥♦ ♦ t ♠♥ ♣r♠trs s sr♦tst♥♥ ts t rsts ♥ ♥ s ♦ ♠♣♠♥tt♦♥ s♥ t ♦♥② rqrs rr♥t♠sr♠♥ts ♥ t ♦s ♦♠♣① ♥ t♠ ♦♥s♠♥ ♦♠♣tt♦♥s s♥
♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦
s ♥ ♦rtss ♥♦ tr② t♥♥ ♦ ♥s ♥ trs♦s s ♦tr trt♦♥ ♦♥tr♦s♠s ♣r♦♣♦rt♦♥♥tr rt♦rs ♦srrs ♥ st♠t♦rs t s♥ tr♥ r♣rs♥ts ♥ ♥ ♠♦r ttrt tr t ♣♦sst② t♦ s ♥ t♦♣♠♥t ♦ ♥ ♥rs r ♠♥s t ♦♠♣t② r♥t rtrsts ♦ ♦♥tr♦ t♦t ♥♥ t ♦♥tr♦ ♦rt♠ ♥ t♦t ♦r♦ss♦♠♠ss♦♥♥ ♦r ♥tt♦♥ ♣r♦rs
s♣t t rt ♣♦t♥tts tr P s ♥ ♣rs♥t ♥ ❬❪ ♦♥② ♦♥♥♦tr ♦r ♣r♦♣♦s ② t s♠ t♦rs s ♥ ♣s ♦♥ t s♠ t♦♣r② rtr rsr s ♥ t♦ ♦♣ ts ♥ ♦ ♦♥tr♦ ♥ t♦ ♠ t ♦♠♣tt
rst r s♣t tt s t♦ rss s t s♦ ♦t t♦rst♥t♦♥ s ♣♥♦♠♥♦♥ ♦rs ♥ ♦ rr♥t rt♦♥ s ♥♦t ♣t♦r sr st♣s t rt ♦t t♦r s ♥♦t ♣♣ ♥ tt t♠ ♥tr♥ t rr♥t rs♣♦♥s ♥r rt♥ ♥rtr ♦t♣t r♠t② ♥s ♥♦t s♥ ♥ ♠♣t t t ♠♥ stt ♥ ♥rq♥t ♣t ♦ t ❯ss ♦♠♣t② r♦♥ ♣rt♦♥s tt r t ♠♥ ♥ ♥①♣t ♦♣rt♦♥s st♥t♦♥ s ♥♦t ♣r♥t ♦r r② s♦ t ♣r♦r♠♥ r r② ♦rs♥♥ ♥ ♣ t t t♦t ♦ss ♦ ♦♥tr♦
❲ ts s♥♥t ss s ♥♦t ♥ ♠♥t♦♥ ♥ ❬❪ t s ♥ rsst s♠♣ s♦t♦♥ ♥ ❬❪ s r② sr ♥ ♣tr ts ♠t♦ ♣rs♥tss♦♠ ♥t rs t ♦rs ♥♦♥ ♦♣t♠ t♦rs ♥ t rqrs t t♥♥ ♦ trs♦ tt s ♠ t ♠♣r ♦♥srt♦♥s ♥ tt ♦ ♥♦t st ♦rr② ♦♣rt♦♥
♥ ts ♣tr ♥♦ ♥tst♥t♦♥ s♦t♦♥ s ♣rs♥t t ①♣♦ts t rt♦♥s♣ t♥ t rr♥t rt♦♥s ♥ ② t t ♥rtr stts s ♦♥ ts tss t tr ♠♦st r♥t s st♦r ♥ t ❯ t♦ r♦♥strt t ♦tr ♦r tt♦ ♥ t♦rs s t s♠ rr♥t rt♦♥ ♥ ♦♥t♥♦s ♣t
♥②ss ♦ st♥t♦♥ ts
P s♠s ♣rs♥t ♥ ❬❪ ♥ ❬❪ r② ♦♥ t rr♥t rt♦♥s st♦r ♥t ❯ ♥ ♣rtr r② t♠ tt ♦t t♦r s ♣♣ t rt rr♥trt♦♥ s ♣t s ①♣♥ ♥ t ♥①t st♦♥s t rr♥t rt♦♥ ♥ ② rt♥ ♦t t♦r ♣♥s ♦♥ t ♠♥ stt ♦♥ t rr♥t ♦♥ t♣♦st♦♥ ♥ ♦♥ t s♣ ♥ ♣rtr ♦t ts ♠♣t ♥ s♥ ♣♥ ♦♥ tttr s ♠♥s tt r② t♠ ♦t t♦r s ♥♦t ♦s♥ ② t ♦st ♥t♦♥♦r ♥♠r ♦ st♣s t rt rr♥t rt♦♥ s ♥♦t ♣t ♥ t ♦♥srt♠ ♥tr ♥ ts s t s ♦♥t♥ ♥ t ❯ ♦ s♥♥t②r♥t r♦♠ t t ♦♥s r② t ♦♥r t t♠ ♥tr ♥ t ♣ts ♥♦t ♣r♦r♠ t r t rr♦r t♥ t ❯ ♥ t t rs♣♦♥s ♦t ♠♥ ♥t ts ♥ ② ts rr♦rs ♦♥ t ♦♥tr♦ ♣♥ ♦♥ t♠♥ ♦♣rt♦♥
♦r t s ♦ rt② ♥ ①♠♣ ♦ t ts ♦ st♥t♦♥ s ♣r♦ ♥ s♠t♦♥ s rr ♦t ♣r♦♥ s♣ st♣ t t strt♣ t ♣rt♦♥s s ♦♥ t ♥♦♠♥ ♠♦ ♥ tr 25 st♣s t ♠♦r ♣rt♦♥ s strt
♥②ss ♦ st♥t♦♥ ts
♠♦s strt♣ ♦s ♥ t ❯ t ♦rrt s ♦ rr♥t rt♦♥tr tt t P s t♦ tr t rr♥t ♥ t s♣ rr♥s t ♦rst♣ k = 2000 t ♦♥tr♦ s ♦st t rr♥t rr♦rs strt ♥rs♥ ♥ t s♣r② rs ③r♦ rst ♣♦t ♦ s♦s tt tr ♥st♥t k = 1000t ♣rt rt♦r strt ♦♦s♥ ♦♥② ♥①s 0 5 ♥ 6 s ♠♥s tt ♦♥② ttr rt rr♥t rt♦♥s r ♣t ♥ t ❯ t ♦tr r r♠♥s♦♥st♥t ♥ q t♦ t st ♣t t t ♥ ♦♥② ♥① 0 ♥ 6 r ♦♣t r♦♠t ♦st ♥t♦♥ ♠♥♠③t♦♥ ♥ t ♦♥tr♦ s ♥♦t t♦ r♥ t rr♥s rs♦♥ ♥ ♦♥ ♦♥sr♥ t s ♦ t rr♥t rt♦♥s ♥ t t♦ ❯♣♦tt ♥ t ♥ s♥ tt tr rt♥ ♥st♥t ♥ ♦t t ❯ t♠♦st ♦ t rr♥t rt♦♥s t s♠ s♥ ♥ ♣rtr ♥ t ❯ ♦ t rt①s t s r ♣♦st t t ♦♥ rt t♦ ♥① 6 ♠r② ♥ t q①s t st♦r s r ♣♦st t ♥①s 0 ♥ 1 ♦r ts rs♦♥ t ♣rtrt♦r ss t s st♦r ♥ t ❯ t♦ t t ♦st ♥t♦♥ ♥ t♦ rt rr♥t rr♦rs t ♥♦ ♥ ♣rtr t ♦♦ss ♥① 6 t♦ ♣r♦ ♥trt♦♥ ♦ t rt rr♥t ♥ 0 t♦ ♣r♦ ♥t rt♦♥ ♦♥ t q①s t ♦tr r♦♥ s st♦r ♥ t ❯ s② t ♦st ♥t♦♥ t♦♥ ♠♥♠♣♦ss ♦r t ♣rt rt♦r t♦ ♦♦s r♥t ♥①
0 1000 2000 3000 4000 5000−200
0
200
Spe
ed [r
pm]
0 1000 2000 3000 4000 5000−1
−0.5
0
d−ax
is [p
.u.]
0 1000 2000 3000 4000 50000
0.5
1
q−ax
is [p
.u.]
Time step [−]
♣ ♥ rr♥ts
01234567
Vol
tage
inde
x [−
]
−0.2
0
0.2
d−ax
is L
UT
[p.u
.]
0 1 2 3 4 5 6 7
0 1000 2000 3000 4000 5000
−0.1
0
0.1
Time step [−]
q−ax
is L
UT
[p.u
.]
❱♦t t♦r ♥①
r ①♠♣ ♦ t stt♦♥ ts
①♠♣ ♣r♦ ♥ ts st♦♥ ♦ t♦ qtt ①♣♥t♦♥ ♦ t♣♥♦♠♥♦♥ ♦ st♥t♦♥ t s ♥ s♦♥ t ♦r ♦ t ♣rt rt♦r♥ ♣rs♥ ♦ st♥♥t t♦r ♥ t s st♦r ♥ t ❯ r ♥♦t r ts ♥ r tt t rr♥t rt♦♥s r ♥♦t r② ♣t t ♦st ♥t♦♥t♦♥ s ♠sr♣rs♥t ♥ t♦ r♦♥ ♦ ♦ t ♦♣t♠ ♦t t♦r♥ t ♦rt♠ ♥trs ts ♦♥t♦♥ t ♦ ♠♣♦ss ♦r ♠ t♦ r♥ t♦♥tr♦ ♦ t ♠♥
♥r ♥②ss ♦♥ t ♣♥♦♠♥♦♥ ♦ st♥t♦♥ s ♥♦t ♣♦ss s♥ ts ts♣♥ ♦♥ ♠tt ♦ r♥t s♣ts s s t ♥ ♦ tst t ♦st ♥t♦♥t
♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦
Pr♦♣♦s ♥tst♥t♦♥ ♠t♦
♣r♦♣♦s ♣t♥ ♠t♦ s sr ♥ ts st♦♥ ♠ s t♦ ♥ trt♦♥s♣ t♥ t rr♥t rt♦♥s ♥ ② t t ♥rtr stts ♥ t♦①♣♦t t ♦r t ❯ ♣t ♥ ts ② t ♦ ♣♦ss t♦ s t ♠♦st r♥ts st♦r ♥ t t t♦ r♦♥strt t ♦trs s ♠t♦ s ①♣t t♦ ♣r♦ ♦♥t♥♦s ❯ ♣t ♥ tr♥ ♠♥s ♥ ♣rs ♥ r rr♥t♣rt♦♥ t♦t ♦r♥ ♥♦♥ ♦♣t♠ ♦t t♦rs
♦♥sr♥ t srt ♠♦ ♦ t ②♥ t rr♥t rt♦♥ ♥ t qrr♥ r♠ s ② t ③t ♦t t♦r ♥ ①♣rss s
∆iz,k = δi
∅ + δiz
r t t♦t rr♥t rt♦♥ ∆iz,k s ①♣rss s t s♠ ♦ t♦ r♥t t♦rs
♥♠② δi∅ = [δi∅d δi∅q ] ♥ δiz = [δizd δizq ] ttr ♥ ①♣rss s
δi∅ = Ai
k
δiz = Bu
z t z ∈ Q
r t ♦r♠r r♣rs♥ts t rr♥t rt♦♥ s ② t ♥ stt ♦ t ♥rtru∅ = u
z = [0 0] t z = 0, 7 ♥ ♦tr ♦rs ts s t ♦t♦♥ ♦ t s②st♠ t♥ ♥♣t ♦t t ♥tr rs♣♦♥s s♦♥ tr♠ δiz s ♥ ② t t♦t t♦r u
z t z ∈ Q ♥ t ♥ s♥ s t ♦r rs♣♦♥s ♦s②∆i
∅ = δi∅
t s ♦rt ♥♦t♥ tt δi∅ ♣♥s ♦♥ t t rr♥ts ♥ s♣ δi∅ =f(ik, ω) ss t s♦♥ tr♠ δiz ♣♥s ♦♥ t rr♥ts tt t t ♥t♥ ♦♥ t rt ♥ qrtr ♦ts tt ♥ t t ♥① ③ ♥t tr r♦t♦r ♣♦st♦♥ ϑ δiz = f(ik, z, ϑ) ♥ ♦tr ♦rs ♥ ♦tt♦r tt ♠♥s ①♥ z t rt rr♥t rt♦♥ ∆i rs t t rr♥tt s♣ ♥ t ♣♦st♦♥ ♥ ♣rtr t s ♣♦ss t♦ ①♣rss t rr♥t rt♦♥s② s♣tt♥ t♠ ♥ t t♦ q ♦♠♣♦♥♥ts ♦t♥♥
∆izd = δi∅d(ik, ω) + δizd(i
k) · ♦s(ϑ− ϑz)
∆izq = δi∅q(ik, ω) + δizq(i
k) · s♥(ϑ− ϑz)
r ϑz s t ♣♦st♦♥ ♦ t t ♦t t♦rs ♥ t αβ ♣♥ t st② stt t ♦♥st♥t rr♥t i
k ♥ s♣ ω t rt♦♥s ♥ r♣rs♥t st s♠ ♦ ♦♥st♥t ♦st ♥ ♥ ♦st♥ tr♠ t ♦♥st♥t ♠♣t r♣♦rts ts stt♦♥
t s ♦rt t♥ tt t s st♦r ♥ t ❯ ♥ r♦rsr t♠ ♥trs ♦r t r♥t ♠♥ stt t② r ♥♦t r♥②♠♦r ♥ t② ♦ str♦♥② r r♦♠ t ♦rrt s ♥ ts s t ♦st♥t♦♥ t♦♥ ♥ ♥ tr♥ t ♦ ♦ t ♥①t ♦t t♦r s s ♦♥ r♦♥s ♥ t♦ ♥ ♥♦rrt ♠♥ ♦♣rt♦♥
♣r♦♣♦s ♣t♥ ♠t♦ s s ♦♥ t ♦♦♥ r♠rs t st♣ ① t rr♥ts t s♣ ♥ t ♣♦st♦♥ t ♥ stt ♦♠♣♦♥♥t δi∅ st s♠ ♦r t t♦rs ∆i
z ss t♦ t r♦t♦r ♥s♦tr♦♣② t t
Pr♦♣♦s ♥tst♥t♦♥ ♠t♦
0 100 200 300−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5d−axis
Electrical angle θ [°]
Current variations [A]
0 100 200 300−0.15
−0.1
−0.05
0.05
0.1q−axis
Electrical angle θ [°]
zid
∅id z
iq
z
iq0
r rr♥t rt♦♥s t st② stt
t♦rs δiz r ♣ ♦♥ ♥ ♣s ♥ t q ♣♥ s t♥ t♦ ♦♠♥ tr♠sr t♦rs ∆i
z ♥ ①♣♦t♥ r♠rs ♥ t♦ r♦♥strt t ♦trs
t s ♦rt ♠♥t♦♥♥ tt ♥ ♦ ♦♥sr♥ t t♦rs ∆iz t t s♠
♥st♥t ♦r t s♠ rr♥tss♣ ♥ ♣♦st♦♥ r♥ t r r ♦♣rt♦♥t rr♥t rt♦♥s r ♦t♥ t r♥t ♥st♥ts ♥tr♦♥ ♥ ♣♣r♦①♠t♦♥♦r ts rs♦♥ st ♦rt♠ ♦r t sq♥ ♥tt♦♥ s t♦ s♥♥ ♦rr t♦ r♥t rq♥t ♣t ♦ t ❯ s ①♣♥ ♥
qtt r♣rs♥tt♦♥ ♦ r♠rs ♥ s ♣r♦ ♥
2,k12
3,k13
5,k1 6,k1
6
1
1,k114
4,k1
6
5
z,k1
z
z∅z Q∈
i
k
δδδ δ
δ δ Δ
Δ
∅Δ =
∅δ
k
Δδδii
i i
i i
i
i
ii
iii
i
i i i
i
i
i
i
r ❱t♦r r♣rs♥tt♦♥
r♥t sq♥s s =< s1, s2, s3 > ♦♠♣♦s ② tr ♠sr t♦rs ♥ s t♦ ♦r ♦♠♣t ❯ ♣t st♠t♥ t ♦tr ♦r ♦r s ♥t ♦♦♥ t sq♥s r ①♣rss ♦♥sr♥ z = 1 s t ♠♥♠♠ ♥① ♦t tr♣ts rsts ♥ s② ①t♥ t♦ z ∈ Q ② ♠♥ ♦ ♣r♦♣r ♥①st♥ ♥ r♦tt♦♥
♦♠♥♥ ♠①♠♠ ♦ tr rr♥t rt♦♥s ♥ ② t t♦rs t s ♣♦ss t♦ ♦t♥ t rt t♦ t ♥ t♦r ♥ tr r♥t ②s s rt♦♥s♣s r s ♥ ♦♥ ♦ t ♣♦ss sq♥s t♦ r♦♥strt t ♠ss♥s strt♥ r♦♠ t ♠♦st r♥t rr♥t rt♦♥s ♥ ♣rtr t ♥ t♦r♥ ♦t♥ ♦♠♥♥ t♦ ♦♣♣♦st t♦rs ①♠♣ ♥ tr♦♥stt t♦rs ①♠♣ ♥ ♦r tr t♦r t ♣s st ♦2/3π ①♠♣ ♥
♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦
i (2),k1δi2i (3),k1δi3
δi5i (5),k1 i (6),k1
δi6 Δi1
i (1),k1δi1δi4
i (4),k1
Δi4
i k
δi∅δi1 δi4+ =[00]+Δi1Δi4
=∅
Δi2
♦ ♦♣♣♦st t♦rs
δi4i (4),k1
δi5i (5),k1 i (6),k1
δi6i (1),k1δi1
Δi2
Δi1
δi∅Δi3 δi2δi3
i (3),k1 i (2),k1
i k
δi1 δi3+ =[00]δi2- +Δi1Δi3
=∅
Δi -Δi2
r ♦♥st t♦rs
i (2),k1δi2δi4
i (4),k1
i (6),k1
δi6 i (1),k1δi1Δi5
Δi1
δi∅Δi3δi3
i (3),k1
i k
δi5i (5),k1
δi1 δi3+ =[00]δi5+
=∅
Δi+Δi1Δi3Δi5+
3
r t♦rs t 2/3π♣s st
r ♦♠♥t♦♥ ♦ δiz rst♥ ♥ t ♥ t♦r [0 0]
q♥
♦♠♣♦s ② tr ♦♥st t t♦rs s =< 1, 2, 3 > s s♦♥ ♥ t①♠♣ ♦ ❯s ♥ ♣t t
∆i∅ = ∆i
1 +∆i3 −∆i
2
∆i4 = 2∆i
∅ −∆i1
∆i5 = 2∆i
∅ −∆i2
∆i6 = 2∆i
∅ −∆i3
rst t rt♦♥ ♦ t ♥ t♦r s ♦♥ ♦♠♥♥ t tr ♦♥st t♦rss ♥ tr tt ② ♥rt♥ t rt♦♥s♣ t♥ t♦ ♦♣♣♦sttt t♦rs ♥ t ♥ t♦r s ♥ t ①♠♣ ♦ t s ♣♦ss t♦ ♥t rr♥t rt♦♥s rt t♦ t tr ♦tr t t♦rs
q♥
♦♠♣♦s ② tr t t♦rs t t♦ ♦♣♣♦st s =< 1, 2, 4 > s s♦♥ ♥t ①♠♣ ♦ ❯s r ♣t t
∆i∅ =
1
2(∆i
1 +∆i4)
∆i3 = ∆i
2 +∆i4 −∆i
∅
∆i5 = 2∆i
∅ −∆i2
∆i6 = 2∆i
∅ −∆i3
rs♣♦♥s t♦ t ♥ t♦r s ♦t♥ ♦♠♥♥ t t♦ ♦♣♣♦st t♦rss ♥ tr tt t t t♦r ♦♥st t♦ t♦ t t♦rs ♥ ts s z = 3 ♥ ♦♥ ② ♥rt♥ t rt♦♥s♣ ①♣rss ♥ ♦t t♦ ♠ss♥ s ♥♠② 5 ♥ 6 r ♦♠♣t ①♣♦t♥ trt♦♥s♣ t♥ t♦ ♦♣♣♦st t♦rs
Pr♦♣♦s ♥tst♥t♦♥ ♠t♦
q♥
♦♠♣♦s ② t♦ t ♦♥st t♦rs ♥ ♥ t♦r s =< 1, 2, 0 > ss♦♥ ♥ t ①♠♣ ♦ ❯s ♥ ♣t t
∆i4 = 2∆i
∅ −∆i1
∆i5 = 2∆i
∅ −∆i2
∆i3 = ∆i
2 +∆i4 −∆i
∅
∆i6 = 2∆i
∅ −∆i3
rst t t♦rs ♥ ♦♣♣♦st♦♥ t t t♦ t t♦rs ♥ ♦♥♥rt♥ t rt♦♥s♣ ①♣rss ♥ ♥ ♦♠♥♥ ♦♥ ♦ t ♥♦♥ z = 2 ♥ ♦♥ r♦♥strt z = 4 t t ♥ t♦r t s ♣♦ss t♦ ♥t t♦r ♦♥st t♦ t ♦♥sr t♦ t t♦rs st s ♦♥①♣♦t♥ t rt♦♥s♣ ♥
q♥
♦♠♣♦s ② t♦ t t♦rs t ♣s st ♦ 2/3π ♥ ♥ t♦r s =< 1, 3, 0 > s s♦♥ ♥ t ①♠♣ ♦ ❯s ♥ ♣t t
∆i2 = ∆i
1 +∆i3 −∆i
∅
∆i4 = 2∆i
∅ −∆i1
∆i6 = 2∆i
∅ −∆i3
∆i5 = 2∆i
∅ −∆i2
tr s r s t♦ ♥ t t♦r t♥ t t♦ ♥♦♥ ttr tt s♠r② t♦ q♥ t ♥①t st♣ ♦♥ssts ♥ ♥♥ t ♠ss♥ t♦rs② ♠♥ ♦ t rt♦♥s♣ tt ♥s t♦ ♦♣♣♦st t♦rs ♥ t ♥
q♥
♦♠♣♦s ② tr t t♦rs t ♠t ♣s st ♦ 2/3π s =< 1, 3, 5 >s s♦♥ ♥ t ①♠♣ ♦ ❯s ♥ ♣t t
∆i∅ =
1
3(∆i
1 +∆i3 +∆i
5)
∆i2 = 2∆i
∅ −∆i5
∆i4 = 2∆i
∅ −∆i1
∆i6 = 2∆i
∅ −∆i3
rst st♣ ♦s ♥♥ t ♥ t♦r ♦♠♣♦♥♥t t t ①♣rss♦♥ ♥ tr tt t ♦tr tr t♦rs r ♦♥ s t ♦♣♣♦st ♦ ♦t ♠sr♥ r♦♥strt s
t ♦tr sq♥s tt ♦ ♥♦t strt t ♠♥t 1 ♥ ♦♥♥t t tsq♥s r♣rs♥t ♥ ♥ ♣rtr ts ♥ ♦♥ ② s♦rt♥ ♥ s♥♥
♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦
4
5 6
0
7
1
23
q♥
5 6
0
7
3
4 1
2
q♥
3
4
5 6
7
1
2
0
q♥
2
4
5 6
7
1
3
0
q♥
0
2
4
6
7
1
3
5
q♥
r ❱♦t t♦r sq♥s
SEQUENCE
IDENTIFICATIONSUBSET
SELECTION
LUT
UPDATE
idx
r q♥ ♥tt♦♥ ♦rt
♦rr t ♠♥ts ♦ t ♥r sq♥ t ts ♣♦♥t t rst ♠♥t s[1] s t♠♥♠♠ ♥① ♥ ♥ t sq♥ tr tt t ♠♥ts ♦ r strt② t q♥tt② s[1] − 1 ♥ ts ② ♦r ♣♦ss tr♣ts ♥ sq♥ s♦t♥ ttr strts t ♥① 1 ♥ ♣s t s♠ s♣t s♣♠♥tst♥ t tr t♦rs ♦ t ♦r♥ sq♥ t s ♦rt ♠♥t♦♥♥ tt ttr♣ts r② s ♠♥♠♥ ♠♥t q t♦ 1 t♥ t strt♦♥ s ♥♦ t s♥s[1]− 1 = 0
q♥ ♥tt♦♥
♣r♦♣r ♦rt♠ ♦r t sq♥ ♥tt♦♥ s ♥ ♦♣ ♦♥sr♥ t♦♦♥ rqr♠♥ts rst t s t♦ r♦♥③ tr t ♦t t♦rs ♦t♥② t ♦st ♥t♦♥ ♠♥♠③t♦♥ ♦r♠ st sq♥ ss t s t♦ r♦rt ♦rr ♥ t t♦rs r ♦s♥ s ♠♣s tt ♦♥sr♥ t ss♥ ♥ ♥ ♦♥② ♦♥ ♥ t♦r r♥t sq♥s r t♦ ♣tt ❯ ♥② t ♦rt♠ ♠st st ♦r r t♠ ♣♣t♦♥s
♦r s sr ♥ ♦rt♠ tr♣ts r stt s ♥ t ♣r♦sst♦♥ t s =< s1, s2, s3 >∈ S ♥ ♦r ♦ t♠ fs s ♥tr♦ ♥♣t ♦t t♦r ♥① ① s ♦♠♣r t s[fs] ♥ fs s ♣t ♦r♥②♥s ♥ ❲♥ tr♣t s ♦♠♣t ♥ ts tr ♠♥ts r ♦♥t ❯ ♣t s ①t ♥s ♥ tr t ♣t s ♣r♦r♠ t rst♠♥t ♦ t ♥t sq♥ s[1] t ♦st ♦t t♦r s sr t♦♦t♥ r ♣t r♥ t ♠♥ ♦♣rt♦♥ ♥ st A ♥s t ♥①s tt t st♣ ♥ s t♦ ♦r♠ st sq♥ t♦s t♦rstt r② ♥ ♦s♥ ② t ♦st ♥t♦♥ ♠♠♥♠③t♦♥ ♥ ♥ tt ♥♦t ♥ sr tr t ❯ ♣t
♦ r t ♦♠♣tt♦♥ r♥ t ♦rt♠ s t♦ st t♦ ♦ tt♦♥ ♦ t r♥t tr♣ts t st♣ ♦r ts rs♦♥t sq♥♥tt♦♥ s ♠ ♦♥sr♥ ♦♥② t sq♥s s ∈ Sidx,j ⊆ S r t ssts ♥ s Si,j ≡ s ∈ S|∃k, s[k] = i∧ s[1] = j s ♠♥s r♥ t rsr t♦ t sq♥s ♥♥ ① ♥ strt♥ t q s s tt sq♥s t tr ts rst ♠♥t s ① ♥ ① 6∈ A ♦r ts rst ♠♥t s q 6= ①♥ q ∈ A ♥
q♥ ♥tt♦♥
♦rt♠ q♥ ♥tt♦♥ ♥♣t ① s t rr♥t ♦t t♦r ♥① Sidx,q s t sst ♦♥t♥♥ ①
♥ strt♥ t q ∈ Q A s t sst ♥♥ t ♥①s tt ♥ st♦ ♦r♠ tr♣t fs∀s
♦r q ∈ Q ♦
♦r s ∈ Sidx,q ♦ q ∈ A ∧ q 6= idx ∨ ① 6∈ t♥ ① == s[fs] t♥ fs ← fs + 1
A ← A∪ ① ♦r q ∈ Q ♦
♦r s ∈ Sidx,q ♦ fs == 3 t♥
♥t♦♥ ❯ ❯P A ← A\s[1]
s♠t r♣rs♥tt♦♥ ♦ t ♦rt♠ s r♣♦rt ♥ s ①♣♥t ♦♥ssts ♦ tr ♠♥ ♣rts rst sst st♦♥ ♦s r♥ t t♦t♥♠r ♦ sq♥ t♦♥s ♦♥sr♥ t ♥ ♥♣t ♥① ① ♦r ♦ t♦rt♠ s ♥ t t ♦ sq♥ ♥tt♦♥ ♥ t ♥①s r♦♠♣r t t sq♥s ♥ t rt s r ♣t ♥② tr sq♥ s ♥t t ❯ r ♣t ♦r♥②
♥ ①♠♣ ♦ t ♦rt♠ ♦♣rt♦♥ s r♣♦rt ♥ ♥ t t t♦t♣t ♦ t ♦st ♥t♦♥ ♠♥♠③t♦♥ r st st♣ ② st♣ ♥ t rt t♦♣rt♦♥ ♦ t ♦rt♠ r r② sr ♥ t ①♠♣ t t rst st♣ t♥♣t ♥① s 1 ♦rt♠ ♣t t ♦ t sq♥s strt♥ t t tt s♦♥ st♣ t ♥♣t s ♥① 4 s ♦ t sq♥s strt♥ t ts ♥①r ♣t ss t sq♥s strt♥ t 1 ♥ t 4 ♥ t s♦♥ ♣♦st♦♥r s ♣t t t ♥①t st♣ t ♥♣t s 0 sq♥s strt♥ t ts♥① ♥ t♦s strt♥ t 4 ♥ ♥ 0 ♥ t s♦♥ ♣♦st♦♥ r ♣t ♥s =< 1, 4, 0 > s ♥♦t ♣♦ss sq♥ t s ♥♦t ♣♦ss t♦ ♣t t ❯ t tsst♣ t t st st♣ t ♥♣t ♥① s 2 t ts ♣♦♥t t sq♥ s =< 4, 0, 2 >s ♥t ♥ t s ♦r t ❯ ♣t rst ♠♥t ♦ t sq♥♥♠② 4 s t♥ sr s ♠♥s tt ts ♥① s ♥♦t ♥②♠♦r ♦rt ❯ ♣t s♠♣ ①♠♣ r♣rs♥t ♥ ♠♦strt t ♦r
1
1 4
1 4 0
1 4 0 2
update: s=<1,-,->
update: s=<1,4,->, s=<4,-,->
update: s=<4,0,->, s=<0,-,->
update: s=<4,0,2>, s=<0,2,->s=<2,-,->
identify: s=<4,0,2>
discard: s[1]=4
r ①♠♣ ♦ t sq♥ ♥tt♦♥
♦ t ♦rt♠ t s ♦rt t♥ tt tr t rst ♠♥t ♦ t ♥tsq♥ s sr 4 ♥ t ①♠♣ tr st ♦ ♦r ♥①s 1 ♥ t①♠♣ s ♦ r♣rs♥t ♥ ss s♥ t ❯ ♣t ♦ ♣r♦r♠ s♥
♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦
♦ ♥① s♣t t sr♥ ♠t♦ ♠♣♦② ♥ t ♦tr ♥ sr♥ t♦r s ♦ rqr t ♥♦ ♦ t ♦ t ♥①s ♦r t st t ♦rr♥ t ♥①s r ♣t ♥s t st A s s♦t♦♥ s ♥♦t ♥ ♠♣♠♥ts t ♦ ♠ t ♦rt♠ ♠♦r ♦♠♣① ♦♥ ♥ts t rqr♠♥t ♦♣♣t② ♥ r t♠ ♣♣t♦♥s ss t ♥②ss ♦ t rsts ♦t♥r♥ t ♣r♠♥r② tsts ♦♥r♠ tt t ♠♣♠♥t s♦t♦♥ s s♥t②♥t ♥ tt t ♦s ♥♦t sr r♦♠ t ♦ ♠♥t♦♥ ss
s♦s t ①s ❯ t♦t ♥ t t ♣r♦♣♦s ♣t♥ ♠t♦ s♣s st♣ tst s ♥ ♣r♦r♠ ♥ t s♠t♦♥ s♦ ♥s r ♥②t②♦♠♣t ♥ ♦♠♣r t t s ♦t♥ t t rr♥t rt♦♥ ♠sr♠♥ts s ♥s t s ♦rt t♥ tt t♦t ♥② ♠t♦ t♦ ♣r♥t♥ ♦ st♥t♦♥ t ♦ss ♦ ♦♥tr♦ ♦rs s t ♥ r♦♥s st r ♥s r ♥♦t s♥s♦s ♥②♠♦r t t ♥ ♦ t tst s ♣♣♥s s♥ trr♥ts ♦ ♥♦t tr t rr♥s ♦♠♠♥ ② t s♣ rt♦r t♦ ♣r♦ trqr t♦rq ♥ ts stt♦♥ t s♣ r② rs ♥t t ♦♠s ③r♦♥ t ♦♥trr② t♥s t♦ t ♣r♦♣♦s ♥tst♥t♦♥ t♥q t ❯ r ♦♥t♥♦s② ♣t ♦t♥♥ ♣rs ♥ r ♣rt♦♥ ♥ tr♥ r♥tst ♣r♦♣r ♦♣rt♦♥ ♦ t ♠♥
−0.02
0
0.02
∆ i q0 [p
.u]
−0.05
0
0.05
∆ i q1 [p
.u]
−0.05
0
0.05
∆ i q2 [p
.u]
−0.05
0
0.05
∆ i q3 [p
.u]
−0.05
0
0.05
∆ i q4 [p
.u]
−0.05
0
0.05
∆ i q5 [p
.u]
0 500 1000−0.05
0
0.05
∆ i q6 [p
.u]
Time step [−]0 500 1000
−0.02
0
0.02
∆ i q7 [p
.u]
Time step [−]
Correct Actual
❯ st♥t♦♥
0
0.01
0.02
∆ i q0 [p
.u]
−0.05
0
0.05
∆ i q1 [p
.u]
−0.05
0
0.05
∆ i q2 [p
.u]
−0.05
0
0.05
∆ i q3 [p
.u]
−0.05
0
0.05
∆ i q4 [p
.u]
−0.05
0
0.05
∆ i q5 [p
.u]
0 500 1000−0.05
0
0.05
∆ i q6 [p
.u]
Time step [−]0 500 1000
0
0.01
0.02
∆ i q7 [p
.u]
Time step [−]
Correct Actual
❯♣t ❯
r t♥ss ♦ t ♣r♦♣♦s ♦rt♠
♦ t♦♥
SynRMMASTER
MACHINE
r ①♣r♠♥t st♣
♣r♦♣♦s s♠ s ♥ ♠♣♠♥t ♦♥ t ①♣r♠♥t st♣ ♦ t s♠♣♥ t♠ ♦ Ts = 200[µs] ♥ t ♠♥ t ♦ t ②♥ r r♣♦rt♥ st② stt ♣r♦r♠♥ ♥ t ♥ ♦♠♣r t
♦ t♦♥
♦t♦r t
P♦ ♣r ♥♠r p 2
Ps rsst♥ R 4.5 Ω
rt ♥t♥ Ld 60 mH
rtr ♥t♥ Lq 190 mH
♦♠♥ rr♥t IN 10 Apk
trt♦♥ P s♠ s♦s t rsts ♦t♥ t ♦♥st♥t s♣n = 100 ❬r♣♠❪ ♥ ♦r r♥t ♦s rr♥t rr♥s r ♥rt ♦♥sr♥♥ ♥strt ♠♥ ss♠♥ ♥r ①♠♠ ♦rq Pr ♠♣rP trt♦r② ♦s② t♦ r♦♥ strt♦♥t t P rs r♦♠t ♦♥ ♥ ♦r ts rs♦♥ t ♠♥ ♦♣rt♦♥ s ♥♦t ♦♣t♠ ♦r tr♥t t trt♦r② rqrs t ♣rs ♥♦ ♦ t ♠♥ ♠♦ t♦t ♦ ♦ s♥ t P s ♥t rs ♠♦r ♦ ♦sss ♥r ♠①♠♠ t♦rq ♣t② t ♦s ♣♥ t s♠ ♦♠♣t② r r♦♠t ♠♦
rst tst s ♥ ♣r♦r♠ t♦ r② t t♥ss ♦ t ♣r♦♣♦s P♥ ♣rtr tr♥st♦♥ t♥ t ♦s ♥ t ♦r ♣rt♦♥ s♥ ♦♥ r♥ t st② stt ♦♣rt♦♥ ♦ t ♠♥ rsts r r♣♦rt♥ r t ♥st♥t ♥ t tr♥st♦♥ s ①t ♥ s② r♦♥sst♣ k = 500 t s ♦rt t♥ tt tr t tr♥st♦♥ t r♣♣ s r♥ ♦t ①s t s ♠♦r ♥t ♥ t qrtr ①s r r♦♥ strt♦♥ss r♠r ♠s♠t t♥ t♥ t t ♥ t ♥♦♠♥ ♠♦ s♦r t ♣rt♦♥ r♥ t P st t ❯ r ♣t t t ♣r♦♣♦s♥tst♥t♦♥ ♠t♦ t t② r ♥♦t s ♦r t ♣rt♦♥
0 250 500 750 1000−0.55
−0.5
−0.45
−0.4
−0.35Load 0.4 [p.u.]
d−ax
is c
urre
nt [p
.u.]
0 250 500 750 10000.35
0.4
0.45
0.5
0.55
q−ax
is c
urre
nt [p
.u.]
Time step [−]
r r♥st♦♥ t♥ P ♥ P
s♦s t q rr♥t ♦r♠s ♦ t P ♥ t ♣r♦♣♦s P t ♥ s♥ tt t ♦ ♦ t ♣r♦r♠♥ ♦ t t♦ s♠s r ♦♠♣r♥ tr♠s ♦ r♣♣ ♥ ts stt♦♥ t ①s ♣rs♥ts ♥ r r♣♣ t♦ t r♦t♦r♥s♦tr♦♣② Lq > Ld t r ♦s r♦♥ strt♦♥ s ♠♦r ♥t♥s ♥ t tsr ♠♥② ♥ t qrtr ①s s ♦♥sq♥ t ♠♦ s ♦r t ♣rt♦♥s s♥♥t② r♥t ♥ t q①s t ①s s ss st t♦ r♦♥ strt♦♥ st ♣♦t ♦ s♦s tt r♦♥ ♠♦ s ♥ t ♣rt♦♥ ss
♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦
♥ ♥rs ♥ t rr♥t r♣♣ ♦♥r♠♥ t ♥②ss ♣rs♥t ♥ ❬❪ s ts r ♥ t qrtr ①s r t ♥t♥ s ♦rst♠t t ♦ ♦st rr♥t r♣♣ s r ♥ t ①s ♣rs♥t♥ t ♦r ♥t♥ ♥♠② t①s ♣♥♥ ♦♥ t str♥t ♦ t r♦♥ stt♦♥ ♥ ♥ tr♥ ♦♥ t ♣r♠tr♠s♠t t rr♥t ♦r♠s ♦ ♣rs♥t t ♦♣♣♦st ♦r t r♣♣s r ♥ t ♦tr ①s s ♥ s♥ t P ♦s ♥♦t ♣rs♥t t s♠t ♥ t rr♥t r♣♣ ♦s ♥♦t ♥ t t ♦
0 500 1000−0.12
−0.1
d−axis current [p.u]
mL=
0.1
[p.u
.]
0 500 10000.08
0.1
0.12
q−axis current [p.u]
0 500 1000
−0.3
−0.2
mL=
0.4
[p.u
.]
0 500 10000.2
0.3
0 500 1000
−0.5
−0.4
−0.3
mL=
0.7
[p.u
.]
0 500 10000.3
0.4
0.5
0 500 1000
−0.7
−0.6
−0.5
mL=
1 [p
.u.]
Time step [−]0 500 1000
0.5
0.6
0.7
Time step [−]
rr♥t ♦r♠s
d−axis q−axis0
0.5
1
1.5
2x 10
−3 Load 0.1 [p.u.]
d−axis q−axis0
0.02
0.04
0.06Load 0.4 [p.u.]
d−axis q−axis0
0.05
0.1
0.15
0.2Load 0.7 [p.u.]
d−axis q−axis0
0.05
0.1
0.15
0.2Load 1 [p.u.]
MB MF
rtr♦♥
r ①♣r♠♥t rsts t 100 ❬r♣♠❪ ♥ r♥t ♦s
t ♦ t ♣rt♦♥ r② s s♦♥ ♥ r t r s♦ ♥♥ t s ♥ r♣rs♥t t t ♥ t ♣rt rr♥t t ♥ s♥tt t ♣rt s s rt s♥ t ♣rt② r♣s t ♦r♠ ♦ t♠sr rr♥t t ♦♥ st♣ ♥ ♥ ♣rt♦♥ r② ♦ t t♦ s♠ss ♦♠♣r ♥ ② ♠♥ ♦ t ♥tr qr rr♦r rtr♦♥ t ♥ s♥ tt ♦r t P t r t ♦ t r t ♣rt♦♥ rr♦r s② t ♠s♠t ♥ t ♠♦ ss t q①s rr♦r s r t♥ t ①s ♦♥ P rr♦rs r ♦♥st♥t ♥ ♦ ♥♦t ♥ ♥ t ♦r tsts t♥ss♦ t ♠♦ r ♣rt♦♥ s ♣rtr② ♥t ♥ t st rsts r② s♦tt t ♣rt♦♥ s r ♥ r② ♦r♥ ♦♥t♦♥
540 545 550 555−0.12
−0.1
−0.08
−0.06d−axis current [p.u]
Load
0.1
[p.u
]
540 545 550 5550.095
0.1
0.105q−axis current [p.u]
540 545 550 555−0.3
−0.28
−0.26
−0.24
Load
0.4
[p.u
]
Time step [−]540 545 550 555
0.26
0.27
0.28
0.29
0.3
Time step [−]
♦ ♦
540 545 550 555−0.48
−0.46
−0.44
−0.42d−axis current [p.u]
Load
0.7
[p.u
]
540 545 550 5550.4
0.45
0.5q−axis current [p.u]
540 545 550 555−0.68
−0.66
−0.64
−0.62
−0.6
−0.58
Load
1 [p
.u]
Time step [−]540 545 550 555
0.62
0.64
0.66
0.68
0.7
Time step [−]
♦
r t ♦ t ♣rt♦♥ r②
♥ s♦ t ♦t♣t ♦ t ♦st ♥t♦♥ ♠♥♠③t♦♥ ♥t ♥♠r ♦ st♥ ♣r st♣ ♦r ♦t t P ♥ t P t ♥ ♥♦t tt st♥ ♣ttr♥ ♦ t t♦ s♠s ♥ t tst t ♦ 0.1[p.u.] s r②
♠tt♦♥s ♥ ♥r♥t st♥t♦♥
s♠r s ♦♥r♠s tt t ♥♦♠♥ ♠♦ s ♦r t ♣rt♦♥ ♥ t Ps s♥t② ♣rs ♥ ts ♦r♥ ♦♥t♦♥ r♥t② t r ♦s t ♦st♥t♦♥ ♦♦ss r♥t ♦t t♦rs ♥ t t♦ s♠s ♥ ♣rtr t P♦s ♥♦t ♣rs♥t ♥s t♥ ♦♣♣♦st t♦rs ♥♦ tr♥st♦♥s tt rqrs ♦♠♠tt♦♥s r ♥♦
0 500 10000
2
4
6MBPCC
Load
0.1
[p.u
]
0 500 10000
2
4
6MFPCC
0 500 10000
2
4
6
Load
0.4
[p.u
]
Time step [−]0 500 1000
0
2
4
6
Time step [−]
♦ ♦
0 500 10000
2
4
6MBPCC
Load
0.1
[p.u
]
0 500 10000
2
4
6MFPCC
0 500 10000
2
4
6
Load
0.4
[p.u
]
Time step [−]0 500 1000
0
2
4
6
Time step [−]
♦
r ♦st ♥t♦♥ ♦♣t
0 500 10000
1
2
3MBPCC
Load
0.1
[p.u
]
0 500 10000
1
2
3MFPCC
0 500 10000
1
2
3
Load
0.4
[p.u
]
Time step [−]0 500 1000
0
1
2
3
Time step [−]
♦ ♦
0 500 10000
1
2
3MBPCC
Load
0.7
[p.u
]
0 500 10000
1
2
3MFPCC
0 500 10000
1
2
3
Load
1 [p
.u]
Time step [−]0 500 1000
0
1
2
3
Time step [−]
♦
r ♠r ♦ ♦♠♠tt♦♥ ♣r st♣
s♦s t ❯ ♦r t tsts t ♦ ♥ t ♥ s♥ tt t ♣t s rq♥t ♥ t ♥ ♦s ♦t♥♥ s♠♦♦t rr♥t rt♦♥ ♦r♠s
♠tt♦♥s ♥ ♥r♥t st♥t♦♥
rsts s♦♥ ♥ t ♣r♦s st♦♥ t t st② ♦ t ♣r♦♣♦sss t ♦♠♣rs♦♥ t t P ♦♥r♠ ts t♥ss rtss ts ♦rt ♠♥t♦♥♥ ♥ t ts ♥♦♥tr r♥ t ♠♣♠♥tt♦♥ ♥ r t stt♦♥s tt r♣rs♥t rtt② ♦r t P
rst t ①sts ♦♠♥t♦♥ ♦ tr ♥①s tt ♦s ♥♦t ♦ t ❯ ♣t♥ ♣rtr ♦r t tr♣ts tt ♥ r♣rs♥t s t ♦♥ ♥ t sq♥s ♦♠♣♦s ② t♦ ♦♣♣♦st t♦rs ♥ ♥ t♦r t s ♥♦t ♣♦ss t♦
♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦
0 500 1000 1500−1
−0.5
0
0.5
1d−axis
Load
0.1
[p.u
.]
0 500 1000 1500−0.2
−0.1
0
0.1
0.2q−axis
0 500 1000 1500−1
−0.5
0
0.5
Load
0.4
[p.u
.]
Time step [−]0 500 1000 1500
−0.5
0
0.5
Time step [−]
♦ ♦
0 500 1000 1500
−1
−0.5
0
0.5
d−axis
Load
0.7
[p.u
.]
0 500 1000 1500
−1
−0.5
0
0.5
q−axis
0 500 1000 1500
−1.5
−1
−0.5
0
0.5
Load
1 [p
.u.]
Time step [−]0 500 1000 1500
−1.5
−1
−0.5
0
0.5
Time step [−]
♦
r rr♥t rt♦♥s ❯
2
5 6
7
3
4 10
r ❯♥st sq♥
r♦♥strt t ♦tr ♥♥♦♥ s ♦r t s ♥ tt ts stt♦♥ t②r♣rs♥t rtt② r♥ t P ♦♣rt♦♥ rs♦♥ ♥ qtt②①♣♥ s ♦♦s ♣rt rt♦r ♥r♥t② r♥s t ♣♦♥t ♦ ♦r s♦s s ♣♦ss t♦ t rr♥ ♥ ♦♥ t rs t t t♥s t♦ r♠♥ ♦s t♦ t ts ② ♥ tt t♦ ♦♣♣♦st t♦rs tt s ♦♣♣♦st rs♣♦♥ss r tr♥t②♥ ♥q② t♦tr t t ♥ t♦r ♦s♥ s t ♦♣t♠
s♥♥t rtt② ♦♥r♥s stt♦♥s ♥ t st♥t♦♥ s ♥♦ ♣r♦♣♦s ♥tst♥t♦♥ s♦t♦♥ s s ♦♥ t ♥tt♦♥ ♦ sq♥s ♦♠♣♦s ② tr♣ts ♦ ♦t t♦rs ♥①s s ♦♥sq♥ ♥♦ sq♥s r♥t t ♣r♦♣♦s ♠t♦ ♥♥♦t ♦r rsts s♦♥ ♥ ♦♥r♠tt r♥ t ♥♦r♠ ♠♥ ♦♣rt♦♥ s♥t ♥♠r ♦ sq♥s r ♥tt♦ r♥t t st ♦♣rt♦♥ ♦ t r ♥ t ♦tr ♥ ♣rtr ♦♣rt♦♥s s s st tr♥s♥ts ♥♦♥ rr♥t rr♦rs ♦ r♣rs♥t ♥ ♦st t♦t ♣r♦♣r ♦♣rt♦♥ ♦ t P Prs② ♥ ♣rs♥ ♦ rr♥t st♣s t s♠♦t t♦r ♦ ♦s♥ ♦r ♥♠r N ♦ ♥tr r♥ ts t♠ ♦♥② ♦♥ ♦ t ❯ s ♣t t ♦trs r ♥♦t s♥ ♥♦ tr♣ts r ♦r♠ ♦rts rs♦♥ tr N st♣s t t stt ♦ t ♠♥ s rst② ♥ t♦r ♣rt♦♥ t ♠♥ t♦ ♥①♣t ♦♣rt♦♥s r t ♦ N t r t rs ♦ ♥stt② t s ♦rt t♥ tt ts stt♦♥r♣rs♥ts rtt② ♥ ♦r t P ♣r♦♣♦s ♥ ❬❪ s♥ tr ♠t♦ ♦♣♣② s① ♥♦♥ ♦♣t♠ t♦rs ♦♥st②
♣rt♦♥ s ② t ♣rt rr♥t rt♦r ♥ t ♣r♦♣♦s Ps s ♦♥ t ♠sr rr♥t rt♦♥s ♥ ts t rr♥t qst♦♥ s t♦ ♣rs ♥ r r♦♠ rq♥② ♥♦ss ♦trs t s st♦r ♥ t ❯♥ t♥ s ♥ t ♣rt♦♥ r ♥♦t r t s t♦ ♥♦t tt r♦♥ s st♦r ♥ t ❯ ts ♦ tr s t♦ ♣t t rr♥t rt♦♥s
♠tt♦♥s ♥ ♥r♥t st♥t♦♥
rt t♦ ♦ ♦t t♦rs rst♥ ♥ s♦♥ ♦ t rr♦r ♥ t ♦ t ts ♦rt ♠♥t♦♥♥ tt t ♣rs♥ ♦ ♦st ♥ t ♠sr♠♥ts ♦s ♥♦t r♣rs♥t rtt② ♦r t ♦♥tr♦ s♠ rs♦♥ s tt r② ♣♦ss ♦♥st♥t ♦st s♠♥t ② t rr♥t strt♦♥ ♥ t ♦tr ♥ rq♥② ♥♦s ♦t t ♦r ♦ t s♠ ♦ ♣r♥t ♥② r♦♥ ♦♣rt♦♥ s ② t ♣rs♥ ♦ ♥♦s ♥ t ♠sr ♦r♠s t ❯ ♥ ♦♣ss tr ss♦t♦♥ ♦s ♦♥ rr♦♥♦s s♣s ♥ t s st♦r ♥ t ❯ t♠♦♥st♥t ♦ t tr s t♦ ♣r♦♣r② s♥ ♥ ♦rr t♦ t t rq♥②♥♦s ♥♦t ♥tr♦♥ ②s ♥ t rr♥t rt♦♥ ♦r♠s s r② ①♣♥ ♥ t str♥ ♥ ② t st♥ ♥ ♦ ② ♣r♦♣r②♣♥ t rr♥t s♠♣♥ s ♥ ♦♥ ② st♥ t s②♥r♦♥③t♦♥ ♦ t♥trr♣t sr r♦t♥ ♥ t ♦tr ♥ ♥ t ①♣r♠♥t st♣ ♦ t ♦rt♦r② s t♦ ♠♥s r s♠t♥♦s② ♦♥tr♦ ♥♠② t ②♥♥ t ♠str t♦ ♣♦r ♦♥rtrs r ♥♦t s♦t s♥ t② sr t r♦♥♦♥♥t♦♥ ss t♦ t s♠♣♥ t♠s s ♥ t t♦ r t s♠ ♦r r♠t♣s tr st r s♦♠ st r♥s ♦r ts rs♦♥s t str♥s♥tr♦ ② t ♦♠♠tt♦♥s ♦ t ♥rtr ♦ t ♠str ♠♥ t t rr♥t♠sr♠♥ts ♦ t ②♥ ♥ t s♠ r♥ t♥ tr rq♥s tstr♥ s ♥♦t ① ♥ rt♥ ♣♦st♦♥ ♦ t s♠♣♥ t♠ ♦ t ②♥ t tss r♦♠ t ♥♥♥ t♦ t ♥ t s r tt t s ♥♦t ♣♦ss t♦ ♦ ts ♥♦ ♦♠♦♣♦r str♥ ② s② ♣♥ t rr♥t s♠♣♥s ♥ ♣r♦♣r ♣♦st♦♥s ♠♥s tt ♣r♦② ♥ t str♥ ♦r ♥ ♦rrs♣♦♥♥ t♦ trr♥t s♠♣♥s t tr♣s rr♥ts t ② t t♥ ♥ tr♣ss②st♠ ♦♥② t♦ rr♥ts r ♠sr t tr s ♦t♥ ② strt♥ t♦tr t♦ s s♦♥ ♥ r ia ib ic r t t rr♥t isa i
sb i
sc r t
s♠♣ s ♥ ∆i s t ♦♠♦♣♦r str♥ ts ♠t♦ s ♠♣② tstr♥ ♦♥ t tr ♣s s②st♠ s t♥ tr♥srr t♦ t q♥t t♦♣s s②st♠ ♥ t ♦♥tr♦ s♠ s ♦♥sq♥t② t s t s s♦♥ ♥♦♥sr♥ t r tr♥s♦r♠t♦♥ ♥ r t ♥ s♥ tt ♥ t t♦♣s rr♥ts ♥ tr♠ ♥ ② t ♦♠♦♣♦r str♥
isa = ia +∆i
isb = ib +∆i
isc = −isa − isb = −ia − ib︸ ︷︷ ︸
ic
−2∆i
isα = 23
[isa − 1
2 isb − 1
2 isc
]= 2
3
[ia − 1
2 ib − 12 ic +
32∆i
]
isβ =√33 [isb − isc] =
√33 [ib − ic −∆i]
r♥t② t tr rr♥ts r ♠sr t str♥ s ♥♦t tr♥srr t♦ t t♦ ♣s s②st♠ ♦r ts rs♦♥ ts s♦t♦♥ s ♥ ♦♣t♥ t ♠♣♠♥tt♦♥
isa = ia +∆i
isb = ib +∆i
isc = ic +∆i
isα = 23
[isa − 1
2 isb − 1
2 isc
]= 2
3
[ia − 1
2 ib − 12 ic]
isβ =√33 [isb − isc] =
√33 [ib − ic]
♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦
♥♦tr ♣♦ss s♦t♦♥ tt ♦s ♥ t♦ ♦♥② rr♥t ♠sr♠♥ts ♦♥ ♦♠♦♣♦r str♥ ♥ t t♦ ♣s s②st♠ ♦♥ssts ♥ ♠sr♥ t♦rr♥t r♥s s s♦♥ ♥ ♥ t ♦♥sq♥t t♦ ♣s tr♥s♦r♠t♦♥rsts s ♥
isa − isb = ia − ib
isb − isc = ib − ic
isα = 23 [(i
sa − isb) + 1/2(isb − isc)] =
23
[ia − 1
2 ib − 12 ic]
isβ =√33 [isb − isc] =
√33 [ib − ic]
♥♦tr str♦♥ ♠tt♦♥ ♦ t ♣r♦♣♦s ♠t♦ s r♣rs♥t ② t ♦♠♣tt♦♥ r♥ t♦t t sq♥ ♥tt♦♥ ♦rt♠ ♦s ♥♦t ♥ ♦♠♣①t♦♥s ♦♥② ♦ ♦♠♣rs♦♥s ♥ t♦♥s r ①t t ♥♠r♦ sq♥s tt t♦ t ss ♥ ♥rs ♦ t tr♥r♦♥ t♠ r t♠ ♣♣t♦♥ s ♦♦s② ♦♥str♥ ② t s♠♣♥ t♠ t ♦♠♣t♦rt♠ s t♦ ①t ♥ t♠ ♥tr ♦r t♥ Ts
r♥ t ♠♣♠♥tt♦♥ ♦ t ♦rt♠ ♥ t ①♣r♠♥t st♣ ♥♠r♦ ♦rr♥s ♦r t Ts = 100[µs] ♦t r② tst s t ② ts ♥ ♦t rs♦♥ ♥ ♦♥ ♦♥ ♦ t ♦rt♠ s ♠ st s ①♣♥♥ t ♥♠r ♦ sq♥s tt t♦ t t st♣ ♣♥s ♦♥t ♥♣t ♥① ♥ ♦♥ t ♠♥ts ♦ t st A s ♦♥sq♥ r♥ rt♥tst t s③ ♦ t st sq♥ sst ♦ ♦r ♦ ♥♦t s t ♦rr♥ ♦ tr t♠ ♣♣t♦♥
♥ ♦rr t♦ ♦ ♥t ♦rr♥ stt♦♥s t♦ st ①♣r♠♥ts ♥r♣r♦ rsts t s♠♣♥ t♠ s ♥ ♥rs t♦ Ts = 200[µs] s♦ s s♦♠ s♥t ts ♦♥ t ♣r♦♣r ♦♣rt♦♥ ♦ t ♣r♦♣♦s P ♥♣rtr s ①♣♥ ♥ t rt♦♥s♣ tt ♥s t rr♥t rt♦♥s♥ tt s s t♦ ♣t t ❯ s s ♦♥ t ss♠♣t♦♥ tt t t sr t♥ t t s♠ ♥st♥t ts ②♣♦tss ♦s ♥♦t ♦ t t♦rs ♦ trr♥t rt♦♥s r ♥♦t ♣ ♦♥ ♥ ♣s r♥ t r ♦♣rt♦♥ t sr st♦r ♥ t ❯ t r♥t ♥st♥ts tt ♠♥s ♥tr♦♥ ♥ ♣♣r♦①♠t♦♥♦s② t s♠r t t♠ ♥trs t♥ t rr♥t rt♦♥ ♠sr♠♥tst ss t ♠♣t ♥tr♦ ② t ♣♣r♦①♠t♦♥ t♦ t r t♠ ♦♥str♥t♥ t ♦♥sq♥t ♦ ♦ Ts = 200[µs] ts ♠♣♠♥tt♦♥ ♦ t P ♦s♥♦t ♦ t st♥t♦♥ ♦♥ r♥ rt♥ ♦♣rt♦♥s ♥ ♣rtr t s ♥♦♥ tt st♥t♦♥ ♦ ♦r r♥ s♣ tsts ♦r st ②♥♠s
trt♣
trt♣
t t ♠♥ strt♣ ♥ ♥♦ rr♥t rt♦♥s ♥ ♠sr ②t t ❯r ♠♣t② ♥ ts s t rr♥t ♣rt♦♥ ♥♥♦t ♦t♥ ♥ ❬❪ ♥ ❬❪ tstrt♣ ♦ t P s ♥♦t rss r t♦ r♥t s♦t♦♥s r ♣r♦♣♦s
rst s② s♦t♦♥ ♦ r♣rs♥t ② ♦s strt♣ ♥ ♣rtrr♥ t rst st♣s t ♠♦ ♦ s t♦ ♣rt t rr♥ts tr tr rt♥ ♥♠r ♦ st♣s ♦r tr t ❯ ♥ ♣t ♦r t rst t♠ t♦♥tr♦ ♦ stt t♦ t ♦r ♣rt♦♥ t s ♦rt t♥ tts♥ ① ♥♠r ♦ st♣s s t rtr♦♥ ♦r t tr♥st♦♥ ♦ ♥♦t st ♦r t ♦♣rt♦♥ ♦ t ♠♥ rs♦♥ s tt t t♠ ♥tr ♥ t♠♦s ♣rt♦♥ s s ♦ st r♥ rt♥ tsts ♦♥ t ❯t♦ ♣t ♦r t ♠♦r ♣rt♦♥ s strt t t ♦ ♥♦t s♦ r♥♦tr tsts ♥ ♦tr ♦rs t ♦ ♦ ts trs♦ s ♥♦t ♥q ♥ t s s ♦♥t ①♣r♥ ♦ t s♥r ss t ♠♥ r s tt ts ♠t♦ ♦rqr t ♥♦ ♦ t ♠♥ ♣r♠tr t♦ ♠s♠t t♥ t♠♦ ♥ t t ♣♥t ♦ rs♦♥② ♣t ♦r ♠t ♥♠r ♦ t♠st♣s ♥ ts ♣rt ♦ t ♦r t s ♥ t♦ ♦♠♣t② ① r② s♦t♦♥tt rqrs t ♥♦ ♦ t ♠♦ ♥ ♦♠♣t② ♠♦r ♦♥tr♦s♠ ♦r ts rs♦♥ t ♠♦s strt♣ s ♥♦t ♥ ♠♣♠♥t
st strt♣ ♣r♦r s ♥ ♦♣ ♥ ♠♣♠♥t s t♦♦t♣t t t ♦t t♦rs ♦r t ♠♥ s strt ♥ ts ② t ❯r ♥ t ♠♥ ♥ strt r♥♥♥ s♥ t ♣rt♦♥ s ♦♥ t st♦rrr♥t rt♦♥ ♠sr♠♥ts sq♥ ♦ ♥①s r♥ ts ♣r♦r s t♦♦ tt t rr♥ts ♥rs ♦r rs r♥ ♥r♦s ♦r t ♠♥♦ ♦ ts sq♥ ♦ ♦♣♣♦st t♦rs s ♣r♦ ♦r t s ♦ rt② ♥♠rs♠t♦♥ s ♥ rr ♦t ♥ t rsts r s♦♥ ♥ t ♥ s♥tt t sq♥ s ♦rr t♦ ♣r♦ ♦♣♣♦st t♦rs 1− 4 2− 5 ♥ 3− 6 ♥ts ② t rr♥t rt♦♥s ♦♣♣♦st rt♦♥s ♦♥ ♥r♦s ♦rt ♠♥ s♥ ♦ t rr♥t rt♦♥ ♥r ♦t t♦r ♣♥s ♦♥t ♣♦st♦♥ ♦ t ♠♥ ♥ ts s ϑme = −π/2 ♥ t ①♠♣ ♥① s♥ ♦r 10 st♣s t♦ r② s♦ t rsts t s ♦rt t♥ tt t s♠♣r♦r ♦ ♦♥ ♣♣②♥ ♥① ♦r st ♦♥ st♣
0 10 20 30 40 50 60 7001234567
Vol
tage
inde
x [−
]
0 10 20 30 40 50 60 70−0.5
0
0.5
d−ax
is [p
.u.]
0 10 20 30 40 50 60 70−0.2
0
0.2
q−ax
is [p
.u.]
Time step [−]
r ♠t strt♣ ♣r♦r
♣r♦♣♦s strt♣ ♣r♦r s ♥ ♠♣♠♥t ♥ tst ♥ t rstsr r♣♦rt ♥ t ♦ t strt♣ ♣r♦r r s♦♥ ♥
♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦
t ♥ s♥ tt tr t ttr s ①t s♣ st♣ s ♥ s rr♥ t♦ ts♣ rt♦r ♦♥st♥t ♦ ♦ 0.3[p.u.] s ♣r♦ ② t ♠str ♠♥ ♥st♦ t st ♣r♦r t ♣rt rt♦r s t♦ r♦♥③ t♦r s tst t♦ r t rr♥t rr♦r ♥ ♥ tr♥ t♦ tr t s♣ rr♥
0 500 1000 1500 20000
50
100
Spe
ed [r
pm]
0 500 1000 1500 2000−5
0
5
d−ax
is [p
.u.]
0 500 1000 1500 2000−5
0
5
q−ax
is [p
.u.]
Time step [−]
♣ ♥ rr♥ts
0 10 20 30 40 50 60 7001234567
Vol
tage
inde
x [−
]
0 10 20 30 40 50 60 70−0.5
0
0.5
d−ax
is [p
.u.]
0 10 20 30 40 50 60 70−0.1
0
0.1
q−ax
is [p
.u.]
Time step [−]
t ♦ t strt♣
r trt♣ ♣r♦r
♦♥s♦♥s
♥ ts ♣tr t ts ♦ ♦t t♦r st♥t♦♥ ♦♥ P s♠s ♥t ♥ ①♠♣ ♦ t ♣♣♥s ♥ ♣rs♥ ♦ ♥rq♥t ❯ ♣t s ♥♣r♦
♥♦ ♣t♥ ♠t♦ s ♥ ♣rs♥t s t♥q ♦s r♦♥strt♥ t rr♥t rt♦♥s s♥ ♦♥② t s rt t♦ t tr ♠♦st r♥t ♦tt♦rs ♣r♦♣♦s s♦t♦♥ s s ♦♥ t rt♦♥s♣ ♥♥ t rr♥t rt♦♥s♥ ② t t ♦t t♦rs ttr ♥ ①♣♦t t♦ r♦♥strt t♦r s ♥ t ❯ strt♥ r♦♠ t ♠♦st r♥t st♦r rr♥t rt♦♥s rt♦♥s♣ ♦s ♥r t ss♠♣t♦♥ tt t t s r t♥ t t s♠♥st♥t r♥t② r♥ t ♠♥ ♦♣rt♦♥ t rr♥t rt♦♥s r ♦t♥t r♥t st♣s ♥tr♦♥ ♥ ♣♣r♦①♠t♦♥ tt s ♥ t t♠ t♥t ♦♥sr s s s♥t② s♦rt ♦r ts rs♦♥ st ♦rt♠ ♦r tsq♥ ♥tt♦♥ s ♥ ♦♣ ♥ ♠♣♠♥t ♠ ♦ t ttr st♦ r♦♥③ tr t ♦t ♥①s ♣ttr♥ ♦r♠s st sq♥ ♦r t ❯♣t ♦rt♠ s ♥ ♦♣ ♦♥sr♥ t strt ♦♥str♥t ♥ ②t r t♠ ♣♣t♦♥ ♥ ♣rtr t ♥♠r ♦ ♣♦ss sq♥s tt s t♦ t s r ♣♥♥ ♦♥ t ♥♣t ♦t ♥①
♣r♦♣♦s ♦rt♠ s ♥ t ② ♠♥ ♦ ①♣r♠♥ts rsts ♥ ♦♠♣r t P s ♦♥ t ♥♦♠♥ ♠♦ ♦ t ②♥ t♦ t tt♥ss ♦ t ♣r♦♣♦s s♠ ♥ ♥r t t ♦ ♣r♠tr rt♦♥s ♥♣rtr t tsts ♥ ♣r♦r♠ t r♥t ♦s t♦ t t ♦r♦ t s♠ t str♦♥ r♦♥ strt♦♥ rsts r② s♦ t ♥ts ♦t ♦r ♣rt♦♥ ♥ ♣rtr t ♣rt♦♥ rr♦r s rt② r ♥ s ♦♥sq♥ t rr♥t ♦r♠s ♣rs♥t r r♣♣
♠tt♦♥s ♦ t ♣r♦♣♦s ♦♥tr♦ s♠ ♥ t ♥ ♣rtr
♦♥s♦♥s
t s ♥ t t ♣rs♥ ♦ stt♦♥s ♥ st♥t♦♥ s ♥♦ ♥ ♥♦ sq♥s ♥ ♥t ♥ s ♦r t ❯ ♣t s t②r♣rs♥ts s♥♥t rtt② ss s♥ t ♣rt♦♥ s s ♦♥ t ♠srrr♥t rt♦♥s t ♣rs♥ ♦ ♥♦ss ♦r str♥s ♦ t t ♣r♦r♠♥ ♦t r ♥ t ♠♣♠♥tt♦♥ tr rr♥t ♠sr♠♥ts ♥ ♦♣ss tr ♦rt ❯ ♥ s t♦ ♦ sss rt t♦ rq♥② ♥♦ss ♥ ♦♠♦♣♦r str♥s ♥tr♦ ② t ♣♦r ♦♥rtr ♦ t ♠str ♠♥ ♥♦trs♥t ss s ♥ ② t ♦♠♣tt♦♥ r♥ ♥ ♦rr t♦ ♦ ♦rr♥ tst s♠♣♥ t♠ s ♥ ♦
♥② ♣r♦r ♦r t strt♣ s ♥ ♣r♦♣♦s ♥ ts ♦♥t♦♥ t ❯r ♠♣t② ♠♥♥ tt t ♣rt rt♦r ♥♥♦t t t t ♦ t t♦t t♦rs rs ♥ ② ♠♦s strt♣ ♥ t♥ tr tt st ♣r♦r s ♥ ♠♣♥t ttr ♦s ♥ t❯ ♦r t ♠♥ strt♣
Chapter 4
♦ Prt ②strss rr♥t
♦♥tr♦
♥ ts ♣tr ♦ Prt ②strss rr♥t ♦♥tr♦ s♠ ♦r ②♥r♦♥♦st♥ ♠♥ s ♣rs♥t ♣r♦♣♦s ♦rt♠ ♣rts t ♣♥t ♦r♦♥sr♥ t ♥t st ♦ t ♣♦r ♦♥rtr stts ♥ t ♦♦ss t ♠♦st st♦t t♦r ♥ ♦rr t♦ ♠♥♠③ t ♦st ♥t♦♥ ttr s s♥ t♦ ♠tr♥t ♦s ♣♥♥ ♦♥ t ♦♣rt♥ ♦♥t♦♥s ♦ t ♠♥ ♥ ♣rtr t①♠♠ ♦rq Pr ♠♣r trt♦r② t ① ❲♥♥ r♦♥ ♥ t ①♠♠♦rq Pr ❱♦t ♦♣rt♦♥ r ♦♥sr ♥ ts ② s♣ r♥ ♦♣rt♦♥s ♥ t rr♥t ♥ ♦t ♠ts r ♥r♥t② ♦sr ss trt ♦st ♥t♦♥ ♦t♣ts rt♥ ♦t t♦r t s♠ t♦r s ♣t ♥t t rr♦r♠♣t ♦s ②♦♥ ① trs♦ ♦r ts rs♦♥ t ♣r♦♣♦s s♠ ♦♥st♦ ②strss ♦♥tr♦
♥tr♦t♦♥
♦♥♥t♦♥ rt ♦rq ♦♥tr♦ s t♥q s ♦♥ t ♦♥tr♦ ♦t t♦rq ♥ t stt♦r ① t♦r ♠♦ t♦t ♥♥ rr♥t ♦♦♣s ♥ t♦ss♠s s♣ rt♦r ♣r♦s t t♦ rr♥s ♥ t ♠♥ ♦♥tr♦r ts t♦♣ t♦rq ♥ ① rr♦rs t♥ ♥ ②strss ♥ ❬❪ ❲ ♥♦♥ ♠rts ♦ tss♠s r t s♠♣ ♥ r♦st ♦♥tr♦ strtr ♥ t st t♦rq ②♥♠ r ♥ ♦♥trst t t t♦rq ♥ rr♥t r♣♣s ♥ t r st♥rq♥② ss s♦rt s♠♣♥ t♠ s rqr t♦ ♣ t t♦rq t♥ t②strss r♦♥ ❬❪
♥ rtr ♦♣♠♥t ♦ t ♦♥♥t♦♥ s♠ t st♥ ♦ t ♥rtr s♥♦t s ♦♥ t t♦rq ♥ ① ②strss ♦♥tr♦r t ♦♥ t ♠♥♠③t♦♥ ♦ ♦st♥t♦♥ s r♣rs♥ts ♠rt♦♥ ♦ t♦rs ♦ Prt ♦♥tr♦ P♥ t ♥ ♥t s P ❬❪
r ♦rs ♣rs♥t s♠s ♦r ②♥ s ♦r ♥st♥ ❬❪ s ♠♥t♣♦♦② trs ♥ ♥r♥t s♣ r♥ ♦♣rt♦♥ r♦r ♥ t ♦♥tr♦
♦ Prt ②strss rr♥t ♦♥tr♦
s♠ ♦r ②♥ ♠♥ s♦ ①♣♦t t ♠♥ ♣ts ♥ ♦♣rt♦♥t♦♥s ♥ ♣rtr ♣♥♥ ♦♥ t s♣ ♦ t ♠♥ t ①♠♠ ♦rqPr ♠♣r P t ① ❲♥♥ ❲ ♥ ①♠♠ ♦rq Pr ❱♦tP❱ trt♦rs t♦ tr s♠ ♦r ②♥ t♦ ♦♥tr♦t ♠♥ ♥ t ♦r♠♥t♦♥ r♦♥s s ♣rs♥t ♥ ❬❪ s♠ s s♦♥ t t② rt♦ rt♦♥ ♥tr♦ ♥ ❬❪ ♦s r♥ t t♦rq r♣♣rt♥♥ t ②♥♠ ♣r♦r♠♥ ♦ t ♦♥♥t♦♥ ss ♥ ♣t stt♦r① rt♦♥ ♦rt♠ s ♦♥ t stts ♦ t t② rt♦ s ♣r♦♣♦s ♦r ❲♦♣rt♦♥
♥ ts ♦r ♥ ♥ ♦♥tr♦ s♠ ♦r s♣ ♦♣rt♦♥ ♦ ②♥♠♥s ♣rs♥t ♠♥ s ♦♥tr♦ t♦ rt② tr t P t ❲ ♥ tP❱ trt♦rs ② ♥tr♦♥ t ♦♣rt♥ ♥ s s ♦s ♦ ♦♣rt♥ ♣♦♥ts♦t t♥ P ♥ P❱ trt♦rs t ♦♥s t ♦r♠r ♦r s♣ ♦rt♥ t s s♣ ♦t ♦r t♥ t ♥♦♠♥ ♦♥ t ♦♥s tt ttr ♦r t st s♣ ♣ t♦ t ♠①♠♠ r s♣
r♥t② r♦♠ ♦♥♥t♦♥ s♠s t ♦rt♠ ♦s ♥♦t ♦♥sr tt♦rq ♥ ① rr♦rs ♦r t s♦♥ ♦ t ♠♦st st ♦t t♦r ♥st trr♥t ♠♣t rr♦r ♥ t st♥ r♦♠ t ♦♣rt♥ ♥ r s t♦ ♦♠♣① rr♦r ♥ tr♥ s s t♦ ♦♠♣t ♦st ♥t♦♥ t♦ ♠♥♠③ t♠ t rr♦r ♠♣t ①s ♥ trs♦ rst tr♠ ♦ t ♦♠♣① rr♦rs t ♦♠♣r♥ t rr♥ ♣r♦ ② t s♣ ♦♥tr♦r ♥ t ♠srrr♥ts s♦♥ ♦♠♣♦♥♥t s ♦t♥ t♥ t st♥ t♥ t t♣♦♥t ♦ ♦r ♥ t ♦♣rt♥ ♥
♥rs♥ t s♣ t tr♥st♦♥ t♥ t P ♥ ❲ s ♦t♥ ②r♦tt♥ t ♦♣rt♥ ♥ r♦♠ t P ♣ t♦ t P❱ trt♦r② ② ♠♥ ♦♥ ♥♥♦t ♦t ♦♦♣ ♦♥♥t♦♥ ♦t ♦♦♣ r s ♦♥ t ♥♦ ♦t s s♣ ♥ tr♥ rqrs t ♠♥ ♣r♠trs ♥r r② ♦r♥♦♥t♦♥ str♥ ♣r♠tr ♥rt♥ts ♥ rt♦♥s ♦ t♦ sr♦rs♥♥ ♦ t r ♣r♦r♠♥ ♦r t♦ ♥♦♥ ② t③t♦♥ ♦ t s ❬❪
♥ ♦rr t♦ ♦r♦♠ ts sss ♥ ts ♣tr t s ♣r♦♣♦s ♥ ♣t ♦rt♠ t♦ r t ♠♥ t s♣ r♥ t ♠♥ ① ♦rt♠ ss ♦♥ t t tt ♥ t ♠ ♦♠s rr t♥ t ♦t t♦♥tr♦ strt② ♥ ♥♦t ♥ ♦t t♦r t♦ r♥ t rr♦r ♥s t♠t r♦r t r♣tt ♦t♦♥ ♦ s♣ ♦♥r♥ rtr♦♥ ♥ ss ♥ ♥①
♥ t ♦ ♦ t ♦t t♦r t♦ ♣♣ s s ♦♥ t ♠♥♠③t♦♥ ♦t ♦st ♥t♦♥ ♣♥s ♦♥ t ♦♠♣① rr♦r ♦♠♣♦s ② t rr♥t ♠♣t rr♦r ♥ t st♥ r♦♠ t ♦♣rt♥ ♥ t ♣r♦♣♦s ♦♥tr♦ s♠s rrr s ♦ Prt ②strss rr♥t ♦♥tr♦ P ss t♦♥s t♦ t ss ♦ s♥ t ♣rts t ♠♥ ♦r ♦♥sr♥ t t♣♦ss ♥rtr stts
t s ♦rt t♥ tt t ♣r♦♣♦s ♦♥tr♦ strt② rs ♦♥ t ♥♦♦ t ♠♥ ♣r♠tr ♦r r② ♦r♥ ♦♥t♦♥
Pr♦♣♦s ♦♥tr♦ s♠
Pr♦♣♦s ♦♥tr♦ s♠
♦r ♦ t ♣r♦♣♦s ♦♥tr♦ s♠ s r♣♦rt ♥ s s ♦♠♣♦s② s♣ rt♦r PIn ♦t ♦♦♣ ❱ ♥ t P ♦ ♦ r♣rs♥ts t ♦r ♦ t s♠
n*
n-
+
MP
HCC
w*=0
*PIn
~SynRM
VL
uz
lim+
z
r ♦♥tr♦ s♠
♣ ♦♦♣
♦t♣t ♦ t P s♣ ♦♥tr♦r s ♥ ♥ ♥ t ♥s t rr♥tt♦r ♠♣t |I∗| ♥ t s♥ ♦ t t♦rq ♠♥ m∗ s q♥tt② r♣rs♥tst rr♥t ♠♣t ♦♠♠♥ t♦ tr t s♣ rr♥ n∗ ♠ts ±µlim ♦ t rt♦r ♣♥ ♦♥ t ①tr♥ ♦t ♦♦♣ ♣r♠tr α ♥ r sr ♥
µ∗ = |I∗| · sign(m∗)
µlim = ±α · IN
r♥ t♥ t rr♥ µ∗ ♥ t t µ st♠t r♦♠ t♠sr rr♥ts r♣rs♥ts t rst ♦♠♣♦♥♥t ♦ t ♦♠♣① rr♦r ♥tr♦ ♥
♣rt♥ ♥
♥r ②♥ ♠♦ s r ♦♥sr ♦r t s ♦ s♠♣t② ♥ ts s tP ♥ t P❱ trt♦rs r strt ♥s ♥ t qrr♥t ♣♥ s ♥♥ ♥ rs♣t② r id ≤ 0 ♥ iq > 0 s ♦r ♠♦t♦r ♦♣rt♦♥
iq = ±id
iq = ±Ld
Lq· id
trt♥ r♦♠ ts t ♦♣ ♥ ts ♦r s t♦ ♦t♥ t rr♥ trt♦r② ② r♦tt♥ t P ♦ ♦ ts t ♦♣rt♥ ♥ w s ♥ ♥ ♥ t♦r♥t ②s rst s♣rt② ♦♥srs t st♥ r♦♠ t P ♥t P❱ stt t wMTPA ♥ wMTPV rs♣t② ♥ t♥ t ts t
♦ Prt ②strss rr♥t ♦♥tr♦
s♠ ♦ t t♦ tr♠s s♦♥ s ♦t♥ strt♥ r♦♠ t ♣♦♥t t♦ ♥st♥ ♦r♠ ♥ ♣rtr sL(β) s t s♦♣ ♦ ♥r ♥ tr♦ t ♦r♥t♥ t P ♥ t P❱ ♥ ♦t t ♥t♦♥s t r♦tt♦♥ s ♦t♥② ♠♥ ♦ t ♣r♠tr β s t ♦t♣t ♦ t ♦t ♦♦♣ sr ♥
w = β · wMTPA + (1− β) · wMTPV
wMTPA = i2d − i2q
wMTPV = i2d −(Lq
Ld
)2
i2q
sL(β) =sign(m∗)
β(Lq
Ld− 1)
− Lq
Ld
w(β) =|iq − sL(β)id|√
1 + sL(β)2
♦r♥ t♦ t t♦ r♥t ♥t♦♥s ♦ t ♦♣rt♥ ♥ t ②strss ♥s ♥ r♣rs♥t ♥ t q rr♥t ♣♥ ♥ ♠♥t ♥ t rs r♣rs♥t t ②strss ♥ ♦♠♣t t ♥ rs♣t② t s♦rt ♥♦t♥ tt s♥ t ♥t♦♥ s♣t t ②strss ♥ s r ♥t rr♦r ♣♥ ts s♣ rsts ♦r♠ ♥ t rr♥t ♣♥ ♣♥♥ ♦♥ t ♣♦♥t♦ ♦r s ♦ rst ♥ ♠♣r♦♣r tr ♦ t rr♥t rr♥ ♥ r♥t♦r ♦ ♦t♥ ♥ t t♦ ①s ♣♥♥ ♦♥ t ♣♦♥t ♦ ♦r ♥ t♦♥trr② ♦s ♦t♥♥ rr s♣ ♥ t ♦ ♣♥♦r ts rs♦♥ tttr s ♥ ♦s♥ ♥ ♠♣♠♥t
−1 −0.8 −0.6 −0.4 −0.2 00
0.2
0.4
0.6
0.8
1
d−axis current [p.u.]
q−a
xis
curr
ent [p
.u.]
=0.25
= 0.5
= 0.75
= 1
=1
=
0.7
=
0
r ②strss ♥ ♥ t q rr♥t ♣♥
st♥ t♥ t t ♣♦♥t ♦ ♦r ♥ t ♦♣rt♥ ♥ r♣rs♥ts ts♦♥ tr♠ ♦ t ♦♠♣① rr♦r ♥ ♥
t s ♦rt ♥♦t♥ tt t ♥t♦♥ ♦ t q♥tt② w ♥ s② ①t♥ t♦t s ♦ strt ♠♥ ♣r♦ tt t P ♥ P❱ ♦ r ♥ ♥♣rtr s♦t♦♥ st ♦r ts s ♦ ♠♣♠♥tt♦♥ ♦♥ssts ♥ ♥r③♥ tt trt♦rs ♦ t ♠♥ s qtt② r♣rs♥t ♥ r tP ♥ t P❱ r ♥r③ ♦♥sr♥ t ♥trst♦♥ t♥ t trs ♥ t rr♥t r♠r♥ ♠t ❯s♥ t ♥r③ trt♦rs ♥tr♦s♥ ♣♣r♦①♠t♦♥ ♥ t ♦♥trr② t s ♣♦♥t ♥ t strt♥ ♣♦♥t ♦ t P❱r t s♠ ♦ t t rs rt ♥t ♦ ts s♠♣ s♦t♦♥ s
Pr♦♣♦s ♦♥tr♦ s♠
♥ ♥ tr♠s ♦ s rst t ♦s ♥♦t rqr t ♦♠♣t ♥♦ ♦ t ♠♥♠♥t rtrsts s♥ ♦♥② t♦ ♣♦♥ts r ♥ ♥♠② t ♥trst♦♥s ♦t t rs t t rr♥t r♠r♥ ss t t ♥r③ rst s♠ ♦rt♠ ♣rs♥t ♦r t ♥r s ♥ s t s♠ ♥t♦♥♦ ♦♣rt♥ ♥ s
−1.5 −1 −0.5 00
0.5
1
1.5
d−axis current [p.u.]
q−a
xis
curr
ent [p
.u.]
MTPA
MTPV
Linearized
MTPA
Linearized
MTPV
(IPM axis convention)
r t ♥ ♥r③ trt♦rs
P ♦
♦r ♦ t s♠ s t P ♦ ♦ r♦♠ t ♦♥tr♦ st t ♦rs s ♥♣t µ∗ r♦♠ t s♣ ♦♥tr♦r t rr♥ ♦ t st♥ r♦♠ t♦♣rt♥ ♥ w∗ = 0 ♥ t ♦♥t β r♦♠ t ♦t ♦♦♣ rtr♠♦r trr♥ts ♥ t ♣♦st♦♥ r ♠sr r♦♠ t ♠♥ ♥ ♣r♦ t♦ t P ♦ ♦t♣t s t ♦t t♦r uj s rt② ♣♣ t♦ t ♣♦r♦♥rtr
♠r② t♦ ❬❪ ♥ rr♦r t♦r E = [Eµ Ew]t s ♥tr♦ r Eµ Ew
r t ♥♦r♠③ rr♦r ♦♠♣♦♥♥ts E s t rr♦r ♠♣t ♥ φE ts ♣s
Eµ =µ∗ − µ
INEw =
w∗ − w
I2N
E =√
E2µ + E2
w φE = tan−1
(Ew
Eµ
)
trt ♦ t ♦♥tr♦ s t♦ ♦sr ♣♥ t ♠♦ ♦ t rr♦rt♥ r♠r♥ ♦ rs Emax
E ≤ Emax
t ♥st♥t E rs t ♠t t ♦♥tr♦ ts ♦♥ t ♥ ♦t ♥ ♦rrt♦ r♥ t ♣♦♥t ♦ ♦r ♥s t ♠t r ♥ ♣rtr t ♦t t♦r t t♥①t st♣ s ♦♦s♥ ♠♦♥ ♦♥ ♦ t t ♣♦ss ♥rtr stt u
z t z ∈ Q♥ ♦rr t♦ r t rr♦r E s♦ tt s ♦sr ♥ st ♦tt♦r uk1 s t♦ r♥t rt♦♥ ♦ E t ♦♥r♥ ♦♥t♦♥ s ♦t♥② ♠♣♦s♥ ♥t rt ♦ t rr♦r ♠♣t r t ♦t st♥s ♦rrt
d
dtE ≤ 0
d
dt
√
E2µ + E2
w =EµEµ + EwEw
E≤ 0
♦ Prt ②strss rr♥t ♦♥tr♦
r♥t rtr ♥ ♦♣t ♦r t ♦ ♦ t ♦t s♣t t♦r ♠♦rt♥ ♦♥ ♠ts ② ♦♦ r♥t rqr♠♥ts r♥ t st♥rq♥② ♦r t st♥ ♦sss ♥ ❬❪ t ♦rt♠ ♦♦ss t s♣t t♦r tt♠♥♠③ ♠①♠♠ ♥t t rt ♦ t rr♦r r♥t② ♥ ts ♦rt ♦♥tr♦ strt② ♣rts t trt♦r② ♦ t ♦r♥ ♣♦♥t ♦r t t ♣♦ss♦t t♦rs u
z ♥ t sts t ♦♥ tt ♣s t rr♦r t♥ t ♠t ♦r t♦♥st t♠
P ♦♣rt♦♥ ♥ sr s ♦♦s t ♥st♥t t rr♥ts ik
♥ t s♣ ωk r ♠sr t ♦ts uk r ♥♦♥ r♦♠ t ♣r♦s st♣ q rr♥ts t t ♥①t s♠♣♥ t♠ i
k1|k r ♣rt t ♥ ♣rt rr♥ts r s t♦ t µk1 wk1 ♥ t rr♦rs Ek1
µ Ek1w
Ek1 s t ♣rt rr♦r ♦srs t♥ t ♥①t ♣♣ ♦t t♦rs t s♠ s ♥ t t st♣ trs t rt ♦ t rr♦r s ♦♠♣t♦r t t ♣♦ss ♥rtr stts uz t z ∈ Q ♥♦♥ t ttr t s ♣♦sst♦ st♠t t rr♦r trt♦rs ♥ t t♠ ∆T z ♥ t ♣♦♥t st② t♥t ♠t Emax ♦s♥ ♦t t♦r s t ♦♥ tt ♥ts t ♠①♠♠ ♦ ∆T z ♥ t tt t ♥st♥t r♦r t ♦st ♥t♦♥ f t♦ ♠♥♠③ s r♣♦rt ♥
f =1
∆T z
♦r t s ♦ rt② t ♦♣rt♦♥ ♦ t ♦rt♠ r r♣rs♥t ♥
μk1
E
< E max
: min fz
= 1/ ΔTz
μ*SPEED LOOP OP. LINE
w *
α β
uk ik
ikΔ = A mik + B m uk
ik1|k = ik+ ikΔ
ik1|k
w k1
wEk1 μE
k1
k1
uk1 = uk uz
uk1
E k1
r ♣rs♥tt♦♥ ♦ t P ♦♣rt♦♥
♦ ∆T z ♥ ♦♠♣t s ♦♦s t E t rr♦r t ♥st♥t k♥ s ♥ Prt♥ t rr♦r t t ♥①t ♥st♥t t s ♣♦ss t♦♦♠♣t t rt E
z = [Eµ Ew]t ①♣rss s ♥ ♠♣t Ez s t
s♣ νz ② t ♦♣rt♥ ♣♦♥t ♠♦s ♥ t rr♦r ♣♥ ♦♥ t rt♦♥ ♦E
z
r♣rs♥ts t ♣♦♥t ♦ ♦r ♥ ts trt♦r② ♥ t rr♦r ♣♥ ss♠♥E = Emax r ϕz s t ♥ t♥ t rr♦r ♥ ts rt ♥ γz s ts♦♠♣♠♥tr② s ①♣rss ♥
Ez =√
(Ezµ)
2 + (Ezw)
2 φEz = tan−1
(
Ezw
Ezµ
)
Pr♦♣♦s ♦♥tr♦ s♠
E
E
wE
E
E
Ez
z
z
Ej
Cz
r rr♦r ♣♥
ϕz = π − γz and γz = φE + φEz
ss♠♥ t s♣ νj ♦♥st♥t t ♣♦♥t ♠♦s ♦♥ t ♦r Cj r♠♥♥t♥ t r♠r♥ ♦r t♠ ∆T j ♦♠♣t ♥
∆T z =Cz
vz=
2Emax · cos(ϕz)
Ez
❱♦t ♦♦♣
②♦♥ t s s♣ t ♠ ♦♠s rr t♥ t s ♥ ts♦♥t♦♥ t P srs ♦r t st ♦t t♦r t♦ st t rr♦r t♥t r♠r♥ t ♥♦ s ♦r ♥ s t♦ st♥s tr ts ♥ssr② t♦ t ① ♦ t ♠♥
♦♥t♦♥ r♣♦rt ♥ s ♥♦t st t♦ s s rtr♦♥ t♦ r♦ttt ♦♣rt♥ ♥ t♦ ♦sr♥ t ttr ♦♥t♦♥ ♠♥s ♥♥ t♦r ttr t rr♦r ♠♣t t ♦s ♥♦t ♠♣② t ♦♥r♥ t♦ t r♠r♥♥ ♦tr ♦rs ♦sr♥ t ♠♣t rt♦♥ ♦♥② s ♥♦t s♥t ♥ t rr♦rrt♦♥ s t♦ ♦♥sr s
s♦s t ♥r s ♥ t rr♦r E s ♦ts t ♠t Emax st ♦t t♦r s t♦ r t rr♦r ♠♣t ♥ t♦ st t ♣♦♥tt♦rs t ♠t r s r♣rs♥ts ♦♥str♥t ♦♥ t ♦ t ♥ γz ♥♥ ♥ t ♥ ①♣rss s ♥ s ♥t♦♥ ♦ ϕz
lim = sin−1(Emax/E) ttr ♠ts t s r ♦r t rr♦r t t ♥①t ♥st♥t Ek1 ♥ t ①♠♣♣t ♥ t rr♦r t s s♠r ♠♣t t t trt♦r② ♦s♥♦t r♦ss t r♠r♥ ♦♥t♦♥ s ♦sr s ♥♦t
γz ∈ π ± ϕzlim
cos(γz) < −cos(ϕzlim)
ξz = EµEzµ + EwE
zw + Ez ·
√
E2 − E2max < 0
♦ Prt ②strss rr♥t ♦♥tr♦
E lim
wE
E
z
z
EzTs
Ek1
r ①♠♣ ♦ t rr♦r s r
♦♥t♦♥ s ②s ♦sr ♥ t ♠♥ s ♦ t s s♣ t ♦s ♥♦t ♦ ♥ t ♠ rs t ♠①♠♠ ♦t t♦ tst r♣rs♥ts t rtr♦♥ t♦ r♦♥③ tr t s ♥ssr② t♦ t ①♦ t ♠♥ ♥ t ♥ s t♦ r t ♠♥ ♦♥ t ❲ ♥ t P❱trt♦rs ♥ ♣rtr t ♦rt♠ ♠♦♥t♦rs t ♦ ξz ♥ t rts t♦♣rt♦♥ ♦ t ♠♥ t♥ ♦♥ t ♣r♠tr η ♦r♥② ♦ ♥ sr s ♦♦s tr s t st ♦♥ ♦t t♦r ♦sr♥ ξz < 0ts ♥t s r♦r ♥rs♥ t ♦ t ♦♥tr cntη ≥ 0 s♦♥ ♦♥trcntcycle s ♣t t r② trt♦♥ tr Ncycle st♣s cntη = 0 t ♠♥s ttr♥ t ♦♥sr ♥tr t P ♥r ♦♥ st ♦t t♦r ♥t ♣r♠tr η s r ♦ q♥tt② ∆η ♥ t ♦♥trr② cntη = Ncycle t ♠♥stt t r② ♥st♥t t st ♦♥ ♦t t♦r uz ♦sr ts s t st ♦rt♠ trs ♥rs♥ t ♦ η tr Ncycle st♣s t ♦♥trs cntη ♥cntcycle r ③r♦ ♥ t ♣r♦ss s r♣t
t s ♦rt ♠♥t♦♥♥ tt t ♣r♦r♠♥ ♦ t ♦rt♠ str♦♥② r② ♦♥ t♣r♦♣r t♥♥ ♦ t ♣r♠trs Ncycle ♥ ∆η
♦rt♠ ♦ rt s r♣♦rt ♥ t s ♦rt t♥ tt η ∈[−1; 1] t ♣♣r ♦♥ ♦ t ♥tr ♠♣s t P ♦♣rt♦♥ t ♦r♠t s ♦r t ♠♥ ♠①♠♠ s♣
trt♥ r♦♠ t ♦ η t♦ r♥t ♦♥ts r ♥tr♦ rst♥♠② α s s t♦ ♠t t s♣ ♦♥tr♦r ♦t♣t µ∗ t♥ t r♥ ±α · IN s♣r♦s② sr ♥
s♦♥ ♦t♣t β s s t♦ r♦tt t ♦♣rt♥ ♥ r♦♠ t P t♦ tP❱ trt♦r② s sr ♥
♦r t s ♦ rt② t ♦r ♦ t ♦t ♦♦♣ ♥ t s ♦ t tr♣r♠tr η α ♥ β r r♣♦rt ♥ ♦r r♥t ♣♦♥ts ♦ ♦r ♥♠② tP t ❲ r♦♥ ♥ t P❱ trt♦r② ♥ t q rr♥t ♣♥
♠♣♠♥tt♦♥ ♥ sts
z = 0:7
< 0z
0flag = 1flag =
cnt + flagcnt=
cnt = Ncyclecycle
cnt = Ncycle
+= -=
cnt = 0
=
cntcycle cntcycle= + 1
=
1
-1
y
y
y
y
n
n
n
n
cnt = 0cnt = 0cycle
>0
= 1=
αβ = 1 + = 0
αβy n
η η
η η η
η
η
η η η η η η η η η η
η
η
η
ηη
r ❱ ♦rt♠ ♦ rt
−1 −0.8 −0.6 −0.4 −0.2 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.1
0.1
0.10.1
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
1
1
d−axis current [p.u]
q−a
xis
cu
rre
nt [p
.u]
Current limit
MTPA
MTPV
MTPAη = 1
= 1 = 1
FWη = (1; 0]
= 1 = η
η =(0; -1]= 1+η = 0
MTPV
BASE POINT
MTPV POINT
r ❱♦t ♦♦♣ ♦♣rt♦♥
♠♣♠♥tt♦♥ ♥ sts
P strt② s ♥ ♠♣♠♥t ♥ ①♣r♠♥t② tst s♦st ①♣r♠♥t st♣ ♠♥ t ♦ t ②♥ ♥ t stt♥ ♣r♠trsr r♣♦rt ♥ ♠♣t ♦ t ②strss ♥ s ♥ ① t♦Emax = 2.5e−2
rst tst s ♥ rr ♦t t♦ r② t rr♥t ♦♥tr♦ strt② ♦r t♦♣rt ♦♥t♦♥s ♥ ♣rtr t ②♥ s ♥ r t t ♠str ♠♦t♦rt s♣ r♠♣ strt♥ r♦♠ ③r♦ t♦ s ②♦♥ t s s♣ P s♣rt♦r ♦ t ②♥ ♠♥ s ♥ rt② strt t♦ t ♠①♠♠ µ∗ = α · IN ♥ ts ② t ♣♦♥t ♦ ♦r strt r♦♠ t s ♣♦♥t ♥ ♠♦♦♥ t ♠①♠♠ rr♥t r♠r♥ r♥ ❲ ♥ ♦♥ t P❱ trt♦r②♥ t stt♥ ♦ ts ①♣r♠♥t t ♦♥tr♦ strt② ♥ tst t t ♠♥♦r♥ t ts ♠ts
rsts r r♣♦rt ♥ r t q①s rr♥ts ♥ t ❱ ♣r♠tr r s♦♥ r t t② ♦ t ♦♥tr♦ strt② ♥ ♦♥r♠ s♦♥ tt t
♦ Prt ②strss rr♥t ♦♥tr♦
s t♦ r t ♠♥ ♥ t ♦ s♣ r♥ r♥t ♦♣rt♦♥s ♥♠②P❲ ♥ P❱ r r② ♥t
0 5 10 15 20−1
−0.5
0
d−axis current [p.u.]
0 5 10 15 200
0.5
0 5 10 15 20−1
0
Time [s]
z
αβ
1
q−axis current [p.u.]
1
VL parameters [−]
rr♥t rs♣♦♥s ♥ ❱ ♣r♠trs
−1 −0.8 −0.6 −0.4 −0.2 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.1
0.1
0.10.1
0.2
0.2
0.20.2
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
1
1
d−axis current [p.u.]
q−ax
is c
urre
nt [p
.u.]
♦♠♣t r♥ ♦♣rt♦♥ ♥ qrr♥t ♣♥
r rst tst ♦♥tr♦ strt② t♦♥
rst tst ♦ r♥ t t② ♦ t ♠♦ t t ♦s ♥♦t t ts②st♠ ♥ ts ♦♠♣t♥ss s♥ t P s♣ rt♦r t♦♥ s ♥ ♥t♥t♦♥②① ♣r ♥②ss s rqr t♦ ♥rst♥ t r ♣r♦r♠♥ ♦ t s②st♠♥ ♥ ♣rtr t ♥trt♦♥ t♥ t s♣ ♥ t ♦t ♦♦♣s s♦♥♦t s ♦st♦♥s ♥ ♥stts
r♣♦rts t rsts ♦t♥ ♣r♦♥ ♦♥st♥t s♣ rr♥ t♦ t②♥ ♥ ts s ♦ t s s♣ ♣r♦♥ ♥ ♥rs♥ r♥ t♦rqt t ♠str ♠♦t♦r ♠♣s t ♠♦♠♥t ♦ t ♣♦♥t ♦ ♦r ♦♥ t Ptrt♦r② r♥ t ♦ t♦rq r♠♣ t q rr♥ts ♦s② tr t Ptrt♦r② ♥ t t ❱ ♦t♣ts η = α = β = 1
0 0.5 1 1.5 2 2.5 30
0.5
1Torque [p.u]
0 0.5 1 1.5 2 2.5 3−1
−0.5
0d−axis current [p.u]
0 0.5 1 1.5 2 2.5 30
0.5
1q−axis current [p.u]
Time [s]
♦rq rr♥ ♥ rr♥t rs♣♦♥s
−1 −0.8 −0.6 −0.4 −0.2 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.1
0.1
0.10.1
0.2
0.2
0.20.2
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
1
1
d−axis current [p.u]
q−ax
is c
urre
nt [p
.u]
P trt♦r② ♥ qrr♥t♣♥
r sts ♦ t t♦rq r♠♣ tst
♥② t ♦♥tr♦ s♠ s ♥ tst t s♣ st♣ ♥t s ♦ t s s♣ t ♥ rr♥ s ②♦♥ t rsts ♦ ts①♣r♠♥t r r♣♦rt ♥ t ♥ s♥ tt t ♣r♠tr η ♥ β strtrs♥ t ♥st♥t t = 1[s] s♥ t r♦tt♦♥ ♦ t ♦♣rt♥ ♥ ts s♦♣ sr② r ♥t t ♦♥s t t P❱ ♥ β = 0 tr ts ♠♦♠♥tη ♦s ♦♥ rs♥ t♦ ♥t s ♣r♦♦♥ t rt♦♥ ♦ α ttr s♥♦ t ♦♥ t rr♥ts s♥ t② r ♦ t α · IN ♥ tr♦r t t
♠♣♠♥tt♦♥ ♥ sts
rr♥t ♠♣t ♦s ♥♦t r t ♠t r
0 0.5 1 1.5 2 2.5 3−1
−0.5
0
d−axis current [p.u.]
0 0.5 1 1.5 2 2.5 30
0.5
1
q−axis current [p.u.]
0 0.5 1 1.5 2 2.5 3−1
0
1
VL−parameter [−]
Time [s]
ηαβ
rr♥t ♥ ❱ ♣r♠tr
−1 −0.8 −0.6 −0.4 −0.2 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.1
0.1
0.10.1
0.2
0.2
0.20.2
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
1
1
d−axis current [p.u]
q−ax
is c
urre
nt [p
.u]
P t♦ P❱ tr♥st♦♥ ♥t qrr♥t ♣♥
r sts ♦ t s♣ st♣ tst
st♥ rq♥② ♦ t ♣r♦♣♦s ♦♥tr♦ s♠ s ♥ t ♥♦♠♣r t tt ♦ trt♦♥ ♣ ❱t♦r ♦t♦♥ ❱ strt② r♥t tsts ♥ ①t t♦ r② t ♦r ♦ t s♠ ♥ r② ♦r♥♦♥t♦♥ rt♦ t♥ t ♥♠r ♦ ♦♠♠tt♦♥s ♦ t t♦ s♠s ♥♥ s ♥ s s ♥① ♦ ♦♠♣rs♦♥ t s ♥ r tt t rt♦ s♥ t r♥ Nsw = 4 − 5.5 ♠♥♥ tt t ♣r♦♣♦s s♠ 4 t♦ 5.5 t♠s♦r ♥♠r ♦ ♦♠♠tt♦♥s
Nsw =NSVM
NMPHCC
s♦s t ①♣r♠♥t tst rr ♦t t♦ t t ♥♠r ♦ ♦♠♠tt♦♥ ♥ t P r♦♥ s♦s t s♣ t rr♥ts ♥ t ♣♦♥t ♦♦r ♥ t q ♣♥ s♦s t ♥♠r ♦ st♥ ♦♥ ♣s ♥t t♦t t ♥ ♥♦t tt t ♦♠♠tt♦♥s r q② strt ♦♥ t tr♣ss
0 0.5 10
100
200
300
Time [s]
Sp
ee
d [rp
m]
0 0.5 1−1
0
1
Time [s]
Cu
rre
nt [p
.u]
−1 −0.5 00
0.2
0.4
0.6
0.8
1
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
d−axis current [p.u.]
q−a
xis
cu
rre
nt [p
.u.]
id
iq
P tst
0 0.5 10
0.2
0.4
0.6
0.8
1
Nmax
=x60000
Timex[s]
Sw
itch
ing
xCo
un
ter
SVM
MP−HCC
0 0.5 10
0.2
0.4
0.6
0.8
1
Singlexphase:xNmax
=x20000
Timex[s]
Sw
itch
ing
xCo
un
ter SVM
MP−HCC
♠r ♦ st♥
r ♦♠♣rs♦♥ t♥ ❱ ♥ P
t s ♦rt t♥ tt t rsts r♣♦rt ♥ ts st♦♥ s♦ r♠r rr♥t r♣♣ rs♦♥ ♦ t ttr s t♦♦ st t ♣rt♦♥ s ♠①♣♦t♥ t ♥♦♠♥ ♠♦ ♦ t ②♥ s ①♣♥ ♥ ts ss ♥
♦ Prt ②strss rr♥t ♦♥tr♦
♥rs r♣♣ ♥ t rr♥t ♦r♠s ss ts s ♥ ②strss s ♦♥tr♦tt ♥r♥t② ♣rs♥ts ♥ r r♣♣ ♦♠♣r t♦ ♦♠♠♦♥ rr♥t ♦♥tr♦ ♦♥srt♦♥s ♠ ♦♥ t ♥♠r ♦ st♥ t tt t ♣r♦♣♦s s♠ ♦r ♥♠r ♦ ♦♠♠tt♦♥s ♦r ts rs♦♥ t rr♥t r♣♣ ♦ ♠tt ② ♥rs♥ t rq♥② t♦t ①♥ t ♥♠r ♦ ♦♠♠tt♦♥s♦t♥ t trt♦♥ ❱ ♦♥tr♦ strt②
♦r ♦ t s②st♠ t Ts = 30[µs] s ♥ t t s♠t♦♥ rsts r r♣♦rt ♥ t ♥ s♥ tt t r♣♣ s s♥♥t②r r♦♠ 30.8% t♦ 14.8% ss t ♥♠r ♦ ♦♠♠tt♦♥s s st ♦r t♥♥ ❱ s♥ t rt♦ s Nsw = 3.3 ♥② t s♦ ♥♦t r♦♠ t st ♣♦t♦ tt t ♠♣t ♦ t rr♦r E s rst② rs ♥ t ♠♦st②r♠♥s ♦ t ♠t Emax ♦r ts rs♦♥ t ②strss ♥ ♠♣t ♥ rs♦♥② r
0 0.05 0.1 0.15 0.2 0.25 0.3−0.6
−0.5
−0.4
d−ax
is [p
.u.]
0 0.05 0.1 0.15 0.2 0.25 0.30.4
0.5
0.6
q−ax
is [p
.u.]
0 0.05 0.1 0.15 0.2 0.25 0.30
0.05
0.1
Err
or [−
]
Time [s]
rr♥ts ♥ rr♦r
−1 −0.8 −0.6 −0.4 −0.2 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.1
0.1
0.10.1
0.2
0.2
0.20.2
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.40.
5
0.5
0.50.
6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
1
1
q−ax
is c
urre
nt [p
.u]
d−axis current [p.u]
P♦♥t ♦ ♦r ♥ t q rr♥t♣♥
r ♦♠♣rs♦♥ t♥ Ts = 100[µs] ♥ Ts = 30[µs]
s♠ tst s ♥ r♣t t ②strss ♥ rsts r s♦♥♥ r ♦♠♣rs♦♥ t s ♦ Ts = 100[µs] s r♣♦rt ♥ t②strss ♥ ♦s rtr rs ♦ t rr♥t r♣♣ ♥ ts tst ts s8.3% t s ♦rt ♠♥t♦♥♥ tt t ♥♠r ♦ ♦♠♠♥tt♦♥ ♥ ts tst sst ♦r t♥ ❱ ♥ Nsw = 1.5
0 0.05 0.1 0.15 0.2 0.25 0.3−0.6
−0.5
−0.4
d−ax
is [p
.u.]
0 0.05 0.1 0.15 0.2 0.25 0.30.4
0.5
0.6
q−ax
is [p
.u.]
0 0.05 0.1 0.15 0.2 0.25 0.30
0.05
0.1
Err
or [−
]
Time [s]
rr♥ts ♥ rr♦r
−1 −0.8 −0.6 −0.4 −0.2 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.1
0.1
0.10.1
0.2
0.2
0.20.2
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
1
1
q−ax
is c
urre
nt [p
.u]
d−axis current [p.u]
P♦♥t ♦ ♦r ♥ t q rr♥t♣♥
r st t ②strss ♥
♦♥s♦♥s
♦♥s♦♥s
♥ ts ♣tr ♥ ♥ ♦♥tr♦ s♠ ♦r s♣ ♦♣rt♦♥ ♦ ②♥ ♠♥s ♣rs♥t s♠ ♦♥s t♦ t ss ♦ ♦ Prt ♦♥tr♦ s♥ t♦rt♠ s s ♦♥ t ♠♥♠③t♦♥ ♦ t ♦st ♥t♦♥ s♦t♦♥ ♦t♥ ②ts ♣r♦ss s t ♦♥ tt ♦s ♣♥ t rr♦r t♥ t ②strss ♥ ♦r t♦♥st t♠
♠♦ s ♦♣ ♦♥sr♥ ♥r ②♥ ♥ ♦t t ♣rt♦♥ ♥ ♥t rr♥ ♥rt♦♥ ♥ t ♥t♦♥ ♦ t ♦♣rt♥ ♥ rtss s♠♣ ♠t♦ t♦ ♥ t ts ♦ r♦♥ strt♦♥ s ♥ ♣r♦♣♦s
♣r♦♣♦s ♦♥tr♦ s♠ ♦s t ①♣♦tt♦♥ ♦ t s②♥r♦♥♦s rt♥ ♠♥ ♣♦t♥ts ♥ ♣rtr t s ♥ s♦♥ t t♥ss ♦ t♦rt♠ ♥ ♦♣rt♥ ♣♦♥ts ♥r P ❲ ♥ P❱ trt♦rs s♣ ♣ts r ② r♦tt♥ t ♦♣rt♥ ♥ ♥ r♥ t P s♣♦♥tr♦r ♠ts ② ♠♥ ♦ ♥ ♥♥♦t ♦t ♦♦♣ ttr rs ♥♦r♠t♦♥♦♥ t rr♦r stts r♦♠ t ♦♥tr♦ strt② t ♥r♥t② r♦♥③s ♥ t ♦t♠t s r ♥ t ♦♣rts ♦r♥②
♥② ♦♠♣rs♦♥ t trt♦♥ ❱ strt② s ♥ ♠ ♥ ♣rtrt ♥♠r ♦ st♥ ♦ t t♦ s♠s s ♥ t ♥ t ♣r♦♣♦ss♠ ♦♥s t♦ ②strss ♦♥tr♦ ① t s♠♣♥ t♠ t ♥♠r ♦ ♦♠♠tt♦♥s s s♥♥t② ♦r t♥ ♥ ❱ s ♦ t♦ rs t s♠♣ t♠ t♦r t rr♥t r♣♣ ttr s s t ♠♦s ♣rt♦♥ tt rs♦♥ t ♥♦♠♥ ♠♦ ♦ t ②♥ ss ②strss ♦♥tr♦ s♠s ♦♠♠♦♥②♣rs♥ts ♥ ♥rs r♣♣
♦♥tr♦ s♠ s ♥ sr ♥ t ♥ r ② ♠♥ ♦ ①♣r♠♥ts rsts ♦t♥ ♦♥ t tst ♥ s♦ ts st② ♥ ts t♥ss ♥♣rtr ♦♦ ♣r♦r♠♥ ♥ t ♥ t ♦ s♣ r♥ ♥ ♥t tr♥st♦♥ t♥ r♥t ♦♣rt♦♥ ♦ t ♠♥
♦♥s♦♥s
♥ t rst ♣rt ♦ ts ♦r t ♠♦ ♦ t s②♥r♦♥♦s rt♥ ♠♥ s ♥♦♣ tr qt♦♥s ♥ ♦t♥ ♦♥sr♥ t ♠♦st ♥rs ♦ tr ♣s tr ♠♥ t ♦② tr ♣s ♠♥ t s♦rt t♥ tt ts ♠♦ ♦ r♣rs♥t t ♦r ♦ r② ♥ ♦ s②♥r♦♥♦s ♠♥ ♥♥ ♣r♠♥♥t ♠♥t ♥ rt♥ ♠♥s ♠♦s ♥ ♦♣ t t ♦♥② ss♠♣t♦♥ ♦ ♥♦ ②strss ♥ ♥♦ ② rr♥tstr tt t qt♦♥s ♥ ♦t♥ ♥tr♦♥ t ②♣♦tss ♦ ♥rt② ♥t♥ r♦♥ strt♦♥ tr ♣s ♠♦ s ♥ t♥ tr♥s♦r♠ ♥ tq♥t t♦ ♣s s②st♠ ♥tr♦♥ ♥r rr♥ r♠ qt♦♥s ♥ ♥② ♦t♥ ♦♥sr♥ stt♦♥r② ♥ s②♥r♦♥♦s② r♦tt♥ rr♥r♠
rsts ♦t♥ ♦r t ♦② ♠♥s ♦t t tr ♣s ♥ t t♦♣s ♥ s t♦ ♦♣ t ♠♦ ♦ t ②♥r♦♥♦s t♥ ♠♥② ♠♣♦s♥ s♣ ♦♥t♦♥ ♦♥ t r♦t♦r rr♥ts ♥ ♣rtr t♥ ♥ r♦t♦rrr♥ts ♥ t ♠♦ ♦ t ♦② ♠♥ t s ♣♦ss t♦ r♣rs♥t ts ♥ ♦♠♥ ♥②ss r♣♦rt ♥ ts ♣tr r t ss ♦ ts ♦r ♥ r str♦♦t t tss
Prt rr♥t ♦♥tr♦ s ♥ sr ♥ ♥②s ♥ t s♦♥ ♣tr♦♠ ♦♥srt♦♥s ♦t t ♦ ♦ t ♣r♦♣r ♦st ♥t♦♥ ♥ s ♦ s♥t♠♥s ♥ ♣r♦ ♦♥sr♥ t s ♦r ♥ ♣rs♥ ♦ r♦t♦r♥s♦tr♦♣② t s ♥ s♦♥ tt t ♦st ♥t♦♥ s ♦♥ t s♦t rr♦r ♦♣rs♥t ♥♦♥ ♦♣t♠ ♦r r♥ ♣rtr ♦♣rt♦♥s rst♥ ♥ ♦ ②♥♠s♥ t ①s t t ♦r ♥t♥ t s ♥ s♦♥ tt ♥tr♦♥ t♥t♦r q t♦ t s♥② rt♦ ♦s ♦r♦♠♥ ts ♦r ❲t ts s♦t♦♥t s s ♦♠♣♥st ♥ t rr♦r ♦♥ t t♦ ①s r r ♥ ♥ ② ♠♥ r ♦ ts s♦t♦♥ s tt t rqrs t ♥♦ ♦ t s♥②rt♦ t s ♦ t ♥t♥ t s ♥ s♦♥ tt s♥ ♦st ♥t♦♥s ♦♥ t sqr rr♦r ♦s ♦t♥♥ s♥t② ♥ ♦r ♦♥ ♦t①s t♦t ♥② rqr♠♥ts ♦♥ t ♥♦ ♦ t ♠♥ ♣r♠trs ♦r tsrs♦♥s ts ♦st ♥t♦♥ s ♥ ♦s♥ ♥ ts ♦r
♦♣rt♦♥ ♦ t ♣rt rt♦r s ♥ sr ♦♥sr♥ t t♦st♣ ♣rt♦♥ rqr t♦ ♦♠♣♥st ♦r t ②s ♥tr♦ ② t t ♠♣♠♥tt♦♥ tr tt ♦t t ♦s ♥ t ♦r ♣rt♦♥ ♥
♦♥s♦♥s
♣rs♥t ♥ t ♦r♠r s t t ♦ ♣r♠tr ♠s♠t ♥ rt♦♥ ♥♥stt ♥ ♣rtr ♥ ②♥r♦♥♦s t♥ ♣♣t♦♥s r♦♥ strt♦♥ s♥ r♦♥s s t ♠♥ rs♣♦♥s ♦r ♠♦ ♥qs
♦r ♣rt♦♥ s ♥ rtr ♥②s ♥ ♣tr tr ♣r♦♥ s♦♠♦♥srt♦♥s ♦♥ t tr♠♥t ts ♦ ♦t t♦r st♥t♦♥ ♦ ♦r♦♠♥stts ♥ t ♦ss ♦ ♦♥tr♦ ♥ ② ♥ ♥rq♥t ❯ ♣t ♥♦ ♣t♥ ♠t♦ s ♥ ♣rs♥t s t♥q ♦s r♦♥strt♥ t rr♥trt♦♥s s♥ ♦♥② t s rt t♦ t tr ♠♦st r♥t ♦t t♦rs rt♦♥s♣ t♥ t t rr♥t rt♦♥s s ♥ s t♦ ♥t② ♣t t❯ st ♦rt♠ ♦r t sq♥ ♥tt♦♥ s ♥ ♦♣ ♥ ♠♣♠♥t ♠ ♦ t ttr s t♦ r♦♥③ tr t ♦t ♥①s ♣ttr♥ ♦r♠s st sq♥ ♦r t ❯ ♣t ♦rt♠ s ♥ ♦♣ ♦♥sr♥t strt ♦♥str♥t ♥ ② t r t♠ ♣♣t♦♥ ♥ ♣rtr t ♥♠r ♦♣♦ss sq♥s tt s t♦ t s r ♣♥♥ ♦♥ t ♥♣t ♦t♥①
♣r♦♣♦s ♦rt♠ s ♥ t ② ♠♥ ♦ ①♣r♠♥ts ♥ t rsts ♥ ♦♠♣r t t♦s ♦ trt♦♥ ♦s ♣rt ♦♥tr♦ t♦t t t♥ss ♦ t ♣r♦♣♦s s♠ ♥ ♥r t t ♦ ♣r♠trrt♦♥s ♥ ♣rtr t tsts ♥ ♣r♦r♠ t r♥t ♦s t♦ tt ♦r ♦ t s♠ t str♦♥ r♦♥ strt♦♥ rsts r② s♦♥t ♥ts ♦ t ♦r ♣rt♦♥ ♥ ♣rtr t ♣rt♦♥ rr♦r s rt②r ♥ s ♦♥sq♥ t rr♥t ♦r♠s ♣rs♥t r r♣♣
♠tt♦♥s ♥ t rtts ♦ t ♣r♦♣♦s ♦♥tr♦ s♠ ♥ t ♥ ♣rtr t ♣rs♥ ♦ stt♦♥s ♥ st♥t♦♥ s ♥♦t s♥stt② t♦ ♥♦ss ♥ str♥s ♥ t ♥rs tr♥r♦♥ t♠ ♦ ts♣rtr ♠♣♠♥tt♦♥ r t ♠♥ sss
♣r♦r ♦r t strt♣ s ♥ ♣r♦♣♦s ♥ ts ♦♥t♦♥ t ❯ r♠♣t② ♠♥♥ tt t ♣rt rt♦r ♥♥♦t t t t ♦ t t♦t t♦rs rs ♥ ② ♠♦s strt♣ ♥ t♥ tr tt st ♣r♦r s ♥ ♠♣♠♥t
st ♣rt ♦ ts ♦r s ♦♠♠t t♦ t ♦♣♠♥t ♦ ♥ ♥ ♦♥tr♦s♠ ♦r s♣ ♦♣rt♦♥ ♦ ②♥r♦♥♦s t♥ ♠♥ s♠♦♥s t♦ ♦t ♦ Prt ♥ ②strss ♦♥tr♦ s♥ t ♦rt♠ s s♦♥ t ♠♥♠③t♦♥ ♦ t ♦st ♥t♦♥ ♥ t rst ♦ ts ♣r♦ss s t ♦♥ tt♦s ♣♥ t rr♥t rr♦r t♥ ① trs♦ ♦r t ♦♥st t♠
♠♦ s ♥ ♦♣ ♦♥sr♥ ♥r ♠♥ ♥ ♦t t ♣rt♦♥♥ ♥ t rr♥ ♥rt♦♥ ♥ t ♥t♦♥ ♦ t ♦♣rt♥ ♥ rtss s♠♣ ♠t♦ t♦ ♥ t ts ♦ r♦♥ strt♦♥ s ♥ ♣r♦♣♦s
t s ♥ s♦♥ tt t ♣r♦♣♦s ♦♥tr♦ s♠ ♦s t ①♣♦tt♦♥ ♦t ②♥r♦♥♦s t♥ ♠♥ ♣♦t♥ts ♥ ♦♣rt♥ ♣♦♥ts ♥r P❲ ♥ P❱ trt♦rs s♣ ♣ts ♥ ② r♦tt♥ t ♦♣rt♥ ♥ ♥ r♥ t P s♣ ♦♥tr♦r ♠ts ② ♠♥ ♦ ♥♥♥♦t ♦t ♦♦♣ ttr rs ♥♦r♠t♦♥ ♦♥ t rr♦r stts r♦♠ t♦♥tr♦ strt② t ♥r♥t② r♦♥③s ♥ t ♦t ♠t s r ♥ t ♦♣rts ♦r♥②
♦♥s♦♥s
♦♠♣rs♦♥ t trt♦♥ ♣ ❱t♦r ♦t♦♥ strt② s ♥ ♦♥♥ ♣rtr t ♥♠r ♦ st♥ ♦ t t♦ s♠s s ♥ t ♥ t♣r♦♣♦s s♠ ♦♥s t♦ ②strss ♦♥tr♦ ① t s♠♣♥ t♠ t ♥♠r ♦♦♠♠tt♦♥s s s♥♥t② ♦r ♦♥sq♥t② t s ♦ t♦ rs t s♠♣t♠ t♦ r t rr♥t r♣♣ ttr s s ② t♦ s♣ts ②strsss s♠s ♥r♥t② ♣rs♥t ♥rs r♣♣ ♥ ss t rr♥t ♣rt♦♥s ♦t♥ s♥ t ♥♦♠♥ ♠♦ tt ♦s ♥♦t ♥ t ts ♦ r♦♥ strt♦♥
tr ♦rs
s tss r♣♦rts ♣rt ♦ t ♦r ♦♥ r♥ t tr ②rs ♦ P sts♥ ♥♦ t t ♥ ♦ ts ♣t r♠② tt tr st r r♦♦♠ ♦r♠♣r♦♠♥ts ♥ rtr ♦♣♠♥ts ②♦♥ tt ♥ ♦♣♥ t♦♣s s r s♦♠tr ♦rs tt ♣rs♦♥② ♦ t♦ ♦ ♠② P s ♥♦t ♠t ♥ t♠
♦r ♥tr♦s rt ♥ts ♥ tr♠s ♦ r② ♦ t ♣rt♦♥ ♥t ♦tr ♥ ts t♥q srs r♦♠ ♦t t♦r st♥t♦♥ tt ♦ t♦♥stt② ts ♣♥♦♠♥♦♥ s ♥♦t t♥ ♥t♦ ♦♥t ♥ ♣r♥t ♥ ts ♦r ♥♦ ♠t♦ t♦ r♥t rq♥t ♥ ♣rs rr♥t rt♦♥s ❯ ♣ts ♥ ♣rs♥t rtss s r② ♠♥t♦♥ s♦♠ rt stt♦♥s ♥ st♥t♦♥ s ♥♦ ♦ ♦r r♥ t ♠♥ ♦♣rt♦♥ r♥t② ♦s ♦♥tr♦ trs ♥ ②s st ♦♣rt♦♥ ♥ ♥r t ts♦ rs♦♥ ♣r♠trs ♠s♠t ♦r ts rs♦♥ t ♦ ♥trst♥ t♦ s ♣rt♦♥ s ♦♥ ♠♦ ♥ t ♣r♠trs r ♥t r♥ t ♠♥♦♣rt♦♥ ♥ ♣rtr t s♠ ♠t♦ s ♦r t ❯ ♣t ♦ ♣tt♦ ♦♣ st ♣r♠tr ♥tt♦♥ ♦rt♠ ♦t♣t ♦ s t♦ t ♦s ♣rt♦♥
♥s t♦ ♠t♦ ♣r♦♣♦s ♥ ts ♦r t ❯ ♦♥t♥s ♣rs ♥♦r♠t♦♥s♦t t t stt ♦ t ♠♥ ♦r ts rs♦♥ t s ♦ t rr♥t rt♦♥s st♦r ♥ t ❯ ♦ s t♦ r♦♥③ tr t ♦t ♠t s ♥r s r♥t tr ♦♠♥ t ♦tr s♦t♦♥s ♦♣ ♥ ts ♦r♦ s t♦ ♦♣ ♦t ♦♦♣ ♦r ① ❲♥♥ ♥ ①♠♠ ♦rq ♣r❱♦t ♦♣rt♦♥s
♦r♣②
❬❪ P ♦rts P ③♠r♦s ♥♥ ♦ ♥ ♦r③Prt ♦♥tr♦ ♥ ♣♦r tr♦♥s ♥ rs r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣
❬❪ ❨♦♥ Pr③ ♥ ♦r③ ♥②ss ♦ ♥t♦♥tr♦st ♠♦♣rt rr♥t ♦♥tr♦ t ♠♦ ♣r♠tr ♠s♠t ♥ tr♣s ♥rtr r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣ ②
❬❪ ❲♥ ❳ ♥ ❨ ❨ ❲♥ ♥ ❳ ♦st ♣rt rr♥t ♦♥tr♦t ♦♥♥ str♥ st♠t♦♥ ♦r ♥t♦♥ ♠♥ rs r♥st♦♥s ♦♥ P♦r tr♦♥s ♦ ♥♦ ♣♣ ♥
❬❪ ❲♥ ❩ ♦♥ ❨ ❨ ❲♥ ♥ ❳ ttrr♦rss t ♣rt rr♥t ♦♥tr♦ s♥ s♦♥♦rr s♥♠♦ str♥ ♦srr ♦r ♥t♦♥ ♠♥ rs r♥st♦♥s ♦♥ P♦r tr♦♥s ♦ PP ♥♦ ♣♣
❬❪ rr ❯ ②s ♥ ♥ r♦st ♣rt rr♥t ♦♥tr♦r♦r ♣♠s♠ rs r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣ ♥
❬❪ ❩ ♥ ♥ ♥ ♣t ♥t♦♥tr♦st ♠♦ ♣rt rr♥t♦♥tr♦ ♦r ♣♠s♠ rs t ♥t♥ rt♦♥ tr P♦r ♣♣t♦♥s ♦ ♥♦ ♣♣
❬❪ ♠ t♦♥♥ ♦♥♠ss♦♥ ♥ ♥ ♦s♥ ♥②ss♦ t ♠♣t ♦ ♦♥♥ ♥tt♦♥ ♦♥ ♠♦ ♣rt rr♥t ♦♥tr♦ ♣♣ t♦♣r♠♥♥t ♠♥t s②♥r♦♥♦s ♠♦t♦rs tr P♦r ♣♣t♦♥s ♦ ♥♦ ♣♣
❬❪ ♥ t ❨ ♥ s♦ ♦r ♣rtrr♥t ♦♥tr♦ ♦r ♥tr♦r ♣r♠♥♥t♠♥t s②♥r♦♥♦s ♠♦t♦r rs s ♦♥rr♥t r♥ tt♦♥ t♥q r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣
♦r♣②
❬❪ ♥ t ❨ ❨ ♥ ❨ ♠♣r♦ ♠♦r ♣rtrr♥t ♦♥tr♦ ♦r s②♥r♦♥♦s rt♥ ♠♦t♦r rs r♥st♦♥s ♦♥♥str tr♦♥s ♦ ♥♦ ♣♣ ♥
❬❪ P ♦rts ♦r③ ♥ ♦rs ② ♦♠♣♥st♦♥ ♥ ♠♦ ♣rt rr♥t ♦♥tr♦ ♦ tr♣s ♥rtr r♥st♦♥s ♦♥ ♥strtr♦♥s ♦ ♥♦ ♣♣
❬❪ ♠ r s③ ♥ ♦r③ ♦st♥ss ♠♣r♦♠♥t ♦ ♣rt rr♥t ♦♥tr♦ s♥ ♣rt♦♥ rr♦r ♦rrt♦♥ ♦r ♣r♠♥♥t♠♥t s②♥r♦♥♦s ♠♥s r♥st♦♥s ♦♥ ♥str tr♦♥s♦ ♥♦ ♣♣ ♥
❬❪ ♦r③ ♥ P ♦rts Prt ♦♥tr♦ ♦ P♦r ♦♥rtrs ♥ trrs ♦♥ ❲② ♦♥s t
❬❪ ♦r③ P♦♥tt P ♦rr P ③♥ P ♦rts ♥ ❯ ♠♠♥♥ Prt rr♥t ♦♥tr♦ ♦ ♦t s♦r ♥rtr r♥st♦♥s♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣
❬❪ ♦♦ rrr♦ r ♦r ♥ ♥stt② t♦ tr♣r♠tr rt♦♥s ♦ ♣rt rr♥t ♦♥tr♦ ♥ ♠t♣s rs ♥ t ♥♥ ♦♥r♥ ♦ t ♥str tr♦♥s ♦t② ♦ ♣♣
❬❪ ❯ ♦♦♥ ♥ Pr Prt♦♥tr♦s rt ♣♦r ♦♥tr♦ t ♥ ♣t ♣r♠tr ♥tt♦♥ t♥q ♦r ♠♣r♦ ♣r♦r♠♥ r♥st♦♥s ♦♥ P♦r tr♦♥s ♦ ♥♦ ♣♣ ♦
❬❪ ♥ P ③♠r♦s rt t♦rq ♦♥tr♦ ♦ ♣♠ ♥rtr ♠♦t♦rs sr② r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣
❬❪ ♦rs♣♦r ♥ Ps ♥♦rss ♣rt rt t♦rq ♦♥tr♦ ♦rs②♥r♦♥♦s rt♥ ♠♥s t r② ♦ ♥ ③r♦ s♣ r♥st♦♥s♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣
❬❪ rt♥ ❱r♥ ♥ rs♥ Prt rt t♦rq ♦♥tr♦ ♦r ①♥ t♦rq r♣♣ rt♦♥ r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣ ♥
❬❪ ♦r③ ♥♥ s♣♥♦③ r♥♦ ♥ ♦s ♣r♦r♠♥ ♦♥tr♦ strts ♦r tr rs ♥ ①♣r♠♥tssss♠♥t r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣
❬❪ ②r P♣♦t♦ ♥ ♦rr ♦ ♣rt rt t♦rq ♦♥tr♦ ♣rt ♦♥♣t ♦rt♠ ♥ ♥②ss r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣ ♥
♦r♣②
❬❪ r♥ P ♦rts ❨③ ♥ ♦r③ Prt t♦rq ♦♥tr♦ ♦♥t♦♥ ♠♥s s ♦♥ stts♣ ♠♦s r♥st♦♥s ♦♥ ♥strtr♦♥s ♦ ♥♦ ♣♣ ♥
❬❪ ❲ ❳ ❳ ❲♥ ❲♥ ❲ ❳ ♥♥ r♥ ♥ ♦r♥③♥t♦♥tr♦st ♠♦ ♣rt t♦rq ♦♥tr♦ t t s♦t♦♥ ♦r♣♠s♠ rs r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣ ♣t
❬❪ Pr♥ ♥ ♦♦♥♥ ♦ ♣rt rt t♦rq ♦♥tr♦ t ♥t♦♥tr♦ st ♦r ♣♠s♠ r s②st♠s ♣rt ①♠♠ t♦rq ♣r ♠♣r ♦♣rt♦♥ r♥st♦♥s ♦♥ ♥str ♥♦r♠ts ♦ ♥♦ ♣♣ ♦
❬❪ ♦ ♣rt rt t♦rq ♦♥tr♦ t ♥t ♦♥tr♦ st ♦r ♣♠s♠ rs②st♠s ♣rt ♥♥ ♦♣rt♦♥ r♥st♦♥s ♦♥ ♥str♥♦r♠ts ♦ ♥♦ ♣♣ ②
❬❪ ♥ ♥ ② ②♥♠ t♦rq ♦♥tr♦ ♦ s②♥r♦♥♦s rt♥♠♦t♦r r♥st♦♥s ♦♥ P♦r tr♦♥s ♦ ♥♦ ♣♣
❬❪ ❳ ❩♥ ♥ ♦♦ r♦st ♥♥ ♦rt♠ s ♦♥t② rt♦ rt♦♥ ♦r rt t♦rq ♦♥tr♦ s②♥r♦♥♦s rt♥ ♠♦t♦r r♥st♦♥s ♦♥ tr♦♥s ♦ ♥♦ ♣♣ ♣r
❬❪ ❨ ♥ ❩ ❩ ♥ rt t♦rq ♦♥tr♦ ♦ ♣r♠♥♥t♠♥t s②♥r♦♥♦s ♠♥ rs t s♠♣ t② rt♦ rt♦r r♥st♦♥s♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣ t
❬❪ ♦♦♥♥ ♥ ♦♥ t ♦r♠t♦♥ ♦ t t strt② ♦r ♦♥r♥ ♥ stt② ♥②ss t ♣♠ ♠♦t♦r r s st② ♥ ♥tr♥t♦♥ ②♠♣♦s♠ ♦♥ ♥s♦rss ♦♥tr♦ ♦r tr rs ♥ Prt ♦♥tr♦ ♦ tr rs ♥ P♦r tr♦♥s P t ♣♣
♥♦♠♥ts
s ss s rtt♥ ♥ ❳♥ tr♦♥ rs♦♥ s t tt♣♣rsr♥♣t