Experimental evaluation of PI tuning techniques for field oriented control of permanent magnet synchronous motors

Ad ances in Elec ical and Elec onic Enginee ing 114 EXPERIMENTAL EVALUATION OF PI TUNING TECHNIQUES FOR FIELD ORIENTED CONTROL OF PERMANENT MAGNET SYNCHRONOUS MOTORS B. Zigmund 3 , A. Te lizzi 1 , X. T. Ga cia 2 , R. Pa lanin 3 , L. Sal a o e 1 1Poli ecnico di Ba i, Dipa imen o di Ele o ecnica ed Ele onica, V. O abona 4, 70125 Ba i, I aly, sal a o [email protected] 2Uni e si y o Glamo gan, School o Elec onics, CF37 1DL Pon yp idd, Wales, UK, [email p o ec ed].uk 3Uni e si y o Zilina, Facul y o Elec ical Enginee ing, Depa men o Mecha onics and Elec onics, Uni e zi na 1, 010 26 Zilina, SK, b [email protected], pa lani[email p o ec ed].sk Summa y This pape p esen s he expe imen al e alua ion o a commonly used me hodology o une he PI egula o s o he Field O ien ed Con ol (FOC) s a egy o Pe manen Magne Synch onous Mo o s (PMSMs). The me hodology used is based on he Absolu e Value Op imum (AVO) and Symme ic Op imum (SO) c i e ions. These me hods employ a simpli ied model o he plan o be con olled. Due o he complexi y and non-linea i ies p esen in he FOC PMSM d i e some di e gence be ween he ideal esponse and he eal esul s will occu . In his pape a compa ison be ween simula ed and expe imen al esul s is ca ied ou o e alua e he goodness o he solu ion ob ained. The di e gences obse ed be ween simula ions and he esul s ob ained in he expe imen al se up a e shown and i is a emp ed o jus i y he causes o hem. O e all i is concluded ha he me hod p o ides a sa is ac o y ini ial commissioning o he PI egula o s o he d i e sys em unde s udy. 1. INTRODUCTION PID egula o s a e widely employed in indus y due o hei sa is ac o y beha iou in mos o he con ol applica ions. One o he mos impo an enginee ing asks du ing he commissioning o con ol sys em is he pa ame ic op imiza ion o he egula o s o ob ain he desi ed con ol esponse [1]. Di e en me hods can be employed o pe o m he uning o he PID egula o s. A possible classi ica ion o hese me hods can be as ollows: • Expe imen al me hods based on he iden i ica ion o ce ain esponse cha ac e is ics o he sys em. The Ziegle -Nichols open and closed loop me hods a e an example o hese me hods [2]. • Ma hema ical model based me hods. These me hods employ a ma hema ical model ha app oxima es he beha iou o he sys em [1, 3, 4]. • Op imiza ion echniques. By means o a me i unc ion ha can be e alua ed in a es , a numbe o solu ions consis ing on di e en se s o pa ame e s is e alua ed. A he end he bes -pe o ming solu ion is ound. Di e en echniques can be used o pe o m he sea ch o he bes solu ion [5, 6]. In he ield o elec ical d i es PI egula o s a e also employed o mo o con ol. The s uc u e and an equi alen ans e unc ion o his con olle a e shown in Fig. 1. The a iables o be con olled a e gene ally posi ion, speed, o que, cu en o ol age. The ac ha he measu emen o hese signals can con ain conside able noise makes he PI s uc u e wi hou he de i a i e pa mo e sui able. One example o applica ion whe e PI egula o s a e employed is he FOC s a egy o PMSM d i es. PI + + 1/s k p i s 1 i s = = k i k p Fig. 1. PI con olle s uc u e and ans e unc ion This pape p esen s he applica ion o wo popula me hods o uning he PI egula o s o he FOC PMSM d i e: he AVO and SO c i e ions [3, 7]. The model o he PMSM is p esen ed oge he wi h he FOC con ol scheme. The AVO and SO c i e ions a e explained and applied o he sys em unde s udy. Finally some simula ed and expe imen al es s a e pe o med in o de o e alua e he solu ion ob ained. 2. PMSM FOC The model o he PMSM in a o a ing d-q ame ixed o he o o is gi en by he ollowing equa ions: sd sd sd L i ψ = + Ψ (1) sq sq sq L i ψ = (2) sd sd s sd sd sq sq di R i L L i d ω = + − (3) ( ) sq sq s sq sq sd sd di R i L L i d ω = + + + Ψ (4) ( ) 3 2 e sd sq sq sd P i i ψ ψ Γ = − (5) whe e sd ψ , sq ψ , sd , sq , sd i and sq i a e espec i ely he mo o luxes, ol ages and cu en s in d-q axes; ω is he elec ical angula speed, e Γ is he elec omagne ic o que, Ψ is he lux o he pe manen magne and P is he numbe o pole pai s. s R is he s a o esis ance and he s a o induc ance can be di ided in o wo di e en componen s sd L and sq L due o he pa icula i ies o he PMSM. The model is comple ed by he mechanical equa ion, which is de ined as: m e l m d J B d ω ω = Γ − Γ − (6) m P ω ω = (7) whe e J is he ine ia o he mo o and coupled load, l Γ is he load o que, B is he ic ion coe icien and  m is Expe imen al e alua ion o PI uning echniques… 115 he mechanical angula speed. Simila ly o induc ion mo o s, in PMSMs a decoupled con ol o he o que and lux magni udes can be achie ed, emula ing a DC mo o , by means o he FOC s a egy. This is done using he d-q ans o ma ion ha sepa a es he componen s d and q o he s a o cu en esponsible o lux and o que p oduc ion espec i ely [8]. Due o he p esence o he cons an lux o he pe manen magne , he e is no need o gene a e lux by means o he i sd cu en , and his cu en can be kep equal o ze o alue, which in u ns dec eases he s a o cu en and inc eases he e iciency o he d i e. The con ol scheme o he FOC s a egy is shown in Fig. 2. PMSM PI PI d,q a,b,c d,q a,b,c PI + - + - + - s + - + L sq i sq + L sd i sd i sd i sd i sq i sa i sb * * d d Fig. 2. FOC con ol scheme o PMSM The con ol sys em is di ided in o h ee di e en loops: he d loop, which con ols he lux, he q loop, which con ols he o que and he speed con ol loop. The d loop pe o ms he con ol o i sd wi h a cu en PI egula o . The e e ence alue is 0. The q-axis con ol sys em con ains wo con ol loops in cascade. The inne loop con ols he o que by means o con olling i sq wi h a cu en PI egula o . The ac ha he o que can be con olled by means o i sq comes om he ollowing simpli ica ion o (5), alid o Su ace Moun ed (SM) PMSM: 3 2 e sq P i Γ = Ψ (8) The e e ence o his inne loop is gi en by he ou e loop, which con ains a speed PI egula o . F om he ol age equa ions o he PMSM model (3) and (4) i can be seen ha d and q axes a e no comple ely independen and he e a e coupling e ms which depend on he cu en om he o he axis. To achie e comple ely independen egula ion i is necessa y o cancel he e ec o hese coupling e ms a he ou pu o he cu en PI egula o . Wi h he use o decoupling i is achie ed he linea iza ion o he con ol sys em as well as highe dynamics. This decoupling ac ion can be seen in Fig. 2 [9, 10]. 3. PI TUNING WITH THE AVO AND SO CRITERIONS In o de o ob ain a sa is ac o y con ol pe o mance i is necessa y o adjus he pa ame e s o he PI egula o s included in he FOC scheme. In he ield o elec ical d i es wo me hods a e equen ly employed in his pa ame ic op imiza ion: he AVO and SO c i e ions [3, 7]. The AVO c i e ion can be applied o design bo h cu en egula o s, while he SO can be employed o design he speed egula o shown in Fig. 2 [3]. These wo me hods employ an app oxima e ans e unc ion o he sys em o be con olled. Some modelling and pa ame e iden i ica ion o he sys em a e he e o e needed. The AVO me hod assumes he sys em’s ans e unc ion in open loop o he ollowing o m [1]: 1 ( ) 2 (1 ) G s s s τ τ Σ Σ =+ (9) On he o he hand, he SO me hod conside s he sys em’s open loop ans e unc ion as ollows: 2 2 1 4 ( ) 8 (1 ) s G s s s τ τ τ Σ Σ Σ + =+ (10) whe e τ Σ is sum o all small delays in loop. Bo h me hods can be applied when he τ Σ is much smalle han he ime cons an o he sys em. The s ep esponse o bo h ans e unc ions in closed loop inco po a ing he PI egula o designed using he AVO and SO c i e ions is as shown in Fig. 3. Table 1 p esen s he main ea u es o he s ep esponses ob ained wi h bo h me hods. 0 1 2 3 4 5 0 0.5 1 1.5 Time (s) Ampli ude AVO SO Fig. 3. AVO and SO cha ac e is ic s ep esponses Tab. 1. S ep esponses o AVO and SO me hods AVO SO Rise Time 4.7 τ Σ 3.1 τ Σ Se ling ime (2%) 8.4 τ Σ 16.5 τ Σ O e shoo 4.3% 43.4% Phase Ma gin 65.5º 37º I bo h me hods a e compa ed i can be said ha he SO is as e ega ding dis u bance ejec ion, which makes i sui able o he speed loop. On he o he hand AVO has a smalle se ling ime and lowe o e shoo , which makes i app op ia e o ha e quicke and mo e accu a e inne loops [3]. Ad ances in Elec ical and Elec onic Enginee ing 116 a) Cu en loop PI (i sd and i sq con ol) In o de o design he egula o an app oxima ed ans e unc ion has o be de ined acco ding o he AVO c i e ion. Fi s o all, he exis ing delays in he sys em need o be aken in o accoun . These delays in he case o a mo o d i e a e due o he digi al implemen a ion o he con ol (which implies he sampling o signals), he use o il e s, he p ocessing o he con ol algo i hm and he use o Pulse Wid h Modula o s (PWM). Fig. 4 shows he block diag am o he cu en con ol in closed loop wi h all he delays conside ed. + - k p i sq i i sq s 1 i i sq s 1 1 s s 1 1 s 2 s 1 R s 1 L sq R s s 1 1 i sq s 1 1 s 2 s i sq i sq * Mo o VSIP ocessing delayPI egula o Sampling Cu en il e Fig. 4. Cu en loop block diag am To make i possible o apply he AVO c i e ion, he eedback delays mus be ans e ed o he o wa d pa h. The esul ing block diag am is shown in Fig. 5. + - k p i s q i i sq s 1 i i sq s 1 1 2 s i sq s 1 R s 1 L sq R s s i sq i sq * 1 1 s 2 i sq s Fig. 5. Simpli ied block diag am o he cu en loop The ollowing app oxima ion can be made due o ac ha sq i s τ τ = : 1 ( ) (1 )(1 ( /2) )(1 ( /2) )(1 ) sq i s s s G s s s s s τ τ τ τ = + + + + (11) ' 1 ( ) ( ) 1 (2 ) sq i s G s G s s τ τ ≈ = + + (12) In o de o adap he ans e unc ion o he de ini ion o he AVO me hod, he PI egula o ime cons an has o be equal o ime cons an o he plan . This in u ns will dec ease o de o open loop ans e unc ion. sq i sq i s L R τ = (13) 1 1 ( ) 2 (1 ) (1 (2 ) ) sq sq sq sq sq i open i i i sq s i p G s L s s s s k τ τ τ τ Σ Σ = = + + + (14) Finally he alues o sq i p k and sq i i k can be calcula ed as ollows: 2 ; ; 2 sq sq sq sqsq sq sq i i i i i sq p s p i i i i L k k k τ τ τ τ τ Σ Σ = + = = (15) The esul ing closed loop ans e unc ion is o he second o de ype. Because second o de ans e unc ions can be app oxima ed by a i s o de ans e unc ion wi h he same se ling ime, he inne loop can be app oxima ed as ollows: 1 ( ) 1 2 sq sq i close i G s s τ Σ = + (16) In o de o une he PI cu en egula o in d axis i can be ollowed he same p ocedu e desc ibed, bu in his case using he induc ance in d axis sd L . b) Speed loop PI The speed loop is designed by means o he SO me hod. As in he cu en loop, he speed loop con ains some delays ha need o be de ined. Fo his loop, i is common o use a sampling ime en imes highe han he sampling ime o he cu en loop. Fig. 6 shows he ans e unc ion o he speed loop wi h all he delays conside ed and inco po a ing he inne q axis cu en loop ans e unc ion de ined in (16). + - k p i s 1 i s 1 1 s s 3 2 P * G close i sq + - P Js T l 1 1 s 2 s 1 1 s 2 i sq s PI egula o P ocessing delay Cu en con ol and mo o Sampling and speed il e Fig. 6. Speed loop block diag am A simila design p ocess as in he AVO me hod is ollowed o adap he ans e unc ion o he SO c i e ion in he speed con ol loop. Fo simplici y he load o que and ic ion e ms a e neglec ed. The simpli ied block diag am passing he eedback delays o he o wa d loop and g ouping all he delay is shown in Fig. 7. + - k p i s 1 i s 1 1 s 3 P 2 2Js 1 1 s 2 s * Fig. 7. Simpli ied speed loop block diag am The sum o all delays in he o wa d loop ω τ Σ is de ined as: 32 2 2 sq sq i i s s ω ω ω τ τ τ τ τ τ Σ Σ = + + − − (17) Expe imen al e alua ion o PI uning echniques… 117 and he esul ing open loop ans e unc ion has he app op ia e o m o apply he SO me hod: 2 2 2 2 1 1 4 ( ) 8( ) (1 ) 2 (1 ) 3 i open i p s s G s s s J s s P k ω ω ω ω ω ω ω ω τ τ τ τ τ τ Σ Σ Σ Σ + + = = + + Ψ (18) Finally he alues o he egula o can be ob ained as ollows: 2 4 ; ; 3 p i p i i k J k k P ω ω ω ω ω ω ω τ τ τ τ Σ Σ = = = Ψ (19) 4. SIMULATED AND EXPERIMENTAL RESULTS The me hods desc ibed in he p e ious sec ion ha e been employed o une he i sd and i sq cu en egula o s (AVO c i e ion) and he speed egula o (SO c i e ion). The esul ing PI egula o s ha e been es ed using a simula ion model and also an expe imen al se up. The expe imen al se up consis s on a Siemens 1KF7 PMSM wi h he cha ac e is ics shown in Table 2, a Dan oss VLT5006 powe con e e and a DSPACE DS1103 con ol boa d. Tab. 2. PMSM cha ac e is ics (Siemens 1KF7) Nominal Ou pu Powe (P n ) 2135W Nominal Speed ( n ) 3000 pm Nominal To que (M n ) 6.8N·m Nominal Cu en (I n ) 4.4A Numbe o pole pai s (P) 4 S a o esis ance (R s ) 1.09 S a o induc ance (L sd and L sq ) 0.0124H Ine ia (J) 4.15e -4 Kg·m 2 Pe manen Magne Flux ( Ψ ) 0.1821Wb The esul ing se o con ol pa ame e s calcula ed a e shown in Table 3. Tab. 3. Calcula ed PI pa ame e s PI egula o p k i k Ou pu Sa u a ion sd i 8.86 778.6 2 2 n I sq i 8.86 778.6 2 2 n I ω 0.0934 3.18 3 DC V Some addi ional in o ma ion ega ding he con ol is shown in Table 4. Tab. 4. Addi ional con ol pa ame e s Sampling ime o cu en ( s τ ) 100s Sampling ime o speed ( s ω τ ) 1ms DC-link Vol age (V DC ) 380 2 Cu en il e ime cons an 500s Speed il e ime cons an 5ms The i s es pe o med consis s on he speed s ep esponse. A compa ison be ween he ideal esponse ob ained employing he AVO me hod and he simula ed and expe imen al esul s is shown in Fig. 8. I can be seen ha simula ed and expe imen al esul s a e e y simila . In bo h cases he o e shoo is smalle han he ideal esponse. This is due o he ic ion e m o he mechanical equa ion (6) ha inc eases he damping o he sys em. I can be also app ecia ed he noise p esen in he expe imen al esponse and a highe se ling ime. 1 1.05 1.1 1.15 1.2 1.25 1.3 -10 0 10 20 30 40 50 60 70 80 ime [s] speed [ ad/s] Ideal Expe imen al Simula ed Fig. 8. Compa a i e s ep esponses The nex es implemen ed is a speed p o ile which pe o ms a speed e e sal. Fig. 9 and Fig. 10 show he simula ed and expe imen al esul s. The a iables shown a e he mo o elec ical speed ( ), he cu en s (i sd and i sq ), and wo phase cu en s (i sa and i sb ). The simula ed and expe imen al esul s a e also e y simila o he speed p o ile es . These esul s illus a e a sa is ac o y acking o he speed p o ile. The load o que in his es is only due o he ic ion. I can be seen how he i sd is kep o ze o excep o he ansien s whe e some cu en peaks a e p oduced. Figu es 10 and 11 show he simula ed and expe imen al esul s espec i ely o he acking o an i sq p o ile. Fo his es he speed PI is elimina ed and he e e ence o i sq is gi en by a p o ile which includes a o que e e sal. I can be seen ha he expe imen al esul s p esen mo e oscilla ion and dis o ion caused by he noise o he eal sys em and he ope a ion o he powe con e e . Ad ances in Elec ical and Elec onic Enginee ing 118 1 1.5 2 2.5 3 -100 0 100 w [ ad/s] 1 1.5 2 2.5 3 -10 0 10 isq [A] 1 1.5 2 2.5 3 -10 0 10 ime [s] isa,isb [A] 1 1.5 2 2.5 3 -0.5 0 0.5 isd [A] 1.5 2 2.5 3 -100 0 100 1.5 2 2.5 3 -10 0 10 1.5 2 2.5 3 -5 0 5 1.5 2 2.5 3 -10 0 10 ime [s] Fig. 9. Simula ed (le ) and expe imen al ( igh ) esul o he speed p o ile es 0 0.2 0.4 0.6 0.8 1 1.2 -4 -3 -2 -1 0 1 2 3 4 ime [s] isq [A] Fig. 10. Simula ed esul s o he i sq p o ile es 0 0.2 0.4 0.6 0.8 1 1.2 -4 -3 -2 -1 0 1 2 3 4 ime [s] isq [A] Fig. 11. Expe imen al esul s o he i sq p o ile es Expe imen al e alua ion o PI uning echniques… 119 The inal es p esen ed has been ca ied ou only in simula ion o illus a e he e ec o he load o que on he speed esponse. I can be seen ha inc easing he load o que esul s on a bigge ise ime due o he sa u a ion o he speed PI egula o ou pu s. The le el o he sa u a ion he e o e in luences he esponse ime o he speed loop. Fig. 12 shows he speed esponse wi h 3 di e en le els o load o que. 1 1.05 1.1 1.15 0 50 100 150 200 ime [s] speed [ ad/s] 100% Tload 50% Tload 0% Tload Fig. 12. Compa a i e s ep esponses o di e en load o que condi ions 5. CONCLUSION This pape p esen s he expe imen al and simula ed pe o mance o he PI uning AVO and SO me hods applied o a FOC PMSM d i e. The p ocedu e o ob ain he pa ame e s o he PI egula o s has been desc ibed o his applica ion. The esul s o he di e en dynamic es s p esen ed show a sa is ac o y con ol esponse o he speed and cu en loops. O e all he uning me hods employed seem o p o ide a alid solu ion o he ini ial commissioning o he d i e sys em unde s udy. In addi ion he simula ion model employed has p o ed o be e y accu a e and he simila i y wi h he expe imen al esul s is e y high. Any di e gences obse ed be ween simula ed and expe imen al esul s a e due o: noise in he eal se up, inaccu acy o he mo o and load pa ame e s and he ope a ion o he con e e . Acknowledgemen s This esea ch p ojec has been suppo ed by a Ma ie Cu ie Ea ly S age Resea ch T aining Fellowship o he Eu opean Communi y’s Six h F amewo k P og amme unde con ac numbe MEST-CT-2004-504243 Elec ical Ene gy Con e sion and Condi ion. REFERENCES [1] K.J. As öm and T. Hägglund: PID Con olle s: Theo y, Design, and Tuning, 2nd ed. Resea ch T iangle Pa k, NC: Ins um. Soc. Ame ., 1995. [2] J.G. Ziegle and N.B. Nichols: Op imum se ings o au oma ic con olle s, T ans. ASME, ol. 64, pp. 759–768, No . 1942. [3] J. H. H. G oss, J. Hamann and G. 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