Guidelines for weighting factors design in model predictive control of power converters and drives

Guidelines o Weigh ing Fac o s Adjus men in Fini e S a e Model P edic i e Con ol o Powe Con e e s and D i es Pa icio Co és, Sami Kou o, B uno La Rocca, René Va gas and José Rod íguez Elec onics Enginee ing Depa men Uni e sidad Técnica Fede ico San a Ma ía Valpa aiso, Chile Email: sami [email p o ec ed]g José I. León, Se gio Vazquez and Leopoldo G. F anquelo Elec onics Enginee ing Depa men Uni e si y o Se ille Se ille, Spain Email:[email p o ec ed] Abs ac —Fini e S a e Model P edic i e Con ol (FS-MPC) has eme ged as a p omising con ol ool o powe con e e s and d i es. One o he majo ad an ages is he possibili y o con ol se e al sys em a iables wi h a single con ol law, by including hem wi h app op ia e weigh ing ac o s. Howe e , a he p esen s a e o he a , hese coe icien s a e de e mined empi ically. The e is no analy ical o nume ical me hod p oposed ye o ob ain an op imal solu ion. In addi ion, he empi ical me hod is no always s aigh o wa d, and no p ocedu es ha e been epo ed. This pape p esen s a i s app oach o a se o guidelines ha educe he unce ain y o his p ocess. Fi s a classi ica ion o di e en ypes o cos unc ions and weigh ing ac o s is p esen ed. Then he di e en s eps o he empi ical p ocess a e explained. Finally, esul s o se e al powe con e e s and d i es applica ions a e p esen ed, which show he e ec i eness o he p oposed guidelines o each app op ia e weigh ing ac o s. I. INTRODUCTION. The con inuous e olu ion and g owing capabili ies o mod- e n mic op ocesso s and signal p ocessing echnologies, has enabled he implemen a ion o mo e sophis ica ed con ol me hods de ised o ul ill he indus y’s inc easing demand o highe pe o mance. P edic i e con ol is one o hese me hods, and has gained ecen ly mo e a en ion specially o powe con e e and d i e applica ions [1]. In essence p edic i e con ol is a g oup o di e en con ol me hods ha sha e one common cha ac e is ic, which is, he use o ma hema ical models o he sys em o p edic u u e beha io s and selec app op ia e con ol ac ions. Se e al p edic i e con ol me hods ha e been applied o powe con e e and d i e sys ems, among hem: Dead Bea Con ol [2]–[8], Model P edic i e Con ol (MPC) [9], [10], Gene alized P edic i e Con ol [11], and Fini e S a e Model P edic i e Con ol (FS-MPC) [12]. FS-MPC can be desc ibed as a pa icula case o MPC which akes in o accoun he inhe en disc e e na u e o he powe con e e swi ching s a es and he digi al implemen a- ion. Since powe con e e s ha e a ini e numbe o swi ching s a es, he MPC op imiza ion p oblem can be simpli ied and educed o he p edic ion o he sys em beha io only o hose possible swi ching s a es. The ini e numbe o sys em p edic ions a e used o e alua e a cos unc ion (also known as quali y o decision unc ion), which usually is composed by he e o s o he con olled a iables. Hence, he swi ching s a e associa ed wi h he p edic ion ha minimizes he cos unc ion is selec ed and gene a ed by he con e e . Wi h his app oach he numbe o calcula ions is g ea ly educed, making eal ime implemen a ions easible wi h cu en mic o- p ocesso echnology. FS-MPC has been success ully applied o a wide ange o powe con e e s and d i es applica ions [12]–[28]. One o he majo ad an ages o FS-MPC is ha se e al con ol a ge s, a iables and cons ain s can be included in a single cos unc ion and simul aneously be con olled. In his way adi ional a iables such as cu en , ol age, o que o lux can be con olled while achie ing addi ional con ol equi emen s like swi ching equency educ ion, common mode ol age educ ion and eac i e powe con ol, o name a ew. This can be accomplished simply by in oducing he addi ional con ol a ge s in he cos unc ion o be e alua ed o he di e en swi ching s a es. Howe e , he combina ion o a iables ha mos likely a e o di e en na u e (di e en uni s and di e en o de s o magni ude) in a single cos unc ion is no a s aigh o wa d ask. Each addi ional e m in he cos unc ion has a co esponding weigh ing ac o , which is used o une he impo ance o cos o ha e m in ela ion o he o he s con ol a ge s. These pa ame e s ha e o be p ope ly designed in o de o achie e he desi ed pe o mance. Un o una ely, he e a e no analy ical o nume ical me hods o con ol design heo ies o adjus hese pa ame e s, and cu en ly hey a e de e mined based on empi ical p ocedu es. Al hough his chal- lenge has no kep back FS-MPC o be applied success ully o se e al powe con e e s, i is highly desi able o es ablish a p ocedu e o de ine some basic guidelines o educe he unce ain y and imp o e he e ec i eness o he uning s age. This pape p esen s a i s app oach o add ess his challenge. Fi s some ep esen a i e examples o FS-MPC cos unc ions a e classi ied acco ding o he na u e o hei e ms, in o de o g oup ypes o weigh ing ac o s ha could be uned simila ly. Then a se o simple guidelines a e analyzed and es ed o e alua e he e olu ion o he sys em pe o mance in ela ion o changes in he weigh ing ac o s. Se e al con e e and d i e con ol applica ions will be s udied o co e a wide a ie y o cos unc ions and weigh ing ac o s. In addi ion, esul s o h ee di e en weigh ing ac o s a e p esen ed o compa e esul s and alida e he me hodology. II. FINITE STATE MODEL PREDICTIVE CONTROL OVERVIEW. Conside he gene ic and simpli ied block diag am o FS- MPC illus a ed in Fig. 1, ha con ols a sys em a iable x h ough a con ol ac ion S, usually he ga ing signals o a con e e . The measu ed a iable x( k)is ed back and used o e alua e a disc e e p edic i e model o unc ion o he sys em p, o ob ain he p edic ed u u e alues o he sys em xp i( k+1) o each possible con ol ac ion Si xp i( k+1) = p{x( k), Si} ∀i= 1, . . . , n. (1) No e ha nco esponds o a ini e numbe o con ol ac ions o swi ching s a es. Then he np edic ions oge he wi h he e e ence a e e alua ed in a cos unc ion g, leading o n di e en cos s g gi= g{x∗, xp i} ∀i= 1, . . . , n. (2) Since he a ge is o con ol a iable x, usually he cos unc ion gis de ined by a measu e o he e o wi h espec o he e e ence. Some example o gene ic cos unc ions a e he absolu e e o , quad a ic e o and mean alue o he e o gi=|x∗−xp i(Si)|, gi= [x∗−xp i(Si)]2,(3) gi=1 TsZTS [x∗( )−xp i( , Si)]d . No e ha he nex con ol ac ion S( k+1)will be he swi ching s a e ha minimizes he cos unc ion g S( k+1) = min Si g{x∗, xp i(Si)} ∀i= 1, . . . , n. (4) I is clea ha FS-MPC akes ad an age o he disc e e na u e o powe con e e s by ela ing he swi ching s a e Fig. 1. FS-MPC gene ic simpli ied con ol diag am. di ec ly o he con ol e o . In addi ion, since he swi ching s a e is di ec ly chosen om he cos unc ion minimiza ion, no linea con olle s and modula o s a e necessa y. This con- ol p inciple has been success ully applied o se e al powe con e e and d i e con ol sys ems, including: ol age sou ce in e e s, mul ile el in e e s, ma ix con e e s, egene a i e ec i ie s and o que con ol o ac mo o s, o name a ew [11]–[28]. III. COST FUNCTION CLASSIFICATION. Al hough he cos unc ion’s main objec i e is o keep ack o a pa icula a iable and con ol he sys em, i is no limi ed o only do so. In ac one o he main ad an ages o FS-MPC is ha he cos unc ion admi s any necessa y e m ha could ep esen a p edic ion o ano he sys em a iable, sys em cons ain o sys em equi emen . This lexibili y enables FS- MPC o achie e easie mo e con ol a ge s ha can ansla e o inc eased sys em pe o mance, e iciency, powe quali y, sa e y and o he possible igu es o me i . Since hese e ms mos likely can be o di e en physical na u e (cu en , ol age, eac i e powe , swi ching losses, o que, lux, e c.) i can lead o coupling e ec s be ween a iables, o o o e es ima e he impo ance o one e m espec he o he s in he cos unc ion, making hei p esence no wo h, hence no con ollable. As men ioned be o e his issue has been commonly deal wi h in MPC by including weigh ing coe icien s o weigh ing ac o s λ, o each e m o he cos unc ion g=λx|x∗−xp|+λy|y∗−yp|+...+λz|z∗−zp|.(5) Depending on he na u e o he di e en e ms in ol ed in he o mula ion o he cos unc ion, hey can be classi ied in di e en g oups. This classi ica ion is necessa y in o de o acili a e he de ini ion o a weigh ing ac o adjus men p ocedu e ha could be applied o simila ypes o cos unc ions o alike e ms. A. Cos unc ions wi hou weigh ing ac o s. In hese kind o cos unc ions, only one, o he componen s o one a iable, a e con olled. This is he simples case, and since only one ype o a iable is con olled, no weigh ing ac o s a e necessa y. Some ep esen a i e examples o his ype o cos unc ions a e ob ained o : p edic i e cu en con ol o a ol age sou ce in e e [14], p edic i e powe con ol o a back o back ac/dc/ac con e e [16], p edic i e ol age con ol o an UPS sys em [24] and p edic i e cu en con ol wi h imposed swi ching equency [18], among o he s. The co esponding cos unc ions a e summa ized in Table I. TABLE I COST FUNCTIONS WITHOUT WEIGHTING FACTORS. Applica ion Cos unc ion Cu en con ol o a VSI |i∗ α−ip α|+|i∗ β−ip β| Powe con ol o ac/dc/ac con e e |Qp|+|P∗−Pp| Vol age con ol o UPS ( ∗ cα − p cα)2+ ( ∗ cβ − p cβ)2 Imposed swi ching equency in a VSI |F(i∗ α−ip α)|+|F(i∗ β−ip β)| No e ha all he e ms in a cos unc ions a e composed o a iables o he same na u e (same uni and o de o magni ude). Mo eo e , some a e a decomposi ion o a single a iable in o wo componen s. The e o e, no weigh ing ac o s and hei co esponding uning a e necessa y. B. Cos unc ions wi h seconda y e ms. Some sys ems ha e a p ima y goal o a mo e impo an con ol objec i e ha mus be achie ed in o de o p o ide a p ope sys em beha io , and addi ional seconda y cons ain s o equi emen s ha should also be accomplished o imp o e sys em pe o mance, e iciency o powe quali y. In his cases he cos unc ion p esen s a p ima y and seconda y e ms, whe e he impo ance o he seconda y e m can a y wi hin a wide ange, depending on he applica ion and i s speci ic needs. Some examples a e: p edic i e cu en con ol wi h educ ion o he swi ching equency o imp o e e iciency [26], p edic i e cu en con ol wi h educ ion o common mode ol ages o p e en mo o damage [28], and p edic i e cu en con ol wi h eac i e powe educ ion o imp o e powe quali y [15], [27]. The co esponding cos unc ions a e lis ed in Table II. The impo ance o he second e m, i.e. how much he swi ching equency, he common mode ol age o he eac i e powe a e educed, will depend on he speci ic needs o he applica ion and will impose a adeo wi h he p ima y con ol objec i e, in his case cu en con ol. No e ha in each cos unc ion a weigh ing ac o λis included wi h he co esponding seconda y e m. Hence, sol ing he adeo can be seen as he weigh ing ac o adjus men in he cos unc ion. C. Cos unc ions wi h equally impo an e ms. Unlike he p e ious case, he e a e sys ems in which se e al a iables need o be con olled simul aneously wi h equal impo ance in o de o con ol he sys em. He e he cos unc ion can include se e al e ms wi h equal impo ance, and i is he job o he weigh ing ac o s o compensa e he di e ence in na u e o he a iables. Such is he case o que and lux con ol o an induc ion machine, whe e bo h a iables need o be con olled accu a ely in o de o ha e p ope sys em pe o mance [17]. An o he example is cu en con ol o an neu al poin clamped in e e , in which he dc-link capaci o ol age balance is a mus in o de o educe ol age dis o ion and a oid sys em damage (exceed he pe mi ed ol age le el o he capaci o s, o he wise o e a ed capaci o s should be used) [26]. Bo h cos unc ions a e included in Table III. TABLE II COST FUNCTIONS WITH SECONDARY TERMS. Applica ion Cos unc ion Swi ching equency educ ion |i∗ α−ip α|+|i∗ β−ip β|+λswnp sw Common mode ol age educ ion |i∗ α−ip α|+|i∗ β−ip β|+λcmVp cm Reac i e powe educ ion |i∗ α−ip α|+|i∗ β−ip β|+λQ|Qp| TABLE III COST FUNCTIONS WITH EQUALLY IMPORTANT TERMS. Applica ion Cos unc ion To que and lux con ol 1 T2 en (T∗ e−Tp e)2+λψ ψ2 sn (|ψs|∗− |ψp s|)2 Capaci o ol age balance 1 isn h|i∗ α−ip α|+|i∗ β−ip β|i+λ∆V Vcn |∆Vp c| IV. WEIGHTING FACTOR ADJUSTMENT. The weigh ing ac o uning p ocedu e will a y depending on which ype o e ms a e p esen in he cos unc ion as classi ied in he p e ious sec ion. A. Fo cos unc ions wi h seconda y e ms. This is he easies case o weigh ing ac o adjus men , since he sys em can be i s con olled using only he p ima y con ol objec i e o e m. This can be e y simply achie ed be neglec ing he seconda y e ms o cing he weigh ing ac o o ze o λ= 0. Hence he i s s ep o he p ocedu e is o con e he cos unc ion wi h seconda y e ms in o a cos unc ion wi hou weigh ing ac o s. This will se he s a ing poin o he measu emen o he beha io o he p ima y a iable. The second s ep is o es ablish measu emen s o igu es o me i ha will be used o e alua e he quali y achie ed by he weigh ing ac o . Fo all he examples gi en in Table II a s aigh o wa d quan i y should be one ela ed o he p ima y a iable, which is cu en e o . Se e al e o measu es o cu en can be de ined, in his wo k he oo mean squa e (RMS) alue o he e o in s eady s a e has been used. A leas one addi ional measu e is necessa y o es ablish he adeo wi h he seconda y e m. Fo he h ee cos unc ions o Table II he co esponding measu es ha we e selec ed a e: he de ice a e age swi ching equency sw, he RMS common mode ol age and he s eady s a e inpu eac i e powe . Once he measu es a e de ined, e alua e he sys em beha io wi h simula ions s a ing wi h λ= 0 and inc ease he alue g adually. Reco d he co esponding measu es o each alue o λ. S op he inc emen s o λonce he measu ed alue o he seconda y e m has eached he desi ed alue o he speci ic applica ion, o keep inc easing λun il he p ima y a iable is no con olled p ope ly. Then plo he esul s and selec a alue o λ ha ul ills he sys em equi emen s o bo h a iables. 1) Resul s o swi ching equency educ ion: The esul s o he p e ious p ocedu e o he i s cos unc ion o Table II a e gi en in Fig. 2(a). He e he seconda y e m is aimed o educe he swi ching equency in a cu en con ol o a NPC con e e applica ion [26]. The seconda y e m np sw co esponds o he p edic ed numbe o swi chings in ol ed when changing om he p esen o he u u e swi ching s a e. Thus by inc easing he associa ed weigh ing ac o λsw i is expec ed ha his e m gains mo e impo ance in he cos unc ion and o ces a educ ion in he swi ching equency, e ec ha can be clea ly obse ed in Fig. 2(a). Howe e , a educ ion in he swi ching equency in oduces highe dis- o ion a ec ing he quali y o he load cu en . This adeo is e y clea in Fig. 2(a) since he cu es ep esen ing each 0 0.02 0.04 0.06 0.08 0.1 0 0.1 0.2 0.3 0.4 0.5 RMS cu en e o [A] Weigh ing ac o λsw 0 200 400 600 800 1000 De ice sw [Hz] (a) 0 0.01 0.02 0.03 0.04 0.06 -20 -10 0 10 20 Load cu en [A] 0.05 0 0.01 0.02 0.03 0.04 0.05 0.06 -400 -200 0 200 400 Time [s] Load ol age [V] λ =0.1 sw λ =0 sw λ =0.05 sw λ =0.1 sw λ =0 sw λ =0.05 sw Time [s] (b) Fig. 2. a) Weigh ing ac o in luence o e he cu en e o and he de ice a e age swi ching equency sw. b) Resul s compa ison o di e en weigh ing ac o s (load cu en and load ol age). measu e ha e opposi e e olu ions o he di e en alues o λsw. A sui able selec ion o λsw would be any alue 0.04 ≤λsw ≤0.06 since he cu en e o is s ill below 10% o he nominal cu en (15[A] in his example) and a educ ion om 1000[Hz] o 500[Hz] is achie ed o he a e age de ice swi ching equency. Finally λsw = 0.05 has been selec ed. Figu e 2(b) shows compa a i e esul s o he sys em wo king wi h h ee di e en alues o λsw, one o hem he selec ed alue. No e how he load cu en p esen s highe dis o ion o he la ge alue o λsw due o he s ong educ ion o he numbe o commu a ions. On he o he hand o λsw = 0 he cu en con ol wo ks a i s bes , howe e a expense o highe swi ching losses. Since he NPC is aimed o medium ol age high powe applica ions whe e losses become impo an , he selec ion o λsw = 0.05 me ges e iciency wi h pe o mance. 2) Resul s o common mode ol age educ ion: The guide- lines ha e been used o une he weigh ing ac o o he second equa ion o Table II, which co esponds o p edic i e cu en con ol o a ma ix con e e [28]. He e he addi ional e m Vp cm is he p edic ed common mode ol age o he di e en swi ching s a es and i will be conside ed an addi ional cos by uning p ope ly he weigh ing ac o λcm. The measu es 0 0.1 0.2 0.3 0.4 0.5 0 0.01 0.02 0.03 0.04 0.05 Weigh ing ac o λcm RMS cu en e o [A] 0 25 50 75 100 125 RMS common mode ol age [V] (a) 0 0.01 0.02 0.03 0.04 -200 -100 0 100 200 Time [s] CM ol age [V] λ =0.05 cm λ =0 cm λ =0.15 cm λ =0.5 cm 0 0.02 0.04 0.06 0.08 0.1 -20 -10 0 10 20 Time[s] Load cu en [A] λ =0.05 cm λ =0 cm λ =0.15 cm λ =0.5 cm (b) Fig. 3. a) Weigh ing ac o in luence o e he cu en e o and he common mode ol age. b) Resul s compa ison o di e en weigh ing ac o s (load cu en and common mode ol age). ha will be used o e alua e he di e en λcm a e he RMS cu en e o and he RMS common mode ol age. Figu e 3(a) shows he esul s ob ained ollowing he p oposed p ocedu e. No e ha , like o he p e ious case, simila e olu ions o bo h measu es a e ob ained, i.e, o highe alues o λcm smalle CM ol age a e ob ained, while he cu en con ol becomes less impo an and looses some pe o mance. The esul shows also ha CM ol ages is a a iable mo e decoupled o he load cu en compa ed o he swi ching equency since he cu en e o emains e y low h oughou he wide ange o λcm. Hence he selec ion o a app op ia e alue is easie , and alues o λcm ≥0.05 will pe o m well. This can be obse ed o he esul s shown in Fig. 3, whe e clea ly a no o ious educ ion o he CM ol ages is achie ed wi hou a ec ing he cu en con ol. 3) Resul s o inpu eac i e powe educ ion: The las cos unc ion o Table II co esponds o a cu en con ol o a ma ix con e e [27] wi h inpu powe ac o co ec ion. The addi ional e m in he cos unc ion is di ec ly he p edic ed inpu eac i e powe Qpwi h i s co esponding weigh ing ac o λQ. The measu es used o une λQa e he RMS cu en e o and he inpu eac i e powe . 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.02 0.04 0.06 0.08 Weigh ing ac o λQ RMS cu en e o [A] 0 300 600 900 1.200 Reac i e powe [W] (a) 0 0.02 0.04 0.06 0.08 -20 -10 0 10 20 Time [s] Load cu en [A] λ =0.05 Q λ =0 Qλ =0.3 Q 0 0.02 0.04 0.06 0.08 -4000 -2000 0 2000 4000 Time [s] Reac i e powe [W] λ =0.05 Q λ =0 Qλ =0.3 Q (b) Fig. 4. a) Weigh ing ac o in luence o e he cu en e o and he inpu eac i e powe . b) Resul s compa ison o di e en weigh ing ac o s (load cu en and inpu eac i e powe ). The esul s o he uning p ocedu e a e depic ed in Fig. 4(a). Since his cos unc ion belongs o he same classi ica ion as he p e ious wo, i is expec ed o p esen simila measu e- men s e olu ion when inc easing λQ. As wi h he p e ious case, he inpu eac i e powe seems o be e y decoupled o he load cu en , hence he cu en e o emains e y low o a wide ange o λQ. I becomes easy o ob ain a sui able alue conside ing λQ≥0.05. This can be co obo a ed wi h he esul s gi en in Fig. 4(b), showing an impo an educ ion o he inpu eac i e powe o λQ= 0.05. The p oposed p ocedu e can be p og ammed by au oma ing and epea ing he simula ion in oducing an inc emen in he weigh ing ac o a e each simula ion. An o he way is o educe he numbe o epe i ions, by applying a b anch and bound algo i hm. Fo his app oach i s selec a couple o ini ial alues o λ, usually wi h di e en o de s o magni ude o co e a e y wide ange λ=0, 0.1, 1 and 10, o example. A quali a i e example o his algo i hm is illus a ed in Fig. 5. Then simula e o hese weigh ing ac o s and ob ain he measu es o bo h e ms, M1and M2 o he p ima y and seconda y e ms espec i ely. Then compa e hese esul s wi h he desi ed maximum e o s admi ed by he applica ion and i hem in o an in e al o wo weigh ing ac o s (0.1≤λ≤1 < λ < S a M1(λ0), M2(λ0)M1(λ0.1), M2(λ0.1)M1(λ1), M2(λ1)M1(λ10), M2(λ10) λ = 0 λ = 0.1 λ = 1 λ = 10 M1(λ0.1), M2(λ0.1)M1(λ1), M2(λ1)M1(λ0.5), M2(λ0.5) λ = 0.5 M1(λ0.25), M2(λ0.25) λ = 0.25 M1(λ0.1), M2(λ0.1)M1(λ0.5), M2(λ0.5) 0.1 0.25 Fig. 5. B anch and bound algo i hm o educe simula ions o ob ain sui able weigh ing ac o s. in he example). Then compu e he measu es o he λin he hal o he new in e al (λ= 0.5in he example) and con inue so on un il you achie e a sui able lambda. No e in Fig. 5 ha each ha d line co esponds o a simula ion and dashed lines co esponds o alues al eady simula ed. This me hod educes he numbe o simula ions necessa y o ob ain a wo king weigh ing ac o . The quali a i e example o Fig. 5 can be ma ched wi h he esul s o he common mode educ ion case o Fig. 3(a). No e ha wi h only 7 simula ions he sea ch o λcm would ha e na owed o an in e al 0.1≤λcm ≤0.25 whe e any λcm would wo k p ope ly. B. Fo cos unc ions wi h equally impo an e ms Fo cos unc ions like hose lis ed in Table III, he p oce- du e needs some mino adjus men s since λis no allowed o be ze o. Ano he di icul y is he di e en na u e o he a iables. Fo example, when con olling o que and lux in an adjus able speed d i e applica ion wi h a nominal o que and lux o 25[Nm] and 1[Wb] espec i ely, he o que e o can ha e di e en o de s o magni udes making bo h a iable no equally impo an in he cos unc ion, a ec ing he sys em pe o mance. Thus he is s ep is o no malize he cos unc- ion. Once no malized, all he e ms will be equally impo an and now λ= 1 can be conside ed as s a ing poin . Usually a sui able λis closely loca ed o 1. No e ha he cos unc ions in Table III ha e al eady included his no maliza ion (nominal alues a e deno ed by subindex n). The second s ep is he same as wi h he p e ious p ocedu e, i.e., measu emen s o igu es o me i ha e o be de ined ha will be used o e alua e he quali y achie ed by he weigh ing ac o . The las s ep is o pe o m he b anch and bound algo i hm o Fig. 5 conside ing a couple o s a ing poin s. Na u ally λ= 1 has o be conside ed, and λ= 0 has o be a oided. When a small in e al o weigh ing ac o s has been eached, meaning by small in e al, ha he e a e no big di e ences in he measu es be ween he uppe and lowe bounds o he in e al, hen he weigh ing ac o has been ob ained. -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 RMS lux e o [Wb] Weigh ing ac o log10(λψ) 0 2 4 6 8 10 RMS o que e o [Nm] (a) 0 10 20 30 To que [Nm] 0.8 0.9 1 1.1 Flux [Wb] 0 0.01 0.02 0.03 0.04 0.05 0.06 λ =10 ψ λ =0.85 ψ λ =0.1 ψ λ =10 ψ λ =0.85 ψ λ =0.1 ψ 0 1.5 3 0 1.5 3 0 1.5 3 -5 0 5 Load cu en [A] 0 0.01 0.02 0.03 0.04 0.05 0.06 λ =10 ψ λ =0.85 ψ λ =0.1 ψ Time [s] Time [s] Time [s] (b) Fig. 6. a) Weigh ing ac o in luence o e he lux and o que e o s. b) Resul s compa ison o di e en weigh ing ac o s ( o que s ep esponse, lux magni ude in s eady s a e and load cu en s). 1) Resul s o o que and lux con ol: The p edic i e o que and lux con ol o an induc ion mo o speed d i e [17], can be implemen ed using he i s cos unc ion in Table III. No e ha he e ms appea al eady no malized. The wo e ms in he cos unc ion a e he o que and lux e o . Hence he measu es ha will be sui able o selec he p ope λψwill be he espec i e RMS e o s. A b anch and bound algo i hm s a ing wi h λψ=0.01,0.1,1,10 and 100 i s ga e he in e al 0.1≤λψ≤1, and hen 0.5≤λψ≤1wi h e y small di e ences. Finally λψ= 0.85 was chosen. Figu e 6(a) shows ex ensi e esul s conside ing much mo e alues o λψ(no e ha he alues a e ep esen ed in log10() scale), o show ha he b anch and bound me hod eally ound a sui able solu ion. Resul s o di e en λψ, including λψ= 0.85 a e gi en in Fig. 6(b) o show he pe o mance achie ed by he FS-MPC. Weigh ing ac o log -16 -14 -12 -10 -8 -6 -4 -2 0 2 0 0.05 0.1 0.15 0.2 RMS cu en e o [A] 0 5 10 15 20 RMS Vol age unbalance [V] 10(λ∆c) (a) 0 0.05 0.1 0.15 0.2 0.25 0.3 0 200 400 600 Time [s] Cap. ol ages [V] λ =0 ∆c λ =0.001 ∆c 0 0.01 0.02 0.03 0.04 0.05 0.06 -400 -200 0 200 400 Time [s] Load ol age [V] λ =100 ∆c λ =0 ∆cλ =0.001 ∆cλ =100 ∆c (b) Fig. 7. a) Weigh ing ac o in luence o e he cu en e o and he dc-link capaci o s unbalance. b) Resul s compa ison o di e en weigh ing ac o s (load cu en and dc-link capaci o ol ages dynamic beha io ). No e ha λψ= 0.85 p esen s he bes combina ion o o que s ep esponse and s eady s a e, lux con ol and load cu en wa e o ms. 2) Resul s o ol age balancing: The p edic i e cu en con ol o a NPC con e e [26], can be implemen ed using he second cos unc ion in Table III. The addi ional e m ∆Vp c co esponds o he p edic ed ol age unbalance o he dc-link capaci o s o he con e e . I his unbalance is no con olled, he dc-link ol ages will d i and in oduce conside able ou - pu ol age dis o ion, no o men ion ha he dc-link capaci o s could ge damaged by o e ol age, unless hey a e o e a ed. No e he e ms appea al eady no malized in he cos unc ion, as indica ed in he i s s ep o he p ocedu e. The measu es ha will be used o e alua e he weigh ing ac o λ∆Va e he RMS cu en e o and he peak ampli ude o he ol age unbalance. A b anch and bound algo i hm s a ing wi h λ∆V=10−4, 10−2, 1, 102and 104 i s ga e he in e al 10−4≤λ∆V≤ 10−2a e his i s e alua ion e y small di e ences we e ob ained. Finally λ∆V= 10−3was e alua ed leading o he same measu es. Hence his alue was chosen. Figu e 7(a) shows ex ensi e esul s conside ing much mo e alues (no e ha he alues a e ep esen ed in log10() scale), o show ha he b anch and bound me hod eally ound a sui able solu ion. Resul s o di e en λψ, including λ∆V= 0.001 a e gi en in Fig. 7(b) o show he pe o mance achie ed by he FS-MPC. No e ha o λ∆V= 0, which no mally is no allowed since i s does no con ol he unbalance p oducing he maximum d i o he dc-link capaci o s, he load ol age only p esen s 5 di e en ol age le els, while 9 le els should appea in he load phase-neu al ol age (since he NPC has 3 le els in he con e e phase-neu al ol age). Only 5 appea since he NPC is no gene a ing 3 ou pu le els, due o he ol age d i o i s capaci o s i is only gene a ing 2 le els. On he o he hand, λ∆V= 100 con ols he ol age unbalance e y accu a ely, i e en makes ol age unbalance so impo an in he cos unc ion ha i disables he gene a ion o hose swi ching s a es ha p oduce unbalance elimina ing ol age le els a he ou pu and inc eases he swi ching equency as can be seen in he load ol age o Fig. 7(b). On he con a y, he selec ed λ∆V= 0.001 p esen s he 9 load ol age le els, con ols he load cu en and keeps he capaci o ol ages balanced. V. CONCLUSION In his pape he design o he weigh ing ac o s used in cos unc ions o Fini e S a e Model P edic i e Con ol has been analyzed. A i s app oach based on an empi ical p ocedu e o ob ain sui able weigh ing ac o s has been p esen ed. Fo cos unc ions wi h a p ima y con ol objec i e and seconda y e ms, he s a ing poin is λ=0, hen es inc emen s o λun il he desi ed beha io is ob ained (b anch and bound can also be used). Fo cos unc ions wi h equally impo an e ms, i s no malize he cos unc ion and se λ= 1, wi h his alue he sys em will be con olled, o ine uning use b anch and bound o mo e sligh ly λa ound 1. A leas wo di e en igu es o me i o sys em pa ame e s ha e o be conside ed, depending on he applica ion, o se le he adeo p esen in he designing choice o he weigh ing ac o s. This con ibu ion is a i s design app oach o educe he unce ain y i he co s unc ion design in FS-MPC o sys ems wi h mo e han one con ol objec i e. 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