ADAPTIVE SELF-TUNING UP MODEL FOR NON-STATIONARY PROCESS SIMULATION

ENGINEERING FOR RURAL DEVELOPMENT Jelgv, 27.-28.05.200. ADAPTIVE SELF-TUNING UP MODEL FOR NON-STATIONARY PROCESS SIMULATION Andris Sniders Ltvi University of Agriculture, Fculty of Engineering, Institute of Agriculturl Energetics ndris.sniders@llu.lv Abstrct. Metodology of non-liner nd non-sttionry process simultion, using MATLAB subprogrm- SIMULINK, is expounded nd justified. A self tuning up model to simulte te trnsient process of non-liner nd non-sttionry electricl eter wit vrible electricl resistnce, s function of eter temperture, nd cngeble et trnsfer coefficient, s function of eter nd ir temperture, is viewed. To compre te trnsient process of simplified sttionry model wit constnt sensitivity nd inerti fctors, nd te rel trnsient process of non-sttionry model wit time dependent sensitivity nd inerti fctors, te simultion model, pplying Lplce trnsforms nd SIMULINK librry components, re composed. Te trnsient process simultion is mde for te step cse input voltge, for te liner cnging input voltge nd for te step nd rndom cnging input voltge. Te nlyses of te simultion results sow tt te rel trnsient process of te electricl eter non-sttionry model differs from te trnsient process of te simplified sttionry model substntilly. Keywords: non-sttionry object, self-tuning model, virtul nlyses, sensitivity fctor, inerti fctor, simultion. Introduction Severl tecnologicl objects, suc s wste wter ertion tnks, electricl mcine systems nd cogenertion plnts (internl combustion engine nd electricl genertor), re typicl non-liner nd non-sttionry objects wit time dependent trnsfer coefficients (gins) nd time constnts [-3]. In tt cse te trnsient crcteristics of te object re described by non-liner nd non-sttionry differentil equtions, mtemticl nlyses of wic re problemtic. Simplifiction of mtemticl models, pplying lineriztion nd freezing of vrible coefficients, put into prctice, cuses n incorrect result. Virtul nlyses, pplying SIMULINK [4-6], mkes it possible to compile te simultion models of suc objects witout simplifictions nd terefore llow to obtin te trnsient crcteristics wit substntilly iger ccurcy becuse of utomtic vritions of te trnsfer coefficients nd of te time constnts during simultion time. For tis purpose te on-line continuous dptive feedbcks or te off-line discrete feedbcks wit utomtic switcbord [3] my be employed. Min metodologicl tsks of tis work re to develop nd to investigte n dptive self-tuning up simultion model of te electricl eting object using dynmic on-line feedbcks for sensitivity nd inerti fctors doption to te vrible trnsient temperture during te wole simultion process. Mterils nd metods. Mtemticl nd simultion models of electricl eting object Te reserc object is two volume electricl termostt, wic consists of n electricl eter (300 W) nd termlly isolted ir mss. Two different models of electricl termostts re compred: ) te sttionry model wit constnt trnsfer coefficients nd invrible time constnts bot of eter nd ir mss; 2) te non-sttionry model wit temperture dependent trnsfer coefficient nd time constnt of eter nd constnt trnsfer coefficient nd time constnt of ir mss Te mtemticl models, simultion block-digrms nd crcteristics of te reserc object re given in te text. Te opertor equtions nd te trnsfer functions of te eter nd te ir mss re composed using mtemticl nlyses, opertor mtemtics nd Lplce trnsforms [4]. Te trnsient process simultion in Windows environment is performed using SIMULINK [5, 6]. Vrible sensitivity nd inerti fctors of te electricl eter re clculted ccording to teir tecnicl prmeters, pplying nlyticl nd empiricl expressions [4, 7, 8]. For simplicity te trnsient et trnsfer process in te electricl eting object is nlysed s lumped process [7], were temperture of te medium cnges uniformly wit time, not position. 92

ENGINEERING FOR RURAL DEVELOPMENT Jelgv, 27.-28.05.200. Ten te trnsient et trnsfer of te electricl eter nd ir mss cn be described by te common differentil equtions. Using Lplce trnsforms to differentil equtions, we obtin te opertor equtions nd te trnsfer functions for trnsient temperture simultion s function of input voltge. Te opertor eqution nd te trnsfer function of te sttionry model for trnsient process simultion in te electricl eter: T 2 τ ( s) s+ ( s) K U ( s), W ( s) τ τ U ( s) K 2 ( s) T s +, () were T 8. min verge time constnt of te eter; τ (s) Lplce trnsform from surfce over-temperture (τ Θ Θ 0 ); Θ, Θ 0 vrible surfce verge temperture nd initil temperture of te eter nd ir mss, C; K 0.007 C V -2 verge trnsfer coefficient of te eter; U 2 (s) Lplce trnsform from input step cse squred voltge U 2, V 2 ; s Lplce vrible, min -. Te opertor eqution nd te trnsfer function of te sttionry model for trnsient process simultion in ir mss: T τ ( s) s+ τ ( s) K τ ( ), W ( s) s τ τ ( s) K ( s) T s + were T 5.5 min verge time constnt of te ir mss; τ (s) Lplce trnsform from ir mss over-temperture (τ Θ Θ 0 ); Θ vrible ir mss verge temperture, C; K 0.4 verge trnsfer coefficient of te ir mss., (2) Actully, te electricl eter is non-sttionry object becuse of te cngeble eting time constnt T f(θ, Θ ) s well s due to te fluent trnsfer coefficient K f(θ, Θ ), wic cnges during te wole trnsient et trnsfer process. Te expression for T clcultion using te prmeters of te ctul electricl eter is s follows [4]: T c m., A (3) ( Θ Θ ) were c 950 J (kg C) - specific et; m 0.45 kg mss; A 0.052 m 2 surfce re; (Θ, Θ ) surfce bot of convection nd rdition et trnsfer coefficient, W (m 2 C) -. Te surfce et trnsfer coefficient cn be clculted by empiricl formul [8]: ( Θ Θ) + 0. Θ Θ.3+ 0.047 02 9, (4) were o 9.3 W (m 2 C) - initil et trnsfer coefficient bot of convection nd rdition. Since T is function of time dependent vribles Θ nd Θ, T is nmed s process inerti fctor. Te expression for K clcultion using te prmeters of te ctul electricl eter is s follows [4]: K Θ, Θ A R Θ (5) ( ) ( ). were R(Θ ) electricl resistnce of electricl eter, Ω. 93

ENGINEERING FOR RURAL DEVELOPMENT Jelgv, 27.-28.05.200. Te nicrome eter resistnce cn be clculted by nlyticl formul: + α n Θ R( Θ ) RΘ. 0 + Θ were α n 0.0004 C - resistnce-temperture coefficient of nicrome eter; R Θ0 6 Ω initil resistnce of nicrome eter, if Θ 0 20 C. Since K is function of time dependent vribles Θ nd Θ, is nmed K s process sensitivity fctor. Te non-sttionry nd non-liner mtemticl model for output temperture Θ (t) simultion in te electricl dryer, ssuming tt te eted ir mss is sttionry object wit constnt prmeters K, T, is s follows: Θ ( s) T K ( Θ, Θ ) ( Θ, Θ ) s + T α n 0 K U s+ 2 ( s) +Θ. (7) were Θ (s) Lplce trnsform of ir mss trnsient temperture. It is unble to solve te given eqution (7) nlyticlly becuse of non-sttionry coefficients nd non-liner input vrible. Furter it will be sown ow to solve problem virtully, using SIMULINK. Te block digrm of te non-sttionry nd non-liner model for simultion of trnsient tempertures Θ (t) nd Θ (t) of te electricl eter, described by eqution (7), is sown in Figure. Te dptive self-tuning up model of te electricl eter consists of te Adptive R-module nd Adptive -module for dynmic clcultion of te et trnsfer coefficient nd electricl resistnce R during te trnsient process of tempertures Θ (t), Θ (t), ccording to expressions (4, 6). Adptive trnsfer function of eter 0 (6) U(t) A, m 2 R, Om U 2, W/(m 2 0 C) Adptive K - module c, J/(kg 0 C) m, kg A, m 2, W/(m 2 0 C) Adptive dynmic feedbck from (Θ, Θ ) + - s Adptive T - module Adptive - module 0 9.3 0.047 0.02 T + - τ (t) Θ (t) Θ (t) Trnsfer function of ir mss K T x s + τ (t) Θ (t) Θ 0, 0 C Θ (t) Adptive dynmic feedbck from R (Θ ) Adptive R- module R θo, Om α n, / 0 C Θ 0, 0 C Fig.. Adptive self - tuning up model of electricl eter wit dptive dynmic feedbcks from temperture dependent et trnsfer coefficient (Θ, Θ ) nd from eter resistnce R (Θ ) Te Adptive R-module cretes dynmic feedbck R(Θ ) for te sensitivity fctor K clcultion. Te Adptive -module forms dynmic feedbck for bot K nd te inerti fctor T 94

ENGINEERING FOR RURAL DEVELOPMENT Jelgv, 27.-28.05.200. clcultion. Te dptive dynmic feedbcks llow clcultion of K nd T step by step during te simultion time. Te dptive self-tuning up trnsfer function of te electricl eter is composed of te following modules: Adptive K -module ; Adptive T -module ; /s integrtor; /T module for T reversion; unit negtive feedbck. 2. Block digrm for trnsient process simultion of sttionry nd non -sttionry models Te simultion block-digrm of te electricl termostt is compiled using stndrd blocks from SIMULINK librries (Figure 2). Te block-digrm consists of severl modules for utomtic clcultions nd simultion: te module for clcultion of te et trnsfer coefficient s function of te eter nd ir mss tempertures f(θ, Θ ) (4); te module for clcultion of te eter resistnce s function of te eter temperture R f(θ ) (6) T te module for clcultion of te inerti fctor T dependent on fixed prmeters nd te fluent et trnsfer coefficient (Θ, Θ ) (3); K x U 2 te module for clcultion of te sensitivity fctor K dependent on fixed prmeters nd fluent prmeters (Θ, Θ ), R(Θ ) (5), nd for K multipliction by input vrible squred voltge U 2 ; Termostt module for output temperture Θ f(t) simultion ccordingly to eqution (7). Te trnsient process simultion is mde for te step cse input voltge, for te liner growing input voltge nd for te step nd rndom input voltge. For pproprite input voltge formtion te following functionl blocks re used: Switc nd Switc 2 mnul switces; U step step cse voltge genertor; U liner liner growing voltge genertor; U rndom rndom normlly distributed voltge genertor; Sturtion for liner growing voltge limittion. Digitl displys for input nd output volumes visuliztion: U s input stedy voltge, V; Θ s, Θ s ir mss stedy temperture for sttionry nd non-sttionry models, C; Θ s, Θ s eter stedy temperture for sttionry nd non-sttionry models, C. Scope nd Scope 2 for visuliztion of input nd output trnsient crcteristics U f(t); Θ f(t); Θ f(t); Θ f(t); Θ f(t). U s, V Heter Air mss U f(t) Scope U step Switc K xu 2 Integrtor /T Θ s, 0 C Air mss Θ0, 0 C Θ s, 0 C U rndom Switc 2 Termostt Scope 2 U liner m, kg Sturtion c, J/(kg 0C) Feedbck from Rf(Θ ) T, min Θ 0, 0 C α n, / 0 C R θo, Ω Θ 0, 0 C Θ s, 0 C Θ s, 0 C S, m 2 0, W/(m 2 0C) Θ f(t) Θ f(t) Feedbck from f(θ, Θ ) Fig. 2. Block digrm for trnsient process simultion of sttionry nd non-sttionry electricl eter models wit clcultion of vrible sensitivity nd inerti fctors during simultion 95

ENGINEERING FOR RURAL DEVELOPMENT Jelgv, 27.-28.05.200. Discussion nd results Te simulted crcteristics of sttionry nd non-sttionry models for step nd liner growing input voltge re sown in Figures 3-5. To form te step cse voltge U 00 V nd liner growing voltge from U 00 V to limited vlue U 2 200 V, te U liner nd Sturtion blocks ve been ctivted. For tis purpose Switc nd Switc 2 sould be set in te lower position (Figure 2). Te configurtion prmeters re s follows: for te U liner block (initil voltge 00 V, initil dely time 40 min, slope 50 V min - ); for te Sturtion block (minimum voltge 0 V, mximum voltge 200 V). Te simultion sows tt te sensitivity fctor K nd te inerti fctor T of te eter re not constnt, but cnge from te mximum vlue K mx 2.5 0-3 C V -2, T mx 4.6 min t initil temperture 20 C to te minimum one K min 6.4 0-3 C V -2, T min 8.5 min t mximum temperture 278 C during te wole trnsient process of eting (Figure 3). Terefore, only te nonsttionry model is ble to ensure correct eting results. K, 0 C/V 2 T, min K 00V f(t) K s00v 9.5 0-3 0 C/V 2 T 00V f(t) Ts00V.3 min K 200V f(t) K 7 0-3 0 C/V 2 const T 200V f(t) T s200v 8.5 min K s200v 6.4 0-3 0 C/V 2 96 T 8. min const Fig. 3. Crcteristics of sensitivity nd inerti fctors for sttionry model (K 7 0-3 C V -2 const nd T 8. min const) nd for non-sttionry self-tuning up model (K 00 V f(t), K 200 V f(t) nd T 00 V f(t), T 200 V f(t)) under two step voltge (U 00 V, U 2 200 V) Te simulted response of te eter temperture for te step cse initil voltge, wic is rpidly growing from te first step 00 V to te limited second step 200 V fter dely time 40 min (necessry for temperture stbiliztion t te first step voltge) demonstrtes, tt te trnsient crcteristic Θ f(t), if te simplified sttionry model is pplied, substntilly differs from tt Θ f(t), wic is obtined using te dptive self-tuning up non-sttionry model (Figure 4). Te rel eting inerti fctor T nd te sensitivity fctor K for te non-sttionry model re iger t low tempertures becuse of lower et trnsfer coefficient. Terefore, te eting proceeds slower, but stedy temperture is iger by te side of tt for te sttionry model. Everyting is opposite t ig tempertures. Te simulted crcteristics of sttionry nd non-sttionry models for step nd rndom input voltge U step nd U rndom re sown in Figures 6-8. To get te step cse voltge U step nd normlly distributed rndom voltge component U rndom, te U step nd U rndom blocks wit Adder sould be ctivted. For tis purpose Switc sould be set in te lower position, but Switc 2 in te over position (Figure 2). Te configurtion prmeters re s follows: for te U step genertor (initil vlue 0 V, finl vlue 200 V, step time 0 min); for te U rndom genertor (mplitude ±20 V, frequency 0.25 min - ). Te simultion sows tt te sensitivity fctor K nd inerti fctor T of te eter fluctute rndomly round te blnce vlue K blnce 6.35 0-3 C V -2, T blnce 8.03 min on ±5 % wit te sme frequency s temperture, but counter pse to temperture oscilltions (Figure 6). Only n dptive self-tuning up model is ble to simulte rndomly fluctuting temperture of te eter nd ir mss correctly. Te simulted response of te eter temperture for te rndomly fluctuting input voltge 200 V ± 0 % wit mplitude ±20 V nd frequency 0.25 min - testifies, tt te output temperture of

ENGINEERING FOR RURAL DEVELOPMENT Jelgv, 27.-28.05.200. te dptive non-sttionry model Θ f(t) is more sensitive to voltge fluctutions s tt Θ f(t) of te non-dptive sttionry model. U, V Θ, 0 C U 00Vconst Θ s 200V 300 0 C Θ s 200V 278 0 C U 50 t liner Θ 00V f(t) Θ 00V f(t) Θ 200V f(t) U 2 200 V const Θ 200V f(t) Θ s 00V 5 0 C Θ s 00V 90 0 C Fig. 4. Simulted crcteristics of eter temperture for sttionry model (Θ f(t)) nd for non-sttionry self-tuning up model (Θ f(t)) under step nd liner growing voltge (U 00 V, U 50 t, U 2 200 V) U, V Θ, 0 C U 50 t liner U 2 200V const Θ s 200V 32 0 C U 00V const Θ 00V f(t) Θ 200V f(t) Θ 200V f(t) Θ s 200V 23 0 C Θ s 00V 58 0 C Θ 00V f(t) Θ s 00V 48 0 C Fig. 5. Simulted crcteristics of ir mss temperture for sttionry model (Θ f(t)) nd for non-sttionry self-tuning up model (Θ f(t)) under step nd liner growing voltge (U 00 V, U 50 t, U 2 200 V) Due to te eting inerti, te rndom fluctutions of te voltge up to ±0 % cuse less temperture devitions of te eter ±6 % (Figure 7) nd still less of te ir mss ±2.5 % (Figure 8). Terefore, te eting inerti of te eter nd of ir mss opertes s filter of input fluctutions. K, 0 C/V 2 K 0.007 0 C/V 2 const K f(t) T, min T f(t) T 8. min const K s 0.00635 0 C/V 2 ± 5% T s 8.03min ± 5% Fig. 6. Crcteristics of sensitivity nd inerti fctors for sttionry model (K 7 0-3 C V -2 const nd T 8. min const) nd for non-sttionry self-tuning up model (K f(t), nd T f(t)) under rndom vrible voltge (U 200 V ± 0 %) 97

ENGINEERING FOR RURAL DEVELOPMENT Jelgv, 27.-28.05.200. U, V Θ, 0 C Θ f(t) Θ f(t) Θ 280 0 C ± 6% U 200V ± 0% U f(t) Fig. 7. Simulted crcteristics of eter temperture for sttionry model (Θ f(t)) nd for non-sttionry self-tuning up model (Θ f(t)) under rndom vrible voltge (U 200 V±0 %) U, V Θ, 0 C U f(t) U 200V ± 0% Θ f(t) Θ f(t) Θ 25 0 C ± 2.5% Fig. 8. Simulted crcteristics of ir mss temperture for sttionry model (Θ f(t)) nd for non-sttionry self-tuning up model (Θ f(t)) under rndom vrible voltge (U 200 V ± 0 %) Conclusions. For trnsient process virtul nlyses wit pproprite ccurcy in non-liner nd non-sttionry tecnologicl objects, te sensitivity nd inerti fctors of wic cnge substntilly on output vribles, te dptive self-tuning up simultion model sould be developed pplying dynmic feedbcks from output vribles for utomtic doption of te functionlly dependent components of te model during te wole simultion process. 2. Te simultion results of te given electricl eter using te dptive self-tuning up model sow, if te input voltge cnges from U 00 V to U 2 200 V, te stedy sensitivity fctor of te eter decreses from K s 00 V 9.5 0-3 C V -2 to K s 200 V 6.4 0-3 C V -2 nd te stedy inerti fctor decreses from T s 00 V.3 min to T s 200 V 8.5 min, wt cuses reltive difference of te simulted stedy finl temperture Θ s up to 25 % in comprison wit te Θ s, wt is obtined using te simplified sttionry model. References. Шнидерс А. Моделирование и энергоэкономное управление системой аэрации сточных вод (Simultion nd Energy-Sving Control of Wste Wter Aertion System). Proceedings of te 3 rd Interntionl Scientific-prcticl conference Экология и сельскохозяйственная техника Санкт-Петербург: СЗНИИМЭСХ, 2002, pp. 294-302. (In Russin). 98

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