APPLICATION OF DISTRIBUTED PARAMETER SYSTEM CONTROL FOR STEEL BILLET INDUCTION HEATING

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1 1. Juraj KAPUSTA, 2. Juraj CAMBER, 3. Gabrel HULKÓ APPLICATION OF DISTRIBUTED PARAMETER SYSTEM CONTROL FOR STEEL BILLET INDUCTION HEATING 1-3. INSTITUTE OF AUTOMATION, MEASUREMENT AND APPLIED INFORMATICS, FACULTY OF MECHANICAL ENGINEERING, SLOVAK UNIVERSITY OF TECHNOLOGY, NÁM. SLOBODY 17, BRATISLAVA, SLOVAKIA ABSTRACT: Nowadays, the achevement of proper steel bllets temperature profle s not the only desgn prorty of nducton heaters for hot formng applcatons. Due to ts hgh operatng costs, ts desgn s constantly mprovng n terms of electrcal and thermal effcency. Therefore the more effcent mult-col desgn starts to be more used n ndustral practce. Numercal model of mentoned heater based on partal dfferental equatons were solved by fnte element method n vrtual software envronment. Prmary goal of computer modelng was to nvestgate the thermal dynamcs of four-module heater workng n steady-state operaton regme. Obtaned data were appled to desgn an advanced control crcut based on dstrbuted parameter systems theory. Ths may open up the opportunty to make further progress n nducton heaters desgn. KEYWORDS: nducton heaters desgn, numercal model, fnte element method, computer modelng INTRODUCTION - MODELING OF CONTINUOUS INDUCTION HEATING Inducton heatng of steel s contactless heatng method that combnes electrcal, magnetc and thermal phenomena. It works on the prncple of energy dstrbuton from nductor col to the heated bllet carred by generated electrcal and magnetc felds. Eddy currents nduced n heated bllet flow through ts body and generate heat due to electrcal resstvty of materal. Descrbed phenomenon s called the Joule effect. The mathematcal descrpton for contnuous nducton heatng of steel bllets can be descrbed by two coupled partal dfferental equatons (PDE) formulatons the electromagnetc part by a tme-harmonc reduced PDE of the magnetc vector potental and the thermal part by PDE for temperature felds [6,8]. Electromagnetc feld can be calculated by magnetc vector potental A as A 1 1 σ ( v A ) + ( μ μ ) 0 r A = J source, (1) t where A represents magnetc vector potental, B = A s magnetc flux densty [T], ω s angular frequency, ε 0 s permttvty of vacuum, ε r relatve permttvty, v [m s -1 ] velocty and J source represents the drve current densty n the col. The ablty of materal to conduct current s specfed by electrcal conductvty σ =1/ρ [S m -1 ], where ρ represents the electrcal resstvty ρ [Ω m]. Electrcal conductvty s temperature dependent materal property. Relatve magnetc permeablty μ r s non-dmensonal parameter whch ndcates the ablty of metal to conduct the magnetc flux better than a vacuum (μ 0 represents the permeablty of vacuum). Ths materal property s both magnetc feld ntensty and temperature dependable (Fgure 1b) and t has a great effect on all basc nducton heatng phenomena. The temperature at whch bllet body becomes nonmagnetc s called the Cure temperature T c (μ r = 1). The drop of relatve permeablty μ r for forgng steel s set to T c =775 [ C]. The phenomenon of non-unform nduced current dstrbuton wthn the bllet cross-secton s called the skn effect. The maxmum value of current densty s located on the surface of the heated body and the current densty decreases from the body surface toward ts center. Current penetraton depth s frequency dependent, the lower current frequency creates greater penetraton depth and vce versa [,6]. In nducton heatng, all three modes of heat transfer (conducton, convecton and radaton) are present [6,8]. Therefore the temperature dstrbuton profle of heated bllet s gven by equaton T ρ ( T)C P( T) + T - ( λ( T) T) = Q j, t v (2) where temperature dependent varables of steel bllet are: densty ρ [kg m -3 ], thermal conductvty λ [W m -1 K -1 ] and specfc heat capacty C P [J kg -1 K -1 ] of forgng steel. Q j represents the Joule losses n bllet generated by eddy currents flow [12]. Joule losses can be rewrtten as copyrght FACULTY of ENGINEERING HUNEDOARA, ROMANIA 337

2 j n 2 Q = J σ( T), (3) TABLE 1. TEMPERATURE-DEPENDENT MATERIAL PROPERTIES Temp (K) σ (S/m).106 λ (W/m.K) CP (J/kg.K) ρ (kg/m3) h (W/m2.K) Both convecton and radaton are responsble for heat loss from bllet surface to the surroundng area. Therefore boundary condton of nward heat flux Q loss =h(t ext T) has been appled. Temperature dependent combned heat transfer coeffcent h carres the nfluences of radaton and convecton to the bllet temperature n one complex varable. Text s external temperature near the bllet surface and T s bllet surface temperature. Fgure 1. a) Detal of model part wth custom mesh, b) Solved steady-state measured by ten vrtual probes The nonlnear, temperature-dependent materal parameters of forgng steel (see Table. 1) have been used n calculaton, Rudnev (2010), Behúlová et. al. (2006). The drop of relatve permeablty μ r was set to value of T c =1020K. Dependng on heated materal, the heatng process of ϕ80mm steel bllets for forgng takes several mnutes to acheve an optmal steady state. Producton rate cycle was set to one properly heated bllet per 25 seconds wth fnal temperature T out 150K. The four col currents I 1 - I [ka] was appled wth real engneerng applcatons n mnd. The bllets were contnuously movng through the nductor col wth velocty of 10mm/s. Three-dmensonal model of nductor col, kndly provded by Roboterm company, was transformed nto 2D axal symmetrc space dmenson. Four col modules n seral connecton has been used. It created nductor wth total of 55 turns and length of 3m. The col nteror was coated by refractory. Custom combnaton of dstrbuted, mapped and free quadratc mesh was appled to acheve good accuracy-to-computng tme rato. Very dense mesh was created on the bllet surface and between cols and bllet. The goal was to acheve a precse calculaton of electro-magnetc phenomena and skn effect. Fgure 1a shows the small part of model and used mesh style. In real producton demands noted above, the steady-state temperature profle has been solved by COMSOL Multphyscs. Obtaned steady-state measured by ten vrtual probes shows Fgure 1b. DYNAMICS INSPECTION OF INDUCTION HEATER Each col {GU } n contnuous modular nducton heater (see Fgure 2) s powered by ts own current {UA (t)} generated by actuatng members {SA }, therefore t s a typcal lumped-nput and dstrbuted-parameter-output system (LDS). It causes the change of bllets thermal profle Y(x,t) along the nductor length. It should be noted that current s not actuatng varable n real-lfe control of nducton heatng col power. In practce PLC s operatng wth col voltage, whch carres the current nto the cols. The change of col voltage gves adequate change of col current as well. Wth model smplfcaton n mnd the col currents represents the actuatng varables of DPS control crcut. 338 Tome XI (Year 2013). Fasccule. ISSN

3 Fgure 2. Contnuous modular nducton heater as LDS. In ths case of numercal smulaton of nducton heatng process, there s necessary to ntroduce a dscrete HLDS model (the same LDS system wth nput block of zero-hold unts {H } ). It s dstrbuted on the nterval of nductor length [0,L] wth output quantty Y(x,k). The model s dscretzed n tme relaton and contnuous n space relaton on the same nterval. In ths case the dstrbuted quanttes are used n ts dscrete form as fnte sequences of quanttes. The dscretzaton n space doman s usually consdered by the computatonal nodes of the numercal model of the controlled DPS over the complex-shape defnton doman. In the lnearzed regon of solved steady-state operaton setpont, the 10% step to each lumpednput varable (col currents) has been appled. The dstrbuted step responses {H (x,k)} =1, had been calculated. The dstrbuted-parameter mpulse characterstcs {G (x,k)} =1 were obtaned by subtractng of shfted dstrbuted-parameter step characterstcs. It has led us to nput-to-output relaton Y( x,k ) = Y ( x,k ) = G (x,k) U ( k ), () =1 = 1 where {Y (x,k)} represents four components of HLDS dstrbuted output. When the vector of lumped varables {U (k)} =1, operates as the nput of HLDS block, the overall output varable Y(x,k) s obtaned. Now let us analyse the dstrbuted transent characterstcs for each pont {x } =1, whch carres the hghest ampltude peaks (see Fgure a). The reduced steady-state dstrbuted parameter transent responses were ntroduced as { R( x, ) = ( x, ) ( x, ) } = 1 K H H H (5) for {H (x, )} =1 0. In the next step of analyss, the dscrete transfer functons {S (x,z)} =1 were assgned to partal dstrbuted parameter transent responses wth maxmal ampltudes at ponts {x } =1 on the nductor length nterval [0,L], n ther proxmty the measurements are performed. Two crucal components of steady-state HLDS dynamcs were obtaned: - tme components of dynamcs {S (x,z)} =1 - space components of dynamcs HR (x, )} =1 Fgure. a) Tme and spatal dstrbuted transent characterstcs to +10% step of each col current b) Four reduced steady - state dstrbuted parameter transent responses. The relaton between lumped nput varables {U (z)} =1, and partal dstrbuted output varables for {x } =1, takes form as follows { ( ) ( )} Y(x,z)=S x,z U z. (6) =1, In respect to used varables, the overall dstrbuted steady-state output varable s gven by equaton Tome XI (Year 2013). Fasccule. ISSN

4 ( ) ( ) R ( ) Y x, = Y x, H x, (7) =1 CONTROL OF INDUCTION HEATER AS DISTRIBUTED PARAMETER SYSTEM Accordng to Fgure 2, the Dstrbuted parameter system (DPS) control crcut has been desgned wth DPS Blockset components n MATLAB & Smulnk nterface. In order to control of nducton heater n a lnearzed regon of the set-pont, the smulaton of dstrbuted parameter feedback control s set up (Fgure 5.). The objectve of control s to hold the proper bllet s temperature profle requred by technology. It s gven by ntal reference varable W(x, ) and the control crcut have to be able to ensure ts optmal approxmaton. a) b) Fgure 5. Control crcut, a) schematcally, b) crcut developed n MATLAB usng DPS Blockset. The SS1 block s solvng Y ( ) mn Y(x,k) - Y (x,k) H R x,. (8) =1 2 and n steady-state (assumng V(x,t) = 0) t holds ( ( Y x, = Y(x, ) H R x,. (9) The SS2 block s calculatng Y ( ) ( ) =1 ( ) mn W(x, ) - W (x, ) H R x,. (10) =1 2 The SS2 block calculaton output s the best approxmaton of reference varable ( ( W x, = W x, H R x,. (11) ( ) ( ) ( ) =1 The mentoned approxmaton problems are beng solved n a strctly normed functon space of dstrbuted parameter quanttes wth the base HR (x, )} =1. In terms of equaton (7), where relaton between {U (k)} =1 and {Y (x,k)} =1 has been demonstrated usng transfer functons {S (x,z)} =1, there s four set of varables Y( x,k), W ( x, ) ( ( (, E( x,k) and U (k) belong to each feedback control crcut. The control synthess of nducton heater model s composed of four dentcal feedback control crcuts. Generally, there can be used the majorty of controllng algorthms (PID, model predctve control, state-space etc.) n the regulator secton R (z), dependng on process control requrements. The conventonal PI algorthm was consdered as optmal for our model control purposes. Parameters of PI regulators were tuned by method of placng the poles. The goal of control synthess was to reach the nput-output balance. In steady-state t can be descrbed by relatons (9) and (11) usng the control error approxmaton ( ( ( E x, =W x, -Y(x, )=0. (12) { ( ) ( ) } Let us consder a control problem of ncreasng the steady-state temperature profle of the bllet by 30K n full col length n the tme of t=100s, Fgure 6a. There s a step change of reference varable performed by actuatng varables of four PI regulators n DPS PID Synthess block. Actuatng varables generated by regulators are changng the col currents, Fgure 6b. From the ntal steadystate temperature profle, the new one s reached n approxmately 00[s]. It corresponds to the consdered bllet transfer over the entre length of nductor col. =1K 30 Tome XI (Year 2013). Fasccule. ISSN

5 a) b) Fgure 6. a) Tme-spatal step response gven by step change at t=100[s], b) Actuatng varables (as percentage of col currents). CONCLUSIONS - SUMMARY The acheved results represent the possbltes of numercal PDE soluton descrbng the process of nducton heatng for control desgn purposes. The control objectve, dscussed n ths artcle, s to keep the heated steel bllet s temperature feld wthn allowable lmts around the set-pont. Non unformty of heated steel propertes may cause the temperature varaton. Thus nducton heatng process s subject to uncertantes n physcal and chemcal propertes of the bllets materal and fluctuatons of the electrc power delvered to the nductor cols. In the next step, there s ambton to desgn proposed control crcut that keeps the process runnng n an emergency settng n case of one nductor module falure. The remanng healthy modules wll hold the desred temperature profle wthn some tolerable lmts, dependng of course on the machne constructon lmtatons. Inductor heater wth ths ablty placed n fully automatc, hgh productvty workplace, offers the advantage of contnuous operaton n case of one nductor module malfuncton. In the case of ths paper the numercal soluton was done usng the vrtual software envronment COMSOL Multphyscs. It s used for nspecton of dynamc characterstcs of the nducton heatng process and buldng the model n form of lumped-nput and dstrbuted-parameter-output system. Interconnectng ths model wth MATLAB & Smulnk va DPS Blockset (Fgure 5b) opens new possbltes for smulaton studes of steel bllet nducton heatng n the feld of control desgn and process optmzaton as well. In the ndustry practce, the number of measurng pyrometers s lmted to the reasonably applcable mnmum due to hgh costs Obtaned results (Fgure 6) ndcate the possblty of replacement of several PLCs and pyrometers wth one ndustral computer equpped by proper software. Ths opens up new possbltes for nducton heatng control for the engneerng practce. ACKNOWLEDGEMENT Research funded by grants VEGA 1/0138/11 Control of dynamc systems represented by numercal structures as dstrbuted parameter systems and APVV Hgh-tech solutons for technologcal processes and mechatronc components as controlled dstrbuted parameter systems and by the European Unon wth the co-fnancng of the European Socal Fund, grant TAMOP-.2.1.B-11/2/KMR REFERENCES [1.] G. Hulkó, et al.: Dstrbuted Parameter Systems Blockset for MATLAB & Smulnk Thrd-Party Product of The MathWorks, Publshng house of STU, Bratslava, ( ) [2.] G. Hulkó, et al.: Control of energy systems as dstrbuted parameter systems wth software support by vrtual software envronments, Proceedngs of the 51st IEEE Conference on Decson and Control, pp Mau, Hawa (2012). [3.] K. Ondrejkovč, et al.: Control of Contnuous Castng Processes as Dstrbuted Parameter Systems, Proceedngs of METEC InSteelCon 2011, 7-th European Contnuous Castng Conference. Düsseldorf (2011). [.] V. Rudnev: Smulaton of Inducton Heatng Pror to Hot Workng and Coatng, In: ASM Handbook Volume 22B - Metal Process Smulaton, pp ASM Internatonal (2010). [5.] G. Hulkó, et al.: Engneerng methods and software support for modellng and desgn of dscretetme control of dstrbuted parameter systems, In: European Journal of Control, Volume 15, (no. 3 ), pp , (2009). [6.] E. Rapoport, Y. Pleshvtseva: Optmal control of nducton heatng processes, Taylor&Francs Group. New York, (2007). [7.] M. Behúlová, B. Mašek, et al.: Statc and Dynamc Inducton Heatng - Experment and Numercal Smulaton, In: MP Materalprüfung, Volume 8, pp , (2006). Tome XI (Year 2013). Fasccule. ISSN

6 [8.] V. Rudnev, D. Loveless, et al.: Handbook of Inducton Heatng, Marcel Dekker, (2003). [9.] G. Hulkó, et al.: Modellng, Control and Desgn of Dstrbuted Parameter Systems wth Demonstratons n MATLAB, Publshng house of STU, Bratslava, (1998). [10.] J. Kapusta, J. Camber, G. Hulkó: Modelng of Inducton Heatng of Steel Bllets for DPS Control Desgn Purposes, Proceedngs of COMSOL Conference, Bangalore, Inda (2012). [11.] J. Kapusta, J. Camber, G. Hulkó: Modelng and Control of Steel Bllet Inducton Heater. Submtted to 1 st IFAC Workshop on Control of Systems Modeled by Partal Dfferental Equatons, Pars, France (2013). [12.] COMSOL Multphyscs User s Gude, COMSOL Inc., ( ) ANNALS of Faculty Engneerng Hunedoara Internatonal Journal of Engneerng copyrght UNIVERSITY POLITEHNICA TIMISOARA, FACULTY OF ENGINEERING HUNEDOARA, 5, REVOLUTIEI, , HUNEDOARA, ROMANIA 32 Tome XI (Year 2013). Fasccule. ISSN