Chapter 2 Mathematical Model of the Safety Factor and Control Problem Formulation

Transcription

1 Chapte 2 Mathematical Model of the Safety Facto and Contol Poblem Fomulation We ae inteested in contolling the safety facto pofile in a tokamak plasma. As the safety facto depends on the atio of the nomalized adius to the poloidal magnetic flux gadient, contolling the gadient of the magnetic flux allows contolling the safety facto pofile. In this chapte we pesent the efeence dynamical model [1] fo the poloidal magnetic flux pofile and its gadient (equivalent to the effective poloidal field magnitude, as defined in [2]), used thoughout the following chaptes, as well as the contol poblem fomulation. Some of the main difficulties encounteed when dealing with this poblem ae also highlighted. 2.1 Inhomogeneous Tanspot of the Poloidal Magnetic Flux The poloidal magnetic flux, denoted ψ(r, Z), is defined as the flux pe adian of the magnetic field B(R, Z) though a disc centeed on the tooidal axis at height Z,having a adius R, see Fig A simplified one-dimensional model fo this poloidal magnetic flux pofile is consideed. Its dynamics is given by the following equation [3]: ψ t = η C 2 2 ψ μ 0 C 3 ρ 2 + η ( ) ρ C2 C 3 ψ μ 0 C3 2 ρ ρ ρ + η V ρ B φ0 j ni (2.1) FC 3 with the geomety defined by: ρ =. φ, πb φ0 C 2 (ρ). = V ρ ρ 2 R 2, C 3(ρ). = V ρ 1 R 2 whee epesents the aveage ove a flux suface, indexed by the equivalent adius ρ. The emaining coefficients ae: φ, the tooidal magnetic flux; B φ0,thevalueof the tooidal magnetic flux at the plasma cente; η, the paallel esistivity of the plasma; and μ 0, the pemeability of fee space. j ni is a souce tem epesenting the F. Bibiesca Agomedo et al., Safety Facto Pofile Contol in a Tokamak, 11 SpingeBiefs in Contol, Automation and Robotics, DOI: / _2, The Autho(s) 2014

2 12 2 Mathematical Model of the Safety Facto and Contol Poblem Fomulation Fig. 2.1 Coodinate system (R, Z) used in this chapte total effective cuent poduced by non-inductive cuent souces. F is the diamagnetic function and V ρ is the spatial deivative of the plasma volume enclosed by the magnetic suface indexed by ρ. A summay of these vaiables can be found in Appendix B. With the following simplifying assumptions: ρ<<r 0 (usually efeed to as the cylindical appoximation, whee R 0 is the majo adius of the plasma); the diamagnetic effect caused by poloidal cuents can be neglected, the coefficients C 2, C 3 and F simplify to: F R 0 B φ0, C 2 (ρ) = C 3 (ρ) = 4π 2 ρ R 0 and the spatial deivative of the enclosed plasma volume becomes: V ρ = 4π 2 ρr 0 Intoducing the nomalized vaiable. = ρ/a, a being the mino adius of the last closed magnetic suface, we obtain the simplified model [1, 4]: ψ t (, t) = η ( (, t) 2 ψ μ 0 a with the bounday condition at the plasma cente: ) ψ + η (, t)r 0 j ni (, t) (2.2) ψ (0, t) = 0 (2.3) one of the two bounday conditions at the plasma edge:

3 2.1 Inhomogeneous Tanspot of the Poloidal Magnetic Flux 13 ψ (1, t) = R 0μ 0 I p (t) 2π o ψ t (1, t) = V loop(t) (2.4) (whee I p is the total plasma cuent and V loop is the tooidal loop voltage) and with the initial condition: ψ(, t 0 ) = ψ 0 () Remak 2.1 The validity of this model (deived fo Toe Supa) can be extended to othe tokamaks by changing the definition of the values C 2, C 3, F and V ρ. 2.2 Peifeal Components Influencing the Poloidal Magnetic Flux The dynamics (2.2) depend on the plasma esistivity (diffusion coefficient), the inductive cuent geneated by the poloidal coils (bounday contol input) and the noninductive cuents (distibuted contol input and nonlineaity), which can be descibed as follows Resistivity and Tempeatue Influence The diffusion tem in the magnetic flux dynamics is povided by the plasma esistivity η, which intoduces a coupling with the tempeatue (main influence) and density pofiles. This paamete is obtained fom the neoclassical conductivity poposed in [5] (appoximate analytic appoach) using the electon themal velocity and Baginskii time, computed fom the tempeatue and density pofiles as in [6]. The tempeatue dynamics ae typically detemined by a esistive-diffusion equation [2], whee the diffusion coefficient depends nonlinealy on the safety facto pofile [7]. A quasi-1d model was poposed in [8] to model the nomalized tempeatue pofiles as scaling laws detemined by the global (0-D) plasma paametes and to constain the tempeatue dynamics by the global enegy consevation. This gey-box model was shown to povide a sufficient accuacy fo the magnetic flux pediction in [1]. The esistive-diffusion time is much faste (moe than 10 times) than the cuent density diffusion time, which motivated lumped contol appoaches based on the sepaation of the timescales and using a linea time-invaiant model [9, 10]. In ou case, we conside that η vaies in time and space, to avoid the stong dependency on the opeating point, but we do not addess specifically the poblem of a coupling with the tempeatue dynamics (thus consideing only the linea time-vaying contibution of the esistivity). As an example, the esistivity calculated with measued tempeatue pofiles is depicted in Fig This plasma shot is chaacteized by powe modulations of the

4 14 2 Mathematical Model of the Safety Facto and Contol Poblem Fomulation (a) (b) Plasma esistivity (Ω x m) Nomalized adius Plasma esistivity (Ω x m) Time (s) Fig. 2.2 a Spatial distibution fo diffeent time instants. b Time evolution at diffeent locations. Plasma esistivity pofiles computed fo Toe Supa shot (modulations on the LH and ICRH antennas) lowe hybid (LH) and ion cycloton adio heating (ICRH) antennas. Note the diffeence of thee odes of magnitude between the plasma cente and its edge on Fig. 2.2a, and the modulated and noisy time-evolution on Fig. 2.2b. The cewels obseved afte 20 s esult fom LH modulations (step inputs) at elatively low powe and illustate an input-to-state coupling effect, as LH is ou main input on the magnetic flux Inductive Cuent Souces The magnetic flux at the bounday t ψ(1, t) is set by the poloidal coils suounding the plasma and the cental solenoid, and constitutes the inductive cuent input. This can be descibed by the classical tansfome model whee the coils geneate the pimay cicuit while the plasma is the seconday, modelled as a single filament. The dynamics of the coils cuent I c (t) and of the plasma cuent I p (t) ae coupled as [11]: [ ] [ ] [ ][ ] [ ] Lc M d Ic Rc 0 Ic Vc = + ML p dt I p 0 R p I p R p I NI whee R c and L c ae the coils esistance and intenal inductance, R p and L p ae the plasma esistance and inductance, M is the matix of mutual inductances, V c is the input voltage applied to the coils and I NI is the cuent geneated by the non-inductive souces. Note that the values of R c and L c ae given fom the coil popeties while M is obtained fom an equilibium code (i.e. CEDRES++ [12]). Consideing the effects of the plasma cuent and inductance vaiations, the loop voltage V loop is obtained fom [13] with:

5 2.2 Peifeal Components Influencing the Poloidal Magnetic Flux 15 [ V loop (t) = 1 d Lp I 2 ] p + M di c I p dt 2 dt In pactice, a local contol law is set on the poloidal coils to adjust the value of V c accoding to a desied value of V loop, which can be measued with a Rogowski coil. If the efeence is set on the plasma cuent I p instead, then V c is such that the coils povide the cuent necessay fo completing the non inductive souces to obtain I p Non-inductive Cuent: Souces and Nonlineaity The non-inductive cuent j ni in the magnetic flux dynamics (2.2) is composed of two types of souces: the contolled inputs and the bootstap effect. The contolled inputs ae the cuent dive (CD) effects associated with neutal beam injection (NBI), taveling waves in the lowe hybid (LH) fequency ange (0.8 8 GHz, the most effective scheme) and electon cycloton (EC) waves. The pecise physical modeling of the CD effects necessitates a complex analysis of the coupling between waves and paticles, which cannot be used fo eal-time contol puposes. We conside instead some semi-empiical models fo the Lowe Hybid Cuent Dive (LHCD) and Electon Cycloton Cuent Dive (ECCD) antennas (NBI is not explicitly included in ou contol schemes but the geneal stategy would emain the same), whee the cuent deposit shape is constained to fit a gaussian bell [1]. The gaussian shape is identified fom expeimental data (LHCD) o obtained fom model simplifications (ECCD), while the amplitude of the deposit comes fom CD efficiency computations involving the density, tempeatue and total cuent of the plasma. Fo example the shape of LHCD deposit can be adequately appoximated by a gaussian cuve with paametes μ, σ and A lh (which depend on the engineeing paametes P lh and N and on the opeating point) as: j lh (, t) = A lh (t)e ( μ(t))2 /(2σ 2 (t)), (, t) [0, 1] [0, T] (2.5) Scaling laws fo the shape paametes can be built based on supathemal electon emission, measued via had X-ay measuements, see fo instance [14] and [15]. The total cuent diven by the LH antenna is then calculated using scaling laws such as those pesented in [16]. It should be noted that the methods pesented in this book can easily be extended to othe cuent deposit shapes (eithe fo use in othe tokamaks o to change the non-inductive cuent dives used). While the impact of I p on the deposit amplitude would induce a nonlineaity (poduct between the state and the contol input), we neglect this effect by consideing an exta loop on the adio-fequency antennas that sets the engineeing inputs accoding to a desied pofile. Such stategy is motivated by the fact that the antennas eact much faste than the plasma and can thus geneate a desied pofile almost instantly.

6 16 2 Mathematical Model of the Safety Facto and Contol Poblem Fomulation The second non-inductive souce of cuent is due to the bootstap effect, induced by paticles tapped in a banana obit. Expessing the physical model deived by [17] in cylindical coodinates and in tems of the magnetic flux, the bootstap cuent is obtained as: j bs (, t) = p er 0 ψ/ { [ 1 p e A 1 p e + p ( i 1 p i p e p i α 1 i T i T i )] } 1 T e A 2 T e whee p e/i is the electon and ion pessue, T e/i is the electon and ion tempeatue, α i depends on the atio of tapped to ciculating paticles x t and A 1/2 (, t) depend on x t and on the effective value of the plasma chage. Maximizing the bootstap effect, as a fee souce of non-inductive cuent, is one of the pime objectives fo lage tokamaks such as ITER, which motivated the bootstap cuent maximization stategy poposed in [18]. As the contol appoaches discussed in this book ae focused on linea time-vaying stategies (on the lumped and PDE models), we will conside small deviations fom an equilibium bootstap distibution (given by the efeence magnetic flux distibution) and ensue the obustness with espect to these deviations athe than addessing the nonlineaity diectly. To summaize, j ni is consideed as the sum of thee components: j ni = j lh + j eccd + j bs 2.3 Contol Poblem Fomulation Based on the pevious desciption of the system dynamics and peifeal components obtained fom a physical analysis of the tokamak plasma, this section discusses the appopiate change of vaiables to fomulate the contol poblem. We also descibe the contol objectives and challenges fo an efficient egulation of the safety-facto pofile, which will be answeed in the following chaptes Equilibium and Regulated Vaiation We define η =. η /μ 0 a 2. and j = μ 0 a 2 R 0 j ni to simplify the notations. An equilibium ψ, if it exists, is defined as a stationay solution of: [ η [ ] ] 0 = ψ + [ ηj ], (0, 1) (2.6) with the bounday conditions:

7 2.3 Contol Poblem Fomulation 17 ψ (0) = 0 ψ (1) = R 0μ 0 I p 2π (2.7) fo a given couple ( ) j, I p, whee, to simplify the notation, fo any function ξ depending on the independent vaiables and t, ξ and ξ t ae used to denote ξ ξ and t, espectively. Remak 2.2 When seeking an equilibium by solving (2.6) (2.7) two cases have to be consideed: (i) theeisnodiftinψ (equivalent to V loop = 0 at all times using the altenative bounday condition ψ t (1, t) = V loop (t) in (2.4)) and theefoe the solution of (2.6) (2.7) veifies: η [ ] ψ + η j = 0 (2.8) In this case, ψ (and its spatial deivative) is independent on the value of η and theefoe the stationay solution exists (i.e. thee is an equilibium of the timevaying system) egadless of the vaiations in η. This is the case we diectly addess in this book. (ii) thee is a adially constant dift in ψ (equivalent to V loop = 0 fo some times when using the altenative bounday condition) and theefoe the solution of (2.6) (2.7) veifies, fo some c(t): η(, t) [ ψ (, t) ] + η(, t)j() = c(t) (2.9) In this case, ψ does not coespond to an equilibium since it will be a function of time and space (in paticula, it will be a function of η(, t) and c(t)), we will call the coesponding ψ(, t) a pseudo-equilibium of the system. It can be shown to veify: with time-deivative: ψ (, t) = 1 ψ t (, t) = ( ) ρ c(t) ρj(ρ) dρ (2.10) η(ρ, t) ( ) ρ ρ η(ρ, t) η(ρ, t)ċ(t) η 2 (ρ, t) c(t) dρ (2.11) This case is not extensively addessed in this book but the esults pesented in Chaps. 4 and 5 will not be seveely affected as long as ċ(t) and η(, t) ae bounded in a suitable way. Since a pseudo-equilibium will exist at each time,

8 18 2 Mathematical Model of the Safety Facto and Contol Poblem Fomulation the obustness esult pesented in Theoem 4.2 can be applied, ewiting the evolution of the system aound this pseudo-equilibium (instead of an actual equilibium) and consideing w = ψ t (, t) (the time-vaying natue of the pseudo-equilibium acts as a state-distubance fo the system). Aound the equilibium (assumed to exist as pe the pevious emak) and neglecting the nonlinea dependence of the bootstap cuent on the state, the dynamics of the system is given by: ψ t = η ] [ ψ + η j, (, t) (0, 1) (0, T) (2.12) with bounday conditions: ψ (0, t) = 0 ψ (1, t) = R 0μ 0 Ĩ p (t) 2π (2.13) and initial condition: ψ(, 0) = ψ 0 () (2.14) whee the dependence of ψ =... ψ ψ, j = j j and η on (, t) is implicit; Ĩ p = Ip I p and 0 < T + is the time hoizon. As the safety facto pofile depends on the magnetic flux gadient, ou focus is on the evolution of z =. ψ/ (equivalent to deviations of the effective poloidal field magnitude aound an equilibium), with input u =. j, defined as: [ ] η(, t) z t (, t) = [z(, t)] + [η(, t)u(, t)], (, t) (0, 1) (0, T) (2.15) with Diichlet bounday conditions: z(0, t) = 0 z(1, t) = R 0μ 0 Ĩ p (t) 2π (2.16) and initial condition: z(, 0) = z 0 () (2.17) whee z 0. = [ ψ 0 ]. Following [19], the following popeties ae assumed to hold in (2.15): P 1 : K η(, t) k > 0 fo all (, t) [0, 1] [0, T) and some positive constants k and K.

9 2.3 Contol Poblem Fomulation 19 Fig. 2.3 Catesian coodinates (x 1, x 2 ), and pola coodinates (,θ)used in this book P 2 : The two-dimensional Catesian epesentations of η and u ae in 1 C 1+α c,α c /2 (Ω [0, T]), 0<α c < 1, whee Ω =. { (x 1, x 2 ) R 2 x1 2 + x2 2 < 1} as shown in Fig P 3 : Ĩ p is in C (1+αc)/2 ([0, T]). Fo completeness puposes, the existence and uniqueness of sufficiently egula solutions (as needed fo the Lyapunov analysis and feedback design puposes in the next chaptes) of the evolution equation is stated, assuming that the popeties P 1 P 3 ae veified. Theoem 2.1 If Popeties P 1 P 3 hold then, fo evey z 0 : [0, 1] R in C 2+α c([0, 1]) (0 <α c < 1) such that z 0 (0) = 0 and z 0 (1) = R 0 μ 0 Ĩ p (0)/2π, the evolution equations (2.15) (2.17) have a unique solution z C 1+α c,1+α c /2 ([0, 1] [0, T]) C 2+α c,1+α c /2 ([0, 1] [0, T]). The poof of this esult is given in [19] and mainly follows fom [20] Inteest of Choosing ψ as the Regulated Vaiable A natual question that may aise at this point is why studying the evolution of the poloidal magnetic flux pofile instead of studying diectly the safety facto pofile. Consideing that the safety facto pofile is elated to the magnetic flux pofile as: q(, t) = B φ 0 a 2 ψ (, t) (2.18) the evolution of the safety facto pofile is then given by: 1 Hee C αc,βc (Ω [0, T]) denotes the space of functions which ae α c -Hölde continuous in Ω, β c -Hölde continuous in [0, T]. P 2 can be stengthened by assuming that η and u ae in C 2,1 (Ω [0, T]) which is the case fo the physical application.

10 20 2 Mathematical Model of the Safety Facto and Contol Poblem Fomulation and, fom (2.2): q t (, t) = q2 (, t) o, in a moe geneal fom: q t (, t) = B φ 0 a 2 ψ 2 (, t)ψ t(, t) = q2 (, t) B φ0 a 2 ψ t(, t) [ [ η(, t) 2 ] ] q(, t) q t (ρ, t) = q2 (ρ, t) μ 0 ρ + q2 (ρ, t) ρ + q2 (, t) B φ0 a 2 [η(, t)u(, t)] (2.19) [ [ ] ] η (ρ, t)ρ C2 (ρ)c 3 (ρ) C3 2(ρ) q(ρ, t) [ η (ρ, t) V ] ρ j ni (ρ, t) FC 3 (ρ) which can be obtained fom (2.1) and the elation: q(ρ, t) = B φ 0 ρ ψ ρ (ρ, t) Equation (2.19) is nonlinea in q (making it difficult to extend esults obtained aound one equilibium to othe equilibia). This can be solved by woking instead with the so-called otational tansfom (denoted ι in [6], which is the invese of the safety facto). Nevetheless, the bounday condition in the z vaiable (i.e. the total plasma cuent) can be diectly (and pecisely) measued using eithe a continuous Rogowski coil o seveal discete magnetic coils aound the plasma (see [6]). Theefoe, in this book, we have chosen to contol the safety facto pofile by contolling the z vaiable. ρ ρ ρ Contol Challenges Contolling the safety facto pofile q in a tokamak is done by contolling the poloidal magnetic flux pofile ψ. In paticula, the desied popeties of the contolle ae: to guaantee the exponential stability, in a given topology, of the solutions of equation (2.15) to zeo (o close enough ) by closing the loop with a contolled input u(, t); to be able to adjust (in paticula, to acceleate) the ate of convegence of the system using the contolled input; to be able to detemine the impact of a lage class of eos motivated by the physical system and to popose a obust feedback design stategy. Actuation eos,

11 2.3 Contol Poblem Fomulation 21 estimation/measuement eos, state distubances and bounday condition eos should be consideed specifically. The poblem unde consideation poses seveal challenges that have to be addessed, some of which ae: diffeent odes of magnitude of the tanspot coefficients depending on the adial position that ae also time-vaying; stong nonlinea shape constaints imposed on the actuatos and satuations on the available paametes; obustness of any poposed contol scheme with espect to numeical poblems (in paticula given the diffeence in magnitude of the tanspot coefficients) and distubances; coupling between the contol applied to the infinite-dimensional system and the bounday condition; the contol algoithms must be implementable in eal-time (estictions on the computational cost). Refeences 1. E. Witant, E. Joffin, S. Bémond, G. Giuzzi, D. Mazon, O. Baana, P. Moeau, A contoloiented model of the cuent contol pofile in tokamak plasma. Plasma Phys. Contol Fusion 49, (2007) 2. F.L. Hinton, R.D. Hazeltine, Theoy of plasma tanspot in tooidal confinement systems. Rev. Mod. Phys. 48(2), (Apil 1976) 3. J. Blum, Numeical Simulation and Optimal Contol in Plasma Physics. Wiley/Gauthie-Villas Seies in Moden Applied Mathematics. (Gauthie-Villas, Wiley, New Yok, 1989) 4. J.F. Ataud et al., The CRONOS suite of codes fo integated tokamak modelling. Nucl. Fusion 50, (2010) 5. S.P. Hishman, R.J. Hawyluk, B. Bige, Neoclassical conductivity of a tokamak plasma. Nucl. Fusion 17(3), (1977) 6. J. Wesson. Tokamaks. Intenational Seies of Monogaphs on Physics, 118, 3d edn. (Oxfod Univesity Pess, USA, 2004) 7. M. Eba, A. Cheubini, V.V. Paail, E. Spingmann, A. Taoni, Development of a non-local model fo tokamak heat tanspot in L-mode, H-mode and tansient egimes. Plasma Phys. Contolled Fusion 39(2), 261 (1997) 8. E. Witant, S. Bémond, Shape identification fo distibuted paamete systems and tempeatue pofiles in tokamaks. in Decision and Contol and Euopean Contol Confeence (CDC-ECC), th IEEE Confeence pp , L. Labode et al., A model-based technique fo integated eal-time pofile contol in the JET tokamak. Plasma Phys. Contol Fusion 47, (2005) 10. D. Moeau, M.L. Walke, J.R. Feon, F. Liu, E. Schuste, J.E. Baton, M.D. Boye, K.H. Buell, S.M. Flanagan, P. Gohil, R.J. Goebne, C. T. Holcomb, D.A. Humpheys, A.W. Hyatt, R.D. Johnson, R.J. La Haye, J. Loh, T.C. Luce, J.M. Pak, B.G. Penaflo, W. Shi, F. Tuco, W. Wehne, and ITPA-IOS goup membes and expets, Integated magnetic and kinetic contol of advanced tokamak plasmas on DIII-D based on data-diven models. Nucl. Fusion 53:063020, F. Kazaian-Vibet et al., Full steady-state opeation in Toe Supa. Plasma Phys. Contol Fusion 38, (1996)

12 22 2 Mathematical Model of the Safety Facto and Contol Poblem Fomulation 12. P. Hetout, C. Boulbe, E. Nadon, J. Blum, S. Bémond, J. Bucalossi, B. Faugeas, V. Gandgiad, P. Moeau, The CEDRES++ equilibium code and its application to ITER, JT-60SA and Toe Supa. Fusion Eng. Des. 86(6 8), (2011) 13. N.J. Fisch, Theoy of cuent dive in plasmas. Rev. Mod. Phys. 59(1), (Januay 1987) 14. F. Imbeaux, Etude de la popagation et de l absoption de l onde hybide dans un plasma de tokamak pa tomogaphie X haute énegie. Ph.D. thesis, Univesité Pais XI, Osay, Fance, O. Baana, D. Mazon, G. Caulie, D. Ganie, M. Jouve, L. Labode, Y. Peyson, Real-time detemination of supatemal electon local emission pofile fom had x-ay measuements on Toe Supa. IEEE Tans. Nucl. Sci. 53, (2006) 16. M. Goniche et al., Lowe hybid cuent dive efficiency on Toe Supa and JET. 16th Topical Confeence on Radio Fequency Powe in Plasmas. Pak City, USA, S.P. Hishman, Finite-aspect-atio effects on the bootstap cuent in tokamaks. Phys. Fluids 31, (1998) 18. A. Gahlawat, E. Witant, M.M. Peet, M. Alami, Bootstap cuent optimization in tokamaks using sum-of-squaes polynomials, Poceedings of the 51st IEEE Confeence on Decision and Contol (Maui, Hawaii, 2012), pp F. Bibiesca Agomedo, C. Pieu, E. Witant, S. Bémond, A stict contol Lyapunov function fo a diffusion equation with time-vaying distibuted coefficients. IEEE Tans. Autom. Contol 58(2), (2013) 20. A. Lunadi, Analytic Semigoups and Optimal Regulaity in Paabolic Poblems, volume 16 of Pogess in nonlinea diffeential equations and thei applications. (Bikhäuse, Basel, 1995)

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