Filament-Circuit Model Analysis of Alcator C-MOD Vertical Stability

Transcription

1 PFC/JA Filament-Circuit Model Analysis of Alcator C-MOD Vertical Stability D. A. Humphreys and I. H. Hutchinson Plasma Fusion Center Massachusetts Institute of Technology Cambridge, MA June, 1989 This paper submitted to Nuclear Fusion. This work was supported by the U. S. Department of Energy Contract No. DE-AC02-78ET Reproduction, translation, publication, use and disposal, in whole or in part by or for the United States government is permitted.

2 Filament-Circuit Model Analysis of Alcator C-Mod Vertical Stability Abstract The rigid displacement, current conserving, multifilament plasma-circuit approach to vertical stability analysis has been applied to the control system design for Alcator C- MOD, a high performance tokamak under construction at MIT. For the purpose of rough benchmarking of the multifilament model, comparison has been made with analytical calculations. For walls conformal to D-shaped C-MOD equilibria, marginal wall positions agree well with analytical results for the onset of ideal instability in corresponding elliptical equilibria with conformal walls. For Alcator C-MOD equilibria, velocity feedback is observed to be necessary for active stabilization. A simple model including one passive and one active coil is shown to ehibit the same phenomenon. A single filament representation of the plasma is inadequate for modeling Alcator C-MOD equilibria, but 3-5 filaments are found to be sufficient to come within 10% of the converged many-filament growth rate. Application of the model to Alcator C-MOD indicates that n=2 equilibria can be stabilized satisfactorily provided active power supplies with response time less than.5 ms are used. 2

3 Introduction Many modern tokamaks possess strongly shaped, highly elongated plasmas in order to eploit the higher betas and current densities achievable with high elongation. Such a vertically elongated plasma is intrinsically unstable to vertical aisymmetric displacements, however, and must be actively controlled in order to maintain equilibrium [1,2,3,9]. Some degree of analysis of vertical stability is necessary for every tokamak possessing an elongated plasma, but it is especially important in the net generation of machines for which elongations near 2 are commonplace. For such devices, wall proimity and conformality are critical factors. In particular, if the vacuum vessel wall cannot be designed close enough and conformal enough to the plasma, there may not be a great deal of stability margin for highly elongated equilibria, and the issue of accurate modeling of the problem becomes etremely important. Because these are resistive wall modes, and machine geometries are in general quite comple, some form of numerical analysis must be used to evaluate the stability of actual tokamak plasmas and design feedback systems. Furthermore, the difficulty of the full MHD and resistive wall problem has resulted in the common use of simpler models to analyze the vertical control problem for real machines. In fact, since the process of design inherently involves iteration, the algorithm must be rapid, reinforcing the need for simple models. Traditionally the most popular simple approach has been to represent the plasma as an array of current-carrying filaments and surrounding conductors as aisymmetric resistive loops [4,5,6,7]. The linearized coupled set of circuit equations for the conductors can then be solved together with the plasma equation of motion. The result is a set of eigenvalues and eigenmodes describing both plasma motion and induced currents. As machines are operated closer to their vertical stability limits at ever-increasing elongation, it becomes very important to have a model which is sufficiently accurate to design a satisfactory control system. At the very least, the model used must yield conservative results, if not accurate ones, so that the lack of passive stability margin can be 3

4 somewhat compensated by sufficient active control margin. Fortunately, there is some reason to believe [8] that the circuit approach provides such conservative results, and for that and other reasons is well-suited to analyzing the vertical stability problem. In the present work we propose to eamine this approach, the further simplifying assumptions often made in using the circuit method, and its application to the design of the Alcator C-MOD tokamak. I. The Vertical Instability Vertical instability in tokamaks arises from the quadrupole magnetic field which must be applied in order to create a yertically elongated plasma. When added to the vertical vacuum field required to maintain radial equilibrium, the quadrupole field results in a vacuum field which is concave toward the outboard side over much of the plasma. This inherently destabilizing configuration of field lines is characterized by having a predominantly negative decay inde [9], n, defined by : R OBZ R OBR B. 8R B,, Oz Regions of negative decay inde must necessarily be formed in order to create elongated equilibria. Typically this curvature will vary widely across the plasma, and the net destabilizing effect is a result of the field curvature profile "sampled" by the current profile over the entire plasma. It should be noted that decay inde is only a measure of curvature, not of destabilizing force. To compare net destabilizing forces in different equilibria, one must sum over the appropriate local field gradient force quantities, not over the local decay indices. The local decay inde is effectively the gradient force term normalized by the local B., so the "average" (weighted by current density) destabilizing gradient force cannot be simply related to the "average" decay inde. The nature of the vertical mode is such that unless a conducting wall is sufficiently close to the plasma in some average sense, implying some degree of conformality, the mode will be unstable on the ideal MHD timescale. In the presence of a resistive wall which is sufficiently close, the mode will grow much more slowly, roughly on the wall L/R timescale. 4

5 II. The Circuit Model The circuit model approach combines physical simplicity and rapid eecution with the potential for a high degree of accuracy in the conductor model. Unfortunately, the weakness of the circuit model is that the plasma itself is not easily treated correctly. Typically the simplifying assumption of rigid uniform vertical displacements is used, along with conservation of plasma current rather than the flu conservation appropriate for an MHD description. Further simplifying the plasma model, many formulations represent the plasma as a single filament rather than an array. Current profile effects are therefore lost, ecept as they are reflected in the equilibrium field gradient at the chosen location for the single filament. Even if many filaments are used, the rigid shift is not necessarily the correct energy minimizing displacement. At best, without allowing filament currents to vary, a multifilament plasma model will simply give a better estimate of the net destabilizing field gradient force across the entire plasma. This is reason enough to use a multifilament model, however, since a priori it is far from clear where a single filament should be located to produce the most accurate results. In the following we shall describe the circuit model form appropriate for a uniform rigid vertical shift and retained finite plasma mass [4,5], modified to include a multifilament plasma specification. The coils representing stabilizing conductors can of course be driven according to control laws modeling a feedback response due to plasma motion. Plasma filament currents are kept fied, so that all plasma-coil induction is due to the motion of the plasma. The filament self inductances are thus effectively infinite. This circuit model yields a set of M voltage equations for the M resistive loops representing the conductors eternal to the plasma. These are of the form: L dlj +ri,+e MJJdIJI, t 1 dt' N +2 19Z = Vappled Where: Ij are the perturbed coil currents (j = 1, M), Mj, is the inter-coil mutual inductance matri, 5

6 Mij is the coil-filament mutual inductance matri, rj is the resistance of coil j, Lj is the inductance of coil j, N is the number of plasma filaments, Vpplied is the voltage applied to coil j. 3 If the coil in question is a driven control coil, the applied voltage is given by a control law depending on the plasma displacement. For passive (undriven) coils this voltage is simply zero. The system of equations is closed by including the equation of motion of the plasma for vertical displacements. This is simply the force balance including the force due to non-zero decay inde (which is destabilizing in the absence of feedback), and the damping force due to eddy currents induced in surrounding conductors. These are denoted FD' and FFddy respectively, and the resulting force equation is: M dt2 = FEddy + FDI Z M N M ii ip j 3 i3 Z N OBR Z27rRIpi OzZ SB Z Where: Ip are the plasma filament currents (i = 1, N), BRi is the radial vacuum field at filament i, We now look for normal mode solutions of the form: z --+ zet*, 1j- I 1 e-' (etc...) The final form of the problem is that of a generalized eigenvalue problem: 6

7 where the vector F contains the perturbed quantities z, C = i, and the fi. The solution of the system thus gives us the normal modes and associated growth or damping rates. This collection of normal modes includes many modes which do not contribute to plasma motions of interest. Many of these represent decay modes of the conductor array with essentially no plasma motion. In addition, if the plasma mass is non-zero, two high frequency oscillatory roots appear which are highly damped on the slow time scale of the instability. These therefore represent transients which can be ignored given sufficient wall stabilization. Depending on the order of the feedback control law, comple conjugate root pairs appear. One of these pairs includes at least one unstable root at zero gain. This pair is strongly affected by feedback (hence referred to here as the "feedback root"), beginning unstable at low gain, possibly becoming stable as gain increases, and often growing unstable again as gain continues to increase. III. Discussion of Various Analytic Models Once one has chosen the circuit model approach to analyzing the vertical stability problem, there are still many possible simplifying (or complicating) variations. A single filament or many filaments can be used to represent the plasma. The plasma mass can be included or neglected. The plasma can be taken to be current conserving or flu conserving. Many considerations contribute to the validity and desirability of such assumptions, and we shall discuss some of these in detail. To illustrate some of these features, we shall use simple cases in which one or two coils are used to represent the conductors in the system. To begin with, consider the case of a single resistive coil in the vicinity of a plasma represented by an array of N filaments which are taken to conserve current rather than flu, so that the circuit equations given in the previous section are applicable: Mi = ScIe + SBZ Leic + Sci + rc I = 0 After Laplace transforming, we obtain a third order characteristic equation s +Ve8 + (Vw VB)S - VbV, = 0 7

8 where v rc/lc, vi, Se/Lem, vz = SB/r. c is thus the characteristic decay time of the coil, vz is proportional to the destabilizing field gradient force term, and V.2 reflects the eddy current stabilization effect. The zero-mass limit of this equation yields one root, which is unstable and consists of the characteristic decay frequency with a modification due to eddy current stabilization. In this limit the divergence corresponding to the ideal MHD stability boundary occurs when v, = vb. In general when mass is retained the roots of this equation consist of a stable comple conjugate pair with imaginary parts on the order of Alfven frequencies, and one real unstable root whose magnitude depends on the degree of stabilization. In this case the ideal stability boundary is somewhat more difficult to define. If we nondimensionalize the characteristic equation by dividing through by vb, defining = 7 =- v/vb, we obtain s/v, e = vcl/vb, X3+ E2 + (7-1) - C = 0 If we now plot the unstable real root as a function of 77, we find a family of curves as e varies [Fig. 1]. The figure shows that although 7 > 1 is no longer such a hard ideal stability boundary as in the massless case, it is still a good approimate boundary for typical cases of interest (10-6 < e < 10-4). In such a case we see the growth rate change from resistive frequencies ( < 1) for 71 Z 1 to Alfven frequencies ( - 1) for 1 ~< is thus also the approimate boundary for validity of the massless approimation. That is, if 17 is sufficiently greater than 1, the system is significantly well stabilized on the Alfven timescale, and the massless approimation will be quite good. Of course, if it should become important to follow the transition to ideal instability, for eample in analyzing highly elongated equilibria, mass must be retained. This is the reason mass has been included in the VSC code and its applications described later. Now consider the two coil massless current conserving plasma case. The equation of motion is now a force balance equation: 0 = SlIl + S SBZ 8

9 and the circuit equations are L 1 i1l+ M12i2 +r1i 1 + Sli = V L 2 i 2 + M124 +r S 2 i 0 where coil 1 is the active control coil, to which control voltage V is applied. Laplace transforming we find a transfer function of the form That is, it contains one zero and two poles. G(s) = z(s) _ rs +1 V(s) As 2 +Bs+C Epressing the passive transfer function in a standard form: G _) Go(rs + 1) (ris + 1)(r2s - 1) where ri- is the unstable pole, we can now add a feedback law V(s) = -Ko 0 (res + 1)z representing proportional-derivative (PD) feedback. The system now appears as in Fig. 2, in which H(s) = Ko(r.s + 1), so that the open loop transfer function is of the form G(s)H(s) = a(rts + 1)(,ras + 1) (Tis + 1)(-r 2 s - 1) The corresponding root-locus equation whose solution gives the location of system poles as a varies is (at )52 + ( a[r. - r.])s + (a + 1) = 0 which can be stabilized by an a which is sufficiently negative, even if r. = 0 (i.e. without derivative feedback). The above has assumed, however, that the system zero, -1/r., is a left-half plane zero (i.e. r, > 0). This is not necessarily the case. Epansion of the full solution shows that the intrinsic system zero is given by 1 r2 9 SIM

10 Now, since Si = I, am", we have S2 Mi, R2BR 2 S1 M(, R1BR1 where " ' " has been used to indicate derivative with respect to z, and BR; is the radial B-field at coil i due to the plasma. Clearly this can be arbitrarily large as the BR-coupling between coil 1 (the control coil) and the plasma is reduced. An etreme eample is placement of coil 1 near the same z-location as the plasma filament, where BR is equal to or nearly zero. In particular, if the coupling between the passive stabilizing structure (represented here by coil 2) and the plasma sufficiently eceeds that between the control coil and the plasma, a right-half plane system zero will result. Allowing r, < 0, defining t = IrZ I, we find for the new characteristic equation (-at.r. - -ri1 2 )s 2 + (-r 1 - r 2 + a[-r. - t.])s + (a + 1) = 0 For -i- > r 2, true for a system which is not etremely well passively stabilized, this cannot now be stabilized without derivative feedback (i.e. if -r = 0). Since vacuum vessels are in general better coupled to the plasma than to control coils, there is usually a maimum elongation which a given machine can stabilize without derivative feedback. This phenomenon is seen in code results to be discussed later. IV. The VSC Code In order to analyze C-MOD equilibria, and PF coil and vacuum vessel designs, a code (known as VSC for Vertical Stability with Circuit model) was written which implements a current-conserving, non-zero plasma mass, multifilament plasma circuit model. All toroidally continuous conductors in the vicinity of the plasma are modeled as arrays of aisymmetric loops with circular poloidal cross section and finite resistivity. Resistivity is defined to be that of the appropriate material, and cross-sections are chosen to conserve the total resistance of each wall section. Coils are located at evenly spaced points along straight segments of the vacuum vessel. EF coils are generally represented by one or two 10

11 model coils of circular cross-section, but the OH stack model is typically composed of a larger coil array. Figure 3 shows the code model of the plasma-conductor system. The filament array representing the plasma is determined by reading the current density profile from ASEQ, a PEST-like equilibrium code [10], and further discretizing that (already discrete) data into a number of filaments much smaller than the entire grid used by the equilibrium code. This yields a plasma model which, though very simple, provides the possibility for very accurate calculation of the net destabilizing force on the plasma. The EF coil currents are taken from the same equilibrium code, and the EF coils themselves are rediscretized much like the plasma. The accuracy of the rediscretized EF coil model is checked by comparing flu values at the plasma resulting from the model with values calculated by the equilibrium code. The code allows the use of a variety of approaches to active feedback control. In particular, an eplicit double pole, single zero control law [5] of the form dt d 2 _ dz can be specified, describing a control voltage Vi(t) applied to coil i as a function of plasma displacement z(t) and velocity i(t). This yields a transfer function of the form V 1 (s) _ Z(s) -C(i + t.s) (1 + tbs)(1 + tes) This form is especially useful in analyzing the constraints of a realistic pulsed power supply. Such supplies possess an inherent time delay in their response to a control signal. If this response time is too long relative to the growth time of the phenomenon being controlled, the power supply will not be able to stabilize that mode. The control law form above allows us to model this intrinsic delay with one of the poles (for eample tb, setting t = 0), and then to determine the maimum possible delay time (tb) for which the system can be stabilized. To support the general equilibrium control analysis, including vertical stability control, a data base is being created consisting of equilibria chosen to sufficiently span the space of C-MOD operation. This data base can be used to interpolate required gain values for controlling both vertical stability and the evolution of equilibria. 11

12 V. Benchmarking and Model Investigations Several alternative methods of analysis are available with which to benchmark the circuit model. Analytic solutions eist for simplified geometries, in particular for marginal wall positions. Full MHD analysis codes (such as GATO) can provide marginal wall positions with more realistic geometries and comple displacement profiles. To benchmark resistive growth rates, either actual machine data or a more detailed tokamak simulation must be used. Here we shall consider marginal wall position comparisons with analytic cases for rough qualitative benchmarking. Haas [11] has demonstrated that there is a maimum distance at which a perfectly conducting wall will stabilize the vertical mode for an elliptical plasma. This provides a rough test of the general reliability of the code algorithm: if a resistive wall is used we would epect to see a similar critical distance manifest itself. This would be the point at which the mode becomes ideal unstable rather than unstable on the slow resistive time scale. For low, cases to be stable on the ideal timescale this criterion requires that: Here r, is the characteristic plasma minor radius, r, is the characteristic conducting wall minor radius, both measured from the magnetic ais along the midplane in the outboard direction, and. = b/a is the plasma elongation. For an elliptical equilibrium with r. of 1.8, this criterion for ideal MHD stability is: ( w <1.97 r. / Assuming that this should be roughly true even for the unusual case of an Alcator C- MOD equilibrium, the code was used to predict the critical distance for onset of ideal MHD instability for two vacuum vessel wall shapes. Figures 4, 5, and 6 show the real parts of all of the system eigenvalues, plotted on an Alfven-like frequency scale to reveal the transition to ideal instability. An array of conductors modeling a given wall shape is enlarged conformally from an initial position at the plasma surface. At a critical value of 12

13 rw/r,, the real parts of two of the modes go rapidly from the resistive frequency regime to an Alfven growth/damping rate. One of these is the unstable root, and this is the critical point we wish to compare with the Haas result. Because the Haas criterion was calculated for a conformal (elliptical) wall, the first vacuum vessel shape chosen was a D-shaped array of conductors approimately conformal to the plasma surface for a typical Alcator C-MOD equilibrium. Using the multifilament plasma model, this array was enlarged from an initial configuration lying on the plasma surface [Fig. 4]. The ideal instability onset point was found to occur at: ( W) = 2.00 Still using a multifilamentary plasma, the critical distance was also determined for the actual C-Mod vessel, sweeping its size using the same algorithm as in the conformal case [Fig. 5]. As epected, the resulting value, ( rr = 1.80 indicates that such a highly non-conformal wall must be closer to the plasma than the conformal wall must be. Nevertheless, even this critical distance is encouragingly close to the Haas prediction. The conformal shell case was repeated using a single filament located at the current profile centroid [Fig. 6]. In this case the result is significantly more optimistic, yielding a critical distance of ( rr = 2.28 This greater optimism for the case of a single filament at the current centroid is consistently observed and emphasizes the need for a multifilament model. Note that as the magnetic ais is further out than the current profile centroid, placing the single filament on the ais yields even more optimistic results. The circuit model can itself be used to investigate the assumptions made in such a model. In particular we shall address two issues raised above: convergence with increasing number of filaments used in the plasma model, and location of a single filament. Figs. 13

14 7 and 8 show growth rates as a function of radial position of a single filament along the plane at the z-position of the current centroid for equilibria eq6 and eq7 (see Table I). Current centroids are denoted by RO in the figures, and in both cases the magnetic ais is located at 66 cm. Multifilament model-derived growth rates are 388 sec' for eq6 and 945 sec- 1 for eq7, whereas the corresponding single filament growth rates are 174 and 461 respectively, for filaments located at the current centroids. It is clear from these eamples that care must be taken when locating a single filament. While for some equilibria placing a filament at the current centroid may yield growth rates relatively close to what one finds with a multifilament model, for others (e.g. eq6 and eq7) this is not the case. In fact, Figs. 7 and 8 illustrate that for eq6 and eq7 there is no location for which a single filament will yield the same growth rate as the multifilament model. The situation is particularly bad for equilibria such as eq7, which are relatively close to the marginal ideal stability point, or equivalently are very highly elongated. Such cases were also illustrated in the benchmarking section, in which the marginal wall position was seen to be strongly dependent on the choice of single or multifilament model. Active analyses also tend to be very sensitive to choice of filament model, since using a single filament is essentially the same as drastically increasing the peakedness of an equilibrium which may not originally have been highly peaked. In particular, this can significantly decrease the coupling between control coil and plasma, even though the vacuum vessel coupling may not be as strongly affected. The result of this difference is seen in Table II, showing tolerable delay times (described in Section IV of the tet) for marginal stability with fied derivative feedback time constant for single and multifilament models of equilibria eq6 and eq7. Having argued that some degree of spatial distribution is important in a filamentary model of a plasma used in circuit models, it is reasonable to ask how many filaments is sufficient. Figs. 9 and 10 show convergence of growth rate values calculated with increasing numbers of filaments modeling the plasma for equilibria eq6 and eq7. Equilibrium eq6 is a double null, elongation 1.8 case, and eq7 is a single null equilibrium with an elongation of 2.0. For these typical C-MOD cases, only 3-5 filaments need be used to get within 10%, and roughly 20 filaments or more should be used to converge to within 1% of the growth 14

15 rate calculated with the maimum number of filaments used, on the order of That only 3-5 filaments are so effective is indicative of the low order of significant multipole moments in the vacuum magnetic field. VI. Alcator C-MOD Vertical Stability Analysis Alcator C-MOD [12] is a high performance tokamak intended to be somewhat prototypical of the Compact Ignition Tokamak (CIT). As such it will provide a test bed for many of the critical issues to be faced in the design and operation of CIT. The essential machine parameters are as follows: major radius R=0.66m, minor radius a=0.21m, maimum toroidal B-field Bt=9T, maimum plasma current I,=3MA, and typical elongation n=1.8. Machine geometry is shown in Figs. 3 and 11. C-MOD will possess a strongly shaped diverted plasma, with maimum elongation in ecess of 2. The compleity of its geometry, including D-shaped single and double null plasmas and a roughly rectangular vacuum vessel with a high degree of diagnostic access producing many interruptions in toroidal current paths, has motivated this study of the vertical control problem. The "standard" mode of the VSC code retaining plasma mass, conserving plasma current, and using full multifilament models, has been used to analyze C-MOD equilibria. Passive growth rates for a variety of equilibria spanning a wide range of parameters are shown in Table I. As elongation increases from n = 1.65 to n = 2.0, the unstable growth rate increases from y = 25 to -y ~ 950 sec~ 1. It is interesting to compare these values with some characteristic resistive decay times of the system. The diffusive penetration time of the vacuum vessel, TD [13], is on the order of TDE _ d 11 ms where a is a rough equivalent radius for a circular model of the vacuum vessel wall, taken to be 50 cm; d is the thickness of the vacuum vessel wall, taken to be 1.27 cm (.5 in); and 1 is the resistivity of the vacuum vessel, taken to be that of stainless steel at room temperature, 70 po - cm. For the EFC control coil, characteristic of copper elements in 15

16 the system in general, the L/R time at room temperature is -L = 80 ms ( R cu The vacuum vessel model itself has many possible decay modes due to the large number of coils used to represent it. One among these of particular interest is the dominant stable midplane-antisymmetric mode. This is the lowest order mode which couples to vertical plasma motion. The current distribution in surrounding conductors for this mode is shown in Fig. 12 for the "standard" C-MOD equilibrium (eq2). The figure shows the entire vacuum vessel above the midplane, as well as a portion of it below. The area of each coil marker in the figure is proportional to the current in that element, and positive or negative currents are denoted by open or filled square markers respectively. As one would epect, the dominant antisymmetric mode has a roughly sing dependence of current distribution on poloidal angle. The corresponding damping rate is () = 10.8 ms R Antisym Comparing this value to the actual growth rates for the range of equilibria in Table I, we observe that C-MOD equilibria do not in a sense take full advantage of the stabilization potential of the system. This is due largely to the plasma shape, irregularity of the vacuum vessel, and the peakedness of actual current profiles. The lack of wall-plasma conformality alone makes it etremely improbable to be able to reduce the growth rate to this optimistic level for plasmas having the elongation of the vacuum vessel (K = 2). By contrast, Fig. 13 shows the current distribution for the feedback root (the dominant mode which is affected by feedback and contains plasma motion) in an actively stabilized case for equilibrium eq2. It is noteworthy that this mode clearly contains high poloidal harmonics, and therefore cannot be accurately modeled by a low order mode. In fact, using a lower order mode to estimate the growth rate will yield an unrealistically well stabilized result, since the characteristic decay time increases with decreasing mode order. It is also of interest to eamine the dependence of growth rate on triangularity. Equilibria eq2, eq4, and eq5 in Table I were chosen requiring all other relevant parameters 16

17 to remain relatively fied while triangularity varies. The data suggest that for the uniform displacement, current conserving, multifilament circuit model, triangularity does not strongly affect the growth rate. Notice that the eq4 single filament calculation yields a growth rate much higher than the single filament result for the other two equilibria having the same plasma parameters. This is another manifestation of the unreliability of single filament results. For eq4, the choice of the current centroid for the single filament location proved to yield a growth rate fortuitously close to that calculated with the multifilament model. A fast circuit model code is ideal for studying the relative passive stabilization effects of various control coil placements. These coils will typically be the closest copper coils well-coupled to the plasma vertical motion, and should therefore have a strong influence on passive stability properties. Many control coil positioning options were investigated during the design of C-MOD, but we shall address the passive stability for two cases alone. They correspond to control on the OH2 portions of the "notched" OH stack, or on coils denoted "EFC" placed in the upper and lower outboard "shoulders" of the vacuum vessel specifically for control purposes [Figs. 3 and 11 ]. Passive growth rates corresponding to these cases are shown in Table I. The stabilizing effects of these two choices for control coil are in general comparable. In light of this, and acknowledging the difficulty of engineering a fast control power supply operating in parallel with the OH2 equilibrium supplies, the outboard coils denoted EFC in Fig. 11 have been chosen for vertical control. Passive analysis can provide a sense of the difficulty of stabilizing given equilibria, and to some etent measure the relative quality of various stabilizing configurations. However, on the basis of these data alone nothing reliable can be said about power supply response time requirements. Active studies are required to determine actual limits on the control system. In particular, while the passive growth rate may not be very sensitive to eact control coil location, the calculated tolerable delay time may depend more strongly on it. Active analysis of tokamak equilibria primarily addresses whether an equilibrium can be stabilized at all given the available geometry and hardware constraints. This can also find application in an iterative procedure attempting to optimize placement of control sensors and coils. The VSC code is capable of implementing flu loops and B-field sensors, 17

18 as well as direct state sensors, which have immediate access to plasma displacement and velocity. However, we shall only address the use of direct state feedback here. Linear power supply dynamics in the form of frequency domain transfer functions can be defined between demands to the power supplies and actual output to a control coil, allowing power supplies to be modeled with a relatively high degree of compleity. This transfer function definition can also be used to represent feedback compensation. For the most part, these etra dynamics have been used to model the response delay time intrinsic to a pulsed power supply, as described in Section II. A single pole has been used to represent the delay, with a single compensating zero representing derivative feedback. In general, elongated C-MOD equilibria have been found to require derivative feedback in order to be stabilized at all, even with zero delay. This is the same phenomenon as that illustrated in the discussion of analytical models above, and has been investigated by other authors [14]. Fig. 14, for eample, shows the result of sweeping gain in eq6 using direct displacement feedback with no derivative term. The unstable root has a growth rate of 388 at zero gain, but becomes more and more stable as gain is increased. In the absence of derivative feedback, however, this root never becomes stable. Fig. 15 shows the result of adding derivative feedback. For this case the mode are stabilized. Addition of a single pole modeling power supply delay makes it more difficult to stabilize, and in fact a given equilibrium will ehibit a maimum tolerable delay which can be stabilized at all. Stability boundary results showing maimum tolerable delays and necessary zero values for eq6 and eq7 are shown in Table II. Using this crude delay model, the maimum tolerable delay time for the maimum elongation equilibrium (eq7; passive growth rate = 945) is found to be approimately 713 = 1 msec for the actual EFC configuration (Fig. 3) with a lead value of 50 msec. The actual fast power supply will have a response time well below.5 msec, ensuring a significant safety margin. It is of interest to note that a previous EFC placement yielded a passive growth rate for this equilibrium of 980 sec 1 and maimum tolerable delay of approimately ra' = 0.4 msec 18

19 with the same lead value. Thus the passive growth rate was affected relatively little by this design change, but the acceptable delay was roughly doubled. The etra dynamics facility is also useful for modeling the higher frequency dynamics of the power supply. Since plasma mass is retained in our model, increasing gain will eventually drive the MHD roots unstable. This destabilization requires the power supply to respond on a microsecond timescale, which is clearly unreasonable. Addition of appropriate filters to the power supply model can prevent this nonphysical destabilization of the MHD roots. The usefulness of the circuit model in determining power requirements is somewhat limited. An initial state must be selected and epanded in eigenfunctions of the system, and some assumption must be made about power supply limits. Using the control coil voltage and current components from the unstable eigenmode calculated by the VSC code, the effective control coil impedance is found. Once the power limit is established, this impedance allows demand currents and voltages to be determined scaled to the displacement. If a reasonable initial displacement can be determined, this procedure provides a test of sufficiency of the power supply specifications. Results indicate that a 1 MVA power supply will stabilize the n = 2 plasma of eq7 with approimately 30 V applied to the EFC coil, yielding about 35 ka for a 5 mm displacement. It should be noted that all of the above calculations have ignored the stabilizing effects of various equilibrium field coil pairs which will routinely be operated so as to be passively stabilizing. The most effective of these pairs is the OH2 upper and lower coils, which are operated independently in order to produce single null plasmas. As noted before, the stabilizing effect of these coils is comparable to that of EFC, whose effect is in turn quite large. Thus, the most pessimistic passive stabilization scenario has been consistently chosen. VII. Conclusions The circuit model approach to vertical stability analysis has proven to be etremely fast in program eecution, easily understandable, and highly versatile. Much information 19

20 can be etracted from the eigenvalue analysis, and the method is useful for feedback control system design as well as analysis. However, great care must be eercised in applying the model. In particular, the use of a single filament to represent the plasma can produce growth rates and active control properties which are significantly different from those of a more realistic multifilament distributed plasma model. With a judicious algorithm for filament placement, only 3-5 filaments are needed to substantially improve on the performance of a single filament. The vertical stability characteristics and control system requirements of Alcator C- MOD have been analyzed using a multifilament circuit model code. Consistent with predictions of simple analytic models, C-MOD has been found to require velocity feedback in order to stabilize highly elongated equilibria. Even for the most highly elongated equilibria (n! ~ 2.0) planned for C-MOD, the final control coil and vacuum vessel designs have been found to tolerate power supply delay times well above the actual power supply response time, and to result in reasonable control system current and voltage demands. This is true even for the most pessimistic of stabilizing structure models. Lacking the wall conformality and closeness of other tokamaks in return for increased diagnostic access and other design considerations, Alcator C-MOD will nevertheless be able to produce high performance plasmas with elongations to rival the highly elongated conformal wall devices operating today. 20

21 REFERENCES [1] Mukhovatov, V.S., Shafranov, V.D., Nucl. Fusion 11 (1971) 605 [2] Laval, G., Pellat, R., Soule, J.S., Phys. Fluids 17 (1974) 835 [3] Rebhan, E., Salat, A., Nucl. Fusion 18 (1978) 1431 [4] Jardin, S.C., Larrabee, D.A., Nucl. Fusion 22 (1982) 1095 [5] Thome, R.J., Pillsbury, R.D., Montgomery, D.B., Politzer, P.A., Wolfe, S.M., Mann, W.R., Langton, W.G., Pribyl, P., MIT Plasma Fusion Center Rept. PFC/RR (1982) [6] Mori, M., Suzuki, N., Shoji, T., Yanagisawa, I., Tani, T., Matsuzaki, Y., Nucl. Fusion 27 (1987) 725 [7] Nagayama, Y., Naito, M., Ueda, Y., Ohki, Y., Miyamoto, K., Nucl. Fusion 24 (1984) 1243 [8] Hutchinson, I.H., submitted to Nucl. Fusion [9] Wesson, J.A., Nucl. Fusion 18 (1978) 87 [10] Strickler, D.J., Miller, J.B., et al ORNL Rept. ORNL/FEDC-83/10 (1983) [11] Haas, F.A., Nucl. Fusion 15 (1975) 407 [12] Hutchinson, I.H., Becker, H., Bonoli, P., Diatchenko, N., Fairfa, S., Fiore, C., Granetz, R., Greenwald, M., Gwinn, D., Hakkarainen, P., Humphreys, D., Irby, J., Lipshultz, B., Marmar, E., Montgomery, D., Park, C., Parker, R., Pierce, N., Pillsbury, R., Porkolab, M., Ramos, J., Rice, J., Schultz, J., Sigmar, D., Silva, F., Takase, Y., Terry, J., Thibeault, E., Wolfe, S., MIT Plasma Fusion Center Rept. PFC/RR [13] Freidberg, J.P., Ideal Magnetohydrodynamics, Plenum Press (1987) 309 [14] Lazarus, E.L., Lister, J.B., Bull. Amer. Phys. Soc. 33 (1988)

22 Figure Captions Fig. 1. Growth rates normalized to the characteristic Alfven frequency ( a s/yb, where s is the actual growth rate), as a function of stabilization factor, 77 _ u.,/zv. a) e = 10-3, b) e = 0.1, c) e = 0.55, where e = vl/vb. e is typically less than Fig. 2. Block diagram of the closed loop feedback system. Fig. 3. Code Model of the Alcator C-MOD machine geometry. Solid squares represent stabilizing conductors, open squares marked with "X" are equilibrium field (EF) coils, and filaments modeling the plasma current distribution are denoted by open circles. EF coils which are considered to be stabilizing are represented as solid squares. Fig. 4. Growth rates as a function of wall distance, illustrating the critical distance for onset of ideal instability for a conformal d-shaped wall and multifilament plasma model. Fig. 5. Growth rates as a function of wall distance, illustrating the critical distance for onset of ideal instability for a C-MOD vacuum vessel model and multifilament plasma model. Fig. 6. Growth rates as a function of wall distance, illustrating the critical distance for onset of ideal instability for a conformal d-shaped wall and single filament plasma model. Fig. 7. Growth rate for a single filament plasma model as a function of filament radial position (equilibrium eq6). Ro denotes the current centroid, at RO=65.Ocm, while the magnetic ais is located at R=66.5cm. Fig. 8. Growth rate for a single filament plasma model as a function of filament radial position (equilibrium eq7). RO denotes the current centroid, at Ro=65.4cm, while the magnetic ais is located at R=66.5cm. Fig. 9. Growth rate as a function of number of plasma filaments for equilibrium eq6,. = 1.8, 6 =.35. Roughly 3-5 filaments are necessary for the calculated growth rate to come within 10% of the converged value. Fig. 10. Growth rate as a function of number of plasma filaments for equilibrium eq7,. = 2.0, 6 =.32. Roughly 3-5 filaments are necessary for the calculated growth rate to come within 10% of the converged value. 22

23 Fig. 11. Alcator C-MOD machine geometry. Arrows indicate equilibrium field (EF) coils and the ohmic heating (OH) stack. EFC is the vertical control coil pair. Note the "notched" OH coil design, consisting of the single large OH1 segment and two independently driven OH2 coils. Fig. 12. Currents induced in the vacuum vessel for the dominant passive antisymmetric mode with equilibrium eq2. The area of each coil marker in the figure is proportional to the current in that element, and positive or negative currents are denoted by open or filled square markers. Fig. 13. Currents induced in the vacuum vessel for the stabilized feedback root (the dominant mode which is affected by feedback and contains plasma motion) with active control of equilibrium eq2. The area of each coil marker in the figure is proportional to the current in that element, and positive or negative currents are denoted by open or filled square markers. Fig. 14. Root-locus plot for eq6 with no derivative feedback (t. = 0). The figure shows that for this equilibrium in the absence of derivative feedback, the plasma is never stabilized. Arrows indicate the direction of motion of the feedback root (the dominant mode affected by feedback containing plasma motion) as gain increases. Note that the plot also contains stationary roots which remain fied as gain varies, representing modes which do not couple to plasma motion. Fig. 15. Root-locus plot for eq6 with derivative feedback (t. $ 0). The figure shows that the addition of derivative feedback stabilizes equilibrium eq6. Arrows indicate the direction of motion of the feedback root (the dominant mode affected by feedback containing plasma motion) as gain increases. Note that the plot also contains stationary roots which remain fied as gain varies, representing modes which do not couple to plasma motion. 23

24 Table Captions Table II. Table of equilibrium data. m is the elongation, 6 is the triangularity, and -y is the passive growth rate for the given equilibrium. "mf" refers to multifilament plasma model, "sf" to single filament model, "vv+efc" to both vacuum vessel and EFC passively stabilizing, and "vv+oh2" to both vacuum vessel and OH2 passively stabilizing. See Fig. 11 for machine geometry. Table II. Table of active feedback law data. t. is the lead compensation (velocity feedback) time constant, and t b is the lag time constant modeling power supply delay time. As in Table I, "mf" and "sf" refer to multifilament and single filament plasma model respectively. For all Table II cases the vacuum vessel is used for passive stabilization and EFC for active control. 24

25 0 + q c i l 00m4z V) N~ +U -o c6 n LO 0M v O 00 v cd cq, IV, 00 v I CY N c LO 00 a) v4 y r4 c 4- C(v (V 0) ) Q Q 25

26 P. 0 a ~0 - Sv 4Q) Sv %I v v Um PM 4 ~ - I49. 26

27 I 0.2 Qc Fig. 1 27

28 V(S) G (s)(s H(s) Fig. 2 28

29 I.l E :-: on gas so: as a U II U U O on 'sg VD: M.e as. "esn so:. 00a000 se a. 000 V U U U mnow so. as WE. a ma * * a m-n..: a I -- I i I I ' R [M] 1.2 Fig. 3 29

30 Ideal MHD unstable r /r rw /rp Fig. 4 30

31 f- a CMOD Configuration 45 0r Ideal 1.50 rw/rp MHD Unstable Fig. 5 31

32 2.00 F ) c Ideal MHD Unstable Fig. 6 rw/rp 32

33 to c~v14~2 0 ') LO 0 0 I O O0 I 0 N a 0 0 [1 ~s] +( 3 33

34 to 0 0 %c 0 0 ') LO i N~ (D IT I [,-:as] +#( 34

35 _0 Cl

36 0- E 0 0 0o0 OD C~*

37 EFI EF2 nip EF3 EFC EFC OH2- OH2- EF3 EFI EF2 Fig

38 DWD D D D Da D ~ cco-- TU Ey I I% 00 N 6: 6 I i I r-" E N

39 * a a a a 0 *-W * a m E 6 co C> I I C> '9 'U-- 39

40 60Q E X X X 0 ow % f U Re y a 4 I Fig

41 E i K U- 4,% Re y %t i w w I wd 20C Fig