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\newcommand{\AI}{\hbox{ARIES-I}}
\newcommand{\AII}{\hbox{ARIES-II}}
\newcommand{\AIII}{\hbox{ARIES-III}}
\newcommand{\TI}{\hbox{TITAN-I}}
\newcommand{\TII}{\hbox{TITAN-II}}
\newcommand{\VTS}{\hbox{V-3Ti-1Si}}
\newcommand{\VCT}{\hbox{V-15Cr-5Ti}}
\newcommand{\HT}{\hbox{HT-9}}
\newcommand{\degC}{$^\circ$C}
\newcommand{\SdegC}{$\,^\circ$C}
\newcommand{\SPL}{MgAl$_2$O$_4$}
\newcommand{\AlO}{Al$_2$O$_3$}
\newcommand{\LNO}{LiNO$_3$}
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\newcommand{\X}{\times}
\newcommand{\Dtf}{\hbox{$\Delta T_f$}}
\newcommand{\Dtw}{\hbox{$\Delta T_w$}}
\newcommand{\WR}{\hbox{W-26Re}}

\title{\bf 12. START-UP AND OPERATIONS}
\author{
\hbox to \hsize{
\hbox to 2.truein{      Robert A.~Krakowski     \hfil } \hfil
\hbox to 2.truein{\hfil Tak-Kuen Mau            \hfil } \hfil
\hbox to 2.truein{\hfil Mohammad Z. Hasan           } } \\
\hbox to \hsize{
\hbox to 2.truein{      Kenneth R.~Schultz      \hfil } \hfil
\hbox to 2.truein{\hfil Giorgio L.~Varsamis     \hfil } \hfil
\hbox to 2.truein{\hfil Clement P.~C.~Wong            } } \\
}
\date{ }
\maketitle

\begin{titlepage}
{%\baselineskip=2.0ex
\tableofcontents
}
\end{titlepage}

\setcounter{chapter}{11}
\setcounter{page}{1}

\chapter{START-UP AND OPERATIONS}

\section{INTRODUCTION}
\label{s-12-1}

The time required for plasma start-up and shut-down for the steady-state \AI\
design is short when compared with the burn period, and is considered to be an
occasional event in the reactor lifetime. The start-up, therefore, can be
performed slowly and under optimal conditions ({\it e.g.}, minimum power,
stress, {\it etc.}). Furthermore, a rapid current ramp-up is undesirable
because of increased power requirements and the demands on the poloidal-field
(PF) coil system to keep the plasma in equilibrium. For the \AI\ study, a
quantitative modeling and assessment of the plasma start-up scenario was
carried out, assuming disruption-free plasma and ensuring compatibility of the
auxiliary-power system needed for start-up with the current-drive system
required for steady-state operation. While a minimum-power requirement or
minimum-energy input has been used frequently as a criterion in designing the
start-up phase, it is of secondary importance in the context of a steady-state 
reactor.

Considerations of the various reactor subsystems, such as first wall, blanket,
divertor, and power conversion also impose constraints on the plant start-up
and partial- and over-power operations. These constraints are mainly due to
the temperature and thermal stress limits of the materials used in the
subsystems, and to the power requirements for magnet cool-down, blanket
heat-up, and coolant circulation.

In this section, both the plasma and engineering aspects of the start-up 
procedure for \AI\ are examined in detail. In particular, Sec.~\ref{s-12-2}
describes a plasma start-up code that is used to simulate the current ramp-up
and plasma ignition scenarios for a tokamak reactor. The coupling of the
fusion plasma with the PF-coil system and the reactor first wall can also be
modeled. Preliminary simulation results indicate that a reasonable start-up
scenario with an \hbox{$\sim$2400~s} duration using auxiliary heating by the 
current-drive system is indeed possible. Section~\ref{s-12-3} contains an
assessment of the engineering constraints on the plant start-up and its various
modes of power operations. No major critical issue has been uncovered by this
study. In \hbox{Sec.~\ref{s-12-4}}, conclusions and recommendations for future
work are given.

\section{PLASMA START-UP}
\label{s-12-2}

In this section, various plasma-physics aspects of the start-up and operation 
of a tokamak reactor are examined in the context of the \AI\ design. A typical
reactor plasma start-up procedure consists of the following sequence of
events: plasma initiation, current initiation, current and density ramp-up,
auxiliary heating to ignition, and approach to steady-state operating
conditions. Specifically, plasma initiation, current start-up, and current
ramp-up have been demonstrated in a notable experiment on PLT~\cite{Job85}
using only lower-hybrid waves. A one-hour-long tokamak discharge sustained
entirely by lower-hybrid waves has recently been reported in
\hbox{TRIAM-1M~\cite{Ito90}}, and a record current-drive efficiency, 
\hbox{$\gamma$\,($\equiv\,n_{e}I_{\phi}R_{T}/P_{CD}$)}, of \hbox{$0.34\X
10^{20}$ A/W-m$^{2}$} was obtained on \hbox{JT-60~\cite{Ima90}}. The data base
for lower-hybrid start-up and current drive, therefore, is strong and the
physics understanding is mature.

Generally, a number of plasma start-up scenarios can be envisaged~\cite{Con86}. 
In \AI, a low-density current ramp-up scenario based on lower-hybrid current
drive is proposed, with a limit on the power input of not more than 20$\,$MW.
Electronic phasing of the lower-hybrid waveguide grille is a crucial element in
this approach. At $0^{\circ}$ phasing and modest power levels, the plasma is
initiated at full radius. The increased power input is balanced against plasma
convective losses through toroidal drift that can be adjusted by the vertical
field~\cite{Per89}. By introducing a non-zero phase shift between adjacent
wave-guides in the grille, the current can be initiated and ramped up. Once the
full current is reached, a programmed fueling and auxiliary-heating schedule
can be activated to bring the plasma to full fusion-power production. 
Transition to fast-wave current drive (FWCD) should be timed to provide 
adequate current ramp-up assist and auxiliary heating to the plasma. 
Throughout the start-up phase, constraints related to plasma-current ramp 
rates \hbox{($\leq\,1\,$MA/s} for disruption mitigation), electron runaway,
poloidal-field ramp-up rates \hbox{($\leq\,1\,$T/s)}, and thermomechanical
responses of the fusion power core are applied.

Only a limited number of start-up scenarios, going from current ramp-up to 
steady state, were considered and the results should be regarded as
preliminary and non-exhaustive. \hbox{Section~\ref{s-12-2-1}} describes the
tokamak-specific features of the start-up simulation code used in the study.
\hbox{Section~\ref{s-12-2-2}} gives preliminary results of a start-up scenario 
leading to the \AI\ reference steady-state parameters. The thermomechanical
response of the first wall for this reference scenario is also presented.
Section~\ref{s-12-2-3} contains a summary of the simulation results and lists a
number of critical issues.

\subsection{Features of Start-Up Code}
\label{s-12-2-1}

Estimates of the \AI\ current and fusion-power ramp-up parameters have been
made using a time-dependent, profile-averaged (zero-dimensional) plasma-circuit
simulation code. This code solves a set of particle and power-balance 
equations for the plasma. These equations are coupled in turn to a set of
circuit equations that describe the magnetic interactions between the plasma
and the various coil systems. The code utilizes a multi-species plasma model
that includes electrons, fuel ions, impurities, and fusion ash. A 
Fokker-Planck formalism is used to describe both charged-particle
fusion-product slowing down and electron-ion energy equi-partition. Assuming
density and temperature profiles of the form 
\hbox{$[1-(r/a)^{2}]^{\alpha_{n},\alpha_{T}}$} and on-axis and edge-plasma
safety factors, respectively $q_{o}$ and $q_{a}$, the equilibrium
current-density and magnetic-field profiles are approximated. In addition to
the basic plasma and field-coil responses ({\it e.g.}, currents, voltages,
densities, temperatures, powers), the code also estimates the first-wall
thermal and mechanical responses. The start-up code, originally developed for
the reversed-field-pinch (RFP) reactor, is described in detail in
Refs.~\cite{Naj90} and \cite{Hag84}; only modifications made to model for the
tokamak are given in this section. All quantities in this section are in MKS
units except for temperature in keV, power in MW, and current in MA.

\subsubsection{Plasma model}
\label{s-12-2-1-1}

In this start-up code, the plasma is modeled as a torus with a circularized 
cross section of minor radius $r_{p}$. For an elongated plasma of half-height
$b$ and equatorial plane half-width $a$ (elongation \hbox{$\kappa = b/a$)}, 
triangularity $\delta$, and major radius \hbox{$R_{T}$}, the equivalent plasma
radius is~\cite{Ehs79}
%
\begin{eqnarray}
\label{e-12-2-1}
r_{p} & = & a \: \kappa\, \left(1-\frac{0.233\,\delta}{A}\right)
\left[ \frac{2\:J_{1}(\delta)}{\delta}\right]^{1/2} \: .
\end{eqnarray}
%
An equivalent first-wall radius is taken as \hbox{$r_{w}=r_{p}/x\ (x \leq 1)$}
when computing the thermomechanical response in \hbox{Sec.~\ref{s-12-3}}.

A cylindrical one-dimensional (1-D) MHD equilibrium solver is used to compute 
the field and current profiles from a given pressure profile. With \hbox{$\mu
\equiv \mu_{o} \vec{j} \cdot \vec{B}/B^{2}$} and $p$ denoting the plasma
pressure, the following equilibrium equations are solved:
%
\begin{eqnarray}
\label{e-12-2-2}
-\frac{\partial B_{\phi}}{\partial r} = \mu_{o}j_{\theta} & = & 
\mu B_{\theta} + \mu_{o}\frac{\partial p}{\partial r}\cdot 
\frac{B_{\phi}}{B^{2}} \: , \\
\label{e-12-2-3}
\frac{1}{r}\frac{\partial}{\partial r}(rB_{\theta}) = \mu_{o}j_{\phi} & = &
\mu B_{\phi} - \mu_{o}\frac{\partial p}{\partial r}\cdot
\frac{B_{\theta}}{B^{2}} \: .
\end{eqnarray}
%
For the RFP case, $\mu (r)$ is specified to satisfy experimental observation
\hbox{$\mu (r) \propto 1 - (r/a)^{8}$}. For the case of a tokamak, \hbox{$\mu
(r)=\mu_{o}j_{\phi}(r)/B_{\phi}$} is used with $j_{\phi}(r)$ being
approximated by \hbox{$[1-(r/r_{p})^{2}]^{\alpha_{J}}$} and
\hbox{$\alpha_{J}=(q_{o}/q_{a})^{-1}$}. While this 1-D approximation is
adequate for the high-aspect-ratio circular RFP, applicability to the \AI\
configuration is less certain. The impact of the approximate profiles on the
start-up results, however, should be small since the equilibrium
current-density and magnetic-field profiles are only used to calculate the
plasma inductance for use in the circuit analysis
(\hbox{Sec.~\ref{s-12-2-1-3}}) and the plasma ohmic dissipation (corrected for
neoclassical trapped-electron effects~\cite{Dne85}). This neoclassical
correction is applied to the classical parallel-field electrical resistivity,
\hbox{$\eta_{\parallel}^{CL}$}, in a tokamak of inverse aspect ratio,
\hbox{$\epsilon \equiv 1/A \equiv a/R_{T}$}, resulting in
\hbox{$\eta_{\parallel}^{NEO}$} given by~\cite{Dne85}
%
\begin{eqnarray}
\label{e-12-2-4}
\frac{\eta_{\parallel}^{CL}}{\eta_{\parallel}^{NEO}} & = & 
\left[ 1-\frac{f_{T}(\epsilon)}{1+\zeta \: \nu_{\ast e}}\right]
\left[ 1-\frac{C_{R} \: f_{T}(\epsilon)}{1+\zeta \: \nu_{\ast e}}\right] \: ,
\end{eqnarray}
%
where $\nu_{\ast e}$ is the electron collisionality and
%
\begin{eqnarray}
f_{T}(\epsilon) & = & 1 - \frac{(1 - \epsilon)^{2}}{(1 - \epsilon^{2})^{1/2}
(1 + 1.46 \, \epsilon^{1/2})} \: , \nonumber \\
C_{R} & = & \frac{0.56}{Z_{eff}}\ \frac{3 - Z_{eff}}{3 + Z_{eff}}
\: , \nonumber \\
\nu_{\ast e} & = & \frac{r_{p}\: (B_{\phi}/B_{\theta})}
{\epsilon^{3/2} \: (T_{e}/m_{e})^{1/2}} \: \nu_{ei} \: , \nonumber \\
\nu_{ei} & = & 9.1 \times 10^{17} \: n_{e} \: ln\Lambda \: T_{e}^{-3/2} 
\: , \nonumber \\
\zeta & = & 0.58 + 0.20\,Z_{eff} \: .
\label{e-12-2-5}
\end{eqnarray} 

The plasma-energy confinement time, $\tau_{E}$, is modeled by the
inverse-quadrature relation between neo-Alcator and Goldston confinement
times, respectively $\tau_{NA}$ and $\tau_{G}$~\cite{Gol84}:
%
\begin{eqnarray}
\label{e-12-2-6}
\tau_{E} & = & \left[\tau^{-2}_{NA} + (H_{G}\tau_{G})^{-2}\right]^{-1/2} \: ,
\end{eqnarray}
%
where
%
\begin{eqnarray}
\tau_{NA} & = & 1.92 \X 10^{21}n_{i} \: a \: R^{2}_{T} \: , \nonumber \\
\tau_{G} & = & 3.7 \X 10^{-5}\kappa^{0.5} \:
\left(\frac{A_i}{1.5}\right)^{0.5} \: I_{\phi} \:
P_{H}^{-0.5} \: R_{T}^{-1.75} \: a^{-0.39} \: ,
\label{e-12-2-7}
\end{eqnarray}
%
and $A_{i}$ is the atomic mass number of the fuel ions and $P_{H}$ is total
heating power. The Goldston confinement multiplier, $H_{G}$, in 
\hbox{Eq.~(\ref{e-12-2-6})} is fixed at 1.5. For purposes of establishing a
fueling algorithm, the ion-particle confinement time, $\tau_{p}$, is taken to
be $4\tau_{E}$.

Synchrotron-radiation power loss, $P_{CYC}$, is calculated using the locally
applied global model described in \hbox{Sec.~2.2}, and is found to scale as 
%
\begin{eqnarray}
\label{e-12-2-8}
P_{CYC} \propto (1-\Re)^{1/2} \, n_{e}^{1/2} \, T_{e}^{5/2} \, B_{\phi}^{5/2} 
\:,
\end{eqnarray}
%
where $\Re$ is the first-wall reflectivity. In \AI, the silicon-carbide
first wall is quite lossy \hbox{($\Re\simeq 0.4$)}, and both $T_{e}$ and
$B_{\phi}$ are high. Thus, $P_{CYC}$ is comparable to the transport power loss
and far exceeds bremsstrahlung radiation losses. Line radiation losses from
carbon, oxygen, and iron impurities, and $Z_{eff}$ are estimated with a 
coronal equilibrium model~\cite{Pos77}.

Two models~\cite{Hsi89,Har88} are used to estimate the pressure-gradient-driven
bootstrap current, $I_{BS}$. Defining the Shafranov beta as \hbox{$\beta_{IO}
\equiv 2 \mu_{o}p_{o}/B^{2}_{\theta}$}, where \hbox{$B_{\theta} =
\mu_{o}I_{\phi}/(2\pi aS)$}, \hbox{$S = [(1 + \kappa^{2})/2]^{1/2}$}, and
$p_{o}$ is the peak pressure, the bootstrap-current fraction, $f_{BS}$, is
given by
%
\begin{eqnarray}
\label{e-12-2-9}
f_{BS} & \equiv & \frac{I_{BS}}{I}{\phi} \: = \: f_{1}\left[ C_{1} -
C_{2}\left(\frac{\alpha_{T}}{\alpha_{p}}\right) - 
C_{3}\left(\frac{\alpha_{T}}{\alpha_{p}}\right)
\frac{n_{e} \, T_{i}}{Z_{eff} \, p_{o}}\right]\frac{\beta_{IO}}{A^{1/2}} \: ,
\end{eqnarray}
%
for the model used in \hbox{Ref.~\cite{Hsi89}}. In this expression 
\hbox{$\alpha_{p}=\alpha_{n} + \alpha_{T}$}, \hbox{$C_{1}=2.1$}, 
\hbox{$C_{2}=1.3$}, \hbox{$C_{3}=1.2$}, and \hbox{$f_{1}=0.24$}.
For the approach recommended in \hbox{Ref.~\cite{Har88}},
%
\begin{eqnarray}
\label{e-12-2-10}
f_{BS} & = & \frac{\beta_{\theta}(1+\alpha_{p})
(\alpha_{n}C_{n}+\alpha_{T}C_{T})}{2\delta^{1/2}}
\int^{1}_{0}\frac{x^{7/2}(1-x^{2})^{\alpha_{p}-1}}
{1-(1-x^{2})^{\alpha_{J}+1}}dx \: ,
\end{eqnarray}
%
where $\beta_{\theta}$ is the poloidal beta,
\hbox{$C_{n}=1.5+3.1/Z_{eff}^{1.26}$}, and
\hbox{$C_{T}=1.3-0.93/Z_{eff}^{0.83}$}. Typically, $f_{BS}$ is small for the
initial start-up conditions assumed and increases as the plasma beta is raised. 
In the simulations, $f_{BS}$ is assumed to increase to the \AI\ design-point
value and, once achieved, is fixed at that value. In order to account for the
pedestal density profile, which is used in the ARIES systems code but not
included in the burn code, $f_{BS}$ is multiplied by a factor of 1.5 with the
constraint that \hbox{$f_{BS}{\,} \leq 0.68$}, which is the steady-state value
used in the systems code.

\subsubsection{Current ramp-up model}
\label{s-12-2-1-2}

The current ramp-up model described in \hbox{Ref.~\cite{Oka86}} is used to
relate the radio-frequency (RF) current-drive power, $P_{CD}$, needed to drive
the non-inductive part of the current transient, $I_{RF}$. With the plasma loop
voltage given by \hbox{$V_{\phi}=P_{\Omega}/I_{\phi}$} in terms of the ohmic
power dissipation, $P_{\Omega}$, and \hbox{$E_{\parallel} \equiv 
V_{\phi}/(2\pi R_{T})$},
%
\begin{eqnarray}
\label{e-12-2-11}
\frac {P_{CD}}{I_{RF}} & = & \frac{1 - \Gamma_{o} E_{\parallel}}{\Gamma_{1}} 
\: ,
\end{eqnarray}
%
where, for lower-hybrid waves~\cite{Oka86},
%
\begin{eqnarray}
\label{e-12-2-12}
\Gamma_{o} = \Gamma_{LH} & = & \frac{9.848 \X 10^{21} \: (1/N^{2}_{Z2} - 
1/N^{2}_{Z1})}
{n_{e}(Z_{eff} + 2) ln \Lambda \: ln(N_{Z1}/N_{Z2})} \: , \\
\label{e-12-2-13}
\Gamma_{1} & = & \frac{\Gamma_{o}} {2\pi R_{T}K} \: , 
\end{eqnarray}
%
$K$ is a constant ($\sim$0.39), and $\Gamma_{1}$ can be interpreted as the
current-drive efficiency in the absence of $E_{\parallel}$. The assumed
square-wave spectrum of parallel refractive indices over which RF electron
interactions occur is defined by \hbox{$N_{Z1} = 2.0$} and \hbox{$N_{Z2} =
1.5$}. These values for the refractive indices are chosen to give a normalized
current-drive efficiency, $\gamma$, to the steady-state value from the
systems code, and to ensure penetration of the wave to the plasma core.

We will show below \hbox{[Eq.~(\ref{e-12-2-22})]} that the self- and mutual 
inductance of coils and plasma can be combined into an equivalent inductance,
$L_{E}$. Then, the RF current driven during start-up is determined by the
following simple circuit equation,
%
\begin{eqnarray}
\label{e-12-2-14}
I_{RF} & = & I_{\phi} + \frac{L_{E}}{R_{p}} \: \frac{dI_{\phi}}{dt} \: ,
\end{eqnarray}
%
where $R_{p}$ is the evolving plasma resistance,
\hbox{$R_{p}\simeq 2R_{T} \: \eta^{NEO}_{\parallel}/r^{2}_{p}$}. A correction
for the bootstrap current is implied in \hbox{Eq.~(\ref{e-12-2-14})}.  The heating
power delivered to the plasma is given by
%
\begin{eqnarray}
\label{e-12-2-15}
P_{H} & = & P_{RF}-L_{E}I_{\phi}\dot{I_{\phi}} \: ,
\end{eqnarray}
%
of which $\sim$99\% is assumed to be deposited in the electrons. Note that the
term $L_{E}I_{\phi}\dot{I}_{\phi}$ in \hbox{Eq.~(\ref{e-12-2-15})} represents
the power that goes into the poloidal-field energy in the plasma.

For low-density lower-hybrid current start-up, it is necessary to monitor the
condition for electron runaway ({\it i.e.}, when electron dynamical friction 
falls below the $eE_{\parallel}$ accelerating force). The runaway condition is
\hbox{$E_{\parallel}/E_{D} \geq 0.01$}, where the Dreicer field, $E_{D}$, is
given by
%
\begin{eqnarray}
\label{e-12-2-16}
E_{D} & \simeq & 10^{-3}\left(\frac{e}{\epsilon_{o}}\right)^{2}ln \Lambda
\: \frac{n_{e}}{T_{e}} \: .
\end{eqnarray}

Once the plasma current reaches the steady-state level, it is assumed that FWCD
takes over as the density and fusion power are simultaneously ramped to the
steady-state condition. Then, an empirical expression for the fast-wave
efficiency, $\gamma_{_{FW}}$, is used for this part of the simulation and is
given by
%
\begin{eqnarray}
\label{e-12-2-17}
\gamma_{_{FW}} & = & 0.72\, \langle{T_{e}}\rangle^{0.77}\,
(0.014+0.235\beta_{\phi}) \: ,
\end{eqnarray}
%
where $\langle{T_{e}}\rangle$ is the volume-averaged electron temperature.

\subsubsection{Plasma circuit model}
\label{s-12-2-1-3}

A wide range of plasma start-up scenarios can be envisaged. These scenarios 
require programming the current-drive power, the fueling rate, and the 
PF-coil voltages. To ensure a smooth approach of the plasma to the design
point in the simulations, a relatively straightforward algorithm is adopted. In
order to maintain the greatest parametric flexibility, the current waveform,
$I_{\phi}(t)$, is specified {\it a priori} in these simulations, although an
option also exists to preprogram the individual coil voltages. An exponential
rise of the plasma current, with a characteristic rise-time, $\tau_{R}$, is
used and assumes the form
%
\begin{eqnarray}
\label{e-12-2-18}
I_{\phi}(t) & = & (I_{\phi}^{MAX}-I_{\phi}^{MIN})(1-e^{-t/\tau_{R}}) 
+ I_{\phi}^{MIN} \: ,
\end{eqnarray}
%
where typically \hbox{$I_{\phi}^{MIN} \simeq 0.2$ MA}, 
and \hbox{$I_{\phi}^{MAX} = 10.4$ MA} 
is the \AI\ steady-state current value.

\begin{figure}[p]
\vspace*{7.5in}
\caption[Fig-2-1]
{\AI\ PF-coil set, as determined by VEQ~\cite{Pen89}, showing the collapsing
of 12 coils into 6 groups.}
\label{f-12-2-1}
\end{figure}

\begin{figure}[t]
\vspace*{3.5in}
\caption[Fig-2-2]
{Equivalent (collapsed) PF-coil set and plasma circuit used in the start-up 
simulation model.}
\label{f-12-2-2}
\end{figure}

To properly model the interaction between the plasma and the PF coils during
start-up, an equivalent-circuit approach is used. In
\hbox{Fig.~\ref{f-12-2-1}}, the entire \AI\ PF-coil set and its locations are
displayed. As defined by the output from the equilibrium code,
VEQ~\cite{Pen89}, the 12 coils in the upper half of the plasma cross section
are collapsed into six groups as shown.  The self- and mutual inductances for
any two groups of coils, designated A and B, each with $n_{A}$ and $n_{B}$
coils, are given by, respectively
%
\begin{eqnarray}
\label{e-12-2-19}
L_{A} & = & \frac{1}{n_{A}^{2}} \: 
\left(\sum^{n_{A}}_{i=1}L_{i}+\sum^{n_{A}}_{i=1}
\sum^{n_{A}}_{j=1,j\neq i}M_{ij} \right) \:, \\
\label{e-12-2-20}
M_{AB} & = & \frac{1}{n_{A}n_{B}} \:
\sum^{n_{A}}_{i=1}\sum^{n_{B}}_{j=1}M_{ij} \: ,
\end{eqnarray}
%
where $L_{i}$ is the self-inductance of the $i$th coil and $M_{ij}$ is the
mutual inductance between the $i$th and $j$th coils within a coil group.
\hbox{Figure~\ref{f-12-2-2}} shows an equivalent circuit for the
\hbox{$n$ (=6)} collapsed coil sets, for which all the inductances can be
computed independently from the VEQ output. The plasma is simply represented as
an additional coil with inductance $L_{p}$ and resistance $R_{p}$. An
equivalent voltage source, $V_{RF}$, is used to model the RF current-drive
power.

To preserve the plasma equilibrium during start-up, the ratio of the coil
current for the $i$th coil group, $I_{gi}$, to the plasma current, 
\hbox{$I_{gi}/I_{\phi} = C_{i}$}, is kept constant. Since the plasma current,
$I_{\phi}$, is known, each of the $C_{i}$ constants can be determined from
VEQ. This allows decoupling of the plasma circuit equation from the rest of
the circuit shown in \hbox{Fig.~\ref{f-12-2-2}}. To illustrate this, one can
write the plasma circuit equation as
%
\begin{eqnarray}
\label{e-12-2-21}
-V_{RF}+I_{\phi}R_{p}+L_{p}\frac{dI_{\phi}}{dt}+\sum^{n}_{i=1}M_{ip}
\frac{dI_{gi}}{dt} & = & 0 \: ,
\end{eqnarray}
%
which reduces to \hbox{Eq.~(\ref{e-12-2-14})} if
%
\begin{eqnarray}
\label{e-12-2-22}
L_{E} & \equiv & L_{p} + \sum^{n}_{i = 1}C_{i}M_{ip} \: 
\end{eqnarray}
%
and \hbox{$V_{RF} \equiv I_{RF}R_{p}$}.

\subsubsection{First-wall model}
\label{s-12-2-1-4}

The \AI\ first wall is a bank of SiC tubes cooled by helium. A cross section of
the start-up code model for the first wall is shown in
\hbox{Fig.~\ref{f-12-2-3}}. \hbox{Table~\ref{t-12-2-1}} lists the main
dimensions and physical properties used to assess the thermomechanical response
of the first-wall tube bank. In addition, a first-wall evaporation and
sputtering model is used to dynamically feed impurities to the plasma. However,
because of the relatively low power and particle fluxes and the first-wall
materials for \AI, impurity feedback is not an important effect. Details of the
first-wall thermomechanical model are given in Ref.~\cite{Wer90}.

\begin{figure}[p]
\vspace*{3.2in}
\caption[Fig-2-3]
{First-wall model used to estimate the transient thermomechanical response for
\AI\ \hbox{($r_{i}=4$ mm, $r_{m}=5.5$ mm, $r_{0}=7$ mm)}.}
\label{f-12-2-3}
\end{figure}

\begin{table}[p]
\caption{}
\centering
{\bf First-Wall Parameters Used in ARIES-I Start-Up Simulations}
\vspace{1.cm}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{l @{\hspace{1.5truein}} c} 
\hline \\
Density, $\rho$ (kg/m$^3$)                  & 3,217 \\
Thermal conductivity, $\kappa$ (W/K-m)      & 15 \\
Heat transfer coefficient, $h$ (W/K-m$^2$)  & 2,198 \\
Heat capacity, $c_{p}$ (J/K-kg)             & 1,200 \\
Thermal expansion coefficient, $\alpha$ ($10^{-6}$/K) & 4.5 \\
Young's modulus, $E$ (GPa)                  & 240 \\
Poisson's ratio, $\nu$                      & 0.30 \\
Inner radius, $r_{i}$ (mm)                  & 4.0 \\
Midpoint radius, $r_{m}$(mm)                & 5.5 \\
Outer radius, $r_{o}$ (mm)                  & 7.0 \\
Coolant temperature, $T_{c}$ (K)            & 923 \\
\\ \hline
\end{tabular}
\label{t-12-2-1}
\end{table}

\subsection{Results of Simulations}
\label{s-12-2-2}

\begin{table}[b]
\caption{}
\centering
{\bf Initial Plasma Conditions Used in 
ARIES-I Start-Up Simulations}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{l @{\hspace{1.25truein}} c}
\hline
\\
Plasma current, $I_{\phi}^{MIN}$ (MA)     &  0.2  \\
Density-weighted average temperature, $T_{i,e}$ (keV)  &  0.4 \\
Average density, $n_{i,e}$ (m$^{-3}$)     & $1.1\X 10^{19}$  \\
Dreicer parameter, $E_{\parallel}/E_{D}$  & 0.01  \\
Volume-averaged beta, $\beta_{\phi}$      & $3.13\X 10^{-6}$  \\
Plasma-current ramp time, $\tau_{R}$ (s)  & 600 \\
Effective charge, $Z_{eff}$               & 1.45 \\
\\ 
\hline
\end{tabular}
\label{t-12-2-2}
\end{table}

Start-up transients, based on a combination of inductive and RF current drive, 
are given for the \AI\ plasma. \hbox{Table~\ref{t-12-2-2}} lists the initial
plasma parameters assumed for the simulations. Strictly speaking, these
parameters, at the end of current initiation, can be determined by solving the
stationary-particle and power-balance equations with a small amount of 
lower-hybrid input power. However, this approach is not used in this study 
because the simulation results are found to be relatively insensitive to the
initial plasma parameters. The initial plasma density is then chosen by 
considering the minimum value allowed by the runaway-electron limit and the 
maximum value governed by the Greenwald limit~\cite{Uck88}. This latter 
density limit, however, is generally violated by $\sim$50\% for full-power 
steady-state conditions and must be re-examined in the context of a DT-burning 
plasma including the $\alpha$-particle pressure. The initial plasma beta is 
constrained to be below the Troyon limit of \hbox{$C_{T}=\beta
Ba/I_{\phi}\simeq 0.03$}. The simulation results presented in this section are
grouped as time histories of currents, voltages, powers, and energies in the
PF-coil circuit (\hbox{Figs.~\ref{f-12-2-4}} and \hbox{\ref{f-12-2-5}}), 
profile-averaged plasma properties (\hbox{Fig.~\ref{f-12-2-6}}), and plasma
powers (\hbox{Fig.~\ref{f-12-2-7}}).

\begin{figure}[t]
\vspace*{4.5in}
\caption[Fig-2-4]
{Transient response of the plasma current and RF voltage for the \AI\
reference start-up scenario.} 
\label{f-12-2-4}
\end{figure}

\begin{figure}[p]
\vspace*{7.5in}
\caption[Fig-2-5]
{Transient response of the PF-coil circuit for the \AI\ reference start-up
scenario showing the current rise rate, coil voltage, and coil input power.}
\label{f-12-2-5}
\end{figure}

\begin{figure}[p]
\vspace*{7.1in}
\caption[Fig-2-6]
{Transients of \AI\ plasma parameters during the reference start-up scenario:
(A)~Electron and ion temperatures, energetic $\alpha$-particle temperature, and
plasma $\beta$; (B)~Electron and fuel ion densities, various ion fractions,
and Dreicer parameter ($He^\star$ and $\alpha^\ast$ represent 
energetic $\alpha$ particles).}
\label{f-12-2-6}
\end{figure}

\begin{figure}[t]
\vspace*{4.0in}
\caption[Fig-2-7]
{Transient response of various plasma power components in the reference \AI\
start-up scenario: $\alpha$-particle heating power ($P_{Q}$), transport power
loss ($P_{COND}$), ohmic dissipation ($P_{\Omega}$), bremsstrahlung ($P_{BR}$)
and synchrotron ($P_{CYC}$) powers, and current-drive heating ($P_{CD}$).}
\label{f-12-2-7}
\end{figure}

As noted in \hbox{Sec.~\ref{s-12-2-1}}, all start-up simulations for ARIES-I 
assume {\it a priori} that the plasma current follows an exponential rise with
a rise-time, $\tau_{R}$, described by \hbox{Eq.~(\ref{e-12-2-18})}, and
calculates the required programming of the PF-coil voltages and powers. 
During the lower-hybrid-driven current-ramp phase, the plasma is fueled at a
rate that is 5\% above the particle loss rate which is characterized by
\hbox{$\tau_{p}=4\tau_{E}$}. To maximize the current-drive efficiency, the
initial ion density, $n_{iR}$, is kept low at \hbox{$n_{iF}/n_{iR}\approx
1-10$}, where $n_{iF}$ is the steady-state value. At \hbox{$t=3.5\tau_{R}$},
when \hbox{$I_{\phi}/I_{\phi}^{MAX} \simeq 0.97$}, the fueling rate is
increased in a controlled fashion to allow the plasma to follow the remaining
current transient to its steady-state design conditions. At the same time,
fast-wave power is switched on to maintain the current and to provide auxiliary
heating to the plasma. Extensive parametric studies have been carried out for
these start-up scenarios using $\tau_{R}$ (characteristic rise-time of plasma
current) and \hbox{$n_{iF}/n_{iR}$} (ratio of steady-state final density to
that of the current ramp-up phase) as variables, primarily to assess the
lower-hybrid and fast-wave power requirements for a reasonable start-up time. 
In most of the cases studied, the gas fueling assumes a 50:50 DT mixture. 
However, to protect the divertor plates during these power transients, ramp-up
of the plasma density to its full value should be as quick as is practical,
while the fusion power level is controlled by adjusting the composition of the
DT fuel mixture. This control technique admits the possibility of a rapid
density rise and a slower power transient with a tritium-lean fuel mixture.
Scenarios involving this feature are also examined in the simulations and are
presented in \hbox{Sec.~\ref{s-12-2-2-3}}.
 
\subsubsection{Reference case study}
\label{s-12-2-2-1}

The temporal response of the \AI\ plasma-coil circuit to a plasma-current
waveform given by \hbox{Eq.~(\ref{e-12-2-18})} is studied first. This reference
\AI\ case uses \hbox{$\tau_{R}=600$~s} and initial plasma parameters given in 
\hbox{Table~\ref{t-12-2-2}}. Specifically, the following initial values are
assumed: \hbox{$I_{\phi}^{MIN}=0.2$~MA}, \hbox{$n_{iR}=1.1\X 10^{19}$ m$^{-3}$}
for an assumed \hbox{$n_{iF}/n_{iR}=10$}, and \hbox{$T_{i}=0.4$~keV}. In 
\hbox{Fig.~\ref{f-12-2-4}}, the time evolution of the plasma current,
$I_{\phi}$, and RF current-drive voltage, $V_{RF}$, is displayed. The rapid
drops of $V_{RF}$ at \hbox{$t\simeq 100$~s} and \hbox{$t=2000$~s} reflect
mainly a decrease in $R_{p}$ due to a rise in electron temperature, $T_{e}$, at
these points, as shown in \hbox{Fig.~\ref{f-12-2-6}(A)}. In
\hbox{Fig.~\ref{f-12-2-5}}, the coil current-rise rates, coil voltages, and
input powers to the six PF-coil groups are given. The current and voltage
transients in this case are well within the design limits of the PF-coil
system.

The time evolution of the relevant plasma parameters during the entire
reference start-up phase for \AI\ is also shown in \hbox{Fig.~\ref{f-12-2-6}}.
Specifically shown in \hbox{Fig.~\ref{f-12-2-6}(A)} are the density-weighted
volume-averaged electron and ion temperatures, respectively, $T_{e}$ and
$T_{i}$, and the total plasma beta, $\beta_{\phi}$, as a function of time. 
During the low-density current-ramp phase \hbox{($0\leq t\leq 2100$~s)},
\hbox{$T_{e}\gg T_{i}$} because all the lower-hybrid power
\hbox{($\sim$20~MW)} is absorbed by electrons that are essentially decoupled
from the fuel ions. At \hbox{$t=2100$~s,} rapid density ramp-up occurs while
the fast-wave power is raised to its full value in a controlled fashion. The
steady-state conditions prescribed for \AI\ are attained at \hbox{$t\simeq
2500$~s}. In \hbox{Fig.~\ref{f-12-2-6}(B)}, the density evolution and the
ion-species fractions, including those of thermal and energetic
$\alpha$-particles, are displayed. The time variation of the Dreicer parameter,
\hbox{$\xi{\,} (=E_{\parallel}/E_{D}$)}, is also shown. In this case, the value
of $\xi$ peaks at the end of the current ramp and is $\sim$0.01, barely within
the runaway limit.

Figure~\ref{f-12-2-7} gives the time evolution of the various power terms in
the plasma power-balance equations for the \AI\ reference start-up scenario.
These include the $\alpha$-particle heating power, the transport power loss,
the ohmic dissipation, the synchrotron and bremsstrahlung powers, and the
current-drive power, $P_{CD}$. The $P_{CD}$ curve represents the programmed
RF-power waveform that is needed to ramp up and sustain the current during the
entire plasma start-up phase. In this reference case, the lower-hybrid
current-drive power requirement for the low-density current ramp-up is found to
be modest \hbox{($\leq 20$~MW)}. For \hbox{$t\geq 2100$~s}, the increase in
$P_{CD}$ reflects the changeover to FWCD. As discussed in
\hbox{Sec.~\ref{s-12-2-1}}, the synchrotron power is comparable to the
transport power throughout the entire start-up and bremsstrahlung radiation
losses are much smaller.

\subsubsection{First-wall response}
\label{s-12-2-2-1a}

The thermal and mechanical responses of the \AI\ first wall to the reference
start-up scenario were also analyzed. The following relevant first-wall power
densities are computed in the code: (A) neutron wall loading, $I_{w}$; (B)
radiated heat flux, \hbox{$q_{RAD}= [P_{BR} + P_{LINE} + P_{CYC}/(1 -
f_{H})]/A_{FW}$}, where $P_{LINE}$ is the line radiation power, $f_{H}$ is the
first-wall hole fraction, and $A_{FW}$ is the nominal first-wall surface area;
and (C)~maximum total heat flux, \hbox{$q_{TOT} = (P_{Q} + P_{\Omega} +
P_{CD})/A_{FW}$}. In principle, $q_{RAD}$ should be used to estimate the
first-wall thermomechanical transient behavior under the assumption that the
divertor is operating during start-up. However, in order to examine a
worst-case scenario (no divertor operation), $q_{TOT}$ is used instead in the
code calculations (all plasma power appearing on the first wall). In
\hbox{Fig.~\ref{f-12-2-8}(A)}, the 1-D thermal response of the \AI\ first wall
(\hbox{Fig.~\ref{f-12-2-3}}), accounting for one-sided surface heat flux, is
given. In particular, the temperatures of the coolant tube at the outer,
middle, and inner radii are displayed. The neutron wall loading, $I_{w}$, and
heat fluxes, $q_{RAD}$ and $q_{TOT}$, are plotted as functions of time in the
lower half of \hbox{Fig.~\ref{f-12-2-8}(B)}. In the same figure, the transient
mechanical response of the first wall is shown in terms of the secondary
(thermal) stress, $\sigma_{\theta}$, at the inner and outer tube radii. The
primary (pressure) stress, $\sigma^{p}_{\theta}$, for the $\sim$10-MPa He
coolant gives rise to loop components of 9.7 and 19.7~MPa at, respectively, the
inner and outer tube radii. These stress levels appear to be well within the
design limits for the SiC coolant-tube structures. A more detailed treatment
of the thermomechanical behavior of the first wall at steady state is given in
\hbox{Sec.~8.3.2}. 

\begin{figure}[p]
\vspace*{6.9in}
\caption[Fig-2-6a]
{Transient thermal and mechanical responses of \AI\ first wall for reference
start-up scenario: (A) Coolant tube temperatures ($T_{i}$, T$_{m}$, and T$_{o}$
at, respectively, inner, mid and outer radii); (B) Neutron wall loading
($I_{w}$), heat fluxes ($q_{RAD}$ and $q_{TOT}$), secondary (thermal) stresses
[$\sigma_{\theta}(r_{i})$ and $\sigma_{\theta}(r_{o})$], and primary (pressure)
stress [$\sigma_{\theta}^{p}(r_{i})$ and $\sigma_{\theta}^{p}(r_{o})$] ($r_i$
and $r_o$ denote, respectively, inner and outer tube radii).}
\label{f-12-2-8}
\end{figure}

\subsubsection{Parametric studies}
\label{s-12-2-2-2}

\begin{figure}[p]
\vspace*{7.3in}
\caption[Fig-2-9]
{Plasma transient response for start-up scenarios with $\tau_{R}=$ 150~s,
200~s, 400~s, and 600~s, at $n_{iF}/n_{iR}=10$: (A) Current-drive power, 
and (B) Ion and electron temperatures.}
\label{f-12-2-9}
\end{figure}

\begin{figure}[b]
\vspace*{4.8in}
\caption[Fig-2-10]
{Time evolution of plasma temperatures in \AI\ start-up
simulations with $\tau_{R}=600$ s and $n_{iF}/n_{iR}=$ 2, 5, and 10.}
\label{f-12-2-10}
\end{figure}

In the start-up simulation results reported up to this point, $\tau_{R}$ has
been fixed at 600~s and the density is kept at the low value of \hbox{$1.1\X
10^{19}$~m$^{-3}$} ({\it i.e.}, \hbox{$n_{iF}/n_{iR}=10$}) during the current
ramp-up phase. In addition, a series of parametric calculations have been
carried out with respect to variations in the prescribed current rise-time and
the density at the ramp-up phase (see \hbox{Fig.~\ref{f-12-2-9}}) in order to
study their impact on the maximum-required current-drive power,
\hbox{$P^{MAX}_{CD}$}, and the plasma evolution.
\hbox{Figure~\ref{f-12-2-9}(A)} depicts the influence of varying $\tau_{R}$ on
the current-drive program to reach the steady state for a fixed ratio of
\hbox{$n_{iF}/n_{iR}=10$} and the initial parameters shown in
\hbox{Table~\ref{t-12-2-2}}. As $\tau_{R}$ is decreased from 600~s,
$\dot{I_{\phi}}$ increases throughout the early phase of the current ramp-up.
Noting that \hbox{$P_{CD}\propto I_{RF}=I_{\phi}+(L_{E}/R_{p})\dot{I_{\phi}}$},
and that $R_{p}$ decreases as electrons are heated during the start-up, a
maximum in $P_{CD}$ occurs. As an extreme example, a $P^{MAX}_{CD}$ of 340~MW
is needed to achieve an exponential rise-time of 150~s for the plasma current.
In \hbox{Fig.~\ref{f-12-2-9}(B)}, the corresponding plasma temperature
evolution is shown for \hbox{$\tau_{R}=$ 150~s}, 200~s, and 600~s. For the
cases with \hbox{$\tau_{R}<600$~s}, a temperature excursion in electrons is
observed in the initial phase of the current ramp-up because of the strong
ohmic heating of electrons from the rapid current rise. In the extreme case of
$\tau_{R}=150$~s, $T_{e}$ peaks beyond 40~keV while $T_{i}$ reaches a maximum
value of 10~keV, resulting in significant $\alpha$-particle production in the
50:50 DT mixture used. Excursions in $T_{e}$ to above $\sim$15~keV prevent core
penetration of the lower-hybrid waves and must be avoided.

Start-up simulations have also been performed using \AI\ parameters for various
values of \hbox{$n_{iF}/n_{iR}$} at \hbox{$n_{iF}=1.1\X 10^{20}$~m$^{-3}$} and
\hbox{$\tau_{R}=600$~s}, and the results are displayed in 
\hbox{Fig.~\ref{f-12-2-10}} as time histories of $T_{i}$ and $T_{e}$. At higher
values of $n_{iR}$, more lower-hybrid power is required to achieve the same
current ramp rate because the current-drive efficiency is inversely
proportional to $n_{e}$. These larger current-drive (and heating) powers lead
to higher temperatures and, in some cases, substantial $\alpha$-particle
production in the current ramp-up phase. Again, for these ``high-density"
start-up scenarios, the question of penetration of the lower-hybrid wave power
to the plasma core warrants a more careful study.

\begin{figure}[t]
\vspace*{4.3in}
\caption[Fig-2-11]
{Dependence of maximum current-drive power during current ramp-up phase on
current rise-time for \hbox{$n_{iF}/n_{iR}=10$}, and on \hbox{$n_{iF}/n_{iR}$}
for \hbox{$\tau_{R}=600$~s}, using \AI\ reference parameters in
\hbox{Table~\protect\ref{t-12-2-2}}. The maximum Dreicer parameter is also
displayed as a function of \hbox{$n_{iF}/n_{iR}$} for 
\hbox{$\tau_{R}=600$~s}.}
\label{f-12-2-11}
\end{figure}

The results of these parametric studies are summarized in 
\hbox{Fig.~\ref{f-12-2-11}}. Here, $P^{MAX}_{CD}$ for the lower-hybrid 
current-ramp phase is plotted as a function of $\tau_{R}$ at 
\hbox{$n_{iF}/n_{iR}=10$}, and also as a function of \hbox{$n_{iF}/n_{iR}$} at
\hbox{$\tau_{R}=600$~s}. For both cases, \hbox{$n_{iF}=1.1\X
10^{20}$~m$^{-3}$}. For reasons given previously, $P^{MAX}_{CD}$ decreases with
increasing $\tau_{R}$ and \hbox{$n_{iF}/n_{iR}$}. Also plotted in
\hbox{Fig.~\ref{f-12-2-11}} is the maximum value of the Dreicer parameter,
$\xi$, as a function of \hbox{$n_{iF}/n_{iR}$} at \hbox{$\tau_{R}=600$~s}. The
linear rise of the Dreicer parameter with \hbox{$n_{iF}/n_{iR}$} can be deduced
by noting that
%
\begin{eqnarray}
\label{e-12-2-23}
|E_{\parallel}^{MAX}|&=& \frac{L_{E}}{2\pi R_{T}}
\left[\dot{I_{\phi}}\right]_{MAX} \:, 
\end{eqnarray}
%
where \hbox{$[\dot{I_{\phi}}]_{MAX}$} is determined by $\tau_{R}$ through
\hbox{Eq.~(\ref{e-12-2-18})}, and by using the definition of $E_{D}$ in 
\hbox{Eq.~(\ref{e-12-2-16})}. It can be shown easily that lowering $\tau_{R}$
from 600~s raises the curve for $\xi$ vertically in the figure and violates the
\hbox{$\xi\leq 0.01$} runaway condition at higher values of 
\hbox{$n_{iF}/n_{iR}$}. Therefore, in order to minimize the lower-hybrid power
requirement and achieve a reasonable current rise-time while avoiding
runaway-electron effects, the reference case presented in
\hbox{Sec.~\ref{s-12-2-2-1}} appears to be an appropriate choice for \AI\ plasma
start-up.

\subsubsection{Control of power ramp-up}
\label{s-12-2-2-3} 

\begin{figure}[p]
\vspace*{6.9in}
\caption[Fig-2-12]
{Comparison of plasma transients between \AI\ start-up scenarios with 50:50 and
99:1 DT fuel mixtures: (A) Plasma temperatures, and (B) Plasma powers for the
99:1 DT case and $P_{CD}$ for the 50:50 DT case [$\alpha$-particle heating
power ($P_{Q}$), transport power loss ($P_{COND}$), ohmic dissipation
($P_{\Omega}$), bremsstrahlung ($P_{BR}$) and synchrotron ($P_{CYC}$) powers,
and current-drive heating ($P_{CD})$].} \label{f-12-2-12}
\end{figure}

As noted in \hbox{Sec.~\ref{s-12-2-2}}, with the full plasma current set at low
density, protection of the divertor plates may require the rapid establishment
of the operating plasma density as the fusion power is ramped up at a slower
rate to its steady-state level. This start-up feature also offers the
additional advantage of minimizing the power transients on the blanket, primary
coolant, steam generator, and turbine systems. To achieve this, a density
ramp-up scenario using a variable DT fuel mixture can be considered. The range
of possibilities in this context would be defined by the 50:50 DT transients
studied in \hbox{Secs.~12.2.2.2} and \hbox{12.2.2.3}, and those of a near 100\%
deuterium (99:1 DT) mixture. The resulting temperature and power transients for
these two start-up scenarios are shown in \hbox{Fig.~\ref{f-12-2-12}}, using
the reference \AI\ parameters of \hbox{Table~\ref{t-12-2-2}}. In
\hbox{Table~\ref{t-12-2-3}}, the steady-state powers at the end of the two
start-up scenarios are compared. Relative to the case of 50:50 DT start-up, the
$\sim$100\% deuterium start-up reduces the neutron and the charged particle
powers by factors of, respectively, 25.6 and 21.2. However, because the
current-drive power is increased by a factor of 2.4 due to the lower $T_{e}$
and because the radiation power loss is smaller, the heat flux to the divertor
plates is reduced by only 50\%. The requirement of 244~MW of fast-wave power to
drive the current in the deuterium start-up case is clearly unacceptable for
\AI\ since only about 100~MW of FWCD power is prescribed for steady-state
operation. A lower FWCD power for this case is possible in principle, with a
somewhat lower target density prior to the injection of tritium.

\begin{table}[p]
\caption{}
\centering
{\bf Steady-State Powers for DT and DD Start-Up}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c c c}
\hline
& \multicolumn{1}{c}{Systems Code}& & \multicolumn{2}{c}{Start-Up Model} \\
\cline{2-2} \cline{4-5}
DT Fuel Mixture                       & 50:50  & & 50:50 & 99:1 \\
\hline
 &  \hbox to .25truein{\hfil} & \hbox to 0.01truecm{\hfil} 
\hbox to .15truein{\hfil}     & \hbox to .15truein{\hfil} \\ 
Electron temperature, $T_{e}$ (keV)  &  \D\Z19.3 &&  \D\Z27.6 &  \Z15.1 \\
Ion temperature, $T_{i}$ (keV)       &  \D\Z20.0 &&  \D\Z21.6 &  \Z12.0 \\
Total thermal power,$^{(a)}$ $P_{TH}$ (MW) & 2,026.9 && 2,325.9 & 361.5 \\
Fusion power, $P_{F}$ (MW)           &   1,925.4 &&   2,224.4 &  \Z90.5 \\
Charged particle power, $P_{Q}$ (MW) &   \D386.7 &&   \D447.0 &  \Z21.1 \\
Neutron power, $P_{n}$ (MW)          &   1,538.7 &&   1,777.4 &  \Z69.4 \\
\quad DT neutron, $P_{nDT}$ (MW)     &   1,538.7 &&   1,775.8 &  \Z67.1 \\
\quad DD neutron, $P_{nDD}$ (MW)     &    ---    && \D\Z\Z1.6 & \Z\Z2.3 \\
FWCD power, $P_{CD}$ (MW)            &  \D\Z96.7 &&   \D101.5 &   243.8 \\
Transport power, $P_{COND}$ (MW)     &   \D244.4 &&   \D259.0 &   126.0 \\
\quad Electron, $P_{CONDe}$ (MW)     &   ---     &&   \D167.9 &  \Z86.2 \\
\quad Ion, $P_{CONDi}$ (MW)          &   ---     &&  \D\Z91.1 &  \Z32.9 \\
Ohmic power, $P_{\Omega}$ (MW)       & \D\Z\Z0.9 && \D\Z\Z0.6 & \Z\Z0.8 \\
Bremsstrahlung power, $P_{BR}$ (MW)  &  \D\Z45.2 &&  \D\Z53.2 &  \Z39.9 \\
Line radiation power, $P_{LINE}$ (MW)      & ---       &&  \D\Z19.9 & \Z16.4 \\
Cyclotron radiation power, $P_{CYC}$ (MW)  &   \D194.7 &&   \D216.7 & \Z90.2 \\
Neutron wall loading, $I_{W}$ (MW/m$^{2}$)  \\
\quad DT (14.06 MeV)                       & \D\Z\Z2.5 && \D\Z\Z2.9 & \Z\Z0.1 \\
\quad DD (2.46 MeV)
       & \Z\Z2.0$\X 10^{-3}$ && \Z\Z2.2$\X 10^{-3}$ & \Z3.8$\X 10^{-3}$ \\
Plasma {\it Q}-value, $Q_{p} = P_{F}/P_{RF}$ & \D\Z19.9 && \D\Z22.8 & \Z\Z1.2 \\
\\ \hline
\multicolumn{4}{l}{$^{(a)}$Without neutron energy multiplication.}\\
\end{tabular}
\label{t-12-2-3}
\end{table}

A more elaborate power ramp-up scheme involving feedback control of the density
fueling rate can also be considered. The key is to maintain the FWCD power to
below $\sim$100~MW during the fusion-power ramp phase under the constraint
%
\begin{eqnarray}
\label{e-12-2-24}
\frac {\gamma_{_{FW}}P_{CD}}{n_{e}(1-f_{BS})} & = & I_{\phi} \: ,
\end{eqnarray}
%
where $\gamma_{_{FW}}$ and $f_{BS}$ are given by, respectively, 
\hbox{Eqs.~(\ref{e-12-2-17})} and \hbox{(\ref{e-12-2-9})} and
\hbox{$I_{\phi}\simeq 10$~MA} in the reference case. The density is dictated
by the fueling rate while the temperature can be partially controlled by the
fuel mixture.  Partial-power operation, demanded by engineering constraints
during start-up, is relatively straightforward (using passive burn-control
techniques involving density and auxiliary heating power) as long as the plasma
is operating in a thermally stable region. However, in regions of low
temperatures \hbox{($< 10$--15~keV)} where the plasma is thermally unstable,
active feedback control will be needed. Because the \AI\ reactor is based on
steady-state current drive, a more detailed analysis is needed to assess the
impact of the requirement of partial-power operation on the overall design.
Safe operation of the divertor during the start-up and partial-power phases
should also be studied.

\subsection{Summary and Critical Issues}
\label{s-12-2-3}

A simple 0-D plasma-circuit code has been used to study start-up scenarios for
\AI. About 20~MW of lower-hybrid power is required to ramp up the current to
its full value at low density \hbox{($\simeq 1.1\X 10^{19}$ m$^{-3}$)} in about
2100~s. At nearly full current, the fuel ion density is raised quickly by
enhanced fueling with a 50:50 DT mixture while the FWCD power of $\sim$100~MW
is applied. This fusion-power ramp-up phase lasts for about 400~s. The final
plasma parameters predicted by the start-up simulation generally agree well
with the steady-state predictions of the ARIES systems code as outlined in
\hbox{Table~\ref{t-12-2-3}}. Some of the discrepancies arise from differences
in the assumed radial density and temperature profile exponents, $\alpha_{n}$
and $\alpha_{T}$. The \AI\ strawman design assumes a ``pedestal'' density
profile with \hbox{$\alpha_{n} = 1.0$}, whereas the start-up simulations use a
parabolic profile with \hbox{$\alpha_{n} = 0.3$} in order to best match the
fusion reactivity profile factor, $\alpha_{DT}$.  Differences in $\alpha_{DT}$
(7\% less in the start-up model), coupled with those in temperature (24~keV
{\it vs} 20~keV, giving a 30\% difference in \hbox{$\langle \sigma v
\rangle_{DT}$)} and the final fuel density (5\% less in the start-up
simulation), account for the $\sim$15\% difference (increase) in the
steady-state fusion power predicted by the start-up model. Although fine tuning
the code should bring the start-up and systems models into better alignment,
the present agreement is considered adequate for the purpose of this study.

It is noted that the simulation results reported in this section represent only
a preliminary assessment of a plausible start-up scenario for \AI. In addition
to examining the impact of longer current-rise times on reducing the start-up
power requirement, ways to better utilize the inherent volt-second capability
of the PF-coil set should be investigated~\cite{Pen89}. Other important
features of the start-up scenario also remain to be examined. In particular,
scenarios involving slower fusion-power ramp-up may be desirable from the
viewpoint of safe operation of the steam generator and turbine systems, while
divertor protection may require a more rapid density buildup. In order to
satisfy both of these needs, a tritium-lean fueling scenario followed by a
slower, full-density power ramp-up may be required. However, the feasibility
of this scheme under the constraints of steady-state \AI\ power requirements
remains to be shown.

\section{PLANT START-UP}
\label{s-12-3}

Operating constraints of the various phases of the plasma and current start-up
for the \AI\ reactor were examined in detail in \hbox{Sec.~\ref{s-12-2}}.
Likewise, the engineering aspects of \AI\ will also place constraints on the
start-up and partial- and over-power operations of the reactor plant. These
constraints are due to the temperature and thermal stress limits of the
first-wall, blanket, divertor, and power-conversion system materials, and to
the power requirements for magnet cool-down, blanket heat-up, and coolant
circulation. In \hbox{Sec.~\ref{s-12-3-1}}, the plant start-up procedure and
the associated constraints imposed on the reactor subsystems are outlined.
Section~\ref{s-12-3-2} gives an assessment of the range of operating parameters
for the various plant components under the conditions of partial- and
over-power operations. The time scale of power ramp-up imposed by these
engineering considerations can in general be met by careful control of the
plasma start-up.

\subsection{Start-Up Requirements}
\label{s-12-3-1}

To start up the \AI\ fusion-power reactor will require a number of
actions involving the engineering systems:
\begin{itemize}
\item Cool down and energize the superconducting magnets.

\item Heat up and pressurize the primary coolant and power conversion systems.

\item Initiate the plasma operations at low power.

\item Balance the power-conversion-system conditions for power operation.

\item Roll the turbines and synchronize with the grid.

\item Increase the plasma power and load the turbines to minimum stable power
levels (about 10\% of full power).

\item Ramp up the plasma and power conversion system to full power.
\end{itemize}
Each of these steps requires some time to accomplish and has the
potential for imposing engineering constraints.

The cool-down and energizing of the superconducting magnets will be governed
by the power limits of the helium refrigeration system and the magnet-current
power supply. With the present \AI\ systems, cool-down and energizing would
take approximately one week. This could be reduced significantly, if desired,
by simply providing larger refrigeration and power supply systems. Since the
magnets will not be de-energized, much less warmed up, for routine shutdowns,
the added expense of larger systems is not justified. The coils will be warmed
only when they are to be removed for annual (at most) blanket replacement.

Heat-up of the blanket, the primary coolant circuit, and the power conversion
system will be done with the main helium circulator pumps ($\sim$20~MW) run at
100\% power, based on HTGR experience~\cite{htg89}, and will take about three
hours. At this point, plasma heating will be initiated using the fast-wave
current-drive and heating system ($\sim$100~MW) to bring the plant to
$\sim$10\% thermal power and to balance the power conversion system.

\begin{table}[b]
\caption{}
\centering
{\bf Start-Up Times (h) for Coal-Fired Power Plants}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c c c c}
\hline
  & \multicolumn{2}{c}{Westinghouse~\cite{Epr82a}}  &
     & \multicolumn{2}{c}{General Electric~\cite{Epr82b}}\\
\cline{2-3}  \cline{5-6}
\hfil {Type of Start-Up} \hfil & Bare Plant & Advanced Plant && Bare Plant 
                                                             & Advanced Plant\\
\hline
\\
Cold                    &  9.1 &  9.9 && 8.1 & 10.\Z\Z \\
After weekend shutdown & --- & --- && 3.6 & 5.5 \\
Warm        &$\sim$2.0\phantom{$\sim$}&$\sim$2.4\phantom{$\sim$}&& --- & --- \\
Hot                     &$\sim$1.4\phantom{$\sim$} & 2.0 && --- & --- \\
\\ \hline
\end{tabular}
\label{t-12-3-1}
\end{table}

In a conventional power plant, steam turbines are massive structures with much
longer thermal time constants than other components, such as the steam
generator. Therefore, the start-up time is primarily the time needed to slowly
heat the turbines to the operating temperature while staying within the thermal
stress limit. Plants with an advanced steam cycle have longer start-up times
because more time is needed to complete the temperature ramp to the higher
throttle temperature required by the advanced design.
\hbox{Table~\ref{t-12-3-1}} shows the start-up times from two previous
studies~\cite{Epr82a,Epr82b} for a bare plant (presently operating) and an
advanced plant using an advanced steam cycle similar to that proposed for \AI.
The longest time for cold start-up is expected to be about 10 hours when the
operating parameters of the system (temperature and pressure) attain their
design values. This is accomplished at about 10\% of full power. Start-up after
weekend shutdown requires about six hours, and warm or hot start-up takes about
two hours. A steam generator can be loaded at 2\% per minute, so unloading
during shutdown can be done at about 1\% per minute. The plant is generally
tripped at 10\% of full power.

The plasma start-up time is on the order of tens of minutes for \AI, as 
mentioned in the previous section. The thermal time constants of the first
wall, blanket, and divertor are very short, on the order of seconds.
Therefore, it is the start-up time for the power conversion system that 
determines the start-up time of \AI. With the magnets already energized, the
total start-up time from cold and hot shutdowns to full power will be,
respectively, about twelve and two hours. The orderly shutdown time will be
nearly two hours.


\subsection{Partial- and Over-Power Operations}
\label{s-12-3-2}

Based on the configuration of the \AI\ reference outboard first-wall and
blanket designs, calculations were performed to estimate the range of
equilibrium partial- and over-power operations that would be permitted without
exceeding any of the design limits of the first-wall and blanket materials.
Three design criteria were used for the evaluation: 

\begin{itemize}
\item The SiC-composite-material maximum temperature should be 
\hbox{$< 1100$\degC},
\item The Li$_{2}$ZrO$_{3}$ operating temperature should be designed to
       be within the temperature window of 400 to 1400\degC, and
\item Structural and thermal 
         stresses have to be designed to SiC-composite-material design limits.  
\end{itemize}

\noindent
In summary, the results of this analysis show that the maximum-acceptable wall
loading is defined by the SiC-composite maximum-allowable temperature; the
minimum-acceptable wall loading is defined by the minimum-allowable temperature
of the solid breeder. The bases for these design limits are presented in
Sec.~8. 

\begin{table}[t]
\caption{}
\centering
{\bf Maximum Wall-Loading and Reference Design Parameters}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c}
\hline
&Maximum Wall Loading&Reference\\
\hline \\
Neutron wall loading (MW/m$^{2}$)& \Z7.35 & \Z3.87 \\
Surface wall loading (MW/m$^{2}$)& \Z1.16 & \Z0.61 \\
First-wall parameters$^{(a)}$\\
\quad Heat transfer coefficient (W/K-m$^{2}$) & 3,690 & 2,208 \\
\quad Coolant and solid interface temperature (\degC) & \D978 & \D938 \\
\quad Solid maximum temperature (\degC)       & 1,096 & 1,001 \\
\quad Pressure drop (kPa)                     & 62.1  & 20.2 \\
\quad Pumping power (MWe)                     & 111   & 19 \\
\\ \hline
\multicolumn{3}{l}{$^{(a)}$At the outboard mid-plane location.}\\
\end{tabular}
\label{t-12-3-2}
\end{table}

Table~\ref{t-12-3-2} shows the operating parameters of the maximum wall-loading
case compared to the reference design at the mid-plane location of the outboard
blanket. As shown, the maximum wall loading could be as much as 1.9 times that
of the reference design. As the wall loading increases, in order to maintain
the same coolant inlet and outlet temperatures, the coolant mass-flow rate will
have to be increased. This will increase the heat transfer coefficient at the
expense of higher pressure drop and much higher pumping power. The maximum wall
temperature will approach the design limit of 1100\degC\ when the wall loading
reaches \hbox{7.35~MW/m$^{2}$}. At this point, the increased thermal stress
across the first wall will have reduced its stress design margin to essentially
1.0, but the structural design limits will not yet have been exceeded. 

Table~\ref{t-12-3-3} shows the operating parameters of the minimum wall-loading
case compared to the reference design at the top or bottom location of the
second breeder zone. As shown, the minimum wall loading can be as low as 0.1
times that of the reference design before the Li$_{2}$ZrO$_{3}$ lower
operating-temperature limit is reached. This is partly due to the relatively
high coolant-inlet temperature of 250\degC. As the wall loading decreases, in
order to maintain the same coolant inlet and outlet temperatures, the coolant
mass-flow rate will have to be decreased. This will drive the coolant to
laminar flow and reduce the heat transfer coefficient and the pressure drop.
The minimum breeder temperature will reach the lower operating limit of
400\degC\ when the wall loading reaches \hbox{0.25~MW/m${^2}$}.

\clearpage

\begin{table}[t]
\caption{}
\centering
{\bf Minimum Wall-Loading and Reference Design Parameters}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c}
\hline
&Minimum Wall Loading&Reference\\
\hline \\
Neutron wall loading (MW/m$^{2}$)       & 0.25 & 2.48 \\
Surface wall loading (MW/m$^{2}$)       & 0.04 & 0.391 \\
Breeder zone parameters$^{(a)}$ \\
\quad Heat transfer coefficient (W/K-m$^{2}$)& 269 & 1,989 \\
\quad Reynolds number                   & 1,185 &11,848 \\
\quad Coolant and solid interface temperature (\degC) & 402 & 460 \\
\quad Solid maximum temperature (\degC) & 411  & 544 \\
\quad Pressure drop (kPa)               & 0.07 & 4.17 \\
\quad Pumping power (MWe)               & 0.03 & 19 \\
\\ \hline
\multicolumn{3}{l}{$^{(a)}$At 2nd breeder zone located at the top or
bottom of the outboard blanket and at the} \\
\multicolumn{3}{l}{\phantom{$^{(a)}$}coolant-inlet location.}\\
\end{tabular}
\label{t-12-3-3}
\end{table}

These results indicate that at equilibrium, the \AI\ reference blanket design
allows a large range (from 10\% to 190\% of the reference design operation) of
partial- and over-power operation. Operation at the 190\% point is not a
practical option because this would leave no design margins for local peaking
factors, {\it etc.} Further, in order to take advantage of the potential for
over-power operation, an oversized helium circulator would be required because
the pumping power goes up significantly with increasing fusion power. 
Similarly, the entire power-conversion system would have to be designed with
excess capacity. Nevertheless, the wide range of blanket operation that is
theoretically possible is an indication of the tolerance for off-nominal
operating conditions that the use of the high-temperature, SiC-composite
blanket materials provides.

\section{SUMMARY AND CONCLUSIONS}
\label{s-12-4}

The plasma physics aspects of the start-up and operation of the \AI\ reactor
have been investigated in detail. Using a 0-D plasma-circuit code, a plausible
start-up scenario going from current initiation to steady state has been
determined. About 20~MW of lower-hybrid power is required to ramp up the plasma
current to its full value at low density \hbox{($\simeq1.1{\X}$
10$^{19}$~m$^{-3}$)} in 2100~s at a characteristic rise-time of 600~s. Another
400~s is required to raise the density and fusion power to their steady-state
levels using FWCD power of $\sim$100~MW. Other lower-hybrid power and fueling
strategies are possible and will be the subject of future work to further
reduce the start-up power requirement while simultaneously meeting a range of
plasma constraints.

There appears to be no major R\&D issue related to lower-hybrid current ramp-up
and current drive specific to \AI\ since the relevant data base is already well
developed. However, the start-up transient codes should be improved by
incorporating more precise physics models, particularly in the areas of current
drive and transport. This may entail the inclusion of self-consistent
plasma-profile evolution in the modeling analysis. Acquiring ample operational
experience in plasma initiation, current start-up, and current ramp-up on a
large tokamak reactor, such as ITER, is most important. Similar experience with
the auxiliary-heating and density-rise phase, smooth transition from
lower-hybrid to fast-wave current drive, and the approach to steady-state burn
will be valuable.

The engineering aspect of the reactor plant start-up and operation is also
examined in the context of the operational constraints imposed by the major
subsystems (first wall, blanket, divertor, and power conversion system) and
material choices. It is found that, as long as the superconducting magnets are
not de-energized during routine shutdowns, it is the start-up time for the
power conversion system that determines the start-up time of the \AI\ plant.
The total start-up time then ranges from twelve hours for cold start-up to two
hours for hot start-up, and the orderly shutdown time is approximately two
hours. The choice of SiC composite as the first-wall structural material and
Li$_{2}$ZrO$_{3}$ as the breeder material permits, in principle, blanket
operation ranging from 10\% to 190\% of full power. No major critical issue
has been uncovered during the course of this investigation.

\clearpage
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\end{document}