Hybrid Temperature-Reactivity PID Controllers for Nuclear Thermal Propulsion Startup

Abstract

In this work, a novel temperature-reactivity controller is introduced to rotate control drums based on a chamber temperature signal. It consists of two components: (1) a reactivity-driven proportional controller converting the temperature demand into a reactivity signal, and (2) a temperature-driven proportional-derivative controller which relies on a derivative term to anticipate sudden changes in demand and limit temperature overshoot. A weighting function is proposed to continuously transition from one component to the other.

Full text

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INL/CON-23-70963-Revision-0
Hybrid Temperature-
Reactivity PID Controllers
for Nuclear Thermal
Propulsion Startup
May 2023
Vincent M Laboure, Stefano Terlizzi, Sebastian Schunert

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INL/CON-23-70963-Revision-0
Hybrid Temperature-Reactivity PID Controllers for
Nuclear Thermal Propulsion Startup
Vincent M Laboure, Stefano Terlizzi, Sebastian Schunert
May 2023
Idaho National Laboratory
Idaho Falls, Idaho 83415
http://www.inl.gov
Prepared for the
U.S. Department of Energy
Under DOE Idaho Operations Office
Contract DE-AC07-05ID14517

Hybrid Temperature-Reactivity PID Controllers for Nuclear Thermal Propulsion Startup
Vincent M. Labouré, Stefano Terlizzi, and Sebastian Schunert
Reactor Physics Methods and Analysis Group, Idaho National Laboratory, 1955 N Fremont Ave, Idaho Falls, ID,
vincent.laboure@inl.gov
In this work, a novel temperature-reactivity controller is
introduced to rotate control drums based on a chamber tem-
perature signal. It consists of two components: (1) a reactivity-
driven proportional controller converting the temperature de-
mand into a reactivity signal, and (2) a temperature-driven
proportional-derivative controller which relies on a deriva-
tive term to anticipate sudden changes in demand and limit
temperature overshoot. A weighting function is proposed to
continuously transition from one component to the other.
I. INTRODUCTION
Given the very short duration over which nuclear thermal
propulsion (NTP) systems must be able to reach nominal
power, precisely following a chamber temperature demand—
using control drums (CDs) as a means to control reactivity—is
challenging to achieve without significant delay or overshoot.
This is due to the inertia of the system, i.e., the intrinsic delay
between the times at which the drums are rotated and the
coolant temperature adjusts to the perturbation.
Previously, a standard proportional integral (PI) controller
for temperature and pressure demands was used in Ref. [
1
],
with delays and oscillations being observed after any sudden
change in demand. With the controller gains adjusted as a
function of total power, near-perfect agreement between the
demands and the responses was obtained [
2
]. Simultaneously,
a limited delay and little to no overshoot were obtained in
Ref. [
3
] by converting the chamber temperature signal to a
power signal to use in a period-generated controller (PGC)
[
4
]. The disadvantage of this latter type of power-following
(i.e., driven by a power signal) controller is that it requires to
provide reactivity coefficients and the rates of change of the
various temperatures affecting reactivity (e.g., fuel or modera-
tor temperatures)—which need to be estimated in real-time.
A different type of power-following controller—a hybrid
power-reactivity PI controller introduced in Ref. [
5
]—relied on
the power and reactivity of the system to compute the CD angle
without attempting to anticipate the future state of the system.
Consequently, it did not require rates of change but showed
similarly good performance as a PGC. Nevertheless, because
it was only tested with a power signal, chamber temperature
overshoots on the order of 7% were obtained on a prototypical
startup sequence. The purpose of this work is to extend this
type of controller by using chamber temperature, rather than
power, as the demand while minimizing delay and overshoot.
This is achieved by combining a proportional derivative (PD)
controller with a temperature signal and a proportional (P)
controller with a reactivity signal (approximately converted
from the temperature signal). A sigmoid-like weighting is
proposed to continuously transition from one controller to the
other.
The resulting hybrid temperature-reactivity controller re-
lies solely on a temperature signal with fixed gains to actuate
the CDs, without requiring reactivity coefficients and tempera-
ture rates of change. It is tested using a simplified multiphysics
model with a point kinetics (PKE) neutronics model, and a
single representative fuel assembly and fuel cooling channel.
For this preliminary work, the mass flow rate and chamber
pressure are assumed given and not controlled.
II. THEORY
In this section, the chamber temperature demand is de-
noted
Td
whereas
T
is the actual chamber temperature of the
system (measured or predicted, in the case of a real system a
or numerical model, respectively). An ideal controller is able
to achieve T=Td.
II.A. Controller with Direct Temperature Signal
A widely used approach is to convert the error,
ε
(
t
)=
Td−T
, into a proportional integral derivative (PID) correction
signal to actuate the drums:
∆θT(t)=Kpε(t)+KiZt
0
ε(t′) dt′+Kd
dε
dt,(1)
where
Kp
,
Ki
, and
Kd
are the gains, usually tuned to achieve
the optimum performance. In this work, they are assumed
constant throughout the simulation. In a nutshell, the propor-
tional, integral and derivative terms respectively react to the
current, past and future performance of the controller. There-
fore, unless the gains are made time-dependent (as in Ref. [
2
]),
a simple PI controller cannot perfectly anticipate any sudden
change in demand. A derivative term adds anticipation and
can also reduce overshoot but, in real systems, it can lead to
instabilities due to instrumentation noise.
II.B.
Controller with Temperature-Inferred Reactivity
Signal
The physical delays between the times at which (1) the
drums are rotated, (2) the power adjusts to the perturbation and
(3) the coolant temperature reacts to this change in power limit
the ability to quickly react to change in demand. Conversely,
a CD rotation results in a virtually immediate change in reac-
tivity, which was the motivation in Ref. [
5
] for introducing a
reactivity informed signal to control the drums. The first step
is to convert the temperature demand to a power demand,
Pd
,
which can be approximately done as follows [3]:
Pd≈˙m(Hout(Td)−Htank ),(2)
where
˙m
is the total mass flow rate of propellant,
Htank
is
the enthalpy in the storage tank and
Hout
(
Td
) is the enthalpy

corresponding to the temperature of the temperature demand.
In Ref. [
3
], this power demand is directly used in a PGC
controller. Alternatively, this can be further converted into a
reactivity demand, ρd, as was done in Ref. [5]:
ρd=Λ
Pdβeff






dPd
dt−X
i
λiCi






+1,(3)
where Λis the neutron mean generation time,
βeff
is the effec-
tive delayed neutron fraction, and
λi
is the decay constant of
the delayed neutron precursor group
i
with concentration
Ci
.
Note that both
βeff/
Λand
Ci
can be obtained experimentally
[6]. The CD correction angle is then computed as:
∆θρ(t)=K′
pe(t)+K′
iZt
0
e(t′) dt′+K′
d
de
dt,(4)
where
e
(
t
)=
ρd−ρ
is the error in reactivity and
K′
p
,
K′
i
, and
K′
d
are the proportional, integral, and derivative gains of the
reactivity-driven PID controller, respectively. Typically,
K′
p
can be chosen as the inverse of the differential CD worth so
parameter tuning is simpler than the controller with a direct
temperature signal. This controller tends to have a better
initial response but it is not guaranteed to converge to the
desired temperature due to the two successive and approximate
conversions from temperature to reactivity signal [5].
II.C. Hybrid Temperature-Reactivity Controller
A third approach to combine the previous two controllers
to benefit from the fast response of the reactivity-driven correc-
tion term while relying on the temperature-driven signal when
the power approaches its nominal value. The resulting con-
troller is illustrated in Figure 1. The only demand is the cham-
ber temperature. It is used directly by the temperature-driven
component and indirectly by the reactivity-driven component
after a conversion using Equations
(2)
and
(3)
is performed.
The CD angle is computed by a weighted sum of both correc-
tions terms:
θ(tn+1)=θi+
n
X
k=1
∆θ(tk)=θi+
n
X
k=1ξ∆θT(tk)+(1 −ξ)∆θρ(tk),
(5)
where
θi
is the initial CD angle and
tk
is the time at the
k
-th
time step.
Multiple weighting functions are possible. In Ref. [
5
],
two weighting functions were proposed for the power whose
analogs for the temperature would be:
ξ=ξlin ≡Td−Ti
Tf−Ti
,(6)
ξ=ξlog ≡
log Td
Ti!
log Tf
Ti!,(7)
where
Ti
is the initial chamber temperature (e.g., at the begin-
ning of the bootstrap phase) and
Tf
is the nominal temperature.
For a power-following controller, both weightings performed
virtually identically because of the outstanding performance
of both components of the hybrid power-reactivity controller
[
5
]. Another difference is that power can vary by two orders
of magnitude or more during bootstrap and thrust build-up
phases whereas temperature varies by at most a factor of 10. In
addition, it was found to be beneficial to adjust the weighting
function to
Tb≡Td
(
tb
) where
tb
is the time at which bootstrap
ends (and thrust build-up begins). Specifically, the idea is to
continuously and monotonically transition from the reactivity
driven component (
ξ
(
Ti
)=0) to the temperature-driven one
(
ξ
(
Tf
)=1) with equal contribution at the end of bootstrap
(
ξ
(
Tb
)=0
.
5). This can be achieved by defining a sigmoid-like
function with finite bounds:
ξ=ξs(Td)=







































0,Td<Ti
1
1+2(Tb−Ti)
Td−Ti
−1s,Ti≤Td<Tb
1
1+2(Tf−Tb)
Td+(Tf−2Tb)−1s,Tb≤Td<Tf
1,Td>Tf
(8)
where
s
is a positive integer controlling the steepness of the
function. Figure 2 plots
ξs
as a function of the temperature
demand for various values of
s
. Increasing the value of
s
results in a faster transition from the reactivity-driven to the
temperature-driven component.
III. RESULTS AND ANALYSIS
To study the performance of the temperature-following
controllers presented in the previous section, a condensed start-
up sequence is considered and a simplified model is derived.
III.A.
Prototypical Bootstrap and Thrust Build-up
Phases
The purpose of the start-up sequence is to assess the per-
formance of the controllers with a rapid increase in temper-
ature with some sudden changes in demand. The core is
assumed to have finished thermal conditioning and prototypi-
cal bootstrap and thrust build-up phases are modeled with a
duration of 10 and 35 s (with the last 15 s at nominal specific
impulse), respectively. Specifically, the temperature demand
is chosen to be:
•Td(t=0 s) =500 K
•Td(t=10 s) =700 K
•Td(t≥30 s) =2700 K,
with a linear increase in between points. The chamber pressure
demand—which is not the focus of this work—is assumed to
be:
•pC(t=0 s) =0.02 ×pC,nom
•pC(t=10 s) =0.13 ×pC,nom
•pC(t=30 s) =0.65 ×pC,nom
•pC(t≥45 s) =pC,nom,

Fig. 1. Schematics of the hybrid temperature-reactivity controller.
with a linear increase in between points, where
pC,nom
is the
nominal chamber pressure.
III.B. Simplified Model
The multiphysics model used for this study is greatly sim-
plified compared to the one from [
5
] to reduce nonlinear effects
and better understand the behavior of the controller itself. It re-
mains based on the Multiphysics Object-Oriented Simulation
Environment (MOOSE) framework [7]. In particular:
•
The neutronics is modeled using a Griffin PKE model
[
8
,
9
], with constant CD reactivity worth (-25 pcm/deg),
fuel temperature reactivity coefficient (-1 pcm/K) and ki-
netics parameters (summarized in Table I) and the initial
reactor is assumed critical with 1 MW initial power. This
implies that the initial state of the reactor is not a thermal
equilibrium, which can be useful to test the robustness of
the controllers presented in this work.
•
A single representative 30-degree fuel assembly slice,
shown in Figure 3 and extruded up at the height of the
active core, is modeled with Bison [
10
]. All the power is
assumed deposited in the fuel and with the only way for
heat to be removed being by convection at the boundary
of the fuel cooling channels (holes in Figure 3).
•
A single representative fuel cooling mechanism preserv-
ing the main hydraulic parameters of the collection of
all the fuel cooling channels (flow area, hydraulic diam-
eter, etc.) is modeled with RELAP-7 [
11
] with an inlet
coolant temperature of 300 K. The H
2
propellant is as-
sumed to be an ideal gas with the specific heat capacity
adjusted to match the expected enthalpy rise over a tem-
perature range of 50–2700 K, as was done in Ref. [
5
].
The mass flow rate is assumed to be proportional to the
chamber pressure. These significant simplifications will
be improved in future work.
TABLE I. Kinetics parameters used in the neutronics model.
λi
and
βi
are the delayed neutron decay constants and fractions,
respectively. Λis the mean generation time.
Delayed Group λi(s−1)βi(pcm) Λ(s)
1 1.3346 ×10−224.6 1.33 ×10−5
2 3.2667 ×10−2128.6
3 1.2094 ×10−1124.3
4 3.0444 ×10−1282.0
5 8.5639 ×10−1121.7
6 2.8764 ×10050.8
Sum 732.1
III.C. Numerical Results
In the section a limited set of results is presented, ex-
hibiting the potential benefits of using a hybrid temperature-
reactivity controller. However, since all the controllers pre-
sented in the previous section are greatly affected by their
gains, only the most relevant values tested are reported. How-
ever, a rigorous optimization has not been conducted at this
point.
III.C.1. Controller with Direct Temperature Signal
If only the proportional term is considered, the propor-
tional gain is set high enough to have a responsive system but
low enough to avoid oscillations and overshoot. It is difficult

500 1000 1500 2000 2500 2700
Chamber temperature (K)
0.0
0.2
0.4
0.6
0.8
1.0
ξ
Sigmoid (s=0)
Sigmoid (s=1)
Sigmoid (s=2)
Sigmoid (s=5)
Linear
Log
Fig. 2. Value of the weighting functions
ξ
for different options.
The black vertical dotted line highlights the temperature at the
end of the bootstrap phase (chosen to be Tb=700 K).
Fig. 3. X-Y view of the representative fuel assembly 30-degree
slice used for the thermal model.
to achieve both requirements due to the inherent time delay
between the time the CDs are rotated and the time the cham-
ber temperature reacts to the perturbation. For instance, in
Figure 4, the controller with
Kp
=5
×
10
−3
deg/K follows the
trend of the demand reasonably well but the controller has an
initial delay—partly due to the initial condition not being in
thermal equilibrium—and a fairly oscillatory behavior with a
temperature overshoot (defined as (
max
(
T
)
−Ti
)
/
(
Tf−Ti
)) of
1.2%. Further increasing the gain would increase oscillations
and overshoot whereas the opposite would further delay the
initial response.
An integral term is found to lead to increased overshoot
and is therefore not considered in this work. Note however
that this is likely due to using constant gains as opposed to
what was done in Ref. [2].
0
50
100
150
200
250
300
Power
deposited in the fuel (MW)
0
1
2
3
4
5
6
7
Chamber pressure (MPa)
40
60
80
100
120
Control Drum angle (deg)
T
c
demand (K)
T
c
(K)
Power (MW)
P
c
(MPa)
Control Drum angle (deg)
0 10 20 30 40 50 60
Time (s)
500
1000
1500
2000
2500
2700
Chamber temperature (K)
Fig. 4. Temperature proportional controller with a gain of
Kp=5×10−3deg/K.
Figure 5 shows the results obtained with a temperature-
driven PD controller (
Kp
=5
×
10
−3
deg/K,
Kd
=5
×
10
−3
deg-
s/K). The derivative term helps anticipate sudden changes in
demand and, despite some initial delay and oscillations during
the bootstrap phase, the controller’s behavior is quite satisfying
for most of the thrust build-up phase. Compared to the case
without derivative term, the temperature overshoot is reduced
to 0.3% and occurs shortly after the chamber pressure demand
(i.e., mass flow rate) reaches its nominal value. The initial
delay, however, is virtually unchanged.
0
50
100
150
200
250
300
Power
deposited in the fuel (MW)
0
1
2
3
4
5
6
7
Chamber pressure (MPa)
40
60
80
100
120
Control Drum angle (deg)
T
c
demand (K)
T
c
(K)
Power (MW)
P
c
(MPa)
Control Drum angle (deg)
0 10 20 30 40 50 60
Time (s)
500
1000
1500
2000
2500
2700
Chamber temperature (K)
Fig. 5. Temperature PD controller with gains of
Kp
=5
×
10−3deg/K and Kd=5×10−3deg-s/K.

III.C.2.
Controller with Temperature-Inferred Reactivity Sig-
nal
The advantage of a reactivity-driven controller is that it
can determine, given a temperature demand, the required re-
activity to achieve the desired ramp and a CD rotation can be
done to immediately insert that amount of reactivity. There-
fore, its initial response could be superior to a controller di-
rectly using a temperature signal. Figure 6 shows the results
obtained using a proportional controller with the gain set to the
inverse of the CD differential worth, i.e.,
K′
p≈
29
.
3 deg/$. The
initial response of the controller is indeed improved. Never-
theless, such a controller is unable to ensure that the chamber
temperature converges to the nominal demand since a zero
error in reactivity does not necessarily imply the same for
temperature. In fact, the coolant enthalpy rise shown in Fig-
ure 6 noticeably trails behind the actual power deposited in the
fuel during the thrust build-up phase which indicates that the
approximation done to convert the temperature demand into
a power demand (see Equation
(2)
) is inaccurate during that
time. As a result, the reactivity signal works on the assumption
that the power matches the enthalpy rise but since the former
is underestimated, the final power and chamber temperature
stabilize about 10% higher than their desired values.
0
50
100
150
200
250
300
350
Power
or enthalpy rise (MW)
0
1
2
3
4
5
6
7
Chamber pressure (MPa)
40
60
80
100
120
Control Drum angle (deg)
T
c
demand (K)
T
c
(K)
Power (MW)
Enthalpy rise (MW)
P
c
(MPa)
Control Drum angle (deg)
0 10 20 30 40 50 60
Time (s)
500
1000
1500
2000
2500
2700
Chamber temperature (K)
Fig. 6. Reactivity proportional controller with a gain of K′
p≈
29.3 deg/$.
III.C.3. Hybrid Temperature-Reactivity Controller
The core idea of this work is to use the the reactivity-
driven controller at early stages and then continuously tran-
sition to the temperature PD controller. The sigmoid-like
weighting function with
s
=5 (see Figure 2) is used to operate
the transition quickly. Figure 7 gives the corresponding results.
As expected, the initial response of the controller is better than
using the temperature PD controller and the chamber temper-
ature stabilizes to the desired value of 2700 K with the same
overshoot of 0.3% as temperature PD controller. A slightly
larger delay is observed around the transition between the
bootstrap and thrust build-up phases which could be further
alleviated by fine tuning of the controller’s gains and of the
expression of the weighting function.
IV. CONCLUSIONS
In this work, a hybrid temperature-reactivity controller
was proposed. It consists of two components: (1) a reactivity-
driven proportional controller converting the temperature de-
mand into a reactivity signal with a decent initial response, and
(2) temperature-driven PD controller which relies on a deriva-
tive term to anticipate sudden changes in demand and limit
temperature overshoot. The novel controller here introduced,
uses constant gains, thereby simplifying the tedious process of
gain tuning, and does not rely on reactivity coefficients and the
rates of change, that could be difficult to estimate in real-time,
as for the PGC controller. Despite having several positive
characteristics, one of its main disadvantages is that derivative
signals are subject to noise in real systems so the instrumen-
tation signal would need to be filtered appropriately to limit
instability. Future work will investigate the effect of noise on
the controller’s performance and the deployment of this novel
controller on a more elaborate multiphysics model, such as
the one considered in Ref. [
5
]. Finally, the control logic will
be extended to incorporate a chamber pressure controller. In
addition, the reactivity will be controlled through both control
drums and structural support control valves.
V. ACKNOWLEDGMENTS
This manuscript was authored by Battelle Energy Al-
liance, LLC under contract no. DE-AC07-05ID14517 with
the U.S. Department of Energy. The U.S. Government retains
and the publisher, by accepting the article for publication, ac-
knowledges that the U.S. Government retains a nonexclusive,
paid-up, irrevocable, worldwide license to publish or repro-
duce the published form of this manuscript, or allow others to
do so, for U.S. Government purposes.
This research made use of the resources of the High Per-
formance Computing Center at Idaho National Laboratory,
which is supported by the Office of Nuclear Energy of the U.S.
Department of Energy and the Nuclear Science User Facilities
under contract no. DE-AC07-05ID14517.
REFERENCES
1.
C. R. JOYNER, T. JENNINGS, D. E. HANKS, and D. J.
LEVACK, “NTP Engine System Design and Modeling,”
in “ASCEND 2022,” Las Vegas, Nevada & Online (Octo-
ber 24-26 2022).
2.
T. JENNINGS, “Latest NTP Engine System Design and
Modeling Results,” (personal communication, January
2023).
3.
V. MANICKAM, M. KRECICKI, and D. KOTLYAR,
“Methodology and Preliminary Results for the Transient
Control of Pressure and Temperature for Nuclear Ther-
mal Propulsion Engines,” in “International Conference on
Physics of Reactors 2022 (PHYSOR 2022),” Pittsburgh,
PA, USA (May 15–20 2022).

0
50
100
150
200
250
300
Power
deposited in the fuel (MW)
0
1
2
3
4
5
6
7
Chamber pressure (MPa)
40
60
80
100
120
Control Drum angle (deg)
T
c
demand (K)
T
c
(K)
Power (MW)
P
c
(MPa)
Control Drum angle (deg)
0 10 20 30 40 50 60
Time (s)
500
1000
1500
2000
2500
2700
Chamber temperature (K)
Fig. 7. Hybrid temperature-reactivity controller with gains of
Kp
=5
×
10
−3
deg/K,
Kd
=5
×
10
−3
deg-s/K, and
K′
p≈
29
.
3 deg/$
using Equation (8) with s=5 for the weighting.
4.
J. A. BERNARD and D. D. LANNING, “Considerations
in the Design and Implementation of Control Laws for the
Digital Operation of Research Reactors,” Nuclear Science
and Engineering,110, 425–444 (1992).
5.
V. M. LABOURÉ, S. SCHUNERT, S. TERLIZZI, Z. M.
PRINCE, J. ORTENSI, C.-S. LIN, L. C. M. CHARLOT,
and M. D. DEHART, “Automated Power-following Con-
trol for Nuclear Thermal Propulsion Startup and Shut-
down Using MOOSE-based Applications,” submitted to
Progress in Nuclear Energy (January 2023).
6.
ÖZER CIFTCIOGLU and M. GECKINLI, “A CAMAC-
based reactivity-meter for nuclear reactors,” Nuclear In-
struments and Methods,177,2, 321–326 (1980).
7.
C. J. PERMANN, D. R. GASTON, D. ANDRS,
R. W. CARLSEN, F. KONG, A. D. LINDSAY, J. M.
MILLER, J. W. PETERSON, A. E. SLAUGHTER, R. H.
STOGNER, and R. C. MARTINEAU, “MOOSE: En-
abling massively parallel multiphysics simulation,” Soft-
wareX,11, 100430 (2020).
8.
Y. WANG, S. SCHUNERT, J. ORTENSI, V. M.
LABOURÉ, M. D. DEHART, Z. M. PRINCE, F. KONG,
J. R. HARTER, P. BALESTRA, and F. N. GLEICHER,
“Rattlesnake: A MOOSE-Based Multiphysics Multi-
scheme Radiation Transport Application,” Nuclear Tech-
nology,207,7, 1047–1072 (2021).
9.
Y. JUNG and C. LEE, “PROTEUS-MOC User Manual,”
Tech. Rep. ANL/NE-18/10, Argonne National Laboratory
(2018).
10.
R. L. WILLIAMSON, J. D. HALES, S. R. NO-
VASCONE, G. PASTORE, K. A. GAMBLE, B. W.
SPENCER, W. JIANG, S. A. PITTS, A. CASAGRANDA,
D. SCHWEN, A. X. ZABRISKIE, A. TOPTAN,
R. GARDNER, C. MATTHEWS, W. F. LIU, and H. L.
CHEN, “BISON: A Flexible Code for Advanced Simula-
tion of the Performance of Multiple Nuclear Fuel Forms,”
Nuclear Technology, pp. 1–27 (2021).
11.
R. A. BERRY, J. W. PETERSON, H. ZHANG, R. C.
MARTINEAU, H. ZHAO, L. ZOU, and D. ANDRS,
RELAP-7 Theory Manual, INL (2014).